window.FINEPROOFS_DISTRIBUTION.problems.push(...[{"i":"dc6ab42003fdccd8","q":"$ABC$  is a triangle with  $AB = 33$ ,  $AC = 21$  and  $BC = m$ , an integer. There are points  $D$ ,  $E$  on the sides  $AB$ ,  $AC$  respectively such that  $AD = DE = EC = n$ , an integer. Find  $m$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.875,"x":0.94196,"p":[[0,29,0.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[4,29,0.1379,0.875,0.12242,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,14],[8,29,0.2759,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[12,29,0.4138,0.89732,0.11425,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16],[16,29,0.5517,0.90625,0.11633,0.85711,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,18],[20,29,0.6897,0.91518,0.11214,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,19],[24,29,0.8276,0.89732,0.12492,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,18],[28,29,0.9655,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[29,29,1.0,0.90179,0.10374,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15]]},{"b":6,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,61,0.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[4,61,0.0656,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[8,61,0.1311,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[12,61,0.1967,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,61,0.2623,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,61,0.3279,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,61,0.3934,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,61,0.459,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,61,0.5246,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,61,0.5902,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,61,0.6557,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,61,0.7213,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,61,0.7869,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,61,0.8525,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,61,0.918,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,61,0.9836,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[61,61,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"88083ef1183740ce","q":"$p(x)$  is the cubic  $x^3 - 3x^2 + 5x$ . If  $h$  is a real root of  $p(x) = 1$  and  $k$  is a real root of  $p(x) = 5$ , find  $h + k$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.91517,"x":1.0,"p":[[0,35,0.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[4,35,0.1143,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[8,35,0.2286,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[12,35,0.3429,0.91517,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22],[16,35,0.4571,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[20,35,0.5714,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[24,35,0.6857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,18,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,18,0.2222,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[8,18,0.4444,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[12,18,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"593f05e3f19ff10d","q":"Let  $f$  be a polynomial with degree at most  $n-1$ . Show that  $$ \\sum_{k=0}^n\\left(\\begin{array}{l}\nn \nk\n\\end{array}\\right)(-1)^k f(k)=0 $$","t":[{"b":0,"e":0.0,"k":"falling","v":0.24107,"x":0.67408,"p":[[0,37,0.0,0.67408,0.35935,0.42857,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,1,0,0,8,0,0,2,0,0,0,0,0,2,0,15],[4,37,0.1081,0.61159,0.36811,0.42857,0.57143,1.0,0.0,1.0,5,13,0,5,0,1,0,0,0,0,0,8,0,0,4,0,0,1,0,0,0,0,13],[8,37,0.2162,0.24107,0.30397,0.0,0.0,0.42857,0.0,1.0,18,1,0,18,0,0,0,0,2,0,0,5,0,0,3,0,0,2,0,0,1,0,1],[12,37,0.3243,0.38839,0.25564,0.24999,0.42857,0.57143,0.0,1.0,6,1,0,6,0,2,0,0,4,0,0,10,0,0,6,0,0,2,0,0,1,0,1],[16,37,0.4324,0.32143,0.27664,0.0,0.42857,0.57143,0.0,1.0,11,1,0,11,0,1,0,0,3,0,0,8,0,0,7,0,0,0,0,0,1,0,1],[20,37,0.5405,0.38391,0.3183,0.0,0.42857,0.57143,0.0,1.0,10,2,0,10,0,1,0,0,2,0,0,7,0,0,7,0,0,0,0,0,3,0,2],[24,37,0.6486,0.33481,0.33807,0.0,0.42857,0.57111,0.0,1.0,13,3,0,13,0,2,0,0,0,0,0,8,0,0,3,0,0,2,0,0,1,0,3],[28,37,0.7568,0.25893,0.22142,0.0,0.2857,0.42857,0.0,0.71429,11,0,0,11,0,3,0,0,4,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[32,37,0.8649,0.33482,0.25904,0.0,0.42857,0.42857,0.0,1.0,9,1,0,9,0,2,0,0,0,0,0,17,0,0,1,0,0,1,0,0,1,0,1],[36,37,0.973,0.31697,0.29393,0.0,0.42857,0.42857,0.0,1.0,12,1,0,12,0,1,0,0,1,0,0,13,0,0,0,0,0,2,0,0,2,0,1],[37,37,1.0,0.35267,0.26482,0.10714,0.42857,0.57143,0.0,0.85714,8,0,0,8,0,4,0,0,1,0,0,9,0,0,5,0,0,4,0,0,1,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.29018,"x":0.61607,"p":[[0,29,0.0,0.46875,0.39646,0.0,0.42857,1.0,0.0,1.0,9,9,0,9,0,2,0,0,2,0,0,7,0,0,1,0,0,1,0,0,1,0,9],[4,29,0.1379,0.61607,0.40159,0.39286,0.57143,1.0,0.0,1.0,7,15,0,7,0,0,0,0,1,0,0,5,0,0,4,0,0,0,0,0,0,0,15],[8,29,0.2759,0.5357,0.32927,0.42857,0.57121,0.85714,0.0,1.0,5,7,0,5,0,1,0,0,1,0,0,8,0,0,8,0,0,0,0,0,2,0,7],[12,29,0.4138,0.52232,0.33808,0.39286,0.57143,0.85714,0.0,1.0,7,4,0,7,0,0,0,0,1,0,0,6,0,0,7,0,0,1,0,0,6,0,4],[16,29,0.5517,0.44642,0.35129,0.0,0.42857,0.60714,0.0,1.0,9,6,0,9,0,0,0,0,1,0,0,11,0,0,3,0,0,1,0,0,1,0,6],[20,29,0.6897,0.29018,0.26362,0.0,0.42857,0.42857,0.0,1.0,13,1,0,13,0,0,0,0,0,0,0,15,0,0,2,0,0,1,0,0,0,0,1],[24,29,0.8276,0.36607,0.3008,0.0,0.42857,0.46431,0.0,1.0,10,2,0,10,0,1,0,0,0,0,0,13,0,0,4,0,0,0,0,0,2,0,2],[28,29,0.9655,0.41517,0.26811,0.39285,0.42857,0.46431,0.0,1.0,6,2,0,6,0,1,0,0,1,0,0,16,0,0,4,0,0,0,0,0,2,0,2],[29,29,1.0,0.35713,0.29232,0.0,0.42857,0.42857,0.0,1.0,10,3,0,10,0,0,0,0,1,0,0,15,0,0,3,0,0,0,0,0,0,0,3]]}]},{"i":"fb5490fab6f69ead","q":"(a) Show that for every  $n\\in\\mathbb{N}$  there is exactly one  $x\\in\\mathbb{R}^+$  so that  $x^n+x^{n+1}=1$ . Call this  $x_n$ .\r\n(b) Find  $\\lim\\limits_{n\\rightarrow+\\infty}x_n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":0.98214,"p":[[0,30,0.0,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[4,30,0.1333,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[8,30,0.2667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,30,0.4,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,30,0.5333,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[20,30,0.6667,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[24,30,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,30,0.9333,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[30,30,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]},{"b":3,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,29,0.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6d21638a9813cd8e","q":"(R.Pirkuliev) Prove the inequality\r\n\\[ \\frac1{\\sqrt {2\\sin A}} \\plus{} \\frac1{\\sqrt {2\\sin B}} \\plus{} \\frac1{\\sqrt {2\\sin C}}\\leq\\sqrt {\\frac {p}{r}},\r\n\\]\r\nwhere  $ p$  and  $ r$  are the semiperimeter and the inradius of triangle  $ ABC$ .","t":[{"b":5,"e":1.0,"k":"volatile","v":0.35714,"x":0.99554,"p":[[0,16,0.0,0.66964,0.40945,0.35714,0.92857,1.0,0.0,1.0,7,16,0,7,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,16],[4,16,0.25,0.65178,0.40711,0.28571,0.92857,1.0,0.0,1.0,6,16,0,6,0,0,0,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,16],[8,16,0.5,0.35714,0.41342,0.0,0.14286,0.78571,0.0,1.0,14,8,0,14,0,4,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,8],[12,16,0.75,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,16,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"rising","v":0.43303,"x":0.70536,"p":[[0,18,0.0,0.53125,0.45769,0.0,0.71429,1.0,0.0,1.0,11,14,0,11,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,14],[4,18,0.2222,0.61607,0.43071,0.0,0.85714,1.0,0.0,1.0,9,15,0,9,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,15],[8,18,0.4444,0.43303,0.45804,0.0,0.14286,1.0,0.0,1.0,16,10,0,16,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,10],[12,18,0.6667,0.47321,0.41563,0.0,0.5,1.0,0.0,1.0,12,9,0,12,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,1,0,9],[16,18,0.8889,0.70536,0.13333,0.57143,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,16,0,0,1,0,4],[18,18,1.0,0.70089,0.13997,0.57143,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,13,0,0,2,0,4]]}]},{"i":"059f4a0fba13ff19","q":"1. Let  $f(x)=x^2+bx+c$ , M = {x | |f(x)|<1}. Prove  $|M|\\leq 2\\sqrt{2}$  (|...| = length of interval(s))","t":[{"b":1,"e":1.0,"k":"flat","v":0.94196,"x":0.99554,"p":[[0,30,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,30,0.1333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,30,0.2667,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,30,0.4,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[16,30,0.5333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,30,0.6667,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,30,0.8,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,30,0.9333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[30,30,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,36,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,36,0.1111,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,36,0.2222,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[12,36,0.3333,0.94196,0.11214,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,22],[16,36,0.4444,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,36,0.5556,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,36,0.6667,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[28,36,0.7778,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,36,0.8889,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[36,36,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"623a35e75f4473d8","q":"1. Let $a, b$ and $c$ be three real numbers such that\n\n$$\n|a| \\geqslant|a+b|,|b| \\geqslant|b+c| \\text { and }|c| \\geqslant|c+a| \\text {. }\n$$\n\nShow that $\\mathrm{a}=\\mathrm{b}=\\mathrm{c}=0$.\n2. Let $a, b, c$ and $d$ be four real numbers such that\n\n$$\n|a| \\geqslant|a+b|,|b| \\geqslant|b+c|,|c| \\geqslant|c+d| \\text { and }|d| \\geqslant|d+a|\n$$\n\nDo we necessarily have $\\mathrm{a}=\\mathrm{b}=\\mathrm{c}=\\mathrm{d}=0$?","t":[{"b":2,"e":0.42857,"k":"flat","v":0.48213,"x":0.58482,"p":[[0,23,0.0,0.52679,0.16917,0.42857,0.42857,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,20,0,0,8,0,0,1,0,0,0,0,3],[4,23,0.1739,0.58482,0.19678,0.42857,0.57143,0.60714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,15,0,0,9,0,0,2,0,0,2,0,4],[8,23,0.3478,0.49107,0.16728,0.42857,0.42857,0.42857,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,27,0,0,2,0,0,0,0,0,0,0,3],[12,23,0.5217,0.48213,0.11151,0.42857,0.42857,0.57111,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,0,0,0,0,0,1],[16,23,0.6957,0.49997,0.14724,0.42857,0.42857,0.571,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,23,0,0,6,0,0,1,0,0,0,0,2],[20,23,0.8696,0.50445,0.14718,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,22,0,0,7,0,0,1,0,0,0,0,2],[23,23,1.0,0.49554,0.14279,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,23,0,0,7,0,0,0,0,0,0,0,2]]},{"b":3,"e":0.42857,"k":"flat","v":0.50448,"x":0.62054,"p":[[0,40,0.0,0.5357,0.18898,0.42857,0.42857,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,21,0,0,6,0,0,1,0,0,0,0,4],[4,40,0.1,0.50893,0.16728,0.42857,0.42857,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,23,0,0,6,0,0,0,0,0,0,0,3],[8,40,0.2,0.55362,0.15868,0.42857,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,11,0,0,3,0,0,1,0,2],[12,40,0.3,0.62054,0.16213,0.53572,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,11,0,0,10,0,0,0,0,3],[16,40,0.4,0.56249,0.16342,0.42857,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,13,0,0,14,0,0,2,0,0,0,0,3],[20,40,0.5,0.60272,0.16258,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,8,0,0,10,0,0,1,0,2],[24,40,0.6,0.55804,0.13054,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,13,0,0,6,0,0,0,0,1],[28,40,0.7,0.50448,0.11832,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,19,0,0,11,0,0,1,0,0,0,0,1],[32,40,0.8,0.55804,0.10926,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,0,5,0,0,1,0,0],[36,40,0.9,0.61161,0.17582,0.42857,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,12,0,0,0,0,3],[40,40,1.0,0.57589,0.15765,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,8,0,0,0,0,2]]}]},{"i":"3d24b9070f2d0dc8","q":"11. (HUN 1) Define sequence $a_{n}$ by $\\sum_{d \\mid n} a_{d}=2^{n}$. Show that $n \\mid a_{n}$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,55,0.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[4,55,0.0727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,55,0.1455,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,55,0.2182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,55,0.2909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,55,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,55,0.4364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,55,0.5818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,55,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,55,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,55,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,55,0.9455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.71429,"k":"flat","v":0.91964,"x":1.0,"p":[[0,35,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,35,0.1143,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,35,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,35,0.3429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,35,0.4571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,35,0.5714,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[32,35,0.9143,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[35,35,1.0,0.93303,0.08738,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19]]}]},{"i":"c08450ac9b75fe01","q":"2. (HUN) If $a, b$, and $c$ are the sides and $\\alpha, \\beta$, and $\\gamma$ the respective angles of the triangle for which $a+b=\\tan \\frac{\\gamma}{2}(a \\tan \\alpha+b \\tan \\beta)$, prove that the triangle is isosceles.","t":[{"b":2,"e":0.2857,"k":"flat","v":0.05804,"x":0.14732,"p":[[0,28,0.0,0.14732,0.13592,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,9,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.12053,0.1488,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,3,0,1,7,0,1,2,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.13839,0.13592,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.08482,0.12299,0.0,0.0,0.17857,0.0,0.28571,21,0,0,21,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.09598,0.13553,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,6,0,1,1,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0625,"x":0.12045,"p":[[0,35,0.0,0.12045,0.14771,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,3,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.09822,0.13092,0.0,0.0,0.1786,0.0,0.42857,19,0,0,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.07813,0.11203,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,1,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.10259,0.1299,0.0,0.0,0.1786,0.0,0.4286,18,0,0,18,0,6,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.11152,0.14608,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.08259,0.11166,0.0,0.0,0.1429,0.0,0.357,19,0,0,19,0,8,0,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"68e1e0223ce67855","q":"2. (CZS) ${ }^{\\mathrm{IMO} 1}$ Let $x_{1} \\geq x_{2} \\geq \\cdots \\geq x_{n}$ and $y_{1} \\geq y_{2} \\geq \\cdots \\geq y_{n}$ be two $n$-tuples of numbers. Prove that  $$ \\sum_{i=1}^{n}\\left(x_{i}-y_{i}\\right)^{2} \\leq \\sum_{i=1}^{n}\\left(x_{i}-z_{i}\\right)^{2} $$  is true when $z_{1}, z_{2}, \\ldots, z_{n}$ denote $y_{1}, y_{2}, \\ldots, y_{n}$ taken in another order.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.85268,"x":1.0,"p":[[0,27,0.0,0.91964,0.24984,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[4,27,0.1481,0.85268,0.28901,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,25],[8,27,0.2963,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,27,0.4444,0.92857,0.22304,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[16,27,0.5926,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,26,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,26,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"746f592993471f29","q":"2. (POL) Let $a, b$, and $c$ be the lengths of a triangle whose area is $S$. Prove that  $$ a^{2}+b^{2}+c^{2} \\geq 4 S \\sqrt{3} $$  In what case does equality hold?","t":[{"b":1,"e":1.0,"k":"rising","v":0.28571,"x":1.0,"p":[[0,37,0.0,0.36607,0.36759,0.14286,0.14286,0.46429,0.14286,1.0,0,8,0,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,37,0.1081,0.51786,0.42521,0.14286,0.14286,1.0,0.14286,1.0,0,14,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[8,37,0.2162,0.28571,0.30929,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,37,0.3243,0.52678,0.39518,0.14286,0.28571,1.0,0.14286,1.0,0,13,0,0,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[16,37,0.4324,0.72768,0.37857,0.28571,1.0,1.0,0.14286,1.0,0,21,0,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[20,37,0.5405,0.84375,0.32608,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[24,37,0.6486,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,37,0.7568,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":0.14286,"k":"falling","v":0.14268,"x":0.44196,"p":[[0,42,0.0,0.44196,0.40463,0.14286,0.14286,1.0,0.14286,1.0,0,11,0,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[4,42,0.0952,0.19643,0.20748,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,42,0.1905,0.1875,0.15746,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,42,0.2857,0.17858,0.15567,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,42,0.381,0.18304,0.09606,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"16cee29aed697ad9","q":"A cyclic quadrilateral  $ABCD$  has side lengths  $AB = 3, BC = AD = 5$ , and  $CD = 8$ . The radius of its circumcircle can be written in the form  $a\\sqrt{b}/c$ , where  $a, b, c$  are positive integers,  $a, c$  are relatively prime, and  $b$  is not divisible by the square of any prime. Find  $a + b + c$ .","t":[{"b":4,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,30,0.0,0.82143,0.21724,0.57143,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,19],[4,30,0.1333,0.94643,0.13243,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[8,30,0.2667,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[12,30,0.4,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,30,0.5333,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,30,0.6667,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[24,30,0.8,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]},{"b":5,"e":1.0,"k":"rising","v":0.83036,"x":1.0,"p":[[0,62,0.0,0.83036,0.22142,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,0,0,20],[4,62,0.0645,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,62,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,62,0.1935,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,62,0.2581,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,62,0.3226,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,62,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,62,0.4516,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,62,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,62,0.5806,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,62,0.6452,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[44,62,0.7097,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,62,0.7742,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,62,0.8387,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,62,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,62,0.9677,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[62,62,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1390608f867346c5","q":"A four-digit number has the following properties:\r\n\r\n(a) It is a perfect square;\r\n\r\n(b) Its first two digits are equal\r\n\r\n(c) Its last two digits are equal.\r\n\r\nFind all such four-digit numbers.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.67856,"x":0.78125,"p":[[0,17,0.0,0.78125,0.17852,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,12,0,0,11,0,6],[4,17,0.2353,0.7321,0.17771,0.71429,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,13,0,0,8,0,4],[8,17,0.4706,0.72767,0.1902,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,10,0,0,6,0,6],[12,17,0.7059,0.67856,0.14726,0.67857,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,18,0,0,6,0,0],[16,17,0.9412,0.68303,0.15865,0.57143,0.71429,0.85704,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,13,0,0,9,0,0],[17,17,1.0,0.71872,0.11567,0.67836,0.71429,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,14,0,0,10,0,0]]},{"b":3,"e":0.85714,"k":"flat","v":0.7366,"x":0.83481,"p":[[0,24,0.0,0.83481,0.12932,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,11,0,9],[4,24,0.1667,0.75,0.13363,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,12,0,2],[8,24,0.3333,0.77231,0.09353,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,13,0,1],[12,24,0.5,0.75,0.10102,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,12,0,0],[16,24,0.6667,0.7366,0.11355,0.67857,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,0,13,0,0],[20,24,0.8333,0.75446,0.09606,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,13,0,0],[24,24,1.0,0.75,0.11294,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,15,0,0]]}]},{"i":"9f4c910ba54b4a44","q":"A function  $f: R \\to  R$  satisfies  $f (x + 1) = f (x) + 1$  for all  $x$ . Given  $a \\in R$ , define the sequence  $(x_n)$  recursively by  $x_0 = a$  and  $x_{n+1} = f (x_n)$  for  $n \\ge  0$ . Suppose that, for some positive integer m, the difference  $x_m - x_0 = k$  is an integer. Prove that the limit  $\\lim_{n\\to \\infty}\\frac{x_n}{n}$   exists and determine its value.","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,42,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,42,0.0952,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,42,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,42,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,42,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,42,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,42,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,50,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,50,0.08,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,50,0.16,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,50,0.24,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,50,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,50,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,50,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,50,0.56,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,50,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,50,0.72,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,50,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,50,0.88,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,50,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[50,50,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ee5d0cabdf4deac2","q":"A grid of size $\\mathrm{n} \\times \\mathrm{n}$ contains $\\mathrm{n}^{2}$ cells. Each cell contains a natural number between 1 and $\\boldsymbol{n}$, such that each integer in the set $\\{1, \\ldots, n\\}$ appears exactly $n$ times in the grid. Show that there exists a column or a row of the grid containing at least $\\sqrt{n}$ different numbers.","t":[{"b":5,"e":0.57143,"k":"falling","v":0.63392,"x":0.89732,"p":[[0,7,0.0,0.89732,0.16457,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,22],[4,7,0.5714,0.83482,0.17896,0.67857,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,5,0,15],[7,7,1.0,0.63392,0.10063,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,6,0,0,4,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,41,0.0,0.83482,0.18935,0.57143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,5,0,16],[4,41,0.0976,0.9107,0.1505,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,22],[8,41,0.1951,0.87053,0.17627,0.71429,1.0,1.0,0.5714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,4,0,19],[12,41,0.2927,0.88392,0.1871,0.67857,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,0,0,23],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1458af45bd74cf5b","q":"A parabola has focus  $F$  and vertex  $V$  , where  $VF = 1$ 0. Let  $AB$  be a chord of length  $100$  that passes through  $F$ . Determine the area of  $\\vartriangle VAB$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.29902,"x":0.97321,"p":[[0,18,0.0,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,18,0.2222,0.44642,0.31491,0.14286,0.42859,0.71429,0.0,1.0,7,2,1,7,0,2,0,0,3,0,0,5,0,0,4,0,0,7,0,0,2,0,2],[8,18,0.4444,0.37947,0.28033,0.14286,0.42857,0.71429,0.0,0.71429,7,0,1,7,0,5,0,0,2,0,0,6,0,0,2,0,0,10,0,0,0,0,0],[12,18,0.6667,0.32588,0.28399,0.0,0.21431,0.57143,0.0,0.71429,9,0,0,9,0,7,0,0,1,0,0,3,0,0,5,0,0,7,0,0,0,0,0],[16,18,0.8889,0.33929,0.25939,0.14286,0.28571,0.57143,0.0,0.71429,6,0,0,6,0,6,0,0,8,0,0,1,0,0,4,0,0,7,0,0,0,0,0],[18,18,1.0,0.29902,0.26337,0.105,0.14286,0.57143,0.0,0.71429,8,0,0,8,0,9,0,0,2,0,0,3,0,0,5,0,0,5,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"flat","v":0.9375,"x":0.99554,"p":[[0,38,0.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,38,0.1053,0.9375,0.13803,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[8,38,0.2105,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,38,0.3158,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,38,0.4211,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,38,0.5263,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,38,0.6316,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,38,0.7368,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,38,0.8421,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,38,0.9474,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[38,38,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"ba09be75824f80a4","q":"A number is called cool if the sum of its digits is multiple of  $17$  and the sum of digits of its successor is multiple of  $17$ . What is the smallest cool number?","t":[{"b":1,"e":1.0,"k":"flat","v":0.94196,"x":0.97321,"p":[[0,9,0.0,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[4,9,0.4444,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,9,0.8889,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[9,9,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":2,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,59,0.0,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,59,0.0678,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,59,0.1356,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,59,0.2034,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,59,0.2712,0.98214,0.04725,1.0,1.0,1.0,0.8571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,59,0.339,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,59,0.4068,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,59,0.4746,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,59,0.5424,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,59,0.6102,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,59,0.678,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,59,0.7458,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,59,0.8136,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,59,0.8814,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,59,0.9492,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[59,59,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"4b61e9d8346fcd3d","q":"A point  $E$  lies on the altitude  $BD$  of triangle  $ABC$ , and  $\\angle AEC=90^\\circ.$  Points  $O_1$  and  $O_2$  are the circumcenters of triangles  $AEB$  and  $CEB$ ; points  $F, L$  are the midpoints of the segments  $AC$  and  $O_1O_2.$  Prove that the points  $L,E,F$  are collinear.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.11598,"x":0.13375,"p":[[0,10,0.0,0.12482,0.04718,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.12483,0.04718,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14,"k":"flat","v":0.10688,"x":0.13357,"p":[[0,20,0.0,0.12018,0.05173,0.14,0.14286,0.14286,0.0,0.143,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.10688,0.06171,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.13357,0.0345,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eefb761f1a24a7ff","q":"A quadratic trinomial  $P(x)$  with the  $x^2$  coefficient of one is such, that  $P(x)$  and  $P(P(P(x)))$  share a root. Prove that  $P(0)*P(1)=0$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,59,0.0,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[4,59,0.0678,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[8,59,0.1356,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[12,59,0.2034,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[16,59,0.2712,0.95089,0.1048,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[20,59,0.339,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[24,59,0.4068,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[28,59,0.4746,0.93302,0.12368,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[32,59,0.5424,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[36,59,0.6102,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[40,59,0.678,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,59,0.7458,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,59,0.8136,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[52,59,0.8814,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,59,0.9492,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[59,59,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":1.0,"k":"flat","v":0.95981,"x":1.0,"p":[[0,37,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,37,0.1081,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[8,37,0.2162,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,37,0.8649,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,37,0.973,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[37,37,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"b5dca50ba6d93ae6","q":"A positive integer is *detestable* if the sum of its digits is a multiple of  $11$ . How many positive integers below  $10000$  are detestable?\n\n*Proposed by Giacomo Rizzo*","t":[{"b":3,"e":1.0,"k":"flat","v":0.83929,"x":0.93304,"p":[[0,31,0.0,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[4,31,0.129,0.83929,0.20748,0.57143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,20],[8,31,0.2581,0.85267,0.20357,0.57143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,21],[12,31,0.3871,0.85268,0.20355,0.57143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,21],[16,31,0.5161,0.90624,0.1772,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,25],[20,31,0.6452,0.85268,0.20355,0.57143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,21],[24,31,0.7742,0.8482,0.23675,0.57143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,22],[28,31,0.9032,0.87946,0.22047,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,24],[31,31,1.0,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.86158,"x":0.94643,"p":[[0,20,0.0,0.86158,0.19723,0.57143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,1,0,21],[4,20,0.2,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[8,20,0.4,0.90622,0.17722,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,25],[12,20,0.6,0.91964,0.16728,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[16,20,0.8,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[20,20,1.0,0.89284,0.21431,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,25]]}]},{"i":"4f574a934e2f1618","q":"An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.","t":[{"b":3,"e":1.0,"k":"volatile","v":0.16965,"x":1.0,"p":[[0,15,0.0,0.16965,0.1448,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,22,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,15,0.2667,0.25893,0.26592,0.14286,0.14286,0.32143,0.0,1.0,3,3,0,3,0,20,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.14286,"k":"flat","v":0.16071,"x":0.22771,"p":[[0,29,0.0,0.16071,0.09279,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,26,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,29,0.1379,0.17411,0.11143,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,23,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.20534,0.15121,0.14286,0.14286,0.14286,0.14286,0.857,0,0,0,0,0,25,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[12,29,0.4138,0.17857,0.07986,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,26,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.17402,0.07775,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.22771,0.13299,0.14286,0.14286,0.32143,0.14286,0.571,0,0,0,0,0,22,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[24,29,0.8276,0.19188,0.09856,0.14286,0.14286,0.14287,0.14,0.42857,0,0,0,0,0,25,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.20982,0.17122,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,26,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[29,29,1.0,0.20527,0.1285,0.14286,0.14286,0.1786,0.14,0.71429,0,0,0,0,0,24,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"b54e06ebe6b8cd72","q":"An integer consists of 7 different digits, and is a multiple of each of its digits.\r\n\r\nWhat digits are in this nubmer?","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,32,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,32,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99553,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e63d8941704c7f36","q":"Angle  $A$  in triangle  $ABC$  is equal to  $120^o$ . Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to  $AB + AC$ .\n\n(V. Protasov)","t":[{"b":1,"e":0.71429,"k":"flat","v":0.63837,"x":0.68747,"p":[[0,26,0.0,0.63837,0.08739,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,14,0,0,1,0,0],[4,26,0.1538,0.68747,0.08332,0.67857,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,23,0,0,0,0,1],[8,26,0.3077,0.64729,0.09441,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,19,0,0,0,0,0],[12,26,0.4615,0.68304,0.09932,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,21,0,0,1,0,1],[16,26,0.6154,0.67854,0.10717,0.57143,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,0,0,0,2],[20,26,0.7692,0.66962,0.10377,0.57143,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,15,0,0,2,0,1],[24,26,0.9231,0.67411,0.09606,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,0,1,0,1],[26,26,1.0,0.68304,0.10555,0.57143,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,19,0,0,0,0,2]]},{"b":7,"e":0.71429,"k":"flat","v":0.62946,"x":0.67857,"p":[[0,43,0.0,0.64284,0.15153,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,14,0,0,15,0,0,1,0,1],[4,43,0.093,0.62946,0.1551,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,0,0,0,13,0,0,15,0,0,2,0,0],[8,43,0.186,0.67857,0.10714,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,23,0,0,2,0,0],[12,43,0.2791,0.66515,0.12171,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,19,0,0,1,0,1],[16,43,0.3721,0.66071,0.12752,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,16,0,0,4,0,0],[20,43,0.4651,0.66963,0.07525,0.57143,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,23,0,0,0,0,0],[24,43,0.5581,0.64282,0.10714,0.57143,0.71429,0.71429,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,15,0,0,2,0,0],[28,43,0.6512,0.67411,0.07349,0.67857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,24,0,0,0,0,0],[32,43,0.7442,0.64728,0.10095,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,17,0,0,1,0,0],[36,43,0.8372,0.65177,0.09408,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,18,0,0,1,0,0],[40,43,0.9302,0.66069,0.09281,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,22,0,0,0,0,0],[43,43,1.0,0.63393,0.11811,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,19,0,0,0,0,0]]}]},{"i":"6a6a83d8f871aacf","q":"Calculate  $\\sin^3 a + \\cos^3 a$  if you know that  $\\sin a+ \\cos a = m$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.70536,"x":0.74554,"p":[[0,40,0.0,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[4,40,0.1,0.71874,0.05629,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[8,40,0.2,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[12,40,0.3,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[16,40,0.4,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[20,40,0.5,0.72321,0.03458,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0],[24,40,0.6,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[28,40,0.7,0.71875,0.02486,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0],[32,40,0.8,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,40,0.9,0.72321,0.06121,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,1,0,1],[40,40,1.0,0.74554,0.09932,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,4]]},{"b":4,"e":0.71429,"k":"flat","v":0.67857,"x":0.74107,"p":[[0,43,0.0,0.70536,0.03458,0.71429,0.71429,0.71429,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[4,43,0.093,0.74094,0.08333,0.71429,0.71429,0.71429,0.71,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[8,43,0.186,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[12,43,0.2791,0.73214,0.05923,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,2,0,1],[16,43,0.3721,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[20,43,0.4651,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[24,43,0.5581,0.67857,0.12877,0.71429,0.71429,0.71429,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0],[28,43,0.6512,0.71874,0.05633,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[32,43,0.7442,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[36,43,0.8372,0.71875,0.02486,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0],[40,43,0.9302,0.73661,0.08828,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,3],[43,43,1.0,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1]]}]},{"i":"0c515a7b146772e7","q":"Consider the sequence  $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$  Find  $n$  such that the \ffirst  $n$  terms sum up to  $2010.$","t":[{"b":2,"e":0.14286,"k":"falling","v":0.25001,"x":0.49545,"p":[[0,36,0.0,0.49545,0.24752,0.28571,0.42857,0.71429,0.14,1.0,0,3,0,0,0,4,0,0,7,0,0,7,0,0,4,0,0,7,0,0,0,0,3],[4,36,0.1111,0.44196,0.22689,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,6,0,0,7,0,0,8,0,0,3,0,0,5,0,0,3,0,0],[8,36,0.2222,0.29469,0.17477,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,9,0,0,12,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[12,36,0.3333,0.28116,0.16562,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,11,0,0,9,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[16,36,0.4444,0.25001,0.12877,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,12,0,0,15,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[20,36,0.5556,0.30357,0.14174,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,14,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[24,36,0.6667,0.25884,0.18369,0.14286,0.21428,0.28571,0.0,0.71429,3,0,0,3,0,13,0,0,9,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[28,36,0.7778,0.26785,0.09942,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,4,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.35249,0.17142,0.28571,0.28571,0.46418,0.14,0.71429,0,0,0,0,0,7,0,0,13,0,0,4,0,0,6,0,0,2,0,0,0,0,0],[36,36,1.0,0.26339,0.13415,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,9,0,0,15,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"falling","v":0.25894,"x":0.42857,"p":[[0,13,0.0,0.42857,0.22868,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,8,0,0,6,0,0,6,0,0,3,0,0,3,0,0],[4,13,0.3077,0.27232,0.18336,0.14286,0.2857,0.32143,0.0,0.71429,4,0,0,4,0,9,0,0,11,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[8,13,0.6154,0.27679,0.15947,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,8,0,0,12,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[12,13,0.9231,0.2767,0.14706,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,7,0,0,13,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[13,13,1.0,0.25894,0.1729,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,9,0,0,8,0,0,7,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"cd58d12f12f7a869","q":"Consider the sequence  $ \\left( x_n \\right)_{n\\ge 1} $  having  $ x_1>1 $  and satisfying the equation  $$  x_1+x_2+\\cdots +x_{n+1} =x_1x_2\\cdots x_{n+1} ,\\quad\\forall n\\in\\mathbb{N} .  $$  Show that this sequence is convergent and find its limit.","t":[{"b":1,"e":1.0,"k":"flat","v":0.85712,"x":1.0,"p":[[0,31,0.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,31,0.129,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[8,31,0.2581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,31,0.3871,0.85712,0.18215,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,1,0,19],[16,31,0.5161,0.89286,0.15152,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,20],[20,31,0.6452,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,35,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,35,0.1143,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,35,0.2286,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,35,0.3429,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,35,0.4571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,35,0.5714,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,35,0.6857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"745e529911f43ba4","q":"Between the states of Alinaesquina and Berlinda, each road connects one city of Alinaesquina to one city of Berlinda. All the roads are in two-ways, and between any two cities, it is possible to travel from one to the other, using only these (possibly more than one) roads. Furthermore, it is known that, from any city of anyone of the two states, the same number of  $k$  roads are going out. We know that  $k\\geq 2$ . Prove that governments of the states can close anyone of the roads, and there will still be a route (possibly through several roads) between any two cities.","t":[{"b":1,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,40,0.0,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,40,0.1,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,40,0.2,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,40,0.3,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,40,0.4,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,40,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,40,0.6,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,40,0.7,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,40,0.8,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,40,0.9,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,40,1.0,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27]]},{"b":2,"e":1.0,"k":"flat","v":0.86598,"x":1.0,"p":[[0,25,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,25,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,25,0.32,0.88392,0.12078,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,15],[12,25,0.48,0.91071,0.1171,0.85711,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,19],[16,25,0.64,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[20,25,0.8,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[24,25,0.96,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[25,25,1.0,0.86598,0.18224,0.85714,0.85714,1.0,0.14,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,12,0,14]]}]},{"i":"7fc1103075c44a8b","q":"Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey.\n\nHow many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?","t":[{"b":2,"e":0.0,"k":"flat","v":0.04911,"x":0.16062,"p":[[0,45,0.0,0.16062,0.16657,0.0,0.14286,0.2857,0.0,0.4286,13,0,0,13,0,9,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.125,0.16269,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,8,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.11161,0.13236,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.14277,0.14286,0.0,0.14286,0.2857,0.0,0.4286,13,0,1,13,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.09821,0.1448,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.08027,0.11254,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.12482,0.14615,0.0,0.14143,0.14286,0.0,0.4286,14,0,0,14,0,13,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.0758,0.0873,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.08464,0.10002,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.0892,0.14614,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.04911,0.11071,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.04911,0.11071,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.05795,"x":0.15179,"p":[[0,23,0.0,0.12054,0.16409,0.0,0.0,0.14287,0.0,0.42857,18,0,0,18,0,7,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.11161,0.11701,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,15,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.15179,0.17474,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,7,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[12,23,0.5217,0.06697,0.10092,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0758,0.10086,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0758,0.13351,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cbb87077dacfbad3","q":"Circles  $S_1$  and  $S_2$  meet at points  $A$  and  $B$ . A line through  $A$  is parallel to the line through the centers of  $S_1$  and  $S_2$  and meets  $S_1$  and  $S_2$  again  $C$  and  $D$  respectively. Circle  $S_3$  having  $CD$  as its diameter meets  $S_1$  and  $S_2$  again at  $P$  and  $Q$  respectively. Prove that lines  $CP$ ,  $DQ$ , and  $AB$  are concurent.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.74554,"x":0.88839,"p":[[0,25,0.0,0.8348,0.23988,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,2,0,18],[4,25,0.16,0.81249,0.27995,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,0,1,0,19],[8,25,0.32,0.88839,0.13709,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,19],[12,25,0.48,0.76786,0.11152,0.71429,0.71429,0.71429,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,6],[16,25,0.64,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[20,25,0.8,0.77679,0.11811,0.71429,0.71429,0.71429,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7],[24,25,0.96,0.74554,0.08552,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,3],[25,25,1.0,0.76786,0.11152,0.71429,0.71429,0.71429,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,6]]},{"b":4,"e":1.0,"k":"flat","v":0.69643,"x":0.9375,"p":[[0,40,0.0,0.82142,0.28794,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,19],[4,40,0.1,0.69643,0.33072,0.64286,0.71429,1.0,0.0,1.0,2,13,0,2,0,3,0,0,2,0,0,1,0,0,0,0,0,11,0,0,0,0,13],[8,40,0.2,0.93303,0.13825,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,25],[12,40,0.3,0.83481,0.16795,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,1,0,15],[16,40,0.4,0.87054,0.15714,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0,18],[20,40,0.5,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[24,40,0.6,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[28,40,0.7,0.91517,0.12299,0.82132,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[32,40,0.8,0.86607,0.13803,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,2,0,16],[36,40,0.9,0.83481,0.14337,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,2,0,13],[40,40,1.0,0.83036,0.15335,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,2,0,13]]}]},{"i":"34c9f7f975ba2ff9","q":"Call the number  $\\overline{a_1a_2... a_m}$  ( $a_1 \\ne 0,a_m \\ne 0$ ) the reverse of the number  $\\overline{a_m...a_2a_1}$ . Prove that the sum between a number  $n$  and its reverse is a multiple of  $81$  if and only if the sum of the digits of  $n$  is a multiple of  $81$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.93748,"x":1.0,"p":[[0,26,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,26,0.1538,0.98212,0.07792,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,26,0.3077,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,26,0.4615,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,26,0.6154,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,26,0.7692,0.93748,0.13337,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[24,26,0.9231,0.95088,0.11072,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[26,26,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]},{"b":7,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,36,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,36,0.1111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,36,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a7971d6531a71d05","q":"Compute the number of solutions to  $1+\\cos(\\theta)+\\cos(2\\theta)+\\ldots+\\cos(2024\\theta) = \\tfrac{1}{2}$  for  $\\theta \\in [0,2\\pi].$","t":[{"b":2,"e":0.85714,"k":"flat","v":0.88393,"x":0.98214,"p":[[0,49,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,49,0.0816,0.89286,0.08748,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11],[8,49,0.1633,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[12,49,0.2449,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[16,49,0.3265,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[20,49,0.4082,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[24,49,0.4898,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[28,49,0.5714,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[32,49,0.6531,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[36,49,0.7347,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[40,49,0.8163,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[44,49,0.898,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[48,49,0.9796,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[49,49,1.0,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12]]},{"b":3,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,38,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,38,0.1053,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,38,0.2105,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,38,0.3158,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,38,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,38,0.5263,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,38,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b9909c996abde9b3","q":"Consider the set: $A = \\{1, 2,..., 100\\}$ Prove that if we take  $11$  different elements from  $A$ , there are  $x, y$  such that  $x \\neq y$  and  $0 < |\\sqrt{x} - \\sqrt{y}| < 1$","t":[{"b":1,"e":0.0,"k":"falling","v":0.04018,"x":0.23214,"p":[[0,38,0.0,0.23214,0.3567,0.0,0.0,0.39286,0.0,1.0,21,3,0,21,0,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,0,0,3],[4,38,0.1053,0.15625,0.29957,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,1],[8,38,0.2105,0.10268,0.25812,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[12,38,0.3158,0.1875,0.35614,0.0,0.0,0.07143,0.0,1.0,24,4,0,24,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,4],[16,38,0.4211,0.04018,0.11971,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,38,0.5263,0.08036,0.24984,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,38,0.6316,0.06249,0.15537,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,38,0.7368,0.05357,0.13716,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,38,0.8421,0.10714,0.2369,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[36,38,0.9474,0.04018,0.12992,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.05357,0.12753,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.23661,"p":[[0,27,0.0,0.15179,0.28557,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,1],[4,27,0.1481,0.23661,0.38731,0.0,0.0,0.32143,0.0,1.0,22,5,0,22,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,5],[8,27,0.2963,0.10268,0.27019,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[12,27,0.4444,0.06696,0.21424,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,27,0.5926,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,27,0.7407,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[27,27,1.0,0.07143,0.17857,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"7dd87a24a393d4a2","q":"Consider the set  $$ \\mathcal{S}=\\{(a, b, c, d, e): 0<a<b<c<d<e<100\\} $$  where  $a, b, c, d, e$  are integers. If  $D$  is the average value of the fourth element of such a tuple in the set, taken over all the elements of  $\\mathcal{S}$ , find the largest integer less than or equal to  $D$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.87946,"x":0.96875,"p":[[0,34,0.0,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[4,34,0.1176,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[8,34,0.2353,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[12,34,0.3529,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[16,34,0.4706,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[20,34,0.5882,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,34,0.7059,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[28,34,0.8235,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[32,34,0.9412,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[34,34,1.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,47,0.0,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[4,47,0.0851,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,47,0.1702,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,47,0.2553,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,47,0.3404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,47,0.4255,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,47,0.5106,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,47,0.5957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,47,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,47,0.8511,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,47,0.9362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[47,47,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"163f74d740988180","q":"Define a sequence  $\\{a_n\\}$   by\\[S_1=1,\\ S_{n+1}=\\frac{(2+S_n)^2}{ 4+S_n} (n=1,\\ 2,\\ 3,\\ \\cdots).\\]\nWhere  $S_n$  the sum of first  $n$  terms of sequence  $\\{a_n\\}$ .\nFor any positive integer  $n$  ,prove that\\[a_{n}\\ge \\frac{4}{\\sqrt{9n+7}}.\\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.80804,"x":1.0,"p":[[0,37,0.0,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,37,0.1081,0.80804,0.35285,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[8,37,0.2162,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[12,37,0.3243,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,37,0.4324,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,37,0.5405,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,37,0.6486,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,37,0.7568,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,37,0.8649,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.14286,"k":"falling","v":0.14277,"x":0.88839,"p":[[0,32,0.0,0.88839,0.28288,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[4,32,0.125,0.17857,0.13832,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[8,32,0.25,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8345bde16c8bc758","q":"Evaluate the sum:  $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,18,0.0,0.82143,0.32733,0.71429,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,23],[4,18,0.2222,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"rising","v":0.79464,"x":1.0,"p":[[0,34,0.0,0.79464,0.31122,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,0,0,19],[4,34,0.1176,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,34,0.2353,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,23],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,34,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[34,34,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"edc24446ed1128cc","q":"Evaluate  $\\dfrac{d}{dx}\\left(\\sin x - \\dfrac{4}{3}\\sin^3 x\\right)$  when  $x=15$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.73658,"x":0.94196,"p":[[0,15,0.0,0.73658,0.28374,0.571,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,7,0,0,0,0,0,4,0,0,6,0,0,0,0,15],[4,15,0.2667,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[8,15,0.5333,0.86159,0.16556,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,4,0,17],[12,15,0.8,0.85266,0.23553,0.71429,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,20],[15,15,1.0,0.84818,0.15547,0.71429,0.85707,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,3,0,15]]},{"b":7,"e":0.28571,"k":"falling","v":0.3482,"x":0.67851,"p":[[0,60,0.0,0.67851,0.23148,0.571,0.64286,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,6,0,0,2,0,8],[4,60,0.0667,0.60711,0.22868,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,6,0,0,2,0,0,7,0,0,10,0,0,3,0,3],[8,60,0.1333,0.56249,0.19212,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,9,0,0,10,0,0,6,0,0,0,0,3],[12,60,0.2,0.44195,0.16114,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,6,0,0,12,0,0,7,0,0,4,0,0,0,0,0],[16,60,0.2667,0.43747,0.15945,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,7,0,0,6,0,0,5,0,0,0,0,0],[20,60,0.3333,0.44643,0.17405,0.28571,0.42857,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,8,0,0,2,0,0,8,0,0,0,0,0],[24,60,0.4,0.39286,0.15971,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,11,0,0,8,0,0,7,0,0,2,0,0,0,0,0],[28,60,0.4667,0.43301,0.15353,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,13,0,0,3,0,0,5,0,0,0,0,0],[32,60,0.5333,0.42857,0.19233,0.2857,0.35714,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,13,0,0,4,0,0,5,0,0,7,0,0,0,0,0],[36,60,0.6,0.3482,0.14694,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,14,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[40,60,0.6667,0.40622,0.15609,0.28571,0.42857,0.4286,0.1429,0.85714,0,0,0,0,0,1,0,0,14,0,0,10,0,0,4,0,0,2,0,0,1,0,0],[44,60,0.7333,0.44634,0.22811,0.28571,0.42857,0.57143,0.14,1.0,0,2,0,0,0,4,0,0,10,0,0,7,0,0,4,0,0,5,0,0,0,0,2],[48,60,0.8,0.41069,0.14171,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,2,0,0,10,0,0,12,0,0,6,0,0,2,0,0,0,0,0],[52,60,0.8667,0.46427,0.13831,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,15,0,0,5,0,0,5,0,0,0,0,0],[56,60,0.9333,0.42856,0.15971,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,10,0,0,4,0,0,5,0,0,0,0,0],[60,60,1.0,0.48214,0.18123,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,9,0,0,6,0,0,6,0,0,0,0,1]]}]},{"i":"c4f7133972218205","q":"Define  $m(n)$  to be the greatest proper natural divisor of  $n\\in \\mathbb{N}$ . Find all  $n \\in \\mathbb{N} $  such that  $n+m(n) $  is a power of  $10$ .\n\n*N. Agakhanov*","t":[{"b":0,"e":1.0,"k":"flat","v":0.89286,"x":0.98214,"p":[[0,25,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,25,0.16,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[8,25,0.32,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,25,0.48,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,25,0.64,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[20,25,0.8,0.91518,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[24,25,0.96,0.90179,0.10374,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15],[25,25,1.0,0.91071,0.10564,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,9,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,9,0.4444,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[9,9,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1f6bf5061d39eea6","q":"Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.16964,"x":0.62054,"p":[[0,37,0.0,0.62054,0.35104,0.25,0.71429,1.0,0.14286,1.0,0,12,0,0,0,8,0,0,2,0,0,3,0,0,1,0,0,6,0,0,0,0,12],[4,37,0.1081,0.55804,0.37348,0.14286,0.57143,1.0,0.14286,1.0,0,11,0,0,0,12,0,0,1,0,0,3,0,0,0,0,0,5,0,0,0,0,11],[8,37,0.2162,0.41071,0.35848,0.14286,0.14286,0.71429,0.0,1.0,1,7,0,1,0,17,0,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,7],[12,37,0.3243,0.40625,0.30328,0.14286,0.21428,0.71429,0.14286,1.0,0,3,0,0,0,16,0,0,1,0,0,4,0,0,0,0,0,8,0,0,0,0,3],[16,37,0.4324,0.25,0.17128,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,22,0,0,0,0,0,8,0,0,0,0,0,2,0,0,0,0,0],[20,37,0.5405,0.21875,0.15146,0.14286,0.14286,0.21429,0.0,0.71429,1,0,0,1,0,23,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[24,37,0.6486,0.20982,0.13825,0.14286,0.14286,0.14287,0.14286,0.71429,0,0,0,0,0,25,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[28,37,0.7568,0.17848,0.12374,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[32,37,0.8649,0.19643,0.11152,0.14286,0.14286,0.17857,0.0,0.4286,1,0,0,1,0,23,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.16964,0.09062,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.21411,0.14295,0.14286,0.14286,0.14287,0.14,0.71429,0,0,0,0,0,25,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.72768,"x":0.92857,"p":[[0,32,0.0,0.75,0.32341,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,6,0,0,0,0,0,0,0,0,2,0,0,7,0,0,0,0,17],[4,32,0.125,0.76786,0.32291,0.64286,1.0,1.0,0.14286,1.0,0,19,0,0,0,5,0,0,0,0,0,3,0,0,0,0,0,5,0,0,0,0,19],[8,32,0.25,0.84821,0.18877,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,17],[12,32,0.375,0.92857,0.14286,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,25],[16,32,0.5,0.73214,0.20124,0.71429,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,19,0,0,0,0,8],[20,32,0.625,0.79464,0.17835,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,17,0,0,0,0,12],[24,32,0.75,0.79463,0.16342,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,19,0,0,0,0,11],[28,32,0.875,0.72768,0.15303,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,22,0,0,1,0,5],[32,32,1.0,0.74107,0.13092,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,25,0,0,0,0,5]]}]},{"i":"6185dc4cc5b78821","q":"Does there exist an integer  $z$  that can be written in two different ways as  $z = x! + y!$ , where  $x, y$  are natural numbers with  $x \\le y$  ?","t":[{"b":3,"e":1.0,"k":"volatile","v":0.30348,"x":0.99107,"p":[[0,6,0.0,0.30348,0.34768,0.0,0.14286,0.32143,0.0,1.0,9,5,0,9,0,10,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,5],[4,6,0.6667,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[6,6,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":6,"e":0.28571,"k":"volatile","v":0.36607,"x":0.74554,"p":[[0,14,0.0,0.41516,0.32802,0.14286,0.28571,0.60682,0.0,1.0,3,6,0,3,0,7,0,0,9,0,0,4,0,0,1,0,0,2,0,0,0,0,6],[4,14,0.2857,0.74554,0.29175,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,6,0,0,0,0,0,3,0,0,2,0,16],[8,14,0.5714,0.5982,0.2976,0.28571,0.42859,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,9,0,0,9,0,0,2,0,0,1,0,0,1,0,10],[12,14,0.8571,0.38393,0.12596,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,3,0,0,0,0,0],[14,14,1.0,0.36607,0.07936,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,16,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6b009ccd91a72732","q":"Each square of a  $100\\times 100$  board is painted with one of  $100$  different colours, so that each colour is used exactly  $100$  times. Show that there exists a row or column of the chessboard in which at least  $10$  colours are used.","t":[{"b":2,"e":1.0,"k":"rising","v":0.70089,"x":1.0,"p":[[0,30,0.0,0.83036,0.29545,0.85714,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[4,30,0.1333,0.72321,0.33681,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,19],[8,30,0.2667,0.70089,0.34137,0.28571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,18],[12,30,0.4,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,30,0.5333,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.28571,"k":"falling","v":0.33036,"x":0.87054,"p":[[0,27,0.0,0.77232,0.31715,0.42857,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,21],[4,27,0.1481,0.83036,0.29545,0.85714,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[8,27,0.2963,0.87054,0.27049,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[12,27,0.4444,0.82143,0.3093,0.82143,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[16,27,0.5926,0.80357,0.31491,0.39286,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,23],[20,27,0.7407,0.85269,0.28003,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,25],[24,27,0.8889,0.33036,0.06622,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.33036,0.06622,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"50209348cd4795b8","q":"Express  $\\sqrt{2 + \\sqrt{3}}$  in the form  $\\frac{a + \\sqrt{b}}{\\sqrt{c}}$ , where  $a$  is a positive integer and  $b$  and  $c$  are square-free positive integers.","t":[{"b":6,"e":0.42857,"k":"flat","v":0.55802,"x":0.73661,"p":[[0,23,0.0,0.73661,0.26271,0.42857,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,3,0,0,0,0,15],[4,23,0.1739,0.73214,0.27606,0.42857,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,11,0,0,3,0,0,1,0,0,0,0,16],[8,23,0.3478,0.71429,0.27199,0.42857,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,12,0,0,2,0,0,2,0,0,1,0,14],[12,23,0.5217,0.71875,0.2461,0.42857,0.64286,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,3,0,0,0,0,13],[16,23,0.6957,0.60714,0.24744,0.42857,0.42857,0.78571,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,15,0,0,4,0,0,3,0,0,0,0,8],[20,23,0.8696,0.55802,0.26812,0.42857,0.42857,0.67857,0.2857,1.0,0,8,0,0,0,0,0,0,7,0,0,13,0,0,4,0,0,0,0,0,0,0,8],[23,23,1.0,0.71429,0.28121,0.42857,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,11,0,0,2,0,0,2,0,0,0,0,15]]},{"b":7,"e":1.0,"k":"falling","v":0.58478,"x":0.77232,"p":[[0,19,0.0,0.77232,0.27166,0.42857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,0,0,0,1,0,18],[4,19,0.2105,0.77232,0.22548,0.57143,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,4,0,0,0,0,15],[8,19,0.4211,0.68304,0.25187,0.42857,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,10,0,0,7,0,0,2,0,0,1,0,11],[12,19,0.6316,0.58478,0.20935,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,8,0,0,10,0,0,5,0,0,1,0,4],[16,19,0.8421,0.59375,0.2055,0.42857,0.57143,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,10,0,0,11,0,0,4,0,0,0,0,5],[19,19,1.0,0.59374,0.22048,0.42857,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,12,0,0,9,0,0,3,0,0,0,0,6]]}]},{"i":"e9d0d939c43c7196","q":"Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^{b}$, where $a$ and $b$ are positive integers with $b \\geqslant 2$ ?","t":[{"b":1,"e":1.0,"k":"flat","v":0.86158,"x":0.99554,"p":[[0,13,0.0,0.86158,0.29558,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,25],[4,13,0.3077,0.92857,0.18898,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[8,13,0.6154,0.92411,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[12,13,0.9231,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[13,13,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":2,"e":1.0,"k":"flat","v":0.79464,"x":0.99107,"p":[[0,10,0.0,0.91964,0.22851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,27],[4,10,0.4,0.79464,0.36759,0.85714,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,21],[8,10,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[10,10,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"20d457442aa75389","q":"Find  $ f(x): (0,\\plus{}\\infty) \\to \\mathbb R$  such that \\[ f(x)\\cdot f(y) \\plus{} f(\\frac{2008}{x})\\cdot f(\\frac{2008}{y})\\equal{}2f(x\\cdot y)\\] and  $ f(2008)\\equal{}1$  for  $ \\forall x \\in (0,\\plus{}\\infty)$ .","t":[{"b":2,"e":0.85714,"k":"falling","v":0.62497,"x":0.88839,"p":[[0,30,0.0,0.86606,0.17476,0.71429,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,18],[4,30,0.1333,0.85714,0.18211,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,4,0,17],[8,30,0.2667,0.87946,0.16793,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,19],[12,30,0.4,0.85266,0.16166,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,16],[16,30,0.5333,0.8817,0.19448,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,4,0,0,3,0,21],[20,30,0.6667,0.88839,0.17762,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,0,0,22],[24,30,0.8,0.86161,0.17672,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,0,0,19],[28,30,0.9333,0.62497,0.15048,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,0,5,0,0,7,0,0],[30,30,1.0,0.65177,0.15948,0.57143,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,1,0,0,15,0,0,5,0,0,9,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.70089,"x":0.79911,"p":[[0,26,0.0,0.73661,0.27689,0.57143,0.78571,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,1,0,0,6,0,0,5,0,0,4,0,12],[4,26,0.1538,0.78121,0.24484,0.57132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,0,4,0,14],[8,26,0.3077,0.79911,0.1984,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,2,0,13],[12,26,0.4615,0.70089,0.27283,0.57143,0.71429,1.0,0.0,1.0,1,10,1,1,0,1,0,0,2,0,0,3,0,0,4,0,0,9,0,0,2,0,10],[16,26,0.6154,0.74106,0.26831,0.53539,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,5,0,0,3,0,0,6,0,0,1,0,14],[20,26,0.7692,0.72768,0.08268,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,1,0,2],[24,26,0.9231,0.74554,0.1461,0.71429,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,23,0,0,2,0,5],[26,26,1.0,0.72321,0.08702,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,26,0,0,1,0,2]]}]},{"i":"b006e6151d1bb272","q":"Find a positive integer  $n$  with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of  $n$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,12,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,12,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,46,0.0,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[4,46,0.087,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,46,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,46,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"71f4c5ee32333ba1","q":"Find all  $(m, n)$  positive integer pairs such that both  $\\frac{3n^2}{m}$  and  $\\sqrt{n^2+m}$  are integers.","t":[{"b":6,"e":0.71429,"k":"falling","v":0.74107,"x":0.9375,"p":[[0,29,0.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[4,29,0.1379,0.8616,0.14054,0.71429,0.9285,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,16],[8,29,0.2759,0.80357,0.12753,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,2,0,9],[12,29,0.4138,0.78571,0.12877,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,1,0,8],[16,29,0.5517,0.7723,0.12302,0.71429,0.71429,0.71429,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,0,0,7],[20,29,0.6897,0.76338,0.11635,0.71429,0.71429,0.71429,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,6],[24,29,0.8276,0.80357,0.13243,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,10],[28,29,0.9655,0.76338,0.13177,0.71429,0.71429,0.71429,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,0,0,0,7],[29,29,1.0,0.74107,0.09061,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,1,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,57,0.0,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[4,57,0.0702,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,57,0.1404,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,57,0.2105,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,57,0.2807,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,57,0.3509,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,57,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,57,0.4912,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,57,0.5614,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[36,57,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,57,0.7018,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,57,0.7719,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,57,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,57,0.9123,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,57,0.9825,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[57,57,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"4b3f669921b9299a","q":"Find all function  $f:\\mathbb{N}^*\\rightarrow \\mathbb{N}^*$  that satisfy: $(f(1))^3+(f(2))^3+...+(f(n))^3=(f(1)+f(2)+...+f(n))^2$","t":[{"b":5,"e":1.0,"k":"rising","v":0.60714,"x":1.0,"p":[[0,32,0.0,0.74554,0.24675,0.42857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,4,0,0,1,0,14],[4,32,0.125,0.64286,0.27199,0.42857,0.5,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,15,0,0,4,0,0,1,0,0,0,0,11],[8,32,0.25,0.60714,0.22588,0.42857,0.5,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,16,0,0,6,0,0,3,0,0,0,0,7],[12,32,0.375,0.82589,0.25439,0.42857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,1,0,21],[16,32,0.5,0.91964,0.21706,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,28],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.71875,"x":1.0,"p":[[0,65,0.0,0.71875,0.26362,0.42857,0.64286,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,1,0,0,1,0,14],[4,65,0.0615,0.78571,0.24744,0.53571,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,3,0,0,1,0,17],[8,65,0.1231,0.85714,0.23419,0.67857,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,1,0,0,0,0,23],[12,65,0.1846,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[16,65,0.2462,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,65,0.3077,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,65,0.3692,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,65,0.4308,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,65,0.4923,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,65,0.5538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,65,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,65,0.6769,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,65,0.7385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,65,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,65,0.8615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,65,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,65,0.9846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[65,65,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2acece7c13e27f27","q":"Find all  $ x, y, z\\in\\mathbb{Z}^+ $  such that \\[ (x-y)(y-z)(z-x)=x+y+z \\]","t":[{"b":4,"e":0.85714,"k":"flat","v":0.91071,"x":0.94642,"p":[[0,16,0.0,0.91517,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16],[4,16,0.25,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[8,16,0.5,0.94642,0.07785,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[12,16,0.75,0.91071,0.09943,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,16],[16,16,1.0,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19]]},{"b":6,"e":1.0,"k":"flat","v":0.91964,"x":0.95088,"p":[[0,29,0.0,0.94195,0.07874,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[4,29,0.1379,0.91964,0.08703,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[8,29,0.2759,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,29,0.4138,0.94643,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[16,29,0.5517,0.93749,0.07937,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[20,29,0.6897,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[24,29,0.8276,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[28,29,0.9655,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[29,29,1.0,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18]]}]},{"i":"2cef56ec3b8c92d0","q":"Find all  $n$  natural numbers such that for each of them there exist  $p , q$  primes such that these terms satisfy. $1.$   $p+2=q$  $2.$   $2^n+p$  and  $2^n+q$  are primes.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.97767,"x":1.0,"p":[[0,5,0.0,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,5,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[5,5,1.0,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":7,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,9,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,9,0.4444,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,9,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[9,9,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5b91396500f910e7","q":"Find all function  $ f: \\mathbb{N}\\rightarrow\\mathbb{N}$  satisfy  $ f(mn)\\plus{}f(m\\plus{}n)\\equal{}f(m)f(n)\\plus{}1$  for all natural number  $ n$","t":[{"b":0,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,33,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,33,0.1212,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,33,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,33,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,33,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,13,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,13,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,13,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"58b7a20be53cb865","q":"Find all functions  $f : R \\to R$  such that for all real  $x,y$ ,  $f(f(x)+y) = f(x^2 -y)+4y f(x)$","t":[{"b":2,"e":0.71429,"k":"rising","v":0.52679,"x":0.97321,"p":[[0,41,0.0,0.66509,0.39558,0.14286,1.0,1.0,0.14,1.0,0,17,0,0,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,17],[4,41,0.0976,0.72311,0.35359,0.39285,0.92857,1.0,0.14,1.0,0,16,0,0,0,7,0,0,1,0,0,1,0,0,2,0,0,0,0,0,5,0,16],[8,41,0.1951,0.64732,0.38628,0.14286,0.85714,1.0,0.14286,1.0,0,14,0,0,0,11,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,14],[12,41,0.2927,0.54017,0.40206,0.14286,0.4998,1.0,0.0,1.0,1,13,0,1,0,13,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,13],[16,41,0.3902,0.52679,0.38038,0.14286,0.4286,1.0,0.14286,1.0,0,11,0,0,0,14,0,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,11],[20,41,0.4878,0.58481,0.40462,0.14286,0.78564,1.0,0.0,1.0,1,13,0,1,0,11,0,0,3,0,0,0,0,0,0,0,0,1,0,0,3,0,13],[24,41,0.5854,0.87946,0.22899,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,21],[28,41,0.6829,0.78125,0.29663,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,5,0,0,1,0,18],[32,41,0.7805,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,41,0.878,0.95534,0.11543,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[40,41,0.9756,0.93304,0.14719,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[41,41,1.0,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":5,"e":0.85714,"k":"falling","v":0.19197,"x":0.83482,"p":[[0,34,0.0,0.83482,0.30537,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,23],[4,34,0.1176,0.71427,0.36597,0.2857,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,18],[8,34,0.2353,0.8125,0.30397,0.57143,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,22],[12,34,0.3529,0.74107,0.37018,0.28571,1.0,1.0,0.14286,1.0,0,20,0,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,20],[16,34,0.4706,0.55804,0.37857,0.14286,0.42857,1.0,0.14286,1.0,0,11,0,0,0,10,0,0,6,0,0,0,0,0,2,0,0,0,0,0,3,0,11],[20,34,0.5882,0.47768,0.3001,0.14286,0.42857,0.60714,0.14286,1.0,0,5,0,0,0,9,0,0,3,0,0,8,0,0,4,0,0,1,0,0,2,0,5],[24,34,0.7059,0.36612,0.26712,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,11,0,0,9,0,0,7,0,0,1,0,0,0,0,0,0,0,4],[28,34,0.8235,0.30804,0.28818,0.14286,0.14286,0.46429,0.14286,1.0,0,3,0,0,0,23,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,3],[32,34,0.9412,0.24981,0.18566,0.14286,0.14286,0.32143,0.14,1.0,0,1,0,0,0,21,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[34,34,1.0,0.19197,0.18423,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"330ee7fba1057052","q":"Find all functions  $f: \\mathbb{Q}\\to\\mathbb{R}$  which fulfill the following conditions: \r\n\r\na)  $f(1)+1>0$ ;\r\n\r\nb)  $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$ , for all  $x,y\\in\\mathbb{Q}$ ;\r\n\r\nc)  $f(x) = 2f(x+1) +x+2$ , for every  $x\\in\\mathbb{Q}$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,42,0.0,0.94643,0.06916,0.85714,1.0,1.0,0.8571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[4,42,0.0952,0.91964,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[8,42,0.1905,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[12,42,0.2857,0.92857,0.08749,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[16,42,0.381,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,42,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,42,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.70981,"x":0.94196,"p":[[0,73,0.0,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,73,0.0548,0.91517,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[8,73,0.1096,0.92857,0.15972,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,22],[12,73,0.1644,0.81249,0.18709,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,7,0,12],[16,73,0.2192,0.83929,0.16656,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,5,0,14],[20,73,0.274,0.79018,0.17491,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,14,0,0,2,0,11],[24,73,0.3288,0.78573,0.20511,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,7,0,11],[28,73,0.3836,0.83928,0.21354,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,3,0,18],[32,73,0.4384,0.81696,0.1439,0.71429,0.85714,1.0,0.5714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,9,0,9],[36,73,0.4932,0.75445,0.19639,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,5,0,0,6,0,9],[40,73,0.5479,0.83257,0.15632,0.71429,0.85707,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,1,0,4,0,13],[44,73,0.6027,0.78793,0.20005,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,5,1,11],[48,73,0.6575,0.70981,0.17673,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,7,0,0,8,0,4],[52,73,0.7123,0.76337,0.18074,0.57143,0.78564,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,0,9,0,7],[56,73,0.7671,0.77677,0.20184,0.57143,0.85707,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,7,0,10],[60,73,0.8219,0.76785,0.20748,0.57143,0.78564,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,5,0,11],[64,73,0.8767,0.72768,0.17985,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,11,0,0,1,0,8],[68,73,0.9315,0.87053,0.14445,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,15],[72,73,0.9863,0.82141,0.17498,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,12,0,10],[73,73,1.0,0.80356,0.17769,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,6,0,11]]}]},{"i":"8b99c73c7c459e86","q":"Find all functions  $f:(0,+\\infty) \\to (0,+\\infty)$  that satisfy  $(i)$   $f(xf(y))=yf(x), \\forall x,y > 0,$  $(ii)$   $\\displaystyle\\lim_{x\\to+\\infty} f(x) = 0.$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.14277,"x":0.15179,"p":[[0,23,0.0,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.14723,"x":0.19196,"p":[[0,68,0.0,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,68,0.1765,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.17411,0.10555,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,68,0.3529,0.19196,0.10479,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,25,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[28,68,0.4118,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,68,0.4706,0.16964,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.16071,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,68,0.5882,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.16955,0.09064,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,68,0.7647,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.18304,0.11425,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[60,68,0.8824,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.16071,0.07784,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"16eca847f38f32a9","q":"Find all integer values of  $a$  such that the quadratic expression  $(x+a)(x+1991) +1$  can be factored as a product  $(x+b)(x+c)$  where  $b,c$  are integers.","t":[{"b":5,"e":1.0,"k":"flat","v":0.32143,"x":0.86607,"p":[[0,39,0.0,0.32143,0.28347,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,20,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4],[4,39,0.1026,0.58929,0.41611,0.14286,0.71429,1.0,0.14286,1.0,0,16,0,0,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,16],[8,39,0.2051,0.69643,0.3989,0.14286,1.0,1.0,0.0,1.0,1,20,0,1,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,20],[12,39,0.3077,0.86607,0.3008,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[16,39,0.4103,0.49554,0.42104,0.14286,0.14286,1.0,0.0,1.0,1,13,0,1,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13],[20,39,0.5128,0.5625,0.4164,0.14286,0.42857,1.0,0.14286,1.0,0,15,0,0,0,15,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,15],[24,39,0.6154,0.5625,0.40396,0.14286,0.4286,1.0,0.14286,1.0,0,14,0,0,0,14,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,14],[28,39,0.7179,0.47322,0.39194,0.14286,0.14293,1.0,0.14286,1.0,0,11,0,0,0,17,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,11],[32,39,0.8205,0.42849,0.37122,0.14286,0.14286,1.0,0.14,1.0,0,9,0,0,0,18,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,9],[36,39,0.9231,0.45071,0.38498,0.14286,0.14286,1.0,0.0,1.0,1,10,0,1,0,16,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,10],[39,39,1.0,0.41964,0.39438,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,10]]},{"b":6,"e":1.0,"k":"rising","v":0.42857,"x":1.0,"p":[[0,41,0.0,0.42857,0.3977,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,10],[4,41,0.0976,0.70982,0.39526,0.14286,1.0,1.0,0.14286,1.0,0,20,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,20],[8,41,0.1951,0.76339,0.38234,0.35714,1.0,1.0,0.0,1.0,1,23,0,1,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,23],[12,41,0.2927,0.64286,0.41033,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,18],[16,41,0.3902,0.83929,0.33455,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[20,41,0.4878,0.92857,0.22588,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[24,41,0.5854,0.92857,0.22588,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"12bfefeede5f38de","q":"Find all functions  $f : R_{>0} \\to  R$  such that  $f \\left(\\frac{x}{y}\\right) = f(x) + f(y) - f(x)f(y)$  for all  $x, y \\in R_{>0}$ . Here,  $R_{>0}$  denotes the set of all positive real numbers.\n\nNguy\u1ec5n Duy Th\u00e1i S\u01a1n","t":[{"b":0,"e":0.14286,"k":"volatile","v":0.27233,"x":0.7099,"p":[[0,6,0.0,0.7099,0.2588,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,9,0,0,3,0,0,3,0,0,4,0,11],[4,6,0.6667,0.27233,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,3,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.33036,0.15335,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,5,0,0,18,0,0,6,0,0,1,0,0,1,0,0,1,0,0]]},{"b":1,"e":1.0,"k":"volatile","v":0.71875,"x":0.95087,"p":[[0,8,0.0,0.84375,0.19352,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,14],[4,8,0.5,0.71875,0.26603,0.42857,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,7,0,0,0,0,0,4,0,0,10,0,8],[8,8,1.0,0.95087,0.09857,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24]]}]},{"i":"4c6e02d94421e9cf","q":"Find all non-negative integer solutions of the equation\n\\[2^x + 3^y = z^2 .\\]","t":[{"b":1,"e":0.57143,"k":"flat","v":0.59819,"x":0.85267,"p":[[0,33,0.0,0.83482,0.2055,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,0,5,0,17],[4,33,0.1212,0.60714,0.14725,0.57143,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,19,0,0,3,0,0,4,0,1],[8,33,0.2424,0.60268,0.10555,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,0,4,0,0,3,0,0],[12,33,0.3636,0.62054,0.12682,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,2,0,0,2,0,2],[16,33,0.4848,0.59821,0.10374,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,22,0,0,6,0,0,0,0,1],[20,33,0.6061,0.59819,0.10972,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,23,0,0,4,0,0,1,0,1],[24,33,0.7273,0.7857,0.15569,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,7,0,8],[28,33,0.8485,0.73213,0.1627,0.57143,0.71429,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,8,0,0,5,0,6],[32,33,0.9697,0.85267,0.15767,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,8,0,14],[33,33,1.0,0.7857,0.14727,0.71429,0.85707,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,11,0,6]]},{"b":7,"e":0.85714,"k":"rising","v":0.77232,"x":0.97768,"p":[[0,10,0.0,0.8125,0.16536,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,7,0,11],[4,10,0.4,0.83929,0.15872,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,7,0,13],[8,10,0.8,0.77232,0.17445,0.67857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,2,0,10],[10,10,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"7e7d0958c8b0b71c","q":"Find all natural numbers  $a>1$ , with the property that every prime divisor of  $a^6-1$  divides also at least one of the numbers  $a^3-1$ ,  $a^2-1$ .\n\n*K. Dochev*","t":[{"b":1,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,54,0.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,54,0.0741,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,54,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,54,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,54,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,54,0.3704,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,54,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,54,0.5185,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,54,0.5926,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,54,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,54,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,54,0.8148,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,54,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,54,0.963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[54,54,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.85714,"k":"flat","v":0.875,"x":0.99107,"p":[[0,21,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,21,0.1905,0.91964,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,18],[8,21,0.381,0.9375,0.1234,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[12,21,0.5714,0.90179,0.16146,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,21],[16,21,0.7619,0.91518,0.1551,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,23],[20,21,0.9524,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[21,21,1.0,0.875,0.1915,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,2,0,21]]}]},{"i":"4d9bd45626c1cfcf","q":"Find all natural numbers  $n> 1$  for which the following applies:\nThe sum of the number  $n$  and its second largest divisor is  $2013$ .\n\n(R. Henner, Vienna)","t":[{"b":3,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,60,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,60,0.0667,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,60,0.1333,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,60,0.2,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,60,0.2667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,60,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,60,0.4,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,60,0.4667,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,60,0.5333,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,60,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,60,0.7333,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,60,0.8,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,60,0.8667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,60,0.9333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,60,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,48,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,48,0.0833,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[8,48,0.1667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,48,0.25,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[16,48,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,48,0.4167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,48,0.5,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,48,0.5833,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,48,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,48,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,48,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,48,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,48,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4eaa5b6be55559ff","q":"Find all ordered triples  $(a,b, c)$  of positive integers which satisfy  $5^a + 3^b - 2^c = 32$","t":[{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,70,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,70,0.0571,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[8,70,0.1143,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,70,0.1714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,70,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,70,0.2857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,70,0.3429,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,70,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,70,0.4571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,70,0.5143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,70,0.5714,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,70,0.6286,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,70,0.6857,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[52,70,0.7429,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[56,70,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,70,0.8571,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[64,70,0.9143,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[68,70,0.9714,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[70,70,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]},{"b":7,"e":0.28571,"k":"falling","v":0.42411,"x":1.0,"p":[[0,61,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,61,0.0656,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,61,0.1311,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,61,0.1967,0.82588,0.26664,0.5354,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,22],[16,61,0.2623,0.79464,0.25739,0.64286,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,6,0,0,0,0,0,6,0,0,0,0,18],[20,61,0.3279,0.67856,0.30514,0.42857,0.78571,1.0,0.143,1.0,0,12,0,0,0,1,0,0,6,0,0,6,0,0,2,0,0,1,0,0,4,0,12],[24,61,0.3934,0.58482,0.29093,0.42857,0.4286,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,12,0,0,3,0,0,3,0,0,1,0,8],[28,61,0.459,0.73659,0.2792,0.42857,0.78564,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,6,0,0,2,0,0,4,0,0,1,0,15],[32,61,0.5246,0.58929,0.25692,0.42857,0.4286,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,14,0,0,4,0,0,2,0,0,0,0,8],[36,61,0.5902,0.55355,0.22798,0.42857,0.42859,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,14,0,0,3,0,0,4,0,0,2,0,4],[40,61,0.6557,0.57143,0.27664,0.42857,0.42857,0.75,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,11,0,0,2,0,0,4,0,0,1,0,7],[44,61,0.7213,0.60267,0.24932,0.42857,0.4286,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,13,0,0,2,0,0,4,0,0,3,0,6],[48,61,0.7869,0.50444,0.2624,0.39286,0.4286,0.60714,0.14286,1.0,0,4,0,0,0,5,0,0,3,0,0,12,0,0,4,0,0,2,0,0,2,0,4],[52,61,0.8525,0.50891,0.23401,0.28571,0.42857,0.71407,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,12,0,0,2,0,0,4,0,0,2,0,3],[56,61,0.918,0.54466,0.22987,0.42857,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,14,0,0,2,0,0,3,0,0,4,0,3],[60,61,0.9836,0.42856,0.17856,0.28571,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,17,0,0,3,0,0,1,0,0,1,0,1],[61,61,1.0,0.42411,0.21572,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,4,0,0,7,0,0,17,0,0,0,0,0,0,0,0,2,0,2]]}]},{"i":"4d3c3e69a27362f2","q":"Find all ordered triplets  $(p,q,r)$  of positive integers such that  $p$  and  $q$  are two (not necessarily distinct) primes,  $r$  is even, and \n\\[p^3+q^2=4r^2+45r+103.\\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.8415,"x":1.0,"p":[[0,60,0.0,0.86161,0.16935,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,12,0,14],[4,60,0.0667,0.87276,0.15335,0.85714,0.85714,1.0,0.357,1.0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,0,0,15,0,13],[8,60,0.1333,0.8415,0.18189,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,8,1,14],[12,60,0.2,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,20],[16,60,0.2667,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,60,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,60,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,60,0.4667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,60,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,60,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,60,0.7333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,60,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,60,0.8667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":0.85714,"k":"flat","v":0.80356,"x":0.9375,"p":[[0,56,0.0,0.83928,0.15465,0.857,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,15,0,10],[4,56,0.0714,0.86163,0.16926,0.85711,0.85714,1.0,0.43,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,10,0,15],[8,56,0.1429,0.82812,0.18717,0.57143,0.85714,1.0,0.5,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,8,0,14],[12,56,0.2143,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[16,56,0.2857,0.83034,0.14913,0.857,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,17,0,8],[20,56,0.3571,0.90179,0.12595,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,13,0,16],[24,56,0.4286,0.85714,0.15567,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,12,0,13],[28,56,0.5,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,17,0,12],[32,56,0.5714,0.83481,0.15612,0.857,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,18,0,8],[36,56,0.6429,0.83481,0.15615,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,16,0,9],[40,56,0.7143,0.80356,0.15467,0.67857,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,14,0,7],[44,56,0.7857,0.83705,0.15693,0.82132,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,13,1,10],[48,56,0.8571,0.83258,0.15832,0.80346,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,14,0,10],[52,56,0.9286,0.8482,0.15949,0.85714,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,13,0,12],[56,56,1.0,0.81694,0.14831,0.71429,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,16,0,7]]}]},{"i":"d8d0604f5bbffc20","q":"Find all pairs  $(p,q)$  of positive primes such that the equation  $3x^2 - px + q = 0$  has two distinct rational roots.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.72321,"x":0.875,"p":[[0,14,0.0,0.82589,0.13709,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,12],[4,14,0.2857,0.875,0.14617,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,1,0,18],[8,14,0.5714,0.85268,0.14054,0.71429,0.78571,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,15],[12,14,0.8571,0.72321,0.07936,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,0,0,2],[14,14,1.0,0.72321,0.06121,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,1,0,1]]},{"b":1,"e":0.71429,"k":"flat","v":0.76339,"x":0.93749,"p":[[0,42,0.0,0.82143,0.15567,0.71429,0.71429,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,1,0,13],[4,42,0.0952,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[8,42,0.1905,0.7991,0.12808,0.71429,0.71429,1.0,0.714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,9],[12,42,0.2857,0.78112,0.14285,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,20,0,0,2,0,8],[16,42,0.381,0.76339,0.10479,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,1,0,5],[20,42,0.4762,0.79464,0.13333,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,1,0,9],[24,42,0.5714,0.82143,0.13832,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,12],[28,42,0.6667,0.91058,0.13263,0.71429,1.0,1.0,0.71,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[32,42,0.7619,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[36,42,0.8571,0.93749,0.11259,0.96425,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[40,42,0.9524,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[42,42,1.0,0.83036,0.14032,0.71429,0.71429,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,13]]}]},{"i":"0ced0ee94ced85b9","q":"Find all polynomial  $P(x)$  with degree  $\\leq n$ and non negative coefficients such that   $$ P(x)P(\\frac{1}{x})\\leq P(1)^2 $$   for all positive  $x$ . Here  $n$  is a natuaral number","t":[{"b":0,"e":1.0,"k":"rising","v":0.74554,"x":1.0,"p":[[0,37,0.0,0.80804,0.29583,0.42857,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,22],[4,37,0.1081,0.74554,0.27603,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,1,0,0,0,0,17],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,37,0.4324,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[20,37,0.5405,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.71875,"x":0.98214,"p":[[0,38,0.0,0.84379,0.24309,0.57143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,0,0,0,1,0,22],[4,38,0.1053,0.71875,0.30823,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,17],[8,38,0.2105,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[12,38,0.3158,0.96429,0.13831,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[16,38,0.4211,0.91071,0.20748,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,27],[20,38,0.5263,0.96427,0.12376,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[24,38,0.6316,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[28,38,0.7368,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,38,0.8421,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[36,38,0.9474,0.91071,0.20748,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,27],[38,38,1.0,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29]]}]},{"i":"e7493b3685d0f1dc","q":"Find all pairs of  $a$ ,  $b$  of positive integers satisfying the equation  $2a^2 = 3b^3$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.68746,"x":0.82589,"p":[[0,55,0.0,0.74999,0.23147,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,6,0,0,2,0,12],[4,55,0.0727,0.70982,0.19719,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,7,0,0,2,0,8],[8,55,0.1455,0.6964,0.17036,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,13,0,0,2,0,5],[12,55,0.2182,0.76782,0.18476,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,10,0,0,1,0,11],[16,55,0.2909,0.6875,0.20025,0.57143,0.57143,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,7,0,0,0,0,8],[20,55,0.3636,0.68746,0.19048,0.57143,0.57143,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,7,0,0,1,0,7],[24,55,0.4364,0.73661,0.1992,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,5,0,0,4,0,9],[28,55,0.5091,0.70534,0.18878,0.57143,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,9,0,0,0,0,8],[32,55,0.5818,0.77676,0.14702,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,3,0,8],[36,55,0.6545,0.76782,0.19152,0.57143,0.78564,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,0,6,0,10],[40,55,0.7273,0.82589,0.18808,0.57143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,2,0,16],[44,55,0.8,0.76337,0.18075,0.57143,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,7,0,0,3,0,10],[48,55,0.8727,0.77679,0.18877,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,9,0,0,1,0,12],[52,55,0.9455,0.78124,0.18554,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,0,2,0,12],[55,55,1.0,0.71429,0.17128,0.57143,0.64286,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,0,2,0,7]]},{"b":3,"e":0.42857,"k":"flat","v":0.56697,"x":0.75893,"p":[[0,19,0.0,0.70088,0.2,0.57143,0.64286,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,5,0,0,4,0,7],[4,19,0.2105,0.683,0.20745,0.571,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,5,0,0,3,0,7],[8,19,0.4211,0.75893,0.19377,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,11,0,0,6,0,8],[12,19,0.6316,0.65625,0.14222,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,13,0,0,4,0,1],[16,19,0.8421,0.56697,0.14053,0.42859,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,17,0,0,3,0,0,0,0,2],[19,19,1.0,0.63392,0.13334,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,12,0,0,2,0,1]]}]},{"i":"26339acff973294a","q":"Find all pairs  $(x,y)$  of positive real numbers such that  $xy$  is an integer and  $x+y = \\lfloor x^2 - y^2 \\rfloor$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.91518,"x":0.98661,"p":[[0,41,0.0,0.91518,0.11214,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,19],[4,41,0.0976,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,41,0.1951,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,41,0.2927,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,41,0.3902,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,41,0.4878,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,41,0.5854,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[28,41,0.6829,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,41,0.7805,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,41,0.878,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,41,0.9756,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[41,41,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.88839,"x":0.95982,"p":[[0,49,0.0,0.88839,0.12745,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,17],[4,49,0.0816,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[8,49,0.1633,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[12,49,0.2449,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[16,49,0.3265,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[20,49,0.4082,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[24,49,0.4898,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[28,49,0.5714,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[32,49,0.6531,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[36,49,0.7347,0.89286,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[40,49,0.8163,0.91071,0.12242,0.82143,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[44,49,0.898,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[48,49,0.9796,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[49,49,1.0,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25]]}]},{"i":"87da525e135d6186","q":"Find all positive integers  $x$  such that the product of all digits of  $x$  is given by  $x^2 - 10 \\cdot x - 22.$","t":[{"b":5,"e":0.71429,"k":"flat","v":0.875,"x":0.96875,"p":[[0,29,0.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[4,29,0.1379,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[8,29,0.2759,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[12,29,0.4138,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,29,0.5517,0.91071,0.1171,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,19],[20,29,0.6897,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[24,29,0.8276,0.875,0.12242,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,14],[28,29,0.9655,0.89732,0.12492,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,18],[29,29,1.0,0.88839,0.11143,0.85711,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,14]]},{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,27,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,27,0.1481,0.97321,0.05578,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,27,0.2963,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,27,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6c4ba93c580745a2","q":"Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.79464,"x":0.91071,"p":[[0,60,0.0,0.875,0.18123,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,2,0,20],[4,60,0.0667,0.91071,0.16269,0.85711,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,2,0,23],[8,60,0.1333,0.83035,0.18363,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,9,0,0,5,0,14],[12,60,0.2,0.87054,0.17261,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,1,0,19],[16,60,0.2667,0.81249,0.16919,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,1,0,13],[20,60,0.3333,0.81249,0.16919,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,14,0,0,3,0,12],[24,60,0.4,0.83036,0.18013,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,11,0,0,4,0,14],[28,60,0.4667,0.85267,0.13115,0.71429,0.85707,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,13],[32,60,0.5333,0.8125,0.1729,0.71429,0.85707,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,12,0,0,6,0,11],[36,60,0.6,0.84822,0.15126,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,4,0,14],[40,60,0.6667,0.83929,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,13,0,0,2,0,15],[44,60,0.7333,0.81696,0.1439,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,12],[48,60,0.8,0.79897,0.15923,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,1,0,11],[52,60,0.8667,0.79464,0.18536,0.71429,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,10,0,0,7,0,10],[56,60,0.9333,0.86161,0.16164,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,2,0,17],[60,60,1.0,0.83481,0.20552,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,9,0,0,0,0,18]]},{"b":6,"e":1.0,"k":"flat","v":0.87946,"x":0.99107,"p":[[0,43,0.0,0.89732,0.16457,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,3,0,21],[4,43,0.093,0.90625,0.16602,0.82132,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,23],[8,43,0.186,0.89284,0.17499,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,1,0,22],[12,43,0.2791,0.87946,0.14335,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,7,0,16],[16,43,0.3721,0.91518,0.16311,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,1,0,24],[20,43,0.4651,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[24,43,0.5581,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,43,0.6512,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,43,0.7442,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[36,43,0.8372,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[40,43,0.9302,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[43,43,1.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30]]}]},{"i":"bf0abc54bb61a304","q":"Find all positive integer triples  $(x, y, z) $  that satisfy the equation   $$ x^4+y^4+z^4=2x^2y^2+2y^2z^2+2z^2x^2-63. $$","t":[{"b":4,"e":0.71429,"k":"falling","v":0.65625,"x":0.85714,"p":[[0,43,0.0,0.85714,0.17128,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,0,0,18],[4,43,0.093,0.69643,0.1171,0.57143,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,19,0,0,0,0,3],[8,43,0.186,0.65625,0.07016,0.57143,0.71429,0.71429,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0],[12,43,0.2791,0.72321,0.13333,0.67857,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,19,0,0,0,0,5],[16,43,0.3721,0.71875,0.12103,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,21,0,0,0,0,4],[20,43,0.4651,0.70981,0.12621,0.57143,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,19,0,0,0,0,4],[24,43,0.5581,0.70088,0.11497,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,20,0,0,0,0,3],[28,43,0.6512,0.67857,0.08748,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,0,0,1],[32,43,0.7442,0.69643,0.07784,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,25,0,0,0,0,1],[36,43,0.8372,0.66963,0.06624,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0],[40,43,0.9302,0.66069,0.0692,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0],[43,43,1.0,0.66964,0.06622,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,4,0.0,0.86607,0.15542,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,0,0,18],[4,4,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9138f8e097e13257","q":"Find all real solutions to the following system of equations.  Carefully justify your answer.\r\n\\[ \\left\\{ \\begin{array}{c} \\displaystyle\\frac{4x^2}{1+4x^2} = y   \\displaystyle\\frac{4y^2}{1+4y^2} = z   \\displaystyle\\frac{4z^2}{1+4z^2} = x \\end{array} \\right. \\]","t":[{"b":0,"e":0.71429,"k":"flat","v":0.80357,"x":0.86607,"p":[[0,13,0.0,0.83929,0.14174,0.71429,0.71429,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,14],[4,13,0.3077,0.83929,0.14174,0.71429,0.71429,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,14],[8,13,0.6154,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[12,13,0.9231,0.80357,0.13243,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,10],[13,13,1.0,0.83929,0.14174,0.71429,0.71429,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,14]]},{"b":3,"e":0.71429,"k":"flat","v":0.82143,"x":0.875,"p":[[0,26,0.0,0.84821,0.15947,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,0,0,16],[4,26,0.1538,0.83929,0.14174,0.71429,0.71429,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,14],[8,26,0.3077,0.875,0.14174,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,18],[12,26,0.4615,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[16,26,0.6154,0.84821,0.14258,0.71429,0.71429,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,15],[20,26,0.7692,0.82143,0.13832,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,12],[24,26,0.9231,0.875,0.14174,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,18],[26,26,1.0,0.84822,0.15947,0.71429,0.85714,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,0,0,16]]}]},{"i":"23a909f005efdf07","q":"Find all positive integers  $n$  such that for all positive integers  $m$ ,  $1<m<n$ , relatively prime to  $n$ ,  $m$  must be a prime number.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,58,0.0,0.94196,0.18161,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[4,58,0.069,0.88839,0.29175,1.0,1.0,1.0,0.0,1.0,3,26,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[8,58,0.1379,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,58,0.2069,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,58,0.2759,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,58,0.3448,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,58,0.4138,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,58,0.4828,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,58,0.5517,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,58,0.6207,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,58,0.6897,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,58,0.7586,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,58,0.8276,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,58,0.8966,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[56,58,0.9655,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[58,58,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":7,"e":0.85714,"k":"flat","v":0.90625,"x":1.0,"p":[[0,91,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,91,0.044,0.90625,0.23313,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,24],[8,91,0.0879,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,91,0.1319,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,91,0.1758,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,91,0.2198,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,91,0.2637,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,91,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,91,0.3516,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,91,0.3956,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,91,0.4396,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,91,0.4835,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,91,0.5275,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,91,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,91,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,91,0.6593,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,91,0.7033,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,91,0.7473,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,91,0.7912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,91,0.8352,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,91,0.8791,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,91,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,91,0.967,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[91,91,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"1c3f7b6ee3954341","q":"Find all real numbers  $a$  for which there exists a function   $f: R \\to R$  asuch that  $x + f(y) =a(y + f(x))$  for all real numbers  $x,y\\in R$ .\n\nI.Voronovich","t":[{"b":0,"e":0.42857,"k":"flat","v":0.61161,"x":0.71875,"p":[[0,94,0.0,0.65178,0.19864,0.42857,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,12,0,0,3,0,4],[4,94,0.0426,0.63393,0.15126,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,19,0,0,1,0,1],[8,94,0.0851,0.64732,0.16745,0.42859,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,17,0,0,2,0,2],[12,94,0.1277,0.71874,0.17307,0.71421,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,12,0,0,11,0,2],[16,94,0.1702,0.66069,0.14618,0.571,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,18,0,0,3,0,1],[20,94,0.2128,0.63393,0.16727,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,17,0,0,1,0,2],[24,94,0.2553,0.61161,0.15663,0.42857,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,18,0,0,2,0,0],[28,94,0.2979,0.65179,0.14698,0.42859,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,19,0,0,4,0,0],[32,94,0.3404,0.64285,0.17857,0.42857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,16,0,0,1,0,3],[36,94,0.383,0.64286,0.17128,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,14,0,0,5,0,1],[40,94,0.4255,0.62945,0.14665,0.42859,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,17,0,0,3,0,0],[44,94,0.4681,0.66072,0.16656,0.42859,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,14,0,0,6,0,1],[48,94,0.5106,0.62947,0.15093,0.42857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,18,0,0,3,0,0],[52,94,0.5532,0.66964,0.13092,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,23,0,0,1,0,1],[56,94,0.5957,0.70981,0.12105,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,24,0,0,2,0,2],[60,94,0.6383,0.70536,0.1234,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,23,0,0,4,0,1],[64,94,0.6809,0.71428,0.11293,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,24,0,0,4,0,1],[68,94,0.7234,0.6875,0.1448,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,22,0,0,2,0,2],[72,94,0.766,0.71429,0.12877,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,21,0,0,6,0,1],[76,94,0.8085,0.70089,0.11495,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,23,0,0,5,0,0],[80,94,0.8511,0.70536,0.13803,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,20,0,0,6,0,1],[84,94,0.8936,0.71875,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0],[88,94,0.9362,0.70982,0.12103,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,21,0,0,7,0,0],[92,94,0.9787,0.71861,0.12619,0.71429,0.71429,0.85704,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,19,0,0,9,0,0],[94,94,1.0,0.69643,0.1171,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,22,0,0,5,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.54017,"x":0.67861,"p":[[0,56,0.0,0.67411,0.21199,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,10,0,0,5,0,5],[4,56,0.0714,0.67861,0.1785,0.53607,0.71429,0.74996,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,13,0,0,5,0,3],[8,56,0.1429,0.5625,0.13333,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,0,13,0,0,0,0,0],[12,56,0.2143,0.58927,0.14174,0.42857,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,3,0,0,15,0,0,1,0,0],[16,56,0.2857,0.61607,0.16146,0.42857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,15,0,0,4,0,0],[20,56,0.3571,0.59819,0.18013,0.42857,0.571,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,11,0,0,2,0,2],[24,56,0.4286,0.54017,0.17399,0.42857,0.42857,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,21,0,0,2,0,0,6,0,0,1,0,2],[28,56,0.5,0.58036,0.16342,0.42857,0.5,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,13,0,0,1,0,1],[32,56,0.5714,0.61607,0.18363,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,12,0,0,3,0,2],[36,56,0.6429,0.58482,0.15714,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,12,0,0,3,0,0],[40,56,0.7143,0.63839,0.15561,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,18,0,0,2,0,1],[44,56,0.7857,0.55357,0.14617,0.42857,0.42857,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,12,0,0,1,0,0],[48,56,0.8571,0.5848,0.16887,0.42857,0.4998,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,12,0,0,2,0,1],[52,56,0.9286,0.57146,0.16362,0.42857,0.4293,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,12,0,0,1,0,1],[56,56,1.0,0.58036,0.15947,0.42857,0.5,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,12,0,0,3,0,0]]}]},{"i":"f3fd8b84ad3497d4","q":"Find all positive rational  $(x,y)$  that satisfy the equation :   $$ yx^y=y+1 $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.95089,"x":0.99107,"p":[[0,21,0.0,0.95089,0.13651,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[4,21,0.1905,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[8,21,0.381,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[12,21,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,21,0.7619,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,21,0.9524,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[21,21,1.0,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28]]},{"b":3,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,17,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,17,0.2353,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[8,17,0.4706,0.95534,0.09745,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,17,0.7059,0.93304,0.1636,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[16,17,0.9412,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[17,17,1.0,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29]]}]},{"i":"1546a3c349f3f2bc","q":"Find all positive integers  $n$  that are quadratic residues modulo all primes greater than  $n$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.47768,"x":0.66964,"p":[[0,16,0.0,0.47768,0.27574,0.39286,0.42857,0.4286,0.14286,1.0,0,6,0,0,0,6,0,0,2,0,0,17,0,0,1,0,0,0,0,0,0,0,6],[4,16,0.25,0.5625,0.34058,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,2,0,0,11,0,0,0,0,0,1,0,0,0,0,11],[8,16,0.5,0.55357,0.32488,0.39286,0.42857,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,1,0,0,12,0,0,0,0,0,2,0,0,1,0,9],[12,16,0.75,0.66964,0.32031,0.42857,0.42857,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,15],[16,16,1.0,0.65179,0.34058,0.42857,0.42857,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,15]]},{"b":3,"e":0.14286,"k":"flat","v":0.37947,"x":0.60268,"p":[[0,28,0.0,0.43304,0.27078,0.25,0.42857,0.4286,0.14286,1.0,0,5,0,0,0,8,0,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,5],[4,28,0.1429,0.43304,0.28679,0.14286,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,10,0,0,3,0,0,12,0,0,0,0,0,1,0,0,2,0,4],[8,28,0.2857,0.5134,0.27166,0.42857,0.42857,0.50002,0.14286,1.0,0,6,0,0,0,4,0,0,2,0,0,18,0,0,0,0,0,1,0,0,1,0,6],[12,28,0.4286,0.57589,0.31234,0.42857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,2,0,0,15,0,0,0,0,0,0,0,0,1,0,10],[16,28,0.5714,0.55357,0.34022,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,2,0,0,12,0,0,0,0,0,0,0,0,0,0,11],[20,28,0.7143,0.60268,0.34392,0.42857,0.42857,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,13],[24,28,0.8571,0.37947,0.20395,0.14286,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,9,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,2],[28,28,1.0,0.55357,0.33264,0.39285,0.42857,1.0,0.14286,1.0,0,10,0,0,0,7,0,0,1,0,0,13,0,0,0,0,0,0,0,0,1,0,10]]}]},{"i":"cfdfb6f7ede78362","q":"Find all rational numbers  $k$  such that  $0 \\le k \\le \\frac{1}{2}$  and  $\\cos k \\pi$  is rational.","t":[{"b":2,"e":1.0,"k":"flat","v":0.80802,"x":1.0,"p":[[0,47,0.0,0.89731,0.17943,0.92857,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,24],[4,47,0.0851,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[8,47,0.1702,0.80802,0.19761,0.57143,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,3,0,15],[12,47,0.2553,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,47,0.3404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,47,0.4255,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,47,0.5106,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,47,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,47,0.6809,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,47,0.766,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[47,47,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.85714,"k":"falling","v":0.62945,"x":0.95089,"p":[[0,50,0.0,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[4,50,0.08,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[8,50,0.16,0.83033,0.20028,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,1,0,18],[12,50,0.24,0.75892,0.19377,0.57143,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,3,0,0,3,0,11],[16,50,0.32,0.73661,0.18935,0.57143,0.64286,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,0,6,0,8],[20,50,0.4,0.73214,0.17405,0.57143,0.71429,0.85714,0.5714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,5,0,0,5,0,7],[24,50,0.48,0.71875,0.17307,0.57143,0.57143,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,3,0,0,6,0,6],[28,50,0.56,0.75893,0.16917,0.57143,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,8,0,0,5,0,8],[32,50,0.64,0.71429,0.17857,0.57143,0.57143,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,0,4,0,7],[36,50,0.72,0.74092,0.18367,0.57143,0.71214,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,5,0,0,3,0,9],[40,50,0.8,0.62945,0.123,0.57143,0.57143,0.57143,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,3,0,0,2,0,2],[44,50,0.88,0.70088,0.15715,0.57143,0.64286,0.75,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,8,0,0,3,0,5],[48,50,0.96,0.65179,0.12846,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,3,0,0,6,0,1],[50,50,1.0,0.67856,0.12878,0.57143,0.64286,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,10,0,0,4,0,2]]}]},{"i":"bdabd25c95112198","q":"Find all real numbers  $x$  such that: \\[ x = \\sqrt{ x - \\frac{1}{x} } + \\sqrt{ 1 - \\frac{1}{x} }  \\]","t":[{"b":0,"e":1.0,"k":"falling","v":0.71869,"x":0.96875,"p":[[0,18,0.0,0.93302,0.15149,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,26],[4,18,0.2222,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[8,18,0.4444,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[12,18,0.6667,0.72764,0.16891,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,11,0,0,2,0,7],[16,18,0.8889,0.76783,0.16659,0.57143,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,0,6,0,8],[18,18,1.0,0.71869,0.16558,0.57143,0.71429,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,10,0,0,5,0,5]]},{"b":4,"e":1.0,"k":"flat","v":0.96874,"x":1.0,"p":[[0,8,0.0,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,8,0.5,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e2f5ec56601b9921","q":"Find all real numbers  $a$  for which there exists a non-constant function  $f :\\Bbb R \\to \\Bbb R$  satisfying the following two equations for all  $x\\in \\Bbb R:$ i)  $f(ax) = a^2f(x)$  and\nii)  $f(f(x)) = a f(x).$","t":[{"b":2,"e":1.0,"k":"flat","v":0.69642,"x":0.83482,"p":[[0,49,0.0,0.69642,0.18471,0.57143,0.71429,0.74996,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,14,0,0,3,0,5],[4,49,0.0816,0.73212,0.17406,0.71429,0.71429,0.74996,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,17,0,0,1,0,7],[8,49,0.1633,0.75446,0.16065,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,16,0,0,5,0,6],[12,49,0.2449,0.73661,0.20858,0.67857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,13,0,0,2,0,9],[16,49,0.3265,0.77677,0.1819,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,13,0,0,3,0,10],[20,49,0.4082,0.75445,0.18978,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,12,0,0,5,0,8],[24,49,0.4898,0.83482,0.19597,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,1,0,17],[28,49,0.5714,0.75893,0.20024,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,13,0,0,4,0,9],[32,49,0.6531,0.77232,0.18161,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,14,0,0,2,0,10],[36,49,0.7347,0.72321,0.17105,0.67857,0.71429,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,16,0,0,2,0,6],[40,49,0.8163,0.80804,0.18766,0.71429,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,3,0,13],[44,49,0.898,0.77677,0.21111,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,3,0,12],[48,49,0.9796,0.77678,0.18877,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,12,0,0,4,0,10],[49,49,1.0,0.74999,0.2113,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,11,0,0,1,0,11]]},{"b":4,"e":0.42857,"k":"flat","v":0.72322,"x":0.80801,"p":[[0,12,0.0,0.78571,0.19233,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,0,2,0,12],[4,12,0.3333,0.79018,0.16746,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,3,0,10],[8,12,0.6667,0.80801,0.20083,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,1,0,15],[12,12,1.0,0.72322,0.24206,0.42857,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,9,0,0,0,0,12]]}]},{"i":"473f6302518e5980","q":"Find all the real numbers  $a$  and  $b$  that satisfy the relation  $2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$","t":[{"b":1,"e":1.0,"k":"rising","v":0.45536,"x":0.99554,"p":[[0,28,0.0,0.78125,0.27429,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,0,0,0,0,0,0,6,0,0,3,0,0,6,0,14],[4,28,0.1429,0.83482,0.25282,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,19],[8,28,0.2857,0.84821,0.19865,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,16],[12,28,0.4286,0.84821,0.26471,0.85714,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,19],[16,28,0.5714,0.86607,0.17474,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,6,0,17],[20,28,0.7143,0.83928,0.24679,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,9,0,16],[24,28,0.8571,0.45536,0.31831,0.14286,0.42857,0.71429,0.0,1.0,1,5,0,1,0,10,0,0,4,0,0,5,0,0,2,0,0,4,0,0,1,0,5],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"rising","v":0.77678,"x":1.0,"p":[[0,19,0.0,0.77678,0.30501,0.67857,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,0,0,0,0,0,0,4,0,0,3,0,0,6,0,15],[4,19,0.2105,0.84821,0.25985,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,21],[8,19,0.4211,0.87946,0.22335,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,20],[12,19,0.6316,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,19,0.8421,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"44bf38a363bc9587","q":"Find all solutions  $x,y,z$   in the positive integers of the equation   $$ 3^x -5^y = z^2 $$","t":[{"b":6,"e":1.0,"k":"flat","v":0.92855,"x":0.98214,"p":[[0,16,0.0,0.92855,0.16755,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,23],[4,16,0.25,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,16,0.5,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[12,16,0.75,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,16,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.90611,"x":0.96875,"p":[[0,29,0.0,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[4,29,0.1379,0.90611,0.21324,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,24],[8,29,0.2759,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,29,0.4138,0.91517,0.18512,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[16,29,0.5517,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,29,0.6897,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[24,29,0.8276,0.91071,0.21053,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,23],[28,29,0.9655,0.95982,0.10853,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[29,29,1.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22]]}]},{"i":"4ccb5743b7dc67cf","q":"Find all triples  $(a,b,c)$  of natural numbers, such that  $LCM(a,b,c)=a+b+c$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.69196,"x":0.76339,"p":[[0,14,0.0,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[4,14,0.2857,0.76339,0.13651,0.71429,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,0,0,7],[8,14,0.5714,0.69196,0.1017,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,27,0,0,0,0,1],[12,14,0.8571,0.73661,0.0724,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,2],[14,14,1.0,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.70982,"x":0.76786,"p":[[0,37,0.0,0.76786,0.11152,0.71429,0.71429,0.71429,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,6],[4,37,0.1081,0.74094,0.08333,0.71429,0.71429,0.71429,0.71,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[8,37,0.2162,0.73661,0.08828,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,3],[12,37,0.3243,0.72768,0.05486,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,1],[16,37,0.4324,0.71428,5e-05,0.71429,0.71429,0.71429,0.714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[20,37,0.5405,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[24,37,0.6486,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[28,37,0.7568,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[32,37,0.8649,0.71874,0.05633,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[36,37,0.973,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[37,37,1.0,0.72767,0.06545,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,1]]}]},{"i":"5af758faeccbd4a1","q":"Find all triples  $ (a,b,c)$  of positive integers such that  $ a^2\\plus{}2^{b\\plus{}1}\\equal{}3^c$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,45,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,45,0.0889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,45,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,45,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,45,0.3556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,45,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,45,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,45,0.7111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[45,45,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,22,0.0,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,22,0.1818,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,22,0.3636,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,22,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"bd7fbe750286021d","q":"Find all values of  $n$  for which all solutions of the equation  $x^3-3x+n=0$  are integers.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.15625,"x":0.25892,"p":[[0,59,0.0,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.18723,0.0754,0.14286,0.14286,0.2857,0.14,0.42857,0,0,0,0,0,23,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.20536,0.14258,0.14286,0.14286,0.14287,0.14286,0.71429,0,0,0,0,0,25,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[12,59,0.2034,0.1875,0.11538,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,59,0.2712,0.25892,0.23265,0.14286,0.14286,0.17857,0.14286,0.85714,0,0,0,0,0,24,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,0],[20,59,0.339,0.17839,0.1072,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,27,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,59,0.4068,0.16072,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.2009,0.14222,0.14286,0.14286,0.14292,0.14286,0.85714,0,0,0,0,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[32,59,0.5424,0.16955,0.10376,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,59,0.6102,0.20518,0.15954,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,59,0.678,0.17848,0.12879,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,59,0.7458,0.18304,0.12992,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,59,0.8136,0.17848,0.12877,0.14286,0.14286,0.14286,0.14,0.857,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,59,0.8814,0.20536,0.10062,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,59,0.9492,0.19197,0.17353,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[59,59,1.0,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.165,"x":0.20089,"p":[[0,31,0.0,0.19197,0.07667,0.14286,0.14286,0.2857,0.14286,0.4286,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.20072,0.1423,0.14286,0.14286,0.14292,0.14,0.85714,0,0,0,0,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[8,31,0.2581,0.165,0.05195,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.20081,0.19189,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[16,31,0.5161,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.20089,0.1551,0.14286,0.14286,0.17857,0.14286,1.0,0,1,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,31,0.7742,0.18295,0.10856,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,31,0.9032,0.17411,0.05905,0.14286,0.14286,0.14287,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.16956,0.06625,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4938d7b3d419e0fc","q":"Find all the natural  $n$  and  $k$  such that  $n^n$  has  $k$  digits and  $k^k$  has  $n$  digits.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.69194,"x":0.90179,"p":[[0,86,0.0,0.79464,0.20806,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,2,0,0,4,0,14],[4,86,0.0465,0.7991,0.18161,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,10,0,10],[8,86,0.093,0.82143,0.20203,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,6,0,15],[12,86,0.1395,0.78124,0.18554,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,1,0,0,10,0,9],[16,86,0.186,0.80803,0.16982,0.57143,0.85714,1.0,0.5714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,10,0,10],[20,86,0.2326,0.83034,0.17657,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,9,0,13],[24,86,0.2791,0.77675,0.18539,0.57143,0.85707,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,7,0,10],[28,86,0.3256,0.69194,0.16795,0.57143,0.57143,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,2,0,0,7,0,4],[32,86,0.3721,0.74998,0.18899,0.57143,0.78564,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,1,0,0,8,0,8],[36,86,0.4186,0.80807,0.16972,0.67857,0.85714,1.0,0.43,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,12,0,9],[40,86,0.4651,0.81696,0.15663,0.67857,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,15,0,8],[44,86,0.5116,0.7991,0.14222,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,16,0,5],[48,86,0.5581,0.85267,0.14054,0.82132,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,13,0,11],[52,86,0.6047,0.83928,0.12242,0.85711,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,20,0,6],[56,86,0.6512,0.84375,0.12556,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,19,0,7],[60,86,0.6977,0.87052,0.1033,0.85714,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,21,0,8],[64,86,0.7442,0.86159,0.09771,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,23,0,6],[68,86,0.7907,0.86159,0.09095,0.85711,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,22,0,6],[72,86,0.8372,0.86159,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3],[76,86,0.8837,0.83033,0.10373,0.857,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,23,0,3],[80,86,0.9302,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[84,86,0.9767,0.90178,0.13092,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,11,0,17],[86,86,1.0,0.875,0.14617,0.85714,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,13,0,14]]},{"b":4,"e":1.0,"k":"flat","v":0.75446,"x":0.9241,"p":[[0,25,0.0,0.78121,0.17495,0.57143,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,10,0,8],[4,25,0.16,0.83928,0.18814,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,8,0,15],[8,25,0.32,0.78122,0.21722,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,2,0,0,3,0,14],[12,25,0.48,0.76783,0.20441,0.57143,0.85707,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,1,0,0,6,0,11],[16,25,0.64,0.75446,0.20277,0.57143,0.78564,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,1,0,0,6,0,10],[20,25,0.8,0.9241,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[24,25,0.96,0.91964,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[25,25,1.0,0.8482,0.15949,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,10,0,13]]}]},{"i":"b17ea006a20a5dc6","q":"Find all triples  $(p,m,n)$  satisfying the equation  $p^m-n^3=8$  where  $p$  is a prime number and  $m,n$  are nonnegative integers.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.88392,"x":0.94642,"p":[[0,66,0.0,0.88839,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[4,66,0.0606,0.90178,0.0974,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,13],[8,66,0.1212,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[12,66,0.1818,0.90625,0.10479,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15],[16,66,0.2424,0.94642,0.07785,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[20,66,0.303,0.89732,0.08171,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,11],[24,66,0.3636,0.88392,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,10],[28,66,0.4242,0.88838,0.09933,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,18,0,11],[32,66,0.4848,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[36,66,0.5455,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[40,66,0.6061,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[44,66,0.6667,0.89732,0.11971,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,15],[48,66,0.7273,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[52,66,0.7879,0.91517,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[56,66,0.8485,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[60,66,0.9091,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[64,66,0.9697,0.93749,0.07937,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[66,66,1.0,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17]]},{"b":5,"e":0.85714,"k":"flat","v":0.87499,"x":0.91071,"p":[[0,28,0.0,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[4,28,0.1429,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,17],[8,28,0.2857,0.87945,0.10171,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,21,0,9],[12,28,0.4286,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[16,28,0.5714,0.88838,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[20,28,0.7143,0.89285,0.08748,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10],[24,28,0.8571,0.87499,0.09279,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,8],[28,28,1.0,0.875,0.07784,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,6]]}]},{"i":"65ffb9f7067c6af4","q":"Find all triples of primes  $(p,q,r)$  satisfying  $3p^{4}-5q^{4}-4r^{2}=26$ .","t":[{"b":1,"e":0.71,"k":"rising","v":0.40179,"x":0.75447,"p":[[0,52,0.0,0.40179,0.23538,0.28571,0.28571,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,4],[4,52,0.0769,0.44197,0.28202,0.28571,0.28571,0.35736,0.2857,1.0,0,6,0,0,0,0,0,0,24,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[8,52,0.1538,0.65179,0.333,0.28571,0.50001,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,15],[12,52,0.2308,0.73661,0.31157,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,5,0,0,0,0,17],[16,52,0.3077,0.75447,0.30143,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,19],[20,52,0.3846,0.48213,0.25442,0.28571,0.35714,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,16,0,0,6,0,0,1,0,0,4,0,0,1,0,4],[24,52,0.4615,0.61159,0.29502,0.28571,0.49979,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,9,0,0,7,0,0,3,0,0,2,0,0,1,0,10],[28,52,0.5385,0.75,0.23145,0.57143,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,8,0,0,0,0,13],[32,52,0.6154,0.71429,0.22304,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,10,0,0,0,0,10],[36,52,0.6923,0.60268,0.20119,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,11,0,0,5,0,0,10,0,0,0,0,4],[40,52,0.7692,0.65622,0.20783,0.42857,0.57143,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,7,0,0,0,0,7],[44,52,0.8462,0.61606,0.21559,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,14,0,0,7,0,0,4,0,0,1,0,6],[48,52,0.9231,0.70982,0.25626,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,4,0,0,0,0,13],[52,52,1.0,0.62944,0.20782,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,12,0,0,7,0,0,7,0,0,0,0,6]]},{"b":2,"e":0.28571,"k":"flat","v":0.29464,"x":0.41964,"p":[[0,63,0.0,0.41964,0.25739,0.28571,0.28571,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,5],[4,63,0.0635,0.39286,0.21429,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,23,0,0,4,0,0,1,0,0,0,0,0,2,0,2],[8,63,0.127,0.31696,0.05906,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,63,0.1905,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,63,0.254,0.36607,0.20497,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[20,63,0.3175,0.31697,0.09268,0.2857,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[24,63,0.381,0.30803,0.05187,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,63,0.4444,0.33928,0.11149,0.28571,0.28571,0.28579,0.2857,0.71429,0,0,0,0,0,0,0,0,25,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[32,63,0.5079,0.3616,0.19228,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[36,63,0.5714,0.30804,0.05187,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[40,63,0.6349,0.38393,0.20341,0.28571,0.28571,0.32143,0.2857,1.0,0,2,0,0,0,0,0,0,24,0,0,2,0,0,3,0,0,0,0,0,1,0,2],[44,63,0.6984,0.35268,0.19556,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[48,63,0.7619,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[52,63,0.8254,0.3125,0.10374,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[56,63,0.8889,0.30803,0.05188,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.3348,0.13649,0.2857,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,26,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[63,63,1.0,0.29464,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b7e304eae3232c36","q":"Find all values of  $a$  such that absolute value of one of the roots of the equation  $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$  is twice of absolute value of the other root.","t":[{"b":1,"e":0.57143,"k":"falling","v":0.46428,"x":0.91964,"p":[[0,65,0.0,0.83036,0.23538,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,4,0,0,2,0,19],[4,65,0.0615,0.73213,0.26905,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,9,0,0,2,0,0,3,0,0,2,0,14],[8,65,0.1231,0.91964,0.19212,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,27],[12,65,0.1846,0.79464,0.27185,0.42857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,1,0,0,0,0,20],[16,65,0.2462,0.54018,0.19475,0.42857,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,15,0,0,7,0,0,3,0,0,1,0,3],[20,65,0.3077,0.57589,0.26362,0.42857,0.42859,0.71429,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,12,0,0,4,0,0,4,0,0,0,0,7],[24,65,0.3692,0.48214,0.22517,0.28571,0.42857,0.46431,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,15,0,0,2,0,0,1,0,0,2,0,3],[28,65,0.4308,0.64732,0.2789,0.42857,0.64286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,8,0,0,2,0,0,5,0,0,1,0,10],[32,65,0.4923,0.70981,0.28005,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,4,0,0,3,0,12],[36,65,0.5538,0.77232,0.27862,0.42859,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,1,0,0,1,0,18],[40,65,0.6154,0.78572,0.28347,0.42859,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,7,0,0,1,0,0,0,0,0,2,0,19],[44,65,0.6769,0.83482,0.23987,0.57143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,2,0,0,0,0,21],[48,65,0.7385,0.76339,0.29366,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,19],[52,65,0.8,0.52232,0.25904,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,11,0,0,6,0,0,4,0,0,5,0,0,0,0,5],[56,65,0.8615,0.54911,0.2372,0.42857,0.42857,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,11,0,0,3,0,0,3,0,0,6,0,2],[60,65,0.9231,0.57143,0.20203,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,12,0,0,7,0,0,5,0,0,2,0,3],[64,65,0.9846,0.46428,0.20203,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,9,0,0,9,0,0,7,0,0,2,0,0,2,0,1],[65,65,1.0,0.48213,0.18123,0.39286,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,11,0,0,5,0,0,7,0,0,1,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.53125,"x":0.92857,"p":[[0,17,0.0,0.88839,0.21939,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,25],[4,17,0.2353,0.92857,0.17857,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,27],[8,17,0.4706,0.77679,0.2922,0.42859,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,5,0,0,4,0,0,2,0,0,1,0,0,1,0,19],[12,17,0.7059,0.78571,0.26964,0.53571,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,1,0,0,0,0,19],[16,17,0.9412,0.58928,0.23076,0.42857,0.57143,0.85714,0.1429,1.0,0,1,0,0,0,1,0,0,5,0,0,7,0,0,6,0,0,3,0,0,9,0,1],[17,17,1.0,0.53125,0.1684,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,11,0,0,8,0,0,7,0,0,2,0,0]]}]},{"i":"8fc5b2946bc917ac","q":"Find prime numbers  $p , q , r$   such that  $p+q^2+r^3=200$ . Give all the possibilities.\n\nRemember that the number  $1$  is not prime.","t":[{"b":1,"e":1.0,"k":"flat","v":0.75893,"x":0.92411,"p":[[0,31,0.0,0.75893,0.1357,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,0,3,0,6],[4,31,0.129,0.78125,0.19228,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,7,0,10],[8,31,0.2581,0.86607,0.13803,0.82143,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,11,0,13],[12,31,0.3871,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[16,31,0.5161,0.87946,0.12428,0.82143,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,10,0,14],[20,31,0.6452,0.87054,0.17261,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,7,0,17],[24,31,0.7742,0.89732,0.12492,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,17],[28,31,0.9032,0.89286,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[31,31,1.0,0.89285,0.11294,0.85708,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,15]]},{"b":5,"e":0.71429,"k":"falling","v":0.71873,"x":0.88839,"p":[[0,35,0.0,0.88839,0.1461,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,0,0,20],[4,35,0.1143,0.72321,0.15542,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,12,0,0,6,0,4],[8,35,0.2286,0.72766,0.15304,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,10,0,0,7,0,4],[12,35,0.3429,0.80789,0.14564,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,12,0,7],[16,35,0.4571,0.75892,0.14032,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,9,0,4],[20,35,0.5714,0.81696,0.14826,0.71429,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,9,0,9],[24,35,0.6857,0.76785,0.18123,0.71429,0.78564,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,9,0,0,9,0,7],[28,35,0.8,0.80357,0.16268,0.71429,0.85707,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,8,0,9],[32,35,0.9143,0.75446,0.17582,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,11,0,0,8,0,6],[35,35,1.0,0.71873,0.1262,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,19,0,0,5,0,2]]}]},{"i":"2b54f72bc2ac0c08","q":"Find all triples  $(a, b, c)$  of positive integers for which   $$ \\begin{cases} a + bc=2010  b + ca = 250\\end{cases} $$","t":[{"b":6,"e":1.0,"k":"flat","v":0.80803,"x":0.91964,"p":[[0,13,0.0,0.80803,0.20079,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,2,0,0,7,0,13],[4,13,0.3077,0.81696,0.22934,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,0,7,0,14],[8,13,0.6154,0.91964,0.15126,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,22],[12,13,0.9231,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[13,13,1.0,0.88838,0.08553,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,9]]},{"b":7,"e":1.0,"k":"flat","v":0.85268,"x":0.88392,"p":[[0,5,0.0,0.875,0.21053,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,19],[4,5,0.8,0.85268,0.05629,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,1],[5,5,1.0,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6]]}]},{"i":"fc9071e7f844e0d7","q":"Find all the solutions of the system\n   $$ \\begin{cases}  y^2 = x^3 - 3x^2 + 2x  x^2 = y^3 - 3y^2 + 2y  \\end{cases} $$","t":[{"b":4,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,50,0.0,0.94643,0.15872,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[4,50,0.08,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,50,0.16,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,50,0.24,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,50,0.32,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,50,0.4,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,50,0.48,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[28,50,0.56,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[32,50,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,50,0.72,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,50,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,50,0.88,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,50,0.96,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[50,50,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,9,0.0,0.92411,0.1988,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,27],[4,9,0.4444,0.92411,0.21124,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[9,9,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9db3117c586009e6","q":"Find all the triples of integers  $ (a, b,c)$  such that:\r\n\\[ \\begin{array}{ccc}a\\plus{}b\\plus{}c &\\equal{}& 24 a^{2}\\plus{}b^{2}\\plus{}c^{2}&\\equal{}& 210 abc &\\equal{}& 440\\end{array}\\]","t":[{"b":2,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,8,0.0,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,8,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,81,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,81,0.0494,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,81,0.0988,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,81,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,81,0.1975,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,81,0.2469,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,81,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,81,0.3457,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,81,0.3951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,81,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,81,0.4938,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,81,0.5432,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,81,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,81,0.642,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,81,0.6914,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,81,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,81,0.7901,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,81,0.8395,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,81,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,81,0.9383,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,81,0.9877,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[81,81,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8f16066f9568a8d1","q":"For a positive integer  $n$ , define the set  $S_n$  as  $S_n =\\{(a, b)|a, b \\in N, lcm[a, b] = n\\}$  . Let  $f(n)$  be the sum of  $\\phi (a)\\phi (b)$  for all  $(a, b) \\in S_n$ .  If a prime  $p$  relatively prime to  $n$  is a divisor of  $f(n)$ , prove that there exists a prime  $q|n$  such that  $p|q^2 - 1$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,7,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,7,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[7,7,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,41,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,41,0.0976,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,41,0.1951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,41,0.2927,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,0.92411,0.13825,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,23],[32,41,0.7805,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,21],[36,41,0.878,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[40,41,0.9756,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[41,41,1.0,0.99094,0.05046,1.0,1.0,1.0,0.71,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"1cef21c42ef4063b","q":"Find all triples  $(a,b,c)$  of positive integers such that the product of any two of them when divided by the third leaves the remainder  $1$ .","t":[{"b":4,"e":0.85714,"k":"flat","v":0.83924,"x":0.91517,"p":[[0,8,0.0,0.875,0.17405,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,8,0,17],[4,8,0.5,0.83924,0.15876,0.857,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,19,0,8],[8,8,1.0,0.91517,0.10013,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,16]]},{"b":5,"e":0.57143,"k":"flat","v":0.7857,"x":0.92411,"p":[[0,44,0.0,0.8973,0.16068,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,10,0,18],[4,44,0.0909,0.88837,0.15866,0.85711,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,12,0,16],[8,44,0.1818,0.84821,0.14258,0.85711,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,17,0,9],[12,44,0.2727,0.7857,0.189,0.82143,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,0,20,0,4],[16,44,0.3636,0.7924,0.20311,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,1,1,0,0,4,0,0,2,0,0,16,0,7],[20,44,0.4545,0.81248,0.16919,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,14,0,8],[24,44,0.5455,0.91963,0.11815,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,20],[28,44,0.6364,0.89286,0.14286,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,19],[32,44,0.7273,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[36,44,0.8182,0.86601,0.16352,0.85711,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,10,0,15],[40,44,0.9091,0.8214,0.1713,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,14,0,9],[44,44,1.0,0.81248,0.16148,0.78561,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,17,0,7]]}]},{"i":"4b0a1532142af004","q":"For a number  $n$  in base  $10$ , let  $f(n)$  be the sum of all numbers possible by removing some digits of  $n$  (including none and all). For example, if  $n = 1234$ ,  $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$ ; this is formed by taking the sums of all numbers obtained when removing no digit from  $n$  (1234), removing one digit from  $n$  (123, 124, 134, 234), removing two digits from  $n$  (12, 13, 14, 23, 24, 34), removing three digits from  $n$  (1, 2, 3, 4), and removing all digits from  $n$  (0). If  $p$  is a 2011-digit integer, prove that  $f(p)-p$  is divisible by  $9$ .\n\nRemark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number  $101$ ,  $1$  appears three times (by removing the first digit, giving  $01$  which is equal to  $1$ , removing the first two digits, or removing the last two digits), so it is counted three times.","t":[{"b":3,"e":0.4286,"k":"falling","v":0.42857,"x":0.63393,"p":[[0,28,0.0,0.63393,0.27879,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,2,0,0,0,0,11],[4,28,0.1429,0.45536,0.10374,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,26,0,0,0,0,0,4,0,0,0,0,0],[8,28,0.2857,0.46875,0.14827,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,22,0,0,0,0,0,7,0,0,0,0,0],[12,28,0.4286,0.51786,0.18123,0.42857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,22,0,0,0,0,0,7,0,0,0,0,2],[16,28,0.5714,0.42857,0.10102,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,28,0,0,0,0,0,2,0,0,0,0,0],[20,28,0.7143,0.45982,0.11143,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,26,0,0,1,0,0,4,0,0,0,0,0],[24,28,0.8571,0.45089,0.11355,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,26,0,0,0,0,0,4,0,0,0,0,0],[28,28,1.0,0.45536,0.1448,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,22,0,0,0,0,0,6,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.62946,"x":1.0,"p":[[0,36,0.0,0.62946,0.26453,0.42857,0.42859,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,15,0,0,0,0,0,6,0,0,0,0,9],[4,36,0.1111,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[8,36,0.2222,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,36,0.3333,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"5a8783c67c789d09","q":"Find all triplets  $(x,y,p)$  of positive integers such that  $p$  be a prime number and  $\\frac{xy^3}{x+y}=p$","t":[{"b":2,"e":0.57143,"k":"flat","v":0.41963,"x":0.64286,"p":[[0,20,0.0,0.64286,0.22304,0.42857,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,6,0,0,7,0,0,2,0,0,11,0,2],[4,20,0.2,0.51789,0.19477,0.39286,0.57121,0.60714,0.1429,0.85714,0,0,0,0,0,1,0,0,7,0,0,7,0,0,9,0,0,4,0,0,4,0,0],[8,20,0.4,0.47766,0.21608,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,9,0,0,9,0,0,5,0,0,3,0,0,3,0,1],[12,20,0.6,0.41963,0.19864,0.28571,0.35714,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,13,0,0,4,0,0,7,0,0,4,0,0,1,0,0],[16,20,0.8,0.52232,0.18073,0.42857,0.42859,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,12,0,0,4,0,0,7,0,0,3,0,0],[20,20,1.0,0.60265,0.12745,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,14,0,0,8,0,0,3,0,0]]},{"b":5,"e":0.857,"k":"rising","v":0.60268,"x":0.85267,"p":[[0,14,0.0,0.63391,0.19542,0.42857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,7,0,0,5,0,0,7,0,0,10,0,0],[4,14,0.2857,0.60268,0.20119,0.42857,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,8,0,0,6,0,0,5,0,0,9,0,0],[8,14,0.5714,0.71429,0.13832,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,11,0,0,12,0,0],[12,14,0.8571,0.84374,0.07457,0.85713,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,23,0,3],[14,14,1.0,0.85267,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,3]]}]},{"i":"3d1d57b9fe6dec5a","q":"Find all triplets  $(a,b,p)$  where  $a,b$  are positive integers and  $p$  is a prime number satisfying:  $\\frac{1}{p}=\\frac{1}{a^2}+\\frac{1}{b^2}$","t":[{"b":5,"e":0.14286,"k":"flat","v":0.28571,"x":0.41517,"p":[[0,38,0.0,0.41517,0.25091,0.14286,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,9,0,0,4,0,0,12,0,0,1,0,0,2,0,0,2,0,2],[4,38,0.1053,0.3615,0.25257,0.14286,0.28571,0.42857,0.14,1.0,0,2,0,0,0,12,0,0,8,0,0,5,0,0,2,0,0,2,0,0,1,0,2],[8,38,0.2105,0.32594,0.19313,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,10,0,0,10,0,0,0,0,0,0,0,0,1,0,1],[12,38,0.3158,0.33482,0.15815,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,13,0,0,12,0,0,0,0,0,0,0,0,0,0,1],[16,38,0.4211,0.32589,0.10249,0.2857,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,5,0,0,13,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.35714,0.16366,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,5,0,0,8,0,0,16,0,0,0,0,0,1,0,0,1,0,0],[24,38,0.6316,0.30357,0.17405,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,8,0,0,11,0,0,0,0,0,0,0,0,0,0,1],[28,38,0.7368,0.30804,0.17536,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,10,0,0,9,0,0,10,0,0,0,0,0,1,0,0,1,0,0],[32,38,0.8421,0.29464,0.1554,0.14286,0.28571,0.42857,0.14286,0.857,0,0,0,0,0,12,0,0,9,0,0,10,0,0,0,0,0,0,0,0,1,0,0],[36,38,0.9474,0.3125,0.12078,0.24999,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,8,0,0,11,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.28571,0.10102,0.24999,0.28571,0.32143,0.14286,0.4286,0,0,0,0,0,8,0,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.19187,"x":0.37053,"p":[[0,42,0.0,0.37053,0.17444,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,15,0,0,11,0,0,1,0,0,0,0,0,1,0,1],[4,42,0.0952,0.36598,0.27426,0.14286,0.28571,0.42857,0.14,1.0,0,4,0,0,0,12,0,0,8,0,0,7,0,0,0,0,0,1,0,0,0,0,4],[8,42,0.1905,0.26774,0.22233,0.14286,0.14286,0.28571,0.14,1.0,0,2,0,0,0,19,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[12,42,0.2857,0.24553,0.15663,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,15,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,42,0.381,0.25,0.15152,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,17,0,0,10,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[20,42,0.4762,0.21428,0.09449,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.20973,0.08744,0.14286,0.14286,0.28571,0.14,0.42857,0,0,0,0,0,19,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.24553,0.12992,0.14286,0.2143,0.28571,0.14286,0.71429,0,0,0,0,0,16,0,0,11,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[32,42,0.7619,0.24543,0.09611,0.14286,0.28571,0.28571,0.14,0.571,0,0,0,0,0,12,0,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,42,0.8571,0.22757,0.10633,0.14286,0.14286,0.28571,0.14,0.571,0,0,0,0,0,17,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,42,0.9524,0.20536,0.07086,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.19187,0.07673,0.14286,0.14286,0.2857,0.14,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"780cb0a05bdcf9b3","q":"Find the functions  $ f:\\mathbb{Z}\\longrightarrow\\mathbb{Z}_{\\ge 0} $  that satisfy the following two conditions: $ \\text{(a)} f(m+n)=f(n)+f(m)+2mn,\\quad\\forall m,n\\in\\mathbb{Z} $  $ \\text{(b)} f(f(1))-f(1) $  is a perfect square\n\n\n*Marin Ionescu*","t":[{"b":3,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,68,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,68,0.0588,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,68,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,68,0.1765,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,68,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,68,0.2941,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,68,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,68,0.4118,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,68,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,68,0.5294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,68,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,68,0.6471,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,68,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,68,0.7647,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,68,0.8235,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,68,0.8824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,31,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,31,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,31,0.2581,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,31,0.6452,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,31,0.7742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"fb9135250221a742","q":"For a positive integer  $n$ , define  $f(n)=n+P(n)$  and  $g(n)=n\\cdot S(n)$ , where  $P(n)$  and  $S(n)$  denote the product and sum of the digits of  $n$ , respectively. Find all solutions to  $f(n)=g(n)$","t":[{"b":3,"e":1.0,"k":"flat","v":0.88393,"x":0.97768,"p":[[0,17,0.0,0.89285,0.12877,0.85711,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[4,17,0.2353,0.88393,0.1357,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,15],[8,17,0.4706,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[12,17,0.7059,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[16,17,0.9412,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[17,17,1.0,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]},{"b":6,"e":1.0,"k":"flat","v":0.89285,"x":0.96428,"p":[[0,61,0.0,0.91964,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,18],[4,61,0.0656,0.90178,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,15],[8,61,0.1311,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,61,0.1967,0.94196,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,23],[16,61,0.2623,0.93302,0.1071,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[20,61,0.3279,0.9375,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[24,61,0.3934,0.89285,0.19233,0.85714,1.0,1.0,0.0,1.0,1,18,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,18],[28,61,0.459,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[32,61,0.5246,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[36,61,0.5902,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[40,61,0.6557,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[44,61,0.7213,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[48,61,0.7869,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[52,61,0.8525,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[56,61,0.918,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[60,61,0.9836,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[61,61,1.0,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18]]}]},{"i":"b8c93d1cccedcabb","q":"Four consecutive even numbers are removed from the set \\[A=\\{ 1, 2, 3, \\cdots, n \\}.\\] If the arithmetic mean of the remaining numbers is  $51.5625$ , which four numbers were removed?","t":[{"b":0,"e":0.85714,"k":"flat","v":0.75893,"x":0.86607,"p":[[0,64,0.0,0.82589,0.18117,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,7,0,13],[4,64,0.0625,0.81696,0.1439,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,14,0,7],[8,64,0.125,0.85268,0.12619,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,13,0,10],[12,64,0.1875,0.78125,0.13825,0.71429,0.78571,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,11,0,5],[16,64,0.25,0.75893,0.12078,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,9,0,3],[20,64,0.3125,0.80804,0.11633,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,13,0,5],[24,64,0.375,0.86607,0.08702,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,23,0,6],[28,64,0.4375,0.78125,0.10705,0.71429,0.78571,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,14,0,2],[32,64,0.5,0.85268,0.09771,0.82143,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,17,0,7],[36,64,0.5625,0.7991,0.1063,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,18,0,2],[40,64,0.625,0.81696,0.08171,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,25,0,0],[44,64,0.6875,0.77232,0.11214,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,16,0,1],[48,64,0.75,0.8125,0.07523,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,23,0,0],[52,64,0.8125,0.79018,0.11837,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,18,0,2],[56,64,0.875,0.82589,0.06902,0.82143,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,23,0,1],[60,64,0.9375,0.79463,0.08706,0.71429,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,20,0,0],[64,64,1.0,0.80804,0.08459,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,23,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.84821,"x":0.94643,"p":[[0,22,0.0,0.84821,0.11811,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,13,0,9],[4,22,0.1818,0.85714,0.14725,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,7,0,14],[8,22,0.3636,0.88839,0.12745,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,15],[12,22,0.5455,0.91518,0.12807,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[16,22,0.7273,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[20,22,0.9091,0.89286,0.11294,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,14],[22,22,1.0,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21]]}]},{"i":"b52f438eebde90bb","q":"Given  $1980$  vectors in the plane, and there are some non-collinear among them. The sum of every  $1979$  vectors is collinear to the vector not included in that sum.  Prove that the sum of all vectors equals to the zero vector.","t":[{"b":4,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,27,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,27,0.1481,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,27,0.2963,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,27,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,27,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,27,0.7407,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,27,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,29,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"94dbfa71d2f93767","q":"Given is a triplet of different positive integers $(a, b, c)$ with $a+b+c=2013$. A step consists of replacing the triplet $(x, y, z)$ with the triplet $(y+z-x, z+x-y, x+y-z)$. Prove that starting from the triplet $(a, b, c)$, after 10 steps, we obtain a triplet that contains at least one negative number.","t":[{"b":4,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,13,0.0,0.95982,0.11426,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[4,13,0.3077,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[8,13,0.6154,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,13,0.9231,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[13,13,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.91517,"x":0.99554,"p":[[0,22,0.0,0.95534,0.10976,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[4,22,0.1818,0.94643,0.12242,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[8,22,0.3636,0.91517,0.16701,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,24],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,22,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[22,22,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c70748b62835e9aa","q":"Function  $ f:\\mathbb{R}\\rightarrow\\mathbb{R} $  such that  $f(xf(y))=yf(x)$  for any  $x,y$  are real numbers. Prove that  $f(-x) = -f(x)$  for all real numbers  $x$ .","t":[{"b":1,"e":0.57143,"k":"falling","v":0.64732,"x":0.92856,"p":[[0,17,0.0,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[4,17,0.2353,0.85714,0.20203,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,2,0,20],[8,17,0.4706,0.92856,0.13836,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[12,17,0.7059,0.83036,0.19704,0.57143,1.0,1.0,0.5714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,0,0,18],[16,17,0.9412,0.70982,0.145,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,17,0,0,0,0,5],[17,17,1.0,0.64732,0.07129,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,18,0.0,0.91518,0.1551,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,24],[4,18,0.2222,0.89286,0.19233,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,23],[8,18,0.4444,0.90625,0.17717,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,24],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[18,18,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"8e00ac93c2a46c6f","q":"Giiven  $\\Delta ABC$ ,  $\\angle CAB=75^{\\circ}$  and  $\\angle ACB=45^{\\circ}$ .  $BC$  is extended to  $T$  so that  $BC=CT$ . Let  $M$  be the midpoint of the segment  $AT$ . Find  $\\angle BMC$ .","t":[{"b":6,"e":0.85714,"k":"flat","v":0.81249,"x":0.91964,"p":[[0,23,0.0,0.81249,0.09061,0.71429,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,16,0,3],[4,23,0.1739,0.82589,0.11143,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,14,0,6],[8,23,0.3478,0.87946,0.13882,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,16,0,12],[12,23,0.5217,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[16,23,0.6957,0.90624,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[20,23,0.8696,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[23,23,1.0,0.89294,0.08745,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10]]},{"b":7,"e":0.71429,"k":"flat","v":0.74107,"x":0.82589,"p":[[0,29,0.0,0.82589,0.12234,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,10,0,8],[4,29,0.1379,0.78125,0.09438,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,3],[8,29,0.2759,0.74107,0.05575,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0],[12,29,0.4138,0.75446,0.07349,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,10,0,0],[16,29,0.5517,0.77231,0.07872,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,1],[20,29,0.6897,0.77678,0.07086,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,14,0,0],[24,29,0.8276,0.76786,0.07784,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,10,0,1],[28,29,0.9655,0.79017,0.07128,0.71429,0.857,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0],[29,29,1.0,0.77232,0.07873,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,1]]}]},{"i":"ae6cf44608cf7404","q":"Gegeven is een kwadratisch polynoom $P(x)$ met twee verschillende re\u00eble nulpunten. Voor alle re\u00eble getallen $a$ en $b$ met $|a|,|b| \\geq 2017$ geldt dat $P\\left(a^{2}+b^{2}\\right) \\geq P(2 a b)$. Bewijs dat minstens \u00e9\u00e9n van de nulpunten van $P$ negatief is.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.56696,"x":0.95536,"p":[[0,19,0.0,0.88393,0.22142,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,24],[4,19,0.2105,0.91964,0.19541,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,27],[8,19,0.4211,0.63393,0.19865,0.42857,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,9,0,0,4,0,0,14,0,0,0,0,4],[12,19,0.6316,0.56696,0.15765,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,14,0,0,4,0,0,12,0,0,0,0,1],[16,19,0.8421,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[19,19,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.82143,"x":1.0,"p":[[0,36,0.0,0.91071,0.18123,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,0,0,25],[4,36,0.1111,0.82143,0.24743,0.67857,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,0,1,0,19],[8,36,0.2222,0.85714,0.23145,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,0,0,22],[12,36,0.3333,0.87054,0.23517,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,0,0,0,0,0,24],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1ded3532b98d1ca8","q":"Given  $2021$  distinct positive integers non divisible by  $2^{1010}$ , show that it's always possible to choose  $3$  of them  $a$ ,  $b$  and  $c$ , such that  $|b^2-4ac|$  is not a perfect square.","t":[{"b":2,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,5,0.0,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[4,5,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[5,5,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.83036,"x":1.0,"p":[[0,36,0.0,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[4,36,0.1111,0.83036,0.23808,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,3,0,18],[8,36,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,36,0.3333,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1ce852b5aec6a6f5","q":"How many positive integers less than  $1000$  have the property that the sum of the digits of each such number is divisible by  $7$  and the number itself is divisible by  $3$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,41,0.0,0.875,0.25191,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,25],[4,41,0.0976,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[8,41,0.1951,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,41,0.2927,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,41,0.5854,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.78125,"x":1.0,"p":[[0,59,0.0,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,59,0.0678,0.78125,0.32534,0.28571,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,22],[8,59,0.1356,0.875,0.26184,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[12,59,0.2034,0.87945,0.24515,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,25],[16,59,0.2712,0.86607,0.25738,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,25],[20,59,0.339,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[24,59,0.4068,0.91964,0.2141,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[28,59,0.4746,0.94196,0.18161,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[32,59,0.5424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,59,0.6102,0.91071,0.21053,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,27],[40,59,0.678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,59,0.7458,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,59,0.8136,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,59,0.8814,0.94196,0.18161,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[56,59,0.9492,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[59,59,1.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"627af51cbf10ed56","q":"Hello Everyone, \n\ni'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level\n\nPlease if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons. \n\nit will be preferred to write the source of the problem.\nplease , show detailed solutions , and please post some challenging Pre-Olympiad problems..\nremember ,, different approaches are most welcome :)\n\nnow .. let's get our hands dirty :lol: :\n\n\nlet  $ f(n)$  denote the sum of the digits of  $ n$ . Now let  $ N \\equal{} {4444^{4444}}$ .\nFind  $ f\\left( {f\\left( {f\\left( N \\right)} \\right)} \\right)$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,20,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac21b362f0c6acf4","q":"Given vector  $\\mathbf{u}=\\left(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3} \\right)\\in\\mathbb{R}^3$  and recursively defined sequence of vectors  $\\{\\mathbf{v}_n\\}_{n\\geq 0}$   $$ \\mathbf{v}_0 = (1,2,3),\\quad \\mathbf{v}_n = \\mathbf{u}\\times\\mathbf{v}_{n-1} $$  Evaluate the value of infinite series  $\\sum_{n=1}^\\infty (3,2,1)\\cdot \\mathbf{v}_{2n}$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.82589,"x":0.87946,"p":[[0,25,0.0,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[4,25,0.16,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7],[8,25,0.32,0.84821,0.07936,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,22,0,4],[12,25,0.48,0.82589,0.05906,0.85713,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0],[16,25,0.64,0.84375,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,1],[20,25,0.8,0.83928,0.04724,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0],[24,25,0.96,0.82589,0.05906,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0],[25,25,1.0,0.83929,0.04725,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0]]},{"b":5,"e":1.0,"k":"flat","v":0.87054,"x":1.0,"p":[[0,50,0.0,0.875,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,6],[4,50,0.08,0.87054,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[8,50,0.16,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[12,50,0.24,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,50,0.32,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[20,50,0.4,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,50,0.48,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[28,50,0.56,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,50,0.64,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,50,0.72,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,50,0.8,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[44,50,0.88,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,50,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[50,50,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1b6d4cf827f0dd24","q":"If  $ \\left|BC\\right| \\equal{} a$ ,  $ \\left|AC\\right| \\equal{} b$ ,  $ \\left|AB\\right| \\equal{} c$ ,  $ 3\\angle A \\plus{} \\angle B \\equal{} 180{}^\\circ$  and  $ 3a \\equal{} 2c$ , then find  $ b$  in terms of  $ a$ . $\\textbf{(A)}\\  \\frac {3a}{2} \\qquad\\textbf{(B)}\\  \\frac {5a}{4} \\qquad\\textbf{(C)}\\  a\\sqrt {2} \\qquad\\textbf{(D)}\\  a\\sqrt {3} \\qquad\\textbf{(E)}\\ \\frac {2a\\sqrt {3} }{3}$","t":[{"b":2,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,47,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,47,0.0851,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,47,0.1702,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,47,0.2553,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,47,0.3404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,47,0.4255,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,47,0.5106,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,47,0.5957,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,47,0.6809,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,47,0.766,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[40,47,0.8511,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,47,0.9362,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[47,47,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,70,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,70,0.0571,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,70,0.1143,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,70,0.1714,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,70,0.2286,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,70,0.2857,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,70,0.3429,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,70,0.4,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,70,0.4571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,70,0.5143,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,70,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,70,0.6286,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[48,70,0.6857,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[52,70,0.7429,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[56,70,0.8,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,70,0.8571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,70,0.9143,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[68,70,0.9714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[70,70,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"0e0dd2e935961ee9","q":"Given three distinct positive numbers  $a,b$  and  $c$ , prove that the quadratic equation   $$ (a+b+c)x^2 +2\\left(\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}\\right) x+\\frac{1}{a} + \\frac{1}{b} +\\frac{1}{c} = 0 $$   has two distinct real roots.","t":[{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,31,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,31,0.129,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,31,0.2581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,31,0.3871,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,31,0.5161,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,18,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,18,0.2222,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[16,18,0.8889,0.92857,0.17496,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[18,18,1.0,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28]]}]},{"i":"554da199243898ed","q":"How many positive integers at most  $420$  leave different remainders when divided by each of  $5$ ,  $6$ , and  $7$ ?\n\n*Proposed by Milan Haiman.*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.17857,"x":0.50893,"p":[[0,15,0.0,0.36607,0.24206,0.14286,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,12,0,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,3],[4,15,0.2667,0.50893,0.30501,0.28571,0.42857,0.67857,0.14286,1.0,0,8,0,0,0,6,0,0,3,0,0,14,0,0,1,0,0,0,0,0,0,0,8],[8,15,0.5333,0.45981,0.22793,0.39286,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,3,0,0,5,0,0,18,0,0,2,0,0,0,0,0,0,0,4],[12,15,0.8,0.24991,0.22308,0.14286,0.14286,0.21429,0.14,1.0,0,2,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[15,15,1.0,0.17857,0.08748,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.20982,"x":0.42857,"p":[[0,71,0.0,0.35268,0.12364,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,4,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[4,71,0.0563,0.42857,0.24744,0.28571,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,7,0,0,3,0,0,17,0,0,1,0,0,0,0,0,0,0,4],[8,71,0.1127,0.39732,0.17029,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,3,0,0,18,0,0,4,0,0,0,0,0,0,0,1],[12,71,0.169,0.38839,0.29285,0.14286,0.42857,0.42857,0.14286,1.0,0,5,0,0,0,14,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,5],[16,71,0.2254,0.42857,0.23958,0.28571,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,5,0,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,4],[20,71,0.2817,0.38839,0.22084,0.25,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,8,0,0,5,0,0,15,0,0,1,0,0,0,0,0,1,0,2],[24,71,0.338,0.40625,0.27458,0.14286,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,10,0,0,5,0,0,11,0,0,1,0,0,0,0,0,1,0,4],[28,71,0.3944,0.37054,0.19516,0.24999,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,14,0,0,2,0,0,0,0,0,1,0,1],[32,71,0.4507,0.24545,0.18298,0.14286,0.14286,0.42857,0.14,1.0,0,1,0,0,0,22,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[36,71,0.507,0.23661,0.18423,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,23,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[40,71,0.5634,0.24982,0.18908,0.14286,0.14286,0.42857,0.14,1.0,0,1,0,0,0,22,0,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[44,71,0.6197,0.22768,0.21387,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[48,71,0.6761,0.30804,0.2372,0.14286,0.21428,0.42857,0.14286,1.0,0,2,0,0,0,16,0,0,6,0,0,7,0,0,0,0,0,0,0,0,1,0,2],[52,71,0.7324,0.29464,0.26949,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,20,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,3],[56,71,0.7887,0.25446,0.12234,0.14286,0.21428,0.42857,0.14286,0.4286,0,0,0,0,0,16,0,0,7,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.25893,0.15746,0.14286,0.14286,0.32143,0.14286,0.85714,0,0,0,0,0,17,0,0,7,0,0,7,0,0,0,0,0,0,0,0,1,0,0],[64,71,0.9014,0.35714,0.26486,0.14286,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,13,0,0,7,0,0,6,0,0,2,0,0,0,0,0,1,0,3],[68,71,0.9577,0.20982,0.11837,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,23,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[71,71,1.0,0.35705,0.25008,0.14286,0.28571,0.42857,0.14,1.0,0,3,0,0,0,13,0,0,4,0,0,10,0,0,2,0,0,0,0,0,0,0,3]]}]},{"i":"54d95e0b381e9b32","q":"In a triangle  $ABC$ , the bisector of  $\\angle BAC$  meets the side  $BC$  at the point  $D$ . Knowing that  $|BD|\\cdot |CD|=|AD|^2$  and  $\\angle ADB=45^{\\circ}$ , determine the angles of triangle  $ABC$ .","t":[{"b":4,"e":0.28571,"k":"flat","v":0.30357,"x":0.59373,"p":[[0,37,0.0,0.45537,0.19703,0.28571,0.35729,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,16,0,0,2,0,0,10,0,0,0,0,0,4,0,0],[4,37,0.1081,0.59373,0.23987,0.28571,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,9,0,0,2,0,0,10,0,1],[8,37,0.2162,0.34822,0.15126,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,26,0,0,2,0,0,2,0,0,0,0,0,2,0,0],[12,37,0.3243,0.32589,0.09606,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[16,37,0.4324,0.33928,0.13716,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,27,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[20,37,0.5405,0.39286,0.18211,0.28571,0.28571,0.46429,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,2,0,0,5,0,0,0,0,0,3,0,0],[24,37,0.6486,0.32143,0.11294,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,28,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[28,37,0.7568,0.33483,0.13651,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,28,0,0,0,0,0,2,0,0,1,0,0,1,0,0],[32,37,0.8649,0.31249,0.10968,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[36,37,0.973,0.30357,0.06916,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[37,37,1.0,0.31696,0.08552,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.42857,"x":0.49999,"p":[[0,5,0.0,0.42857,0.22016,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,20,0,0,3,0,0,3,0,0,3,0,0,1,0,2],[4,5,0.8,0.48214,0.22232,0.28571,0.35714,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,16,0,0,1,0,0,8,0,0,1,0,0,6,0,0],[5,5,1.0,0.49999,0.20516,0.28571,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,1,0,0,12,0,0,1,0,0,5,0,0]]}]},{"i":"4aacd829373b729a","q":"In a triangle  $ABC$ , the median from  $B$  to  $CA$  is perpendicular to the median from  $C$  to  $AB$ . \nIf the median from  $A$  to  $BC$  is  $30$ , determine  $\\frac{BC^2 + CA^2 + AB^2}{100}$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.66518,"x":0.90625,"p":[[0,29,0.0,0.85267,0.25125,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,0,0,23],[4,29,0.1379,0.90625,0.25407,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[8,29,0.2759,0.73213,0.27837,0.42859,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,4,0,0,2,0,14],[12,29,0.4138,0.75446,0.27947,0.42857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,0,0,0,1,0,17],[16,29,0.5517,0.66518,0.19434,0.57143,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,9,0,0,13,0,0,2,0,4],[20,29,0.6897,0.6964,0.18473,0.57143,0.64286,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,4,0,0,8,0,4],[24,29,0.8276,0.69195,0.20551,0.57143,0.64286,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,0,0,0,13,0,0,4,0,0,7,0,5],[28,29,0.9655,0.73658,0.2205,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,1,0,0,8,0,0,3,0,0,10,0,7],[29,29,1.0,0.73661,0.18933,0.57143,0.71429,0.85714,0.286,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,4,0,0,7,0,7]]},{"b":7,"e":1.0,"k":"rising","v":0.79908,"x":1.0,"p":[[0,48,0.0,0.79908,0.26933,0.57132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,2,0,0,1,0,19],[4,48,0.0833,0.94196,0.18509,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[8,48,0.1667,0.91071,0.1915,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,25],[12,48,0.25,0.8705,0.21243,0.57143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,0,0,23],[16,48,0.3333,0.91071,0.17035,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,25],[20,48,0.4167,0.90625,0.17717,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,24],[24,48,0.5,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[28,48,0.5833,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[32,48,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,48,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,48,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,48,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,48,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]}]},{"i":"c21daefdd8a74c48","q":"In a triangle  $ABC$ , the bisector of the angle  $A$  intersects the excircle that is tangential to side  $[BC]$  at two points  $D$  and  $E$  such that  $D\\in [AE]$ . Prove that,  $$ \\frac{|AD|}{|AE|}\\leq \\frac{|BC|^2}{|DE|^2}. $$","t":[{"b":3,"e":0.57143,"k":"flat","v":0.56696,"x":0.65619,"p":[[0,38,0.0,0.56696,0.15764,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,4,0,0,18,0,0,5,0,0,1,0,1],[4,38,0.1053,0.625,0.26426,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,5,0,0,10,0,0,3,0,0,1,0,8],[8,38,0.2105,0.59375,0.0724,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,4,0,0,1,0,0],[12,38,0.3158,0.62946,0.11769,0.57143,0.57143,0.60714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,5,0,0,1,0,2],[16,38,0.4211,0.65619,0.15513,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,7,0,0,1,0,4],[20,38,0.5263,0.57589,0.07563,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,4,0,0,0,0,0],[24,38,0.6316,0.57588,0.02486,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0],[28,38,0.7368,0.57588,0.02486,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0],[32,38,0.8421,0.57137,0.06186,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,29,0,0,2,0,0,0,0,0],[36,38,0.9474,0.58036,0.03458,0.57143,0.57143,0.57143,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[38,38,1.0,0.56696,0.02486,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.52677,"x":0.66071,"p":[[0,30,0.0,0.57133,0.17879,0.57143,0.57143,0.57143,0.14,1.0,0,2,0,0,0,2,0,0,1,0,0,3,0,0,20,0,0,3,0,0,1,0,2],[4,30,0.1333,0.66071,0.17768,0.57143,0.57143,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,4,0,0,4,0,4],[8,30,0.2667,0.61605,0.16537,0.571,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,4,0,0,2,0,3],[12,30,0.4,0.57594,0.1872,0.42857,0.57143,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,11,0,0,12,0,0,4,0,0,1,0,3],[16,30,0.5333,0.54911,0.06297,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,6,0,0,25,0,0,1,0,0,0,0,0],[20,30,0.6667,0.52677,0.08327,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,21,0,0,1,0,0,0,0,0],[24,30,0.8,0.53125,0.07349,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0],[28,30,0.9333,0.55804,0.09006,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,2,0,0,27,0,0,2,0,0,0,0,0],[30,30,1.0,0.55803,0.04164,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0]]}]},{"i":"1075137eba0498a3","q":"In a triangle $A B C$ right-angled at $C$, the median through $B$ bisects the angle between $B A$ and the bisector of $\\angle B$. Prove that\n\n$$\n\\frac{5}{2}<\\frac{A B}{B C}<3\n$$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.70083,"x":0.86606,"p":[[0,51,0.0,0.86606,0.18877,0.85711,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,11,0,15],[4,51,0.0784,0.83035,0.18363,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,6,0,0,11,0,11],[8,51,0.1569,0.84817,0.14705,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,7,0,13],[12,51,0.2353,0.70083,0.15309,0.57143,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,15,0,0,4,0,3],[16,51,0.3137,0.83481,0.12431,0.71429,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,13,0,8],[20,51,0.3922,0.76338,0.1976,0.57143,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,3,0,0,10,0,8],[24,51,0.4706,0.73212,0.15466,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,12,0,0,7,0,4],[28,51,0.549,0.80357,0.14617,0.71429,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,8,0,8],[32,51,0.6275,0.8214,0.1473,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,10,0,9],[36,51,0.7059,0.79017,0.15965,0.71429,0.85707,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,0,10,0,7],[40,51,0.7843,0.80355,0.17769,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,0,11,0,9],[44,51,0.8627,0.76338,0.18766,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,10,0,0,8,0,7],[48,51,0.9412,0.81247,0.09743,0.71429,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,17,0,3],[51,51,1.0,0.75441,0.16846,0.57143,0.78564,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,11,0,5]]},{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.96875,"p":[[0,20,0.0,0.88839,0.14167,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,16],[4,20,0.2,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[8,20,0.4,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[12,20,0.6,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[16,20,0.8,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[20,20,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"33a10f82f7325560","q":"If  $\\zeta$  denote the Riemann Zeta Function, and  $s>1$  then   $$ \\sum \\limits_{k=1}^{\\infty} \\frac{1}{1+k^s}\\geq \\frac{\\zeta (s)}{1+\\zeta (s)} $$","t":[{"b":2,"e":0.71429,"k":"falling","v":0.51339,"x":0.84821,"p":[[0,10,0.0,0.84821,0.29653,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,25],[4,10,0.4,0.79018,0.34438,0.64286,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,22],[8,10,0.8,0.53568,0.17856,0.42857,0.42859,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,16,0,0,8,0,0,2,0,0,2,0,2],[10,10,1.0,0.51339,0.13296,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,20,0,0,7,0,0,4,0,0,0,0,1]]},{"b":7,"e":1.0,"k":"rising","v":0.66964,"x":1.0,"p":[[0,16,0.0,0.75893,0.32818,0.42857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,20],[4,16,0.25,0.83929,0.31084,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,3,0,23],[8,16,0.5,0.66964,0.37701,0.28571,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,17],[12,16,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d4904cc26ebced89","q":"In acute  $\\vartriangle ABC$ , let points  $D$ ,  $E,$  and  $F$  be the feet of the altitudes of the triangle from  $A$ ,  $B$ ,and  $C$ , respectively. The area of  $\\vartriangle AEF$  is  $1$ , the area of  $\\vartriangle CDE$  is  $2$ , and the area of  $\\vartriangle BF D$  is  $2 -\\sqrt3$ . What is the area of  $\\vartriangle DEF$ ?","t":[{"b":0,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,17,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,17,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,17,0.4706,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,17,0.7059,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,17,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[17,17,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":2,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,23,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,23,0.1739,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,23,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,23,0.5217,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3cced1e8032a4870","q":"In each field of 2009*2009 table you can write either 1 or -1.\r\nDenote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column.\r\nIs it possible to write the numbers in such a way that \r $ \\sum_{i\\equal{}1}^{2009}{Ai}\\plus{}\t\\sum_{i\\equal{}1}^{2009}{Bi}\\equal{}0$ ?","t":[{"b":0,"e":0.0,"k":"flat","v":0.16964,"x":0.32588,"p":[[0,18,0.0,0.32588,0.21198,0.2857,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,0,0,0,20,0,0,0,0,0,2,0,0,5,0,0,0,0,0],[4,18,0.2222,0.3125,0.17655,0.28571,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,0,0,0,22,0,0,0,0,0,4,0,0,2,0,0,0,0,0],[8,18,0.4444,0.21875,0.11837,0.24999,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.17857,0.13832,0.0,0.28571,0.28571,0.0,0.28571,12,0,0,12,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.16964,0.14032,0.0,0.28571,0.28571,0.0,0.28571,13,0,0,13,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.19642,0.13243,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.21429,"x":0.32143,"p":[[0,20,0.0,0.25446,0.17762,0.24999,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,1,0,0,21,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[4,20,0.2,0.32143,0.22016,0.2857,0.28571,0.32143,0.0,0.71429,6,0,0,6,0,0,0,0,18,0,0,1,0,0,2,0,0,5,0,0,0,0,0],[8,20,0.4,0.25893,0.10374,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,0,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.27678,0.15126,0.2857,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,0,0,0,22,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[16,20,0.8,0.21429,0.17857,0.0,0.28571,0.28571,0.0,0.71429,11,0,1,11,0,1,0,0,15,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[20,20,1.0,0.24553,0.10853,0.28571,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,0,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d748957d86b31c99","q":"Let  $ a$ ,  $ b$ , and  $ c$  be non-negative real numbers and let  $ x$ ,  $ y$ , and  $ z$  be positive real numbers such that  $ a\\plus{}b\\plus{}c\\equal{}x\\plus{}y\\plus{}z$ . Prove that\r\n\\[ \\dfrac{a^3}{x^2}\\plus{}\\dfrac{b^3}{y^2}\\plus{}\\dfrac{c^3}{z^2} \\ge a\\plus{}b\\plus{}c.\\]\r\n*Hery Susanto, Malang*","t":[{"b":0,"e":1.0,"k":"flat","v":0.85714,"x":0.95982,"p":[[0,27,0.0,0.85714,0.26726,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,1,0,24],[4,27,0.1481,0.94642,0.13244,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[8,27,0.2963,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[12,27,0.4444,0.94196,0.14223,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,27],[16,27,0.5926,0.92409,0.16748,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,26],[20,27,0.7407,0.875,0.21053,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,6,0,20],[24,27,0.8889,0.95087,0.1269,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[27,27,1.0,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28]]},{"b":7,"e":0.85714,"k":"flat","v":0.79464,"x":0.96429,"p":[[0,23,0.0,0.79464,0.2878,0.53572,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,20],[4,23,0.1739,0.875,0.23891,0.96425,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,1,0,24],[8,23,0.3478,0.91964,0.17835,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,26],[12,23,0.5217,0.88838,0.20436,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,1,0,24],[16,23,0.6957,0.88393,0.20652,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,2,0,23],[20,23,0.8696,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[23,23,1.0,0.90176,0.18712,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,0,0,25]]}]},{"i":"2fa75c8b23045963","q":"Let  $ a,b\\in (0,1) $  and a continuous function  $ f:[0,1]\\longrightarrow\\mathbb{R} $  with the property that  $$  \\int_0^x f(t)dt=\\int_0^{ax} f(t)dt +\\int_0^{bx} f(t)dt,\\quad\\forall x\\in [0,1] .  $$  **a)** Show that if  $ a+b<1, $  then  $ f=0. $ **b)** Show that if  $ a+b=1, $  then  $ f $  is constant.","t":[{"b":6,"e":1.0,"k":"flat","v":0.69642,"x":0.9933,"p":[[0,11,0.0,0.82589,0.20743,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,8,0,0,1,0,17],[4,11,0.3636,0.69642,0.23892,0.42857,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,9,0,0,0,0,10],[8,11,0.7273,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[11,11,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"rising","v":0.69643,"x":1.0,"p":[[0,31,0.0,0.73661,0.26027,0.42857,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,5,0,0,1,0,14],[4,31,0.129,0.69643,0.24419,0.42857,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,8,0,0,3,0,0,8,0,0,1,0,10],[8,31,0.2581,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ee2fdbef14e8aa87","q":"Let  $ G $  be a finite group of odd order having, at least, three elements. For  $ a\\in G $  denote  $ n(a) $  as the number of ways  $ a $  can be written as a product of two distinct elements of  $ G. $ Prove that  $ \\sum_{\\substack{a\\in Ga\\neq\\text{id}}} n(a) $  is a perfect square.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.15179,"x":0.60714,"p":[[0,26,0.0,0.60714,0.35892,0.28571,0.42857,1.0,0.14286,1.0,0,14,0,0,0,6,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,14],[4,26,0.1538,0.36161,0.2923,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,15,0,0,6,0,0,4,0,0,2,0,0,0,0,0,1,0,4],[8,26,0.3077,0.2142,0.16755,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[12,26,0.4615,0.28571,0.26,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,21,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,3],[16,26,0.6154,0.20089,0.1063,0.14286,0.14286,0.17857,0.14286,0.42857,0,0,0,0,0,24,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.17411,0.09268,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,26,0.9231,0.16964,0.09061,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[26,26,1.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.33036,"x":0.62946,"p":[[0,57,0.0,0.62946,0.31916,0.39286,0.42857,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,7,0,0,10,0,0,0,0,0,1,0,0,0,0,13],[4,57,0.0702,0.61161,0.35398,0.28571,0.42857,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,8,0,0,5,0,0,1,0,0,0,0,0,0,0,14],[8,57,0.1404,0.47321,0.2683,0.28571,0.42857,0.42857,0.14286,1.0,0,6,0,0,0,2,0,0,11,0,0,12,0,0,1,0,0,0,0,0,0,0,6],[12,57,0.2105,0.43304,0.2382,0.28571,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,2,0,0,13,0,0,12,0,0,0,0,0,1,0,0,0,0,4],[16,57,0.2807,0.375,0.15872,0.2857,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,13,0,0,12,0,0,3,0,0,0,0,0,0,0,1],[20,57,0.3509,0.36161,0.10092,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,13,0,0,15,0,0,2,0,0,0,0,0,0,0,0],[24,57,0.4211,0.33036,0.14914,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,12,0,0,11,0,0,1,0,0,0,0,0,1,0,0],[28,57,0.4912,0.39286,0.15152,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,8,0,0,18,0,0,2,0,0,0,0,0,0,0,1],[32,57,0.5614,0.38393,0.22142,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,5,0,0,12,0,0,12,0,0,0,0,0,0,0,0,0,0,3],[36,57,0.6316,0.38839,0.1197,0.28571,0.42857,0.42857,0.1429,0.85714,0,0,0,0,0,1,0,0,11,0,0,18,0,0,1,0,0,0,0,0,1,0,0],[40,57,0.7018,0.38839,0.06423,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.37054,0.07016,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.37946,0.07668,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,19,0,0,1,0,0,0,0,0,0,0,0],[52,57,0.9123,0.37946,0.07668,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,19,0,0,1,0,0,0,0,0,0,0,0],[56,57,0.9825,0.36607,0.07936,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,12,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.35714,0.09449,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,16,0,0,13,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"d2f3fc6e803abb5a","q":"Let  $ \\left( x_n\\right)_{n\\ge 1} $  be a sequence of real numbers of the interval  $ [1,\\infty) . $  Suppose that the sequence  $ \\left( \\left[ x_n^k\\right]\\right)_{n\\ge 1} $  is convergent for all natural numbers  $ k. $  Prove that  $ \\left( x_n\\right)_{n\\ge 1} $  is convergent.\n\nHere,  $ [\\beta ] $  means the greatest integer smaller than  $ \\beta . $","t":[{"b":3,"e":1.0,"k":"flat","v":0.9732,"x":1.0,"p":[[0,29,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.9732,"x":1.0,"p":[[0,35,0.0,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[4,35,0.1143,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,35,0.2286,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"91c64da44c4ad38f","q":"Let  $ S$  be the set of nonnegative integers. Find all functions  $ f,g,h: S\\rightarrow S$  such that\r\n\r $ f(m\\plus{}n)\\equal{}g(m)\\plus{}h(n),$  for all  $ m,n\\in S$ , and\r\n\r $ g(1)\\equal{}h(1)\\equal{}1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.90178,"x":1.0,"p":[[0,82,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,82,0.0488,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,82,0.0976,0.91071,0.09942,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,16],[12,82,0.1463,0.90178,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[16,82,0.1951,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[20,82,0.2439,0.91518,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[24,82,0.2927,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[28,82,0.3415,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[32,82,0.3902,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[36,82,0.439,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[40,82,0.4878,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,82,0.5366,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,82,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,82,0.6341,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,82,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,82,0.7317,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,82,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,82,0.8293,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[72,82,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,82,0.9268,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,82,0.9756,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[82,82,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,39,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,39,0.1026,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,39,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,39,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,39,0.4103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,39,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,39,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,39,0.7179,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,39,0.8205,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,39,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d749ef3da8fc4374","q":"Let  $ p$  be a prime and let \\[ l_k(x,y)\\equal{}a_kx\\plus{}b_ky \\;(k\\equal{}1,2,...,p^2)\\ .\\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair  $ (\\xi,\\eta)$  of integers, not both divisible by  $ p$ , the values  $ l_k(\\xi,\\eta), \\;1\\leq k\\leq p^2 $ , represent every residue class  $ \\textrm{mod} \\;p$  exactly  $ p$  times. Prove that the set of pairs  $ \\{(a_k,b_k): 1\\leq k \\leq p^2 \\}$  is identical  $ \\textrm{mod} \\;p$  with the set  $ \\{(m,n): 0\\leq m,n \\leq p\\minus{}1 \\}.$","t":[{"b":3,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,45,0.0,0.91071,0.17035,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,23],[4,45,0.0889,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[8,45,0.1778,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,45,0.2667,0.94643,0.15465,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[16,45,0.3556,0.88839,0.18808,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,2,0,22],[20,45,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,45,0.5333,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,45,0.7111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[45,45,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,20,0.0,0.92857,0.16366,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,26],[4,20,0.2,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"57bb6d6140fe0271","q":"Let  $-2 < x_1 < 2$  be a real number and define  $x_2, x_3, \\ldots$  by  $x_{n+1} = x_n^2-2$  for  $n \\geq 1$ . Assume that no  $x_n$  is  $0$  and define a number  $A$ ,  $0 \\leq A \\leq 1$  in the following way: The  $n^{\\text{th}}$  digit after the decimal point in the binary representation of  $A$  is a  $0$  if  $x_1x_2\\cdots x_n$  is positive and  $1$  otherwise. Prove that  $A = \\frac{1}{\\pi}\\cos^{-1}\\left(\\frac{x_1}{2}\\right)$ .\n\n*Evan O' Dorney.*","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,37,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,37,0.1081,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.91071,"x":0.98214,"p":[[0,36,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,36,0.1111,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,36,0.2222,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,36,0.3333,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[16,36,0.4444,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[20,36,0.5556,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[24,36,0.6667,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[28,36,0.7778,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[32,36,0.8889,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[36,36,1.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]}]},{"i":"79ffd8c4ca1400b3","q":"Let  $0<a<b$  and let  $f:[a,b]\\to\\mathbb R$  be a continuous function with  $\\int^b_af(t)dt=0$ . Show that  $$ \\int^b_a\\int^b_af(x)f(y)\\ln(x+y)dxdy\\le0. $$","t":[{"b":2,"e":1.0,"k":"rising","v":0.69642,"x":1.0,"p":[[0,21,0.0,0.76784,0.27376,0.42857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,1,0,0,0,0,18],[4,21,0.1905,0.69642,0.32094,0.39286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,6,0,0,3,0,0,3,0,0,2,0,0,1,0,15],[8,21,0.381,0.75893,0.28446,0.42857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,2,0,0,0,0,18],[12,21,0.5714,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,21,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,21,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[21,21,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"volatile","v":0.66071,"x":1.0,"p":[[0,5,0.0,0.66071,0.27607,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,11,0,0,0,0,0,6,0,0,0,0,11],[4,5,0.8,0.69196,0.30117,0.42857,0.71429,1.0,0.0,1.0,1,14,1,1,0,0,0,0,2,0,0,11,0,0,0,0,0,4,0,0,0,0,14],[5,5,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d483a9aa3c7e96cf","q":"Let  $A$  be a real  $n\\times n$  matrix satisfying  $$ A+A^{\\text T}=I, $$  where  $A^{\\text T}$  denotes the transpose of  $A$  and  $I$  the  $n\\times n$  identity matrix. Show that  $\\det A>0$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,29,0.0,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,29,0.1379,0.88839,0.25935,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[8,29,0.2759,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,31,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,31,0.129,0.93304,0.17852,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[8,31,0.2581,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,31,0.7742,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,31,0.9032,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1992a048bd935f01","q":"Let  $(x_n)_{n\\geq1}$  be a sequence that verifies:   $$ x_1=1, \\quad x_2=7, \\quad x_{n+1}=x_n+3x_{n-1}, \\forall n \\geq 2. $$  Prove that for every prime number  $p$  the number  $x_p-1$  is divisible by  $3p.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.90178,"x":1.0,"p":[[0,11,0.0,0.90178,0.12595,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,19],[4,11,0.3636,0.91295,0.12719,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,1,20],[8,11,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[11,11,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.81696,"x":0.94196,"p":[[0,36,0.0,0.91071,0.12753,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,21],[4,36,0.1111,0.94196,0.10631,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[8,36,0.2222,0.82143,0.12372,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,6,0,9],[12,36,0.3333,0.81696,0.12993,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,3,0,10],[16,36,0.4444,0.82143,0.14725,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,4,0,11],[20,36,0.5556,0.89955,0.13226,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,1,0,1,0,20],[24,36,0.6667,0.87499,0.14177,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,3,0,17],[28,36,0.7778,0.89732,0.12492,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,18],[32,36,0.8889,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[36,36,1.0,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22]]}]},{"i":"9295b6736fa8fbd9","q":"Let  $ z_1,z_2,z_3 $  be nonzero complex numbers and pairwise distinct, having the property that  $\\left( z_1+z_2\\right)^3 =\\left( z_2+z_3\\right)^3 =\\left( z_3+z_1\\right)^3. $  Show that  $ \\left| z_1-z_2\\right| =\\left| z_2-z_3\\right| =\\left| z_3-z_1\\right| . $","t":[{"b":3,"e":0.71429,"k":"flat","v":0.82589,"x":0.95535,"p":[[0,28,0.0,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[4,28,0.1429,0.90179,0.13092,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,20],[8,28,0.2857,0.85714,0.13832,0.71429,0.85707,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,2,0,15],[12,28,0.4286,0.83482,0.13415,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,3,0,12],[16,28,0.5714,0.83927,0.12752,0.71429,0.857,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,6,0,11],[20,28,0.7143,0.82589,0.12234,0.71429,0.78564,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,9],[24,28,0.8571,0.87054,0.13054,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,5,0,15],[28,28,1.0,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20]]},{"b":4,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,57,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,57,0.0702,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,57,0.1404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,57,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,57,0.2807,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,57,0.3509,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,57,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,57,0.4912,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,57,0.5614,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,57,0.6316,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,57,0.7018,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,57,0.7719,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,57,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,57,0.9123,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,57,0.9825,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[57,57,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ef35f94c2a46194d","q":"Let  $A=20132013...2013$  be formed by joining  $2013$ ,  $165$  times. Prove that  $2013^2 \\mid A$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.34376,"x":0.38839,"p":[[0,37,0.0,0.38393,0.1357,0.28571,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,15,0,0,1,0,0,0,0,0,0,0,1],[4,37,0.1081,0.37946,0.1411,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,15,0,0,1,0,0,0,0,0,0,0,1],[8,37,0.2162,0.34821,0.07087,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.37504,0.08567,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[16,37,0.4324,0.35269,0.07973,0.28571,0.35729,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.38839,0.07349,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0,0,0,0],[24,37,0.6486,0.36161,0.07129,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.36161,0.07128,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.34376,0.08645,0.28571,0.28586,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,15,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.36161,0.07974,0.28571,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.37052,0.10628,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,12,0,0,15,0,0,3,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.33928,"x":0.37947,"p":[[0,37,0.0,0.37947,0.14555,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,12,0,0,16,0,0,1,0,0,0,0,0,0,0,1],[4,37,0.1081,0.34821,0.09407,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.3616,0.09439,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[12,37,0.3243,0.375,0.07784,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[16,37,0.4324,0.35714,0.07143,0.28571,0.35714,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.36161,0.07973,0.28571,0.35714,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[24,37,0.6486,0.33928,0.06916,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.35269,0.07128,0.28571,0.28586,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,17,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.375,0.06916,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.36161,0.07129,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.35715,0.07142,0.28571,0.35729,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b345878bf593f36f","q":"Let  $ABC$  be a triangle and  $h_a$  be the altitude through  $A$ . Prove that \\[ (b+c)^2 \\geq a^2 + 4h_a ^2 .  \\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.77677,"x":0.92856,"p":[[0,25,0.0,0.77677,0.32916,0.57132,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,1,0,20],[4,25,0.16,0.92856,0.15155,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,25],[8,25,0.32,0.86606,0.24207,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[12,25,0.48,0.87946,0.22899,0.82132,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,22],[16,25,0.64,0.87945,0.14775,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,19],[20,25,0.8,0.89285,0.15152,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,2,0,20],[24,25,0.96,0.81696,0.18977,0.71429,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,14,0,0,3,0,13],[25,25,1.0,0.87053,0.18336,0.71429,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,18]]},{"b":6,"e":1.0,"k":"flat","v":0.84375,"x":1.0,"p":[[0,22,0.0,0.85713,0.30306,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,25],[4,22,0.1818,0.84375,0.29312,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,22],[8,22,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,22,0.9091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[22,22,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"a2b53850e2ead381","q":"Let  $A$  and  $B$  be variable points on  $x-$ axis and  $y-$ axis respectively such that the line segment  $AB$  is in the first quadrant and of a fixed length  $2d$  . Let  $C$  be the mid-point of  $AB$  and  $P$  be a point such that**(a)**  $P$  and the origin are on the opposite sides of  $AB$  and,**(b)**  $PC$  is a line segment of length  $d$  which is perpendicular to  $AB$  .\n\nFind the locus of  $P$  .","t":[{"b":5,"e":1.0,"k":"flat","v":0.875,"x":0.99107,"p":[[0,40,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,40,0.1,0.93749,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,40,0.2,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,40,0.3,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[16,40,0.4,0.89284,0.06187,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[20,40,0.5,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[24,40,0.6,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[28,40,0.7,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[32,40,0.8,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[36,40,0.9,0.875,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[40,40,1.0,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9]]},{"b":7,"e":0.857,"k":"flat","v":0.88392,"x":0.97321,"p":[[0,11,0.0,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,11,0.3636,0.94195,0.07018,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[8,11,0.7273,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[11,11,1.0,0.88393,0.05576,0.85714,0.85714,0.85714,0.8571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6]]}]},{"i":"a6a20f12f633d053","q":"Let  $A$  be a real  $n \\times n$  matrix and suppose that for every positive integer  $m$  there exists a real symmetric matrix  $B$  such that  $$ 2021B = A^m+B^2. $$  Prove that  $|\\text{det} A| \\leq 1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,38,0.0,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[4,38,0.1053,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,38,0.2105,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,8,0,21],[12,38,0.3158,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,12,0.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[4,12,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5f2d1d7edc97ff55","q":"Let  $ABC$  be a triangle such that  $|BC|=7$  and  $|AB|=9$ . If  $m(\\widehat{ABC}) = 2m(\\widehat{BCA})$ , then what is the area of the triangle? $ \n\\textbf{(A)}\\ 14\\sqrt 5\n\\qquad\\textbf{(B)}\\ 30\n\\qquad\\textbf{(C)}\\ 10\\sqrt 6\n\\qquad\\textbf{(D)}\\ 20 \\sqrt 2\n\\qquad\\textbf{(E)}\\ 12 \\sqrt 3\n$","t":[{"b":5,"e":0.71429,"k":"flat","v":0.56696,"x":0.74107,"p":[[0,49,0.0,0.56696,0.21275,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,15,0,0,1,0,0,11,0,0,0,0,3],[4,49,0.0816,0.63393,0.16342,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,19,0,0,0,0,2],[8,49,0.1633,0.70536,0.18189,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,19,0,0,0,0,6],[12,49,0.2449,0.66964,0.16146,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,24,0,0,0,0,2],[16,49,0.3265,0.66071,0.15047,0.64286,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,22,0,0,0,0,2],[20,49,0.4082,0.625,0.20748,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,9,0,0,0,0,0,18,0,0,0,0,3],[24,49,0.4898,0.6875,0.13092,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,25,0,0,0,0,2],[28,49,0.5714,0.64286,0.15972,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,20,0,0,0,0,2],[32,49,0.6531,0.61607,0.18363,0.42857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,15,0,0,0,0,3],[36,49,0.7347,0.74107,0.13092,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,25,0,0,0,0,5],[40,49,0.8163,0.59822,0.20024,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,8,0,0,0,0,0,20,0,0,0,0,1],[44,49,0.898,0.65179,0.13803,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,26,0,0,0,0,0],[48,49,0.9796,0.66964,0.10374,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,27,0,0,0,0,0],[49,49,1.0,0.6875,0.10972,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.46429,"x":0.62052,"p":[[0,50,0.0,0.58929,0.22517,0.42857,0.57144,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,14,0,0,0,0,0,12,0,0,0,0,4],[4,50,0.08,0.61607,0.18363,0.42857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,15,0,0,0,0,3],[8,50,0.16,0.48214,0.16656,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,23,0,0,0,0,0,6,0,0,0,0,1],[12,50,0.24,0.62052,0.19434,0.42857,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,13,0,0,0,0,4],[16,50,0.32,0.47321,0.16146,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,24,0,0,0,0,0,5,0,0,0,0,1],[20,50,0.4,0.52679,0.19704,0.42857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,19,0,0,0,0,0,9,0,0,0,0,2],[24,50,0.48,0.5,0.15972,0.42857,0.42857,0.50002,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,23,0,0,0,0,0,7,0,0,0,0,1],[28,50,0.56,0.50897,0.14696,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,21,0,0,0,0,0,10,0,0,0,0,0],[32,50,0.64,0.50893,0.14698,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,21,0,0,0,0,0,10,0,0,0,0,0],[36,50,0.72,0.48214,0.15047,0.42857,0.42857,0.50002,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,22,0,0,0,0,0,8,0,0,0,0,0],[40,50,0.8,0.5625,0.14258,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,15,0,0,0,0,0],[44,50,0.88,0.46429,0.11845,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,5,0,0,0,0,0],[48,50,0.96,0.50446,0.15146,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,20,0,0,0,0,0,10,0,0,0,0,0],[50,50,1.0,0.5,0.14286,0.42857,0.42857,0.42858,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,6,0,0,0,0,1]]}]},{"i":"d16c5ee016cb9b34","q":"Let  $ABC$  be a scalene triangle,  $\\Gamma$  its circumscribed circle and  $H$  the point where the altitudes of triangle  $ABC$  meet. The circumference with center at  $H$  passing through  $A$  cuts  $\\Gamma$  at a second point  $D$ . In the same way, the circles with center at  $H$  and passing through  $B$  and  $C$  cut  $\\Gamma$  again at points  $E$  and  $F$ , respectively. Prove that  $H$  is also the point in which the altitudes of the triangle  $DEF$  meet.","t":[{"b":4,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,31,0.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[4,31,0.129,0.95089,0.19103,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,14,0.0,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[4,14,0.2857,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[8,14,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,14,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[14,14,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"165d52cdc2e4f8fc","q":"Let  $A=\\{1,2,3,4\\}$ , and  $f$  and  $g$  be randomly chosen (not necessarily distinct) functions from  $A$  to  $A$ . The probability that the range of  $f$  and the range of  $g$  are disjoint is  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m$ .","t":[{"b":1,"e":0.14286,"k":"falling","v":0.32143,"x":0.75893,"p":[[0,74,0.0,0.53572,0.30929,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,9],[4,74,0.0541,0.49107,0.26471,0.42857,0.42857,0.42857,0.14286,1.0,0,6,0,0,0,5,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,6],[8,74,0.1081,0.61607,0.28221,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,11],[12,74,0.1622,0.52679,0.26351,0.42857,0.42857,0.42858,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,7],[16,74,0.2162,0.75893,0.32031,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,20],[20,74,0.2703,0.54464,0.30186,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,9],[24,74,0.3243,0.66071,0.30671,0.42857,0.42857,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,14],[28,74,0.3784,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,74,0.4324,0.41071,0.06916,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[40,74,0.5405,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[44,74,0.5946,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[48,74,0.6486,0.39286,0.09449,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,4,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[52,74,0.7027,0.39286,0.15567,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,1],[56,74,0.7568,0.39286,0.09449,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,4,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.40179,0.08328,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[64,74,0.8649,0.34821,0.12846,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,9,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[68,74,0.9189,0.375,0.11152,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,6,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[72,74,0.973,0.34821,0.12846,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,9,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[74,74,1.0,0.32143,0.13832,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,12,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"falling","v":0.25893,"x":0.8125,"p":[[0,89,0.0,0.58036,0.31931,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,11],[4,89,0.0449,0.46429,0.30929,0.14286,0.42857,0.42858,0.14286,1.0,0,7,0,0,0,10,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,7],[8,89,0.0899,0.57143,0.37115,0.14286,0.42857,1.0,0.14286,1.0,0,13,0,0,0,10,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,13],[12,89,0.1348,0.59821,0.38538,0.14286,0.42857,1.0,0.0,1.0,1,14,0,1,0,9,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,14],[16,89,0.1798,0.46429,0.33312,0.14286,0.42857,0.57143,0.14286,1.0,0,8,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,8],[20,89,0.2247,0.73214,0.37754,0.35714,1.0,1.0,0.14286,1.0,0,21,0,0,0,8,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,21],[24,89,0.2697,0.75,0.33312,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,20],[28,89,0.3146,0.66964,0.32031,0.42857,0.42857,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,15],[32,89,0.3596,0.76786,0.30671,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,20],[36,89,0.4045,0.5625,0.37786,0.14286,0.42857,1.0,0.14286,1.0,0,13,0,0,0,11,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,13],[40,89,0.4494,0.66072,0.37415,0.35714,1.0,1.0,0.14286,1.0,0,17,0,0,0,8,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,17],[44,89,0.4944,0.77232,0.34967,0.42857,1.0,1.0,0.0,1.0,1,22,0,1,0,4,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,22],[48,89,0.5393,0.8125,0.33204,0.85714,1.0,1.0,0.14286,1.0,0,24,0,0,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,24],[52,89,0.5843,0.40179,0.08328,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[56,89,0.6292,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[60,89,0.6742,0.41518,0.05486,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[64,89,0.7191,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.33929,0.13243,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,10,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.28571,0.14286,0.14286,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.26786,0.14174,0.14286,0.14286,0.42857,0.14286,0.42857,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.30357,0.14174,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,14,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.29464,0.14258,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,15,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.25893,0.14032,0.14286,0.14286,0.42857,0.14286,0.42857,0,0,0,0,0,19,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.3125,0.14032,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,13,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9d1f2c379c85e35d","q":"Let  $ABC$  be a triangle of perimeter  $100$  and  $I$  be the point of intersection of its bisectors. Let  $M$  be the midpoint of side  $BC$ . The line parallel to  $AB$  drawn by $ I$  cuts the median  $AM$  at point  $P$  so that  $\\frac{AP}{PM} =\\frac73$ . Find the length of side  $AB$ .","t":[{"b":4,"e":0.42857,"k":"flat","v":0.34812,"x":0.62723,"p":[[0,39,0.0,0.39286,0.25505,0.14286,0.42857,0.42858,0.0,1.0,2,2,0,2,0,8,0,0,3,0,0,12,0,0,2,0,0,2,0,0,1,0,2],[4,39,0.1026,0.45088,0.28595,0.14286,0.42857,0.71429,0.0,1.0,1,1,0,1,0,9,0,0,5,0,0,3,0,0,3,0,0,6,0,0,4,0,1],[8,39,0.2051,0.45982,0.2618,0.24999,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,8,0,0,5,0,0,5,0,0,6,0,0,4,0,0,2,0,2],[12,39,0.3077,0.42411,0.23551,0.14286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,3,0,0,8,0,0,5,0,0,5,0,0,2,0,0],[16,39,0.4103,0.34812,0.22578,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,5,0,0,8,0,0,2,0,0,4,0,0,1,0,0],[20,39,0.5128,0.3884,0.25812,0.14286,0.35714,0.46429,0.0,1.0,1,1,0,1,0,10,0,0,5,0,0,8,0,0,2,0,0,2,0,0,3,0,1],[24,39,0.6154,0.41071,0.25191,0.24999,0.42857,0.57143,0.0,1.0,2,1,0,2,0,6,0,0,7,0,0,6,0,0,5,0,0,3,0,0,2,0,1],[28,39,0.7179,0.37945,0.24381,0.14286,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,10,0,0,3,0,0,6,0,0,6,0,0,3,0,0,2,0,0],[32,39,0.8205,0.52232,0.22759,0.42857,0.50001,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,3,0,0,10,0,0,6,0,0,7,0,0,1,0,2],[36,39,0.9231,0.62723,0.2278,0.42857,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,7,0,0,5,0,0,4,1,4],[39,39,1.0,0.52081,0.16832,0.42857,0.4286,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,13,0,0,9,0,1,2,0,0,2,0,1]]},{"b":7,"e":0.42857,"k":"flat","v":0.35706,"x":0.66517,"p":[[0,50,0.0,0.48661,0.23107,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,6,0,0,9,0,0,4,0,0,5,0,0,3,0,1],[4,50,0.08,0.40625,0.28372,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,8,0,0,4,0,0,6,0,0,4,0,0,2,0,0,4,0,1],[8,50,0.16,0.35706,0.22312,0.14286,0.35714,0.46431,0.0,0.85714,3,0,1,3,0,7,0,0,6,0,0,8,0,0,4,0,0,3,0,0,1,0,0],[12,50,0.24,0.4866,0.21682,0.42857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,3,0,0,13,0,0,5,0,0,3,0,0,3,0,1],[16,50,0.32,0.55357,0.23622,0.42857,0.57141,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,8,0,0,6,0,0,4,0,0,5,0,2],[20,50,0.4,0.64732,0.23954,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,9,0,0,5,0,0,4,0,0,5,0,6],[24,50,0.48,0.54911,0.17536,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,7,0,0,14,0,0,4,0,0,2,0,1],[28,50,0.56,0.5357,0.2369,0.39286,0.42857,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,7,0,0,9,0,0,3,0,0,7,0,0,3,0,2],[32,50,0.64,0.57141,0.21724,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,7,0,0,5,0,0,6,0,0,6,0,1],[36,50,0.72,0.66517,0.17718,0.53539,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,6,0,0,9,0,0,7,0,2],[40,50,0.8,0.58482,0.18335,0.42859,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,8,0,0,8,0,0,8,0,0,5,0,0],[44,50,0.88,0.57141,0.23958,0.42857,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,8,0,0,10,0,0,2,0,0,3,0,4],[48,50,0.96,0.5491,0.16408,0.42857,0.57143,0.57143,0.1429,1.0,0,2,0,0,0,1,0,0,0,0,0,12,0,0,13,0,0,4,0,0,0,0,2],[50,50,1.0,0.58927,0.21651,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,8,0,0,9,0,0,7,0,0,2,0,3]]}]},{"i":"56c75ec87ab60b3e","q":"Let  $ABC$  be a triangle with  $AB=10$ ,  $AC=11$ , and circumradius  $6$ .  Points  $D$  and  $E$  are located on the circumcircle of  $\\triangle ABC$  such that  $\\triangle ADE$  is equilateral.  Line segments  $\\overline{DE}$  and  $\\overline{BC}$  intersect at  $X$ .  Find  $\\tfrac{BX}{XC}$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.65625,"x":0.70982,"p":[[0,46,0.0,0.68303,0.09932,0.57143,0.71429,0.71429,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,15,0,0,5,0,0],[4,46,0.087,0.69196,0.11904,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,17,0,0,6,0,0],[8,46,0.1739,0.66964,0.0974,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,14,0,0,4,0,0],[12,46,0.2609,0.68737,0.07519,0.67536,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,22,0,0,2,0,0],[16,46,0.3478,0.66951,0.06613,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0],[20,46,0.4348,0.68749,0.07525,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,25,0,0,1,0,0],[24,46,0.5217,0.67857,0.11294,0.67857,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,21,0,0,3,0,0],[28,46,0.6087,0.66964,0.06622,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0],[32,46,0.6957,0.65625,0.10012,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,17,0,0,2,0,0],[36,46,0.7826,0.67857,0.07986,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,20,0,0,2,0,0],[40,46,0.8696,0.67411,0.08171,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,22,0,0,1,0,0],[44,46,0.9565,0.68304,0.09268,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,20,0,0,3,0,0],[46,46,1.0,0.70982,0.08364,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,24,0,0,4,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.64286,"x":0.77232,"p":[[0,26,0.0,0.64286,0.11294,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,18,0,0,1,0,0],[4,26,0.1538,0.69196,0.09523,0.67857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,20,0,0,4,0,0],[8,26,0.3077,0.75,0.09449,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,12,0,0],[12,26,0.4615,0.74554,0.07771,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,9,0,0],[16,26,0.6154,0.76785,0.09278,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,15,0,0],[20,26,0.7692,0.77232,0.07873,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,14,0,0],[24,26,0.9231,0.75446,0.10248,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,14,0,0],[26,26,1.0,0.74105,0.10376,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,0,11,0,0]]}]},{"i":"a3f3f92c5282d60d","q":"Let  $ABC$  be a triangle and  $D$  the foot of the altitude from  $A$ . Let  $E$  and  $F$  lie on a line passing through  $D$  such that  $AE$  is perpendicular to  $BE$ ,  $AF$  is perpendicular to  $CF$ , and  $E$  and  $F$  are different from  $D$ . Let  $M$  and  $N$  be the midpoints of the segments  $BC$  and  $EF$ , respectively. Prove that  $AN$  is perpendicular to  $NM$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,20,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,20,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,32,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,32,0.125,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,32,0.25,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,32,0.375,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,32,0.5,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,32,0.875,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,32,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"653473cf01f788dc","q":"Let  $ABC$  be a triangle in which  $\\angle ABC = 60^{\\circ}$ . Let  $I$  and  $O$  be the incentre and circumcentre of  $ABC$ , respectively. Let  $M$  be the midpoint of the arc  $BC$  of the circumcircle of  $ABC$ , which does not contain the point  $A$ . Determine  $\\angle BAC$  given that  $MB = OI$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.48659,"x":0.57141,"p":[[0,40,0.0,0.51786,0.13243,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,4,0,0,18,0,0,4,0,0,0,0,0],[4,40,0.1,0.48659,0.16697,0.28571,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,4,0,0,12,0,0,6,0,0,0,0,0],[8,40,0.2,0.5268,0.14031,0.42857,0.50001,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,13,0,0,7,0,0,9,0,0,0,0,0],[12,40,0.3,0.53572,0.15567,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,12,0,0,6,0,0,11,0,0,0,0,0],[16,40,0.4,0.5179,0.13713,0.42857,0.42859,0.60714,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,17,0,0,6,0,0,8,0,0,0,0,0],[20,40,0.5,0.52232,0.11633,0.42857,0.4286,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,7,0,0,7,0,0,0,0,0],[24,40,0.6,0.53571,0.14286,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,12,0,0,9,0,0,9,0,0,0,0,0],[28,40,0.7,0.57141,0.13362,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,14,0,0,4,0,0,14,0,0,0,0,0],[32,40,0.8,0.54909,0.12428,0.42857,0.57121,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,15,0,0,7,0,0,10,0,0,0,0,0],[36,40,0.9,0.55802,0.16888,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,12,0,0,3,0,0,15,0,0,0,0,0],[40,40,1.0,0.55359,0.12748,0.42857,0.57141,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,12,0,0,9,0,0,10,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.41964,"x":0.5,"p":[[0,72,0.0,0.49999,0.12876,0.42857,0.57143,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,3,0,0,10,0,0,15,0,0,3,0,0,0,0,0],[4,72,0.0556,0.47768,0.16602,0.28571,0.50001,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,6,0,0,13,0,0,2,0,0,0,0,1],[8,72,0.1111,0.45982,0.12234,0.28571,0.50001,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[12,72,0.1667,0.42857,0.12877,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,9,0,0,10,0,0,1,0,0,0,0,0],[16,72,0.2222,0.46875,0.12492,0.28571,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,9,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[20,72,0.2778,0.46429,0.11845,0.39286,0.50001,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,8,0,0,8,0,0,16,0,0,0,0,0,0,0,0],[24,72,0.3333,0.5,0.11845,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,6,0,0,5,0,0,20,0,0,1,0,0,0,0,0],[28,72,0.3889,0.44642,0.13715,0.39286,0.4286,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,7,0,0,10,0,0,14,0,0,0,0,0,0,0,0],[32,72,0.4444,0.46875,0.12492,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[36,72,0.5,0.46874,0.12992,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,3,0,0,19,0,0,0,0,0,0,0,0],[40,72,0.5556,0.45982,0.13236,0.28571,0.50001,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,6,0,0,15,0,0,1,0,0,0,0,0],[44,72,0.6111,0.42858,0.13362,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,4,0,0,14,0,0,0,0,0,0,0,0],[48,72,0.6667,0.4732,0.12594,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,4,0,0,19,0,0,0,0,0,0,0,0],[52,72,0.7222,0.45534,0.12594,0.28571,0.49979,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,6,0,0,16,0,0,0,0,0,0,0,0],[56,72,0.7778,0.45089,0.13415,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,3,0,0,17,0,0,0,0,0,0,0,0],[60,72,0.8333,0.44643,0.17035,0.28571,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,1,0,0,6,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[64,72,0.8889,0.42409,0.11563,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,11,0,0,11,0,0,10,0,0,0,0,0,0,0,0],[68,72,0.9444,0.46874,0.11424,0.42857,0.49979,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,9,0,0,16,0,0,0,0,0,0,0,0],[72,72,1.0,0.41964,0.12846,0.28571,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,14,0,0,6,0,0,12,0,0,0,0,0,0,0,0]]}]},{"i":"8b36ec510343631d","q":"Let  $ABC$  be a triangle with  $\\angle A=90^{\\circ}$  and  $AB=AC$ . Let  $D$  and  $E$  be points on the segment  $BC$  such that  $BD:DE:EC = 1:2:\\sqrt{3}$ . Prove that  $\\angle DAE= 45^{\\circ}$","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.05357,"p":[[0,21,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.03125,"x":0.07143,"p":[[0,36,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fde4db10bc69f4ea","q":"Let  $ABC$  be a triangle with  $AB=AC$ . Also, let  $D\\in[BC]$  be a point such that  $BC>BD>DC>0$ , and let  $\\mathcal{C}_1,\\mathcal{C}_2$  be the circumcircles of the triangles  $ABD$  and  $ADC$  respectively. Let  $BB'$  and  $CC'$  be diameters in the two circles, and let  $M$  be the midpoint of  $B'C'$ . Prove that the area of the triangle  $MBC$  is constant (i.e. it does not depend on the choice of the point  $D$ ).\r\n\r\n*Greece*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.38393,"x":0.42411,"p":[[0,30,0.0,0.41965,0.04971,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0],[4,30,0.1333,0.39733,0.05904,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.41523,0.04166,0.42857,0.42857,0.42857,0.2857,0.43,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.38393,0.07524,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.38393,0.07524,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.40625,0.08073,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,24,0,0,2,0,0,0,0,0,0,0,0],[24,30,0.8,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.39732,0.05906,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.41072,0.05923,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.39732,"x":0.42396,"p":[[0,33,0.0,0.39737,0.05909,0.42857,0.42857,0.42857,0.2857,0.43,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.41964,0.07087,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0,0,0,0],[8,33,0.2424,0.41964,0.04971,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0],[12,33,0.3636,0.39732,0.06902,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.41969,0.07088,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0,0,0,0],[20,33,0.6061,0.41965,0.07936,0.42857,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,25,0,0,1,0,0,1,0,0,0,0,0],[24,33,0.7273,0.39732,0.08553,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,22,0,0,0,0,0,1,0,0,0,0,0],[28,33,0.8485,0.42396,0.07509,0.42857,0.42857,0.42857,0.2857,0.71,0,0,0,0,0,0,0,0,4,0,0,26,0,0,1,0,0,1,0,0,0,0,0],[32,33,0.9697,0.40178,0.05576,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.41518,0.08268,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,24,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"45a264fcda1cd695","q":"Let  $ABC$  be a triangle with height  $AH$ .  $P$  lies on the circle over 3 midpoint of  $AB,BC,CA$  ( $P \\notin BC$ ). Prove that the line connect 2 center of  $(PBH)$  and  $(PCH)$  go through a fixed point.\n(where  $(XYZ)$  be a circumscribed circle of triangle  $XYZ$ )","t":[{"b":1,"e":1.0,"k":"flat","v":0.81696,"x":0.98214,"p":[[0,35,0.0,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[4,35,0.1143,0.86607,0.24206,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,23],[8,35,0.2286,0.81696,0.29502,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,5,0,0,1,0,0,0,0,0,3,0,0,0,0,22],[12,35,0.3429,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,35,0.4571,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,35,0.5714,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[24,35,0.6857,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,35,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,35,0.9143,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[35,35,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":0.98661,"p":[[0,35,0.0,0.9375,0.17473,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[4,35,0.1143,0.95089,0.14555,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[8,35,0.2286,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,35,0.3429,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,35,0.4571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,35,0.5714,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,35,0.6857,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,35,0.8,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[32,35,0.9143,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[35,35,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]}]},{"i":"a9e4e8d969d10b36","q":"Let  $ABC$  be an acute triangle with altitude  $AD$  ( $D \\in BC$ ). The line through  $C$  parallel to  $AB$  meets the perpendicular bisector of  $AD$  at  $G$ . Show that  $AC = BC$  if and only if  $\\angle AGC = 90^{\\circ}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.22321,"x":0.44643,"p":[[0,55,0.0,0.40179,0.46898,0.0,0.07143,1.0,0.0,1.0,16,12,0,16,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[4,55,0.0727,0.36607,0.44598,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,10],[8,55,0.1455,0.3125,0.43805,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[12,55,0.2182,0.25446,0.38752,0.0,0.0,0.28571,0.0,1.0,19,6,0,19,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[16,55,0.2909,0.39286,0.47515,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[20,55,0.3636,0.3125,0.43805,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[24,55,0.4364,0.25445,0.4099,0.0,0.0,0.35702,0.0,1.0,21,7,0,21,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,7],[28,55,0.5091,0.26786,0.40208,0.0,0.0,0.32143,0.0,1.0,19,7,0,19,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[32,55,0.5818,0.22321,0.38289,0.0,0.0,0.28571,0.0,1.0,21,6,0,21,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[36,55,0.6545,0.35714,0.44607,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,10],[40,55,0.7273,0.38839,0.45208,0.0,0.14286,1.0,0.0,1.0,15,11,0,15,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[44,55,0.8,0.40178,0.47034,0.0,0.0,1.0,0.0,1.0,17,12,0,17,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[48,55,0.8727,0.44643,0.47615,0.0,0.14286,1.0,0.0,1.0,15,13,0,15,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,13],[52,55,0.9455,0.33482,0.40502,0.0,0.14288,0.57145,0.0,1.0,14,8,0,14,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,8],[55,55,1.0,0.375,0.45841,0.0,0.07143,1.0,0.0,1.0,16,11,0,16,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,11]]},{"b":7,"e":0.0,"k":"flat","v":0.14732,"x":0.48661,"p":[[0,37,0.0,0.33036,0.45518,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[4,37,0.1081,0.22321,0.41178,0.0,0.0,0.03571,0.0,1.0,24,7,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,37,0.2162,0.29009,0.42183,0.0,0.0,0.57143,0.0,1.0,19,8,0,19,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[12,37,0.3243,0.32134,0.43452,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[16,37,0.4324,0.14732,0.33021,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[20,37,0.5405,0.48661,0.47361,0.0,0.28571,1.0,0.0,1.0,14,14,0,14,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,14],[24,37,0.6486,0.32589,0.45769,0.0,0.0,1.0,0.0,1.0,20,10,0,20,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[28,37,0.7568,0.29018,0.41876,0.0,0.0,0.46429,0.0,1.0,18,8,0,18,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[32,37,0.8649,0.37053,0.46134,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[36,37,0.973,0.25,0.40564,0.0,0.0,0.28571,0.0,1.0,21,7,0,21,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[37,37,1.0,0.36607,0.46694,0.0,0.0,1.0,0.0,1.0,19,11,0,19,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,11]]}]},{"i":"110845a9fa00bc4b","q":"Let  $ABC$  be an acute-angled  triangle with  $AB<AC,$   and let  $D$  be the foot of its altitude from $A.$  Let  $R$  and  $Q$  be the centroids of  triangles  $ABD$  and  $ACD$ , respectively. Let  $P$  be a point on the line segment  $BC$  such that  $P \\neq D$  and points  $P$   $Q$   $R$  and  $D$  are concyclic .Prove that the lines  $AP$   $BQ$  and  $CR$   are concurrent.","t":[{"b":4,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,23,0.0,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[4,23,0.1739,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,23,0.3478,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,23,0.8696,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.92411,"x":0.98661,"p":[[0,32,0.0,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[4,32,0.125,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[8,32,0.25,0.92411,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[12,32,0.375,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,32,0.5,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[20,32,0.625,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,32,0.75,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,32,0.875,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,32,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"f5dcbe66aa2d29bf","q":"Let  $ABCD$  be a convex quadrilateral whose diagonals intersect at  $O.$  Given that \\[\\overrightarrow{AB}+\\overrightarrow{AD}+\\overrightarrow{AO}=\\overrightarrow{BC}+\\overrightarrow{DC}+\\overrightarrow{OC},\\]prove that  $ABCD$  is a parallelogram.","t":[{"b":5,"e":1.0,"k":"flat","v":0.84374,"x":1.0,"p":[[0,27,0.0,0.92409,0.16748,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,26],[4,27,0.1481,0.84374,0.20316,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,4,0,17],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.57143,"k":"falling","v":0.56686,"x":0.95089,"p":[[0,37,0.0,0.95089,0.16213,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[4,37,0.1081,0.77006,0.28839,0.57132,1.0,1.0,0.071,1.0,0,17,0,0,1,0,0,0,4,0,0,2,0,0,2,0,0,5,0,0,1,0,17],[8,37,0.2162,0.77679,0.24984,0.67857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,6,0,0,4,0,14],[12,37,0.3243,0.79463,0.23942,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,3,0,0,4,0,15],[16,37,0.4324,0.83481,0.23987,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,3,0,18],[20,37,0.5405,0.73661,0.29038,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,2,0,0,1,0,16],[24,37,0.6486,0.76343,0.26143,0.57143,0.85707,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,5,0,13],[28,37,0.7568,0.56686,0.246,0.42857,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,6,0,0,6,0,0,8,0,0,5,0,0,2,0,4],[32,37,0.8649,0.66515,0.21012,0.42859,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,8,0,0,6,0,4],[36,37,0.973,0.64733,0.2112,0.42964,0.71429,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,9,0,0,4,0,4],[37,37,1.0,0.56695,0.2355,0.42857,0.4286,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,6,0,0,11,0,0,4,0,0,4,0,0,3,0,4]]}]},{"i":"08de118298fe47a3","q":"Let  $ABC$  be an isosceles triangle with  $\\angle A = 90^{\\circ}$ . Points  $D$  and  $E$  are selected on sides  $AB$  and  $AC$ , and points  $X$  and  $Y$  are the feet of the altitudes from  $D$  and  $E$  to side  $BC$ .  Given that  $AD = 48\\sqrt2$  and  $AE = 52\\sqrt2$ , compute  $XY$ .\n\n*Proposed by Evan Chen*","t":[{"b":1,"e":1.0,"k":"falling","v":0.36161,"x":0.82143,"p":[[0,53,0.0,0.73661,0.40107,0.28571,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,22],[4,53,0.0755,0.50893,0.40238,0.28571,0.35714,1.0,0.0,1.0,7,12,0,7,0,0,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,12],[8,53,0.1509,0.48214,0.44428,0.0,0.28571,1.0,0.0,1.0,11,13,0,11,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,13],[12,53,0.2264,0.55803,0.40777,0.28571,0.42857,1.0,0.0,1.0,6,14,0,6,0,0,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,14],[16,53,0.3019,0.82143,0.32143,0.85714,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,24],[20,53,0.3774,0.74107,0.34707,0.42857,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,20],[24,53,0.4528,0.67857,0.35535,0.39286,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,17],[28,53,0.5283,0.69196,0.39465,0.39286,1.0,1.0,0.0,1.0,5,19,0,5,0,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,19],[32,53,0.6038,0.69197,0.38814,0.28571,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,19],[36,53,0.6792,0.70536,0.36932,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,18],[40,53,0.7547,0.50446,0.38793,0.28571,0.42857,1.0,0.0,1.0,7,11,0,7,0,0,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,11],[44,53,0.8302,0.46428,0.37287,0.2857,0.35714,1.0,0.0,1.0,7,9,0,7,0,0,0,0,9,0,0,6,0,0,0,0,0,1,0,0,0,0,9],[48,53,0.9057,0.36161,0.3694,0.0,0.28571,0.42857,0.0,1.0,11,7,0,11,0,0,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,7],[52,53,0.9811,0.47321,0.3489,0.28571,0.28571,1.0,0.0,1.0,4,9,0,4,0,0,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,9],[53,53,1.0,0.375,0.35848,0.0,0.28571,0.42857,0.0,1.0,9,7,0,9,0,0,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,7]]},{"b":6,"e":0.2857,"k":"falling","v":0.25,"x":0.75,"p":[[0,41,0.0,0.67411,0.3867,0.28571,1.0,1.0,0.0,1.0,4,18,0,4,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,18],[4,41,0.0976,0.62946,0.41321,0.28571,1.0,1.0,0.0,1.0,6,17,0,6,0,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,17],[8,41,0.1951,0.58929,0.42671,0.28571,0.71429,1.0,0.0,1.0,7,16,0,7,0,0,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,16],[12,41,0.2927,0.61607,0.42624,0.28571,1.0,1.0,0.0,1.0,7,17,0,7,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,17],[16,41,0.3902,0.75,0.37627,0.39286,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,21],[20,41,0.4878,0.6875,0.38538,0.28571,1.0,1.0,0.0,1.0,4,18,0,4,0,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,1,0,18],[24,41,0.5854,0.41964,0.38289,0.0,0.28571,0.78571,0.0,1.0,10,8,0,10,0,0,0,0,7,0,0,5,0,0,1,0,0,1,0,0,0,0,8],[28,41,0.6829,0.36161,0.37455,0.0,0.28571,0.42857,0.0,1.0,12,7,0,12,0,0,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,7],[32,41,0.7805,0.26339,0.2969,0.0,0.28571,0.42857,0.0,1.0,14,3,0,14,0,0,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,3],[36,41,0.878,0.26786,0.21943,0.0,0.28571,0.42857,0.0,1.0,10,1,0,10,0,0,0,0,10,0,0,11,0,0,0,0,0,0,0,0,0,0,1],[40,41,0.9756,0.29019,0.25626,0.0,0.28571,0.42857,0.0,1.0,10,2,0,10,0,0,0,0,10,0,0,9,0,0,1,0,0,0,0,0,0,0,2],[41,41,1.0,0.25,0.17128,0.10714,0.28571,0.28571,0.0,0.71429,8,0,0,8,0,1,0,0,16,0,0,6,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"e28300e9581c011d","q":"Let  $ABCD$  be a rectangle. Construct equilateral triangles  $BCX$  and  $DCY$ , in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line  $AX$  intersects line  $CD$  on  $P$ , and line  $AY$  intersects line  $BC$  on  $Q$ . Prove that triangle  $APQ$  is equilateral.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.83929,"x":0.91964,"p":[[0,35,0.0,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[4,35,0.1143,0.84375,0.17261,0.71429,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,0,0,16],[8,35,0.2286,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[12,35,0.3429,0.83929,0.17035,0.71429,0.78571,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,1,0,15],[16,35,0.4571,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[20,35,0.5714,0.83929,0.17035,0.71429,0.78571,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,1,0,15],[24,35,0.6857,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[28,35,0.8,0.84374,0.17262,0.71429,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,0,0,16],[32,35,0.9143,0.90179,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,21],[35,35,1.0,0.83929,0.19805,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,13,0,0,0,0,17]]},{"b":2,"e":1.0,"k":"flat","v":0.80804,"x":0.91071,"p":[[0,37,0.0,0.86161,0.17307,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,18],[4,37,0.1081,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[8,37,0.2162,0.87054,0.17627,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,1,0,19],[12,37,0.3243,0.88839,0.13709,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,19],[16,37,0.4324,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20],[20,37,0.5405,0.86594,0.14272,0.71429,1.0,1.0,0.71,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[24,37,0.6486,0.88379,0.14049,0.71429,1.0,1.0,0.71,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19],[28,37,0.7568,0.84821,0.14258,0.71429,0.71429,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,15],[32,37,0.8649,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[36,37,0.973,0.80804,0.16602,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,19,0,0,0,0,12],[37,37,1.0,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19]]}]},{"i":"f5033cbd73a41889","q":"Let  $ABCD$  be a square inscribed in a circle  $\\omega$   and let  $P$  be a point on the shorter arc  $AB$  of  $\\omega$ . Let  $CP\\cap BD = R$  and  $DP \\cap  AC = S.$  \nShow that triangles  $ARB$  and  $DSR$  have equal areas.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.54018,"x":0.70536,"p":[[0,34,0.0,0.625,0.25443,0.42857,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,10,0,0,1,0,0,5,0,0,9,0,3],[4,34,0.1176,0.60714,0.22588,0.42857,0.5,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,14,0,0,3,0,0,4,0,0,5,0,4],[8,34,0.2353,0.5759,0.26841,0.42857,0.4286,0.75,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,13,0,0,0,0,0,6,0,0,3,0,5],[12,34,0.3529,0.62054,0.26633,0.42857,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,0,0,0,12,0,0,1,0,0,6,0,0,4,0,6],[16,34,0.4706,0.57143,0.25505,0.42857,0.4286,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,2,0,0,13,0,0,0,0,0,4,0,0,8,0,2],[20,34,0.5882,0.70536,0.19865,0.42857,0.78571,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,6,0,0,14,0,2],[24,34,0.7059,0.62052,0.1976,0.42857,0.57121,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,5,0,0,9,0,1],[28,34,0.8235,0.60268,0.23887,0.42857,0.42857,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,15,0,0,0,0,0,3,0,0,11,0,1],[32,34,0.9412,0.54018,0.21646,0.42857,0.42857,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,16,0,0,1,0,0,2,0,0,7,0,1],[34,34,1.0,0.54018,0.23347,0.42857,0.42859,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,16,0,0,0,0,0,5,0,0,6,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.62499,"x":0.70981,"p":[[0,15,0.0,0.69196,0.22619,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,8,0,0,2,0,0,6,0,0,9,0,5],[4,15,0.2667,0.68303,0.23347,0.42857,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,5,0,0,11,0,4],[8,15,0.5333,0.62499,0.26184,0.57132,0.71429,0.75,0.0,1.0,3,1,0,3,0,0,0,0,3,0,0,1,0,0,3,0,0,14,0,0,7,0,1],[12,15,0.8,0.70533,0.20498,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,15,0,0,9,0,2],[15,15,1.0,0.70981,0.23003,0.71429,0.85707,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,0,0,0,1,0,0,7,0,0,17,0,1]]}]},{"i":"075dd8891807c5f7","q":"Let  $AC$  and  $BD$  be two chords of a circle with center  $O$  such that they intersect at right angles inside the circle at the point  $M$ . Suppose  $K$  and  $L$  are midpoints of the chords  $AB$  and  $CD$  respectively. Prove that  $OKML$  is a parallelogram.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04009,"x":0.20973,"p":[[0,34,0.0,0.14732,0.18029,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,34,0.1176,0.13831,0.18723,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[8,34,0.2353,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.20973,0.15969,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,18,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[16,34,0.4706,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.02679,"x":0.09376,"p":[[0,25,0.0,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.09376,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.07144,0.07144,0.0,0.07143,0.14286,0.0,0.143,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.04902,0.06774,0.0,0.0,0.14286,0.0,0.143,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f40b3fb12a538852","q":"Let  $A_1,A_2,...$  be a sequence of sets such that for any positive integer  $i$ , there are only finitely many values of  $j$  such that  $A_j\\subseteq A_i$ . Prove that there is a sequence of positive integers  $a_1,a_2,...$  such that for any pair  $(i,j)$  to have  $a_i\\mid a_j\\iff A_i\\subseteq A_j$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.85714,"x":0.88392,"p":[[0,29,0.0,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[4,29,0.1379,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[8,29,0.2759,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[12,29,0.4138,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[16,29,0.5517,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[20,29,0.6897,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[24,29,0.8276,0.86606,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[28,29,0.9655,0.85714,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[29,29,1.0,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5]]},{"b":6,"e":1.0,"k":"flat","v":0.85714,"x":0.87946,"p":[[0,41,0.0,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[4,41,0.0976,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[8,41,0.1951,0.85714,0.0,0.85714,0.85714,0.85714,0.85714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[12,41,0.2927,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[16,41,0.3902,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[20,41,0.4878,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[24,41,0.5854,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[28,41,0.6829,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[32,41,0.7805,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[36,41,0.878,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[40,41,0.9756,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[41,41,1.0,0.86607,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2]]}]},{"i":"12bf40fe1be964ff","q":"Let  $N\\ge2$  be a positive integer. Given a sequence of natural numbers  $a_1,a_2,a_3,\\dots,a_{N+1}$  such that for every integer  $1\\le i\\le j\\le N+1$ ,  $$ a_ia_{i+1}a_{i+2}\\dots a_j \\not\\equiv1\\mod{N} $$  Prove that there exist a positive integer  $k\\le N+1$  such that  $\\gcd(a_k, N) \\neq 1$","t":[{"b":2,"e":1.0,"k":"flat","v":0.95535,"x":1.0,"p":[[0,63,0.0,0.95535,0.11539,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[4,63,0.0635,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,63,0.127,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,63,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,63,0.254,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,63,0.3175,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,63,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,63,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,63,0.5079,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,63,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,63,0.6349,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,63,0.6984,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,63,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,63,0.8254,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,63,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,63,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[63,63,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,55,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,55,0.0727,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,55,0.1455,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,55,0.2182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,55,0.2909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,55,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,55,0.4364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,55,0.5818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,55,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,55,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,55,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,55,0.9455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3c311df715bc52e4","q":"Let  $M$  be the midpoint of side  $AC$  of the triangle  $ABC$ . Let  $P$  be a point on the side  $BC$ . If  $O$  is the point of intersection of  $AP$  and  $BM$  and  $BO = BP$ , determine the ratio  $\\frac{OM}{PC}$  .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.26339,"x":0.27678,"p":[[0,27,0.0,0.26786,0.05923,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.26785,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.26785,0.05922,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.26339,0.0724,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.25446,"x":0.3125,"p":[[0,50,0.0,0.3125,0.1448,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,50,0.08,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.30357,0.12753,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,50,0.4,0.26786,0.05923,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.28125,0.145,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,50,0.56,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.26339,0.06298,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.27679,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b47562531e0ea4f1","q":"Let  $M=\\{1,2,...,n\\}$ . Prove that the number of pairs  $(A,a)$ , where  $A\\subset M$  and  $a$  is a permutation of  $M$ , for which  $a(A)\\cap A=\\emptyset $ , is equal to  $n!.F_{n+1}$ , where  $F_{n+1}$  is the  $n+1$  member of the Fibonacci sequence.","t":[{"b":1,"e":1.0,"k":"rising","v":0.66509,"x":1.0,"p":[[0,30,0.0,0.66509,0.46385,0.0,1.0,1.0,0.0,1.0,9,21,0,9,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[4,30,0.1333,0.87946,0.31966,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[8,30,0.2667,0.79018,0.39767,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[12,30,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,30,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"rising","v":0.21429,"x":1.0,"p":[[0,37,0.0,0.70536,0.43877,0.14286,1.0,1.0,0.0,1.0,6,22,0,6,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22],[4,37,0.1081,0.47321,0.46762,0.0,0.14286,1.0,0.0,1.0,10,14,0,10,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[8,37,0.2162,0.21429,0.38299,0.0,0.0,0.14287,0.0,1.0,21,6,1,21,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[12,37,0.3243,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"cd52e9ca2fdaf1a6","q":"Let  $O = (0,0), A = (0,a), and B = (0,b)$ , where  $0<b<a$  are reals. Let  $\\Gamma$  be a circle with diameter  $\\overline{AB}$  and let  $P$  be any other point on  $\\Gamma$ . Line  $PA$  meets the x-axis again at  $Q$ . Prove that angle  $\\angle BQP = \\angle BOP$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,22,0.0,0.96874,0.08556,1.0,1.0,1.0,0.714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[4,22,0.1818,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[8,22,0.3636,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,22,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,22,0.7273,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,16,0.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[4,16,0.25,0.92411,0.1636,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,1,0,25],[8,16,0.5,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,16,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,16,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"2042ade9d5d53082","q":"Let  $P$  and  $Q$  be distinct points in the plane of a triangle  $ABC$  such that  $AP : AQ = BP : BQ = CP : CQ$ . Prove that the line  $PQ$  passes through the circumcenter of the triangle.","t":[{"b":1,"e":0.0,"k":"falling","v":0.02679,"x":0.65179,"p":[[0,37,0.0,0.65179,0.44598,0.14286,1.0,1.0,0.0,1.0,7,19,0,7,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,19],[4,37,0.1081,0.25,0.36246,0.0,0.0,0.42858,0.0,1.0,19,5,0,19,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,5],[8,37,0.2162,0.11607,0.27534,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[12,37,0.3243,0.11161,0.19799,0.0,0.0,0.2857,0.0,0.71429,23,0,0,23,0,0,0,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[16,37,0.4324,0.08928,0.18471,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[20,37,0.5405,0.09822,0.19704,0.0,0.0,0.03572,0.0,0.71429,24,0,0,24,0,1,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[24,37,0.6486,0.0625,0.13334,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,37,0.8649,0.08928,0.18123,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[36,37,0.973,0.0625,0.13333,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[37,37,1.0,0.125,0.1915,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,1,0,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.57143,"x":1.0,"p":[[0,32,0.0,0.78124,0.36941,0.57143,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,22],[4,32,0.125,0.72768,0.39827,0.39287,1.0,1.0,0.0,1.0,5,19,0,5,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,19],[8,32,0.25,0.57143,0.45316,0.10714,0.85714,1.0,0.0,1.0,8,16,0,8,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,16],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,0.98438,0.04276,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,28],[32,32,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"228657e4c4cc8591","q":"Let  $P(x)$  be a polynomial with at most  $n{}$  real coefficeints. Prove that if  $P(x)$  has integer values for  $n+1$  consecutive values of the argument, then  $P(m)\\in\\mathbb{Z},\\forall m\\in\\mathbb{Z}.$","t":[{"b":4,"e":1.0,"k":"flat","v":0.91964,"x":0.96429,"p":[[0,28,0.0,0.92411,0.19228,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,24],[4,28,0.1429,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[8,28,0.2857,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[12,28,0.4286,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[16,28,0.5714,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[20,28,0.7143,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[24,28,0.8571,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,28,1.0,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17]]},{"b":5,"e":1.0,"k":"flat","v":0.9375,"x":0.97321,"p":[[0,16,0.0,0.94643,0.13243,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[4,16,0.25,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,16,0.5,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,16,0.75,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,16,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]}]},{"i":"47c8dc370de01fa8","q":"Let  $P(X),Q(X)$  be monic irreducible polynomials with rational coefficients. suppose that  $P(X)$  and  $Q(X)$  have roots  $\\alpha$  and  $\\beta$  respectively, such that  $\\alpha + \\beta $  is rational. Prove that  $P(X)^2-Q(X)^2$  has a rational root.\n\n*Bogdan Enescu*","t":[{"b":0,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,31,0.0,0.95536,0.14914,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[4,31,0.129,0.92857,0.14725,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,25],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,35,0.0,0.96427,0.10719,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,0.92411,0.14719,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,24],[12,35,0.3429,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[16,35,0.4571,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,35,0.5714,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,35,0.6857,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,35,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,35,0.9143,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[35,35,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"4ca38860d72dc052","q":"Let  $P(x)$  be a nonzero polynomial such that  $(x-1)P(x+1)=(x+2)P(x)$  for every real  $x$ , and  $\\left(P(2)\\right)^2 = P(3)$ . Then  $P(\\tfrac72)=\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,32,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,32,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,32,0.875,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[32,32,1.0,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23]]},{"b":5,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,48,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,48,0.0833,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,48,0.1667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,48,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,48,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,48,0.4167,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,48,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,48,0.5833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,48,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,48,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,48,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,48,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,48,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"cdd781f6ebcb3832","q":"Let  $PAB$  and  $PBC$  be two similar right-angled triangles (in the same plane) with  $\\angle PAB = \\angle PBC = 90^\\circ$  such that  $A$  and  $C$  lie on opposite sides of the line  $PB$ . If  $PC = AC$ , calculate the ratio  $\\frac{PA}{AB}$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.62051,"x":0.82589,"p":[[0,22,0.0,0.76785,0.24936,0.57143,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,5,0,0,3,0,14],[4,22,0.1818,0.71875,0.2435,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,3,0,0,3,0,11],[8,22,0.3636,0.76339,0.22477,0.57143,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,3,0,0,1,0,14],[12,22,0.5455,0.82589,0.22794,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,3,0,0,5,0,17],[16,22,0.7273,0.70534,0.18878,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,8,0,0,4,0,6],[20,22,0.9091,0.64284,0.20825,0.5354,0.57143,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,5,0,0,9,0,0,7,0,0,4,0,4],[22,22,1.0,0.62051,0.18074,0.42857,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,7,0,0,10,0,0,4,0,0,9,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.69195,"x":0.99554,"p":[[0,14,0.0,0.69195,0.23987,0.4286,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,3,0,0,2,0,10],[4,14,0.2857,0.79463,0.24469,0.57132,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,2,0,0,4,0,16],[8,14,0.5714,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[12,14,0.8571,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[14,14,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"03307e43b6105251","q":"Let  $a>0$  and  $x_1,x_2,x_3$  be real numbers with  $x_1+x_2+x_3=0$ . Prove that  $$ \\log_2\\left(1+a^{x_1}\\right)+\\log_2\\left(1+a^{x_2}\\right)+\\log_2\\left(1+a^{x_3}\\right)\\ge3. $$","t":[{"b":1,"e":1.0,"k":"flat","v":0.83479,"x":0.98214,"p":[[0,23,0.0,0.83479,0.18939,0.57143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,2,0,17],[4,23,0.1739,0.95982,0.08918,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[8,23,0.3478,0.9375,0.11811,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[12,23,0.5217,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,23,0.6957,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,23,0.8696,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[23,23,1.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23]]},{"b":2,"e":1.0,"k":"flat","v":0.86161,"x":0.99107,"p":[[0,14,0.0,0.86607,0.19541,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,2,0,20],[4,14,0.2857,0.86161,0.18029,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,2,0,19],[8,14,0.5714,0.90625,0.16213,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,22],[12,14,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[14,14,1.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"e0fe83baf702e102","q":"Let  $\\alpha$  be a complex number such that both  $\\alpha$  and  $\\alpha+1$  have modulus  $1$ . If for a positive integer  $n$ ,  $1+\\alpha$  is an  $n$ -th root of unity, then show that  $\\alpha$  is also an  $n$ -th root of unity and  $n$  is a multiple of  $6$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,27,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,27,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,27,0.5926,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,27,0.7407,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.85714,"k":"flat","v":0.95535,"x":0.99107,"p":[[0,34,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,34,0.1176,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,34,0.2353,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,34,0.3529,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,34,0.4706,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,34,0.5882,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,34,0.7059,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,34,0.8235,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,34,0.9412,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[34,34,1.0,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]}]},{"i":"d549e74835f8f73c","q":"Let  $a_1, a_2, \\ldots, a_n$  be real numbers.Prove that you can select  $\\varepsilon _1, \\varepsilon _2, \\ldots, \\varepsilon _n\\in\\{-1,1\\}$  such that  $$ \\left( \\sum_{i=1}^{n}a_{i}\\right)^2 +\\left( \\sum_{i=1}^{n}\\varepsilon _ia_{i}\\right)^2 \\leq(n+1)\\left( \\sum_{i=1}^{n}a^2_{i}\\right). $$","t":[{"b":0,"e":0.28571,"k":"flat","v":0.20089,"x":0.30803,"p":[[0,26,0.0,0.20089,0.08645,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,26,0.4615,0.24544,0.06437,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.24089,0.06648,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.28125,0.14054,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,26,0.9231,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.24554,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.20536,"x":0.29911,"p":[[0,25,0.0,0.22768,0.1551,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,25,0.16,0.29911,0.13054,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,25,0.32,0.25893,0.05576,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.22321,0.07087,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,14,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.23205,0.06928,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.25446,0.1504,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,12,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,25,0.96,0.27669,0.19871,0.14286,0.28571,0.28571,0.14,1.0,0,2,0,0,0,12,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[25,25,1.0,0.20536,0.07087,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b00ff236fbe6f389","q":"Let  $\\Delta ABC$  be a triangle with angle  $\\angle CAB=60^{\\circ}$ , let  $D$  be the intersection point of the angle bisector at  $A$  and the side  $BC$ , and let  $r_B,r_C,r$  be the respective radii of the incircles of  $ABD$ ,  $ADC$ ,  $ABC$ . Let  $b$  and  $c$  be the lengths of sides  $AC$  and  $AB$  of the triangle. Prove that\n\\[ \\frac{1}{r_B} +\\frac{1}{r_C} ~=~ 2\\cdot\\left( \\frac1r +\\frac1b +\\frac1c\\right)\\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,23,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,23,0.1739,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,23,0.3478,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,23,0.5217,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[16,23,0.6957,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,27,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,27,0.1481,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,27,0.2963,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[12,27,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"723683772fc73adf","q":"Let  $a_1, a_2, \\ldots, a_n$  be  $n$  positive integers, and let  $b_1, b_2, \\ldots, b_m$  be  $m$  positive integers such that  $a_1 a_2 \\cdots a_n = b_1 b_2 \\cdots b_m$ . Prove that a rectangular table with  $n$  rows and  $m$  columns can be filled with positive integer entries in such a way that\n\n* the product of the entries in the  $i$ -th row is  $a_i$  (for each  $i \\in \\left\\{1,2,\\ldots,n\\right\\}$ );\n\n* the product of the entries in the  $j$ -th row is  $b_j$  (for each  $i \\in \\left\\{1,2,\\ldots,m\\right\\}$ ).","t":[{"b":1,"e":1.0,"k":"flat","v":0.94183,"x":1.0,"p":[[0,34,0.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[4,34,0.1176,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[8,34,0.2353,0.94183,0.11242,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,34,0.5882,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,34,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[34,34,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.82142,"x":1.0,"p":[[0,23,0.0,0.89285,0.11294,0.857,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,15],[4,23,0.1739,0.93737,0.10092,0.85714,1.0,1.0,0.71,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[8,23,0.3478,0.88839,0.13709,0.82132,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,17],[12,23,0.5217,0.82142,0.14286,0.71429,0.85714,1.0,0.5714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,10,0,9],[16,23,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ca823e56c63ad67d","q":"Let  $a_1, a_2, a_3, \\ldots$  be an infinite sequence of positive integers such that  $a_1=4$ ,  $a_2=12$ , and for all positive integers  $n$ , \\[a_{n+2}=\\gcd\\left(a_{n+1}^2-4,a_n^2+3a_n \\right).\\] Find, with proof, a formula for  $a_n$  in terms of  $n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.81696,"x":0.98214,"p":[[0,52,0.0,0.83482,0.25281,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,4,0,0,0,0,21],[4,52,0.0769,0.85714,0.20825,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,0,0,0,20],[8,52,0.1538,0.81696,0.20277,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,7,0,0,1,0,16],[12,52,0.2308,0.85268,0.16554,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,1,0,17],[16,52,0.3077,0.90179,0.19377,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,24],[20,52,0.3846,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[24,52,0.4615,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[28,52,0.5385,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,52,0.6154,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[36,52,0.6923,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[40,52,0.7692,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,52,0.8462,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[48,52,0.9231,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[52,52,1.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26]]},{"b":6,"e":1.0,"k":"rising","v":0.77677,"x":0.99107,"p":[[0,61,0.0,0.78571,0.28572,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,4,0,0,0,0,19],[4,61,0.0656,0.82588,0.20435,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,11,0,0,1,0,16],[8,61,0.1311,0.77677,0.25986,0.67857,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,8,0,0,0,0,16],[12,61,0.1967,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,61,0.2623,0.91964,0.18536,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,25],[20,61,0.3279,0.94643,0.17768,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[24,61,0.3934,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[28,61,0.459,0.92411,0.15146,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,25],[32,61,0.5246,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[36,61,0.5902,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[40,61,0.6557,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[44,61,0.7213,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[48,61,0.7869,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,61,0.8525,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[56,61,0.918,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[60,61,0.9836,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[61,61,1.0,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28]]}]},{"i":"b31cbdef59dce9f7","q":"Let  $a,b,c$  be three integers for which the sum\n\\[ \\frac{ab}{c}+ \\frac{ac}{b}+ \\frac{bc}{a}\\]\nis integer.\nProve that each of the three numbers \n\\[ \\frac{ab}{c}, \\quad \\frac{ac}{b},\\quad \\frac{bc}{a}\\] \nis integer.\n\n(Proposed by Gerhard J. Woeginger)","t":[{"b":2,"e":1.0,"k":"rising","v":0.47768,"x":1.0,"p":[[0,15,0.0,0.47768,0.43538,0.0,0.42857,1.0,0.0,1.0,12,11,0,12,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,11],[4,15,0.2667,0.65179,0.45167,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,18],[8,15,0.5333,0.88384,0.26376,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,25],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.57143,"k":"rising","v":0.56696,"x":0.81696,"p":[[0,21,0.0,0.56696,0.45384,0.0,0.71429,1.0,0.0,1.0,11,15,0,11,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,15],[4,21,0.1905,0.73661,0.37134,0.49998,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,2,0,19],[8,21,0.381,0.81696,0.36111,0.85714,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,23],[12,21,0.5714,0.62946,0.42085,0.10714,0.85714,1.0,0.0,1.0,8,14,0,8,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,14],[16,21,0.7619,0.73659,0.2262,0.57143,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,19,0,3],[20,21,0.9524,0.77233,0.17442,0.67857,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,0,21,0,2],[21,21,1.0,0.74107,0.16917,0.57143,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,2,0,0,20,0,0]]}]},{"i":"66a2581e9caf8510","q":"Let  $a,b,c,d$  be positive real numbers such that  $ 2(a+b+c+d)\\ge abcd $ . Prove that \\[ a^2+b^2+c^2+d^2 \\ge abcd .\\]","t":[{"b":0,"e":1.0,"k":"rising","v":0.55357,"x":1.0,"p":[[0,21,0.0,0.55357,0.36553,0.14286,0.57143,1.0,0.0,1.0,1,10,0,1,0,9,0,0,3,0,0,2,0,0,4,0,0,1,0,0,2,0,10],[4,21,0.1905,0.55804,0.38359,0.14286,0.57143,1.0,0.0,1.0,3,11,0,3,0,7,0,0,3,0,0,2,0,0,2,0,0,3,0,0,1,0,11],[8,21,0.381,0.80804,0.29797,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,20],[12,21,0.5714,0.83034,0.21854,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,5,0,17],[16,21,0.7619,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,21,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[21,21,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":1.0,"k":"rising","v":0.47768,"x":1.0,"p":[[0,13,0.0,0.47768,0.37048,0.14286,0.35714,1.0,0.0,1.0,3,9,0,3,0,8,0,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,9],[4,13,0.3077,0.48214,0.40838,0.14286,0.28571,1.0,0.0,1.0,5,11,0,5,0,9,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,11],[8,13,0.6154,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[12,13,0.9231,0.95536,0.15746,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"27cce33f41d13b20","q":"Let  $a,b,c,n$  be positive integers such that the following conditions hold\n(i) numbers  $a,b,c,a+b+c$  are pairwise coprime,\n(ii) number  $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$  is a perfect  $n$ -th power.\nProve, that the product  $abc$  can be expressed as a difference of two perfect  $n$ -th powers.","t":[{"b":4,"e":0.571,"k":"flat","v":0.39283,"x":0.52677,"p":[[0,32,0.0,0.51782,0.20437,0.28571,0.57121,0.60714,0.28571,1.0,0,2,0,0,0,0,0,0,11,0,0,2,0,0,11,0,0,6,0,0,0,0,2],[4,32,0.125,0.49553,0.26241,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,2,0,0,14,0,0,0,0,0,8,0,0,3,0,0,1,0,4],[8,32,0.25,0.39283,0.16749,0.28571,0.28571,0.571,0.14286,1.0,0,1,0,0,0,1,0,0,18,0,0,4,0,0,8,0,0,0,0,0,0,0,1],[12,32,0.375,0.46428,0.12877,0.28571,0.49999,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,7,0,0,15,0,0,1,0,0,0,0,0],[16,32,0.5,0.4955,0.11834,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,3,0,0,22,0,0,0,0,0,0,0,0],[20,32,0.625,0.45979,0.21047,0.28571,0.49979,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,11,0,0,3,0,0,13,0,0,0,0,0,2,0,1],[24,32,0.75,0.48211,0.17033,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,10,0,0,1,0,0,18,0,0,0,0,0,2,0,0],[28,32,0.875,0.52677,0.08328,0.53539,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,6,0,0,24,0,0,0,0,0,0,0,0],[32,32,1.0,0.48654,0.16308,0.42857,0.571,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,6,0,0,18,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.42857,"k":"flat","v":0.5312,"x":0.57584,"p":[[0,29,0.0,0.57584,0.2382,0.28571,0.57143,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,9,0,0,2,0,0,10,0,0,6,0,0,0,0,5],[4,29,0.1379,0.54904,0.08071,0.571,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,24,0,0,0,0,0,1,0,0],[8,29,0.2759,0.56243,0.15946,0.42859,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,9,0,0,19,0,0,0,0,0,0,0,3],[12,29,0.4138,0.56243,0.10061,0.571,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,24,0,0,1,0,0,0,0,1],[16,29,0.5517,0.54018,0.05906,0.57142,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0],[20,29,0.6897,0.56247,0.10677,0.57142,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,25,0,0,1,0,0,0,0,1],[24,29,0.8276,0.56692,0.10403,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,26,0,0,1,0,0,0,0,1],[28,29,0.9655,0.57584,0.10403,0.57132,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,26,0,0,0,0,0,1,0,1],[29,29,1.0,0.5312,0.10851,0.53539,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,6,0,0,22,0,0,2,0,0,0,0,0]]}]},{"i":"4fddc35503e8bcea","q":"Let  $a_1,a_2 ,\\ldots, a_n$  be an arithmetic progression of integers such that  $i|a_i$  for  $i=1, 2,\\ldots ,n-1$  and  $n\\nmid a_n$ . Prove that  $n$  is a prime power.","t":[{"b":3,"e":1.0,"k":"flat","v":0.76339,"x":0.99554,"p":[[0,32,0.0,0.94196,0.18509,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[4,32,0.125,0.76339,0.32461,0.60714,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,6,0,0,0,0,0,0,0,0,4,0,0,2,0,18],[8,32,0.25,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,32,0.375,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,32,0.5,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[20,32,0.625,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,32,0.75,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,32,0.875,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,32,1.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,30,0.0,0.94196,0.16698,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,28],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,30,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,30,0.5333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e2f3496b4c62bec5","q":"Let  $f: \\mathbb R \\to \\mathbb R$  be a function which is differentiable at  $0$ . Define another function  $g: \\mathbb R \\to \\mathbb R$  as follows:   $$ g(x) = \\begin{cases}\nf(x)\\sin\\left(\\frac 1x\\right) ~ &\\text{if} ~ x \\neq 0 \n\n0 &\\text{if} ~ x = 0.\n \\end{cases} $$  Suppose that  $g$  is also differentiable at  $0$ . Prove that \\[g'(0) = f'(0) = f(0) = g(0) = 0.\\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.87946,"x":0.95089,"p":[[0,29,0.0,0.95089,0.11071,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,24],[4,29,0.1379,0.87946,0.12931,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,14,0,13],[8,29,0.2759,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[12,29,0.4138,0.91964,0.13333,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,18],[16,29,0.5517,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[20,29,0.6897,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[24,29,0.8276,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[28,29,0.9655,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[29,29,1.0,0.90624,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12]]},{"b":7,"e":0.857,"k":"flat","v":0.90178,"x":0.97321,"p":[[0,24,0.0,0.92857,0.18898,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,24],[4,24,0.1667,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,24,0.3333,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[12,24,0.5,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[16,24,0.6667,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[20,24,0.8333,0.90624,0.11072,0.85714,0.85714,1.0,0.5714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,15,0,15],[24,24,1.0,0.90178,0.0974,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,13]]}]},{"i":"cb53d50fc3d766b0","q":"Let  $f:[0,1]\\to[0,1]$  be a differentiable function such that  $|f'(x)|\\ne1$  for all  $x\\in[0,1]$ . Prove that there exist unique  $\\alpha,\\beta\\in[0,1]$  such that  $f(\\alpha)=\\alpha$  and  $f(\\beta)=1-\\beta$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":0.99107,"p":[[0,23,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,23,0.1739,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,23,0.3478,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,23,0.5217,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,23,0.6957,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,23,0.8696,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[23,23,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,33,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,33,0.1212,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,33,0.2424,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,33,0.3636,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,33,0.4848,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,33,0.8485,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]}]},{"i":"d8fea7851afb1cd9","q":"Let  $f(x) = (x^2+3x+2)^{\\cos(\\pi x)}$ . Find the sum of all positive integers  $n$  for which \\[\\left| \\sum_{k=1}^n \\log_{10} f(k) \\right| = 1.\\]","t":[{"b":1,"e":0.85714,"k":"flat","v":0.8616,"x":0.91517,"p":[[0,48,0.0,0.88839,0.08553,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,9],[4,48,0.0833,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,48,0.1667,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[12,48,0.25,0.8616,0.06667,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,3],[16,48,0.3333,0.87499,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[20,48,0.4167,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[24,48,0.5,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[28,48,0.5833,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[32,48,0.6667,0.86607,0.07087,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,4],[36,48,0.75,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[40,48,0.8333,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[44,48,0.9167,0.87053,0.08268,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,6],[48,48,1.0,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6]]},{"b":6,"e":0.85714,"k":"flat","v":0.83036,"x":0.91964,"p":[[0,44,0.0,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[4,44,0.0909,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[8,44,0.1818,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[12,44,0.2727,0.83036,0.16917,0.85714,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,22,0,6],[16,44,0.3636,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[20,44,0.4545,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[24,44,0.5455,0.87053,0.07457,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,5],[28,44,0.6364,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[32,44,0.7273,0.85268,0.05629,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,1],[36,44,0.8182,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[40,44,0.9091,0.86161,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,2],[44,44,1.0,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2]]}]},{"i":"8477b79416144814","q":"Let  $c$  be a given real number. Find all polynomials  $P$  with real coefficients such that:  $(x + 1)P(x - 1) - (x - 1)P(x) = c$  for all  $x \\in R$","t":[{"b":2,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,46,0.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,46,0.087,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,46,0.1739,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,46,0.2609,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,46,0.3478,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[20,46,0.4348,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,10,0.0,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[4,10,0.4,0.9375,0.1234,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25],[8,10,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[10,10,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"840ce9c123b4de29","q":"Let  $f:Z \\to N -\\{0\\}$  such that: $f(x + y)f(x-y) = (f(x)f(y))^2$  and  $f(1)\\ne 1$ .\n\nProvethat  $\\log_{f(1)}f(z)$  is a perfect square for every integer  $z$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,33,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,33,0.1212,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,33,0.2424,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,33,0.3636,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,33,0.4848,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,33,0.6061,0.97991,0.05985,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,28],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.84817,"x":0.99554,"p":[[0,45,0.0,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[4,45,0.0889,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[8,45,0.1778,0.94195,0.07018,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[12,45,0.2667,0.84817,0.12345,0.857,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,17,0,8],[16,45,0.3556,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[20,45,0.4444,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[24,45,0.5333,0.9375,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[28,45,0.6222,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[32,45,0.7111,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[36,45,0.8,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[40,45,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,45,0.9778,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[45,45,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"7e8ebd2c4073533f","q":"Let  $f(n)=\\displaystyle\\sum_{k=2}^\\infty \\dfrac{1}{k^n\\cdot k!}.$   Calculate  $\\displaystyle\\sum_{n=2}^\\infty f(n)$ .","t":[{"b":1,"e":0.42857,"k":"falling","v":0.42857,"x":0.83036,"p":[[0,15,0.0,0.72322,0.27879,0.42857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,2,0,15],[4,15,0.2667,0.83036,0.23808,0.75,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,6,0,18],[8,15,0.5333,0.74107,0.26351,0.42857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,6,0,13],[12,15,0.8,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.80357,"x":0.96875,"p":[[0,12,0.0,0.80357,0.25692,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,18],[4,12,0.3333,0.8125,0.24598,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,6,0,17],[8,12,0.6667,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[12,12,1.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]}]},{"i":"c965c5bad7e45d0a","q":"Let  $f:(0,\\infty)\\to\\mathbb R$  be a differentiable function. Assume that  $$ \\lim_{x\\to\\infty}\\left(f(x)+\\frac{f'(x)}x\\right)=0. $$  Prove that  $$ \\lim_{x\\to\\infty}f(x)=0. $$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.70536,"x":0.83036,"p":[[0,37,0.0,0.83036,0.20025,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,11,0,0,0,0,17],[4,37,0.1081,0.79463,0.22852,0.71429,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,9,0,0,1,0,15],[8,37,0.2162,0.80356,0.18815,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,14,0,0,1,0,13],[12,37,0.3243,0.80356,0.15467,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,18,0,0,1,0,11],[16,37,0.4324,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0],[20,37,0.5405,0.71429,0.07986,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0,4,0,0],[24,37,0.6486,0.72321,0.08702,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,3,0,1],[28,37,0.7568,0.70536,0.07086,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0,2,0,0],[32,37,0.8649,0.72321,0.04971,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0],[36,37,0.973,0.71875,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0],[37,37,1.0,0.73214,0.07784,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,1,0,2]]},{"b":4,"e":0.71429,"k":"falling","v":0.58927,"x":0.90179,"p":[[0,35,0.0,0.90179,0.14032,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,21],[4,35,0.1143,0.83929,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,13,0,0,2,0,15],[8,35,0.2286,0.75446,0.1197,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,0,2,0,5],[12,35,0.3429,0.66963,0.12596,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,21,0,0,1,0,1],[16,35,0.4571,0.66069,0.20439,0.5354,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,13,0,0,1,0,5],[20,35,0.5714,0.62054,0.19103,0.42857,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,9,0,0,9,0,0,8,0,0,1,0,4],[24,35,0.6857,0.58927,0.17768,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,10,0,0,7,0,0,10,0,0,1,0,2],[28,35,0.8,0.66516,0.25156,0.42857,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,10,0,0,5,0,0,5,0,0,0,0,10],[32,35,0.9143,0.62497,0.21355,0.42857,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,11,0,0,7,0,0,7,0,0,0,0,6],[35,35,1.0,0.59374,0.2055,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,9,0,0,4,0,0,11,0,0,1,0,3]]}]},{"i":"db4fdc2d7dddcf01","q":"Let  $d$  be the tangent at  $B$  to the circumcircle of the acute scalene  triangle  $ABC$ . Let  $K$  be the orthogonal projection of the orthocenter,  $H$ , of triangle  $ABC$  to the line  $d$  and  $L$  the midpoint of the side  $AC$ . Prove that the triangle  $BKL$  is isosceles.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.06696,"x":0.11152,"p":[[0,21,0.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.10714,0.1515,0.0,0.14286,0.14286,0.0,0.857,13,0,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,21,0.5714,0.09349,0.06767,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.09357,"x":0.11599,"p":[[0,24,0.0,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.11599,0.05572,0.14286,0.14286,0.14286,0.0,0.143,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.09357,0.06773,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.1025,0.06412,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.09358,0.06773,0.0,0.14286,0.14286,0.0,0.143,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b1230b332867d2b3","q":"Let  $f(x),g(x)$  be two polynomials with integer coefficients. It is known that for infinitely many prime  $p$ , there exist integer  $m_p$  such that  $$ f(a) \\equiv g(a+m_p) \\pmod p $$  holds for all  $a \\in \\mathbb{Z}.$  Prove that there exists a rational number  $r$  such that  $$ f(x)=g(x+r). $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.96433,"x":1.0,"p":[[0,41,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,41,0.0976,0.96433,0.11825,1.0,1.0,1.0,0.43,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[8,41,0.1951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,41,0.2927,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,18,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,18,0.2222,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"90ad1ad80f06163f","q":"Let  $f:[0,1]\\times[0,1]\\to\\mathbb R$  be a continuous function. Find the limit  $$ \\lim_{n\\to\\infty}\\left(\\frac{(2n+1)!}{(n!)^2}\\right)^2\\int^1_0\\int^1_0(xy(1-x)(1-y))^nf(x,y)\\text dx\\text dy. $$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.51339,"x":0.67857,"p":[[0,18,0.0,0.67857,0.15152,0.57143,0.57143,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,6,0,0,7,0,2],[4,18,0.2222,0.62945,0.10631,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,6,0,0,2,0,1],[8,18,0.4444,0.62054,0.14555,0.57143,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,21,0,0,2,0,0,5,0,1],[12,18,0.6667,0.51339,0.1063,0.42857,0.4286,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,17,0,0,12,0,0,2,0,0,1,0,0],[16,18,0.8889,0.55802,0.15303,0.42857,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,8,0,0,17,0,0,3,0,0,0,0,2],[18,18,1.0,0.58033,0.11259,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,23,0,0,4,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"rising","v":0.60266,"x":0.91518,"p":[[0,12,0.0,0.65625,0.1551,0.57143,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,6,0,0,3,0,3],[4,12,0.3333,0.66518,0.15407,0.57143,0.57143,0.75,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,3,0,0,5,0,3],[8,12,0.6667,0.60266,0.11143,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,25,0,0,4,0,0,1,0,1],[12,12,1.0,0.91518,0.13296,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,19]]}]},{"i":"73b2c02753808144","q":"Let  $f(x)$  be a polynomial of degree  $3$  with real coefficients satisfying  $|f(x)| = 12$  for  $x = 1, 2, 3, 5, 6, 7$ . Find  $|f(0)|$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.58034,"x":0.67857,"p":[[0,47,0.0,0.67857,0.16751,0.57143,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,12,0,0,0,0,5],[4,47,0.0851,0.61158,0.11426,0.5713,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,13,0,0,1,0,0],[8,47,0.1702,0.63839,0.10705,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,14,0,0,0,0,1],[12,47,0.2553,0.58034,0.09407,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,21,0,0,7,0,0,0,0,0],[16,47,0.3404,0.60268,0.08552,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,10,0,0,0,0,0],[20,47,0.4255,0.63393,0.12339,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,16,0,0,0,0,1],[24,47,0.5106,0.60712,0.0945,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,12,0,0,0,0,0],[28,47,0.5957,0.63839,0.09438,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,18,0,0,0,0,0],[32,47,0.6809,0.61607,0.0974,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,14,0,0,0,0,0],[36,47,0.766,0.62495,0.09945,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,13,0,0,1,0,0],[40,47,0.8511,0.62499,0.09943,0.57143,0.64286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,16,0,0,0,0,0],[44,47,0.9362,0.66518,0.09852,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,16,0,0,1,0,1],[47,47,1.0,0.65625,0.07016,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.60714,"x":0.68749,"p":[[0,41,0.0,0.68749,0.20342,0.57143,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,13,0,0,10,0,0,0,0,7],[4,41,0.0976,0.65177,0.07937,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,19,0,0,0,0,0],[8,41,0.1951,0.66072,0.13243,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,21,0,0,0,0,1],[12,41,0.2927,0.60714,0.14286,0.57143,0.64286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,1,0,0,12,0,0,16,0,0,0,0,0],[16,41,0.3902,0.65625,0.07016,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0],[20,41,0.4878,0.66518,0.07668,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,19,0,0,1,0,0],[24,41,0.5854,0.65625,0.07016,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0],[28,41,0.6829,0.63839,0.07129,0.57143,0.57143,0.71429,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,15,0,0,0,0,0],[32,41,0.7805,0.64731,0.0713,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0],[36,41,0.878,0.66071,0.06916,0.57143,0.71429,0.71429,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0],[40,41,0.9756,0.64284,0.07144,0.57143,0.64286,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0],[41,41,1.0,0.66071,0.06916,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0]]}]},{"i":"53aea27accc1031a","q":"Let  $f:\\mathbb{R}\\to\\mathbb{R}^+$  be a continuous and periodic function. Prove that for all  $\\alpha\\in\\mathbb{R}$  the following inequality holds: $\\int_0^T\\frac{f(x)}{f(x+\\alpha)}dx\\ge T$ ,\n\nwhere  $T$  is the period of  $f(x)$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.17857,"x":0.34375,"p":[[0,25,0.0,0.24545,0.3355,0.0,0.14286,0.32143,0.0,1.0,14,4,0,14,0,8,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,4],[4,25,0.16,0.26786,0.29827,0.0,0.14286,0.42857,0.0,1.0,11,3,0,11,0,8,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,3],[8,25,0.32,0.24997,0.27661,0.0,0.14286,0.42857,0.0,1.0,10,2,0,10,0,10,0,0,2,0,0,5,0,0,2,0,0,1,0,0,0,0,2],[12,25,0.48,0.17857,0.24223,0.0,0.07143,0.32143,0.0,1.0,16,1,0,16,0,6,0,0,2,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[16,25,0.64,0.29464,0.27418,0.0,0.35714,0.42857,0.0,1.0,11,2,0,11,0,2,0,0,3,0,0,13,0,0,0,0,0,1,0,0,0,0,2],[20,25,0.8,0.31249,0.28668,0.0,0.35714,0.42857,0.0,1.0,10,2,0,10,0,4,0,0,2,0,0,10,0,0,2,0,0,2,0,0,0,0,2],[24,25,0.96,0.29911,0.26573,0.0,0.28571,0.42857,0.0,1.0,9,2,0,9,0,4,0,0,4,0,0,12,0,0,0,0,0,1,0,0,0,0,2],[25,25,1.0,0.34375,0.2621,0.10714,0.42857,0.42857,0.0,1.0,8,2,0,8,0,2,0,0,2,0,0,16,0,0,1,0,0,1,0,0,0,0,2]]},{"b":4,"e":0.0,"k":"flat","v":0.16071,"x":0.25893,"p":[[0,38,0.0,0.22767,0.26929,0.0,0.14286,0.42857,0.0,1.0,13,2,0,13,0,6,0,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,2],[4,38,0.1053,0.16071,0.25443,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,7,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[8,38,0.2105,0.18749,0.23264,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,14,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[12,38,0.3158,0.1875,0.26108,0.0,0.07143,0.32143,0.0,1.0,16,1,0,16,0,7,0,0,1,0,0,4,0,0,1,0,0,2,0,0,0,0,1],[16,38,0.4211,0.22768,0.29851,0.0,0.14286,0.21432,0.0,1.0,11,3,0,11,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,3],[20,38,0.5263,0.20089,0.26453,0.0,0.14286,0.2857,0.0,1.0,12,2,0,12,0,11,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[24,38,0.6316,0.19196,0.27342,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,7,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,2],[28,38,0.7368,0.21429,0.27894,0.0,0.14286,0.42857,0.0,1.0,14,2,0,14,0,8,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,2],[32,38,0.8421,0.24106,0.29109,0.0,0.14286,0.42857,0.0,1.0,13,2,0,13,0,7,0,0,2,0,0,5,0,0,1,0,0,2,0,0,0,0,2],[36,38,0.9474,0.19643,0.28065,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,7,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,2],[38,38,1.0,0.25893,0.32229,0.0,0.14286,0.42857,0.0,1.0,12,3,0,12,0,10,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,3]]}]},{"i":"cdabaa27362fa1a2","q":"Let  $k$  be a fixed real number. Find all functions  $f: R \\to R$  such that  $f(x)+ (f(y))^2 = kf(x + y^2)$  for all real numbers  $x$  and  $y$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,35,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,35,0.1143,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,35,0.2286,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,35,0.3429,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.90625,"x":0.99554,"p":[[0,76,0.0,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[4,76,0.0526,0.94195,0.10635,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[8,76,0.1053,0.90625,0.16213,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,21],[12,76,0.1579,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,76,0.2105,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,76,0.2632,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,76,0.3158,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,76,0.3684,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[32,76,0.4211,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[36,76,0.4737,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,76,0.5263,0.94196,0.12807,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[44,76,0.5789,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[48,76,0.6316,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[52,76,0.6842,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[56,76,0.7368,0.96428,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[60,76,0.7895,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,76,0.8421,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,76,0.8947,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[72,76,0.9474,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[76,76,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]}]},{"i":"88dc0c05db8e2824","q":"Let  $m, n$  integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides  $n$","t":[{"b":1,"e":1.0,"k":"flat","v":0.97768,"x":0.99554,"p":[[0,9,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,9,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,9,0.8889,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[9,9,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,30,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,30,0.4,0.98661,0.07457,1.0,1.0,1.0,0.5714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,30,0.5333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"62c2187247e92543","q":"Let  $n$  be a five digit number (whose first digit is non-zero) and let  $m$  be the four digit number formed from  $n$  by removing its middle digit. Determine all  $n$  such that  $n/m$  is an integer.","t":[{"b":3,"e":1.0,"k":"flat","v":0.90623,"x":0.95982,"p":[[0,76,0.0,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[4,76,0.0526,0.91071,0.12242,0.82143,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[8,76,0.1053,0.91517,0.12807,0.85711,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[12,76,0.1579,0.94642,0.09945,0.96429,1.0,1.0,0.714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[16,76,0.2105,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[20,76,0.2632,0.94196,0.10631,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[24,76,0.3158,0.91518,0.11769,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,20],[28,76,0.3684,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[32,76,0.4211,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[36,76,0.4737,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[40,76,0.5263,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22],[44,76,0.5789,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[48,76,0.6316,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[52,76,0.6842,0.90623,0.13653,0.857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,19],[56,76,0.7368,0.91963,0.11259,0.857,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[60,76,0.7895,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[64,76,0.8421,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[68,76,0.8947,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[72,76,0.9474,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[76,76,1.0,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22]]},{"b":5,"e":1.0,"k":"flat","v":0.85268,"x":0.9375,"p":[[0,48,0.0,0.85268,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,5,0,14],[4,48,0.0833,0.91517,0.11769,0.85711,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,20],[8,48,0.1667,0.92857,0.10714,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[12,48,0.25,0.88839,0.13236,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,3,0,18],[16,48,0.3333,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22],[20,48,0.4167,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[24,48,0.5,0.91517,0.12299,0.82132,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[28,48,0.5833,0.88392,0.12078,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,15],[32,48,0.6667,0.91071,0.12753,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,21],[36,48,0.75,0.89731,0.12493,0.71429,1.0,1.0,0.714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,18],[40,48,0.8333,0.88838,0.12745,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,17],[44,48,0.9167,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[48,48,1.0,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20]]}]},{"i":"896321d87f0d05fe","q":"Let  $n \\geq 2$  be a positive integer. Call a sequence  $a_1, a_2, \\cdots , a_k$  of integers an  $n$ *-chain* if  $1 = a_2 < a_ 2 < \\cdots < a_k =n$ ,  $a_i$  divides  $a_{i+1}$  for all  $i$ ,  $1 \\leq i \\leq k-1$ . Let  $f(n)$  be the number of  $n$ *-chains* where  $n \\geq 2$ . For example,  $f(4) = 2$  corresponds to the  $4$ -chains  $\\{1,4\\}$  and  $\\{1,2,4\\}$ . \n\nProve that  $f(2^m \\cdot 3)  = 2^{m-1} (m+2)$  for every positive integer  $m$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.61606,"x":0.76337,"p":[[0,5,0.0,0.76337,0.2893,0.57132,1.0,1.0,0.0,1.0,1,17,1,1,0,1,0,0,1,0,0,3,0,0,5,0,0,4,0,0,0,0,17],[4,5,0.8,0.61606,0.2299,0.42859,0.57143,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,1,0,0,10,0,0,9,0,0,2,0,0,4,0,5],[5,5,1.0,0.68304,0.19475,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,7,0,0,5,0,5]]},{"b":4,"e":1.0,"k":"rising","v":0.66964,"x":0.94643,"p":[[0,14,0.0,0.70534,0.31932,0.42857,0.85714,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,7,0,0,2,0,0,2,0,0,0,0,16],[4,14,0.2857,0.66964,0.36323,0.28571,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,4,0,0,5,0,0,0,0,0,1,0,0,1,0,16],[8,14,0.5714,0.91518,0.22829,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,27],[12,14,0.8571,0.92409,0.20822,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,28],[14,14,1.0,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27]]}]},{"i":"3f2cb85082eafc00","q":"Let  $n$  be a fixed positive integer. To any choice of real numbers satisfying \\[0\\le x_{i}\\le 1,\\quad i=1,2,\\ldots, n,\\] there corresponds the sum \\[\\sum_{1\\le i<j\\le n}|x_{i}-x_{j}|.\\] Let  $S(n)$  denote the largest possible value of this sum. Find  $S(n)$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,10,0.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,10,0.4,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[8,10,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[10,10,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,29,0.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[4,29,0.1379,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,29,0.2759,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,29,0.4138,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,29,0.5517,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,29,0.9655,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[29,29,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"68fba21ee66f1a0f","q":"Let  $n \\geq 2$  be a positive integer. On a spaceship, there are  $n$  crewmates. At most one accusation of\nbeing an imposter can occur from one crewmate to another crewmate. Multiple accusations are\nthrown, with the following properties:\n\u2022 Each crewmate made a different number of accusations.\n\u2022 Each crewmate received a different number of accusations.\n\u2022 A crewmate does not accuse themself.\nProve that no two crewmates made accusations at each other.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.875,"x":0.96875,"p":[[0,21,0.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,21,0.1905,0.90629,0.15802,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,8,0,20],[8,21,0.381,0.94642,0.0928,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[12,21,0.5714,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[16,21,0.7619,0.875,0.18123,0.85714,0.85714,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,12],[20,21,0.9524,0.88838,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[21,21,1.0,0.90177,0.06623,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10]]},{"b":7,"e":1.0,"k":"flat","v":0.89732,"x":0.94642,"p":[[0,28,0.0,0.89732,0.15663,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,20],[4,28,0.1429,0.94642,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[8,28,0.2857,0.91963,0.08703,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[12,28,0.4286,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[16,28,0.5714,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[20,28,0.7143,0.92856,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[24,28,0.8571,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[28,28,1.0,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17]]}]},{"i":"a78147f27a2b9e2f","q":"Let  $n$  be an integer with  $n \\ge 2$ . Show that  $n$  does not divide  $2^{n}-1$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.02679,"p":[[0,38,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.03571,"p":[[0,35,0.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"60c74a3d242444a0","q":"Let  $n$  be an integer.  If the tens digit of  $n^2$  is 7, what is the units digit of  $n^2$ ?","t":[{"b":4,"e":0.57143,"k":"falling","v":0.62945,"x":0.97768,"p":[[0,14,0.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[4,14,0.2857,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,14,0.5714,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[12,14,0.8571,0.70088,0.18337,0.57143,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,5,0,0,4,0,6],[14,14,1.0,0.62945,0.16698,0.57143,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,22,0,0,0,0,0,4,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,44,0.0,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[4,44,0.0909,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,44,0.1818,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,44,0.2727,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,44,0.3636,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,44,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,44,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,44,0.6364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6e879a5b346f8093","q":"Let  $n$  be the product of the first  $10$  primes, and let   $$ S=\\sum_{xy\\mid n} \\varphi(x) \\cdot y, $$  where  $\\varphi(x)$  denotes the number of positive integers less than or equal to  $x$  that are relatively prime to  $x$ , and the sum is taken over ordered pairs  $(x, y)$  of positive integers for which  $xy$  divides  $n$ . Compute  $\\tfrac{S}{n}.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,51,0.0,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[4,51,0.0784,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,51,0.1569,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[12,51,0.2353,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,51,0.3137,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,51,0.3922,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,51,0.4706,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,51,0.549,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,51,0.6275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,51,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,51,0.7843,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,51,0.8627,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,51,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[51,51,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,53,0.0,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[4,53,0.0755,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,53,0.1509,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,53,0.2264,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,53,0.3019,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[20,53,0.3774,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,53,0.4528,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,53,0.5283,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,53,0.6038,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,53,0.6792,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,53,0.7547,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,53,0.8302,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,53,0.9057,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,53,0.9811,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[53,53,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"abacec3c96f07826","q":"Let  $p$  and  $q$  be odd prime numbers and  $a$  a positive integer so that  $p|a^q+1$  and  $q|a^p+1$ . Show that  $p|a+1$  or  $q|a+1$ .\n\n*Authored by Nikola Velov*","t":[{"b":1,"e":0.71429,"k":"flat","v":0.72766,"x":0.82587,"p":[[0,25,0.0,0.74998,0.18558,0.57143,0.78564,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,9,0,7],[4,25,0.16,0.72766,0.13055,0.67857,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,11,0,0,13,0,0],[8,25,0.32,0.77676,0.14702,0.71429,0.78571,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,11,0,5],[12,25,0.48,0.80354,0.15468,0.67857,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,14,0,7],[16,25,0.64,0.81693,0.16461,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,10,0,10],[20,25,0.8,0.74106,0.20653,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,8,0,0,9,0,6],[24,25,0.96,0.82587,0.17031,0.82132,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,15,0,9],[25,25,1.0,0.7857,0.15974,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,17,0,4]]},{"b":2,"e":0.57143,"k":"flat","v":0.59813,"x":0.73212,"p":[[0,58,0.0,0.72318,0.14701,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,13,0,1],[4,58,0.069,0.73212,0.14175,0.57143,0.78564,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,16,0,0],[8,58,0.1379,0.6696,0.12078,0.57143,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,0,7,0,0],[12,58,0.2069,0.65165,0.14252,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,14,0,0,3,0,1],[16,58,0.2759,0.65174,0.10679,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,8,0,0,5,0,0],[20,58,0.3448,0.6294,0.11772,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,5,0,0,3,0,1],[24,58,0.4138,0.63838,0.12865,0.57143,0.57143,0.71429,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,6,0,0,4,0,1],[28,58,0.4828,0.66962,0.11539,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,8,0,0,7,0,0],[32,58,0.5517,0.66514,0.1217,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,13,0,0,5,0,0],[36,58,0.6207,0.62498,0.10564,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,8,0,0,3,0,0],[40,58,0.6897,0.63836,0.11837,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,4,0,0,6,0,0],[44,58,0.7586,0.63838,0.12362,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,8,0,0,5,0,0],[48,58,0.8276,0.67403,0.12496,0.57143,0.57143,0.857,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,5,0,0,9,0,0],[52,58,0.8966,0.59813,0.07526,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,5,0,0,1,0,0],[56,58,0.9655,0.59817,0.15336,0.57132,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,3,0,0,17,0,0,6,0,0,4,0,0],[58,58,1.0,0.63388,0.11813,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,11,0,0,1,0,1]]}]},{"i":"0c31b32f6afdf347","q":"Let  $n>1$  be any integer. Define  $f,g$  as functions from  $\\{0,1,2,\\cdots,n-1 \\}$  to  $\\{0,1,2,\\cdots,n-1\\}$  defined as \n\\begin{align*} \n&f(i)=2i \\pmod{n}  \n&g(i)=2i+1 \\pmod{n} \\end{align*}\nShow that for any integers  $\\ell,m \\in \\{0,1,2,\\cdots,n-1 \\}$  , there are infinitely many compositions of  $f,g$  that map  $\\ell$  to  $m$","t":[{"b":1,"e":0.85714,"k":"flat","v":0.70536,"x":0.83929,"p":[[0,18,0.0,0.70536,0.27184,0.42857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,3,0,0,0,0,14],[4,18,0.2222,0.77679,0.27418,0.42857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,19],[8,18,0.4444,0.79018,0.24739,0.4286,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,5,0,0,1,0,17],[12,18,0.6667,0.82143,0.24222,0.64286,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,3,0,0,2,0,19],[16,18,0.8889,0.81249,0.2299,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,3,0,0,5,0,16],[18,18,1.0,0.83929,0.23076,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,2,0,0,4,0,19]]},{"b":4,"e":1.0,"k":"flat","v":0.70089,"x":0.86607,"p":[[0,25,0.0,0.7991,0.24185,0.42859,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,7,0,15],[4,25,0.16,0.74107,0.25364,0.42857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,6,0,12],[8,25,0.32,0.8616,0.22442,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,4,0,21],[12,25,0.48,0.70089,0.28651,0.42857,0.78571,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,4,0,12],[16,25,0.64,0.77232,0.27166,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0,18],[20,25,0.8,0.79464,0.22851,0.64286,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,3,0,0,8,0,13],[24,25,0.96,0.79911,0.24964,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,1,0,0,4,0,17],[25,25,1.0,0.86607,0.27879,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,25]]}]},{"i":"1581e24adb88a16a","q":"Let  $p$  prime and  $m$  a positive integer. Determine all pairs  $( p,m)$  satisfying the equation:  $ p(p+m)+p=(m+1)^3$","t":[{"b":2,"e":0.28571,"k":"flat","v":0.19196,"x":0.24107,"p":[[0,39,0.0,0.19196,0.10479,0.14286,0.21428,0.28571,0.0,0.28571,5,0,0,5,0,11,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.19643,0.09942,0.14286,0.2143,0.28571,0.0,0.28571,4,0,0,4,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.2008,0.10635,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,9,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.19643,0.10564,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,10,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.19643,0.09942,0.14286,0.21428,0.28571,0.0,0.28571,4,0,0,4,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.20535,0.11258,0.14286,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,6,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.21429,0.07986,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.21875,0.10091,0.14286,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,7,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.24107,0.07523,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.21875,0.07973,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.24107,0.07523,0.1429,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.16071,"x":0.21875,"p":[[0,32,0.0,0.1875,0.11538,0.14286,0.2857,0.28571,0.0,0.28571,7,0,0,7,0,8,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.20982,0.11285,0.14286,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,5,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.21875,0.10705,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,5,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.20536,0.10677,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,8,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.19197,0.09851,0.14286,0.1429,0.28571,0.0,0.28571,4,0,0,4,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.18304,0.10853,0.14286,0.1429,0.28571,0.0,0.28571,6,0,0,6,0,11,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.16071,0.1171,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,10,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.16071,0.11152,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,12,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.20089,0.18161,0.14286,0.1429,0.2857,0.0,1.0,7,1,0,7,0,10,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"b5eb12763680d7ba","q":"Let  $n\\ge 3$  be an integer, and suppose  $x_1,x_2,\\cdots ,x_n$  are positive real numbers such that  $x_1+x_2+\\cdots +x_n=1.$  Prove that   $$ x_1^{1-x_2}+x_2^{1-x_3}\\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2. $$  *~Sutanay Bhattacharya*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.67856,"x":0.83929,"p":[[0,40,0.0,0.83929,0.1915,0.57143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,1,0,18],[4,40,0.1,0.67856,0.39448,0.57132,0.92857,1.0,0.0,1.0,7,16,0,7,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,1,0,16],[8,40,0.2,0.78571,0.20516,0.57143,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,0,0,15],[12,40,0.3,0.75,0.18898,0.57143,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,7,0,0,0,0,11],[16,40,0.4,0.73214,0.1915,0.57143,0.57143,1.0,0.5714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,0,1,0,10],[20,40,0.5,0.75446,0.1794,0.57143,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,6,0,0,4,0,9],[24,40,0.6,0.69642,0.16657,0.57143,0.64286,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,0,1,0,6],[28,40,0.7,0.70982,0.17307,0.57143,0.57143,0.85704,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,6,0,0,2,0,7],[32,40,0.8,0.79018,0.20198,0.57143,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,0,0,15],[36,40,0.9,0.75,0.19233,0.57143,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,5,0,0,1,0,11],[40,40,1.0,0.74107,0.18013,0.57143,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,7,0,0,2,0,9]]},{"b":3,"e":0.57143,"k":"flat","v":0.65178,"x":0.81249,"p":[[0,23,0.0,0.79464,0.24206,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,2,0,16],[4,23,0.1739,0.81249,0.29975,0.57143,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,21],[8,23,0.3478,0.70536,0.14698,0.57143,0.71429,0.75,0.5714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,10,0,0,4,0,4],[12,23,0.5217,0.77231,0.18511,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,0,0,0,12],[16,23,0.6957,0.69195,0.16017,0.57143,0.57143,0.75,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,3,0,5],[20,23,0.8696,0.71875,0.16164,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,9,0,0,3,0,6],[23,23,1.0,0.65178,0.12845,0.57143,0.57143,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,6,0,0,3,0,2]]}]},{"i":"88b18515c607407d","q":"Let  $p>3$  be a prime number. Prove that the product of all primitive roots between 1 and  $p-1$  is congruent 1 modulo  $p$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,31,0.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[4,31,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,32,0.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[4,32,0.125,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,32,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,32,0.375,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,32,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,32,0.625,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1013786b788876aa","q":"Let  $p$ ,  $q$ , and  $r$  be the three roots of the polynomial  $x^3 -2x^2 + 3x - 2023$ . Suppose that the polynomial  $x^3 + Bx^2 +Mx + T$  has roots  $p + q$ ,  $p + r$ , and  $q + r$  for real numbers  $B$ ,  $M$ , and  $T$ . Compute  $B -M + T$ .","t":[{"b":6,"e":0.85714,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,81,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,81,0.0494,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,81,0.0988,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,81,0.1481,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[16,81,0.1975,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,81,0.2469,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,81,0.2963,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[28,81,0.3457,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,81,0.3951,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,81,0.4444,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[40,81,0.4938,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[44,81,0.5432,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[48,81,0.5926,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[52,81,0.642,0.89732,0.17582,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,15],[56,81,0.6914,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[60,81,0.7407,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[64,81,0.7901,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[68,81,0.8395,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,81,0.8889,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[76,81,0.9383,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[80,81,0.9877,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[81,81,1.0,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]},{"b":7,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,35,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,35,0.1143,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,35,0.2286,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,35,0.3429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,35,0.4571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,35,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,35,0.6857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"00a3f1084ffada52","q":"Let $A, B, C$ be the angles of an acute-angled triangle. Prove the inequality\n\n$$\n\\sin A+\\sin B>\\cos A+\\cos B+\\cos C\n$$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,14,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,29,0.0,0.06696,0.08737,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b41bf1feae9b1a75","q":"Let $a, b, c, d$ be positive integers such that $a b=c d$. Prove that $a+b+c+d$ is not prime.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,25,0.0,0.05804,0.18851,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,25,0.16,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,25,0.96,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04464,0.16146,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.08929,"p":[[0,21,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.04464,0.17655,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[16,21,0.7619,0.04464,0.16146,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,21,0.9524,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.08929,0.24419,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,1]]}]},{"i":"9bf303c9ca8104ea","q":"Let $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\\cdots+a_{0}$ be a polynomial with real coefficients, $a_{n} \\neq 0$. Suppose every root of $P$ is a root of unity, but $P(1) \\neq 0$. Show that the coefficients of $P$ are symmetric; that is, show that $a_{n}=a_{0}, a_{n-1}=a_{1}, \\ldots$","t":[{"b":0,"e":1.0,"k":"flat","v":0.85713,"x":0.99107,"p":[[0,19,0.0,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[4,19,0.2105,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,19,0.4211,0.85713,0.06186,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,29,0,2],[12,19,0.6316,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,19,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[19,19,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":2,"e":1.0,"k":"flat","v":0.88839,"x":0.99554,"p":[[0,42,0.0,0.90625,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[4,42,0.0952,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[8,42,0.1905,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[12,42,0.2857,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,42,0.381,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[20,42,0.4762,0.88839,0.1504,0.85714,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,12],[24,42,0.5714,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,42,0.6667,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[32,42,0.7619,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,42,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,42,0.9524,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[42,42,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c0407ac08f01f6cf","q":"Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, with $P \\neq 0$, and let $d=\\operatorname{deg} P$. Prove that there exist polynomials $A(x)$ and $B(x)$, not both zero, such that $\\max \\{\\operatorname{deg} A, \\operatorname{deg} B\\} \\leq d / 2$ and $P(x) \\mid A(x)+Q(x) \\cdot B(x)$.","t":[{"b":1,"e":0.4286,"k":"flat","v":0.90179,"x":1.0,"p":[[0,10,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,10,0.4,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[8,10,0.8,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[10,10,1.0,0.90179,0.12595,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,17]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,58,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,58,0.069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,58,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,58,0.2069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,58,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,58,0.3448,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,58,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,58,0.4828,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,58,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,58,0.6207,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,58,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,58,0.7586,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,58,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,58,0.8966,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,58,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[58,58,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ca95ee6a6172438b","q":"Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3, \\ldots, 1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.28571,"x":0.30357,"p":[[0,11,0.0,0.28571,0.12372,0.28571,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,11,0.3636,0.30357,0.09942,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,11,0.7273,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.27678,"x":0.28571,"p":[[0,14,0.0,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.27678,0.13333,0.28571,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,14,0.5714,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"219aef709073425a","q":"Let $a, b, c, d$ be positive integers such that $a \\geq b \\geq c \\geq d$. Prove that the equation $x^{4}-a x^{3}-b x^{2}-c x-d=0$ has no integer solution.","t":[{"b":1,"e":0.57143,"k":"falling","v":0.52677,"x":0.78557,"p":[[0,33,0.0,0.78557,0.20521,0.57143,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,0,2,0,14],[4,33,0.1212,0.56249,0.10677,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,25,0,0,1,0,0,0,0,1],[8,33,0.2424,0.54911,0.10779,0.57142,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,24,0,0,0,0,0,0,0,1],[12,33,0.3636,0.53125,0.11425,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,0,0,0,0,0,1],[16,33,0.4848,0.54464,0.05576,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0],[20,33,0.6061,0.53571,0.09449,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,8,0,0,22,0,0,0,0,0,1,0,0],[24,33,0.7273,0.54905,0.10169,0.53539,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,23,0,0,0,0,0,0,0,1],[28,33,0.8485,0.55357,0.04725,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[32,33,0.9697,0.53125,0.07349,0.5357,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,7,0,0,24,0,0,0,0,0,0,0,0],[33,33,1.0,0.52677,0.06621,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"falling","v":0.54909,"x":0.79463,"p":[[0,30,0.0,0.79463,0.21999,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,5,0,14],[4,30,0.1333,0.75,0.22304,0.57143,0.57143,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,14],[8,30,0.2667,0.54909,0.11355,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,4,0,0,25,0,0,0,0,0,0,0,1],[12,30,0.4,0.54911,0.11904,0.57142,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,5,0,0,23,0,0,1,0,0,0,0,1],[16,30,0.5333,0.56696,0.06667,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,1,0,0],[20,30,0.6667,0.55356,0.04724,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[24,30,0.8,0.58482,0.11495,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,0,0,2],[28,30,0.9333,0.56695,0.04351,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0],[30,30,1.0,0.59819,0.17291,0.57143,0.57143,0.57143,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,3,0,0,22,0,0,1,0,0,0,0,4]]}]},{"i":"49f6cb43cddbfbd0","q":"Let $n \\geq 2$ be a positive integer. A subset of positive integers $S$ is said to be comprehensive if for every integer $0 \\leq x<n$, there is a subset of $S$ whose sum has remainder $x$ when divided by $n$. Note that the empty set has sum 0 . Show that if a set $S$ is comprehensive, then there is some (not necessarily proper) subset of $S$ with at most $n-1$ elements which is also comprehensive.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.29008,"x":0.54018,"p":[[0,18,0.0,0.29008,0.32044,0.0,0.14286,0.28571,0.0,1.0,9,3,0,9,0,8,0,0,8,0,0,1,0,0,0,0,0,1,0,0,2,0,3],[4,18,0.2222,0.54018,0.29824,0.28571,0.42859,0.71429,0.0,1.0,1,6,0,1,0,1,0,0,12,0,0,3,0,0,1,0,0,7,0,0,1,0,6],[8,18,0.4444,0.50882,0.31132,0.28571,0.49979,0.75,0.0,1.0,3,5,0,3,0,1,0,0,10,0,0,2,0,0,6,0,0,2,0,0,3,0,5],[12,18,0.6667,0.49553,0.34066,0.2857,0.35714,0.85714,0.0,1.0,3,7,0,3,0,2,0,0,11,0,0,5,0,0,0,0,0,1,0,0,3,0,7],[16,18,0.8889,0.37052,0.1631,0.2857,0.28571,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,18,0,0,8,0,0,4,0,0,0,0,0,0,0,1],[18,18,1.0,0.37053,0.17807,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,7,0,0,1,0,0,0,0,0,0,0,2]]},{"b":3,"e":1.0,"k":"rising","v":0.31696,"x":0.99554,"p":[[0,32,0.0,0.31696,0.32485,0.0,0.2857,0.42857,0.0,1.0,9,4,0,9,0,6,0,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,4],[4,32,0.125,0.66962,0.30606,0.28571,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,7,0,0,1,0,0,1,0,0,7,0,0,5,0,9],[8,32,0.25,0.59822,0.30396,0.28571,0.4286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,11,0,0,6,0,0,2,0,0,2,0,0,1,0,10],[12,32,0.375,0.58036,0.29001,0.28571,0.57143,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,10,0,0,2,0,0,7,0,0,3,0,0,2,0,7],[16,32,0.5,0.65177,0.33108,0.42857,0.71429,1.0,0.0,1.0,3,10,0,3,0,1,0,0,3,0,0,3,0,0,4,0,0,4,0,0,4,0,10],[20,32,0.625,0.88393,0.19377,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,0,3,0,21],[24,32,0.75,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,32,0.875,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,32,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"bacbec5f98ecc719","q":"Let $n \\geqslant 0$ be an integer, and let $a_{0}, a_{1}, \\ldots, a_{n}$ be real numbers. Show that there exists $k \\in\\{0,1, \\ldots, n\\}$ such that\n\n$$\na_{0}+a_{1} x+a_{2} x^{2}+\\cdots+a_{n} x^{n} \\leqslant a_{0}+a_{1}+\\cdots+a_{k}\n$$\n\nfor all real numbers $x \\in[0,1]$.\n\n#","t":[{"b":2,"e":0.0,"k":"flat","v":0.12945,"x":0.18749,"p":[[0,26,0.0,0.12945,0.21234,0.0,0.0,0.28571,0.0,0.71429,22,0,0,22,0,0,0,0,6,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[4,26,0.1538,0.15177,0.234,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,0,0,0,6,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[8,26,0.3077,0.14732,0.25123,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,1,0,0,6,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[12,26,0.4615,0.18749,0.27065,0.0,0.0,0.28571,0.0,0.71429,20,0,0,20,0,0,0,0,5,0,0,1,0,0,1,0,0,5,0,0,0,0,0],[16,26,0.6154,0.14283,0.22298,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,0,0,0,7,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[20,26,0.7692,0.16071,0.25692,0.0,0.0,0.2857,0.0,1.0,20,1,0,20,0,1,0,0,7,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[24,26,0.9231,0.14284,0.22301,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,0,0,0,0],[26,26,1.0,0.18304,0.26782,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,0,0,0,8,0,0,1,0,0,0,0,0,3,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"flat","v":0.07143,"x":0.24107,"p":[[0,41,0.0,0.14286,0.24223,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,0,0,0,6,0,0,0,0,0,0,0,0,4,0,0,0,0,0],[4,41,0.0976,0.10714,0.22588,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,41,0.1951,0.17411,0.22227,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[12,41,0.2927,0.07143,0.15567,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,1,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,41,0.3902,0.16964,0.26107,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,0,0,0,7,0,0,1,0,0,1,0,0,2,0,0,0,0,1],[20,41,0.4878,0.18304,0.31183,0.0,0.0,0.28571,0.0,1.0,22,2,0,22,0,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,0,0,2],[24,41,0.5854,0.17411,0.26901,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,0,0,0,5,0,0,0,0,0,1,0,0,5,0,0,0,0,0],[28,41,0.6829,0.22766,0.28314,0.0,0.0,0.35704,0.0,1.0,17,1,0,17,0,0,0,0,7,0,0,0,0,0,5,0,0,2,0,0,0,0,1],[32,41,0.7805,0.10268,0.21498,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,41,0.878,0.20981,0.22862,0.0,0.21428,0.28571,0.0,0.71429,15,0,0,15,0,1,0,0,9,0,0,1,0,0,5,0,0,1,0,0,0,0,0],[40,41,0.9756,0.24107,0.27534,0.0,0.14285,0.46429,0.0,0.71429,16,0,0,16,0,0,0,0,7,0,0,1,0,0,3,0,0,5,0,0,0,0,0],[41,41,1.0,0.16963,0.20956,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,0,0,0,12,0,0,0,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"26932d77d83b712e","q":"Let $n \\geq 3$ be a positive integer. Prove that given any $n$ angles $0<\\theta_{1}, \\theta_{2}, \\ldots, \\theta_{n}<$ $180^{\\circ}$, such that their sum is $180(n-2)$ degrees, there exists a convex $n$-gon having exactly those angles, in that order.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,18,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d7a2492a972e16ed","q":"Let $n \\geqslant 1$. We have a finite number of bottles, each containing an amount of water less than 1 liter, such that the total amount of water is $\\frac{\\mathfrak{n}}{2}$ liters. We also have $n$ empty buckets.\n\nShow that it is possible to empty each bottle into a bucket (we can empty several bottles into the same bucket, but not empty a bottle partially into one bucket and partially into another) in such a way that there is at most 1 liter of water in each bucket.","t":[{"b":3,"e":1.0,"k":"rising","v":0.50893,"x":0.98661,"p":[[0,41,0.0,0.51339,0.37941,0.14286,0.42857,1.0,0.0,1.0,4,10,0,4,0,5,0,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,10],[4,41,0.0976,0.60714,0.33503,0.28571,0.64286,1.0,0.0,1.0,3,9,0,3,0,0,0,0,6,0,0,6,0,0,1,0,0,3,0,0,4,0,9],[8,41,0.1951,0.6875,0.35072,0.42857,0.85707,1.0,0.0,1.0,2,15,0,2,0,3,0,0,2,0,0,3,0,0,4,0,0,1,0,0,2,0,15],[12,41,0.2927,0.52677,0.34522,0.25,0.57121,0.85714,0.0,1.0,5,6,0,5,0,3,0,0,2,0,0,5,0,0,4,0,0,4,0,0,3,0,6],[16,41,0.3902,0.5223,0.34368,0.28571,0.42857,0.89275,0.0,1.0,3,8,0,3,0,4,0,0,5,0,0,5,0,0,5,0,0,0,0,0,2,0,8],[20,41,0.4878,0.57143,0.36943,0.14286,0.71429,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,1,0,0,3,0,0,2,0,0,4,0,0,6,0,7],[24,41,0.5854,0.50893,0.37786,0.14286,0.42857,0.89286,0.0,1.0,5,8,0,5,0,5,0,0,4,0,0,3,0,0,2,0,0,2,0,0,3,0,8],[28,41,0.6829,0.59821,0.42173,0.14286,0.78571,1.0,0.0,1.0,6,15,0,6,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,15],[32,41,0.7805,0.61161,0.40757,0.14286,0.78571,1.0,0.0,1.0,5,13,0,5,0,5,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,13],[36,41,0.878,0.73661,0.38649,0.39286,1.0,1.0,0.0,1.0,4,20,0,4,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,20],[40,41,0.9756,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[41,41,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":5,"e":0.4286,"k":"flat","v":0.55357,"x":0.87053,"p":[[0,27,0.0,0.55357,0.37585,0.25,0.71429,0.89286,0.0,1.0,5,8,0,5,0,3,0,0,5,0,0,2,0,0,0,0,0,5,0,0,4,0,8],[4,27,0.1481,0.81248,0.33205,0.82132,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,22],[8,27,0.2963,0.74554,0.31891,0.53571,0.92857,1.0,0.0,1.0,1,16,0,1,0,3,0,0,1,0,0,3,0,0,1,0,0,5,0,0,2,0,16],[12,27,0.4444,0.87052,0.24838,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[16,27,0.5926,0.79463,0.31732,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,19],[20,27,0.7407,0.87053,0.17627,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,7,0,17],[24,27,0.8889,0.64285,0.28121,0.42857,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,10,0,0,1,0,0,4,0,0,3,0,9],[27,27,1.0,0.62499,0.30041,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,12,0,0,2,0,0,1,0,0,1,0,11]]}]},{"i":"17f0452079e89bea","q":"Let $n \\geqslant 2$ be a positive integer and $a_{1}, a_{2}, \\ldots, a_{n}$ be real numbers such that  $$ a_{1}+a_{2}+\\cdots+a_{n}=0 $$  Define the set $A$ by  $$ A=\\left\\{(i, j)\\left|1 \\leqslant i<j \\leqslant n,\\left|a_{i}-a_{j}\\right| \\geqslant 1\\right\\} .\\right. $$  Prove that, if $A$ is not empty, then  $$ \\sum_{(i, j) \\in A} a_{i} a_{j}<0 $$  (China)","t":[{"b":4,"e":0.28571,"k":"flat","v":0.13831,"x":0.27232,"p":[[0,65,0.0,0.24554,0.15251,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,5,0,0,14,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,65,0.0615,0.20981,0.16742,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,6,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[8,65,0.1231,0.16732,0.1737,0.0,0.14286,0.2857,0.0,0.64286,11,0,0,11,0,12,0,0,4,0,0,3,0,0,1,1,0,0,0,0,0,0,0],[12,65,0.1846,0.24107,0.18707,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,16,0,0,3,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[16,65,0.2462,0.18304,0.1931,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,0,12,0,0,6,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[20,65,0.3077,0.16518,0.12931,0.14286,0.14286,0.14287,0.0,0.57143,5,0,0,5,0,21,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,65,0.3692,0.13831,0.12619,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,18,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,65,0.4308,0.15616,0.12557,0.105,0.14286,0.1786,0.0,0.4286,8,0,0,8,0,16,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.20981,0.19554,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,13,0,0,4,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[36,65,0.5538,0.21874,0.15143,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,18,0,0,4,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[40,65,0.6154,0.17401,0.17027,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,13,0,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[44,65,0.6769,0.20535,0.17471,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,16,0,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[48,65,0.7385,0.23214,0.17405,0.14286,0.14286,0.32143,0.0,0.7143,5,0,0,5,0,13,0,0,6,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[52,65,0.8,0.20981,0.22297,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,14,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[56,65,0.8615,0.24107,0.18013,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,13,0,0,8,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[60,65,0.9231,0.21874,0.18203,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,14,0,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[64,65,0.9846,0.27232,0.17627,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,9,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[65,65,1.0,0.25,0.19885,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,8,0,0,7,0,0,8,0,0,1,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.24998,"x":0.39498,"p":[[0,71,0.0,0.26553,0.16875,0.14286,0.28571,0.28571,0.0,1.0,3,1,0,3,0,6,0,0,19,0,1,2,0,0,0,0,0,0,0,0,0,0,1],[4,71,0.0563,0.24998,0.23416,0.14286,0.14286,0.32143,0.0,1.0,5,1,0,5,0,15,0,0,4,0,0,4,0,0,2,0,0,0,0,0,1,0,1],[8,71,0.1127,0.28561,0.18562,0.14286,0.2857,0.32143,0.0,0.85714,2,0,0,2,0,11,0,0,11,0,0,3,0,0,4,0,0,0,0,0,1,0,0],[12,71,0.169,0.30799,0.20545,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,12,0,0,6,0,0,6,0,0,4,0,0,1,0,0,1,0,0],[16,71,0.2254,0.26116,0.14574,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,13,0,0,11,0,1,4,0,0,1,0,0,1,0,0,0,0,0],[20,71,0.2817,0.36606,0.24983,0.14286,0.28571,0.42858,0.0,1.0,2,2,0,2,0,8,0,0,7,0,0,9,0,0,2,0,0,1,0,0,1,0,2],[24,71,0.338,0.31246,0.21847,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,9,0,0,5,0,0,4,0,0,1,0,0,0,0,1],[28,71,0.3944,0.35487,0.17626,0.25,0.28571,0.44643,0.14286,0.85714,0,0,0,0,0,8,0,0,10,0,0,6,0,1,6,0,0,0,0,0,1,0,0],[32,71,0.4507,0.3125,0.23808,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,10,0,0,9,0,0,5,0,0,1,0,0,2,0,0,1,0,1],[36,71,0.507,0.33908,0.19165,0.14286,0.28571,0.46418,0.0,0.71429,1,0,0,1,0,10,0,0,7,0,0,6,0,0,6,0,0,2,0,0,0,0,0],[40,71,0.5634,0.29463,0.1921,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,10,0,0,7,0,0,8,0,0,3,0,0,0,0,0,1,0,0],[44,71,0.6197,0.31246,0.2002,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,10,0,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,1],[48,71,0.6761,0.37947,0.17353,0.28571,0.35714,0.42858,0.14286,0.85714,0,0,0,0,0,5,0,0,11,0,0,10,0,0,3,0,0,2,0,0,1,0,0],[52,71,0.7324,0.38839,0.19638,0.2857,0.28571,0.46429,0.14286,0.85714,0,0,0,0,0,6,0,0,11,0,0,7,0,0,3,0,0,4,0,0,1,0,0],[56,71,0.7887,0.33044,0.12971,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,10,0,2,10,0,0,3,0,0,0,0,0,0,0,0],[60,71,0.8451,0.35268,0.18893,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,6,0,0,16,0,0,5,0,0,0,0,0,4,0,0,1,0,0],[64,71,0.9014,0.39498,0.16951,0.28571,0.42857,0.4642,0.14,0.71429,0,0,0,0,0,5,0,0,8,0,2,9,0,0,4,1,0,3,0,0,0,0,0],[68,71,0.9577,0.36607,0.19865,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,11,0,0,9,0,0,2,0,0,0,0,0,3,0,0],[71,71,1.0,0.37498,0.17402,0.2857,0.35714,0.4286,0.14286,0.71429,0,0,0,0,0,6,0,0,10,0,0,10,0,0,2,0,0,4,0,0,0,0,0]]}]},{"i":"bb5393529c64ff55","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be an arithmetic progression of integers such that $i$ divides $a_{i}$ for $i=1,2, \\ldots, n-1$ and $n$ does not divide $a_{n}$. Prove that $n$ is a power of a prime.","t":[{"b":4,"e":1.0,"k":"flat","v":0.9241,"x":0.97768,"p":[[0,40,0.0,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[4,40,0.1,0.9241,0.12364,0.82132,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[8,40,0.2,0.92857,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[12,40,0.3,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,40,0.4,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[20,40,0.5,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,40,0.6,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[28,40,0.7,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[32,40,0.8,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[36,40,0.9,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[40,40,1.0,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25]]},{"b":5,"e":1.0,"k":"flat","v":0.96639,"x":1.0,"p":[[0,27,0.0,0.96639,0.08962,1.0,1.0,1.0,0.71,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,28],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,27,0.7407,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,27,0.8889,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[27,27,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"3697030618a99587","q":"Let $n$ be a natural number. Prove that the binary representation of the integer $n\\left(2^{n}-1\\right)$ contains exactly $n$ occurrences of the digit 1.","t":[{"b":3,"e":1.0,"k":"rising","v":0.55802,"x":1.0,"p":[[0,29,0.0,0.55802,0.44372,0.10714,0.64264,1.0,0.0,1.0,8,15,0,8,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,15],[4,29,0.1379,0.84375,0.29957,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,24],[8,29,0.2759,0.95981,0.16461,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.571,"k":"flat","v":0.18741,"x":0.87054,"p":[[0,50,0.0,0.49108,0.42847,0.10714,0.28571,1.0,0.0,1.0,8,12,0,8,0,5,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,12],[4,50,0.08,0.81696,0.34113,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,24],[8,50,0.16,0.87054,0.30169,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[12,50,0.24,0.70089,0.41551,0.14286,1.0,1.0,0.0,1.0,2,21,0,2,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[16,50,0.32,0.6875,0.40945,0.14286,1.0,1.0,0.0,1.0,2,19,0,2,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,19],[20,50,0.4,0.73214,0.39407,0.24999,1.0,1.0,0.0,1.0,2,21,0,2,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,21],[24,50,0.48,0.58482,0.39827,0.14286,0.57143,1.0,0.14286,1.0,0,14,0,0,0,12,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,14],[28,50,0.56,0.47768,0.4159,0.14286,0.14286,1.0,0.0,1.0,3,12,0,3,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,12],[32,50,0.64,0.3259,0.32583,0.14286,0.14286,0.46431,0.0,1.0,4,4,0,4,0,17,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,4],[36,50,0.72,0.20982,0.23686,0.14286,0.14286,0.14286,0.0,1.0,4,2,0,4,0,22,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[40,50,0.8,0.18741,0.20027,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,22,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[44,50,0.88,0.22322,0.21706,0.14286,0.14286,0.1786,0.0,0.85714,4,0,0,4,0,20,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[48,50,0.96,0.43749,0.34057,0.14286,0.42857,0.71429,0.0,1.0,7,5,0,7,0,4,0,0,2,0,0,5,0,0,5,0,0,4,0,0,0,0,5],[50,50,1.0,0.37722,0.28695,0.14286,0.42857,0.57143,0.0,1.0,4,1,0,4,1,9,0,0,1,0,0,6,0,0,4,0,0,4,0,0,2,0,1]]}]},{"i":"5f7e6ae70384f50e","q":"Let $p$ be an odd prime and $x_{1}, \\ldots, x_{p}$ be integers.\nWe assume that for all $k \\geqslant 1$ integer, we have\n\n$$\n\\mathrm{p} \\mid \\mathrm{x}_{1}^{\\mathrm{k}}+\\cdots+\\mathrm{x}_{\\mathrm{p}}^{\\mathrm{k}}\n$$\n\nShow that the integers $x_{1}, \\ldots, x_{p}$ are all congruent modulo $p$.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.0,"x":0.84375,"p":[[0,17,0.0,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,17,0.2353,0.60714,0.47246,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,18],[8,17,0.4706,0.53571,0.49487,0.0,1.0,1.0,0.0,1.0,14,17,0,14,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[17,17,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.5,"x":1.0,"p":[[0,25,0.0,0.78125,0.4134,1.0,1.0,1.0,0.0,1.0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[4,25,0.16,0.5,0.5,0.0,0.5,1.0,0.0,1.0,16,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[8,25,0.32,0.78125,0.4134,1.0,1.0,1.0,0.0,1.0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[12,25,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b5e7d2928cc47810","q":"Let $p \\geq 5$ be a prime and let $r$ be the number of ways of placing $p$ checkers on a $p \\times p$ checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that $r$ is divisible by $p^{5}$. Here, we assume that all the checkers are identical.","t":[{"b":2,"e":1.0,"k":"falling","v":0.37054,"x":0.97768,"p":[[0,99,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,99,0.0404,0.875,0.25692,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,24],[8,99,0.0808,0.87946,0.28372,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[12,99,0.1212,0.93302,0.17125,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[16,99,0.1616,0.89286,0.26244,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[20,99,0.202,0.89732,0.2506,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,25],[24,99,0.2424,0.9375,0.2111,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[28,99,0.2828,0.88839,0.22794,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,23],[32,99,0.3232,0.78125,0.33309,0.64286,1.0,1.0,0.0,1.0,1,19,0,1,0,4,0,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,19],[36,99,0.3636,0.79463,0.32916,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,20],[40,99,0.404,0.83928,0.30671,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,22],[44,99,0.4444,0.74106,0.36323,0.46397,1.0,1.0,0.14286,1.0,0,19,0,0,0,8,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,19],[48,99,0.4848,0.80803,0.30641,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,21],[52,99,0.5253,0.78572,0.33312,0.64286,1.0,1.0,0.14286,1.0,0,21,0,0,0,5,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,21],[56,99,0.5657,0.53572,0.38132,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,14,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,10],[60,99,0.6061,0.62054,0.38401,0.14286,0.71429,1.0,0.14286,1.0,0,15,0,0,0,10,0,0,1,0,0,4,0,0,0,0,0,2,0,0,0,0,15],[64,99,0.6465,0.62947,0.38442,0.14286,0.78571,1.0,0.14286,1.0,0,14,0,0,0,11,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,14],[68,99,0.6869,0.73214,0.36553,0.35714,1.0,1.0,0.14286,1.0,0,19,0,0,0,8,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,19],[72,99,0.7273,0.62054,0.37048,0.14286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,1,0,0,5,0,0,1,0,0,0,0,0,3,0,13],[76,99,0.7677,0.73658,0.32755,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,2,0,0,1,0,0,2,0,0,2,0,0,5,0,15],[80,99,0.8081,0.625,0.40366,0.14286,0.85714,1.0,0.14286,1.0,0,14,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,14],[84,99,0.8485,0.58035,0.4071,0.14286,0.71429,1.0,0.0,1.0,2,13,0,2,0,11,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,13],[88,99,0.8889,0.40179,0.3719,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,20,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,8],[92,99,0.9293,0.56695,0.38046,0.14286,0.64264,1.0,0.14286,1.0,0,10,0,0,0,13,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,10],[96,99,0.9697,0.46875,0.35216,0.14286,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,15,0,0,0,0,0,5,0,0,1,0,0,1,0,0,4,0,6],[99,99,1.0,0.37054,0.33093,0.14286,0.14286,0.50002,0.14286,1.0,0,5,0,0,0,20,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,5]]},{"b":6,"e":0.71429,"k":"falling","v":0.31687,"x":0.98214,"p":[[0,110,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,110,0.0364,0.87054,0.28652,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[8,110,0.0727,0.82589,0.28735,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,20],[12,110,0.1091,0.94642,0.17768,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[16,110,0.1455,0.9375,0.20806,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[20,110,0.1818,0.84821,0.29867,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,23],[24,110,0.2182,0.85268,0.28901,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,22],[28,110,0.2545,0.88392,0.26351,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,25],[32,110,0.2909,0.78571,0.3481,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,21],[36,110,0.3273,0.875,0.28516,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[40,110,0.3636,0.799,0.3133,0.67857,1.0,1.0,0.14,1.0,0,20,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,20],[44,110,0.4,0.7321,0.29398,0.53539,0.85707,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,4,0,0,4,0,0,3,0,0,3,0,14],[48,110,0.4364,0.65624,0.37941,0.14286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,10,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,15],[52,110,0.4727,0.54017,0.3655,0.14286,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,13,0,0,1,0,0,1,0,0,1,0,0,5,0,0,3,0,8],[56,110,0.5091,0.58027,0.37627,0.14286,0.71429,1.0,0.14,1.0,0,11,0,0,0,12,0,0,1,0,0,1,0,0,0,0,0,6,0,0,1,0,11],[60,110,0.5455,0.41964,0.37276,0.14286,0.14286,0.89286,0.14286,1.0,0,8,0,0,0,19,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,8],[64,110,0.5818,0.5625,0.38949,0.14286,0.42859,1.0,0.14286,1.0,0,12,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,12],[68,110,0.6182,0.42401,0.37546,0.14286,0.14286,0.85714,0.14,1.0,0,7,0,0,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,7],[72,110,0.6545,0.48213,0.36202,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,15,0,0,0,0,0,4,0,0,1,0,0,3,0,0,1,0,8],[76,110,0.6909,0.51339,0.37772,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,14,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,10],[80,110,0.7273,0.43304,0.39526,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,20,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,10],[84,110,0.7636,0.43738,0.35709,0.14286,0.14286,0.74996,0.14,1.0,0,6,0,0,0,18,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,6],[88,110,0.8,0.39732,0.34577,0.14286,0.14286,0.71429,0.14286,1.0,0,5,0,0,0,19,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,5],[92,110,0.8364,0.39276,0.34815,0.14286,0.14286,0.74996,0.14,1.0,0,5,0,0,0,20,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,5],[96,110,0.8727,0.33929,0.30671,0.14286,0.14286,0.57143,0.0,1.0,1,3,0,1,0,20,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,3],[100,110,0.9091,0.31687,0.32293,0.14286,0.14286,0.17857,0.14,1.0,0,4,0,0,0,24,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,4],[104,110,0.9455,0.36604,0.29218,0.14286,0.14286,0.60714,0.14286,1.0,0,2,0,0,0,19,0,0,0,0,0,2,0,0,3,0,0,5,0,0,1,0,2],[108,110,0.9818,0.54911,0.35375,0.14286,0.57144,0.85714,0.14286,1.0,0,7,0,0,0,11,0,0,2,0,0,3,0,0,0,0,0,4,0,0,5,0,7],[110,110,1.0,0.45982,0.36374,0.14286,0.2143,0.85704,0.14286,1.0,0,7,0,0,0,16,0,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,7]]}]},{"i":"c7c0efb2db1725b4","q":"Let $n \\in \\mathbb{N}^{*}$. We call $S$ the set of sequences of $2n$ digits containing $n$ zeros and as many ones. Two sequences in $S$ are neighbors if it suffices to change the position of one digit in one to obtain the other. For example, 11100010 and 10110010 are neighbors since by moving the first 0 in the second sequence two \"steps\" to the right, we obtain the first sequence. Let $T$ be a dense subset of minimal cardinality. Show that $\\frac{1}{n^{2}+1}|S| \\leq|T| \\leq \\frac{1}{n+1}|S|$. Note: we denote $|A|$ the cardinality of $A$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.30357,"x":0.99554,"p":[[0,38,0.0,0.30357,0.16269,0.28571,0.28571,0.28571,0.0,0.85714,3,0,3,3,0,0,0,0,25,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[4,38,0.1053,0.64284,0.30305,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,8,0,0,3,0,0,6,0,0,1,0,0,2,0,11],[8,38,0.2105,0.66962,0.27995,0.42857,0.64286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,3,0,0,6,0,0,4,0,0,1,0,11],[12,38,0.3158,0.62054,0.30849,0.28571,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,8,0,0,5,0,0,4,0,0,2,0,0,2,0,10],[16,38,0.4211,0.75443,0.28401,0.571,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,5,0,0,1,0,0,3,0,0,3,0,0,5,0,14],[20,38,0.5263,0.74106,0.28221,0.53539,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,3,0,0,3,0,0,4,0,0,3,0,14],[24,38,0.6316,0.77679,0.26949,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,1,0,0,2,0,17],[28,38,0.7368,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[32,38,0.8421,0.90179,0.20341,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,25],[36,38,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[38,38,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]},{"b":6,"e":1.0,"k":"rising","v":0.35713,"x":0.96429,"p":[[0,29,0.0,0.35713,0.15566,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[4,29,0.1379,0.66964,0.32229,0.28571,0.78571,1.0,0.0,1.0,2,10,0,2,0,0,0,0,7,0,0,1,0,0,3,0,0,3,0,0,6,0,10],[8,29,0.2759,0.6875,0.29111,0.42857,0.64286,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,3,0,0,6,0,0,2,0,0,1,0,13],[12,29,0.4138,0.74552,0.28957,0.57132,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,7,0,0,0,0,0,5,0,0,3,0,0,1,0,16],[16,29,0.5517,0.58926,0.30041,0.28571,0.571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,3,0,0,6,0,0,1,0,0,0,0,10],[20,29,0.6897,0.76338,0.29798,0.42859,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,7,0,0,2,0,0,1,0,0,2,0,0,3,0,17],[24,29,0.8276,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[28,29,0.9655,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[29,29,1.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]}]},{"i":"d2d97d8b1556a2b4","q":"Let $n$ be a positive integer, and consider a sequence $a_{1}, a_{2}, \\ldots, a_{n}$ of positive integers. Extend it periodically to an infinite sequence $a_{1}, a_{2}, \\ldots$ by defining $a_{n+i}=a_{i}$ for all $i \\geqslant 1$. If  $$ a_{1} \\leqslant a_{2} \\leqslant \\cdots \\leqslant a_{n} \\leqslant a_{1}+n $$  and  $$ a_{a_{i}} \\leqslant n+i-1 \\quad \\text { for } i=1,2, \\ldots, n $$  prove that  $$ a_{1}+\\cdots+a_{n} \\leqslant n^{2} . $$  (Germany)","t":[{"b":0,"e":1.0,"k":"rising","v":0.37722,"x":0.66515,"p":[[0,78,0.0,0.37722,0.1823,0.28571,0.42857,0.42858,0.0,0.71429,4,0,0,4,0,0,0,0,8,0,0,13,0,1,4,0,0,2,0,0,0,0,0],[4,78,0.0513,0.56696,0.26362,0.42857,0.71429,0.71429,0.0,1.0,3,1,0,3,0,1,0,0,3,0,0,3,0,0,5,0,0,12,0,0,4,0,1],[8,78,0.1026,0.50442,0.27428,0.28571,0.57121,0.71429,0.0,1.0,4,2,0,4,0,0,0,0,7,0,0,2,0,0,7,0,0,9,0,0,1,0,2],[12,78,0.1538,0.56692,0.30406,0.42857,0.57121,0.75,0.0,1.0,4,6,0,4,0,0,0,0,2,0,0,7,0,0,7,0,0,4,0,0,2,0,6],[16,78,0.2051,0.56249,0.28107,0.39286,0.71429,0.71429,0.0,1.0,4,2,0,4,0,0,0,0,4,0,0,2,0,0,5,0,0,12,0,0,3,0,2],[20,78,0.2564,0.55802,0.35058,0.28571,0.57121,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,8,0,0,2,0,0,2,0,0,3,0,0,5,0,7],[24,78,0.3077,0.62945,0.31914,0.39393,0.57143,1.0,0.0,1.0,3,9,0,3,0,0,0,0,5,0,0,1,0,0,8,0,0,3,0,0,3,0,9],[28,78,0.359,0.56262,0.26697,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,8,0,0,2,0,0,4,0,0,8,0,0,4,0,3],[32,78,0.4103,0.64283,0.25001,0.571,0.71429,0.75,0.0,1.0,1,4,0,1,0,1,0,0,4,0,0,1,0,0,5,0,0,12,0,0,4,0,4],[36,78,0.4615,0.59818,0.25364,0.42857,0.57143,0.85704,0.0,1.0,1,3,0,1,0,0,0,0,6,0,0,5,0,0,5,0,0,6,0,0,6,0,3],[40,78,0.5128,0.60264,0.25689,0.42859,0.71429,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,2,0,0,4,0,0,6,0,0,12,0,0,1,0,4],[44,78,0.5641,0.59817,0.27765,0.28571,0.57121,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,7,0,0,4,0,0,4,0,0,6,0,0,3,0,6],[48,78,0.6154,0.66515,0.24383,0.57132,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,4,0,0,1,0,0,7,0,0,9,0,0,5,0,5],[52,78,0.6667,0.59371,0.24252,0.42857,0.71429,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,6,0,0,4,0,0,4,0,0,13,0,0,0,0,4],[56,78,0.7179,0.61606,0.25862,0.42857,0.71429,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,4,0,0,3,0,0,11,0,0,2,0,5],[60,78,0.7692,0.53122,0.29284,0.28571,0.49979,0.71429,0.0,1.0,2,4,0,2,0,2,0,0,7,0,0,5,0,0,3,0,0,6,0,0,3,0,4],[64,78,0.8205,0.64731,0.26961,0.57132,0.64286,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,1,0,0,9,0,0,6,0,0,3,0,7],[68,78,0.8718,0.61161,0.32189,0.39286,0.71429,0.85714,0.0,1.0,4,6,0,4,0,1,0,0,3,0,0,2,0,0,1,0,0,12,0,0,3,0,6],[72,78,0.9231,0.54016,0.33452,0.28571,0.64286,0.85714,0.0,1.0,4,4,0,4,0,2,0,0,7,0,0,1,0,0,2,0,0,6,0,0,6,0,4],[76,78,0.9744,0.62058,0.23034,0.42965,0.71429,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,5,0,0,5,0,0,11,0,0,5,0,2],[78,78,1.0,0.5625,0.33681,0.28571,0.71429,0.85714,0.0,1.0,5,5,0,5,0,1,0,0,4,0,0,3,0,0,1,0,0,9,0,0,4,0,5]]},{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.63838,"p":[[0,84,0.0,0.38393,0.2299,0.28571,0.42857,0.46429,0.0,1.0,5,1,0,5,0,0,0,0,9,0,0,10,0,0,4,0,0,3,0,0,0,0,1],[4,84,0.0476,0.60713,0.28347,0.42857,0.71429,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,2,0,0,5,0,0,3,0,0,9,0,0,5,0,4],[8,84,0.0952,0.54462,0.25614,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,8,0,0,0,0,0,8,0,0,8,0,0,3,0,2],[12,84,0.1429,0.5491,0.29038,0.28571,0.57143,0.71429,0.0,1.0,3,4,0,3,0,0,0,0,7,0,0,3,0,0,6,0,0,6,0,0,3,0,4],[16,84,0.1905,0.54017,0.26901,0.28571,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,8,0,0,3,0,0,5,0,0,7,0,0,3,0,3],[20,84,0.2381,0.5714,0.27433,0.42857,0.57143,0.71429,0.0,1.0,3,3,0,3,0,0,0,0,4,0,0,4,0,0,7,0,0,7,0,0,4,0,3],[24,84,0.2857,0.50444,0.27194,0.28571,0.57121,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,8,0,0,3,0,0,7,0,0,4,0,0,4,0,2],[28,84,0.3333,0.51785,0.26183,0.28571,0.4286,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,7,0,0,8,0,0,6,0,0,4,0,0,1,0,4],[32,84,0.381,0.52679,0.2911,0.28571,0.50001,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,9,0,0,4,0,0,4,0,0,6,0,0,1,0,5],[36,84,0.4286,0.58477,0.24317,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,3,0,0,5,0,0,8,0,0,9,0,0,2,0,3],[40,84,0.4762,0.52677,0.25111,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,8,0,0,7,0,0,2,0,0,8,0,0,3,0,2],[44,84,0.5238,0.48212,0.24678,0.28571,0.571,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,9,0,0,3,0,0,7,0,0,8,0,0,1,0,1],[48,84,0.5714,0.63838,0.27196,0.42857,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,3,0,0,4,0,0,5,0,0,9,0,0,3,0,6],[52,84,0.619,0.51339,0.23381,0.28571,0.57143,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,9,0,0,4,0,0,5,0,0,9,0,0,2,0,1],[56,84,0.6667,0.46429,0.24223,0.28571,0.42859,0.60714,0.0,1.0,2,2,0,2,0,1,0,0,10,0,0,5,0,0,6,0,0,6,0,0,0,0,2],[60,84,0.7143,0.49552,0.24218,0.28571,0.4998,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,12,0,0,2,0,0,4,0,0,9,0,0,2,0,1],[64,84,0.7619,0.53568,0.24743,0.28571,0.57121,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,10,0,0,3,0,0,6,0,0,8,0,0,1,0,3],[68,84,0.8095,0.38393,0.25111,0.28571,0.28571,0.57143,0.0,0.85714,6,0,0,6,0,0,0,0,11,0,0,4,0,0,4,0,0,6,0,0,1,0,0],[72,84,0.8571,0.37945,0.24119,0.28571,0.28571,0.57111,0.0,1.0,4,1,0,4,0,0,0,0,17,0,0,2,0,0,2,0,0,6,0,0,0,0,1],[76,84,0.9048,0.42854,0.28792,0.28571,0.28571,0.60714,0.0,1.0,5,3,0,5,0,0,0,0,12,0,0,2,0,0,5,0,0,5,0,0,0,0,3],[80,84,0.9524,0.26339,0.26027,0.0,0.28571,0.32143,0.0,0.85714,12,0,0,12,0,1,0,0,11,0,0,2,0,0,1,0,0,4,0,0,1,0,0],[84,84,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4d01da2531a8906c","q":"Let $n$ be a positive integer. Let $P_{n}=\\left\\{2^{n}, 2^{n-1} \\cdot 3,2^{n-2} \\cdot 3^{2}, \\ldots, 3^{n}\\right\\}$. For each subset $X$ of $P_{n}$, we write $S_{X}$ for the sum of all elements of $X$, with the convention that $S_{\\emptyset}=0$ where $\\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \\leq y \\leq 3^{n+1}-2^{n+1}$. Prove that there is a subset $Y$ of $P_{n}$ such that $0 \\leq y-S_{Y}<2^{n}$","t":[{"b":3,"e":0.28571,"k":"falling","v":0.28571,"x":0.73214,"p":[[0,19,0.0,0.69195,0.30328,0.5354,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,4,0,0,1,0,0,5,0,0,6,0,0,0,0,13],[4,19,0.2105,0.73214,0.2519,0.53571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,4,0,0,3,0,12],[8,19,0.4211,0.46426,0.30092,0.25,0.42857,0.60714,0.0,1.0,3,5,0,3,0,5,0,0,2,0,0,11,0,0,3,0,0,3,0,0,0,0,5],[12,19,0.6316,0.40625,0.18249,0.42857,0.42857,0.42857,0.0,1.0,2,1,0,2,0,3,0,0,2,0,0,19,0,0,5,0,0,0,0,0,0,0,1],[16,19,0.8421,0.34821,0.17835,0.14286,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,7,0,0,3,0,0,17,0,0,2,0,0,0,0,0,1,0,0],[19,19,1.0,0.28571,0.15568,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,13,0,0,6,0,0,9,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"falling","v":0.37054,"x":0.64286,"p":[[0,32,0.0,0.57137,0.24223,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,2,0,0,5,0,0,8,0,0,8,0,0,2,0,3],[4,32,0.125,0.62946,0.2547,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,11,0,0,6,0,0,3,0,0,3,0,7],[8,32,0.25,0.54018,0.27371,0.39286,0.42857,0.75,0.14286,1.0,0,5,0,0,0,4,0,0,4,0,0,9,0,0,6,0,0,1,0,0,3,0,5],[12,32,0.375,0.64286,0.2369,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,9,0,0,11,0,0,2,0,0,1,0,8],[16,32,0.5,0.46427,0.1821,0.42857,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,23,0,0,2,0,0,2,0,0,0,0,2],[20,32,0.625,0.45089,0.17536,0.42857,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,19,0,0,5,0,0,0,0,0,0,0,2],[24,32,0.75,0.41072,0.16269,0.42857,0.42857,0.42857,0.0,1.0,2,1,0,2,0,1,0,0,2,0,0,24,0,0,2,0,0,0,0,0,0,0,1],[28,32,0.875,0.41964,0.1234,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,25,0,0,2,0,0,0,0,0,1,0,0],[32,32,1.0,0.37054,0.12299,0.42857,0.42857,0.42857,0.0,0.42857,2,0,0,2,0,2,0,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6597d20197c4ea45","q":"Let $n>0$ be an integer, and $a, b, c$ be strictly positive integers such that\n\n$$\n(a+b c)(b+a c)=19^{n}\n$$\n\nProve that $n$ is even.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.25,"x":0.44186,"p":[[0,30,0.0,0.44186,0.20329,0.28571,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,6,0,0,5,0,0,8,0,0,7,0,0,5,0,0,1,0,0],[4,30,0.1333,0.42408,0.19717,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,7,0,0,7,0,0,6,0,0,6,0,0,0,0,0],[8,30,0.2667,0.36597,0.24475,0.14286,0.28571,0.4642,0.14,0.85714,0,0,0,0,0,13,0,0,5,0,0,6,0,0,3,0,0,1,0,0,4,0,0],[12,30,0.4,0.35266,0.19878,0.14286,0.28571,0.57111,0.0,0.71429,1,0,0,1,0,9,0,0,8,0,0,5,0,0,6,0,0,3,0,0,0,0,0],[16,30,0.5333,0.375,0.24157,0.14286,0.28571,0.46429,0.0,0.85714,2,0,0,2,0,7,0,0,9,0,0,6,0,0,2,0,0,3,0,0,3,0,0],[20,30,0.6667,0.37054,0.20782,0.14286,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,10,0,0,7,0,0,6,0,0,5,0,0,3,0,0,1,0,0],[24,30,0.8,0.27232,0.19352,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,11,0,0,8,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[28,30,0.9333,0.30357,0.19148,0.14286,0.2857,0.42857,0.14286,0.857,0,0,0,0,0,14,0,0,9,0,0,3,0,0,4,0,0,1,0,0,1,0,0],[30,30,1.0,0.25,0.20516,0.14286,0.21429,0.28571,0.0,0.85714,5,0,0,5,0,11,0,0,10,0,0,3,0,0,0,0,0,2,0,0,1,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.23661,"x":0.50445,"p":[[0,85,0.0,0.35268,0.14279,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,8,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[4,85,0.0471,0.50445,0.2448,0.28571,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,3,0,0,6,0,0,8,0,0,4,0,0,4,0,1],[8,85,0.0941,0.42848,0.20216,0.28571,0.42857,0.46431,0.14,0.85714,0,0,0,0,0,5,0,0,6,0,0,13,0,0,3,0,0,2,0,0,3,0,0],[12,85,0.1412,0.39731,0.18465,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,9,0,0,8,0,0,8,0,0,1,0,0,1,0,0],[16,85,0.1882,0.45533,0.2065,0.28571,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,5,0,0,10,0,0,5,0,0,1,0,0],[20,85,0.2353,0.44196,0.26332,0.24999,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,8,0,0,6,0,0,7,0,0,3,0,0,5,0,0,0,0,3],[24,85,0.2824,0.48661,0.22263,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,5,0,0,10,0,0,4,0,0,6,0,0,2,0,1],[28,85,0.3294,0.42856,0.22303,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,9,0,0,7,0,0,5,0,0,3,0,0,3,0,0],[32,85,0.3765,0.49106,0.22285,0.28571,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,3,0,0,7,0,0,8,0,0,6,0,0,4,0,0,3,0,1],[36,85,0.4235,0.39285,0.27663,0.14286,0.28571,0.57143,0.0,1.0,2,2,0,2,0,9,0,0,7,0,0,3,0,0,4,0,0,4,0,0,1,0,2],[40,85,0.4706,0.48214,0.23351,0.2857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,6,0,0,3,0,0,10,0,0,2,0,0],[44,85,0.5176,0.45089,0.19269,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,5,0,0,8,0,0,9,0,0,4,0,0,1,0,0],[48,85,0.5647,0.38391,0.20651,0.14286,0.42857,0.46418,0.0,0.71429,1,0,0,1,0,9,0,0,2,0,0,12,0,0,3,0,0,5,0,0,0,0,0],[52,85,0.6118,0.42409,0.22723,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,9,0,0,5,0,0,5,0,0,5,0,0,2,0,0],[56,85,0.6588,0.46429,0.18558,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,13,0,0,5,0,0,4,0,0,2,0,0],[60,85,0.7059,0.42854,0.17126,0.28571,0.42857,0.57141,0.0,0.85714,1,0,0,1,0,2,0,0,7,0,0,11,0,0,9,0,0,1,0,0,1,0,0],[64,85,0.7529,0.37498,0.21352,0.14286,0.28571,0.57111,0.14286,0.85714,0,0,0,0,0,9,0,0,9,0,0,5,0,0,5,0,0,2,0,0,2,0,0],[68,85,0.8,0.33025,0.20965,0.14286,0.28571,0.42858,0.0,0.85714,1,0,0,1,0,12,0,0,6,0,0,6,0,0,4,0,0,2,0,0,1,0,0],[72,85,0.8471,0.42857,0.20825,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,6,0,0,8,0,0,5,0,0,7,0,0,0,0,0],[76,85,0.8941,0.37053,0.21086,0.14286,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,8,0,0,6,0,0,5,0,0,3,0,0,1,0,0],[80,85,0.9412,0.35714,0.19885,0.14286,0.28571,0.46431,0.14286,0.71429,0,0,0,0,0,11,0,0,6,0,0,7,0,0,4,0,0,4,0,0,0,0,0],[84,85,0.9882,0.24098,0.12086,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,17,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[85,85,1.0,0.23661,0.13175,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,17,0,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"131b86abdcbd5eeb","q":"Let $n>1$ be a given integer. Prove that infinitely many terms of the sequence $\\left(a_{k}\\right)_{k \\geqslant 1}$, defined by  $$ a_{k}=\\left\\lfloor\\frac{n^{k}}{k}\\right\\rfloor $$  are odd. (For a real number $x,\\lfloor x\\rfloor$ denotes the largest integer not exceeding $x$.) (Hong Kong)","t":[{"b":0,"e":0.0,"k":"falling","v":0.00446,"x":0.20089,"p":[[0,51,0.0,0.20089,0.29203,0.0,0.14286,0.14286,0.0,1.0,11,3,0,11,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[4,51,0.0784,0.14723,0.1838,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[8,51,0.1569,0.17411,0.24415,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,16,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[12,51,0.2353,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,51,0.3137,0.10259,0.08167,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,51,0.3922,0.12946,0.20316,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,17,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[24,51,0.4706,0.12054,0.18595,0.0,0.14286,0.14286,0.0,0.85714,14,0,0,14,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[28,51,0.549,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,51,0.6275,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,51,0.7843,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.16945,"p":[[0,68,0.0,0.14732,0.20666,0.0,0.14286,0.14286,0.0,1.0,11,1,3,11,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,68,0.0588,0.14732,0.16164,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,68,0.1176,0.16945,0.23538,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,17,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[12,68,0.1765,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.11161,0.17762,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,68,0.2941,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.0892,0.11706,0.0,0.07,0.14286,0.0,0.57143,16,0,0,16,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,68,0.4118,0.09822,0.17655,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,68,0.4706,0.12946,0.17261,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,68,0.5294,0.09374,0.12677,0.0,0.07143,0.14286,0.0,0.571,16,0,0,16,0,14,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,68,0.5882,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a79ed83c94599c1a","q":"Let $x$ be a real number such that $0<x<\\frac{\\pi}{2}$. Prove that\n\n$$\n\\cos ^{2}(x) \\cot (x)+\\sin ^{2}(x) \\tan (x) \\geq 1 \\text {. }\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.88393,"x":0.98214,"p":[[0,19,0.0,0.92411,0.1988,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,27],[4,19,0.2105,0.88393,0.25364,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,26],[8,19,0.4211,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[12,19,0.6316,0.91071,0.24157,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[16,19,0.8421,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[19,19,1.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]},{"b":3,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,20,0.0,0.94196,0.18851,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[4,20,0.2,0.86161,0.29339,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[8,20,0.4,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[12,20,0.6,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[16,20,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"092917cb78064966","q":"Let $n$ be an odd natural number such that the numbers $\\varphi(n)$ and $\\varphi(n+1)$ are powers of two ( $\\varphi(n)$ is the number of natural numbers not greater than $n$ and coprime with $n$ ). Prove that $n+1$ is a power of two or $n=5$.\n\n(Marco Radovanovi\u0107)","t":[{"b":4,"e":0.28571,"k":"flat","v":0.29911,"x":0.48213,"p":[[0,54,0.0,0.29911,0.09006,0.28571,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,54,0.0741,0.45533,0.1403,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,12,0,0,7,0,0,4,0,0,0,0,0],[8,54,0.1481,0.41069,0.12749,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,12,0,0,5,0,0,2,0,0,0,0,0],[12,54,0.2222,0.47322,0.2034,0.28571,0.42857,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,15,0,0,4,0,0,3,0,0,8,0,0,2,0,0],[16,54,0.2963,0.44643,0.16656,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,11,0,0,3,0,0,5,0,0,1,0,0],[20,54,0.3704,0.43304,0.13115,0.28571,0.42857,0.46431,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,14,0,0,5,0,0,3,0,0,0,0,0],[24,54,0.4444,0.41514,0.16111,0.28571,0.35714,0.571,0.14286,0.71429,0,0,0,0,0,1,0,0,15,0,0,6,0,0,6,0,0,4,0,0,0,0,0],[28,54,0.5185,0.41515,0.10922,0.28571,0.42857,0.4642,0.2857,0.57143,0,0,0,0,0,0,0,0,11,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[32,54,0.5926,0.4107,0.15871,0.28571,0.28571,0.4642,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,7,0,0,3,0,0,5,0,0,0,0,0],[36,54,0.6667,0.45087,0.17167,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,8,0,0,5,0,0,5,0,0,1,0,0],[40,54,0.7407,0.4418,0.16864,0.28571,0.42857,0.57143,0.0,0.85714,1,0,1,1,0,0,0,0,10,0,0,9,0,0,9,0,0,2,0,0,1,0,0],[44,54,0.8148,0.48213,0.15046,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,15,0,0,5,0,0,5,0,0,1,0,0],[48,54,0.8889,0.42855,0.14723,0.28571,0.42857,0.571,0.0,0.85714,1,0,0,1,0,0,0,0,8,0,0,14,0,0,8,0,0,0,0,0,1,0,0],[52,54,0.963,0.43749,0.14257,0.28571,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,12,0,0,5,0,0,4,0,0,0,0,0],[54,54,1.0,0.40179,0.12596,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,12,0,0,4,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.29911,"x":0.4241,"p":[[0,15,0.0,0.29911,0.07457,0.28571,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,0,0,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.4241,0.13113,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,14,0,0,4,0,0,3,0,0,0,0,0],[8,15,0.5333,0.40625,0.12931,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,11,0,0,5,0,0,2,0,0,0,0,0],[12,15,0.8,0.39735,0.11142,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,14,0,0,4,0,0,1,0,0,0,0,0],[15,15,1.0,0.31699,0.15459,0.2857,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,0,0,0,21,0,0,5,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"89163326e255281d","q":"Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.","t":[{"b":1,"e":0.0,"k":"flat","v":0.125,"x":0.25446,"p":[[0,13,0.0,0.22768,0.18509,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,0,0,0,19,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[4,13,0.3077,0.25446,0.22794,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,0,0,0,14,0,0,3,0,0,0,0,0,4,0,0,0,0,0],[8,13,0.6154,0.24553,0.18638,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,0,0,0,18,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[12,13,0.9231,0.125,0.17768,0.0,0.0,0.28571,0.0,0.71429,20,0,0,20,0,0,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[13,13,1.0,0.17411,0.20119,0.0,0.14285,0.28571,0.0,0.71429,16,0,0,16,0,0,0,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.23662,"x":0.30803,"p":[[0,18,0.0,0.25,0.19233,0.10714,0.28571,0.28571,0.0,0.71429,8,0,0,8,0,1,0,0,20,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[4,18,0.2222,0.23662,0.18766,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,2,0,0,16,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[8,18,0.4444,0.25447,0.21939,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,0,0,0,17,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[12,18,0.6667,0.27232,0.17985,0.2857,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,0,0,0,23,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[16,18,0.8889,0.30358,0.1915,0.2857,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,0,0,0,21,0,0,2,0,0,0,0,0,4,0,0,0,0,0],[18,18,1.0,0.30803,0.14773,0.28571,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,0,0,0,27,0,0,0,0,0,0,0,0,3,0,0,0,0,0]]}]},{"i":"7ea7a9922bd78a1b","q":"Let $p>5$ be a prime number. Show that there exists a prime number $q<p$ and a positive integer $n$ such that $p$ divides $n^{2}-q$.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.14277,"x":0.52232,"p":[[0,48,0.0,0.48214,0.23891,0.42857,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,6,0,0,0,0,0,16,0,0,2,0,0,5,0,0,0,0,3],[4,48,0.0833,0.52232,0.41282,0.14286,0.28574,1.0,0.0,1.0,1,13,0,1,0,15,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,13],[8,48,0.1667,0.26786,0.23623,0.14286,0.14286,0.28571,0.14286,1.0,0,2,0,0,0,22,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[12,48,0.25,0.36607,0.35524,0.14286,0.14286,0.5,0.14286,1.0,0,7,0,0,0,22,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,7],[16,48,0.3333,0.24099,0.25367,0.14286,0.14286,0.14286,0.14,1.0,0,3,0,0,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[20,48,0.4167,0.2009,0.20782,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,48,0.5,0.17411,0.08552,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.20527,0.21113,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[32,48,0.6667,0.17857,0.15567,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,48,0.75,0.1875,0.21261,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,48,0.8333,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,48,0.9167,0.17857,0.15567,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[48,48,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"falling","v":0.1383,"x":0.57134,"p":[[0,51,0.0,0.57134,0.25769,0.42857,0.42857,0.71429,0.14,1.0,0,7,0,0,0,2,0,0,0,0,0,19,0,0,0,0,0,4,0,0,0,0,7],[4,51,0.0784,0.28125,0.28456,0.14286,0.14286,0.1786,0.14286,1.0,0,4,0,0,0,24,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[8,51,0.1569,0.28561,0.27897,0.14286,0.14286,0.21429,0.14,1.0,0,3,0,0,0,24,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,3],[12,51,0.2353,0.17857,0.09449,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,51,0.3137,0.24554,0.25313,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[20,51,0.3922,0.3125,0.23266,0.14286,0.14286,0.42858,0.0,1.0,1,1,0,1,0,17,0,0,0,0,0,9,0,0,1,0,0,3,0,0,0,0,1],[24,51,0.4706,0.21875,0.1988,0.14286,0.14286,0.14287,0.0,1.0,2,1,0,2,0,23,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[28,51,0.549,0.20982,0.19556,0.14286,0.14286,0.14287,0.0,1.0,2,1,0,2,0,24,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[32,51,0.6275,0.24554,0.16841,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,22,0,0,1,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[36,51,0.7059,0.20973,0.17855,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[40,51,0.7843,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.16063,0.06918,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f9eed2359fc29711","q":"Let $x$, $y$ and $z$ be strictly positive real numbers such that\n\n$$\nx+y+z \\geqslant \\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}\n$$\n\nShow that\n\n$$\n\\frac{x}{y}+\\frac{y}{z}+\\frac{z}{x} \\geqslant \\frac{1}{x y}+\\frac{1}{y z}+\\frac{1}{z x}\n$$","t":[{"b":6,"e":0.28571,"k":"rising","v":0.5,"x":0.85714,"p":[[0,58,0.0,0.5,0.37457,0.14286,0.28571,0.85714,0.0,1.0,3,7,0,3,0,7,0,0,7,0,0,2,0,0,0,0,0,0,0,0,6,0,7],[4,58,0.069,0.70982,0.33973,0.28571,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0,5,0,15],[8,58,0.1379,0.78571,0.29881,0.85711,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,12,0,13],[12,58,0.2069,0.83482,0.28596,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0,4,0,21],[16,58,0.2759,0.80804,0.29797,0.85714,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,9,0,16],[20,58,0.3448,0.78125,0.31539,0.53571,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0,5,0,18],[24,58,0.4138,0.78571,0.27664,0.53571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,4,0,0,1,0,0,0,0,0,8,0,15],[28,58,0.4828,0.75893,0.30397,0.53571,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,4,0,0,2,0,0,1,0,0,0,0,0,10,0,13],[32,58,0.5517,0.75893,0.32623,0.39286,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0,7,0,16],[36,58,0.6207,0.85714,0.2342,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,9,0,18],[40,58,0.6897,0.72768,0.32803,0.42857,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,4,0,0,4,0,0,2,0,0,0,0,0,5,0,15],[44,58,0.7586,0.74552,0.33643,0.57132,0.85714,1.0,0.0,1.0,2,15,0,2,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,15],[48,58,0.8276,0.73213,0.31289,0.53539,0.85714,1.0,0.0,1.0,1,11,0,1,0,3,0,0,2,0,0,2,0,0,1,0,0,2,0,0,10,0,11],[52,58,0.8966,0.70534,0.2788,0.39286,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,2,0,0,1,0,0,1,0,0,13,0,7],[56,58,0.9655,0.65625,0.32705,0.28571,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,1,0,0,9,0,9],[58,58,1.0,0.67857,0.29451,0.28571,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,9,0,0,0,0,0,0,0,0,2,0,0,16,0,4]]},{"b":7,"e":0.14286,"k":"falling","v":0.16518,"x":0.91518,"p":[[0,111,0.0,0.61607,0.39999,0.14286,0.85714,1.0,0.0,1.0,2,13,0,2,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,5,0,13],[4,111,0.036,0.80357,0.26426,0.82143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,1,0,0,9,0,15],[8,111,0.0721,0.77232,0.27862,0.5714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,6,0,0,1,0,0,3,0,0,1,0,0,6,0,15],[12,111,0.1081,0.83036,0.23266,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,14,0,13],[16,111,0.1441,0.75446,0.29931,0.42859,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,4,0,0,3,0,0,0,0,0,2,0,0,7,0,14],[20,111,0.1802,0.79464,0.30501,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,19],[24,111,0.2162,0.75446,0.3199,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,0,10,0,13],[28,111,0.2523,0.91518,0.1984,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,22],[32,111,0.2883,0.68304,0.38752,0.24999,0.85714,1.0,0.0,1.0,4,13,0,4,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,8,0,13],[36,111,0.3243,0.75891,0.35073,0.57132,1.0,1.0,0.0,1.0,3,17,0,3,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,6,0,17],[40,111,0.3604,0.81696,0.29501,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,18],[44,111,0.3964,0.6116,0.37155,0.28571,0.64284,1.0,0.0,1.0,3,12,0,3,0,4,0,0,2,0,0,6,0,0,1,0,0,1,0,0,3,0,12],[48,111,0.4324,0.71875,0.36331,0.28571,1.0,1.0,0.0,1.0,2,17,0,2,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,4,0,17],[52,111,0.4685,0.63839,0.36593,0.28571,0.85707,1.0,0.0,1.0,2,11,0,2,0,5,0,0,3,0,0,3,0,0,0,0,0,2,0,0,6,0,11],[56,111,0.5045,0.80357,0.30462,0.85711,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0,8,0,17],[60,111,0.5405,0.5625,0.42848,0.14286,0.78564,1.0,0.0,1.0,4,13,0,4,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,13],[64,111,0.5766,0.67857,0.3607,0.39286,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,3,0,0,6,0,0,0,0,0,0,0,0,2,0,16],[68,111,0.6126,0.67411,0.35934,0.28571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,6,0,0,2,0,0,0,0,0,1,0,0,3,0,15],[72,111,0.6486,0.69643,0.36553,0.28571,0.85714,1.0,0.0,1.0,1,14,0,1,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,7,0,14],[76,111,0.6847,0.58036,0.42549,0.10714,0.78571,1.0,0.0,1.0,8,12,0,8,0,3,0,0,0,0,0,3,0,0,0,0,0,2,0,0,4,0,12],[80,111,0.7207,0.70973,0.36345,0.28571,0.85714,1.0,0.0,1.0,1,15,0,1,0,5,0,0,4,0,0,0,0,0,0,0,0,1,0,0,6,0,15],[84,111,0.7568,0.65179,0.42097,0.14286,1.0,1.0,0.0,1.0,5,17,0,5,0,5,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,17],[88,111,0.7928,0.65625,0.36572,0.28571,0.85714,1.0,0.0,1.0,1,12,0,1,0,6,0,0,4,0,0,1,0,0,0,0,0,2,0,0,6,0,12],[92,111,0.8288,0.46427,0.38959,0.14286,0.28571,0.85714,0.0,1.0,6,7,0,6,0,7,0,0,4,0,0,1,0,0,2,0,0,1,0,0,4,0,7],[96,111,0.8649,0.37946,0.39867,0.14286,0.14286,0.89286,0.0,1.0,6,8,0,6,0,14,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,8],[100,111,0.9009,0.35259,0.38633,0.14286,0.14286,0.75,0.0,1.0,6,7,0,6,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,7],[104,111,0.9369,0.21427,0.31742,0.0,0.14286,0.14286,0.0,1.0,12,4,0,12,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[108,111,0.973,0.24554,0.33548,0.0,0.14286,0.1786,0.0,1.0,10,5,0,10,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[111,111,1.0,0.16518,0.22619,0.10714,0.14286,0.14286,0.0,1.0,8,2,0,8,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"5e169edfbf0f810c","q":"Let \\( a, b, c > 0 \\). Show that\n\n\\[\n\\sqrt{\\frac{a}{b+c}} + \\sqrt{\\frac{b}{c+a}} + \\sqrt{\\frac{c}{a+b}} > 2.\n\\]","t":[{"b":5,"e":0.857,"k":"flat","v":0.75446,"x":0.98214,"p":[[0,26,0.0,0.87946,0.31966,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,26,0.1538,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,26,0.3077,0.91964,0.22286,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[12,26,0.4615,0.91964,0.22286,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[16,26,0.6154,0.91058,0.11732,0.85714,1.0,1.0,0.71,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,19],[20,26,0.7692,0.84821,0.10062,0.71429,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,16,0,7],[24,26,0.9231,0.75446,0.22654,0.71429,0.85714,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,13,0,4],[26,26,1.0,0.78571,0.23146,0.71429,0.85714,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,13,0,7]]},{"b":7,"e":0.71429,"k":"flat","v":0.76784,"x":0.95536,"p":[[0,24,0.0,0.88393,0.3102,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,24,0.1667,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,24,0.3333,0.91964,0.23673,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[12,24,0.5,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[16,24,0.6667,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[20,24,0.8333,0.78125,0.11837,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,0,12,0,3],[24,24,1.0,0.76784,0.09282,0.71429,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,9,0,2]]}]},{"i":"1f4ca01d14ddbbb4","q":"Let Akbar and Birbal together have  $n$  marbles, where  $n > 0$ .\nAkbar says to Birbal, \u201c If I give you some marbles then you will have twice as many marbles as I will have.\u201d \nBirbal says to Akbar, \u201c If I give you some marbles then you will have thrice as many marbles as I will have.\u201d\nWhat is the minimum possible value of  $n$  for which the above statements are true?","t":[{"b":3,"e":0.42857,"k":"flat","v":0.69643,"x":0.91071,"p":[[0,91,0.0,0.90179,0.20958,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,26],[4,91,0.044,0.73214,0.28515,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,17],[8,91,0.0879,0.78572,0.27664,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,20],[12,91,0.1319,0.71429,0.28571,0.42857,0.7143,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16],[16,91,0.1758,0.71433,0.27659,0.42857,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,2,0,0,0,0,15],[20,91,0.2198,0.75893,0.27765,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,0,0,0,0,18],[24,91,0.2637,0.78125,0.27429,0.42857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,19],[28,91,0.3077,0.80357,0.26184,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,0,0,20],[32,91,0.3516,0.74107,0.27993,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,1,0,0,0,0,17],[36,91,0.3956,0.75893,0.27765,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,0,0,0,0,18],[40,91,0.4396,0.77679,0.26471,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,3,0,0,0,0,18],[44,91,0.4835,0.76786,0.2714,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,0,0,18],[48,91,0.5275,0.69643,0.27606,0.42857,0.57144,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,2,0,0,0,0,14],[52,91,0.5714,0.72322,0.28107,0.42857,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,16],[56,91,0.6154,0.73659,0.27226,0.42857,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,2,0,0,0,0,16],[60,91,0.6593,0.80357,0.2714,0.42859,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,21],[64,91,0.7033,0.91071,0.20748,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,27],[68,91,0.7473,0.85714,0.24743,0.85715,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,24],[72,91,0.7912,0.71429,0.28571,0.42857,0.7143,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16],[76,91,0.8352,0.76786,0.27141,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,0,0,18],[80,91,0.8791,0.83036,0.24598,0.64286,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,3,0,0,0,0,21],[84,91,0.9231,0.76786,0.28064,0.42857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,19],[88,91,0.967,0.80804,0.26149,0.4286,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,1,0,0,1,0,20],[91,91,1.0,0.79464,0.26949,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,20]]},{"b":4,"e":0.42857,"k":"falling","v":0.44197,"x":0.90179,"p":[[0,39,0.0,0.90179,0.2299,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,27],[4,39,0.1026,0.69643,0.28516,0.42857,0.42857,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,15],[8,39,0.2051,0.64286,0.27664,0.42857,0.42857,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,12],[12,39,0.3077,0.50893,0.19212,0.42857,0.42857,0.42857,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,4],[16,39,0.4103,0.51786,0.1948,0.42857,0.42857,0.42858,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,2,0,0,0,0,4],[20,39,0.5128,0.5,0.17496,0.42857,0.42857,0.4286,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,2,0,0,0,0,3],[24,39,0.6154,0.51786,0.20748,0.42857,0.42857,0.42858,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,5],[28,39,0.7179,0.49107,0.15542,0.42857,0.42857,0.42858,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,3,0,0,0,0,2],[32,39,0.8205,0.46429,0.13832,0.42857,0.42857,0.42857,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,2],[36,39,0.9231,0.46429,0.13832,0.42857,0.42857,0.42857,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,2],[39,39,1.0,0.44197,0.10326,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"ecb155ea9e27ed53","q":"Let $x, y$ and $z$ be strictly positive real numbers such that $x y + y z + z x = 3$.\nProve that\n\n$$\n\\frac{x+3}{y+z}+\\frac{y+3}{z+x}+\\frac{z+3}{x+y}+3 \\geqslant 27 \\frac{(\\sqrt{x}+\\sqrt{y}+\\sqrt{z})^{2}}{(x+y+z)^{3}}\n$$","t":[{"b":0,"e":0.571,"k":"flat","v":0.70978,"x":0.90625,"p":[[0,98,0.0,0.70978,0.26364,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,8,0,0,1,0,11],[4,98,0.0408,0.85714,0.16752,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,5,0,16],[8,98,0.0816,0.87053,0.18336,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,4,0,19],[12,98,0.1224,0.83927,0.2044,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,0,4,0,17],[16,98,0.1633,0.89731,0.15253,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,20],[20,98,0.2041,0.87499,0.15465,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,17],[24,98,0.2449,0.79018,0.23141,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,3,0,0,5,0,14],[28,98,0.2857,0.89284,0.1557,0.82143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,20],[32,98,0.3265,0.90625,0.18766,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,24],[36,98,0.3673,0.89286,0.16366,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,0,0,22],[40,98,0.4082,0.88392,0.1692,0.82143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,4,0,20],[44,98,0.449,0.77675,0.17839,0.57143,0.71429,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,4,0,10],[48,98,0.4898,0.81249,0.18015,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,5,0,13],[52,98,0.5306,0.79461,0.16345,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,5,0,10],[56,98,0.5714,0.72768,0.20628,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,6,0,0,5,0,8],[60,98,0.6122,0.78125,0.18893,0.67857,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,7,0,0,8,0,9],[64,98,0.6531,0.7723,0.21087,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,7,0,0,3,0,12],[68,98,0.6939,0.82143,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,8,0,11],[72,98,0.7347,0.76786,0.21943,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,8,0,0,4,0,11],[76,98,0.7755,0.71874,0.25377,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,0,0,0,7,0,0,8,0,0,7,0,7],[80,98,0.8163,0.73214,0.20748,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,11,0,0,3,0,8],[84,98,0.8571,0.79462,0.23403,0.71429,0.857,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,3,0,14],[88,98,0.898,0.76338,0.2276,0.71429,0.78571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,7,0,9],[92,98,0.9388,0.79909,0.21388,0.71429,0.85707,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,7,0,11],[96,98,0.9796,0.79909,0.15095,0.71429,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,6,0,9],[98,98,1.0,0.79017,0.12363,0.71429,0.78564,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,12,0,4]]},{"b":7,"e":0.57143,"k":"flat","v":0.64284,"x":0.85266,"p":[[0,160,0.0,0.66068,0.29179,0.57132,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,3,0,0,0,0,0,8,0,0,8,0,0,1,0,9],[4,160,0.025,0.80801,0.18769,0.67857,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,2,0,14],[8,160,0.05,0.81249,0.18015,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,4,0,13],[12,160,0.075,0.83487,0.17158,0.71429,0.85714,1.0,0.43,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,5,0,14],[16,160,0.1,0.82587,0.1881,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,5,0,14],[20,160,0.125,0.83929,0.20438,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,1,0,18],[24,160,0.15,0.85266,0.18726,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,6,0,16],[28,160,0.175,0.81695,0.19962,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,2,0,16],[32,160,0.2,0.77679,0.26471,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,2,0,0,6,0,0,3,0,0,5,0,14],[36,160,0.225,0.74553,0.19144,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,6,0,0,4,0,9],[40,160,0.25,0.79911,0.20159,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,0,3,0,14],[44,160,0.275,0.75892,0.2126,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,5,0,0,2,0,12],[48,160,0.3,0.75889,0.22713,0.67857,0.857,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,9,0,8],[52,160,0.325,0.75444,0.23485,0.57143,0.78571,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,11,0,0,3,0,0,5,0,11],[56,160,0.35,0.76784,0.23892,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0,7,0,11],[60,160,0.375,0.78571,0.26487,0.57143,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,5,0,14],[64,160,0.4,0.7232,0.23942,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,0,0,0,9,0,0,6,0,0,6,0,8],[68,160,0.425,0.72753,0.31616,0.57143,0.78564,1.0,0.0,1.0,4,12,0,4,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,4,0,12],[72,160,0.45,0.66951,0.2214,0.57143,0.71214,0.75,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,1,0,0,12,0,0,9,0,0,3,0,5],[76,160,0.475,0.71428,0.12371,0.57143,0.71429,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,8,0,0,12,0,0],[80,160,0.5,0.70087,0.15305,0.57143,0.64286,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,0,5,0,4],[84,160,0.525,0.66518,0.16982,0.57143,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,14,0,0,10,0,0,3,0,3],[88,160,0.55,0.70529,0.18882,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,3,0,0,2,0,8],[92,160,0.575,0.68304,0.15458,0.57143,0.71429,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,10,0,0,6,0,2],[96,160,0.6,0.64284,0.10715,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,0,2,0,1],[100,160,0.625,0.74981,0.14735,0.57143,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,7,0,5],[104,160,0.65,0.71872,0.15767,0.57143,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,5,0,5],[108,160,0.675,0.69192,0.11908,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,18,0,0,2,0,2],[112,160,0.7,0.72321,0.13333,0.57143,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,16,0,0,3,0,4],[116,160,0.725,0.71873,0.17308,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,12,0,0,6,0,4],[120,160,0.75,0.6875,0.1729,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,14,0,0,2,0,4],[124,160,0.775,0.67407,0.12996,0.57143,0.57143,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,9,0,0,4,0,2],[128,160,0.8,0.75445,0.12994,0.71429,0.71429,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,15,0,0,7,0,4],[132,160,0.825,0.70979,0.11002,0.71421,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,19,0,0,5,0,1],[136,160,0.85,0.73661,0.14334,0.67857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,14,0,0,6,0,4],[140,160,0.875,0.74552,0.13237,0.71429,0.71429,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,15,0,0,6,0,4],[144,160,0.9,0.75893,0.16917,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,16,0,0,4,0,7],[148,160,0.925,0.75446,0.12492,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,0,6,0,4],[152,160,0.95,0.75893,0.12078,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,0,6,0,4],[156,160,0.975,0.77676,0.13804,0.71429,0.71429,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,6,0,6],[160,160,1.0,0.79907,0.133,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,5,0,8]]}]},{"i":"9c1c55437ba412d1","q":"Let $n \\geqslant 3$ be an integer, show that there exist two integers $x$ and $y$ such that $7 x^{2}+y^{2}=2^{n}$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.17411,"x":0.39732,"p":[[0,40,0.0,0.35714,0.36246,0.0,0.14286,0.71429,0.0,1.0,10,5,0,10,0,7,0,0,0,0,0,6,0,0,0,0,0,4,0,0,0,0,5],[4,40,0.1,0.18304,0.27254,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,12,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[8,40,0.2,0.20982,0.3174,0.0,0.14286,0.14286,0.0,1.0,12,4,0,12,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[12,40,0.3,0.31688,0.39731,0.0,0.14286,0.71429,0.0,1.0,14,6,0,14,0,7,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,6],[16,40,0.4,0.27223,0.32806,0.0,0.14286,0.32143,0.0,1.0,9,3,0,9,0,14,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,3],[20,40,0.5,0.39732,0.41455,0.0,0.14286,0.89286,0.0,1.0,9,8,0,9,0,10,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,8],[24,40,0.6,0.24106,0.3203,0.0,0.14286,0.42858,0.0,1.0,14,2,0,14,0,9,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,2],[28,40,0.7,0.20982,0.32924,0.0,0.0,0.14286,0.0,1.0,17,3,0,17,0,8,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,3],[32,40,0.8,0.23215,0.28959,0.0,0.14286,0.17868,0.0,1.0,11,1,0,11,0,13,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,1],[36,40,0.9,0.20089,0.274,0.0,0.14286,0.14286,0.0,1.0,10,2,0,10,0,17,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[40,40,1.0,0.17411,0.24675,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,15,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,1]]},{"b":5,"e":0.14286,"k":"falling","v":0.12054,"x":0.48661,"p":[[0,26,0.0,0.42857,0.35355,0.14286,0.14286,0.71429,0.0,1.0,3,5,0,3,0,14,0,0,0,0,0,1,0,0,1,0,0,8,0,0,0,0,5],[4,26,0.1538,0.48661,0.41933,0.0,0.42857,1.0,0.0,1.0,9,10,0,9,0,5,0,0,0,0,0,3,0,0,0,0,0,5,0,0,0,0,10],[8,26,0.3077,0.43304,0.3838,0.14286,0.14286,0.71429,0.0,1.0,6,6,0,6,0,11,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,6],[12,26,0.4615,0.24554,0.31183,0.0,0.14286,0.32143,0.0,1.0,12,2,0,12,0,11,0,0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,2],[16,26,0.6154,0.13393,0.20806,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,14,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,26,0.7692,0.12946,0.22968,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[24,26,0.9231,0.12054,0.18249,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,12,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[26,26,1.0,0.14732,0.28004,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,2]]}]},{"i":"8baf7f0549e45f96","q":"Let be a function  $ f $  of class  $ \\mathcal{C}^1[a,b] $  whose derivative is positive. Prove that there exists a real number  $ c\\in (a,b) $  such that  $$  f(f(b))-f(f(a))=(f'(c))^2(b-a) .  $$","t":[{"b":3,"e":0.85714,"k":"rising","v":0.45088,"x":0.96875,"p":[[0,21,0.0,0.51786,0.39082,0.24999,0.28571,1.0,0.0,1.0,3,11,0,3,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0,2,0,11],[4,21,0.1905,0.45088,0.34829,0.14286,0.35714,0.857,0.0,1.0,3,7,0,3,0,7,0,0,6,0,0,7,0,0,0,0,0,0,0,0,2,0,7],[8,21,0.381,0.55357,0.3567,0.28571,0.42857,1.0,0.0,1.0,3,11,0,3,0,2,0,0,7,0,0,6,0,0,2,0,0,1,0,0,0,0,11],[12,21,0.5714,0.625,0.37244,0.28571,0.85714,1.0,0.0,1.0,1,13,0,1,0,4,0,0,9,0,0,1,0,0,0,0,0,0,0,0,4,0,13],[16,21,0.7619,0.54018,0.37582,0.14286,0.42857,1.0,0.0,1.0,1,10,0,1,0,9,0,0,5,0,0,3,0,0,0,0,0,1,0,0,3,0,10],[20,21,0.9524,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[21,21,1.0,0.91964,0.17835,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,22]]},{"b":4,"e":0.28571,"k":"rising","v":0.32143,"x":0.71427,"p":[[0,21,0.0,0.39732,0.36897,0.14286,0.28571,0.57145,0.0,1.0,6,8,1,6,0,6,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,8],[4,21,0.1905,0.45089,0.37134,0.14286,0.28571,0.89286,0.0,1.0,4,8,0,4,0,6,0,0,10,0,0,1,0,0,1,0,0,0,0,0,2,0,8],[8,21,0.381,0.32143,0.31542,0.0,0.28571,0.42857,0.0,1.0,9,4,0,9,0,2,0,0,12,0,0,4,0,0,0,0,0,0,0,0,1,0,4],[12,21,0.5714,0.58929,0.369,0.28571,0.5,1.0,0.0,1.0,2,13,0,2,0,2,0,0,11,0,0,1,0,0,1,0,0,2,0,0,0,0,13],[16,21,0.7619,0.69643,0.36553,0.28571,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,18],[20,21,0.9524,0.70535,0.36932,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,5,0,16],[21,21,1.0,0.71427,0.32733,0.28571,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,9,0,0,0,0,0,1,0,0,2,0,0,5,0,14]]}]},{"i":"e6b9e0d7bda2332d","q":"Let $n$ be a strictly positive integer. Show that there exist $n$ pairwise distinct integers $r_{1}, \\ldots, r_{n}$ such that each $r_{i}$ divides $r_{1}+\\cdots+r_{n}$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.26784,"x":0.43302,"p":[[0,24,0.0,0.28124,0.34714,0.0,0.07143,0.57143,0.0,1.0,16,4,0,16,0,1,0,0,5,0,0,0,0,0,6,0,0,0,0,0,0,0,4],[4,24,0.1667,0.36161,0.38793,0.0,0.21429,0.57143,0.0,1.0,14,6,0,14,0,2,0,0,2,0,0,0,0,0,7,0,0,1,0,0,0,0,6],[8,24,0.3333,0.43302,0.39201,0.0,0.57143,0.57143,0.0,1.0,13,7,0,13,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,7],[12,24,0.5,0.43302,0.40639,0.0,0.57143,0.67857,0.0,1.0,13,8,0,13,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,8],[16,24,0.6667,0.26784,0.34763,0.0,0.0,0.57143,0.0,1.0,17,3,0,17,0,3,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,3],[20,24,0.8333,0.35713,0.39769,0.0,0.14286,0.57143,0.0,1.0,13,7,0,13,0,5,0,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,7],[24,24,1.0,0.35712,0.36244,0.0,0.28571,0.57143,0.0,1.0,13,5,0,13,0,1,0,0,4,0,0,0,0,0,9,0,0,0,0,0,0,0,5]]},{"b":7,"e":0.57143,"k":"flat","v":0.183,"x":0.46875,"p":[[0,26,0.0,0.24552,0.33356,0.0,0.0,0.57143,0.0,1.0,18,3,0,18,0,2,0,0,2,0,0,0,0,0,7,0,0,0,0,0,0,0,3],[4,26,0.1538,0.36607,0.34615,0.0,0.42857,0.57143,0.0,1.0,12,4,0,12,0,2,0,0,2,0,0,0,0,0,12,0,0,0,0,0,0,0,4],[8,26,0.3077,0.40622,0.35911,0.0,0.57121,0.57143,0.0,1.0,12,5,0,12,0,0,0,0,2,0,0,0,0,0,13,0,0,0,0,0,0,0,5],[12,26,0.4615,0.183,0.24014,0.0,0.0,0.35704,0.0,0.57143,18,0,0,18,0,3,0,0,3,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[16,26,0.6154,0.32141,0.30928,0.0,0.28571,0.57143,0.0,1.0,13,2,0,13,0,1,0,0,3,0,0,1,0,0,12,0,0,0,0,0,0,0,2],[20,26,0.7692,0.46875,0.37836,0.0,0.57143,0.67857,0.0,1.0,9,8,0,9,0,1,0,0,4,0,0,0,0,0,10,0,0,0,0,0,0,0,8],[24,26,0.9231,0.18304,0.29501,0.0,0.0,0.2857,0.0,1.0,20,2,0,20,0,3,0,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,2],[26,26,1.0,0.19634,0.2918,0.0,0.0,0.28571,0.0,1.0,18,2,0,18,0,3,0,0,5,0,0,0,0,0,3,0,0,1,0,0,0,0,2]]}]},{"i":"b6b55e41dc30d1bc","q":"Let $n$ be a positive integer. Define a sequence by setting $a_{1}=n$ and, for each $k>1$, letting $a_{k}$ be the unique integer in the range $0 \\leq a_{k} \\leq k-1$ for which $a_{1}+a_{2}+\\cdots+a_{k}$ is divisible by $k$. (For instance, when $n=9$ the obtained sequence is $9,1,2,0,3,3,3, \\ldots$ ) Prove that for any $n$ the sequence $a_{1}, a_{2}, \\ldots$ eventually becomes constant.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.72317,"x":0.73658,"p":[[0,11,0.0,0.72317,0.19868,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,6,0,0,4,0,8],[4,11,0.3636,0.72767,0.16887,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,5,0,0,10,0,4],[8,11,0.7273,0.73658,0.17539,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,4,0,0,8,0,6],[11,11,1.0,0.73214,0.18814,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,6,0,0,4,0,8]]},{"b":6,"e":1.0,"k":"flat","v":0.59821,"x":0.70982,"p":[[0,30,0.0,0.69642,0.21651,0.57132,0.64286,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,6,0,0,1,0,9],[4,30,0.1333,0.70982,0.1838,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,9,0,0,2,0,7],[8,30,0.2667,0.66517,0.16214,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,14,0,0,2,0,3],[12,30,0.4,0.70981,0.14055,0.57143,0.71429,0.71429,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,16,0,0,2,0,4],[16,30,0.5333,0.65176,0.147,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,13,0,0,2,0,2],[20,30,0.6667,0.63838,0.15966,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,7,0,0,2,0,3],[24,30,0.8,0.69197,0.12931,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,17,0,0,3,0,2],[28,30,0.9333,0.67411,0.1394,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,14,0,0,1,0,3],[30,30,1.0,0.59821,0.14479,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,12,0,0,8,0,0,2,0,1]]}]},{"i":"66ebac107b461eca","q":"Let the incircle  $\\omega $  of  $\\triangle ABC $  touch  $AC $  and  $AB $  at points  $E $  and  $F $  respectively. Points  $X $ ,  $Y $  of  $\\omega $  are such that  $\\angle BXC=\\angle BYC=90^{\\circ} $ . Prove that  $EF $  and  $XY $  meet on the medial line of  $ABC $ .","t":[{"b":5,"e":0.14286,"k":"flat","v":0.24553,"x":0.49553,"p":[[0,56,0.0,0.34373,0.19185,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,5,0,0,19,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[4,56,0.0714,0.43302,0.22155,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,5,0,0,6,0,0,14,0,0,2,0,0,2,0,0,1,0,2],[8,56,0.1429,0.4688,0.23482,0.39286,0.42857,0.4286,0.14286,1.0,0,4,0,0,0,3,0,0,5,0,0,18,0,0,0,0,0,2,0,0,0,0,4],[12,56,0.2143,0.44643,0.20124,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,7,0,0,18,0,0,0,0,0,2,0,0,1,0,2],[16,56,0.2857,0.44643,0.22232,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,4,0,0,5,0,0,17,0,0,1,0,0,2,0,0,0,0,3],[20,56,0.3571,0.49553,0.29877,0.28571,0.42857,0.71429,0.0,1.0,1,6,0,1,0,5,0,0,4,0,0,12,0,0,1,0,0,2,0,0,1,0,6],[24,56,0.4286,0.39732,0.23886,0.24999,0.42857,0.42858,0.0,1.0,1,2,0,1,0,7,0,0,5,0,0,14,0,0,0,0,0,2,0,0,1,0,2],[28,56,0.5,0.41508,0.20945,0.28571,0.42857,0.42857,0.14,1.0,0,1,0,0,0,7,0,0,3,0,0,16,0,0,1,0,0,3,0,0,1,0,1],[32,56,0.5714,0.43755,0.22286,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,17,0,0,0,0,0,2,0,0,0,0,3],[36,56,0.6429,0.41955,0.28789,0.14286,0.42857,0.46429,0.0,1.0,1,4,0,1,0,10,0,0,2,0,0,11,0,0,2,0,0,1,0,0,1,0,4],[40,56,0.7143,0.36606,0.24983,0.14286,0.42857,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,2,0,0,11,0,0,1,0,0,3,0,0,0,0,2],[44,56,0.7857,0.24553,0.1394,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,18,0,0,7,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[48,56,0.8571,0.25438,0.18814,0.14286,0.14286,0.42857,0.0,1.0,2,1,0,2,0,16,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[52,56,0.9286,0.25443,0.16661,0.14286,0.14288,0.32164,0.0,0.71429,1,0,0,1,0,17,0,0,6,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[56,56,1.0,0.32589,0.28174,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,19,0,0,4,0,0,2,0,0,1,0,0,3,0,0,0,0,3]]},{"b":7,"e":0.14286,"k":"falling","v":0.21875,"x":0.48663,"p":[[0,65,0.0,0.37053,0.24186,0.25,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,8,0,0,12,0,0,7,0,0,1,0,0,1,0,0,0,0,3],[4,65,0.0615,0.3883,0.17228,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,4,0,0,18,0,0,0,0,0,4,0,0,0,0,0],[8,65,0.1231,0.48663,0.24963,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,2,0,0,9,0,0,12,0,0,1,0,0,3,0,0,1,0,4],[12,65,0.1846,0.4375,0.22569,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,5,0,0,5,0,0,16,0,0,1,0,0,1,0,0,2,0,2],[16,65,0.2462,0.34821,0.16341,0.28571,0.42857,0.42857,0.0,0.857,2,0,0,2,0,4,0,0,8,0,0,16,0,0,1,0,0,0,0,0,1,0,0],[20,65,0.3077,0.45982,0.30458,0.14286,0.42857,0.71429,0.0,1.0,1,5,0,1,0,8,0,0,4,0,0,9,0,0,1,0,0,3,0,0,1,0,5],[24,65,0.3692,0.43302,0.22155,0.28571,0.42857,0.4286,0.14286,1.0,0,2,0,0,0,6,0,0,4,0,0,15,0,0,1,0,0,4,0,0,0,0,2],[28,65,0.4308,0.39285,0.17495,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,4,0,0,4,0,0,20,0,0,1,0,0,1,0,0,0,0,1],[32,65,0.4923,0.42848,0.20837,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,3,0,0,6,0,0,16,0,0,0,0,0,4,0,0,1,0,1],[36,65,0.5538,0.36161,0.15966,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,4,0,0,5,0,0,19,0,0,0,0,0,2,0,0,0,0,0],[40,65,0.6154,0.41069,0.23621,0.24999,0.42857,0.4286,0.14286,1.0,0,2,0,0,0,8,0,0,5,0,0,12,0,0,2,0,0,2,0,0,1,0,2],[44,65,0.6769,0.41965,0.18877,0.42857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,4,0,0,2,0,0,20,0,0,1,0,0,3,0,0,0,0,1],[48,65,0.7385,0.36607,0.17835,0.14286,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,8,0,0,1,0,0,19,0,0,1,0,0,1,0,0,1,0,0],[52,65,0.8,0.45982,0.22794,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,3,0,0,6,0,0,17,0,0,0,0,0,2,0,0,1,0,3],[56,65,0.8615,0.3125,0.20652,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,15,0,0,4,0,0,10,0,0,0,0,0,2,0,0,0,0,1],[60,65,0.9231,0.41969,0.20183,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,5,0,0,4,0,0,19,0,0,0,0,0,2,0,0,0,0,2],[64,65,0.9846,0.32122,0.22026,0.14286,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,15,0,0,5,0,0,7,0,0,2,0,0,0,0,0,3,0,0],[65,65,1.0,0.21875,0.1988,0.14286,0.14286,0.2857,0.0,1.0,3,1,0,3,0,20,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"b01b2d6716b3a8ee","q":"Let n be an integer which is greater than 1, not divisible by 1997. \r\nLet  $ a_m\\equal{}m\\plus{}\\frac{mn}{1997}$  for all m=1,2,..,1996\r $ b_m\\equal{}m\\plus{}\\frac{1997m}{n}$  for all m=1,2,..,n-1\r\nWe arrange the terms of two sequence  $ (a_i), (b_j)$  in the ascending order to form a new sequence  $ c_1\\le c_2\\le ...\\le c_{1995\\plus{}n}$ \r\nProve that  $ c_{k\\plus{}1}\\minus{}c_k<2$  for all k=1,2,...,1994+n","t":[{"b":3,"e":0.71429,"k":"flat","v":0.56695,"x":0.7321,"p":[[0,54,0.0,0.67411,0.30771,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,1,0,0,4,0,0,2,0,0,6,0,0,4,0,10],[4,54,0.0741,0.66962,0.30397,0.53539,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,3,0,0,2,0,0,5,0,0,7,0,0,2,0,10],[8,54,0.1481,0.70982,0.24087,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,1,0,0,2,0,0,7,0,0,6,0,0,7,0,7],[12,54,0.2222,0.65177,0.30917,0.42859,0.71429,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,3,0,0,2,0,0,4,0,0,5,0,0,7,0,7],[16,54,0.2963,0.59819,0.3489,0.28571,0.64286,1.0,0.0,1.0,5,9,0,5,0,0,0,0,4,0,0,2,0,0,5,0,0,5,0,0,2,0,9],[20,54,0.3704,0.62052,0.35285,0.39286,0.64286,1.0,0.0,1.0,3,10,0,3,0,3,0,0,2,0,0,5,0,0,3,0,0,1,0,0,5,0,10],[24,54,0.4444,0.57142,0.27433,0.42857,0.57121,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,3,0,0,9,0,0,5,0,0,5,0,0,1,0,6],[28,54,0.5185,0.56695,0.27776,0.42857,0.57143,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,2,0,0,6,0,0,9,0,0,4,0,0,2,0,5],[32,54,0.5926,0.64729,0.27661,0.57132,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,2,0,0,0,0,0,12,0,0,2,0,0,5,0,7],[36,54,0.6667,0.62942,0.23653,0.4286,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,5,0,0,8,0,0,6,0,0,5,0,4],[40,54,0.7407,0.70084,0.22971,0.57132,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,8,0,0,3,0,8],[44,54,0.8148,0.72317,0.22854,0.57132,0.78571,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,3,0,0,9,0,7],[48,54,0.8889,0.7321,0.2252,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,5,0,0,5,0,9],[52,54,0.963,0.71864,0.21579,0.571,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,5,0,0,5,0,8],[54,54,1.0,0.70529,0.16733,0.5713,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,9,0,0,6,0,4]]},{"b":5,"e":0.0,"k":"flat","v":0.54909,"x":0.67856,"p":[[0,55,0.0,0.64731,0.25997,0.5354,0.57143,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,2,0,0,4,0,0,9,0,0,4,0,0,5,0,6],[4,55,0.0727,0.67853,0.25756,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,1,0,0,8,0,0,7,0,0,3,0,8],[8,55,0.1455,0.67856,0.2113,0.57143,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,9,0,0,1,0,7],[12,55,0.2182,0.66508,0.32278,0.42857,0.71429,1.0,0.14,1.0,0,13,0,0,0,4,0,0,3,0,0,5,0,0,3,0,0,3,0,0,1,0,13],[16,55,0.2909,0.64286,0.26726,0.42857,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,8,0,0,3,0,0,4,0,0,5,0,7],[20,55,0.3636,0.65177,0.28558,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,1,0,0,3,0,0,5,0,0,7,0,0,3,0,0,3,0,9],[24,55,0.4364,0.57588,0.2435,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,4,0,0,6,0,0,7,0,0,7,0,0,3,0,3],[28,55,0.5091,0.58034,0.31122,0.28571,0.57143,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,5,0,0,4,0,0,6,0,0,3,0,0,3,0,7],[32,55,0.5818,0.54909,0.29257,0.28571,0.57121,0.71429,0.0,1.0,1,6,0,1,0,3,0,0,6,0,0,5,0,0,5,0,0,5,0,0,1,0,6],[36,55,0.6545,0.57587,0.31029,0.39286,0.71429,0.71429,0.0,1.0,4,5,0,4,0,1,0,0,3,0,0,3,0,0,4,0,0,10,0,0,2,0,5],[40,55,0.7273,0.65624,0.29636,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,5,0,0,3,0,0,5,0,0,4,0,0,4,0,9],[44,55,0.8,0.55802,0.24578,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,3,0,0,10,0,0,3,0,0,4,0,3],[48,55,0.8727,0.67411,0.30563,0.39286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,5,0,0,2,0,0,3,0,0,4,0,0,5,0,10],[52,55,0.9455,0.62945,0.30901,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,3,0,0,6,0,0,4,0,0,3,0,0,2,0,10],[55,55,1.0,0.56246,0.25984,0.28571,0.571,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,7,0,0,6,0,0,4,0,0,6,0,0,3,0,4]]}]},{"i":"9301a2b4b6229606","q":"Let the incircle  $k$  of the triangle  $ABC$  touch its side  $BC$  at  $D$ . Let the line  $AD$  intersect  $k$  at  $L \\neq D$  and denote the excentre of  $ABC$  opposite to  $A$  by  $K$ . Let  $M$  and  $N$  be the midpoints of  $BC$  and  $KM$  respectively.\n\nProve that the points  $B, C, N,$  and  $L$  are concyclic.","t":[{"b":1,"e":0.0,"k":"flat","v":0.02232,"x":0.43748,"p":[[0,86,0.0,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,86,0.0465,0.39283,0.24221,0.25,0.35714,0.571,0.0,0.85714,2,0,0,2,0,6,0,0,8,0,0,7,0,0,4,0,0,1,0,0,4,0,0],[8,86,0.093,0.37497,0.3046,0.10714,0.42857,0.60714,0.0,0.85714,8,0,0,8,0,5,0,0,2,0,0,5,0,0,4,0,0,4,0,0,4,0,0],[12,86,0.1395,0.38159,0.27182,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,8,0,0,5,0,0,1,0,1,7,0,0,3,0,0,3,0,0],[16,86,0.186,0.40623,0.29688,0.14286,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,9,0,0,2,0,0,4,0,0,4,0,0,4,0,0,5,0,0],[20,86,0.2326,0.43748,0.25984,0.2857,0.42857,0.60714,0.0,0.85714,3,0,0,3,0,4,0,0,6,0,0,6,0,0,5,0,0,4,0,0,4,0,0],[24,86,0.2791,0.35926,0.28548,0.14286,0.28571,0.57111,0.0,1.0,4,1,0,4,0,11,0,0,2,0,0,5,0,0,4,1,0,1,0,0,3,0,1],[28,86,0.3256,0.35713,0.30092,0.10714,0.28571,0.60714,0.0,0.85714,8,0,0,8,0,5,0,0,4,0,0,5,0,0,2,0,0,4,0,0,4,0,0],[32,86,0.3721,0.19418,0.24885,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,6,0,0,4,0,0,1,0,1,4,0,0,0,0,0,0,0,1],[36,86,0.4186,0.16293,0.21219,0.0,0.14286,0.1786,0.0,0.71429,15,0,0,15,0,9,0,0,2,0,1,0,0,0,4,0,0,1,0,0,0,0,0],[40,86,0.4651,0.25,0.30514,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,4,0,0,2,0,0,5,0,0,1,0,0,2,0,0,2,0,1],[44,86,0.5116,0.24552,0.26055,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,5,0,0,6,0,0,3,0,0,3,0,0,1,0,0,2,0,0],[48,86,0.5581,0.15849,0.23941,0.0,0.0,0.1786,0.0,0.85714,17,0,0,17,0,7,0,0,3,0,0,2,0,0,0,1,0,0,0,0,2,0,0],[52,86,0.6047,0.1875,0.22711,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,5,0,0,6,0,0,1,0,0,3,0,0,2,0,0,0,0,0],[56,86,0.6512,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,86,0.6977,0.08929,0.16656,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[64,86,0.7442,0.10266,0.19635,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[68,86,0.7907,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,86,0.8372,0.07588,0.12358,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,86,0.8837,0.0982,0.18704,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[80,86,0.9302,0.05349,0.11146,0.0,0.0,0.035,0.0,0.4286,24,0,0,24,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[84,86,0.9767,0.0625,0.08702,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[86,86,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.45088,"p":[[0,92,0.0,0.13393,0.15947,0.0,0.14286,0.17857,0.0,0.57143,15,0,0,15,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,92,0.0435,0.45088,0.30952,0.2857,0.42859,0.85714,0.0,0.85714,6,0,0,6,0,1,0,0,7,0,0,4,0,0,5,0,0,0,0,0,9,0,0],[8,92,0.087,0.35266,0.27193,0.14286,0.35714,0.57111,0.0,0.85714,6,0,0,6,0,7,0,0,3,0,0,6,0,0,5,0,0,2,0,0,3,0,0],[12,92,0.1304,0.39284,0.30721,0.14286,0.35714,0.71429,0.0,0.85714,6,0,0,6,0,7,0,0,3,0,0,3,0,0,4,0,0,4,0,0,5,0,0],[16,92,0.1739,0.24108,0.24074,0.0,0.1429,0.32143,0.0,0.85714,10,0,0,10,0,7,0,0,7,0,0,4,0,0,0,0,0,3,0,0,1,0,0],[20,92,0.2174,0.31693,0.2939,0.0,0.28571,0.57143,0.0,0.85714,11,0,0,11,0,5,0,0,0,0,0,5,0,0,6,0,0,3,0,0,2,0,0],[24,92,0.2609,0.2366,0.28926,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,8,0,0,2,0,0,3,0,0,2,0,0,0,0,0,4,0,0],[28,92,0.3043,0.29013,0.26356,0.10714,0.14288,0.571,0.0,0.85714,8,0,0,8,0,9,0,0,3,0,0,3,0,0,6,0,0,1,0,0,2,0,0],[32,92,0.3478,0.30801,0.30742,0.0,0.14288,0.571,0.0,0.85714,10,0,0,10,0,7,0,0,3,0,0,3,0,0,2,0,0,3,0,0,4,0,0],[36,92,0.3913,0.25436,0.27139,0.0,0.14286,0.57111,0.0,0.85714,12,0,0,12,0,6,0,0,5,0,0,0,0,0,5,0,0,3,0,0,1,0,0],[40,92,0.4348,0.29015,0.31231,0.0,0.14286,0.57111,0.0,0.85714,12,0,0,12,0,6,0,0,3,0,0,1,0,0,4,0,0,2,0,0,4,0,0],[44,92,0.4783,0.26334,0.26021,0.0,0.14288,0.4642,0.0,0.85714,12,0,0,12,0,5,0,0,2,0,0,5,0,0,6,0,0,1,0,0,1,0,0],[48,92,0.5217,0.20982,0.303,0.0,0.0,0.28571,0.0,0.85714,18,0,0,18,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,0,4,0,0],[52,92,0.5652,0.09822,0.1448,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,10,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[56,92,0.6087,0.16062,0.22799,0.0,0.0,0.17857,0.0,0.71429,17,0,0,17,0,7,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,0],[60,92,0.6522,0.19195,0.29145,0.0,0.0,0.42857,0.0,0.85714,20,0,0,20,0,2,0,0,1,0,0,4,0,0,1,0,0,1,0,0,3,0,0],[64,92,0.6957,0.19196,0.28928,0.0,0.0,0.21429,0.0,0.85714,18,0,0,18,0,6,0,0,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0],[68,92,0.7391,0.15624,0.22966,0.0,0.0,0.17857,0.0,0.85714,17,0,0,17,0,7,0,0,3,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[72,92,0.7826,0.08929,0.18123,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,5,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[76,92,0.8261,0.11159,0.18805,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[80,92,0.8696,0.12051,0.18929,0.0,0.0,0.17857,0.0,0.571,21,0,0,21,0,3,0,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[84,92,0.913,0.08929,0.18123,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,5,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[88,92,0.9565,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[92,92,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"563afd521aa514fd","q":"Let $x, y, z$ be three positive real numbers, such that $x \\leqslant 1$. Prove that:\n\n$$\nx y + y + 2 z \\geqslant 4 \\sqrt{x y z}\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.77679,"x":1.0,"p":[[0,34,0.0,0.77679,0.2922,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,20],[4,34,0.1176,0.79017,0.3174,0.42857,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,21],[8,34,0.2353,0.79911,0.27861,0.42857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,21],[12,34,0.3529,0.91071,0.20748,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,27],[16,34,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,34,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,34,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,34,0.9412,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[34,34,1.0,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29]]},{"b":2,"e":1.0,"k":"flat","v":0.79911,"x":1.0,"p":[[0,29,0.0,0.875,0.23623,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,25],[4,29,0.1379,0.875,0.23623,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,25],[8,29,0.2759,0.875,0.23623,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,25],[12,29,0.4138,0.79911,0.2693,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,20],[16,29,0.5517,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a89b398fd7eb0200","q":"Let the incircle of an acute triangle  $\\triangle ABC$  touches  $BC,CA$ , and  $AB$  at points  $D,E$ , and  $F$ , respectively. Place point  $K$  on the side  $AB$  so that  $DF$  bisects  $\\angle ADK$ , and place point  $L$  on the side  $AB$  so that  $EF$  bisects  $\\angle BEL$ .\n[list=a]\n[*]Prove that  $\\triangle ALE\\sim\\triangle AEB$ .\n[*]Prove that  $FK=FL$ .\n[/list]","t":[{"b":1,"e":0.4286,"k":"falling","v":0.30802,"x":0.77229,"p":[[0,150,0.0,0.77229,0.16701,0.71429,0.857,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,10,0,0,13,0,4],[4,150,0.0267,0.74552,0.28062,0.67856,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,2,0,0,14,0,8],[8,150,0.0533,0.76784,0.19151,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,9,0,0,13,0,5],[12,150,0.08,0.61149,0.25076,0.42857,0.71429,0.74996,0.14,1.0,0,2,0,0,0,4,0,0,2,0,0,4,0,0,3,0,0,11,0,0,6,0,2],[16,150,0.1067,0.68299,0.26424,0.5354,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,3,0,0,5,0,0,4,0,0,9,0,6],[20,150,0.1333,0.71427,0.22869,0.4286,0.78564,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,7,0,0,2,0,0,5,0,0,10,0,6],[24,150,0.16,0.67408,0.19961,0.571,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,9,0,0,9,0,2],[28,150,0.1867,0.69637,0.24422,0.571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,5,0,0,10,0,5],[32,150,0.2133,0.7053,0.24209,0.571,0.78571,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,2,0,0,9,0,7],[36,150,0.24,0.73659,0.22049,0.67857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,11,0,0,6,0,7],[40,150,0.2667,0.74106,0.18709,0.71429,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,11,0,0,9,0,5],[44,150,0.2933,0.68748,0.2683,0.4286,0.85714,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,6,0,0,3,0,0,2,0,0,14,0,4],[48,150,0.32,0.67852,0.24224,0.4286,0.78564,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,4,0,0,5,0,0,2,0,0,12,0,4],[52,150,0.3467,0.71424,0.20519,0.57143,0.78564,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,6,0,0,14,0,2],[56,150,0.3733,0.64725,0.22866,0.53539,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,4,0,0,6,0,0,7,0,0,9,0,2],[60,150,0.4,0.69193,0.22899,0.42859,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,6,0,0,2,0,0,4,0,0,15,0,2],[64,150,0.4267,0.49551,0.20197,0.39286,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,8,0,0,10,0,0,2,0,0,4,0,0],[68,150,0.4533,0.42408,0.24866,0.2857,0.42857,0.57143,0.0,1.0,4,1,0,4,0,2,0,0,7,0,0,5,0,0,9,0,0,3,0,0,1,0,1],[72,150,0.48,0.50441,0.13822,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,10,0,0,13,0,0,3,0,0,1,0,0],[76,150,0.5067,0.4374,0.22581,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,4,0,0,5,0,0,8,0,0,7,0,0,4,0,0,2,0,0],[80,150,0.5333,0.41516,0.17626,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,10,0,0,8,0,0,9,0,0,1,0,0,1,0,0],[84,150,0.56,0.4866,0.19839,0.28571,0.4286,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,9,0,0,8,0,0,8,0,0,3,0,0,2,0,1],[88,150,0.5867,0.41084,0.18804,0.28571,0.42857,0.4642,0.0,1.0,2,1,0,2,0,1,0,0,8,0,0,13,0,0,6,0,0,1,0,0,0,0,1],[92,150,0.6133,0.43747,0.17471,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,8,0,0,10,0,0,7,0,0,3,0,0,1,0,0],[96,150,0.64,0.43745,0.16338,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,6,0,0,10,0,0,11,0,0,2,0,0,0,0,0],[100,150,0.6667,0.35714,0.16367,0.2857,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,1,0,0,16,0,0,7,0,0,4,0,0,2,0,0,0,0,0],[104,150,0.6933,0.38835,0.19634,0.28571,0.42857,0.46418,0.0,1.0,3,1,0,3,0,1,0,0,9,0,0,11,0,0,7,0,0,0,0,0,0,0,1],[108,150,0.72,0.39283,0.17494,0.2857,0.35714,0.46418,0.0,0.71429,1,0,0,1,0,2,0,0,13,0,0,8,0,0,4,0,0,4,0,0,0,0,0],[112,150,0.7467,0.45088,0.17535,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,10,0,0,6,0,0,1,0,0,3,0,0],[116,150,0.7733,0.44191,0.17981,0.28571,0.42857,0.571,0.0,1.0,1,1,0,1,0,0,0,0,9,0,0,12,0,0,8,0,0,0,0,0,1,0,1],[120,150,0.8,0.39731,0.20432,0.2857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,4,0,0,7,0,0,9,0,0,8,0,0,0,0,0,2,0,0],[124,150,0.8267,0.36157,0.17486,0.2857,0.42857,0.571,0.0,0.57143,2,0,0,2,0,5,0,0,8,0,0,8,0,0,9,0,0,0,0,0,0,0,0],[128,150,0.8533,0.39289,0.16366,0.2857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,4,0,0,8,0,0,15,0,0,4,0,0,0,0,0,0,0,1],[132,150,0.88,0.34817,0.21106,0.14286,0.42857,0.571,0.0,0.57143,6,0,0,6,0,3,0,0,4,0,0,9,0,0,10,0,0,0,0,0,0,0,0],[136,150,0.9067,0.39731,0.15457,0.28571,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,3,0,0,8,0,0,10,0,0,10,0,0,0,0,0,0,0,0],[140,150,0.9333,0.38838,0.16839,0.28571,0.42857,0.4642,0.0,0.71429,2,0,0,2,0,3,0,0,6,0,0,13,0,0,7,0,0,1,0,0,0,0,0],[144,150,0.96,0.38839,0.14827,0.28571,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,1,0,0,8,0,0,14,0,0,7,0,0,0,0,0,0,0,0],[148,150,0.9867,0.30802,0.18247,0.25,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,2,0,0,9,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[150,150,1.0,0.40177,0.16916,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,9,0,0,9,0,0,8,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.2857,"k":"falling","v":0.33472,"x":0.75445,"p":[[0,127,0.0,0.74996,0.14727,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,11,0,0,11,0,3],[4,127,0.0315,0.67856,0.29234,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,5,0,0,2,0,0,5,0,0,7,0,8],[8,127,0.063,0.68303,0.29609,0.42857,0.71429,1.0,0.1429,1.0,0,10,0,0,0,2,0,0,5,0,0,4,0,0,1,0,0,5,0,0,5,0,10],[12,127,0.0945,0.66069,0.31894,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,2,0,0,3,0,0,3,0,0,2,0,0,6,0,0,5,0,9],[16,127,0.126,0.72766,0.22407,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,8,0,0,9,0,6],[20,127,0.1575,0.75445,0.22084,0.67846,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,5,0,0,12,0,7],[24,127,0.189,0.68304,0.25187,0.53572,0.71429,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,4,0,0,2,0,0,7,0,0,12,0,3],[28,127,0.2205,0.64729,0.23415,0.5354,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,3,0,0,6,0,0,5,0,0,12,0,1],[32,127,0.252,0.65177,0.24469,0.42857,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,6,0,0,6,0,0,4,0,0,7,0,5],[36,127,0.2835,0.68303,0.24152,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,9,0,0,12,0,2],[40,127,0.315,0.5357,0.24222,0.39286,0.4998,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,8,0,0,2,0,0,8,0,0,6,0,0],[44,127,0.3465,0.60253,0.24433,0.42857,0.57143,0.85714,0.14,1.0,0,3,0,0,0,3,0,0,2,0,0,5,0,0,9,0,0,4,0,0,6,0,3],[48,127,0.378,0.49999,0.27892,0.28571,0.42859,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,5,0,0,7,0,0,3,0,0,4,0,0,5,0,2],[52,127,0.4094,0.59816,0.25615,0.42857,0.64286,0.75,0.0,1.0,2,2,0,2,0,1,0,0,2,0,0,5,0,0,6,0,0,8,0,0,6,0,2],[56,127,0.4409,0.56247,0.28107,0.28571,0.57143,0.85714,0.0,1.0,2,2,0,2,0,2,0,0,5,0,0,3,0,0,6,0,0,5,0,0,7,0,2],[60,127,0.4724,0.55356,0.23622,0.39286,0.57121,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,7,0,0,5,0,0,3,0,0,9,0,0],[64,127,0.5039,0.61159,0.25563,0.4286,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,4,0,0,7,0,0,6,0,0,5,0,4],[68,127,0.5354,0.49547,0.24478,0.28571,0.571,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,5,0,0,5,0,0,8,0,0,5,0,0,3,0,1],[72,127,0.5669,0.52673,0.21851,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,6,0,0,9,0,0,6,0,0,4,0,0],[76,127,0.5984,0.44193,0.2681,0.2857,0.42857,0.71429,0.0,0.85714,3,0,0,3,0,4,0,0,6,0,0,7,0,0,3,0,0,4,0,0,5,0,0],[80,127,0.6299,0.41961,0.24726,0.2857,0.28571,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,11,0,0,3,0,0,7,0,0,2,0,0,1,0,2],[84,127,0.6614,0.38838,0.23209,0.24999,0.42857,0.57111,0.0,0.85714,3,0,0,3,0,5,0,0,6,0,0,9,0,0,4,0,0,3,0,0,2,0,0],[88,127,0.6929,0.49103,0.26228,0.2857,0.571,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,4,0,0,4,0,0,6,0,0,7,0,0,3,0,1],[92,127,0.7244,0.49549,0.24737,0.28571,0.4286,0.60714,0.14286,1.0,0,2,0,0,0,4,0,0,6,0,0,9,0,0,5,0,0,2,0,0,4,0,2],[96,127,0.7559,0.39283,0.25503,0.25,0.28571,0.57111,0.0,0.85714,3,0,0,3,0,5,0,0,9,0,0,5,0,0,4,0,0,2,0,0,4,0,0],[100,127,0.7874,0.43301,0.25121,0.2857,0.42857,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,9,0,0,4,0,0,6,0,0,4,0,0,2,0,1],[104,127,0.8189,0.37497,0.20121,0.2857,0.28571,0.4286,0.14286,0.85714,0,0,0,0,0,6,0,0,13,0,0,6,0,0,4,0,0,0,0,0,3,0,0],[108,127,0.8504,0.4508,0.22329,0.2857,0.4286,0.57111,0.0,0.85714,1,0,0,1,0,3,0,0,10,0,0,3,0,0,9,0,0,3,0,0,3,0,0],[112,127,0.8819,0.41066,0.25936,0.24999,0.42857,0.571,0.0,0.85714,3,0,0,3,0,5,0,0,6,0,0,8,0,0,4,0,0,1,0,0,5,0,0],[116,127,0.9134,0.40179,0.18707,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,8,0,0,9,0,0,4,0,0,5,0,0,0,0,0],[120,127,0.9449,0.33472,0.23591,0.14286,0.2857,0.4642,0.0,0.85714,3,0,0,3,0,8,0,0,11,0,0,2,0,0,2,0,0,5,0,0,1,0,0],[124,127,0.9764,0.33926,0.22514,0.14289,0.28571,0.571,0.0,0.71429,4,0,0,4,0,7,0,0,7,0,0,5,0,0,5,0,0,4,0,0,0,0,0],[127,127,1.0,0.36604,0.20494,0.2857,0.2857,0.57111,0.0,0.71429,2,0,0,2,0,5,0,0,12,0,0,3,0,0,6,0,0,4,0,0,0,0,0]]}]},{"i":"5ff8ea3cd4695aa8","q":"Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer  $\\mathbb{Z}$ . To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least  $2024$  divisors. Can Nirajan always escape?\n\n*( Proposed by Vlad Sp\u01cetaru, Romania)*","t":[{"b":0,"e":1.0,"k":"volatile","v":0.26339,"x":0.99107,"p":[[0,16,0.0,0.26339,0.30952,0.10714,0.14286,0.42857,0.0,1.0,8,4,0,8,0,14,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,4],[4,16,0.25,0.76321,0.34029,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,5,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,16],[8,16,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,16,0.75,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,16,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"rising","v":0.39286,"x":0.98214,"p":[[0,22,0.0,0.39286,0.39448,0.14286,0.14286,0.89286,0.0,1.0,5,8,0,5,0,15,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,8],[4,22,0.1818,0.89732,0.17582,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,15],[8,22,0.3636,0.83482,0.22899,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0,12,0,14],[12,22,0.5455,0.85714,0.22588,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,9,0,17],[16,22,0.7273,0.91963,0.13334,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,18],[20,22,0.9091,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[22,22,1.0,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30]]}]},{"i":"4bc7ee6940bb243c","q":"Let $p>2$ be a prime number and $1+\\frac{1}{2^{3}}+\\frac{1}{3^{3}}+\\cdots+\\frac{1}{(p-1)^{3}}=\\frac{m}{n}$ where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.28571,"x":0.375,"p":[[0,47,0.0,0.375,0.23623,0.28571,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,47,0.0851,0.36607,0.21706,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[8,47,0.1702,0.375,0.21943,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,27,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[12,47,0.2553,0.36607,0.2111,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[16,47,0.3404,0.36161,0.21124,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[20,47,0.4255,0.32589,0.1394,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[24,47,0.5106,0.32142,0.10102,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,28,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[28,47,0.5957,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.34821,0.16728,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,2,0,0,2,0,0],[36,47,0.766,0.2991,0.07457,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,47,0.8511,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.28571,"x":0.47766,"p":[[0,51,0.0,0.47766,0.28928,0.28571,0.28571,0.57143,0.2857,1.0,0,7,0,0,0,0,0,0,20,0,0,2,0,0,3,0,0,0,0,0,0,0,7],[4,51,0.0784,0.42411,0.27078,0.28571,0.28571,0.28571,0.2857,1.0,0,5,0,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0,1,0,5],[8,51,0.1569,0.38393,0.22142,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,26,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[12,51,0.2353,0.42411,0.26119,0.28571,0.28571,0.35714,0.2857,1.0,0,5,0,0,0,0,0,0,24,0,0,0,0,0,3,0,0,0,0,0,0,0,5],[16,51,0.3137,0.33482,0.17353,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[20,51,0.3922,0.41518,0.23244,0.28571,0.28571,0.46429,0.2857,1.0,0,3,0,0,0,0,0,0,23,0,0,1,0,0,2,0,0,3,0,0,0,0,3],[24,51,0.4706,0.41964,0.25238,0.28571,0.28571,0.35714,0.2857,1.0,0,4,0,0,0,0,0,0,24,0,0,0,0,0,3,0,0,0,0,0,1,0,4],[28,51,0.549,0.34821,0.18536,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[32,51,0.6275,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,51,0.7059,0.28571,1e-05,0.2857,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,51,0.7843,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.3125,0.10972,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[48,51,0.9412,0.2991,0.07457,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[51,51,1.0,0.2991,0.07457,0.2857,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"4966efe41e13234e","q":"Let's call integer square-free if it's not divisible by  $p^2$  for any prime  $p$ . You are given a square-free integer  $n>1$ , which has exactly  $d$  positive divisors. Find the largest number of its divisors that you can choose, such that  $a^2 + ab - n$  isn't a square of an integer for any  $a, b$  among chosen divisors.\n\n*(Proposed by Oleksii Masalitin)*","t":[{"b":6,"e":0.85714,"k":"flat","v":0.63839,"x":0.89286,"p":[[0,43,0.0,0.63839,0.25997,0.28571,0.71429,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,9,0,0,0,0,0,5,0,0,10,0,0,1,0,7],[4,43,0.093,0.85267,0.14936,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,9,0,13],[8,43,0.186,0.85267,0.14501,0.71429,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,5,0,14],[12,43,0.2791,0.78125,0.12869,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,6,0,6],[16,43,0.3721,0.87054,0.16115,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,3,0,18],[20,43,0.4651,0.85714,0.15567,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,6,0,15],[24,43,0.5581,0.87053,0.14445,0.71429,0.85714,1.0,0.5714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,15],[28,43,0.6512,0.89286,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[32,43,0.7442,0.79909,0.15512,0.71429,0.78571,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,7,0,9],[36,43,0.8372,0.77232,0.13767,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,9,0,5],[40,43,0.9302,0.79464,0.13333,0.71429,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,7,0,7],[43,43,1.0,0.76339,0.10479,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,7,0,3]]},{"b":7,"e":1.0,"k":"rising","v":0.63839,"x":0.99107,"p":[[0,52,0.0,0.63839,0.26723,0.28571,0.71429,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,9,0,0,1,0,0,4,0,0,10,0,0,0,0,8],[4,52,0.0769,0.87946,0.14334,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,17],[8,52,0.1538,0.83482,0.16016,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,3,0,14],[12,52,0.2308,0.83481,0.15615,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,5,0,13],[16,52,0.3077,0.82589,0.1665,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,5,0,13],[20,52,0.3846,0.84821,0.14698,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,7,0,13],[24,52,0.4615,0.84821,0.16342,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,5,0,15],[28,52,0.5385,0.87052,0.15305,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,7,0,16],[32,52,0.6154,0.87946,0.14773,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,3,0,18],[36,52,0.6923,0.88393,0.13092,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,7,0,16],[40,52,0.7692,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[44,52,0.8462,0.91964,0.11259,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[48,52,0.9231,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[52,52,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"2dccd8db68fcc153","q":"N numbers are marked in the set  $\\{1,2,...,2000\\}$  so that any pair of the numbers  $(1,2),(2,4),...,(1000,2000)$  contains at least one marked number. Find the least possible value of  $N$ .\n\nI.Gorodnin","t":[{"b":4,"e":1.0,"k":"flat","v":0.88838,"x":0.97321,"p":[[0,48,0.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,48,0.0833,0.88838,0.17764,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,8,0,19],[8,48,0.1667,0.94196,0.13767,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[12,48,0.25,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[16,48,0.3333,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[20,48,0.4167,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,48,0.5,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[28,48,0.5833,0.89286,0.2342,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,22],[32,48,0.6667,0.91071,0.21053,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,23],[36,48,0.75,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[40,48,0.8333,0.91964,0.21706,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[44,48,0.9167,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[48,48,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":5,"e":0.85714,"k":"flat","v":0.70534,"x":0.98214,"p":[[0,60,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,60,0.0667,0.87946,0.1992,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,6,0,20],[8,60,0.1333,0.88839,0.23072,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,24],[12,60,0.2,0.74999,0.29882,0.53539,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,6,0,0,1,0,0,3,0,0,1,0,0,5,0,15],[16,60,0.2667,0.78125,0.3113,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,0,6,0,17],[20,60,0.3333,0.89732,0.18293,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,4,0,22],[24,60,0.4,0.74554,0.30037,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,3,0,0,1,0,0,3,0,0,3,0,0,5,0,14],[28,60,0.4667,0.84821,0.21706,0.78561,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0,0,0,6,0,18],[32,60,0.5333,0.90179,0.19045,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,4,0,23],[36,60,0.6,0.70534,0.24469,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,15,0,0,0,0,0,2,0,11],[40,60,0.6667,0.96875,0.08553,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[44,60,0.7333,0.95088,0.15407,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[48,60,0.8,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,60,0.8667,0.94642,0.09284,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[56,60,0.9333,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[60,60,1.0,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]}]},{"i":"d723f083c7a535be","q":"Let n be an integer such that $\\mathrm{n} \\geqslant 2$. We denote d as the greatest divisor of n different from n. We assume that $\\mathrm{d}>1$. Prove that $\\mathrm{n}+\\mathrm{d}$ is not a power of 2.","t":[{"b":0,"e":1.0,"k":"rising","v":0.8482,"x":1.0,"p":[[0,45,0.0,0.8482,0.21112,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,0,0,20],[4,45,0.0889,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,45,0.1778,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,45,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,45,0.3556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,45,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,45,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,45,0.7111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[45,45,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":0.42857,"k":"falling","v":0.50445,"x":0.93751,"p":[[0,32,0.0,0.93751,0.17101,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,28],[4,32,0.125,0.84375,0.22406,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,5,0,0,1,0,20],[8,32,0.25,0.87947,0.2055,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,0,0,23],[12,32,0.375,0.79911,0.22548,0.71429,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,0,0,16],[16,32,0.5,0.58926,0.20439,0.42857,0.4998,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,15,0,0,4,0,0,7,0,0,1,0,4],[20,32,0.625,0.60713,0.21129,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,4,0,0,0,0,6],[24,32,0.75,0.57586,0.16934,0.42857,0.4998,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,4,0,0,8,0,0,3,0,1],[28,32,0.875,0.58482,0.20316,0.42857,0.50001,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,14,0,0,3,0,0,8,0,0,2,0,3],[32,32,1.0,0.50445,0.13355,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,22,0,0,5,0,0,4,0,0,0,0,1]]}]},{"i":"5a3077cf9ca07284","q":"Let $x$ and $y$ be positive real numbers.\na) Prove: if $x^{3}-y^{3} \\geq 4 x$, then $x^{2}>2 y$.\nb) Prove: if $x^{5}-y^{3} \\geq 2 x$, then $x^{3} \\geq 2 y$.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.75878,"x":0.875,"p":[[0,23,0.0,0.75878,0.1871,0.71429,0.8571,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,9,0,0,17,0,2],[4,23,0.1739,0.85267,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[8,23,0.3478,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[12,23,0.5217,0.85714,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[16,23,0.6957,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[20,23,0.8696,0.87052,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[23,23,1.0,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4]]},{"b":3,"e":0.71429,"k":"falling","v":0.59821,"x":0.82588,"p":[[0,37,0.0,0.82588,0.10558,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,19,0,4],[4,37,0.1081,0.59821,0.25862,0.57143,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,2,0,0,0,0,0,4,0,0,15,0,0,6,0,0],[8,37,0.2162,0.6607,0.18814,0.67857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,20,0,0,3,0,1],[12,37,0.3243,0.66518,0.15815,0.67857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,0,0,0,6,0,0,22,0,0,1,0,1],[16,37,0.4324,0.62946,0.23381,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,0,0,0,0,0,0,2,0,0,22,0,0,2,0,1],[20,37,0.5405,0.65625,0.18851,0.67857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,22,0,0,0,0,2],[24,37,0.6486,0.68304,0.12745,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,23,0,0,1,0,1],[28,37,0.7568,0.70535,0.12339,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,26,0,0,2,0,1],[32,37,0.8649,0.67854,0.15154,0.67857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,7,0,0,21,0,0,2,0,1],[36,37,0.973,0.66062,0.18148,0.71421,0.71429,0.71429,0.14,0.85714,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,21,0,0,4,0,0],[37,37,1.0,0.66071,0.20124,0.67857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,1,0,0,4,0,0,21,0,0,0,0,3]]}]},{"i":"bb9a04247d36b070","q":"Let $x_{1}, x_{2}, \\ldots, x_{n}$ be real numbers with arithmetic mean $X$. Prove that there is a positive integer $K$ such that the arithmetic mean of each of the lists $\\left\\{x_{1}, x_{2}, \\ldots, x_{K}\\right\\},\\left\\{x_{2}, x_{3}, \\ldots, x_{K}\\right\\}$, $\\ldots,\\left\\{x_{K-1}, x_{K}\\right\\},\\left\\{x_{K}\\right\\}$ is not greater than $X$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.04018,"x":0.04911,"p":[[0,11,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.03125,"x":0.06696,"p":[[0,20,0.0,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a2a286a196241ff","q":"Let $x_{1}, x_{2}, \\ldots, x_{n}$ be positive real numbers, and let\n\n$$\nS=x_{1}+x_{2}+\\cdots+x_{n}\n$$\n\nProve that\n\n$$\n\\left(1+x_{1}\\right)\\left(1+x_{2}\\right) \\cdots\\left(1+x_{n}\\right) \\leq 1+S+\\frac{S^{2}}{2!}+\\frac{S^{3}}{3!}+\\cdots+\\frac{S^{n}}{n!}\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.78123,"x":1.0,"p":[[0,33,0.0,0.78123,0.3194,0.82132,0.85714,1.0,0.0,1.0,4,13,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,13],[4,33,0.1212,0.91964,0.18189,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,21],[8,33,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,33,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,33,0.4848,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.85714,"k":"volatile","v":0.63839,"x":0.9375,"p":[[0,12,0.0,0.9375,0.13333,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,22],[4,12,0.3333,0.83036,0.29545,0.85714,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,17],[8,12,0.6667,0.63839,0.39121,0.28571,0.85714,1.0,0.0,1.0,6,13,0,6,0,0,0,0,3,0,0,4,0,0,0,0,0,2,0,0,4,0,13],[12,12,1.0,0.85714,0.10714,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,20,0,7]]}]},{"i":"64fce50273e93e38","q":"On a line, there are 400 blue points and 200 green points. Show that one can find a segment that contains exactly 200 blue points and 100 green points.","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.87054,"p":[[0,57,0.0,0.87054,0.29093,1.0,1.0,1.0,0.0,1.0,2,25,2,2,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[4,57,0.0702,0.70089,0.356,0.42857,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,1,0,0,7,0,0,2,0,0,0,0,0,1,0,17],[8,57,0.1404,0.54018,0.3655,0.24999,0.57143,1.0,0.0,1.0,6,9,0,6,0,2,0,0,2,0,0,3,0,0,8,0,0,1,0,0,1,0,9],[12,57,0.2105,0.47767,0.38234,0.0,0.42857,0.85714,0.0,1.0,9,6,0,9,0,2,0,0,1,0,0,5,0,0,3,0,0,2,0,0,4,0,6],[16,57,0.2807,0.46426,0.35714,0.14286,0.42857,0.64286,0.0,1.0,7,7,0,7,0,3,0,0,1,0,0,8,0,0,5,0,0,0,0,0,1,0,7],[20,57,0.3509,0.60266,0.33832,0.42857,0.57143,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,1,0,0,4,0,0,7,0,0,0,0,0,7,0,7],[24,57,0.4211,0.47765,0.34738,0.25,0.49979,0.64286,0.0,1.0,7,6,0,7,0,1,0,0,4,0,0,4,0,0,8,0,0,0,0,0,2,0,6],[28,57,0.4912,0.40177,0.31224,0.14286,0.42857,0.57143,0.0,1.0,7,4,0,7,0,3,0,0,4,0,0,6,0,0,7,0,0,1,0,0,0,0,4],[32,57,0.5614,0.4375,0.34057,0.14286,0.42857,0.60714,0.0,1.0,7,5,0,7,0,3,0,0,2,0,0,9,0,0,3,0,0,1,0,0,2,0,5],[36,57,0.6316,0.44633,0.36385,0.14214,0.42857,0.67857,0.0,1.0,7,8,0,7,0,3,0,0,3,0,0,9,0,0,2,0,0,0,0,0,0,0,8],[40,57,0.7018,0.43741,0.29446,0.2857,0.42857,0.57143,0.0,1.0,5,4,0,5,0,1,0,0,6,0,0,10,0,0,4,0,0,1,0,0,1,0,4],[44,57,0.7719,0.47321,0.30185,0.2857,0.42857,0.60714,0.0,1.0,3,5,0,3,0,3,0,0,6,0,0,8,0,0,4,0,0,2,0,0,1,0,5],[48,57,0.8421,0.29463,0.3193,0.0,0.14286,0.57111,0.0,1.0,12,2,0,12,0,6,0,0,1,0,0,4,0,0,5,0,0,0,0,0,2,0,2],[52,57,0.9123,0.15615,0.22118,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,1,0,0,4,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[56,57,0.9825,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.28572,"x":0.82143,"p":[[0,55,0.0,0.82143,0.3093,0.67857,1.0,1.0,0.0,1.0,2,22,1,2,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,22],[4,55,0.0727,0.72768,0.37348,0.42857,1.0,1.0,0.0,1.0,3,20,0,3,0,2,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,20],[8,55,0.1455,0.58928,0.42521,0.14286,0.71421,1.0,0.0,1.0,6,15,0,6,0,5,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,15],[12,55,0.2182,0.54911,0.37646,0.28571,0.50001,1.0,0.0,1.0,7,10,0,7,0,0,0,0,2,0,0,7,0,0,3,0,0,2,0,0,1,0,10],[16,55,0.2909,0.40179,0.39194,0.0,0.42857,0.60714,0.0,1.0,12,7,0,12,0,3,0,0,0,0,0,3,0,0,6,0,0,1,0,0,0,0,7],[20,55,0.3636,0.55804,0.31615,0.39286,0.5,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,4,0,0,8,0,0,5,0,0,1,0,0,3,0,7],[24,55,0.4364,0.56696,0.34531,0.39286,0.5,1.0,0.0,1.0,3,9,0,3,0,4,0,0,1,0,0,8,0,0,3,0,0,2,0,0,2,0,9],[28,55,0.5091,0.36158,0.29446,0.0,0.42857,0.57111,0.0,1.0,9,2,0,9,0,2,0,0,3,0,0,8,0,0,6,0,0,1,0,0,1,0,2],[32,55,0.5818,0.59829,0.35799,0.39286,0.57143,1.0,0.0,1.0,5,9,0,5,0,1,0,0,2,0,0,4,0,0,6,0,0,0,0,0,5,0,9],[36,55,0.6545,0.49107,0.3976,0.10714,0.42857,1.0,0.0,1.0,8,10,0,8,0,3,0,0,1,0,0,7,0,0,1,0,0,2,0,0,0,0,10],[40,55,0.7273,0.4732,0.33204,0.24999,0.42857,0.57143,0.0,1.0,6,6,0,6,0,2,0,0,2,0,0,8,0,0,7,0,0,0,0,0,1,0,6],[44,55,0.8,0.39732,0.29393,0.2857,0.42857,0.46431,0.0,1.0,7,4,0,7,0,0,0,0,6,0,0,11,0,0,4,0,0,0,0,0,0,0,4],[48,55,0.8727,0.41517,0.27746,0.25,0.42857,0.57143,0.0,1.0,7,2,0,7,0,1,0,0,1,0,0,11,0,0,8,0,0,1,0,0,1,0,2],[52,55,0.9455,0.3571,0.29011,0.10714,0.28571,0.57143,0.0,1.0,8,1,0,8,0,3,0,0,7,0,0,2,0,0,7,0,0,2,0,0,2,0,1],[55,55,1.0,0.28572,0.22303,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,6,0,0,4,0,0,7,0,0,6,0,0,1,0,0,0,0,0]]}]},{"i":"5ff97a689bc8d43b","q":"Prove that the number permutations  $ \\alpha$  of  $ \\{1,2,\\dots,n\\}$  s.t. there does not exist  $ i<j<n$  s.t.  $ \\alpha(i)<\\alpha(j\\plus{}1)<\\alpha(j)$  is equal to the number of partitions of that set.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.66071,"x":0.95536,"p":[[0,27,0.0,0.78124,0.24482,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,1,0,0,9,0,12],[4,27,0.1481,0.66071,0.31491,0.39286,0.78571,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,5,0,0,1,0,0,5,0,0,2,0,0,8,0,8],[8,27,0.2963,0.80801,0.27344,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,0,5,0,17],[12,27,0.4444,0.83929,0.28959,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,3,0,22],[16,27,0.5926,0.85713,0.22017,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,20],[20,27,0.7407,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[24,27,0.8889,0.89731,0.1864,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,24],[27,27,1.0,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29]]},{"b":4,"e":0.57143,"k":"flat","v":0.65622,"x":0.82143,"p":[[0,26,0.0,0.65622,0.28541,0.571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,5,0,0,0,0,0,12,0,0,2,0,0,0,0,11],[4,26,0.1538,0.82143,0.2369,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,6,0,16],[8,26,0.3077,0.79462,0.19543,0.67857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,4,0,0,11,0,9],[12,26,0.4615,0.71873,0.1838,0.57143,0.85707,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,14,0,0,0,0,0,15,0,2],[16,26,0.6154,0.75447,0.18977,0.57143,0.85714,0.85714,0.1429,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,16,0,4],[20,26,0.7692,0.69642,0.17769,0.57143,0.57143,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,0,0,0,9,0,4],[24,26,0.9231,0.77677,0.16729,0.57143,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,16,0,5],[26,26,1.0,0.74553,0.16262,0.57143,0.85707,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,13,0,4]]}]},{"i":"2d4a6c5c57f9aea8","q":"Let \\( R \\) be an integral domain such that every non-constant polynomial in \\( R[x] \\) has a root in \\( R \\). Let \\( P \\) and \\( Q \\) be two prime ideals in \\( R \\). Prove that for any \\( a, b \\in R \\) such that  $ab\\in P+Q$ , either \\( b \\in P + Q \\) or \\( a \\in P + Q \\).","t":[{"b":2,"e":1.0,"k":"flat","v":0.90178,"x":0.95536,"p":[[0,38,0.0,0.90178,0.1729,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,12,0,18],[4,38,0.1053,0.90625,0.06785,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[8,38,0.2105,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,38,0.3158,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[16,38,0.4211,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[20,38,0.5263,0.94196,0.07016,0.85714,1.0,1.0,0.8571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[24,38,0.6316,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[28,38,0.7368,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[32,38,0.8421,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[36,38,0.9474,0.92411,0.07129,0.85714,0.85714,1.0,0.8571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[38,38,1.0,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12]]},{"b":4,"e":0.85714,"k":"flat","v":0.875,"x":0.92856,"p":[[0,39,0.0,0.92411,0.13356,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,19],[4,39,0.1026,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[8,39,0.2051,0.88839,0.10555,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,21,0,10],[12,39,0.3077,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[16,39,0.4103,0.875,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[20,39,0.5128,0.91071,0.06917,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[24,39,0.6154,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[28,39,0.7179,0.92856,0.07144,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[32,39,0.8205,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[36,39,0.9231,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[39,39,1.0,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]}]},{"i":"58934b77c89786ff","q":"Let the sequence of real numbers  $(a_n),n=1,2,3...$  with  $a_1=2$  and  $a_n=\\left(\\frac{n+1}{n-1} \\right)\\left(a_1+a_2+...+a_{n-1} \\right),n\\geq 2$ . \nFind the term  $a_{2013}$ .","t":[{"b":4,"e":0.57143,"k":"flat","v":0.52232,"x":0.625,"p":[[0,61,0.0,0.59375,0.21461,0.42857,0.57143,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,12,0,0,8,0,0,4,0,0,1,0,5],[4,61,0.0656,0.56696,0.10403,0.42857,0.57143,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,15,0,0,8,0,0,0,0,0],[8,61,0.1311,0.5625,0.13333,0.53571,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,5,0,0,16,0,0,7,0,0,1,0,0],[12,61,0.1967,0.5625,0.12846,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,11,0,0,9,0,0,11,0,0,0,0,0],[16,61,0.2623,0.54909,0.11355,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,7,0,0,17,0,0,6,0,0,0,0,0],[20,61,0.3279,0.57589,0.13115,0.53572,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,7,0,0,16,0,0,7,0,0,0,0,1],[24,61,0.3934,0.57589,0.13115,0.53571,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,6,0,0,14,0,0,9,0,0,1,0,0],[28,61,0.459,0.52232,0.16213,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,13,0,0,11,0,0,5,0,0,0,0,1],[32,61,0.5246,0.57143,0.11294,0.42857,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,15,0,0,7,0,0,1,0,0],[36,61,0.5902,0.58929,0.13716,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,13,0,0,8,0,0,1,0,1],[40,61,0.6557,0.55804,0.10326,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,15,0,0,7,0,0,0,0,0],[44,61,0.7213,0.57143,0.11294,0.57143,0.57143,0.60714,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,4,0,0,18,0,0,8,0,0,0,0,0],[48,61,0.7869,0.57143,0.14286,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,7,0,0,14,0,0,8,0,0,0,0,1],[52,61,0.8525,0.55804,0.14445,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,6,0,0,13,0,0,10,0,0,0,0,0],[56,61,0.918,0.57143,0.15152,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,7,0,0,13,0,0,8,0,0,2,0,0],[60,61,0.9836,0.625,0.14174,0.57143,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,14,0,0,1,0,1],[61,61,1.0,0.5625,0.15542,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,10,0,0,13,0,0,6,0,0,1,0,1]]},{"b":6,"e":0.42857,"k":"flat","v":0.53125,"x":0.875,"p":[[0,42,0.0,0.67411,0.17582,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,11,0,0,1,0,5],[4,42,0.0952,0.68304,0.23347,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,9,0,0,5,0,0,7,0,0,1,0,9],[8,42,0.1905,0.79464,0.25238,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,2,0,0,1,0,18],[12,42,0.2857,0.76339,0.25904,0.53571,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,0,0,0,2,0,16],[16,42,0.381,0.84375,0.2212,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,0,1,0,20],[20,42,0.4762,0.83929,0.22232,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,0,0,20],[24,42,0.5714,0.875,0.22517,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,24],[28,42,0.6667,0.55804,0.10326,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,18,0,0,4,0,0,1,0,0],[32,42,0.7619,0.58482,0.12556,0.53571,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,15,0,0,8,0,0,0,0,1],[36,42,0.8571,0.58034,0.16342,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,8,0,0,14,0,0,7,0,0,0,0,2],[40,42,0.9524,0.53125,0.10248,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,11,0,0,16,0,0,4,0,0,0,0,0],[42,42,1.0,0.58036,0.13803,0.42859,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,13,0,0,9,0,0,0,0,1]]}]},{"i":"c2996bec32f51463","q":"Prove that, for all real numbers $a, b, c > 0$ and any real number $t \\geqslant 0$, we have\n\n$$\n\\frac{a}{b+t c}+\\frac{b}{c+t a}+\\frac{c}{a+t b} \\geqslant \\frac{3}{1+t} .\n$$","t":[{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.04911,"p":[[0,15,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.11161,"p":[[0,40,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.11161,0.20119,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,40,0.7,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"209c89ed25150da1","q":"On the side  $ AB$  of a cyclic quadrilateral  $ ABCD$  there is a point  $ X$  such that diagonal  $ BD$  bisects  $ CX$  and diagonal  $ AC$  bisects  $ DX$ . What is the minimum possible value of  $ AB\\over CD$ ?\r\n\r\n*Proposed by S. Berlov*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.2142,"p":[[0,144,0.0,0.13839,0.12619,0.0,0.14286,0.28571,0.0,0.28571,13,0,4,13,0,7,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,144,0.0278,0.18303,0.1394,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,10,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,144,0.0556,0.1875,0.12078,0.14286,0.21428,0.28571,0.0,0.42857,7,0,0,7,0,9,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,144,0.0833,0.17857,0.11845,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,11,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,144,0.1111,0.16518,0.13883,0.0,0.14286,0.28571,0.0,0.4286,10,0,0,10,0,10,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,144,0.1389,0.16518,0.12428,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,144,0.1667,0.16519,0.13416,0.0,0.14286,0.28571,0.0,0.42857,10,0,0,10,0,9,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,144,0.1944,0.14277,0.14286,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,144,0.2222,0.1875,0.1448,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,11,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,144,0.25,0.16071,0.15047,0.0,0.14286,0.28571,0.0,0.4286,12,0,0,12,0,8,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,144,0.2778,0.17857,0.12372,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,144,0.3056,0.12045,0.10777,0.0,0.14286,0.14287,0.0,0.28571,12,0,0,12,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,144,0.3333,0.13393,0.10677,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,144,0.3611,0.16518,0.1017,0.14286,0.14286,0.28571,0.0,0.28571,6,0,0,6,0,15,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,144,0.3889,0.18303,0.15251,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,14,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[60,144,0.4167,0.17857,0.12372,0.14286,0.14286,0.28571,0.0,0.42857,6,0,0,6,0,15,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,144,0.4444,0.2142,0.12882,0.14286,0.21428,0.28571,0.0,0.57143,4,0,0,4,0,12,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[68,144,0.4722,0.1517,0.12847,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[72,144,0.5,0.12946,0.13054,0.0,0.14286,0.2857,0.0,0.28571,15,0,0,15,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,144,0.5278,0.16518,0.09522,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,144,0.5556,0.15178,0.11811,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,13,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,144,0.5833,0.09822,0.09062,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,144,0.6111,0.14731,0.14496,0.0,0.14286,0.28571,0.0,0.571,12,0,0,12,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[92,144,0.6389,0.15625,0.11495,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,11,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,144,0.6667,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,144,0.6944,0.11161,0.12234,0.0,0.14286,0.14289,0.0,0.42857,15,0,0,15,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,144,0.7222,0.09821,0.10972,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,144,0.75,0.1384,0.14054,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,9,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[112,144,0.7778,0.12504,0.11721,0.0,0.14286,0.14286,0.0,0.43,12,0,0,12,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[116,144,0.8056,0.125,0.12752,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,9,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[120,144,0.8333,0.18302,0.11966,0.14286,0.14286,0.2857,0.0,0.571,5,0,0,5,0,15,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[124,144,0.8611,0.08928,0.11152,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,144,0.8889,0.11607,0.10972,0.0,0.14286,0.1429,0.0,0.28571,13,0,0,13,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,144,0.9167,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[136,144,0.9444,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,144,0.9722,0.04911,0.12169,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[144,144,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.00447,"x":0.27232,"p":[[0,194,0.0,0.16072,0.15047,0.0,0.14286,0.28571,0.0,0.4286,13,0,4,13,0,5,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,194,0.0206,0.27232,0.11495,0.14289,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,9,0,0,19,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,194,0.0412,0.21429,0.16751,0.10714,0.28571,0.28571,0.0,0.71429,8,0,0,8,0,7,0,0,12,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[12,194,0.0619,0.20089,0.11769,0.14286,0.2143,0.28571,0.0,0.42857,5,0,0,5,0,11,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,194,0.0825,0.21873,0.14275,0.14286,0.2857,0.28571,0.0,0.571,6,0,0,6,0,8,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,194,0.1031,0.1875,0.12595,0.14286,0.14286,0.28571,0.0,0.42857,6,0,0,6,0,13,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,194,0.1237,0.17858,0.12878,0.0,0.21428,0.28571,0.0,0.42857,9,0,0,9,0,7,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,194,0.1443,0.1875,0.1448,0.0,0.21428,0.28571,0.0,0.57143,9,0,0,9,0,7,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,194,0.1649,0.17857,0.13832,0.0,0.14286,0.28571,0.0,0.4286,9,0,0,9,0,9,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,194,0.1856,0.16071,0.13243,0.0,0.14286,0.28571,0.0,0.4286,10,0,0,10,0,10,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,194,0.2062,0.20537,0.14699,0.10714,0.2857,0.28571,0.0,0.57143,8,0,0,8,0,6,0,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,194,0.2268,0.18303,0.13474,0.0,0.21428,0.28571,0.0,0.42857,9,0,0,9,0,7,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,194,0.2474,0.17857,0.12877,0.0,0.21428,0.28571,0.0,0.42857,9,0,0,9,0,7,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,194,0.268,0.1741,0.12234,0.0,0.21428,0.28571,0.0,0.28571,9,0,0,9,0,7,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,194,0.2887,0.16964,0.12595,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,9,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,194,0.3093,0.12053,0.12428,0.0,0.14286,0.1786,0.0,0.42857,14,0,0,14,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,194,0.3299,0.16518,0.14334,0.0,0.14288,0.28571,0.0,0.42857,12,0,0,12,0,5,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,194,0.3505,0.12946,0.13053,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,8,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,194,0.3711,0.16518,0.12931,0.0,0.21428,0.28571,0.0,0.28571,11,0,0,11,0,5,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,194,0.3918,0.09375,0.12169,0.0,0.0,0.17857,0.0,0.28571,19,0,0,19,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,194,0.4124,0.14732,0.13592,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,194,0.433,0.14286,0.12877,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,6,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,194,0.4536,0.17411,0.13236,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,194,0.4742,0.14732,0.13115,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,8,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,194,0.4948,0.14732,0.14053,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,4,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,194,0.5155,0.125,0.15872,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,10,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[104,194,0.5361,0.12045,0.15197,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,6,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[108,194,0.5567,0.14732,0.13592,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,194,0.5773,0.13393,0.12846,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,194,0.5979,0.16071,0.13243,0.0,0.2143,0.28571,0.0,0.28571,12,0,0,12,0,4,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,194,0.6186,0.15178,0.12846,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,194,0.6392,0.14732,0.15355,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,4,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[128,194,0.6598,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,194,0.6804,0.07143,0.10101,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,194,0.701,0.125,0.12242,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,8,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,194,0.7216,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,194,0.7423,0.09375,0.12168,0.0,0.0,0.17857,0.0,0.28571,19,0,0,19,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,194,0.7629,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,194,0.7835,0.09822,0.12078,0.0,0.0,0.1786,0.0,0.28571,18,0,0,18,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,194,0.8041,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[160,194,0.8247,0.10268,0.12492,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[164,194,0.8454,0.13839,0.12619,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,7,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,194,0.866,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[172,194,0.8866,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[176,194,0.9072,0.0625,0.13803,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[180,194,0.9278,0.04902,0.10471,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[184,194,0.9485,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[188,194,0.9691,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[192,194,0.9897,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[194,194,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1953594f8d6748c1","q":"Prove that there do not exist pairwise distinct complex numbers $a, b, c$, and $d$ such that\n\n$$\na^{3}-b c d=b^{3}-c d a=c^{3}-d a b=d^{3}-a b c .\n$$","t":[{"b":4,"e":0.28571,"k":"falling","v":0.03125,"x":0.73661,"p":[[0,41,0.0,0.41518,0.48096,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,12],[4,41,0.0976,0.57143,0.43006,0.14286,0.57143,1.0,0.0,1.0,7,15,0,7,0,3,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,15],[8,41,0.1951,0.58482,0.42462,0.0,0.78571,1.0,0.0,1.0,9,13,0,9,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,0,3,0,13],[12,41,0.2927,0.65625,0.39263,0.42857,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,17],[16,41,0.3902,0.73661,0.37476,0.42857,1.0,1.0,0.0,1.0,4,20,0,4,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,20],[20,41,0.4878,0.53125,0.4808,0.0,0.78571,1.0,0.0,1.0,14,15,0,14,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,15],[24,41,0.5854,0.21875,0.36243,0.0,0.0,0.32143,0.0,1.0,21,5,0,21,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[28,41,0.6829,0.08036,0.20183,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,41,0.7805,0.05804,0.19186,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,41,0.878,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.23215,0.15047,0.14286,0.2857,0.32164,0.0,0.42857,6,0,0,6,0,8,0,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.22322,0.18877,0.0,0.28571,0.42857,0.0,0.4286,12,0,0,12,0,2,0,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.20982,"x":0.99107,"p":[[0,127,0.0,0.40625,0.46444,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,10],[4,127,0.0315,0.33929,0.46941,0.0,0.0,1.0,0.0,1.0,21,10,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[8,127,0.063,0.23661,0.38896,0.0,0.0,0.42857,0.0,1.0,22,6,0,22,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[12,127,0.0945,0.20982,0.38628,0.0,0.0,0.17857,0.0,1.0,23,6,0,23,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,127,0.126,0.21429,0.39123,0.0,0.0,0.10714,0.0,1.0,24,6,0,24,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,6],[20,127,0.1575,0.27232,0.39506,0.0,0.0,0.5,0.0,1.0,18,6,0,18,0,4,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,6],[24,127,0.189,0.3125,0.41255,0.0,0.0,0.53571,0.0,1.0,18,7,0,18,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,7],[28,127,0.2205,0.35268,0.45314,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,10],[32,127,0.252,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[36,127,0.2835,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,127,0.315,0.94196,0.18509,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[44,127,0.3465,0.9241,0.19228,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,25],[48,127,0.378,0.79464,0.35881,0.71429,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,21],[52,127,0.4094,0.79462,0.348,0.71421,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,21],[56,127,0.4409,0.75893,0.32818,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,3,0,0,0,0,0,1,0,0,7,0,0,1,0,17],[60,127,0.4724,0.81696,0.32583,0.71429,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,1,0,22],[64,127,0.5039,0.60268,0.43848,0.0,0.85714,1.0,0.0,1.0,9,16,0,9,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,16],[68,127,0.5354,0.88393,0.3102,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[72,127,0.5669,0.9375,0.2141,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[76,127,0.5984,0.94196,0.20159,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[80,127,0.6299,0.83482,0.30328,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,24],[84,127,0.6614,0.91518,0.21683,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,25],[88,127,0.6929,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[92,127,0.7244,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[96,127,0.7559,0.85714,0.31944,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[100,127,0.7874,0.58929,0.41764,0.0,0.71429,1.0,0.0,1.0,9,13,0,9,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,0,0,13],[104,127,0.8189,0.50447,0.41952,0.0,0.50001,1.0,0.0,1.0,10,11,0,10,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,0,0,0,11],[108,127,0.8504,0.5982,0.41101,0.21429,0.71429,1.0,0.0,1.0,8,13,0,8,0,0,0,0,3,0,0,1,0,0,2,0,0,4,0,0,1,0,13],[112,127,0.8819,0.52679,0.44239,0.0,0.57143,1.0,0.0,1.0,11,13,0,11,0,0,0,0,3,0,0,2,0,0,0,0,0,3,0,0,0,0,13],[116,127,0.9134,0.59821,0.3984,0.14286,0.71429,1.0,0.0,1.0,6,12,0,6,0,3,0,0,2,0,0,1,0,0,1,0,0,6,0,0,1,0,12],[120,127,0.9449,0.46874,0.42293,0.0,0.64286,0.78571,0.0,1.0,13,8,0,13,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,8],[124,127,0.9764,0.4865,0.38284,0.0,0.42857,0.78571,0.0,1.0,9,8,0,9,0,1,0,0,1,0,0,7,0,0,1,0,0,5,0,0,0,0,8],[127,127,1.0,0.39732,0.18808,0.42857,0.42857,0.42857,0.0,1.0,4,1,0,4,0,0,0,0,1,0,0,25,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"a9fd2dabb7e92258","q":"Say that $\\frac{a}{b}$ is a positive rational number in simplest form, with $a \\neq 1$. Further, say that $n$ is an integer such that:\n\n$$\n\\frac{1}{n}>\\frac{a}{b}>\\frac{1}{n+1}\n$$\n\nShow that when $\\frac{a}{b}-\\frac{1}{n+1}$ is written in simplest form, its numerator is smaller than $a$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,12,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,12,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,12,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,12,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,20,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,20,0.2,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2f237d4b6b4842a1","q":"Real numbers  $a,b,c$  satisfy  $\\tfrac{1}{ab} = b+2c, \\tfrac{1}{bc} = 2c+3a, \\tfrac{1}{ca}=3a+b.$  Then,  $(a+b+c)^3$  can be written as  $\\tfrac{m}{n}$  for relatively prime positive integers  $m$  and  $n.$  Find  $m+n.$","t":[{"b":0,"e":0.85714,"k":"flat","v":0.80803,"x":0.92857,"p":[[0,80,0.0,0.87054,0.22689,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,23],[4,80,0.05,0.85268,0.19393,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,5,0,18],[8,80,0.1,0.83929,0.21651,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,0,0,0,3,0,19],[12,80,0.15,0.89286,0.13832,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,12,0,16],[16,80,0.2,0.80803,0.21011,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,3,0,16],[20,80,0.25,0.82589,0.17029,0.57143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,12,0,11],[24,80,0.3,0.84375,0.18336,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,12,0,13],[28,80,0.35,0.90625,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[32,80,0.4,0.88839,0.10555,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,19,0,11],[36,80,0.45,0.89732,0.10853,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,19,0,12],[40,80,0.5,0.91071,0.11152,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,15],[44,80,0.55,0.91518,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,13,0,17],[48,80,0.6,0.86161,0.10999,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,22,0,7],[52,80,0.65,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[56,80,0.7,0.88839,0.13236,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,0,12],[60,80,0.75,0.90625,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[64,80,0.8,0.88392,0.10374,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,0,9],[68,80,0.85,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[72,80,0.9,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[76,80,0.95,0.91518,0.09354,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,15],[80,80,1.0,0.88839,0.12234,0.85714,0.85714,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,18,0,12]]},{"b":6,"e":0.85714,"k":"rising","v":0.73214,"x":0.94196,"p":[[0,81,0.0,0.73214,0.23351,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,16,0,0,0,0,0,2,0,12],[4,81,0.0494,0.86161,0.20973,0.78571,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,4,0,20],[8,81,0.0988,0.85268,0.19393,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,5,0,18],[12,81,0.1481,0.84821,0.21998,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,0,8,0,17],[16,81,0.1975,0.92857,0.14286,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,24],[20,81,0.2469,0.94196,0.09354,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[24,81,0.2963,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[28,81,0.3457,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[32,81,0.3951,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[36,81,0.4444,0.89285,0.12372,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,13],[40,81,0.4938,0.89286,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,18,0,12],[44,81,0.5432,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[48,81,0.5926,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[52,81,0.642,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[56,81,0.6914,0.91963,0.09412,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[60,81,0.7407,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[64,81,0.7901,0.91071,0.1171,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,14,0,16],[68,81,0.8395,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[72,81,0.8889,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[76,81,0.9383,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[80,81,0.9877,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[81,81,1.0,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9]]}]},{"i":"950c8bfdaae14e9d","q":"Prove there are no integers $a$ and $b$ satisfying the following conditions:\ni) $16 a-9 b$ is a prime number\n\nii) $\\quad a b$ is a perfect square\n\niii) $a+b$ is a perfect square","t":[{"b":5,"e":0.571,"k":"falling","v":0.60267,"x":0.95534,"p":[[0,50,0.0,0.85267,0.21574,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,17],[4,50,0.08,0.91071,0.12753,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[8,50,0.16,0.95534,0.09746,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,50,0.24,0.93303,0.13825,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[16,50,0.32,0.91517,0.14223,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[20,50,0.4,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[24,50,0.48,0.94642,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[28,50,0.56,0.90176,0.15751,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,22],[32,50,0.64,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[36,50,0.72,0.8839,0.16922,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,3,0,20],[40,50,0.8,0.77229,0.19189,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,0,6,0,9],[44,50,0.88,0.7723,0.18511,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,6,0,0,13,0,6],[48,50,0.96,0.65625,0.17806,0.4286,0.71429,0.74996,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,10,0,0,6,0,2],[50,50,1.0,0.60267,0.17762,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,4,0,0,10,0,0,4,0,1]]},{"b":7,"e":0.71429,"k":"flat","v":0.74107,"x":0.97768,"p":[[0,26,0.0,0.88839,0.17399,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,20],[4,26,0.1538,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,26,0.3077,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[12,26,0.4615,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[16,26,0.6154,0.88839,0.13709,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,19],[20,26,0.7692,0.90625,0.13175,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,21],[24,26,0.9231,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[26,26,1.0,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3]]}]},{"i":"b087e9f73f203d20","q":"Let set  $A=\\{1,2,\\ldots,n\\} ,$  and  $X,Y$  be two subsets (not necessarily distinct) of  $A.$  Define that  $\\textup{max} X$  and  $\\textup{min} Y$  represent the greatest element of  $X$  and the least element of  $Y,$  respectively. Determine the number of two-tuples  $(X,Y)$  which satisfies  $\\textup{max} X>\\textup{min} Y.$","t":[{"b":3,"e":0.85714,"k":"flat","v":0.76784,"x":0.85268,"p":[[0,32,0.0,0.85268,0.17307,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,8,0,15],[4,32,0.125,0.84817,0.18196,0.78571,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,9,0,15],[8,32,0.25,0.79464,0.18189,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,9,0,10],[12,32,0.375,0.78571,0.17128,0.57143,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,13,0,7],[16,32,0.5,0.76784,0.13245,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,18,0,1],[20,32,0.625,0.80803,0.09852,0.85711,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,25,0,0],[24,32,0.75,0.78125,0.11837,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,22,0,0],[28,32,0.875,0.83035,0.08328,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,26,0,1],[32,32,1.0,0.78125,0.12869,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,23,0,0]]},{"b":4,"e":0.57143,"k":"falling","v":0.5714,"x":0.83929,"p":[[0,8,0.0,0.83929,0.17405,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,10,0,13],[4,8,0.5,0.82589,0.18466,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,6,0,14],[8,8,1.0,0.5714,0.12877,0.53539,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,20,0,0,1,0,0,2,0,1]]}]},{"i":"a754d03ac3b57ad4","q":"On a line are given  $n$  blue and  $n$  red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors. \n\n(O. Musin)","t":[{"b":1,"e":0.0,"k":"falling","v":0.09375,"x":0.25,"p":[[0,20,0.0,0.25,0.43301,0.0,0.0,0.25,0.0,1.0,24,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,20,0.2,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,20,0.4,0.09821,0.2911,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,20,0.6,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,20,0.8,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[20,20,1.0,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3]]},{"b":2,"e":0.0,"k":"flat","v":0.04911,"x":0.14286,"p":[[0,10,0.0,0.14286,0.33882,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[4,10,0.4,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,10,0.8,0.12945,0.31207,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[10,10,1.0,0.04911,0.18766,0.0,0.0,0.0,0.0,1.0,29,1,1,29,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"8dba18a1bcc3032e","q":"Show that  $19^{93} - 13^{99}$  is a positive integer divisible by  $162$ .","t":[{"b":2,"e":1.0,"k":"falling","v":0.7589,"x":1.0,"p":[[0,38,0.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[4,38,0.1053,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,38,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,38,0.3158,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,38,0.5263,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,38,0.6316,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,38,0.7368,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,38,0.8421,0.93303,0.16746,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[36,38,0.9474,0.88391,0.17658,0.85711,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,17],[38,38,1.0,0.7589,0.23539,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,5,0,0,16,0,5]]},{"b":7,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,24,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,24,0.1667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,24,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,24,0.5,0.94643,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,24,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"8a41f1d570e00177","q":"Let triangle $A B C$ satisfy $2 B C=A B+A C$ and have incenter $I$ and circumcircle $\\omega$. Let $D$ be the intersection of $A I$ and $\\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $A D$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.4732,"x":1.0,"p":[[0,43,0.0,0.81249,0.27535,0.53539,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,2,0,0,0,0,21],[4,43,0.093,0.75893,0.29974,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,8,0,0,0,0,0,1,0,0,0,0,19],[8,43,0.186,0.72321,0.27879,0.42857,0.78571,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,2,0,0,1,0,15],[12,43,0.2791,0.58035,0.24468,0.42857,0.42857,0.74996,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,21,0,0,0,0,0,2,0,0,1,0,7],[16,43,0.3721,0.57143,0.25,0.42857,0.42857,0.67857,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,22,0,0,1,0,0,0,0,0,0,0,8],[20,43,0.4651,0.53572,0.22303,0.42857,0.42857,0.42858,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,6],[24,43,0.5581,0.4732,0.19045,0.42857,0.42857,0.42857,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,25,0,0,2,0,0,0,0,0,0,0,3],[28,43,0.6512,0.54911,0.22047,0.42857,0.42857,0.60714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,22,0,0,1,0,0,2,0,0,1,0,5],[32,43,0.7442,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,43,0.8372,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[40,43,0.9302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[43,43,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.42857,"k":"falling","v":0.43748,"x":0.91964,"p":[[0,38,0.0,0.91964,0.19212,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,27],[4,38,0.1053,0.89286,0.22588,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,26],[8,38,0.2105,0.78125,0.29877,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,8,0,0,0,0,0,0,0,0,1,0,20],[12,38,0.3158,0.76786,0.28738,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,1,0,0,0,0,19],[16,38,0.4211,0.52679,0.22711,0.42857,0.42857,0.57143,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,20,0,0,2,0,0,2,0,0,0,0,5],[20,38,0.5263,0.48661,0.17075,0.42857,0.42857,0.42857,0.1429,1.0,0,2,0,0,0,1,0,0,1,0,0,23,0,0,2,0,0,3,0,0,0,0,2],[24,38,0.6316,0.45089,0.12931,0.42857,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,22,0,0,3,0,0,2,0,0,1,0,0],[28,38,0.7368,0.45759,0.18546,0.42857,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,1,21,0,0,0,0,0,3,0,0,0,0,2],[32,38,0.8421,0.46429,0.17496,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,6,0,0,21,0,0,0,0,0,3,0,0,0,0,2],[36,38,0.9474,0.43748,0.12843,0.42857,0.42857,0.42857,0.2857,0.857,0,0,0,0,0,0,0,0,7,0,0,20,0,0,2,0,0,2,0,0,1,0,0],[38,38,1.0,0.4554,0.09739,0.42857,0.42857,0.42858,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,25,0,0,2,0,0,3,0,0,0,0,0]]}]},{"i":"d66c196ed013310c","q":"Rectangles  $BCC_1B_2,$   $CAA_1C_2,$  and  $ABB_1A_2$  are erected outside an acute triangle  $ABC.$  Suppose that \\[\\angle BC_1C+\\angle CA_1A+\\angle AB_1B=180^{\\circ}.\\] Prove that lines  $B_1C_2,$   $C_1A_2,$  and  $A_1B_2$  are concurrent.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.45097,"x":0.96428,"p":[[0,52,0.0,0.45097,0.26284,0.28571,0.4286,0.57143,0.0,1.0,4,1,3,4,0,3,0,0,3,0,0,7,0,0,8,0,0,4,0,0,2,0,1],[4,52,0.0769,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[8,52,0.1538,0.89286,0.22868,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,24],[12,52,0.2308,0.86607,0.21706,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,20],[16,52,0.3077,0.88392,0.18013,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,18],[20,52,0.3846,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[24,52,0.4615,0.84375,0.24053,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,5,0,18],[28,52,0.5385,0.88392,0.1448,0.82132,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,7,0,17],[32,52,0.6154,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[36,52,0.6923,0.83035,0.21852,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,0,5,0,16],[40,52,0.7692,0.73654,0.29477,0.571,0.857,1.0,0.0,1.0,2,12,0,2,0,1,0,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,12],[44,52,0.8462,0.74999,0.23959,0.67857,0.78571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,7,0,9],[48,52,0.9231,0.83034,0.18365,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,3,0,15],[52,52,1.0,0.68735,0.23265,0.57143,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,3,0,0,6,0,0,7,0,0,9,0,4]]},{"b":1,"e":1.0,"k":"rising","v":0.49106,"x":0.95089,"p":[[0,110,0.0,0.49106,0.2257,0.42857,0.42857,0.60714,0.0,0.85714,3,0,3,3,0,1,0,0,1,0,0,12,0,0,7,0,0,5,0,0,3,0,0],[4,110,0.0364,0.88836,0.17034,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,5,0,20],[8,110,0.0727,0.89285,0.20516,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,20],[12,110,0.1091,0.84372,0.26091,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,19],[16,110,0.1455,0.88392,0.2243,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,21],[20,110,0.1818,0.94195,0.123,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[24,110,0.2182,0.90178,0.20025,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,23],[28,110,0.2545,0.88392,0.20654,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,22],[32,110,0.2909,0.76784,0.29613,0.57132,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,15],[36,110,0.3273,0.87943,0.19602,0.85711,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,19],[40,110,0.3636,0.83473,0.25557,0.71429,1.0,1.0,0.14,1.0,0,19,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,19],[44,110,0.4,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[48,110,0.4364,0.90179,0.21558,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,23],[52,110,0.4727,0.89732,0.20897,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,4,0,23],[56,110,0.5091,0.90623,0.13179,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,19],[60,110,0.5455,0.91071,0.23077,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[64,110,0.5818,0.91517,0.19186,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,23],[68,110,0.6182,0.91517,0.22829,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[72,110,0.6545,0.89729,0.16463,0.85713,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,21],[76,110,0.6909,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[80,110,0.7273,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[84,110,0.7636,0.88839,0.19475,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,21],[88,110,0.8,0.87499,0.19482,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,10,0,17],[92,110,0.8364,0.90179,0.1729,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,5,0,21],[96,110,0.8727,0.91058,0.13736,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,20],[100,110,0.9091,0.84375,0.24054,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,5,0,19],[104,110,0.9455,0.89283,0.17133,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,20],[108,110,0.9818,0.90179,0.18707,0.85714,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,21],[110,110,1.0,0.90402,0.16903,0.85714,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,1,19]]}]},{"i":"efe91fcf6fd298f8","q":"Lines  $b$  and  $c$  passing through vertices  $B$  and  $C$  of triangle  $ABC$  are perpendicular to sideline  $BC$ . The perpendicular bisectors to  $AC$  and  $AB$  meet  $b$  and  $c$  at points  $P$  and  $Q$  respectively. Prove that line  $PQ$  is perpendicular to median  $AM$  of triangle  $ABC$ .\n\n(D. Prokopenko)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.11161,"x":0.15143,"p":[[0,33,0.0,0.13822,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.15143,0.04979,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.07143,"x":0.14286,"p":[[0,23,0.0,0.10705,0.10711,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,18,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.09357,0.09172,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.08027,0.07929,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f19607c179917476","q":"Show that for any real numbers  $x, y > 1$ , we have   $$ \\frac{x^2}{y - 1}+ \\frac{y^2}{x - 1} \\ge  8 $$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.17848,"x":0.27678,"p":[[0,107,0.0,0.1875,0.15746,0.14286,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,16,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,107,0.0374,0.23214,0.17405,0.14286,0.14286,0.42857,0.0,0.57143,6,0,0,6,0,13,0,0,1,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[8,107,0.0748,0.20982,0.13825,0.14286,0.14286,0.32143,0.0,0.4286,3,0,0,3,0,19,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,107,0.1121,0.21874,0.15143,0.14286,0.14286,0.42857,0.0,0.571,3,0,0,3,0,19,0,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[16,107,0.1495,0.18295,0.15253,0.14214,0.14286,0.21432,0.0,0.4286,7,0,0,7,0,17,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[20,107,0.1869,0.21429,0.15152,0.14286,0.14286,0.42857,0.0,0.57143,3,0,0,3,0,20,0,0,0,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[24,107,0.2243,0.20089,0.13767,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,0,18,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,107,0.2617,0.24554,0.14826,0.14286,0.14288,0.42857,0.0,0.42857,3,0,0,3,0,14,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[32,107,0.2991,0.23215,0.15047,0.14286,0.14286,0.42857,0.0,0.42857,3,0,0,3,0,17,0,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[36,107,0.3364,0.18749,0.14476,0.14286,0.14286,0.1786,0.0,0.571,5,0,0,5,0,19,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[40,107,0.3738,0.23214,0.13243,0.14286,0.14286,0.32143,0.0,0.42857,2,0,0,2,0,16,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[44,107,0.4112,0.23661,0.14987,0.14286,0.14286,0.42857,0.0,0.42857,4,0,0,4,0,13,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[48,107,0.4486,0.23214,0.13243,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,18,0,0,6,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[52,107,0.486,0.26339,0.19269,0.14286,0.14288,0.28571,0.0,1.0,1,1,0,1,0,16,0,0,8,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[56,107,0.5234,0.24107,0.15746,0.14286,0.2143,0.28571,0.0,0.71429,3,0,0,3,0,13,0,0,10,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[60,107,0.5607,0.19643,0.12752,0.14286,0.1429,0.28571,0.0,0.57143,5,0,0,5,0,13,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,107,0.5981,0.1874,0.13091,0.14286,0.14286,0.2857,0.0,0.571,4,0,0,4,0,19,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[68,107,0.6355,0.17848,0.12374,0.14214,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,107,0.6729,0.25447,0.13709,0.14286,0.2857,0.32143,0.0,0.57143,2,0,0,2,0,12,0,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[76,107,0.7103,0.19196,0.1411,0.14286,0.14286,0.28571,0.0,0.57143,6,0,0,6,0,14,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[80,107,0.7477,0.20534,0.14254,0.14286,0.14286,0.28571,0.0,0.571,5,0,0,5,0,14,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[84,107,0.785,0.25893,0.11538,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,8,0,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[88,107,0.8224,0.24991,0.13839,0.14286,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,7,0,0,20,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[92,107,0.8598,0.21428,0.10714,0.14286,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,6,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,107,0.8972,0.19196,0.14987,0.10714,0.21428,0.28571,0.0,0.71429,8,0,0,8,0,8,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[100,107,0.9346,0.27678,0.13803,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,4,0,0,24,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[104,107,0.972,0.21866,0.08745,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,11,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[107,107,1.0,0.23661,0.10479,0.14286,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,7,0,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.19197,"x":0.3125,"p":[[0,10,0.0,0.19197,0.15815,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,16,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.25893,0.14032,0.14286,0.14286,0.42857,0.0,0.4286,1,0,0,1,0,16,0,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.27232,0.16115,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,12,0,0,3,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[10,10,1.0,0.3125,0.1448,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,13,0,0,1,0,0,17,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"c0773451a0fbb30b","q":"Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs  $+, -, \\times$  and a special sign  $\\pm$ . Usage of  $\\pm$  is equivalent to usage of  $+$  and  $-$  in all possible combinations. For instance, the expression  $5 \\pm 1$  results in  $\\{4, 6\\}$ , while  $(2 \\pm 0.5) \\pm 0.5$  results in  $\\{1, 2, 3\\}$ .\nCan Peter construct an expression if the numbers on the blackboard are :\n(a)  $1, 2, 4$  ? *( $2$  points)*\n(b) any  $100$  distinct real numbers ? *( $6$  points)*","t":[{"b":2,"e":0.571,"k":"rising","v":0.27677,"x":0.64283,"p":[[0,134,0.0,0.27677,0.11255,0.2857,0.28571,0.28571,0.0,0.57143,3,0,1,3,0,0,0,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,134,0.0299,0.5982,0.24856,0.42857,0.57143,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,7,0,0,4,0,0,11,0,0,3,0,0,0,0,7],[8,134,0.0597,0.56247,0.23941,0.28571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,8,0,0,0,0,0,13,0,0,6,0,0,0,0,4],[12,134,0.0896,0.4375,0.20806,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,19,0,0,0,0,0,9,0,0,2,0,0,0,0,2],[16,134,0.1194,0.55353,0.19805,0.49968,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,0,0,0,18,0,0,3,0,0,0,0,3],[20,134,0.1493,0.55357,0.25939,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,10,0,0,0,0,0,11,0,0,5,0,0,0,0,5],[24,134,0.1791,0.54911,0.18935,0.53571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,1,0,0,17,0,0,5,0,0,0,0,2],[28,134,0.209,0.55803,0.23244,0.28571,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,10,0,0,1,0,0,12,0,0,4,0,0,1,0,4],[32,134,0.2388,0.53571,0.22868,0.28571,0.57143,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,11,0,0,1,0,0,13,0,0,3,0,0,0,0,4],[36,134,0.2687,0.54014,0.19144,0.28571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,9,0,0,0,0,0,16,0,0,5,0,0,0,0,2],[40,134,0.2985,0.50442,0.17121,0.28571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,1,0,0,17,0,0,3,0,0,0,0,1],[44,134,0.3284,0.56695,0.20973,0.42857,0.57143,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,2,0,0,16,0,0,3,0,0,0,0,4],[48,134,0.3582,0.50446,0.18893,0.28571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,13,0,0,6,0,0,0,0,1],[52,134,0.3881,0.46874,0.15662,0.28571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,0,0,0,16,0,0,3,0,0,0,0,0],[56,134,0.4179,0.52032,0.24338,0.28571,0.57121,0.71429,0.22222,1.0,0,4,0,0,0,0,0,1,12,0,0,2,0,0,8,0,0,5,0,0,0,0,4],[60,134,0.4478,0.54015,0.23347,0.28571,0.57143,0.57143,0.0,1.0,1,4,0,1,0,0,0,0,8,0,0,1,0,0,16,0,0,2,0,0,0,0,4],[64,134,0.4776,0.48214,0.22799,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,16,0,0,0,0,0,10,0,0,3,0,0,0,0,3],[68,134,0.5075,0.52232,0.21903,0.28571,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,13,0,0,4,0,0,0,0,3],[72,134,0.5373,0.41066,0.174,0.28571,0.28571,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,19,0,0,2,0,0,9,0,0,1,0,0,0,0,1],[76,134,0.5672,0.50892,0.23129,0.28571,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,14,0,0,0,0,0,11,0,0,3,0,0,1,0,3],[80,134,0.597,0.51783,0.24936,0.28571,0.57121,0.60714,0.0,1.0,1,4,0,1,0,0,0,0,11,0,0,2,0,0,10,0,0,4,0,0,0,0,4],[84,134,0.6269,0.45534,0.18012,0.28571,0.49979,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,1,0,0,13,0,0,2,0,0,0,0,1],[88,134,0.6567,0.55801,0.16505,0.57142,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,1,0,0,21,0,0,5,0,0,0,0,1],[92,134,0.6866,0.55352,0.15871,0.571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,1,0,0,23,0,0,1,0,0,0,0,2],[96,134,0.7164,0.52232,0.18423,0.28571,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,1,0,0,13,0,0,7,0,0,0,0,1],[100,134,0.7463,0.54908,0.11903,0.5713,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,2,0,0,21,0,0,5,0,0,0,0,0],[104,134,0.7761,0.56242,0.08702,0.571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,26,0,0,3,0,0,0,0,0],[108,134,0.806,0.58927,0.09943,0.57143,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,0,9,0,0,0,0,0],[112,134,0.8358,0.60709,0.10716,0.57143,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,7,0,0,0,0,1],[116,134,0.8657,0.58923,0.08565,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,0,4,0,0,1,0,0],[120,134,0.8955,0.58926,0.08565,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,7,0,0,0,0,0],[124,134,0.9254,0.57589,0.12103,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,1,0,0,21,0,0,6,0,0,1,0,0],[128,134,0.9552,0.61161,0.08918,0.57143,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,11,0,0,0,0,0],[132,134,0.9851,0.64283,0.09451,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,13,0,0,0,0,1],[134,134,1.0,0.6071,0.07145,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,9,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.24553,"x":0.6339,"p":[[0,128,0.0,0.24553,0.12992,0.2857,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,0,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,128,0.0312,0.57134,0.27213,0.28571,0.57143,0.71429,0.14,1.0,0,7,0,0,0,1,0,0,10,0,0,1,0,0,10,0,0,3,0,0,0,0,7],[8,128,0.0625,0.52677,0.1729,0.28571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,17,0,0,5,0,0,0,0,1],[12,128,0.0938,0.58482,0.17627,0.57143,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,1,0,0,17,0,0,6,0,0,1,0,2],[16,128,0.125,0.53571,0.19562,0.28571,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,14,0,0,5,0,0,2,0,1],[20,128,0.1562,0.48212,0.18122,0.28571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,15,0,0,3,0,0,0,0,1],[24,128,0.1875,0.48659,0.22548,0.28571,0.57143,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,11,0,0,0,0,0,13,0,0,4,0,0,0,0,2],[28,128,0.2188,0.55357,0.23077,0.28571,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,10,0,0,2,0,0,10,0,0,6,0,0,0,0,4],[32,128,0.25,0.56249,0.24468,0.28571,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,11,0,0,0,0,0,11,0,0,5,0,0,0,0,5],[36,128,0.2812,0.50445,0.19227,0.28571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,1,0,0,16,0,0,2,0,0,0,0,2],[40,128,0.3125,0.49553,0.23954,0.28571,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,13,0,0,0,0,0,11,0,0,4,0,0,0,0,3],[44,128,0.3438,0.4732,0.23538,0.28571,0.49979,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,14,0,0,1,0,0,11,0,0,2,0,0,0,0,3],[48,128,0.375,0.4866,0.20782,0.28571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,1,0,0,11,0,0,4,0,0,0,0,2],[52,128,0.4062,0.54014,0.21048,0.28571,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,15,0,0,4,0,0,0,0,3],[56,128,0.4375,0.49998,0.19232,0.28571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,2,0,0,15,0,0,2,0,0,0,0,2],[60,128,0.4688,0.51337,0.22263,0.28571,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,9,0,0,2,0,0,15,0,0,2,0,0,0,0,3],[64,128,0.5,0.53567,0.18557,0.28571,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,1,0,0,14,0,0,6,0,0,1,0,1],[68,128,0.5312,0.48214,0.23351,0.28571,0.35716,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,15,0,0,1,0,0,6,0,0,6,0,0,1,0,2],[72,128,0.5625,0.51784,0.21053,0.28571,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,11,0,0,1,0,0,15,0,0,2,0,0,0,0,3],[76,128,0.5938,0.5402,0.27135,0.28571,0.57143,0.57143,0.2857,1.0,0,7,0,0,0,0,0,0,13,0,0,2,0,0,10,0,0,0,0,0,0,0,7],[80,128,0.625,0.60266,0.26423,0.49968,0.57143,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,7,0,0,0,0,0,13,0,0,4,0,0,0,0,7],[84,128,0.6562,0.56696,0.25376,0.28571,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,12,0,0,3,0,0,0,0,6],[88,128,0.6875,0.62946,0.2392,0.53572,0.57143,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,6,0,0,2,0,0,11,0,0,6,0,0,0,0,7],[92,128,0.7188,0.62498,0.23077,0.57142,0.57143,0.64286,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,18,0,0,0,0,0,1,0,7],[96,128,0.75,0.62497,0.26667,0.39286,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,8,0,0,1,0,0,12,0,0,2,0,0,0,0,9],[100,128,0.7812,0.54458,0.24073,0.28571,0.57143,0.60714,0.1429,1.0,0,4,0,0,0,1,0,0,10,0,0,0,0,0,13,0,0,3,0,0,1,0,4],[104,128,0.8125,0.6339,0.24469,0.571,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,1,0,0,14,0,0,3,0,0,0,0,8],[108,128,0.8438,0.5625,0.23941,0.28571,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,10,0,0,1,0,0,12,0,0,4,0,0,0,0,5],[112,128,0.875,0.47766,0.21312,0.2857,0.5712,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,13,0,0,1,0,0,9,0,0,7,0,0,0,0,1],[116,128,0.9062,0.51786,0.29613,0.28571,0.28571,0.67857,0.2857,1.0,0,8,0,0,0,0,0,0,17,0,0,2,0,0,5,0,0,0,0,0,0,0,8],[120,128,0.9375,0.44641,0.17767,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,3,0,0,11,0,0,2,0,0,0,0,1],[124,128,0.9688,0.3482,0.11809,0.2857,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[128,128,1.0,0.32589,0.08918,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,3,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"87e207141d59eaa5","q":"Show that the numerator of \\[ \\frac{2^{p-1}}{p+1} - \\left(\\sum_{k = 0}^{p-1}\\frac{\\binom{p-1}{k}}{(1-kp)^2}\\right) \\] is a multiple of  $p^3$  for any odd prime  $p$ .\n\n*Proposed by Yang Liu*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.92411,"x":1.0,"p":[[0,87,0.0,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,87,0.046,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,87,0.092,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,87,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,87,0.1839,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,87,0.2299,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,87,0.2759,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,87,0.3218,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,87,0.3678,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[36,87,0.4138,0.95089,0.15815,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[40,87,0.4598,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,87,0.5057,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[48,87,0.5517,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[52,87,0.5977,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[56,87,0.6437,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[60,87,0.6897,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[64,87,0.7356,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[68,87,0.7816,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[72,87,0.8276,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[76,87,0.8736,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[80,87,0.9195,0.92857,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[84,87,0.9655,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[87,87,1.0,0.92411,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,22]]},{"b":2,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,52,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,52,0.0769,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,52,0.1538,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,52,0.2308,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,52,0.3077,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[20,52,0.3846,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[24,52,0.4615,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[28,52,0.5385,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,52,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,52,0.6923,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e32f719df17674af","q":"Show that there exists a positive integer  $N$  such that for all integers  $a>N$ , there exists a contiguous substring of the decimal expansion of  $a$ , which is divisible by  $2011$ .\nNote. A contiguous substring of an integer  $a$  is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of  $a$ .","t":[{"b":0,"e":0.57143,"k":"flat","v":0.38393,"x":0.55804,"p":[[0,28,0.0,0.51786,0.14174,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,18,0,0,4,0,0,9,0,0,0,0,0],[4,28,0.1429,0.55804,0.15303,0.42857,0.42859,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,17,0,0,3,0,0,11,0,0,0,0,1],[8,28,0.2857,0.47768,0.20705,0.42857,0.42857,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,16,0,0,1,0,0,4,0,0,4,0,0],[12,28,0.4286,0.49552,0.18205,0.42857,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,14,0,0,3,0,0,6,0,0,1,0,1],[16,28,0.5714,0.47321,0.19377,0.42857,0.42857,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,5,0,0,13,0,0,3,0,0,9,0,0,0,0,0],[20,28,0.7143,0.52679,0.12595,0.42857,0.42857,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,18,0,0,7,0,0,6,0,0,1,0,0],[24,28,0.8571,0.44196,0.17627,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,11,0,0,4,0,0,3,0,0,2,0,0],[28,28,1.0,0.38393,0.18013,0.28571,0.28571,0.46429,0.0,0.71429,1,0,0,1,0,3,0,0,13,0,0,7,0,0,4,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.4286,"k":"flat","v":0.47321,"x":0.54907,"p":[[0,30,0.0,0.49999,0.12371,0.42857,0.42857,0.5711,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,23,0,0,3,0,0,5,0,0,1,0,0],[4,30,0.1333,0.50893,0.19541,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,18,0,0,4,0,0,4,0,0,1,0,2],[8,30,0.2667,0.52232,0.17717,0.42857,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,20,0,0,3,0,0,6,0,0,0,0,2],[12,30,0.4,0.54907,0.13882,0.42857,0.49979,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,0,7,0,0,2,0,0],[16,30,0.5333,0.5,0.11845,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,2,0,0,7,0,0,0,0,0],[20,30,0.6667,0.50442,0.11282,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,5,0,0,6,0,0,0,0,0],[24,30,0.8,0.52219,0.14538,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,18,0,0,3,0,0,10,0,0,0,0,0],[28,30,0.9333,0.50893,0.13333,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,17,0,0,7,0,0,5,0,0,1,0,0],[30,30,1.0,0.47321,0.10972,0.42857,0.42857,0.42857,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,27,0,0,1,0,0,3,0,0,1,0,0]]}]},{"i":"b467e236f3ddbcb3","q":"Show that the sequence  $\\{a_{n}\\}_{n \\ge 1}$  defined by  $a_{n}=\\lfloor n\\sqrt{2}\\rfloor$  contains an infinite number of integer powers of  $2$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.3125,"x":0.99554,"p":[[0,39,0.0,0.86161,0.21572,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,5,0,19],[4,39,0.1026,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,39,0.2051,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,39,0.3077,0.9241,0.18552,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[16,39,0.4103,0.95534,0.12599,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[20,39,0.5128,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[24,39,0.6154,0.88393,0.26107,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,22],[28,39,0.7179,0.875,0.23891,0.85714,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,6,0,21],[32,39,0.8205,0.8348,0.25534,0.85711,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,8,0,17],[36,39,0.9231,0.42856,0.35355,0.10714,0.42857,0.75,0.0,1.0,8,4,0,8,0,3,0,0,4,0,0,4,0,0,4,0,0,1,0,0,4,0,4],[39,39,1.0,0.3125,0.27994,0.0,0.21429,0.57143,0.0,0.85714,9,0,0,9,0,7,0,0,2,0,0,3,0,0,7,0,0,2,0,0,2,0,0]]},{"b":5,"e":1.0,"k":"flat","v":0.8616,"x":0.98661,"p":[[0,9,0.0,0.8616,0.26362,0.85711,1.0,1.0,0.0,1.0,2,21,2,2,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,21],[4,9,0.4444,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,9,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[9,9,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"a8eb2a400823d9c7","q":"Sequence  $(a_n)$  is defined as  $a_{n+1}-2a_n+a_{n-1}=7$  for every  $n\\geq 2$ , where  $a_1 = 1, a_2=5$ . What is  $a_{17}$ ? $ \n\\textbf{(A)}\\ 895\n\\qquad\\textbf{(B)}\\ 900\n\\qquad\\textbf{(C)}\\ 905\n\\qquad\\textbf{(D)}\\ 910\n\\qquad\\textbf{(E)}\\ \\text{None of the above}\n$","t":[{"b":2,"e":0.857,"k":"flat","v":0.91071,"x":0.95089,"p":[[0,67,0.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,67,0.0597,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,67,0.1194,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,67,0.1791,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,67,0.2388,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[20,67,0.2985,0.91071,0.06916,0.85714,0.85714,1.0,0.8571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[24,67,0.3582,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[28,67,0.4179,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[32,67,0.4776,0.93303,0.07129,0.85714,1.0,1.0,0.8571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[36,67,0.5373,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[40,67,0.597,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[44,67,0.6567,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[48,67,0.7164,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[52,67,0.7761,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[56,67,0.8358,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[60,67,0.8955,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[64,67,0.9552,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[67,67,1.0,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17]]},{"b":5,"e":0.85714,"k":"flat","v":0.91964,"x":0.97321,"p":[[0,60,0.0,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[4,60,0.0667,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[8,60,0.1333,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[12,60,0.2,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[16,60,0.2667,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[20,60,0.3333,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[24,60,0.4,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[28,60,0.4667,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[32,60,0.5333,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[36,60,0.6,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[40,60,0.6667,0.93303,0.07129,0.85714,1.0,1.0,0.8571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[44,60,0.7333,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[48,60,0.8,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[52,60,0.8667,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[56,60,0.9333,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[60,60,1.0,0.97321,0.05576,1.0,1.0,1.0,0.8571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]}]},{"i":"0d21b18ff49327eb","q":"Prove that there are no rational numbers  $x,y,z$  with  $x+y+z=0$  and  $x^2+y^2+z^2=100$ .","t":[{"b":1,"e":0.85714,"k":"rising","v":0.54909,"x":0.79017,"p":[[0,32,0.0,0.54909,0.29037,0.2857,0.71429,0.85702,0.0,0.85714,1,0,0,1,0,4,0,0,7,0,0,3,0,0,0,0,0,6,0,0,11,0,0],[4,32,0.125,0.63392,0.24206,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,2,0,0,0,0,0,13,0,0,10,0,0],[8,32,0.25,0.74554,0.08552,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,0,9,0,0],[12,32,0.375,0.79017,0.07128,0.71429,0.85707,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0],[16,32,0.5,0.78571,0.07143,0.71429,0.78571,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,16,0,0],[20,32,0.625,0.75446,0.06422,0.71429,0.71429,0.857,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,9,0,0],[24,32,0.75,0.76786,0.06916,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,12,0,0],[28,32,0.875,0.77679,0.07087,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,14,0,0],[32,32,1.0,0.79017,0.07973,0.71429,0.78564,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,15,0,1]]},{"b":6,"e":0.71429,"k":"flat","v":0.62945,"x":0.79016,"p":[[0,25,0.0,0.65625,0.16312,0.64286,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,6,0,0,0,0,0,19,0,0,5,0,0],[4,25,0.16,0.62945,0.29635,0.39286,0.78564,0.85714,0.0,0.85714,2,0,0,2,0,3,0,0,3,0,0,2,0,0,0,0,0,6,0,0,16,0,0],[8,25,0.32,0.75445,0.08174,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,11,0,0],[12,25,0.48,0.77231,0.07872,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,14,0,0],[16,25,0.64,0.77231,0.07015,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,13,0,0],[20,25,0.8,0.75,0.06186,0.71429,0.71429,0.75,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,8,0,0],[24,25,0.96,0.79016,0.07127,0.71429,0.857,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0],[25,25,1.0,0.76784,0.07787,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,13,0,0]]}]},{"i":"da6e02fc2d38faa9","q":"Show that the only polynomial of odd degree satisfying  $p(x^2-1) = p(x)^2 - 1$  for all  $x$  is  $p(x) = x$","t":[{"b":4,"e":0.0,"k":"flat","v":0.10268,"x":0.28125,"p":[[0,49,0.0,0.17848,0.0799,0.14286,0.14286,0.28571,0.0,0.28571,2,0,0,2,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.28125,0.17672,0.24999,0.28571,0.28571,0.0,1.0,2,1,0,2,0,6,0,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,49,0.1633,0.28125,0.18723,0.14286,0.28571,0.28571,0.0,1.0,1,1,0,1,0,9,0,0,19,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[12,49,0.2449,0.1875,0.10972,0.14286,0.14286,0.28571,0.0,0.4286,5,0,0,5,0,13,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.22768,0.13767,0.14286,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,7,0,0,18,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[20,49,0.4082,0.22768,0.15093,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,9,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[24,49,0.4898,0.16518,0.12931,0.0,0.21428,0.28571,0.0,0.28571,11,0,0,11,0,5,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.20536,0.15542,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,12,0,0,10,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[32,49,0.6531,0.24554,0.22934,0.14286,0.14286,0.28571,0.0,1.0,4,2,0,4,0,15,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[36,49,0.7347,0.20973,0.15969,0.14214,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,10,0,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[40,49,0.8163,0.13384,0.12845,0.0,0.14286,0.1786,0.0,0.42857,12,0,0,12,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.14732,0.145,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,49,0.9796,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.14946,"x":0.3125,"p":[[0,79,0.0,0.22322,0.07087,0.14286,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,14,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,79,0.0506,0.3125,0.20652,0.24999,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,8,0,0,20,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[8,79,0.1013,0.18303,0.11425,0.14286,0.21428,0.28571,0.0,0.28571,7,0,0,7,0,9,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,79,0.1519,0.29901,0.20006,0.14286,0.28571,0.28571,0.0,1.0,2,1,0,2,0,7,0,0,17,0,0,3,0,0,1,0,0,0,0,0,1,0,1],[16,79,0.2025,0.21875,0.10705,0.14286,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,5,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,79,0.2532,0.21875,0.17122,0.14286,0.28571,0.28571,0.0,0.85714,6,0,0,6,0,9,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[24,79,0.3038,0.18751,0.10972,0.14286,0.21428,0.28571,0.0,0.286,6,0,0,6,0,10,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,79,0.3544,0.17848,0.12879,0.105,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,10,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,79,0.4051,0.24107,0.20025,0.14286,0.21428,0.28571,0.0,0.85714,5,0,0,5,0,11,0,0,11,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[36,79,0.4557,0.18527,0.14158,0.0,0.2857,0.28571,0.0,0.57143,9,0,0,9,1,5,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,79,0.5063,0.23214,0.18123,0.14286,0.2857,0.28571,0.0,0.85714,5,0,0,5,0,10,0,0,14,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[44,79,0.557,0.14946,0.11894,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,1,12,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,79,0.6076,0.17412,0.12235,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,10,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,79,0.6582,0.26116,0.19773,0.14286,0.2857,0.28571,0.0,1.0,3,1,0,3,1,7,0,0,19,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[56,79,0.7089,0.23661,0.14555,0.14286,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,12,0,0,16,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[60,79,0.7595,0.22321,0.11811,0.14286,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,9,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,79,0.8101,0.2142,0.10108,0.14286,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,8,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,79,0.8608,0.29464,0.06121,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[72,79,0.9114,0.27233,0.09006,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,7,0,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[76,79,0.962,0.25446,0.06902,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[79,79,1.0,0.22768,0.08645,0.14286,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,9,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d439fb3501b1b318","q":"Say that an ordered triple  $(a, b, c)$  is *pleasing* if\n\n(a)  $a$ ,  $b$ , and  $c$  are in the set  $\\{ 1, 2, \\dots, 17 \\}$ , and\n\n(b) both  $b - a$  and  $c - b$  are greater than  $3$ , and at least one of them is equal to  $4$ .\n\nHow many pleasing triples are there?","t":[{"b":3,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,83,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,83,0.0482,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,83,0.0964,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,83,0.1446,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,83,0.1928,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,83,0.241,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,83,0.2892,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,83,0.3373,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,83,0.3855,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,83,0.4337,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,83,0.4819,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,83,0.5301,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,83,0.5783,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,83,0.6265,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,83,0.6747,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,83,0.7229,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,83,0.7711,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,83,0.8193,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,83,0.8675,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,83,0.9157,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,83,0.9639,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[83,83,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,46,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,46,0.087,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,46,0.1739,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,46,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c6cde8125717d617","q":"Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is  $2011.$","t":[{"b":0,"e":1.0,"k":"flat","v":0.85268,"x":1.0,"p":[[0,22,0.0,0.85268,0.34346,1.0,1.0,1.0,0.0,1.0,3,27,2,3,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,22,0.1818,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,22,0.3636,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,22,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.83929,"x":1.0,"p":[[0,45,0.0,0.83929,0.32488,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,24],[4,45,0.0889,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,45,0.1778,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,45,0.2667,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[16,45,0.3556,0.92411,0.24996,1.0,1.0,1.0,0.0,1.0,2,29,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[20,45,0.4444,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,45,0.5333,0.95089,0.17717,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[28,45,0.6222,0.90178,0.22142,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,25],[32,45,0.7111,0.91071,0.20748,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,26],[36,45,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,45,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,45,0.9778,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[45,45,1.0,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27]]}]},{"i":"c6aa5e2a2b26d348","q":"Santa Claus has at least $n$ gifts for $n$ children. For $i \\in\\{1,2, \\ldots, n\\}$, the $i$-th child considers $x_{i}>0$ of these items to be desirable. Assume that\n\n$$\n\\frac{1}{x_{1}}+\\ldots+\\frac{1}{x_{n}} \\leq 1\n$$\n\nProve that Santa Claus can give each child a gift that this child likes.","t":[{"b":0,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,22,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,22,0.1818,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[8,22,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,32,0.0,0.82143,0.33503,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,23],[4,32,0.125,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21],[8,32,0.25,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"20dc2a7c34717f01","q":"Solve over non-negative integers the system   $$  \\begin{cases} x+y+z^2=xyz,  z\\leq min(x,y). \\end{cases}   $$","t":[{"b":2,"e":1.0,"k":"falling","v":0.76786,"x":1.0,"p":[[0,36,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,36,0.1111,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,36,0.2222,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,36,0.5556,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[24,36,0.6667,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[28,36,0.7778,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[32,36,0.8889,0.89731,0.18979,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,23],[36,36,1.0,0.76786,0.21354,0.71429,0.78571,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,9,0,0,6,0,10]]},{"b":5,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,56,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,56,0.0714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,56,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,56,0.3571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,56,0.4286,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[28,56,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,56,0.5714,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,56,0.6429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,56,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,56,0.7857,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,56,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"20720e458ddb86e6","q":"Snow White and the Seven Dwarves are living in their house in the forest. On each of 16 consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine.\n\nProve that, on one of these 16 days, all seven dwarves were collecting berries.","t":[{"b":0,"e":1.0,"k":"rising","v":0.40625,"x":0.96429,"p":[[0,57,0.0,0.40625,0.2055,0.28571,0.42857,0.42857,0.0,1.0,1,2,1,1,0,1,0,0,13,0,0,12,0,0,1,0,0,2,0,0,0,0,2],[4,57,0.0702,0.46652,0.21869,0.42857,0.42857,0.4286,0.07143,1.0,0,3,0,0,1,2,0,0,2,0,0,22,0,0,0,0,0,1,0,0,1,0,3],[8,57,0.1404,0.58036,0.31122,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,11],[12,57,0.2105,0.58927,0.25939,0.42857,0.42857,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,21,0,0,1,0,0,0,0,0,0,0,9],[16,57,0.2807,0.55357,0.27607,0.42857,0.42857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,7,0,0,16,0,0,0,0,0,0,0,0,1,0,8],[20,57,0.3509,0.78571,0.28347,0.42857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,9,0,0,0,0,0,1,0,0,0,0,20],[24,57,0.4211,0.73659,0.25533,0.42857,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,6,0,0,0,0,14],[28,57,0.4912,0.81697,0.2321,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,5,0,0,3,0,17],[32,57,0.5614,0.73214,0.27375,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,4,0,0,2,0,14],[36,57,0.6316,0.85714,0.22868,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,1,0,22],[40,57,0.7018,0.78125,0.26722,0.42859,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,2,0,0,1,0,18],[44,57,0.7719,0.77679,0.26471,0.4286,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,1,0,18],[48,57,0.8421,0.87054,0.2,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,4,0,20],[52,57,0.9123,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[56,57,0.9825,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[57,57,1.0,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24]]},{"b":3,"e":0.71429,"k":"flat","v":0.35714,"x":0.60268,"p":[[0,52,0.0,0.41964,0.20183,0.28571,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,15,0,0,13,0,0,1,0,0,0,0,0,0,0,3],[4,52,0.0769,0.48213,0.20438,0.42857,0.42857,0.42858,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,19,0,0,1,0,0,3,0,0,0,0,3],[8,52,0.1538,0.5,0.25,0.28571,0.42857,0.42857,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,16,0,0,1,0,0,0,0,0,0,0,6],[12,52,0.2308,0.60268,0.28288,0.42857,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,17,0,0,0,0,0,0,0,0,1,0,10],[16,52,0.3077,0.5,0.26726,0.28571,0.42857,0.42858,0.14286,1.0,0,6,0,0,0,2,0,0,7,0,0,16,0,0,0,0,0,0,0,0,1,0,6],[20,52,0.3846,0.49107,0.2141,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,22,0,0,0,0,0,0,0,0,1,0,4],[24,52,0.4615,0.52232,0.2412,0.42857,0.42857,0.5,0.14286,1.0,0,5,0,0,0,1,0,0,4,0,0,19,0,0,0,0,0,2,0,0,1,0,5],[28,52,0.5385,0.48214,0.21053,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,21,0,0,0,0,0,1,0,0,0,0,4],[32,52,0.6154,0.47321,0.21558,0.42857,0.42857,0.42857,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,24,0,0,0,0,0,0,0,0,0,0,4],[36,52,0.6923,0.44643,0.22799,0.28571,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,14,0,0,13,0,0,0,0,0,1,0,0,0,0,4],[40,52,0.7692,0.44196,0.20316,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,9,0,0,18,0,0,0,0,0,1,0,0,0,0,3],[44,52,0.8462,0.43303,0.20356,0.28571,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,13,0,0,15,0,0,0,0,0,1,0,0,0,0,3],[48,52,0.9231,0.41518,0.12037,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,1],[52,52,1.0,0.35714,0.07143,0.28571,0.35714,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eb24a94db50a3043","q":"Prove the inequality\r\n\\[ \\sqrt {a^{1 \\minus{} a}b^{1 \\minus{} b}c^{1 \\minus{} c}} \\le \\frac {1}{3}\r\n\\]\r\nholds for all positive real numbers  $ a$ ,  $ b$  and  $ c$  with  $ a \\plus{} b \\plus{} c \\equal{} 1$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.92857,"x":0.99107,"p":[[0,36,0.0,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[4,36,0.1111,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,36,0.2222,0.92857,0.14286,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,25],[12,36,0.3333,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,36,0.4444,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[20,36,0.5556,0.94196,0.14665,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[24,36,0.6667,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,36,0.7778,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[32,36,0.8889,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[36,36,1.0,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28]]},{"b":2,"e":1.0,"k":"flat","v":0.92857,"x":0.98214,"p":[[0,40,0.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[4,40,0.1,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[8,40,0.2,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[12,40,0.3,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,40,0.4,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,40,0.5,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[24,40,0.6,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[28,40,0.7,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[32,40,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,40,0.9,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[40,40,1.0,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29]]}]},{"i":"c9c32cd00c828962","q":"Some language has only three letters -  $A, B$  and  $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all  $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?","t":[{"b":5,"e":0.42857,"k":"flat","v":0.3616,"x":0.57143,"p":[[0,53,0.0,0.3616,0.18552,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,26,0,0,2,0,0,1,0,0,0,0,0,2,0,1],[4,53,0.0755,0.57143,0.25,0.42857,0.5,0.75,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,11,0,0,5,0,0,3,0,0,4,0,4],[8,53,0.1509,0.47321,0.25862,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,2,0,0,13,0,0,7,0,0,2,0,0,3,0,0,1,0,4],[12,53,0.2264,0.43972,0.24229,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,19,0,0,6,0,0,1,0,0,0,1,0,2,0,3],[16,53,0.3019,0.41964,0.18877,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,5,0,0,5,0,0,2,0,0,1,0,1],[20,53,0.3774,0.39732,0.198,0.28571,0.28571,0.46431,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,2,0,0,5,0,0,1,0,0,0,0,2],[24,53,0.4528,0.45089,0.26027,0.28571,0.28571,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,18,0,0,2,0,0,5,0,0,0,0,0,3,0,3],[28,53,0.5283,0.43303,0.23551,0.28571,0.28571,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,19,0,0,6,0,0,2,0,0,0,0,0,2,0,3],[32,53,0.6038,0.39731,0.19798,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,20,0,0,7,0,0,2,0,0,0,0,0,1,0,2],[36,53,0.6792,0.37946,0.15815,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,20,0,0,8,0,0,1,0,0,1,0,0,2,0,0],[40,53,0.7547,0.37946,0.16982,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,18,0,0,7,0,0,4,0,0,0,0,0,2,0,0],[44,53,0.8302,0.42857,0.23146,0.28571,0.28571,0.4286,0.14286,1.0,0,3,0,0,0,1,0,0,17,0,0,7,0,0,2,0,0,1,0,0,1,0,3],[48,53,0.9057,0.45088,0.22047,0.28571,0.28571,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,18,0,0,3,0,0,4,0,0,3,0,0,3,0,1],[52,53,0.9811,0.38397,0.1208,0.28571,0.28571,0.46536,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,6,0,0,8,0,0,0,0,0,0,0,0],[53,53,1.0,0.4107,0.18122,0.28571,0.28571,0.4642,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,6,0,0,5,0,0,1,0,0,1,0,1]]},{"b":7,"e":0.28571,"k":"flat","v":0.33482,"x":0.58927,"p":[[0,72,0.0,0.35714,0.1675,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,26,0,0,2,0,0,0,0,0,2,0,0,2,0,0],[4,72,0.0556,0.58927,0.2714,0.39286,0.57121,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,7,0,0,7,0,0,2,0,0,5,0,0,6,0,4],[8,72,0.1111,0.51786,0.26905,0.28571,0.42857,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,8,0,0,2,0,0,1,0,0,3,0,5],[12,72,0.1667,0.54468,0.29543,0.28571,0.42857,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,11,0,0,7,0,0,2,0,0,1,0,0,4,0,6],[16,72,0.2222,0.52232,0.21902,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,9,0,0,8,0,0,8,0,0,1,0,0,4,0,2],[20,72,0.2778,0.49107,0.23941,0.28571,0.42857,0.46429,0.2857,1.0,0,3,0,0,0,0,0,0,11,0,0,13,0,0,1,0,0,0,0,0,4,0,3],[24,72,0.3333,0.51339,0.2942,0.28571,0.35714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,15,0,0,3,0,0,3,0,0,1,0,0,4,0,5],[28,72,0.3889,0.51784,0.2714,0.28571,0.42857,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,11,0,0,8,0,0,2,0,0,1,0,0,6,0,3],[32,72,0.4444,0.41518,0.16506,0.28571,0.42857,0.4286,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,12,0,0,4,0,0,0,0,0,1,0,1],[36,72,0.5,0.43754,0.21998,0.28571,0.28571,0.46536,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,6,0,0,2,0,0,1,0,0,4,0,1],[40,72,0.5556,0.53571,0.2369,0.28571,0.5,0.60714,0.2857,1.0,0,3,0,0,0,0,0,0,10,0,0,6,0,0,8,0,0,1,0,0,4,0,3],[44,72,0.6111,0.45089,0.23449,0.28571,0.35714,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,15,0,0,5,0,0,5,0,0,2,0,0,2,0,2],[48,72,0.6667,0.52229,0.249,0.28571,0.42857,0.60714,0.2857,1.0,0,4,0,0,0,0,0,0,11,0,0,8,0,0,5,0,0,1,0,0,3,0,4],[52,72,0.7222,0.47321,0.23265,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,14,0,0,8,0,0,4,0,0,1,0,0,2,0,3],[56,72,0.7778,0.48214,0.21053,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,11,0,0,3,0,0,2,0,0,4,0,1],[60,72,0.8333,0.44643,0.20124,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,7,0,0,5,0,0,3,0,0,0,0,2],[64,72,0.8889,0.41518,0.15303,0.28571,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,14,0,0,12,0,0,2,0,0,3,0,0,1,0,0],[68,72,0.9444,0.34375,0.1063,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,23,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[72,72,1.0,0.33482,0.06785,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a7824d7819b19e1","q":"Suppose  $a$  is a complex number such that\n\\[a^2+a+\\frac{1}{a}+\\frac{1}{a^2}+1=0\\]\nIf  $m$  is a positive integer, find the value of\n\\[a^{2m}+a^m+\\frac{1}{a^m}+\\frac{1}{a^{2m}}\\]","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,28,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,19,0.0,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,19,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,19,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,19,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,19,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"72041030d4b764b7","q":"Real numbers  $a, b, c$  are not equal  $0$  and are solution of the system: $\\begin{cases} a^2 + a =  b^2  b^2 + b = c^2  c^2  +c = a^2 \\end{cases}$ Prove that  $(a - b)(b - c)(c - a) = 1$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.68304,"x":0.80804,"p":[[0,19,0.0,0.77679,0.2549,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,6,0,0,7,0,12],[4,19,0.2105,0.80804,0.21902,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,4,0,0,4,0,15],[8,19,0.4211,0.77232,0.22263,0.67857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,9,0,0,3,0,12],[12,19,0.6316,0.73658,0.25283,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,1,0,0,7,0,0,3,0,0,7,0,10],[16,19,0.8421,0.68304,0.25187,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,2,0,0,0,0,0,6,0,0,10,0,0,5,0,6],[19,19,1.0,0.79451,0.26475,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,7,0,0,6,0,14]]},{"b":4,"e":0.85714,"k":"flat","v":0.74545,"x":0.81696,"p":[[0,20,0.0,0.74553,0.30037,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,1,0,0,2,0,0,3,0,0,2,0,0,7,0,13],[4,20,0.2,0.74545,0.262,0.57143,0.85714,1.0,0.14,1.0,0,10,0,0,0,3,0,0,0,0,0,2,0,0,5,0,0,4,0,0,8,0,10],[8,20,0.4,0.79018,0.2575,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,4,0,0,8,0,13],[12,20,0.6,0.8125,0.23538,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,4,0,0,4,0,16],[16,20,0.8,0.78123,0.18894,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,10,0,8],[20,20,1.0,0.81696,0.20277,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,9,0,12]]}]},{"i":"5c84fa64c7096d5b","q":"Solve in  $\\Bbb{N}^*$  the equation   $$  4^a \\cdot 5^b - 3^c \\cdot 11^d = 1. $$","t":[{"b":4,"e":0.28571,"k":"falling","v":0.29464,"x":0.90624,"p":[[0,90,0.0,0.7232,0.23129,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,9,0,0,5,0,8],[4,90,0.0444,0.79911,0.14664,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,4,0,9],[8,90,0.0889,0.90624,0.11633,0.857,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,18],[12,90,0.1333,0.85267,0.14936,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,6,0,14],[16,90,0.1778,0.87052,0.14883,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,15],[20,90,0.2222,0.625,0.30252,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,8,0,0,1,0,0,1,0,0,6,0,0,7,0,6],[24,90,0.2667,0.62946,0.28089,0.39286,0.71429,0.85704,0.0,1.0,1,6,0,1,0,1,0,0,6,0,0,3,0,0,1,0,0,11,0,0,3,0,6],[28,90,0.3111,0.75891,0.24074,0.71429,0.857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,7,0,0,9,0,9],[32,90,0.3556,0.64277,0.26743,0.5,0.71429,0.85714,0.14,1.0,0,5,0,0,0,3,0,0,5,0,0,0,0,0,3,0,0,12,0,0,4,0,5],[36,90,0.4,0.56694,0.27545,0.28571,0.64286,0.71429,0.0,1.0,1,4,0,1,0,3,0,0,6,0,0,1,0,0,5,0,0,11,0,0,1,0,4],[40,90,0.4444,0.66947,0.26107,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,1,0,0,6,0,0,8,0,0,4,0,7],[44,90,0.4889,0.70088,0.28429,0.57132,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,4,0,0,1,0,0,4,0,0,6,0,0,6,0,9],[48,90,0.5333,0.58026,0.2945,0.28571,0.57143,0.85714,0.14,1.0,0,5,0,0,0,4,0,0,7,0,0,2,0,0,4,0,0,5,0,0,5,0,5],[52,90,0.5778,0.60714,0.27664,0.28571,0.71429,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,10,0,0,1,0,0,2,0,0,10,0,0,2,0,6],[56,90,0.6222,0.67857,0.28794,0.53571,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,5,0,0,1,0,0,4,0,0,6,0,0,6,0,8],[60,90,0.6667,0.61607,0.27994,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,10,0,0,0,0,0,4,0,0,7,0,0,4,0,6],[64,90,0.7111,0.75,0.25754,0.67857,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,7,0,0,6,0,11],[68,90,0.7556,0.52231,0.30641,0.28571,0.49979,0.71429,0.0,1.0,2,4,0,2,0,2,0,0,11,0,0,1,0,0,1,0,0,8,0,0,3,0,4],[72,90,0.8,0.58929,0.27837,0.28571,0.71429,0.75,0.14286,1.0,0,4,0,0,0,3,0,0,8,0,0,1,0,0,2,0,0,10,0,0,4,0,4],[76,90,0.8444,0.65177,0.2765,0.42857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,6,0,0,3,0,0,3,0,0,8,0,0,4,0,7],[80,90,0.8889,0.61604,0.25615,0.28571,0.71429,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,8,0,0,2,0,0,2,0,0,11,0,0,4,0,4],[84,90,0.9333,0.49107,0.3008,0.2857,0.28571,0.75,0.0,1.0,1,5,0,1,0,0,0,0,19,0,0,0,0,0,1,0,0,3,0,0,3,0,5],[88,90,0.9778,0.34373,0.17443,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,25,0,0,0,0,0,2,0,0,2,0,0,0,0,1],[90,90,1.0,0.29464,0.18877,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,2,0,0,24,0,0,1,0,0,0,0,0,0,0,0,1,0,1]]},{"b":6,"e":1.0,"k":"flat","v":0.68298,"x":0.875,"p":[[0,82,0.0,0.68298,0.20746,0.57143,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,12,0,0,10,0,0,0,0,7],[4,82,0.0488,0.83923,0.17776,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,4,0,15],[8,82,0.0976,0.80357,0.24419,0.71429,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,7,0,12],[12,82,0.1463,0.8125,0.18707,0.71429,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,13,0,0,4,0,12],[16,82,0.1951,0.84375,0.13054,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,8,0,11],[20,82,0.2439,0.84821,0.13803,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,5,0,13],[24,82,0.2927,0.81696,0.17215,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,7,0,11],[28,82,0.3415,0.82142,0.18898,0.71429,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,11,0,0,3,0,14],[32,82,0.3902,0.83928,0.15872,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,6,0,13],[36,82,0.439,0.81247,0.16537,0.71429,0.857,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,10,0,9],[40,82,0.4878,0.82603,0.19438,0.71429,0.85714,1.0,0.29,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,5,0,14],[44,82,0.5366,0.77232,0.16698,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,8,0,7],[48,82,0.5854,0.73211,0.2044,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,11,0,0,5,0,7],[52,82,0.6341,0.7232,0.17106,0.67857,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,15,0,0,4,0,5],[56,82,0.6829,0.8348,0.152,0.71429,0.857,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,4,0,13],[60,82,0.7317,0.79464,0.18536,0.71429,0.78571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,13,0,0,7,0,9],[64,82,0.7805,0.875,0.13716,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,8,0,15],[68,82,0.8293,0.80803,0.14987,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,4,0,10],[72,82,0.878,0.7455,0.1417,0.67857,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,14,0,0,5,0,5],[76,82,0.9268,0.77676,0.1426,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,0,2,0,8],[80,82,0.9756,0.71425,0.13367,0.57143,0.71429,0.71429,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,16,0,0,4,0,3],[82,82,1.0,0.74552,0.14168,0.71429,0.71429,0.74996,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,20,0,0,3,0,5]]}]},{"i":"2ed586f7b20831c8","q":"Solve the system of equations:\r\n\r $\r\n\\begin{matrix} \r\nx^2 + x - 1 = y  \r\ny^2 + y - 1 = z  \r\nz^2 + z - 1 = x.\r\n\\end{matrix}\r\n$","t":[{"b":4,"e":0.2857,"k":"rising","v":0.24982,"x":0.47321,"p":[[0,33,0.0,0.24982,0.19571,0.14286,0.21428,0.28571,0.0,1.0,4,1,0,4,0,12,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[4,33,0.1212,0.32141,0.26725,0.14286,0.28571,0.28571,0.0,1.0,4,3,0,4,0,6,0,0,16,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[8,33,0.2424,0.36607,0.3071,0.14286,0.28571,0.42857,0.0,1.0,3,5,0,3,0,9,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,5],[12,33,0.3636,0.27232,0.27747,0.14286,0.14286,0.32143,0.0,1.0,7,3,0,7,0,10,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,3],[16,33,0.4848,0.34375,0.29851,0.14286,0.28571,0.28571,0.0,1.0,2,5,0,2,0,10,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[20,33,0.6061,0.33482,0.33808,0.14286,0.2143,0.32143,0.0,1.0,5,6,0,5,0,11,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,6],[24,33,0.7273,0.35259,0.27897,0.24999,0.28571,0.32143,0.0,1.0,3,4,0,3,0,5,0,0,16,0,0,3,0,0,0,0,0,1,0,0,0,0,4],[28,33,0.8485,0.33036,0.24074,0.24999,0.28571,0.42857,0.0,1.0,4,2,0,4,0,4,0,0,14,0,0,6,0,0,0,0,0,2,0,0,0,0,2],[32,33,0.9697,0.47321,0.29761,0.2857,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,6,0,0,9,0,0,7,0,0,1,0,0,3,0,0,0,0,6],[33,33,1.0,0.44196,0.29744,0.24999,0.42857,0.50002,0.0,1.0,2,5,0,2,0,6,0,0,5,0,0,11,0,0,0,0,0,3,0,0,0,0,5]]},{"b":5,"e":0.28571,"k":"flat","v":0.12501,"x":0.37053,"p":[[0,140,0.0,0.21428,0.21429,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,11,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[4,140,0.0286,0.37053,0.27862,0.2857,0.28571,0.42857,0.0,1.0,4,3,0,4,0,3,0,0,14,0,0,5,0,0,0,0,0,2,0,0,1,0,3],[8,140,0.0571,0.35713,0.27198,0.14286,0.28571,0.42857,0.0,1.0,3,3,0,3,0,7,0,0,11,0,0,4,0,0,2,0,0,2,0,0,0,0,3],[12,140,0.0857,0.26786,0.23077,0.14286,0.28571,0.28571,0.0,1.0,5,2,0,5,0,8,0,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[16,140,0.1143,0.32589,0.23753,0.14286,0.28571,0.28571,0.0,1.0,1,3,0,1,0,8,0,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[20,140,0.1429,0.27678,0.23673,0.14286,0.28571,0.28571,0.0,1.0,6,2,0,6,0,6,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,2],[24,140,0.1714,0.31696,0.25187,0.14286,0.28571,0.42857,0.0,1.0,5,2,0,5,0,5,0,0,13,0,0,4,0,0,1,0,0,2,0,0,0,0,2],[28,140,0.2,0.31696,0.25935,0.14286,0.28571,0.42857,0.0,1.0,4,3,0,4,0,7,0,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[32,140,0.2286,0.29464,0.25489,0.14286,0.28571,0.28571,0.0,1.0,4,3,0,4,0,8,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[36,140,0.2571,0.30347,0.17411,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,8,0,0,14,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[40,140,0.2857,0.24991,0.16757,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[44,140,0.3143,0.28125,0.29339,0.14286,0.2857,0.28571,0.0,1.0,7,4,0,7,0,7,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[48,140,0.3429,0.26339,0.18595,0.14286,0.28571,0.28571,0.0,1.0,3,1,0,3,0,9,0,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[52,140,0.3714,0.19643,0.12753,0.14286,0.14286,0.28571,0.0,0.4286,6,0,0,6,0,11,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,140,0.4,0.20987,0.19885,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,12,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[60,140,0.4286,0.20536,0.11258,0.14286,0.2857,0.28571,0.0,0.42857,5,0,0,5,0,9,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,140,0.4571,0.18303,0.1931,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[68,140,0.4857,0.21875,0.20198,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,14,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[72,140,0.5143,0.25446,0.26422,0.14286,0.2857,0.28571,0.0,1.0,7,3,0,7,0,8,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[76,140,0.5429,0.20089,0.09354,0.14286,0.14286,0.28571,0.0,0.4286,2,0,0,2,0,16,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,140,0.5714,0.19196,0.09182,0.14286,0.14286,0.28571,0.0,0.28571,3,0,0,3,0,15,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,140,0.6,0.23661,0.12169,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,9,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[88,140,0.6286,0.20536,0.11259,0.14286,0.21428,0.28571,0.0,0.4286,4,0,0,4,0,12,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,140,0.6571,0.20982,0.19228,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,11,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[96,140,0.6857,0.2008,0.15513,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,13,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[100,140,0.7143,0.24553,0.20589,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,13,0,0,10,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[104,140,0.7429,0.23661,0.17353,0.14286,0.2857,0.28571,0.0,1.0,4,1,0,4,0,9,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[108,140,0.7714,0.17411,0.12234,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,13,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,140,0.8,0.19196,0.19434,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,7,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[116,140,0.8286,0.19196,0.18073,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,12,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[120,140,0.8571,0.20982,0.10705,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,10,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,140,0.8857,0.1741,0.1223,0.14286,0.14286,0.28571,0.0,0.571,6,0,0,6,0,15,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[128,140,0.9143,0.16964,0.13092,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,12,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[132,140,0.9429,0.15625,0.12037,0.0,0.14286,0.28571,0.0,0.28571,10,0,0,10,0,9,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,140,0.9714,0.15625,0.18681,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[140,140,1.0,0.12501,0.12243,0.0,0.14286,0.2857,0.0,0.286,14,0,0,14,0,8,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"34c09768b1b60dc9","q":"Show that for nonnegative real numbers  $a,b$  and integers  $n\\ge 2$ ,\n\\[\\frac{a^n+b^n}{2}\\ge\\left(\\frac{a+b}{2}\\right)^n\\]\nWhen does equality hold?","t":[{"b":1,"e":1.0,"k":"rising","v":0.50894,"x":0.81249,"p":[[0,19,0.0,0.50894,0.37954,0.14286,0.42857,1.0,0.14286,1.0,0,11,0,0,0,14,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,11],[4,19,0.2105,0.58036,0.40396,0.14286,0.42859,1.0,0.14286,1.0,0,15,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,15],[8,19,0.4211,0.74099,0.38716,0.1429,1.0,1.0,0.14,1.0,0,22,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,22],[12,19,0.6316,0.77229,0.2547,0.67846,0.8571,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,3,0,0,10,0,11],[16,19,0.8421,0.6116,0.30142,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,7,0,0,0,0,0,5,0,0,2,0,0,5,0,0,9,0,4],[19,19,1.0,0.81249,0.22428,0.82132,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4,0,0,14,0,10]]},{"b":4,"e":0.14286,"k":"falling","v":0.29446,"x":0.66964,"p":[[0,34,0.0,0.56241,0.39769,0.14286,0.42857,1.0,0.14,1.0,0,14,0,0,0,13,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,14],[4,34,0.1176,0.66964,0.3787,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,9,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,17],[8,34,0.2353,0.53125,0.38172,0.14286,0.42857,1.0,0.14286,1.0,0,11,0,0,0,14,0,0,0,0,0,3,0,0,1,0,0,3,0,0,0,0,11],[12,34,0.3529,0.32589,0.25313,0.14286,0.14286,0.42857,0.14286,1.0,0,2,0,0,0,18,0,0,1,0,0,8,0,0,0,0,0,3,0,0,0,0,2],[16,34,0.4706,0.41063,0.23901,0.14286,0.42857,0.60714,0.14,1.0,0,1,0,0,0,11,0,0,1,0,0,11,0,0,1,0,0,7,0,0,0,0,1],[20,34,0.5882,0.29446,0.19224,0.14286,0.1429,0.42857,0.14,0.71429,0,0,0,0,0,18,0,0,1,0,0,9,0,0,1,0,0,3,0,0,0,0,0],[24,34,0.7059,0.3125,0.19045,0.14286,0.21429,0.42857,0.14286,0.71429,0,0,0,0,0,16,0,0,1,0,0,11,0,0,1,0,0,3,0,0,0,0,0],[28,34,0.8235,0.31696,0.18466,0.14286,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,15,0,0,1,0,0,13,0,0,0,0,0,3,0,0,0,0,0],[32,34,0.9412,0.30785,0.19282,0.14286,0.1429,0.42857,0.14,0.71429,0,0,0,0,0,17,0,0,0,0,0,11,0,0,1,0,0,3,0,0,0,0,0],[34,34,1.0,0.34375,0.22829,0.14286,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,15,0,0,1,0,0,11,0,0,0,0,0,4,0,0,0,0,1]]}]},{"i":"ccc72af3aaa5d79f","q":"Suppose medians  $m_a$  and  $m_b$  of a triangle are orthogonal. Prove that:\r\n\r\na.) Using medians of that triangle it is possible to construct a rectangular triangle.\r\n\r\nb.) The following inequality: \\[5(a^2+b^2-c^2) \\geq 8ab,\\] is valid, where  $a,b$  and  $c$  are side length of the given triangle.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.93302,"x":0.99107,"p":[[0,50,0.0,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,50,0.08,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,50,0.16,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,50,0.24,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,50,0.32,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,50,0.4,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,50,0.48,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,50,0.56,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[32,50,0.64,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[36,50,0.72,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,50,0.8,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,50,0.88,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,50,0.96,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[50,50,1.0,0.93302,0.11841,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.80357,"x":0.99107,"p":[[0,34,0.0,0.93748,0.11263,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[4,34,0.1176,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[8,34,0.2353,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,34,0.3529,0.92411,0.13825,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,22],[16,34,0.4706,0.94642,0.15465,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,28],[20,34,0.5882,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[24,34,0.7059,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[28,34,0.8235,0.9375,0.18189,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,28],[32,34,0.9412,0.87054,0.21237,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,4,0,21],[34,34,1.0,0.80357,0.23076,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,5,0,0,3,0,16]]}]},{"i":"f1aad49ec501d423","q":"Show that there do not exist strictly positive real numbers $x, y, z$ such that\n\n$$\n\\left(2 x^{2}+y z\\right)\\left(2 y^{2}+x z\\right)\\left(2 z^{2}+x y\\right)=26 x^{2} y^{2} z^{2}\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.79017,"x":0.94643,"p":[[0,36,0.0,0.94643,0.19149,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[4,36,0.1111,0.85267,0.25376,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,20],[8,36,0.2222,0.86159,0.2487,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,22],[12,36,0.3333,0.85268,0.28004,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,22],[16,36,0.4444,0.81696,0.284,0.71429,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,17],[20,36,0.5556,0.87945,0.15615,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,18],[24,36,0.6667,0.83479,0.23178,0.82132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,15],[28,36,0.7778,0.79017,0.33879,0.67857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,20],[32,36,0.8889,0.88393,0.21852,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,20],[36,36,1.0,0.90179,0.2126,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,23]]},{"b":7,"e":0.42857,"k":"flat","v":0.6875,"x":0.98661,"p":[[0,41,0.0,0.94196,0.18161,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[4,41,0.0976,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,41,0.1951,0.75888,0.35256,0.571,1.0,1.0,0.0,1.0,3,19,0,3,0,2,0,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,19],[12,41,0.2927,0.81695,0.26783,0.82143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,2,0,0,8,0,16],[16,41,0.3902,0.82587,0.27604,0.857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,0,7,0,18],[20,41,0.4878,0.88839,0.23072,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,23],[24,41,0.5854,0.84821,0.26229,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,20],[28,41,0.6829,0.81249,0.30187,0.57143,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,0,0,0,1,0,0,5,0,0,0,0,0,3,0,20],[32,41,0.7805,0.70982,0.35978,0.53572,0.85714,1.0,0.0,1.0,4,14,0,4,0,2,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,14],[36,41,0.878,0.6875,0.36498,0.42857,0.85714,1.0,0.0,1.0,3,13,0,3,0,4,0,0,0,0,0,3,0,0,1,0,0,2,0,0,6,0,13],[40,41,0.9756,0.74552,0.362,0.53539,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,0,4,0,18],[41,41,1.0,0.84813,0.23155,0.71429,1.0,1.0,0.14,1.0,0,18,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,18]]}]},{"i":"de44bb882ee7dd7b","q":"Some towns are connected by roads, with at most one road between any two towns. Let  $v$  be the number of towns and  $e$  be the number of roads. Prove that\n\n     $(a)$  if  $e<v-1$ , then there are two towns such that one cannot travel between them;\n     $(b)$  if  $2e>(v-1)(v-2)$ , then one can travel between any two towns.","t":[{"b":3,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,27,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,27,0.7407,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":0.2857,"k":"falling","v":0.33034,"x":0.97768,"p":[[0,31,0.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,31,0.129,0.45088,0.22618,0.28571,0.42857,0.42858,0.2857,1.0,0,4,0,0,0,0,0,0,14,0,0,11,0,0,3,0,0,0,0,0,0,0,4],[8,31,0.2581,0.37946,0.08459,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[12,31,0.3871,0.33034,0.0832,0.28571,0.28571,0.32164,0.2857,0.571,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[16,31,0.5161,0.33928,0.06916,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.39284,0.14724,0.28571,0.35714,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,16,0,0,11,0,0,4,0,0,0,0,0,0,0,1],[24,31,0.7742,0.34375,0.07873,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,20,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[28,31,0.9032,0.375,0.08564,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[31,31,1.0,0.37501,0.11151,0.28571,0.28586,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,11,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"180b9348a0948ad2","q":"Suppose $P(x)$ is a polynomial with real coefficients satsfying the condition $P(\\cos \\theta+\\sin \\theta)=$ $P(\\cos \\theta-\\sin \\theta)$, for every real $\\theta$. Prove that $P(x)$ can be expressed in the form\n\n$$\nP(x)=a_{0}+a_{1}\\left(1-x^{2}\\right)^{2}+a_{2}\\left(1-x^{2}\\right)^{4}+\\cdots+a_{n}\\left(1-x^{2}\\right)^{2 n}\n$$\n\nfor some real numbers $a_{0}, a_{1}, a_{2}, \\ldots, a_{n}$ and nonnegative integer $n$.","t":[{"b":2,"e":0.1429,"k":"falling","v":0.82588,"x":0.97768,"p":[[0,22,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,22,0.1818,0.86159,0.25875,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,23],[8,22,0.3636,0.94196,0.15916,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,26],[12,22,0.5455,0.90179,0.19377,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,23],[16,22,0.7273,0.95982,0.08918,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[20,22,0.9091,0.86606,0.21412,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,4,0,20],[22,22,1.0,0.82588,0.18466,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,12,0,11]]},{"b":6,"e":1.0,"k":"flat","v":0.71873,"x":1.0,"p":[[0,48,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,48,0.0833,0.94643,0.15465,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,28],[8,48,0.1667,0.75,0.26,0.4286,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,8,0,0,3,0,0,2,0,0,5,0,13],[12,48,0.25,0.73661,0.26752,0.42857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,1,0,0,3,0,14],[16,48,0.3333,0.71873,0.25626,0.42859,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,7,0,0,6,0,0,2,0,0,3,0,12],[20,48,0.4167,0.82589,0.21939,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,0,0,0,5,0,17],[24,48,0.5,0.77232,0.25966,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,0,0,0,3,0,16],[28,48,0.5833,0.80362,0.23616,0.57142,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,1,0,0,5,0,16],[32,48,0.6667,0.76786,0.26184,0.4286,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,7,0,0,2,0,0,2,0,0,4,0,15],[36,48,0.75,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[40,48,0.8333,0.90179,0.18707,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,9,0,20],[44,48,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,48,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"61d780836f367be5","q":"Suppose  $ABCD$  is a square piece of cardboard with side length  $a$ . On a plane are two parallel lines  $\\ell_1$  and  $\\ell_2$ , which are also  $a$  units apart. The square  $ABCD$  is placed on the plane so that sides  $AB$  and  $AD$  intersect  $\\ell_1$  at  $E$  and  $F$  respectively. Also, sides  $CB$  and  $CD$  intersect  $\\ell_2$  at  $G$  and  $H$  respectively. Let the perimeters of  $\\triangle AEF$  and  $\\triangle CGH$  be  $m_1$  and  $m_2$  respectively. \r\n\r\nProve that no matter how the square was placed,  $m_1+m_2$  remains constant.","t":[{"b":3,"e":0.0,"k":"falling","v":0.25892,"x":1.0,"p":[[0,120,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,120,0.0333,0.87945,0.22621,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,0,1,0,24],[8,120,0.0667,0.91071,0.17768,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,1,0,24],[12,120,0.1,0.91518,0.20472,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,27],[16,120,0.1333,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,120,0.1667,0.96428,0.14286,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[24,120,0.2,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[28,120,0.2333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,120,0.2667,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[36,120,0.3,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[40,120,0.3333,0.84374,0.24837,0.67846,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,1,0,0,1,0,22],[44,120,0.3667,0.7142,0.31558,0.39286,0.85714,1.0,0.14,1.0,0,16,0,0,0,1,0,0,7,0,0,3,0,0,1,0,0,4,0,0,0,0,16],[48,120,0.4,0.68302,0.31081,0.42859,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,3,0,0,5,0,0,1,0,0,5,0,0,4,0,11],[52,120,0.4333,0.65177,0.37104,0.42857,0.78571,1.0,0.0,1.0,4,14,0,4,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,0,2,0,14],[56,120,0.4667,0.62945,0.36045,0.28571,0.64286,1.0,0.0,1.0,3,13,0,3,0,0,0,0,8,0,0,3,0,0,2,0,0,1,0,0,2,0,13],[60,120,0.5,0.66961,0.28891,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,5,0,0,5,0,0,2,0,0,7,0,0,2,0,10],[64,120,0.5333,0.71872,0.28681,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,6,0,0,2,0,0,3,0,0,2,0,14],[68,120,0.5667,0.67408,0.33548,0.42857,0.71429,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,5,0,0,3,0,0,3,0,0,2,0,13],[72,120,0.6,0.70968,0.30426,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,1,0,0,3,0,0,4,0,0,4,0,0,4,0,0,1,0,14],[76,120,0.6333,0.60259,0.35678,0.28571,0.57144,1.0,0.0,1.0,2,12,0,2,0,2,0,0,8,0,0,4,0,0,0,0,0,3,0,0,1,0,12],[80,120,0.6667,0.6607,0.35129,0.42857,0.71429,1.0,0.0,1.0,3,14,0,3,0,1,0,0,3,0,0,5,0,0,3,0,0,2,0,0,1,0,14],[84,120,0.7,0.52232,0.30849,0.2857,0.4286,0.71429,0.0,1.0,2,6,0,2,0,3,0,0,6,0,0,7,0,0,2,0,0,5,0,0,1,0,6],[88,120,0.7333,0.48213,0.3549,0.28571,0.42857,1.0,0.0,1.0,5,9,0,5,0,1,0,0,8,0,0,8,0,0,1,0,0,0,0,0,0,0,9],[92,120,0.7667,0.63391,0.3387,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,3,0,0,5,0,0,1,0,0,7,0,0,2,0,10],[96,120,0.8,0.54909,0.36265,0.28571,0.4998,1.0,0.0,1.0,4,9,0,4,0,3,0,0,5,0,0,4,0,0,2,0,0,3,0,0,2,0,9],[100,120,0.8333,0.5134,0.31715,0.28571,0.4286,0.71429,0.0,1.0,4,5,0,4,0,1,0,0,6,0,0,7,0,0,1,0,0,6,0,0,2,0,5],[104,120,0.8667,0.60712,0.31135,0.42857,0.57143,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,3,0,0,5,0,0,6,0,0,3,0,0,3,0,8],[108,120,0.9,0.55799,0.32015,0.2857,0.4998,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,9,0,0,4,0,0,3,0,0,4,0,0,1,0,8],[112,120,0.9333,0.5624,0.31743,0.28571,0.42857,1.0,0.0,1.0,2,9,0,2,0,1,0,0,6,0,0,9,0,0,2,0,0,3,0,0,0,0,9],[116,120,0.9667,0.31694,0.25685,0.14286,0.28571,0.46418,0.0,1.0,7,1,0,7,0,4,0,0,10,0,0,3,0,0,5,0,0,1,0,0,1,0,1],[120,120,1.0,0.25892,0.22708,0.0,0.2857,0.4286,0.0,0.857,9,0,0,9,0,6,0,0,6,0,0,7,0,0,2,0,0,1,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.27212,"x":1.0,"p":[[0,76,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,76,0.0526,0.83034,0.22144,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,2,0,18],[8,76,0.1053,0.82142,0.23692,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,6,0,0,1,0,18],[12,76,0.1579,0.82141,0.24745,0.67857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,3,0,0,2,0,19],[16,76,0.2105,0.79461,0.26951,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,5,0,16],[20,76,0.2632,0.81696,0.25313,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,4,0,0,3,0,18],[24,76,0.3158,0.80802,0.28034,0.57132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,1,0,0,2,0,20],[28,76,0.3684,0.74553,0.32485,0.53571,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,4,0,0,1,0,0,1,0,0,5,0,0,1,0,17],[32,76,0.4211,0.7321,0.29615,0.571,0.78571,1.0,0.0,1.0,2,14,0,2,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,0,2,0,14],[36,76,0.4737,0.73658,0.34091,0.57132,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,2,0,0,0,0,0,5,0,0,3,0,0,1,0,17],[40,76,0.5263,0.77231,0.29204,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,1,0,0,1,0,0,7,0,0,2,0,16],[44,76,0.5789,0.70535,0.28333,0.57143,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,2,0,0,4,0,0,8,0,0,2,0,11],[48,76,0.6316,0.76335,0.26394,0.57143,0.78564,1.0,0.0,1.0,1,14,0,1,0,0,0,0,2,0,0,2,0,0,4,0,0,7,0,0,2,0,14],[52,76,0.6842,0.65625,0.346,0.39286,0.71429,1.0,0.0,1.0,2,13,0,2,0,3,0,0,3,0,0,3,0,0,3,0,0,4,0,0,1,0,13],[56,76,0.7368,0.65622,0.37773,0.28571,0.71429,1.0,0.0,1.0,5,14,0,5,0,1,0,0,3,0,0,0,0,0,4,0,0,4,0,0,1,0,14],[60,76,0.7895,0.6741,0.32972,0.49967,0.71429,1.0,0.0,1.0,2,10,0,2,0,3,0,0,3,0,0,0,0,0,3,0,0,6,0,0,5,0,10],[64,76,0.8421,0.62049,0.35654,0.28571,0.71429,1.0,0.0,1.0,3,12,0,3,0,3,0,0,3,0,0,3,0,0,3,0,0,5,0,0,0,0,12],[68,76,0.8947,0.5938,0.32566,0.28571,0.5712,0.895,0.0,1.0,2,8,0,2,0,3,0,0,4,0,0,4,0,0,4,0,0,4,0,0,3,0,8],[72,76,0.9474,0.29015,0.20969,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,10,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[76,76,1.0,0.27212,0.20942,0.14214,0.28571,0.42857,0.0,1.0,6,1,0,6,0,6,0,0,11,0,0,6,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"f313f004e59ed7b0","q":"Show that for any non-negative integer  $n$ , the number  $2^{2n+1}$  cannot be expressed as a sum of four non-zero square numbers.","t":[{"b":2,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,44,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,44,0.0909,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,44,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,44,0.2727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,44,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,44,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,44,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,44,0.6364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,37,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,37,0.1081,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a8777544ea483764","q":"Solve\r\n\\[ \\left\\{ \\begin{array}{l}\r\n \\log_2 x\\plus{}\\log_4 y\\plus{}\\log_4 z\\equal{}2  \r\n \\log_3 y\\plus{}\\log_9 z\\plus{}\\log_9 x\\equal{}2  \r\n \\log_4 z\\plus{}\\log_{16} x\\plus{}\\log_{16} y\\equal{}2  \r\n \\end{array} \\right.\\]","t":[{"b":1,"e":0.71429,"k":"falling","v":0.64732,"x":0.93303,"p":[[0,80,0.0,0.93303,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[4,80,0.05,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20],[8,80,0.1,0.79911,0.17445,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,0,2,0,11],[12,80,0.15,0.74106,0.10971,0.71429,0.71429,0.74996,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,0,0,7,0,1],[16,80,0.2,0.72321,0.14258,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,25,0,0,2,0,3],[20,80,0.25,0.71875,0.12619,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,27,0,0,0,0,3],[24,80,0.3,0.71875,0.10403,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,29,0,0,0,0,2],[28,80,0.35,0.74554,0.08552,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,3],[32,80,0.4,0.70982,0.16554,0.71429,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,25,0,0,0,0,4],[36,80,0.45,0.74554,0.11143,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,0,1,0,4],[40,80,0.5,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[44,80,0.55,0.71429,0.09449,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,29,0,0,1,0,1],[48,80,0.6,0.67857,0.11294,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,29,0,0,0,0,0],[52,80,0.65,0.66964,0.16917,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,25,0,0,0,0,2],[56,80,0.7,0.68304,0.12234,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,27,0,0,0,0,1],[60,80,0.75,0.6875,0.12595,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,28,0,0,0,0,1],[64,80,0.8,0.71429,0.15567,0.71429,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,25,0,0,0,0,4],[68,80,0.85,0.71429,0.07143,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,1],[72,80,0.9,0.64732,0.15561,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,27,0,0,0,0,0],[76,80,0.95,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[80,80,1.0,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"falling","v":0.72321,"x":0.91964,"p":[[0,11,0.0,0.89731,0.12994,0.71429,1.0,1.0,0.714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,19],[4,11,0.3636,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[8,11,0.7273,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[11,11,1.0,0.74554,0.08552,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,3]]}]},{"i":"4242817faf0eabc8","q":"Solve the system of simultaneous equations\n\\[\\sqrt x - \\frac 1y - 2w + 3z = 1,\\]\\[x + \\frac{1}{y^2} - 4w^2 - 9z^2 = 3,\\]\\[x \\sqrt x - \\frac{1}{y^3} - 8w^3 + 27z^3 = -5,\\]\\[x^2 + \\frac{1}{y^4} - 16w^4 - 81z^4 = 15.\\]","t":[{"b":1,"e":0.28571,"k":"falling","v":0.16741,"x":0.98661,"p":[[0,257,0.0,0.75,0.31542,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,4,0,0,4,0,0,1,0,0,2,0,0,1,0,18],[4,257,0.0156,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,257,0.0311,0.77232,0.35689,0.67857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,20],[12,257,0.0467,0.81696,0.34113,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,22],[16,257,0.0623,0.77232,0.34043,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,18],[20,257,0.0778,0.8125,0.33586,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[24,257,0.0934,0.72757,0.37702,0.57132,1.0,1.0,0.0,1.0,4,17,0,4,0,3,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,17],[28,257,0.1089,0.80804,0.32656,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,21],[32,257,0.1245,0.75,0.34069,0.64286,1.0,1.0,0.0,1.0,2,17,0,2,0,2,0,0,3,0,0,1,0,0,0,0,0,4,0,0,3,0,17],[36,257,0.1401,0.875,0.26904,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,24],[40,257,0.1556,0.8125,0.28669,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,4,0,0,3,0,19],[44,257,0.1712,0.81696,0.28624,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,19],[48,257,0.1868,0.81246,0.30815,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,2,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,19],[52,257,0.2023,0.87043,0.2587,0.96429,1.0,1.0,0.14,1.0,0,24,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,24],[56,257,0.2179,0.75,0.37796,0.57143,1.0,1.0,0.0,1.0,3,21,0,3,0,4,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,21],[60,257,0.2335,0.79911,0.31916,0.75,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,4,0,20],[64,257,0.249,0.79018,0.29447,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,18],[68,257,0.2646,0.8125,0.33776,0.82143,1.0,1.0,0.0,1.0,3,21,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,21],[72,257,0.2802,0.92409,0.17854,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[76,257,0.2957,0.83033,0.3143,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,22],[80,257,0.3113,0.80802,0.27805,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,4,0,0,0,0,20],[84,257,0.3268,0.69643,0.37244,0.42857,1.0,1.0,0.0,1.0,3,17,0,3,0,3,0,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,17],[88,257,0.3424,0.78116,0.33707,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,4,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,20],[92,257,0.358,0.68302,0.37413,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,4,0,0,4,0,0,1,0,0,1,0,0,2,0,0,2,0,16],[96,257,0.3735,0.59374,0.41513,0.14286,0.71429,1.0,0.0,1.0,5,14,0,5,0,6,0,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,14],[100,257,0.3891,0.5848,0.35956,0.14289,0.64286,1.0,0.0,1.0,1,10,0,1,0,8,0,0,3,0,0,1,0,0,3,0,0,4,0,0,2,0,10],[104,257,0.4047,0.49999,0.3677,0.14286,0.42859,0.89286,0.0,1.0,2,8,0,2,0,10,0,0,3,0,0,2,0,0,4,0,0,0,0,0,3,0,8],[108,257,0.4202,0.36606,0.38288,0.0,0.14288,0.64286,0.0,1.0,9,6,0,9,0,9,0,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,6],[112,257,0.4358,0.38839,0.39161,0.14286,0.14286,0.78571,0.0,1.0,7,8,0,7,0,11,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,8],[116,257,0.4514,0.62054,0.38731,0.14289,0.71429,1.0,0.0,1.0,2,14,0,2,0,7,0,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,14],[120,257,0.4669,0.49553,0.40086,0.14286,0.42857,1.0,0.0,1.0,4,10,0,4,0,10,0,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,10],[124,257,0.4825,0.32141,0.33692,0.14286,0.14286,0.46418,0.0,1.0,7,4,0,7,0,12,0,0,3,0,0,2,0,0,1,0,0,2,0,0,1,0,4],[128,257,0.4981,0.33929,0.35129,0.0,0.14286,0.57143,0.0,1.0,9,5,0,9,0,9,0,0,1,0,0,3,0,0,4,0,0,1,0,0,0,0,5],[132,257,0.5136,0.35713,0.30722,0.14286,0.2857,0.4642,0.0,1.0,5,4,0,5,0,9,0,0,4,0,0,6,0,0,3,0,0,1,0,0,0,0,4],[136,257,0.5292,0.34821,0.3443,0.0,0.2857,0.46431,0.0,1.0,9,5,0,9,0,5,0,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,5],[140,257,0.5447,0.36608,0.3272,0.14286,0.2857,0.46429,0.0,1.0,4,5,0,4,0,11,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,5],[144,257,0.5603,0.26786,0.25443,0.10714,0.14286,0.42857,0.0,1.0,8,2,0,8,0,9,0,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,2],[148,257,0.5759,0.39732,0.32875,0.14286,0.28571,0.57143,0.0,1.0,3,5,0,3,0,11,0,0,4,0,0,4,0,0,3,0,0,1,0,0,1,0,5],[152,257,0.5914,0.37947,0.32851,0.14286,0.28571,0.57143,0.0,1.0,6,4,0,6,0,8,0,0,3,0,0,5,0,0,3,0,0,2,0,0,1,0,4],[156,257,0.607,0.16741,0.14353,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,10,0,1,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[160,257,0.6226,0.21875,0.20198,0.0,0.14286,0.42857,0.0,0.85714,9,0,0,9,0,10,0,0,3,0,0,9,0,0,0,0,0,0,0,0,1,0,0],[164,257,0.6381,0.3125,0.25862,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,7,0,0,6,0,0,9,0,0,0,0,0,1,0,0,2,0,1],[168,257,0.6537,0.26338,0.2425,0.14286,0.14288,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,4,0,0,4,0,0,4,0,0,1,0,0,0,0,1],[172,257,0.6693,0.26339,0.27689,0.10714,0.14288,0.32143,0.0,1.0,8,3,0,8,0,9,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[176,257,0.6848,0.19643,0.21943,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,11,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[180,257,0.7004,0.24552,0.23751,0.0,0.21428,0.42857,0.0,0.85714,10,0,0,10,0,6,0,0,7,0,0,5,0,0,2,0,0,0,0,0,2,0,0],[184,257,0.71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0,1,0,0,2,0,0,1,0,28],[108,232,0.4655,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[112,232,0.4828,0.95535,0.13092,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[116,232,0.5,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[120,232,0.5172,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[124,232,0.5345,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[128,232,0.5517,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[132,232,0.569,0.95089,0.18073,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[136,232,0.5862,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[140,232,0.6034,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[144,232,0.6207,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[148,232,0.6379,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[152,232,0.6552,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[156,232,0.6724,0.94196,0.17807,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[160,232,0.6897,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[164,232,0.7069,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[168,232,0.7241,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[172,232,0.7414,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[176,232,0.7586,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[180,232,0.7759,0.92411,0.19556,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[184,232,0.7931,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[188,232,0.8103,0.91072,0.2519,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[192,232,0.8276,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[196,232,0.8448,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[200,232,0.8621,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[204,232,0.8793,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[208,232,0.8966,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[212,232,0.9138,0.92848,0.20858,1.0,1.0,1.0,0.14,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[216,232,0.931,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[220,232,0.9483,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[224,232,0.9655,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[228,232,0.9828,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[232,232,1.0,0.87054,0.28652,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,24]]}]},{"i":"6c83e66a621f18dd","q":"Suppose the elements of $A$ are either bounded below or bounded above. Show that if $S$ tiles $A$, then it does so uniquely, i.e., there is a unique tiling of $A$ by $S$.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.82143,"x":0.92857,"p":[[0,19,0.0,0.82143,0.19885,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,9,0,0,6,0,13],[4,19,0.2105,0.92857,0.17128,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,26],[8,19,0.4211,0.89286,0.17128,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,21],[12,19,0.6316,0.91964,0.1234,0.82132,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[16,19,0.8421,0.91071,0.12242,0.82132,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[19,19,1.0,0.84373,0.14448,0.71429,0.78564,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,14]]},{"b":5,"e":0.85714,"k":"flat","v":0.87499,"x":0.94195,"p":[[0,17,0.0,0.87499,0.14617,0.85711,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,13,0,13],[4,17,0.2353,0.875,0.15872,0.82143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,8,0,16],[8,17,0.4706,0.91517,0.11214,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,19],[12,17,0.7059,0.90625,0.11071,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,17],[16,17,0.9412,0.90175,0.14039,0.82132,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20],[17,17,1.0,0.94195,0.09355,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22]]}]},{"i":"cf5a66e3d14e5286","q":"Suppose that  $k,m,n$  are positive integers with  $k \\le n$ . Prove that:\n\\[\\sum_{r=0}^m \\dfrac{k \\binom{m}{r} \\binom{n}{k}}{(r+k) \\binom{m+n}{r+k}} = 1\\]","t":[{"b":2,"e":0.71429,"k":"flat","v":0.70089,"x":0.91517,"p":[[0,82,0.0,0.79911,0.16311,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,0,0,12],[4,82,0.0488,0.80357,0.18123,0.67857,0.71429,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,10,0,0,0,0,14],[8,82,0.0976,0.91071,0.15047,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,23],[12,82,0.1463,0.90624,0.14558,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,1,0,22],[16,82,0.1951,0.81696,0.17941,0.71429,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,0,0,15],[20,82,0.2439,0.90624,0.14558,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,1,0,22],[24,82,0.2927,0.84821,0.25489,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,0,0,22],[28,82,0.3415,0.86606,0.20185,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,0,1,0,20],[32,82,0.3902,0.77677,0.27419,0.57143,0.85714,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,0,0,16],[36,82,0.439,0.85714,0.22868,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,20],[40,82,0.4878,0.83926,0.17772,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,0,0,17],[44,82,0.5366,0.84374,0.18338,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,2,0,17],[48,82,0.5854,0.9107,0.15875,0.92857,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,24],[52,82,0.6341,0.91071,0.16656,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,24],[56,82,0.6829,0.91517,0.15096,0.92857,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,24],[60,82,0.7317,0.80804,0.16602,0.71429,0.71429,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,0,0,13],[64,82,0.7805,0.83482,0.17536,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,0,0,16],[68,82,0.8293,0.73661,0.11355,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,0,0,4],[72,82,0.878,0.73661,0.0724,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,2],[76,82,0.9268,0.72768,0.10326,0.71429,0.71429,0.71429,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,0,0,3],[80,82,0.9756,0.70089,0.14445,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,25,0,0,3,0,1],[82,82,1.0,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1]]},{"b":3,"e":0.71429,"k":"flat","v":0.66963,"x":0.86607,"p":[[0,49,0.0,0.75893,0.15335,0.71429,0.71429,0.78571,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,0,0,0,8],[4,49,0.0816,0.82589,0.21048,0.71429,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,9,0,0,0,0,17],[8,49,0.1633,0.79911,0.18509,0.57143,0.71429,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,14],[12,49,0.2449,0.74106,0.17291,0.71429,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,20,0,0,0,0,7],[16,49,0.3265,0.86607,0.16342,0.71429,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,1,0,18],[20,49,0.4082,0.78125,0.14279,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,0,0,9],[24,49,0.4898,0.73659,0.14336,0.67857,0.71429,0.71429,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,17,0,0,1,0,6],[28,49,0.5714,0.74107,0.14032,0.71429,0.71429,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,18,0,0,1,0,6],[32,49,0.6531,0.69642,0.15048,0.71429,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,26,0,0,0,0,2],[36,49,0.7347,0.69196,0.06298,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,25,0,0,1,0,0],[40,49,0.8163,0.67857,0.07143,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,22,0,0,1,0,0],[44,49,0.898,0.66963,0.09741,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,22,0,0,1,0,0],[48,49,0.9796,0.70982,0.06667,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,3,0,0],[49,49,1.0,0.68304,0.13236,0.71429,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,1,0,0]]}]},{"i":"29adb817ff1baa53","q":"Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.","t":[{"b":3,"e":0.0,"k":"flat","v":0.02232,"x":0.04464,"p":[[0,23,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.01339,"x":0.04018,"p":[[0,13,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"836bc3687c188076","q":"Show that there is a number $1<\\mathrm{b}<1993$ such that if 1994 is written in base $\\mathrm{b}$ then all its digits are the same. Show that there is no number $1<b<1992$ such that if 1993 is written in base $\\mathrm{b}$ then all its digits are the same.","t":[{"b":1,"e":0.857,"k":"flat","v":0.88392,"x":0.94196,"p":[[0,23,0.0,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[4,23,0.1739,0.94195,0.09359,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[8,23,0.3478,0.94196,0.08646,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[12,23,0.5217,0.88839,0.09932,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,12],[16,23,0.6957,0.90177,0.13095,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,18],[20,23,0.8696,0.88392,0.14914,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,18],[23,23,1.0,0.91517,0.12299,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,19]]},{"b":2,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,17,0.0,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[4,17,0.2353,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[8,17,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,17,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,17,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[17,17,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2d97f823a2b3546e","q":"Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.","t":[{"b":3,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,74,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,74,0.0541,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,74,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,74,0.1622,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,74,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,74,0.2703,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,74,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,74,0.3784,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,74,0.4324,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,74,0.4865,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,74,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,74,0.5946,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[48,74,0.6486,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,74,0.7027,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,74,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,74,0.8108,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,74,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,74,0.9189,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,74,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[74,74,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,68,0.0,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,68,0.0588,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,68,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,68,0.1765,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,68,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,68,0.2941,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,68,0.3529,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,68,0.4118,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,68,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,68,0.5294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,68,0.5882,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[44,68,0.6471,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,68,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,68,0.7647,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,68,0.8235,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,68,0.8824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"51171030c87377e2","q":"Suppose the sequence of nonnegative integers $a_{1}, a_{2}, \\ldots, a_{1997}$ satisfies  $$ a_{i}+a_{j} \\leq a_{i+j} \\leq a_{i}+a_{j}+1 $$  for all $i, j \\geq 1$ with $i+j \\leq 1997$. Show that there exists a real number $x$ such that $a_{n}=\\lfloor n x\\rfloor$ for all $1 \\leq n \\leq 1997$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.07134,"x":0.13393,"p":[[0,81,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,81,0.0494,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,81,0.0988,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,81,0.1481,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,81,0.1975,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,81,0.2469,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,81,0.2963,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,81,0.3457,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,81,0.4444,0.09598,0.06588,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,1,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,81,0.4938,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,81,0.5432,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,81,0.642,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,81,0.7407,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,81,0.7901,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01777,"x":0.10705,"p":[[0,56,0.0,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.143,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.04911,0.06786,0.0,0.0,0.14286,0.0,0.143,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.143,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de7586597360d661","q":"Show that there are infinitely many positive integer numbers  $n$  such that  $n^2+1$  has two positive divisors whose difference is  $n$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.54464,"x":0.69197,"p":[[0,18,0.0,0.66517,0.2615,0.42857,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,10,0,0,4,0,0,2,0,0,4,0,9],[4,18,0.2222,0.67856,0.26726,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,9,0,0,4,0,0,5,0,0,2,0,10],[8,18,0.4444,0.69197,0.22335,0.42857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,10,0,0,2,0,8],[12,18,0.6667,0.59824,0.23803,0.42857,0.57121,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,11,0,0,4,0,0,8,0,0,1,0,5],[16,18,0.8889,0.58929,0.25939,0.42857,0.4286,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,14,0,0,2,0,0,4,0,0,1,0,7],[18,18,1.0,0.54464,0.25614,0.42857,0.42857,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,15,0,0,0,0,0,7,0,0,0,0,5]]},{"b":3,"e":1.0,"k":"flat","v":0.72322,"x":0.88392,"p":[[0,26,0.0,0.74094,0.26352,0.42857,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,3,0,0,1,0,15],[4,26,0.1538,0.72322,0.27418,0.42857,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,1,0,0,1,0,15],[8,26,0.3077,0.74553,0.24675,0.53571,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,6,0,0,7,0,10],[12,26,0.4615,0.88392,0.18015,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,21],[16,26,0.6154,0.8482,0.21112,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,3,0,19],[20,26,0.7692,0.81695,0.1864,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,5,0,13],[24,26,0.9231,0.77229,0.21089,0.57143,0.78564,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,4,0,12],[26,26,1.0,0.84821,0.2141,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,3,0,18]]}]},{"i":"adb09ad8262027ad","q":"Suppose   $\\theta_{i}\\in(-\\frac{\\pi}{2},\\frac{\\pi}{2}), i = 1,2,3,4$ . Prove that, there exist  $x\\in \\mathbb{R}$ , satisfying two inequalities \\begin{eqnarray*} \\cos^2\\theta_1\\cos^2\\theta_2-(\\sin\\theta\\sin\\theta_2-x)^2 &\\geq& 0,  \\cos^2\\theta_3\\cos^2\\theta_4-(\\sin\\theta_3\\sin\\theta_4-x)^2 & \\geq & 0 \\end{eqnarray*}\r\nif and only if \\[ \\sum^4_{i=1}\\sin^2\\theta_i\\leq2(1+\\prod^4_{i=1}\\sin\\theta_i + \\prod^4_{i=1}\\cos\\theta_i). \\]","t":[{"b":2,"e":0.57143,"k":"flat","v":0.59374,"x":0.77232,"p":[[0,73,0.0,0.59374,0.16409,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,8,0,0,9,0,0,1,0,2],[4,73,0.0548,0.6875,0.20958,0.53571,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,12,0,0,1,0,7],[8,73,0.1096,0.60714,0.19885,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,8,0,0,2,0,0,14,0,0,2,0,2],[12,73,0.1644,0.6875,0.2299,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,9,0,0,2,0,8],[16,73,0.2192,0.66518,0.16213,0.57143,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,16,0,0,1,0,3],[20,73,0.274,0.71429,0.12877,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,21,0,0,2,0,3],[24,73,0.3288,0.75891,0.16918,0.67857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,14,0,0,1,0,9],[28,73,0.3836,0.69196,0.14334,0.67857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,20,0,0,1,0,3],[32,73,0.4384,0.71874,0.13116,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,20,0,0,3,0,3],[36,73,0.4932,0.7232,0.147,0.71429,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,21,0,0,0,0,5],[40,73,0.5479,0.74553,0.16263,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,16,0,0,2,0,7],[44,73,0.6027,0.72321,0.14258,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,21,0,0,0,0,5],[48,73,0.6575,0.73659,0.16017,0.71429,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,18,0,0,0,0,7],[52,73,0.7123,0.68304,0.14166,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,20,0,0,0,0,3],[56,73,0.7671,0.70536,0.14258,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,21,0,0,0,0,4],[60,73,0.8219,0.77232,0.17445,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,17,0,0,2,0,9],[64,73,0.8767,0.73214,0.13716,0.71429,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,22,0,0,2,0,4],[68,73,0.9315,0.72768,0.17627,0.71429,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,18,0,0,2,0,6],[72,73,0.9863,0.65624,0.15511,0.57143,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,18,0,0,1,0,2],[73,73,1.0,0.71875,0.15765,0.71429,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,20,0,0,2,0,4]]},{"b":5,"e":0.57143,"k":"flat","v":0.57589,"x":0.64731,"p":[[0,9,0.0,0.57589,0.19061,0.42857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,15,0,0,2,0,0,11,0,0,1,0,2],[4,9,0.4444,0.64731,0.17852,0.57143,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,13,0,0,8,0,0,0,0,5],[8,9,0.8889,0.58482,0.12556,0.42859,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,8,0,0,10,0,0,13,0,0,0,0,0],[9,9,1.0,0.60714,0.12877,0.57143,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,6,0,0,10,0,0,14,0,0,1,0,0]]}]},{"i":"857c6c26164aa7f6","q":"Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.","t":[{"b":2,"e":0.85714,"k":"rising","v":0.04911,"x":0.94196,"p":[[0,73,0.0,0.04911,0.15407,0.0,0.0,0.0,0.0,0.85714,26,0,1,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,73,0.0548,0.28571,0.39123,0.0,0.07143,0.60714,0.0,1.0,16,5,0,16,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,5],[8,73,0.1096,0.24097,0.34891,0.0,0.0,0.57111,0.0,1.0,18,2,0,18,0,5,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,2],[12,73,0.1644,0.25893,0.36147,0.0,0.14286,0.21429,0.0,1.0,14,3,0,14,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,3],[16,73,0.2192,0.34822,0.38785,0.0,0.14286,0.75,0.0,1.0,13,4,0,13,0,5,0,0,1,0,0,3,0,0,0,0,0,2,0,0,4,0,4],[20,73,0.274,0.58926,0.38257,0.14286,0.78564,0.85714,0.0,1.0,4,7,0,4,0,7,0,0,0,0,0,1,0,0,1,0,0,3,0,0,9,0,7],[24,73,0.3288,0.5357,0.40564,0.14286,0.71429,0.85714,0.0,1.0,7,7,0,7,0,6,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,7],[28,73,0.3836,0.44643,0.41149,0.0,0.28571,0.85714,0.0,1.0,9,7,0,9,0,7,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,7],[32,73,0.4384,0.57589,0.42028,0.14286,0.85714,1.0,0.0,1.0,6,10,0,6,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,8,0,10],[36,73,0.4932,0.52229,0.38895,0.14286,0.57121,0.85714,0.0,1.0,5,7,0,5,0,8,0,0,0,0,0,1,0,0,4,0,0,1,0,0,6,0,7],[40,73,0.5479,0.625,0.42069,0.14286,0.85714,1.0,0.0,1.0,6,12,0,6,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,12],[44,73,0.6027,0.48213,0.41764,0.10714,0.42836,0.89286,0.0,1.0,8,8,0,8,0,7,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,8],[48,73,0.6575,0.52232,0.40972,0.0,0.71429,0.85714,0.0,1.0,9,6,0,9,0,4,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,6],[52,73,0.7123,0.4775,0.42961,0.105,0.21428,1.0,0.0,1.0,8,9,0,8,0,8,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,9],[56,73,0.7671,0.54018,0.38915,0.14286,0.57144,0.89286,0.0,1.0,3,8,0,3,0,10,0,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,8],[60,73,0.8219,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[64,73,0.8767,0.92857,0.08748,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[68,73,0.9315,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[72,73,0.9863,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[73,73,1.0,0.91517,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,17]]},{"b":7,"e":0.14286,"k":"flat","v":0.01339,"x":0.32588,"p":[[0,53,0.0,0.1517,0.28108,0.0,0.0,0.14286,0.0,1.0,19,2,1,19,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[4,53,0.0755,0.32588,0.3817,0.0,0.14286,0.74996,0.0,1.0,12,4,0,12,0,8,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,4],[8,53,0.1509,0.14732,0.28456,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[12,53,0.2264,0.22321,0.32327,0.0,0.0,0.32143,0.0,1.0,17,2,0,17,0,5,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,2],[16,53,0.3019,0.11607,0.21558,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,53,0.3774,0.29464,0.36932,0.0,0.14286,0.60714,0.0,1.0,13,3,0,13,0,9,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,3],[24,53,0.4528,0.15179,0.27418,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,1],[28,53,0.5283,0.12945,0.2584,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0],[32,53,0.6038,0.10714,0.23145,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[36,53,0.6792,0.11161,0.24675,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[40,53,0.7547,0.15624,0.28202,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,1],[44,53,0.8302,0.22321,0.35703,0.0,0.0,0.21429,0.0,1.0,19,3,0,19,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,3],[48,53,0.9057,0.14731,0.30194,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[52,53,0.9811,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[53,53,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8bd1ac0f15249bda","q":"The altitude  $CH$  of a right triangle  $ABC$ , with  $\\angle{C}=90$ , cut the angles bisectors  $AM$  and  $BN$  at  $P$  and  $Q$ , and let  $R$  and  $S$  be the midpoints of  $PM$  and  $QN$ . Prove that  $RS$  is parallel to the hypotenuse of  $ABC$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,89,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,89,0.0449,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,89,0.0899,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,89,0.1348,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,89,0.1798,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,89,0.2247,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,89,0.2697,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,89,0.3146,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,89,0.3596,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,89,0.4045,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,89,0.4494,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,89,0.4944,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,89,0.5393,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,89,0.5843,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,89,0.6292,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,89,0.6742,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,89,0.7191,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04455,"p":[[0,58,0.0,0.03571,0.12877,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,58,0.069,0.04455,0.1765,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,58,0.1379,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"532a13b5b94cb8a6","q":"Suppose  $A,B\\in M_3(\\mathbb C)$ ,  $B\\ne0$ , and  $AB=0$ . Prove that there exists  $D\\in M_3(\\mathbb C)$  with  $D\\ne0$  such that  $AD=DA=0$ .","t":[{"b":2,"e":0.42857,"k":"flat","v":0.35268,"x":0.41518,"p":[[0,19,0.0,0.38393,0.06622,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.40625,0.08073,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,24,0,0,0,0,0,1,0,0,0,0,0],[8,19,0.4211,0.35268,0.07974,0.28571,0.35714,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.40179,0.05576,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.37946,0.06785,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.41518,0.04164,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.375,"x":0.40178,"p":[[0,33,0.0,0.375,0.07784,0.28571,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,10,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.38393,0.07523,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.39732,0.05906,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.38839,0.06423,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.375,0.06916,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.39732,0.05906,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.38393,0.06622,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.39732,0.05906,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.39732,0.05906,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.40178,0.06622,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0491959c8adaf8a","q":"The difference between the maximal and the minimal diagonals of the regular  $n$ -gon equals to its side (  $n > 5$  ).  Find  $n$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,68,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,68,0.0588,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,68,0.1176,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,68,0.1765,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,68,0.2353,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,68,0.2941,0.94643,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[24,68,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,68,0.4118,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,68,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,68,0.5294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,68,0.5882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,68,0.6471,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,68,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,68,0.7647,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,68,0.8235,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,68,0.8824,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,88,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,88,0.0455,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,88,0.0909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,88,0.1364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,88,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,88,0.2273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,88,0.2727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,88,0.3182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,88,0.3636,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,88,0.4091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,88,0.4545,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,88,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,88,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,88,0.5909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,88,0.6364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,88,0.6818,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,88,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,88,0.7727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,88,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,88,0.8636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,88,0.9091,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,88,0.9545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,88,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"eae3cc62efa701f7","q":"Solve the system  $$ \\begin{cases} x+\\log\\left(x+\\sqrt{x^2+1}\\right)=y \ny+\\log\\left(y+\\sqrt{y^2+1}\\right)=z \nz+\\log\\left(z+\\sqrt{z^2+1}\\right)=x \\end{cases} $$","t":[{"b":3,"e":0.42857,"k":"rising","v":0.23214,"x":0.44643,"p":[[0,66,0.0,0.23214,0.15047,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,23,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[4,66,0.0606,0.23214,0.13243,0.14286,0.14286,0.42857,0.14286,0.4286,0,0,0,0,0,22,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.35705,0.14299,0.14286,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,9,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0],[12,66,0.1818,0.37054,0.14223,0.35714,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,8,0,0,0,0,0,22,0,0,1,0,0,1,0,0,0,0,0],[16,66,0.2424,0.37054,0.14223,0.35714,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,8,0,0,0,0,0,22,0,0,1,0,0,1,0,0,0,0,0],[20,66,0.303,0.44643,0.15872,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,25,0,0,0,0,0,3,0,0,0,0,1],[24,66,0.3636,0.43304,0.11564,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,28,0,0,0,0,0,1,0,0,1,0,0],[28,66,0.4242,0.38393,0.12595,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0],[32,66,0.4848,0.4375,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[36,66,0.5455,0.43304,0.10403,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,27,0,0,1,0,0,2,0,0,0,0,0],[40,66,0.6061,0.4375,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[44,66,0.6667,0.4375,0.13333,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,0,0,0,25,0,0,0,0,0,4,0,0,0,0,0],[48,66,0.7273,0.42857,0.07143,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[52,66,0.7879,0.44197,0.05486,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0],[56,66,0.8485,0.4329,0.07513,0.42857,0.42857,0.42857,0.14286,0.71,0,0,0,0,0,1,0,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0],[60,66,0.9091,0.42857,0.07143,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[64,66,0.9697,0.43754,0.0497,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[66,66,1.0,0.41965,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.1517,"x":0.40179,"p":[[0,47,0.0,0.24107,0.16917,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,23,0,0,0,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[4,47,0.0851,0.22321,0.14698,0.14286,0.14286,0.21429,0.14286,0.71429,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[8,47,0.1702,0.40179,0.19377,0.35714,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,0,0,0,20,0,0,0,0,0,3,0,0,0,0,1],[12,47,0.2553,0.27679,0.17105,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,19,0,0,0,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[16,47,0.3404,0.28571,0.15972,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,17,0,0,0,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[20,47,0.4255,0.2633,0.19925,0.14286,0.14286,0.42857,0.14,0.71429,0,0,0,0,0,22,0,0,1,0,0,5,0,0,0,0,0,4,0,0,0,0,0],[24,47,0.5106,0.22321,0.12846,0.14286,0.14286,0.42857,0.14286,0.42857,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.17857,0.09449,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.22321,0.14698,0.14286,0.14286,0.21429,0.14286,0.71429,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[36,47,0.766,0.22768,0.1984,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,26,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[40,47,0.8511,0.16062,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,47,0.9362,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"27e15bbfdc1174df","q":"The function  $f(x,y)$  satisfies:  $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$  for all non-negative integers  $x,y$ . Find  $f(4,1981)$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.95089,"x":1.0,"p":[[0,37,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,37,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,37,0.4324,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,37,0.5405,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,37,0.8649,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,37,0.973,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[37,37,1.0,0.95089,0.12682,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27]]},{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,24,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,24,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,24,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"fd70f7051864aebe","q":"The circle inscribed in the triangle  $ABC$  is tangent to side  $AC$  at point  $B_1$ , and to side  $BC$  at point  $A_1$ . On the side  $AB$  there is a point  $K$  such that  $AK = KB_1, BK = KA_1$ . Prove that  $ \\angle ACB\\ge 60$","t":[{"b":5,"e":1.0,"k":"flat","v":0.73213,"x":0.9464,"p":[[0,147,0.0,0.92856,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[4,147,0.0272,0.7857,0.29015,0.53539,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,5,0,0,2,0,0,1,0,0,3,0,18],[8,147,0.0544,0.79016,0.31132,0.57132,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,3,0,19],[12,147,0.0816,0.76338,0.25407,0.53571,0.857,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,3,0,0,4,0,14],[16,147,0.1088,0.90179,0.2683,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[20,147,0.1361,0.84374,0.30798,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,23],[24,147,0.1633,0.8616,0.22155,0.71429,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,7,0,0,2,0,20],[28,147,0.1905,0.73213,0.31894,0.42857,0.85707,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,4,0,14],[32,147,0.2177,0.88835,0.16657,0.82143,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,20],[36,147,0.2449,0.91963,0.20184,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[40,147,0.2721,0.88396,0.17646,0.71429,1.0,1.0,0.43,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,2,0,21],[44,147,0.2993,0.9464,0.1225,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[48,147,0.3265,0.79909,0.27863,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,1,0,0,3,0,0,1,0,0,3,0,0,6,0,16],[52,147,0.3537,0.91963,0.13334,0.85711,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,21],[56,147,0.381,0.82142,0.28123,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,2,0,19],[60,147,0.4082,0.83033,0.27535,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,19],[64,147,0.4354,0.79909,0.26694,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,16],[68,147,0.4626,0.85267,0.23551,0.85711,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,6,0,19],[72,147,0.4898,0.88383,0.20373,0.857,1.0,1.0,0.14,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,21],[76,147,0.517,0.91741,0.15269,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,1,0,7,0,21],[80,147,0.5442,0.83487,0.28812,0.85714,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,20],[84,147,0.5714,0.83034,0.23809,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,16],[88,147,0.5986,0.89285,0.21724,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,6,0,21],[92,147,0.6259,0.86159,0.2328,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,20],[96,147,0.6531,0.79903,0.21093,0.57132,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,1,0,0,9,0,12],[100,147,0.6803,0.86604,0.23404,0.857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,7,0,19],[104,147,0.7075,0.74999,0.30306,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,4,0,0,0,0,0,1,0,0,8,0,0,3,0,14],[108,147,0.7347,0.85268,0.23001,0.82132,1.0,1.0,0.143,1.0,0,19,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,0,5,0,19],[112,147,0.7619,0.81249,0.24338,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,5,0,0,5,0,16],[116,147,0.7891,0.82588,0.25936,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,7,0,0,1,0,19],[120,147,0.8163,0.79909,0.25471,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,0,0,17],[124,147,0.8435,0.7857,0.25755,0.57132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,3,0,0,5,0,15],[128,147,0.8707,0.8482,0.23404,0.85714,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,0,7,0,18],[132,147,0.898,0.9107,0.1777,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,23],[136,147,0.9252,0.83033,0.22992,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,0,5,0,16],[140,147,0.9524,0.85264,0.25629,0.82132,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,3,0,21],[144,147,0.9796,0.80354,0.26907,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,8,0,15],[147,147,1.0,0.91069,0.15875,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,22]]},{"b":6,"e":1.0,"k":"flat","v":0.65175,"x":0.98884,"p":[[0,160,0.0,0.875,0.21053,0.85711,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,19],[4,160,0.025,0.84373,0.25093,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,3,0,20],[8,160,0.05,0.72543,0.29457,0.57132,0.78564,1.0,0.0,1.0,1,12,0,1,0,2,0,1,0,0,0,3,0,0,3,0,0,6,0,0,4,0,12],[12,160,0.075,0.74999,0.31543,0.57132,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,4,0,15],[16,160,0.1,0.69642,0.32488,0.42857,0.78564,1.0,0.0,1.0,3,12,0,3,0,0,0,0,2,0,0,5,0,0,1,0,0,5,0,0,4,0,12],[20,160,0.125,0.73214,0.33834,0.42857,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,2,0,0,6,0,0,1,0,0,0,0,0,3,0,17],[24,160,0.15,0.71419,0.31558,0.4286,0.78564,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,0,4,0,0,1,0,0,6,0,0,3,0,13],[28,160,0.175,0.65175,0.34057,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,1,0,0,0,0,0,6,0,0,3,0,0,3,0,0,5,0,10],[32,160,0.2,0.66516,0.35104,0.42857,0.71429,1.0,0.0,1.0,4,13,0,4,0,0,0,0,2,0,0,6,0,0,1,0,0,4,0,0,2,0,13],[36,160,0.225,0.68748,0.34707,0.42857,0.85714,1.0,0.0,1.0,3,13,0,3,0,1,0,0,3,0,0,3,0,0,3,0,0,1,0,0,5,0,13],[40,160,0.25,0.87052,0.21238,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,8,0,18],[44,160,0.275,0.92409,0.13359,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,22],[48,160,0.3,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[52,160,0.325,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[56,160,0.35,0.90177,0.19706,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,21],[60,160,0.375,0.94642,0.0928,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[64,160,0.4,0.93302,0.12873,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,23],[68,160,0.425,0.93079,0.16123,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,1,0,0,0,26],[72,160,0.45,0.91964,0.16342,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,24],[76,160,0.475,0.92854,0.13839,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[80,160,0.5,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[84,160,0.525,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[88,160,0.55,0.94641,0.11157,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[92,160,0.575,0.93749,0.13337,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,26],[96,160,0.6,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[100,160,0.625,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[104,160,0.65,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[108,160,0.675,0.89286,0.19885,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,21],[112,160,0.7,0.92855,0.1429,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,25],[116,160,0.725,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[120,160,0.75,0.93748,0.1285,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[124,160,0.775,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[128,160,0.8,0.95981,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[132,160,0.825,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[136,160,0.85,0.93301,0.13363,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[140,160,0.875,0.95087,0.12687,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[144,160,0.9,0.95087,0.12177,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[148,160,0.925,0.98884,0.03621,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,29],[152,160,0.95,0.94642,0.12243,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[156,160,0.975,0.94195,0.15513,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[160,160,1.0,0.92408,0.13363,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,22]]}]},{"i":"643881324baefaf1","q":"Solve the system in reals:  $\\frac{4-a}{b}=\\frac{5-b}{a}=\\frac{10}{a^2+b^2}$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.17856,"x":0.25892,"p":[[0,57,0.0,0.21866,0.11842,0.14286,0.21428,0.28571,0.0,0.42857,3,0,0,3,0,13,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.22304,0.13344,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,18,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,57,0.1404,0.19196,0.15407,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,14,0,0,10,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,57,0.2105,0.23652,0.14116,0.14286,0.21428,0.28571,0.0,0.71429,2,0,0,2,0,14,0,0,11,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[16,57,0.2807,0.25,0.11294,0.14286,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,15,0,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.23652,0.16607,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,17,0,0,7,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[24,57,0.4211,0.21428,0.11294,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,19,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.24105,0.1765,0.14286,0.2143,0.28571,0.0,0.71429,5,0,0,5,0,11,0,0,9,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[32,57,0.5614,0.23661,0.15406,0.14286,0.2143,0.28571,0.0,0.71429,3,0,0,3,0,13,0,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[36,57,0.6316,0.22312,0.15547,0.14286,0.14286,0.28571,0.0,0.71429,3,0,0,3,0,15,0,0,11,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[40,57,0.7018,0.18304,0.09606,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,18,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.20088,0.14219,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,14,0,0,10,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[48,57,0.8421,0.25892,0.1801,0.14286,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,9,0,0,9,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[52,57,0.9123,0.23214,0.14174,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,19,0,0,10,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[56,57,0.9825,0.20534,0.11254,0.14286,0.14286,0.28571,0.0,0.571,1,0,0,1,0,20,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[57,57,1.0,0.17856,0.10709,0.14286,0.14286,0.14287,0.0,0.571,2,0,0,2,0,23,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.17411,"x":0.2232,"p":[[0,11,0.0,0.2232,0.17832,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,18,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,11,0.3636,0.20089,0.12299,0.14286,0.14286,0.28571,0.0,0.4286,4,0,0,4,0,15,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.17411,0.08552,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,19,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.20972,0.1287,0.14286,0.14286,0.2857,0.0,0.571,2,0,0,2,0,19,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"5c024bd73a60ee13","q":"The Bank of Bath issues coins with an  $H$  on one side and a  $T$  on the other. Harry has  $n$  of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly  $k>0$  coins showing  $H$ , then he turns over the  $k$ th coin from the left; otherwise, all coins show  $T$  and he stops. For example, if  $n=3$  the process starting with the configuration  $THT$  would be  $THT \\to HHT  \\to HTT \\to TTT$ , which stops after three operations.\n\n(a) Show that, for each initial configuration, Harry stops after a finite number of operations.\n\n(b) For each initial configuration  $C$ , let  $L(C)$  be the number of operations before Harry stops. For example,  $L(THT) = 3$  and  $L(TTT) = 0$ . Determine the average value of  $L(C)$  over all  $2^n$  possible initial configurations  $C$ .\n\n*Proposed by David Altizio, USA*","t":[{"b":1,"e":0.85714,"k":"rising","v":0.29027,"x":0.71875,"p":[[0,33,0.0,0.29027,0.26622,0.0,0.28571,0.42857,0.0,0.86,11,0,0,11,0,3,0,0,4,0,0,8,0,0,1,0,0,4,0,0,1,0,0],[4,33,0.1212,0.38393,0.36672,0.14286,0.14286,0.85714,0.0,1.0,6,3,0,6,0,13,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,3],[8,33,0.2424,0.45536,0.34151,0.14286,0.42857,0.71429,0.0,1.0,3,3,0,3,0,11,0,0,1,0,0,3,0,0,0,0,0,7,0,0,4,0,3],[12,33,0.3636,0.69196,0.29038,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,10,0,0,9,0,6],[16,33,0.4848,0.71875,0.26841,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,6,0,0,17,0,3],[20,33,0.6061,0.63393,0.2922,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,1],[24,33,0.7273,0.66518,0.29148,0.67857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,9,0,0,12,0,3],[28,33,0.8485,0.625,0.28738,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,1],[32,33,0.9697,0.69196,0.27919,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,11,0,0,11,0,4],[33,33,1.0,0.66072,0.2335,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,20,0,0,6,0,1]]},{"b":5,"e":0.0,"k":"falling","v":0.0625,"x":0.60714,"p":[[0,58,0.0,0.29911,0.27516,0.0,0.28571,0.42857,0.0,1.0,9,1,1,9,0,3,0,0,10,0,0,5,0,0,0,0,0,2,0,0,2,0,1],[4,58,0.069,0.40179,0.35072,0.14286,0.14286,0.71429,0.0,1.0,4,5,1,4,0,13,0,0,1,0,0,2,0,0,1,0,0,6,0,0,0,0,5],[8,58,0.1379,0.37946,0.37391,0.14286,0.14286,0.71429,0.0,1.0,6,6,0,6,0,13,0,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,6],[12,58,0.2069,0.48213,0.40993,0.14286,0.42857,0.89286,0.0,1.0,7,8,0,7,0,8,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,8],[16,58,0.2759,0.54911,0.4105,0.14286,0.57143,1.0,0.0,1.0,5,12,0,5,0,7,0,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,12],[20,58,0.3448,0.60714,0.39448,0.14286,0.78571,1.0,0.0,1.0,4,12,0,4,0,5,0,0,2,0,0,3,0,0,0,0,0,2,0,0,4,0,12],[24,58,0.4138,0.42839,0.36436,0.14286,0.2857,0.75,0.0,1.0,4,6,0,4,0,11,0,0,2,0,0,4,0,0,0,0,0,3,0,0,2,0,6],[28,58,0.4828,0.51338,0.33855,0.14286,0.42857,0.85714,0.0,1.0,3,6,0,3,0,6,0,0,2,0,0,6,0,0,3,0,0,3,0,0,3,0,6],[32,58,0.5517,0.34821,0.34615,0.14286,0.14286,0.60714,0.0,1.0,7,4,0,7,0,12,0,0,0,0,0,3,0,0,2,0,0,3,0,0,1,0,4],[36,58,0.6207,0.27679,0.32525,0.10714,0.14286,0.32143,0.0,1.0,8,4,0,8,0,14,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,4],[40,58,0.6897,0.29017,0.29983,0.14286,0.14286,0.42857,0.0,1.0,7,2,0,7,0,13,0,0,1,0,0,5,0,0,1,0,0,1,0,0,2,0,2],[44,58,0.7586,0.20089,0.274,0.0,0.14286,0.17857,0.0,1.0,13,1,0,13,0,11,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,1],[48,58,0.8276,0.29464,0.2878,0.14286,0.14286,0.42857,0.0,1.0,5,2,0,5,0,15,0,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,2],[52,58,0.8966,0.13393,0.2141,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,13,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[56,58,0.9655,0.10714,0.11845,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,16,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[58,58,1.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bfebdaddf3ffd10a","q":"Suppose we list the decimal representations of the positive even numbers from left to right. Determine the  $2015^{th}$  digit in the list.","t":[{"b":5,"e":0.71429,"k":"flat","v":0.65624,"x":0.70089,"p":[[0,36,0.0,0.6875,0.06622,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0],[4,36,0.1111,0.66516,0.14111,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0],[8,36,0.2222,0.67409,0.06425,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0],[12,36,0.3333,0.66058,0.06906,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0],[16,36,0.4444,0.6875,0.05576,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[20,36,0.5556,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[24,36,0.6667,0.65624,0.07018,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0],[28,36,0.7778,0.67409,0.06425,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0],[32,36,0.8889,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[36,36,1.0,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.60714,"x":0.70534,"p":[[0,59,0.0,0.6875,0.05576,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[4,59,0.0678,0.60714,0.20203,0.57143,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,3,0,0,0,0,0,1,0,0,5,0,0,22,0,0,0,0,0],[8,59,0.1356,0.6875,0.05576,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[12,59,0.2034,0.64731,0.14279,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,23,0,0,0,0,0],[16,59,0.2712,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[20,59,0.339,0.6875,0.05576,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[24,59,0.4068,0.6875,0.05576,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[28,59,0.4746,0.67397,0.06415,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0],[32,59,0.5424,0.69196,0.05187,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0],[36,59,0.6102,0.67411,0.06423,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0],[40,59,0.678,0.70534,0.03463,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[44,59,0.7458,0.67411,0.06423,0.57143,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0],[48,59,0.8136,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[52,59,0.8814,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[56,59,0.9492,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[59,59,1.0,0.70088,0.04168,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0]]}]},{"i":"c085488aaa6e62bf","q":"The function  $f(n)$  is defined on the positive integers and takes non-negative integer values.  $f(2)=0,f(3)>0,f(9999)=3333$  and for all  $m,n:$  \\[ f(m+n)-f(m)-f(n)=0 \\text{ or } 1. \\] Determine  $f(1982)$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.75446,"x":0.94196,"p":[[0,61,0.0,0.75893,0.24598,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,4,0,0,4,0,13],[4,61,0.0656,0.875,0.1915,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,4,0,20],[8,61,0.1311,0.89286,0.18898,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,4,0,22],[12,61,0.1967,0.89286,0.18558,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,5,0,21],[16,61,0.2623,0.80804,0.25155,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,3,0,17],[20,61,0.3279,0.88838,0.19477,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,4,0,22],[24,61,0.3934,0.86161,0.23279,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,0,2,0,22],[28,61,0.459,0.9375,0.14258,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,6,0,24],[32,61,0.5246,0.91518,0.19186,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,24],[36,61,0.5902,0.91964,0.16728,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,23],[40,61,0.6557,0.94196,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[44,61,0.7213,0.88839,0.21939,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,22],[48,61,0.7869,0.91518,0.15916,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,3,0,23],[52,61,0.8525,0.88393,0.23266,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,5,0,22],[56,61,0.918,0.90625,0.16602,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,4,0,22],[60,61,0.9836,0.89286,0.17496,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,21],[61,61,1.0,0.75446,0.32189,0.42857,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,1,0,0,5,0,0,1,0,0,1,0,0,5,0,16]]},{"b":4,"e":0.85714,"k":"flat","v":0.70982,"x":0.91964,"p":[[0,88,0.0,0.7232,0.2623,0.42857,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,9,0,0,2,0,0,3,0,0,4,0,12],[4,88,0.0455,0.87946,0.2055,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,5,0,21],[8,88,0.0909,0.79018,0.22583,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,2,0,0,5,0,14],[12,88,0.1364,0.79464,0.20183,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,6,0,12],[16,88,0.1818,0.89732,0.19638,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,1,0,24],[20,88,0.2273,0.83036,0.18707,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,3,0,15],[24,88,0.2727,0.8125,0.24856,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,0,2,0,18],[28,88,0.3182,0.90179,0.15335,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,21],[32,88,0.3636,0.85714,0.25254,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,20],[36,88,0.4091,0.91517,0.16312,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,23],[40,88,0.4545,0.85714,0.22588,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,0,6,0,19],[44,88,0.5,0.91072,0.17034,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,5,0,22],[48,88,0.5455,0.85267,0.20974,0.82143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,6,0,18],[52,88,0.5909,0.89732,0.1996,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,5,0,22],[56,88,0.6364,0.79016,0.24741,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,3,0,0,3,0,16],[60,88,0.6818,0.85268,0.20355,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,11,0,15],[64,88,0.7273,0.81695,0.25813,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,0,8,0,15],[68,88,0.7727,0.84374,0.22121,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,5,0,18],[72,88,0.8182,0.83036,0.24074,0.82143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,9,0,15],[76,88,0.8636,0.84821,0.24984,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,2,0,0,3,0,21],[80,88,0.9091,0.91964,0.14698,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,22],[84,88,0.9545,0.8125,0.23538,0.85714,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,13,0,12],[88,88,1.0,0.70982,0.25376,0.42859,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,5,0,0,6,0,9]]}]},{"i":"cf1a4377b24a40e2","q":"The numbers from  $1$  to  $2010$  inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.","t":[{"b":1,"e":0.2857,"k":"flat","v":0.61161,"x":0.81696,"p":[[0,16,0.0,0.69196,0.28372,0.60714,0.78571,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,8,0,0,10,0,6],[4,16,0.25,0.77679,0.28557,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,0,8,0,14],[8,16,0.5,0.81696,0.2321,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,4,0,0,7,0,15],[12,16,0.75,0.73214,0.29613,0.42857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,3,0,0,3,0,0,0,0,0,2,0,0,11,0,10],[16,16,1.0,0.61161,0.30563,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,8,0,0,3,0,0,0,0,0,5,0,0,7,0,6]]},{"b":4,"e":0.42857,"k":"rising","v":0.54464,"x":0.94196,"p":[[0,29,0.0,0.69195,0.31965,0.28571,0.857,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,5,0,0,9,0,9],[4,29,0.1379,0.73214,0.30878,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,2,0,0,2,0,16],[8,29,0.2759,0.74545,0.33085,0.42857,1.0,1.0,0.14,1.0,0,19,0,0,0,4,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,19],[12,29,0.4138,0.86161,0.22724,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,2,0,22],[16,29,0.5517,0.54464,0.27994,0.42857,0.42857,0.78571,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,17,0,0,0,0,0,1,0,0,0,0,8],[20,29,0.6897,0.79464,0.24727,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,3,0,0,2,0,17],[24,29,0.8276,0.91071,0.20439,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,26],[28,29,0.9655,0.83482,0.25532,0.42857,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,22],[29,29,1.0,0.94196,0.16698,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,28]]}]},{"i":"8a010f60742fb793","q":"The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.  Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{A B}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{B A}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{A B}=N_{B A}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13393,"x":0.20536,"p":[[0,51,0.0,0.14732,0.13592,0.14286,0.14286,0.14286,0.0,0.85714,4,0,0,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,51,0.0784,0.17402,0.09271,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,51,0.1569,0.16964,0.11538,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,25,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,51,0.2353,0.20536,0.2111,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,51,0.3137,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,51,0.3922,0.1383,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,51,0.4706,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,51,0.549,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,51,0.6275,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,51,0.7843,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.11607,"x":0.23205,"p":[[0,31,0.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.23205,0.25446,0.14286,0.14286,0.14287,0.0,1.0,2,3,0,2,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,31,0.2581,0.1517,0.06122,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.16072,0.07784,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab163ea7d2e1104a","q":"The function  $f$  from the set  $\\mathbb{N}$  of positive integers into itself is defined by the equality \\[f(n)=\\sum_{k=1}^{n} \\gcd(k,n),\\qquad n\\in \\mathbb{N}.\\]\na) Prove that  $f(mn)=f(m)f(n)$  for every two relatively prime  ${m,n\\in\\mathbb{N}}$ .\n\nb) Prove that for each  $a\\in\\mathbb{N}$  the equation  $f(x)=ax$  has a solution.\n\nc) Find all  ${a\\in\\mathbb{N}}$  such that the equation  $f(x)=ax$  has a unique solution.","t":[{"b":2,"e":0.57143,"k":"falling","v":0.61158,"x":0.99554,"p":[[0,51,0.0,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,51,0.0784,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[8,51,0.1569,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,51,0.2353,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[16,51,0.3137,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,51,0.3922,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,51,0.4706,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,51,0.549,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,51,0.6275,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,51,0.7059,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[40,51,0.7843,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[44,51,0.8627,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[48,51,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[51,51,1.0,0.61158,0.10854,0.57143,0.57143,0.57143,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,3,0,1]]},{"b":7,"e":0.85714,"k":"flat","v":0.95089,"x":1.0,"p":[[0,39,0.0,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[4,39,0.1026,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[8,39,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,39,0.3077,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,39,0.4103,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,39,0.5128,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,39,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,39,0.7179,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,39,0.8205,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[36,39,0.9231,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[39,39,1.0,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]}]},{"i":"92b8e1dbf25f65fc","q":"Suppose, medians  $m_a$  and  $m_b$  of a triangle are orthogonal. Prove that:\n(a) The medians of the triangle correspond to the sides of a right-angled triangle.\n(b) If  $a,b,c$  are the side-lengths of the triangle, then, the following inequality holds:\\[5(a^2+b^2-c^2)\\geq 8ab\\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,41,0.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[4,41,0.0976,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,41,0.1951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,41,0.2927,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.94195,"x":1.0,"p":[[0,27,0.0,0.94195,0.12303,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,0.97544,0.07278,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,1,28],[12,27,0.4444,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,27,0.5926,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[27,27,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"9f928b0e8f0cb83b","q":"The positive integers $\\mathrm{a}, \\mathrm{b}, \\mathrm{c}, \\mathrm{d}, \\mathrm{p}, \\mathrm{q}$ satisfy $\\mathrm{ad}-\\mathrm{bc}=1 \\mathrm{and} \\mathrm{a} / \\mathrm{b}>\\mathrm{p} / \\mathrm{q}>\\mathrm{c} / \\mathrm{d}$. Show that $\\mathrm{q}>=\\mathrm{b}+$ $d$ and that if $q=b+d$, then $p=a+c$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,32,0.0,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[4,32,0.125,0.91518,0.12807,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[8,32,0.25,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[12,32,0.375,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,32,0.5,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,32,0.875,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,32,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.8973,"x":0.98661,"p":[[0,66,0.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[4,66,0.0606,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[8,66,0.1212,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[12,66,0.1818,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,66,0.2424,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[20,66,0.303,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[24,66,0.3636,0.96427,0.11299,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[28,66,0.4242,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[32,66,0.4848,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,66,0.5455,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[40,66,0.6061,0.8973,0.15666,0.82132,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,3,0,21],[44,66,0.6667,0.91964,0.14258,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[48,66,0.7273,0.90625,0.14555,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,1,0,22],[52,66,0.7879,0.90624,0.13179,0.82143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[56,66,0.8485,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[60,66,0.9091,0.94196,0.12807,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,24],[64,66,0.9697,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[66,66,1.0,0.92857,0.13363,0.96429,1.0,1.0,0.5714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,24]]}]},{"i":"7e747cf652dbac64","q":"Suppose that $j(0) j(1) \\cdots j(n-1)$ is a valid juggling sequence. For $i=0,1, \\ldots, n-1$, Let $a_{i}$ denote the remainder of $j(i)+i$ when divided by $n$. Prove that $\\left(a_{0}, a_{1}, \\ldots, a_{n-1}\\right)$ is a permutation of $(0,1, \\ldots, n-1)$.","t":[{"b":2,"e":0.1429,"k":"flat","v":0.23214,"x":0.36607,"p":[[0,36,0.0,0.36607,0.20183,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,10,0,0,4,0,0,4,0,0,5,0,0,0,0,0],[4,36,0.1111,0.23214,0.12752,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,13,0,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,36,0.2222,0.26786,0.14174,0.14286,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,10,0,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[12,36,0.3333,0.30357,0.15465,0.2857,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,6,0,0,18,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[16,36,0.4444,0.31249,0.23264,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,9,0,0,9,0,0,7,0,0,2,0,0,0,0,0,0,0,2],[20,36,0.5556,0.23661,0.13175,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,12,0,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[24,36,0.6667,0.30339,0.19165,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,9,0,0,8,0,0,7,0,0,3,0,0,2,0,0,0,0,0],[28,36,0.7778,0.26339,0.16016,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,10,0,0,12,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[32,36,0.8889,0.29464,0.18877,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,10,0,0,8,0,0,5,0,0,5,0,0,1,0,0,0,0,0],[36,36,1.0,0.27679,0.16728,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,9,0,0,11,0,0,6,0,0,2,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.26786,"x":0.34821,"p":[[0,33,0.0,0.31696,0.23887,0.14286,0.28571,0.32143,0.0,1.0,2,2,0,2,0,10,0,0,12,0,0,2,0,0,3,0,0,1,0,0,0,0,2],[4,33,0.1212,0.26786,0.16269,0.14286,0.28571,0.32143,0.0,0.57143,5,0,0,5,0,5,0,0,14,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[8,33,0.2424,0.32143,0.19561,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,11,0,0,7,0,0,2,0,0,1,0,0,0,0,1],[12,33,0.3636,0.30357,0.16269,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,12,0,0,7,0,0,1,0,0,2,0,0,0,0,0],[16,33,0.4848,0.27232,0.13054,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,11,0,0,16,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[20,33,0.6061,0.32141,0.14723,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,16,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[24,33,0.7273,0.31696,0.11701,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,18,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[28,33,0.8485,0.30354,0.11145,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,21,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[32,33,0.9697,0.32588,0.1299,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,17,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[33,33,1.0,0.34821,0.13803,0.28571,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,10,0,0,13,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"1ce9a7d11aafb858","q":"The real-valued function  $f$  is defined for  $0 \\le x \\le 1, f(0) = 0, f(1) = 1$ , and  $\\frac{1}{2} \\le \\frac{ f(z) - f(y)}{f(y) - f(x)} \\le 2$  for all  $0 \\le x < y < z \\le 1$  with  $z - y = y -x$ . Prove that  $\\frac{1}{7} \\le  f (\\frac{1}{3} ) \\le \\frac{4}{7}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,20,0.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,20,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,20,0.4,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,58,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,58,0.069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,58,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,58,0.2069,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,58,0.2759,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,58,0.3448,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,58,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,58,0.4828,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,58,0.5517,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,58,0.6207,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,58,0.6897,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[44,58,0.7586,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,58,0.8276,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[52,58,0.8966,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,58,0.9655,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[58,58,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"af03e7a634dd49bf","q":"The real numbers $x_{1}, \\ldots, x_{2011}$ satisfy\n\n$$\nx_{1}+x_{2}=2 x_{1}^{\\prime}, \\quad x_{2}+x_{3}=2 x_{2}^{\\prime}, \\quad \\ldots, \\quad x_{2011}+x_{1}=2 x_{2011}^{\\prime}\n$$\n\nwhere $x_{1}^{\\prime}, x_{2}^{\\prime}, \\ldots, x_{2011}^{\\prime}$ is a permutation of $x_{1}, x_{2}, \\ldots, x_{2011}$. Prove that $x_{1}=x_{2}=\\cdots=x_{2011}$.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.76339,"x":0.99554,"p":[[0,90,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,90,0.0444,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,90,0.0889,0.92411,0.20511,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,27],[12,90,0.1333,0.92411,0.20511,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[16,90,0.1778,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[20,90,0.2222,0.92857,0.21724,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,28],[24,90,0.2667,0.83705,0.28641,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,1,0,0,0,1,0,0,4,0,0,2,0,21],[28,90,0.3111,0.89286,0.25254,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[32,90,0.3556,0.84375,0.29093,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,22],[36,90,0.4,0.89286,0.23146,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,25],[40,90,0.4444,0.875,0.24679,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,24],[44,90,0.4889,0.86161,0.27313,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,24],[48,90,0.5333,0.86607,0.3071,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,26],[52,90,0.5778,0.76339,0.37561,0.64286,1.0,1.0,0.0,1.0,5,20,0,5,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,20],[56,90,0.6222,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[60,90,0.6667,0.83482,0.31766,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,22],[64,90,0.7111,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[68,90,0.7556,0.79911,0.33285,0.67857,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,22],[72,90,0.8,0.83482,0.33713,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,24],[76,90,0.8444,0.80804,0.20705,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,2,0,15],[80,90,0.8889,0.82143,0.18558,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,0,6,0,13],[84,90,0.9333,0.85268,0.17672,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,5,0,16],[88,90,0.9778,0.82589,0.20743,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,6,0,15],[90,90,1.0,0.77009,0.24402,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,1,5,0,0,3,0,0,2,0,15]]},{"b":5,"e":1.0,"k":"flat","v":0.76784,"x":1.0,"p":[[0,32,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,32,0.125,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,32,0.25,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,32,0.375,0.91071,0.23077,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[16,32,0.5,0.86607,0.22851,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,22],[20,32,0.625,0.88393,0.20025,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,5,0,21],[24,32,0.75,0.85265,0.26844,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,1,0,0,0,0,24],[28,32,0.875,0.76784,0.30253,0.49968,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,8,0,0,0,0,0,2,0,0,2,0,0,2,0,18],[32,32,1.0,0.92857,0.16367,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,25]]}]},{"i":"2c5fcdc9baf345ef","q":"The integers  $a_0, a_1, a_2, a_3,\\ldots$  are defined as follows: $a_0 = 1$ ,  $a_1 = 3$ , and  $a_{n+1} = a_n + a_{n-1}$  for all  $n \\ge 1$ .\nFind all integers  $n \\ge 1$  for which  $na_{n+1} + a_n$  and  $na_n + a_{n-1}$  share a common factor greater than  $1$ .","t":[{"b":0,"e":0.2857,"k":"flat","v":0.27232,"x":0.37053,"p":[[0,112,0.0,0.27232,0.06546,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,112,0.0357,0.34375,0.17445,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[8,112,0.0714,0.3125,0.12595,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,112,0.1071,0.33928,0.17405,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[16,112,0.1429,0.30361,0.04736,0.28571,0.28571,0.28571,0.2857,0.43,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,112,0.1786,0.37053,0.20782,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[24,112,0.2143,0.31696,0.1461,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[28,112,0.25,0.32589,0.12993,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[32,112,0.2857,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,112,0.3214,0.28571,0.07143,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,112,0.3571,0.33036,0.07523,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[44,112,0.3929,0.30357,0.07784,0.2857,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,112,0.4286,0.33482,0.13176,0.2857,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[52,112,0.4643,0.33482,0.13175,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[56,112,0.5,0.2991,0.09689,0.28571,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,0,0,0,28,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[60,112,0.5357,0.34375,0.17807,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,1,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[64,112,0.5714,0.29911,0.05486,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,112,0.6071,0.3125,0.14032,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[72,112,0.6429,0.29464,0.04972,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,112,0.6786,0.29464,0.04971,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,112,0.7143,0.32144,0.06186,0.28571,0.28571,0.32164,0.2857,0.4286,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[84,112,0.75,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[88,112,0.7857,0.30803,0.06298,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[92,112,0.8214,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,112,0.8571,0.30357,0.05923,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[100,112,0.8929,0.30803,0.05187,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[104,112,0.9286,0.2991,0.05486,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[108,112,0.9643,0.30803,0.13415,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[112,112,1.0,0.30357,0.04725,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.30803,"x":0.50446,"p":[[0,144,0.0,0.30803,0.08828,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,27,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,144,0.0278,0.32141,0.07982,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[8,144,0.0556,0.37946,0.20705,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[12,144,0.0833,0.33035,0.14914,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,29,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[16,144,0.1111,0.48214,0.29397,0.28571,0.28571,0.53571,0.2857,1.0,0,7,0,0,0,0,0,0,19,0,0,5,0,0,0,0,0,0,0,0,1,0,7],[20,144,0.1389,0.41071,0.25692,0.28571,0.28571,0.32143,0.2857,1.0,0,5,0,0,0,0,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[24,144,0.1667,0.41071,0.25939,0.28571,0.28571,0.42857,0.14286,1.0,0,5,0,0,0,1,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,5],[28,144,0.1944,0.43303,0.28231,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[32,144,0.2222,0.35714,0.19562,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[36,144,0.25,0.33928,0.18472,0.2857,0.28571,0.28571,0.0,1.0,1,2,0,1,0,0,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[40,144,0.2778,0.50446,0.31539,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,20,0,0,2,0,0,1,0,0,0,0,0,0,0,9],[44,144,0.3056,0.36607,0.25238,0.2857,0.28571,0.28571,0.0,1.0,2,4,0,2,0,0,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[48,144,0.3333,0.38384,0.31231,0.2857,0.28571,0.42858,0.0,1.0,5,5,0,5,0,1,0,0,17,0,0,2,0,0,1,0,0,0,0,0,1,0,5],[52,144,0.3611,0.33928,0.13243,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[56,144,0.3889,0.3125,0.1357,0.2857,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,0,0,0,23,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[60,144,0.4167,0.30803,0.12428,0.2857,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,1,0,0,25,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[64,144,0.4444,0.30803,0.0724,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[68,144,0.4722,0.33483,0.08458,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,20,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[72,144,0.5,0.35714,0.18898,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,1,0,0,19,0,0,9,0,0,0,0,0,0,0,0,0,0,2],[76,144,0.5278,0.34821,0.14258,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,5,0,0,2,0,0,0,0,0,0,0,1],[80,144,0.5556,0.3616,0.15966,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,22,0,0,8,0,0,0,0,0,0,0,0,1,0,1],[84,144,0.5833,0.33036,0.10374,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,18,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[88,144,0.6111,0.32143,0.06186,0.28571,0.28571,0.32143,0.2857,0.4286,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[92,144,0.6389,0.33036,0.07524,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[96,144,0.6667,0.33927,0.0856,0.28571,0.28571,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,19,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[100,144,0.6944,0.32589,0.08917,0.28571,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,3,0,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[104,144,0.7222,0.34375,0.07016,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[108,144,0.75,0.32588,0.08914,0.28571,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,1,0,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[112,144,0.7778,0.3125,0.07524,0.2857,0.28571,0.32143,0.14286,0.4286,0,0,0,0,0,2,0,0,22,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[116,144,0.8056,0.375,0.10565,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,13,0,0,16,0,0,1,0,0,1,0,0,0,0,0],[120,144,0.8333,0.3616,0.0797,0.28571,0.35729,0.42857,0.2857,0.571,0,0,0,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[124,144,0.8611,0.35714,0.11294,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,19,0,0,12,0,0,0,0,0,0,0,0,1,0,0],[128,144,0.8889,0.34373,0.08642,0.28571,0.28571,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[132,144,0.9167,0.34376,0.09353,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,21,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[136,144,0.9444,0.32143,0.07143,0.28571,0.28571,0.42857,0.1429,0.4286,0,0,0,0,0,1,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[140,144,0.9722,0.33035,0.1097,0.28571,0.28571,0.28571,0.2857,0.857,0,0,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[144,144,1.0,0.41519,0.16114,0.28571,0.35729,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,9,0,0,1,0,0,6,0,0,0,0,0]]}]},{"i":"520a8a22b3787b92","q":"The real numbers $r_{1}, r_{2}, \\ldots, r_{2019}$ satisfy the conditions\n$r_{1}+r_{2}+\\ldots+r_{2019}=0$ (1) and $r_{1}^{2}+r_{2}^{2}+\\ldots+r_{2019}^{2}=1$ (2).\nLet $a=\\min \\left(r_{1}, r_{2}, \\ldots, r_{2019}\\right)$ and $b=\\max \\left(r_{1}, r_{2}, \\ldots, r_{2019}\\right)$. Prove: $a b \\leq \\frac{-1}{2019}$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,15,0.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,15,0.2667,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,26,0.0,0.95089,0.19102,1.0,1.0,1.0,0.1429,1.0,0,30,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,26,0.1538,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[8,26,0.3077,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"63df2fa18e5aa2a6","q":"The *weird* mean of two numbers  $ a$  and  $ b$  is defined as  $ \\sqrt {\\frac {2a^2 + 3b^2}{5}}$ .  $ 2009$  positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these  $ 2009$  numbers must be equal.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.15625,"x":0.26339,"p":[[0,21,0.0,0.26339,0.19597,0.24999,0.28571,0.28571,0.0,1.0,6,1,0,6,0,2,0,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,21,0.1905,0.1741,0.13709,0.0,0.2857,0.28571,0.0,0.28571,12,0,0,12,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.17411,0.1504,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,3,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,21,0.5714,0.15625,0.16115,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,2,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.18303,0.1439,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,5,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,21,0.9524,0.16964,0.12078,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,8,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.18295,0.11974,0.105,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.20536,"x":0.29464,"p":[[0,70,0.0,0.20982,0.13825,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,3,0,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,70,0.0571,0.24107,0.14913,0.24999,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,1,0,0,21,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,70,0.1143,0.20536,0.13333,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,1,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,70,0.1714,0.23214,0.14617,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,2,0,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,70,0.2286,0.24991,0.07998,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.28125,0.10403,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,4,0,0,24,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[24,70,0.3429,0.26786,0.1171,0.28571,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,70,0.4,0.26785,0.1171,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[32,70,0.4571,0.24107,0.13092,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,6,0,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,70,0.5143,0.26339,0.11355,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,3,0,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,70,0.5714,0.29464,0.10677,0.28571,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,2,0,0,26,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[44,70,0.6286,0.25,0.11294,0.28571,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,2,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,70,0.6857,0.24552,0.11967,0.24999,0.28571,0.28571,0.0,0.571,4,0,0,4,0,4,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,70,0.7429,0.25446,0.1504,0.14286,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,4,0,0,19,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[56,70,0.8,0.26339,0.10779,0.28571,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[60,70,0.8571,0.26786,0.09942,0.28571,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,70,0.9143,0.26786,0.09279,0.28571,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,2,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,70,0.9714,0.23661,0.09182,0.24999,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,5,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.25893,0.11538,0.25,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,6,0,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"25b11037f9fb7786","q":"The sum of  $\\lfloor x \\rfloor$  for all real numbers  $x$  satisfying the equation  $16 + 15x + 15x^2 = \\lfloor x \\rfloor ^3$  is:","t":[{"b":2,"e":1.0,"k":"flat","v":0.95982,"x":0.99107,"p":[[0,83,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,83,0.0482,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,83,0.0964,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,83,0.1446,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[16,83,0.1928,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,83,0.241,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,83,0.2892,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,83,0.3373,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,83,0.3855,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,83,0.4337,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,83,0.4819,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,83,0.5301,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[48,83,0.5783,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[52,83,0.6265,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,83,0.6747,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[60,83,0.7229,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[64,83,0.7711,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[68,83,0.8193,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[72,83,0.8675,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[76,83,0.9157,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[80,83,0.9639,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[83,83,1.0,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25]]},{"b":7,"e":0.85714,"k":"flat","v":0.93304,"x":1.0,"p":[[0,76,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,76,0.0526,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,76,0.1053,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,76,0.1579,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,76,0.2105,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,76,0.2632,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,76,0.3158,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[28,76,0.3684,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[32,76,0.4211,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[36,76,0.4737,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,76,0.5263,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[44,76,0.5789,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,76,0.6316,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[52,76,0.6842,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[56,76,0.7368,0.94196,0.10012,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[60,76,0.7895,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[64,76,0.8421,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[68,76,0.8947,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[72,76,0.9474,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[76,76,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"bed21a53b6024d79","q":"The following sum of three four digits numbers is divisible by  $75$ ,  $7a71 + 73b7 + c232$ , where  $a, b, c$  are decimal digits. Find the necessary conditions in  $a, b, c$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.6339,"x":0.77677,"p":[[0,72,0.0,0.75892,0.1357,0.57143,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,21,0,0],[4,72,0.0556,0.76785,0.13243,0.57143,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0],[8,72,0.1111,0.77677,0.12848,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,23,0,0],[12,72,0.1667,0.74105,0.14033,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,19,0,0],[16,72,0.2222,0.76339,0.13175,0.57143,0.85714,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,21,0,0],[20,72,0.2778,0.72767,0.13996,0.57143,0.857,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,17,0,0],[24,72,0.3333,0.68749,0.14033,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,13,0,0],[28,72,0.3889,0.73214,0.14174,0.57143,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,18,0,0],[32,72,0.4444,0.70535,0.14258,0.57143,0.57143,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,15,0,0],[36,72,0.5,0.67855,0.13832,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,12,0,0],[40,72,0.5556,0.68749,0.18707,0.57143,0.57143,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,15,0,0],[44,72,0.6111,0.68749,0.14033,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,13,0,0],[48,72,0.6667,0.7232,0.14259,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,17,0,0],[52,72,0.7222,0.6339,0.11813,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0],[56,72,0.7778,0.65175,0.12846,0.57143,0.57143,0.857,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0],[60,72,0.8333,0.64286,0.12372,0.57143,0.57143,0.64286,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0],[64,72,0.8889,0.67411,0.13474,0.57143,0.57143,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,1,0,0,11,0,0],[68,72,0.9444,0.70536,0.14698,0.57143,0.57143,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,13,0,1],[72,72,1.0,0.6607,0.13244,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0]]},{"b":3,"e":0.57143,"k":"falling","v":0.61604,"x":0.80354,"p":[[0,123,0.0,0.78125,0.13355,0.78561,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,24,0,0],[4,123,0.0325,0.75444,0.13475,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,20,0,0],[8,123,0.065,0.74105,0.14033,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,19,0,0],[12,123,0.0976,0.69642,0.14173,0.57143,0.57143,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0],[16,123,0.1301,0.7232,0.14259,0.57143,0.85707,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,17,0,0],[20,123,0.1626,0.78571,0.12371,0.78561,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,24,0,0],[24,123,0.1951,0.76783,0.13245,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0],[28,123,0.2276,0.75887,0.13575,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,21,0,0],[32,123,0.2602,0.72319,0.14258,0.57143,0.857,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,17,0,0],[36,123,0.2927,0.77676,0.12847,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,23,0,0],[40,123,0.3252,0.78569,0.12371,0.78561,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,24,0,0],[44,123,0.3577,0.79015,0.11838,0.82132,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,24,0,0],[48,123,0.3902,0.76783,0.13245,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0],[52,123,0.4228,0.74997,0.13835,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,20,0,0],[56,123,0.4553,0.77677,0.12848,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,23,0,0],[60,123,0.4878,0.77675,0.17835,0.857,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,25,0,0],[64,123,0.5203,0.80354,0.11154,0.857,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,26,0,0],[68,123,0.5528,0.76336,0.13176,0.57143,0.85707,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,21,0,0],[72,123,0.5854,0.73212,0.14177,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,18,0,0],[76,123,0.6179,0.7857,0.12371,0.78561,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,24,0,0],[80,123,0.6504,0.74551,0.1371,0.57143,0.85707,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,19,0,0],[84,123,0.6829,0.79462,0.11814,0.85711,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,25,0,0],[88,123,0.7154,0.73211,0.14174,0.57143,0.857,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,18,0,0],[92,123,0.748,0.67854,0.13834,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,12,0,0],[96,123,0.7805,0.71427,0.14284,0.57143,0.71421,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0],[100,123,0.813,0.75892,0.1357,0.57143,0.85714,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,21,0,0],[104,123,0.8455,0.66069,0.13245,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0],[108,123,0.878,0.66963,0.13569,0.57143,0.57143,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,11,0,0],[112,123,0.9106,0.74998,0.13833,0.57143,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,20,0,0],[116,123,0.9431,0.6473,0.13356,0.57143,0.57143,0.85704,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,9,0,0],[120,123,0.9756,0.65176,0.12848,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0],[123,123,1.0,0.61604,0.10375,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,5,0,0]]}]},{"i":"82dfbd6e5c73e0f4","q":"The sequence  $a_i$  is defined as  $a_1 = 2, a_2 = 3$ , and $a_{n+1} = 2a_{n-1}$  or  $a_{n+1} = 3a_n - 2a_{n-1}$  for all integers  $n \\ge 2$ .\nProve that no term in  $a_i$  is in the range  $[1612, 2012]$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.02678,"x":0.78125,"p":[[0,79,0.0,0.11607,0.23266,0.0,0.0,0.14286,0.0,1.0,22,1,2,22,0,5,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[4,79,0.0506,0.74999,0.31543,0.67857,0.85714,1.0,0.0,1.0,3,13,0,3,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,7,0,13],[8,79,0.1013,0.78125,0.30926,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,4,0,16],[12,79,0.1519,0.70982,0.32436,0.57143,0.78571,1.0,0.0,1.0,4,12,0,4,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,4,0,12],[16,79,0.2025,0.68301,0.30668,0.42859,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,3,0,0,5,0,0,3,0,0,6,0,0,1,0,12],[20,79,0.2532,0.61605,0.3719,0.39286,0.71429,0.89286,0.0,1.0,7,8,0,7,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,6,0,8],[24,79,0.3038,0.58035,0.42248,0.0,0.71429,1.0,0.0,1.0,9,12,0,9,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,12],[28,79,0.3544,0.50443,0.39766,0.0,0.571,0.89286,0.0,1.0,9,8,0,9,0,1,0,0,3,0,0,2,0,0,5,0,0,0,0,0,4,0,8],[32,79,0.4051,0.50877,0.40072,0.0,0.57143,0.89286,0.0,1.0,10,8,0,10,0,0,0,0,3,0,0,0,0,0,6,0,0,2,0,0,3,0,8],[36,79,0.4557,0.47766,0.43829,0.0,0.35714,1.0,0.0,1.0,10,11,0,10,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,11],[40,79,0.5063,0.41517,0.39666,0.0,0.35714,0.85704,0.0,1.0,13,5,0,13,0,0,0,0,3,0,0,2,0,0,3,0,0,2,0,0,4,0,5],[44,79,0.557,0.3482,0.34057,0.0,0.2857,0.57143,0.0,1.0,12,2,0,12,0,2,0,0,4,0,0,2,0,0,5,0,0,2,0,0,3,0,2],[48,79,0.6076,0.24105,0.25611,0.0,0.2143,0.4286,0.0,0.85714,14,0,0,14,0,2,0,0,6,0,0,3,0,0,5,0,0,1,0,0,1,0,0],[52,79,0.6582,0.56693,0.34715,0.2857,0.64286,0.85714,0.0,1.0,6,5,0,6,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,0,6,0,5],[56,79,0.7089,0.25,0.30093,0.0,0.0,0.57143,0.0,0.85714,17,0,0,17,0,2,0,0,1,0,0,2,0,0,5,0,0,4,0,0,1,0,0],[60,79,0.7595,0.32589,0.3867,0.0,0.07143,0.60714,0.0,1.0,16,5,0,16,0,1,0,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,5],[64,79,0.8101,0.33035,0.3719,0.0,0.21428,0.60714,0.0,1.0,14,4,0,14,0,2,0,0,5,0,0,0,0,0,3,0,0,2,0,0,2,0,4],[68,79,0.8608,0.23659,0.30431,0.0,0.0,0.57143,0.0,1.0,18,1,0,18,0,1,0,0,3,0,0,0,0,0,6,0,0,3,0,0,0,0,1],[72,79,0.9114,0.31694,0.3383,0.0,0.28571,0.57143,0.0,1.0,14,3,0,14,0,0,0,0,6,0,0,1,0,0,6,0,0,1,0,0,1,0,3],[76,79,0.962,0.04464,0.13092,0.0,0.0,0.0,0.0,0.57143,28,0,1,28,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[79,79,1.0,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.00446,"x":0.76339,"p":[[0,68,0.0,0.1875,0.29111,0.0,0.0,0.32143,0.0,1.0,19,2,2,19,0,4,0,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,2],[4,68,0.0588,0.62053,0.37731,0.28571,0.71429,1.0,0.0,1.0,6,12,0,6,0,0,0,0,3,0,0,2,0,0,3,0,0,5,0,0,1,0,12],[8,68,0.1176,0.6741,0.35934,0.39286,0.85714,1.0,0.0,1.0,5,10,0,5,0,0,0,0,3,0,0,1,0,0,1,0,0,4,0,0,8,0,10],[12,68,0.1765,0.75441,0.27257,0.57143,0.78564,1.0,0.0,1.0,1,13,0,1,0,2,0,0,0,0,0,0,0,0,7,0,0,6,0,0,3,0,13],[16,68,0.2353,0.76339,0.28259,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,0,0,0,5,0,0,4,0,0,3,0,15],[20,68,0.2941,0.62944,0.35867,0.28571,0.71429,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,3,0,0,1,0,0,2,0,0,5,0,0,7,0,8],[24,68,0.3529,0.66962,0.35073,0.42857,0.78571,1.0,0.0,1.0,4,13,0,4,0,0,0,0,3,0,0,2,0,0,6,0,0,1,0,0,3,0,13],[28,68,0.4118,0.55351,0.3531,0.39285,0.57143,0.89275,0.0,1.0,7,8,0,7,0,0,0,0,1,0,0,2,0,0,11,0,0,2,0,0,1,0,8],[32,68,0.4706,0.51785,0.369,0.10714,0.57143,0.85714,0.0,1.0,8,6,0,8,0,1,0,0,2,0,0,1,0,0,8,0,0,2,0,0,4,0,6],[36,68,0.5294,0.49104,0.32524,0.28571,0.57141,0.71429,0.0,1.0,7,4,0,7,0,0,0,0,4,0,0,2,0,0,8,0,0,6,0,0,1,0,4],[40,68,0.5882,0.48661,0.40226,0.0,0.57143,0.89286,0.0,1.0,11,8,0,11,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,1,0,8],[44,68,0.6471,0.49998,0.41187,0.0,0.57121,0.89286,0.0,1.0,11,8,0,11,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,3,0,8],[48,68,0.7059,0.43303,0.42028,0.0,0.4286,0.89286,0.0,1.0,14,8,0,14,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,2,0,8],[52,68,0.7647,0.19196,0.29366,0.0,0.0,0.32143,0.0,0.85714,20,0,0,20,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,0,3,0,0],[56,68,0.8235,0.07589,0.18552,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[60,68,0.8824,0.07143,0.21129,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[64,68,0.9412,0.09372,0.23579,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[68,68,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"36facd68c2ca19cb","q":"There are  $6$  irrational numbers. Prove that there are always three of them, suppose  $a,b,c$  such that  $a+b$ , $b+c$ , $c+a$  are irrational numbers.\n\n*(Erken)*","t":[{"b":1,"e":0.57143,"k":"falling","v":0.53121,"x":0.96874,"p":[[0,17,0.0,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[4,17,0.2353,0.96874,0.12238,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[8,17,0.4706,0.86607,0.23403,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,0,0,0,23],[12,17,0.7059,0.7232,0.29438,0.5354,0.78571,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,5,0,0,7,0,0,1,0,0,1,0,15],[16,17,0.9412,0.53121,0.17581,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,8,0,0,17,0,0,0,0,0,1,0,2],[17,17,1.0,0.5491,0.2055,0.42857,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,9,0,0,16,0,0,0,0,0,0,0,4]]},{"b":3,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,16,0.0,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[4,16,0.25,0.92857,0.15567,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[8,16,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,16,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"84152085430e74d6","q":"The positive integers $a, b, c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^{2}+b^{2}=c^{2}$. Prove that $b+c$ is a square of an integer.","t":[{"b":0,"e":1.0,"k":"rising","v":0.42839,"x":1.0,"p":[[0,35,0.0,0.42839,0.39136,0.14286,0.14286,1.0,0.14,1.0,0,10,0,0,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,10],[4,35,0.1143,0.57143,0.4165,0.14286,0.57143,1.0,0.14286,1.0,0,15,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,15],[8,35,0.2286,0.63393,0.41793,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18],[12,35,0.3429,0.60714,0.42107,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":1.0,"k":"rising","v":0.55357,"x":1.0,"p":[[0,25,0.0,0.58929,0.40994,0.14286,0.71429,1.0,0.14286,1.0,0,15,0,0,0,14,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,15],[4,25,0.16,0.58036,0.42248,0.14286,0.71429,1.0,0.14286,1.0,0,16,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16],[8,25,0.32,0.72321,0.36759,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,7,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,20],[12,25,0.48,0.55357,0.4222,0.14286,0.28574,1.0,0.14286,1.0,0,15,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4b7ba6e1f1ec4b2b","q":"The wording is just ever so slightly different, however the problem is identical.\n\nProblem 3. Determine all functions  $f: \\mathbb{N} \\to \\mathbb{N}$  such that  $n^2 + f(n)f(m)$  is a multiple of  $f(n) + m$  for all natural numbers  $m, n$ .","t":[{"b":0,"e":0.42857,"k":"falling","v":0.27232,"x":0.71874,"p":[[0,150,0.0,0.51338,0.24447,0.39286,0.5,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,3,0,0,8,0,0,8,0,0,1,0,0,6,0,1],[4,150,0.0267,0.625,0.24679,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,6,0,0,8,0,0,1,0,0,11,0,2],[8,150,0.0533,0.63392,0.22285,0.53571,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,5,0,0,11,0,0,0,0,0,12,0,1],[12,150,0.08,0.6339,0.25985,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,6,0,0,7,0,0,0,0,0,13,0,2],[16,150,0.1067,0.66514,0.27343,0.53539,0.85714,0.85714,0.0,1.0,2,2,0,2,0,1,0,0,1,0,0,4,0,0,6,0,0,0,0,0,16,0,2],[20,150,0.1333,0.70088,0.2269,0.57132,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,0,0,0,17,0,2],[24,150,0.16,0.62052,0.21312,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,0,2,0,0,12,0,0],[28,150,0.1867,0.67408,0.23483,0.42859,0.78564,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,1,0,0,13,0,3],[32,150,0.2133,0.69193,0.21756,0.57132,0.857,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,1,0,0,15,0,2],[36,150,0.24,0.71874,0.19394,0.57143,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,0,0,0,15,0,3],[40,150,0.2667,0.59374,0.18934,0.53569,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,6,0,0,14,0,0,3,0,0,7,0,0],[44,150,0.2933,0.57141,0.19884,0.42857,0.57143,0.74996,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,11,0,0,8,0,0,2,0,0,8,0,0],[48,150,0.32,0.58482,0.24578,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,8,0,0,6,0,0,0,0,0,11,0,1],[52,150,0.3467,0.63838,0.25502,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,7,0,0,7,0,0,0,0,0,10,0,4],[56,150,0.3733,0.66514,0.20396,0.5354,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,10,0,0,1,0,0,10,0,3],[60,150,0.4,0.61607,0.25111,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,4,0,0,0,0,0,7,0,0,6,0,0,2,0,0,12,0,1],[64,150,0.4267,0.66515,0.1976,0.571,0.57143,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,2,0,0,9,0,3],[68,150,0.4533,0.59821,0.27066,0.42857,0.57143,0.85714,0.0,1.0,1,1,0,1,0,3,0,0,1,0,0,8,0,0,5,0,0,0,0,0,13,0,1],[72,150,0.48,0.61607,0.24337,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,7,0,0,7,0,0,1,0,0,12,0,1],[76,150,0.5067,0.66516,0.23854,0.42857,0.64286,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,6,0,0,6,0,0,1,0,0,11,0,4],[80,150,0.5333,0.62499,0.24157,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,7,0,0,8,0,0,3,0,0,8,0,3],[84,150,0.56,0.58034,0.25738,0.42857,0.4998,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,10,0,0,4,0,0,1,0,0,8,0,3],[88,150,0.5867,0.61604,0.24074,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,6,0,0,9,0,0,0,0,0,12,0,1],[92,150,0.6133,0.7098,0.1803,0.57143,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,1,0,0,18,0,0],[96,150,0.64,0.62943,0.24186,0.42857,0.57143,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,6,0,0,10,0,0,0,0,0,11,0,2],[100,150,0.6667,0.54907,0.23174,0.42857,0.49979,0.74996,0.0,1.0,1,1,0,1,0,2,0,0,0,0,0,13,0,0,7,0,0,1,0,0,7,0,1],[104,150,0.6933,0.62498,0.24679,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,5,0,0,7,0,0,0,0,0,15,0,0],[108,150,0.72,0.60266,0.25935,0.42857,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,1,0,0,7,0,0,7,0,0,2,0,0,11,0,1],[112,150,0.7467,0.62498,0.26666,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,1,0,0,5,0,0,8,0,0,0,0,0,11,0,3],[116,150,0.7733,0.62499,0.28291,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,6,0,0,5,0,0,0,0,0,11,0,4],[120,150,0.8,0.61158,0.27719,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,3,0,0,2,0,0,4,0,0,8,0,0,1,0,0,10,0,3],[124,150,0.8267,0.64286,0.22303,0.42859,0.64286,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,5,0,0,7,0,0,3,0,0,12,0,1],[128,150,0.8533,0.51339,0.33476,0.14286,0.5,0.85714,0.0,1.0,3,3,0,3,0,6,0,0,3,0,0,4,0,0,4,0,0,0,0,0,9,0,3],[132,150,0.88,0.46875,0.29284,0.14286,0.42857,0.85714,0.0,0.85714,2,0,0,2,0,8,0,0,1,0,0,7,0,0,5,0,0,0,0,0,9,0,0],[136,150,0.9067,0.50893,0.28779,0.28571,0.42857,0.85714,0.0,1.0,2,1,0,2,0,4,0,0,4,0,0,7,0,0,5,0,0,0,0,0,9,0,1],[140,150,0.9333,0.47763,0.31258,0.24999,0.571,0.85704,0.0,0.85714,6,0,0,6,0,2,0,0,3,0,0,4,0,0,7,0,0,1,0,0,9,0,0],[144,150,0.96,0.46875,0.31183,0.14286,0.42857,0.85714,0.14286,1.0,0,2,0,0,0,11,0,0,3,0,0,7,0,0,0,0,0,1,0,0,8,0,2],[148,150,0.9867,0.57586,0.3204,0.42857,0.57143,0.85714,0.0,1.0,5,2,0,5,0,0,0,0,2,0,0,6,0,0,4,0,0,1,0,0,12,0,2],[150,150,1.0,0.27232,0.26812,0.0,0.2143,0.42857,0.0,0.85714,9,0,0,9,0,7,0,0,7,0,0,4,0,0,0,0,0,2,0,0,3,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.5803,"x":0.74106,"p":[[0,65,0.0,0.5803,0.17105,0.42859,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,6,0,0,16,0,0,0,0,0,7,0,0],[4,65,0.0615,0.70534,0.19213,0.57143,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,0,0,0,17,0,1],[8,65,0.1231,0.65175,0.23674,0.571,0.57143,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,4,0,0,10,0,0,0,0,0,14,0,1],[12,65,0.1846,0.63838,0.2448,0.42859,0.57143,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,4,0,0,8,0,0,0,0,0,14,0,1],[16,65,0.2462,0.61159,0.25564,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,3,0,0,4,0,0,12,0,0,0,0,0,7,0,4],[20,65,0.3077,0.67855,0.21428,0.5354,0.857,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,0,0,0,16,0,1],[24,65,0.3692,0.66071,0.26426,0.53571,0.71421,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,0,3,0,0,8,0,0,0,0,0,12,0,4],[28,65,0.4308,0.67854,0.22017,0.571,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,5,0,0,9,0,0,1,0,0,12,0,3],[32,65,0.4923,0.7232,0.21109,0.57143,0.85707,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,1,0,0,7,0,0,2,0,0,17,0,2],[36,65,0.5538,0.71874,0.26363,0.57143,0.85714,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,1,0,0,1,0,0,9,0,0,0,0,0,13,0,6],[40,65,0.6154,0.60712,0.25754,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,4,0,0,10,0,0,0,0,0,9,0,3],[44,65,0.6769,0.70087,0.20628,0.57143,0.857,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,0,0,0,17,0,1],[48,65,0.7385,0.74106,0.18012,0.57143,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,0,0,0,18,0,2],[52,65,0.8,0.60265,0.20434,0.571,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,5,0,0,15,0,0,1,0,0,9,0,0],[56,65,0.8615,0.66964,0.18707,0.57143,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,12,0,0,1,0,0,14,0,0],[60,65,0.9231,0.66962,0.21853,0.57143,0.57143,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,1,0,0,13,0,0,0,0,0,14,0,1],[64,65,0.9846,0.66516,0.23037,0.57143,0.857,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,0,0,0,9,0,0,0,0,0,17,0,0],[65,65,1.0,0.67411,0.20897,0.57143,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,0,0,0,13,0,0,1,0,0,15,0,0]]}]},{"i":"5e196c1f403bb329","q":"The natural number  $A$  has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from  $1$  to  $A$ . Find  $A$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,8,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,8,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,39,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,39,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,39,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,39,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,39,0.4103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,39,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,39,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,39,0.7179,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,39,0.8205,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,39,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a4495d5e8d586c6c","q":"There are  $N$  red cards and  $N$  blue cards. Each card has a positive integer between  $1$  and  $N$  (inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.","t":[{"b":2,"e":0.14286,"k":"rising","v":0.03125,"x":0.1875,"p":[[0,26,0.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.15179,0.1234,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,17,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.17411,0.11143,0.14286,0.14286,0.28571,0.0,0.42857,6,0,0,6,0,14,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.17411,0.11143,0.14286,0.14286,0.28571,0.0,0.42857,6,0,0,6,0,14,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.18303,0.08171,0.14286,0.14286,0.28571,0.0,0.28571,2,0,0,2,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.16955,0.10376,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,17,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.1875,0.0974,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.02679,"x":0.17857,"p":[[0,16,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.17857,0.10714,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,15,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.16518,0.11355,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,14,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.11607,0.09062,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.14286,0.11294,0.0,0.14286,0.17857,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e44495066ccd96ca","q":"There are  $n$  boys and  $n$  girls in a school class, where  $n$  is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.\nProve that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).\n\n*Proposed by Stephan Wagner, Austria*","t":[{"b":3,"e":0.28571,"k":"falling","v":0.7991,"x":1.0,"p":[[0,53,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,53,0.0755,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,53,0.1509,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,53,0.2264,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,53,0.3019,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,53,0.3774,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,53,0.4528,0.93304,0.1636,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[28,53,0.5283,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,53,0.6038,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[36,53,0.6792,0.93304,0.15146,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,25],[40,53,0.7547,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,53,0.8302,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[48,53,0.9057,0.83482,0.22899,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,4,0,18],[52,53,0.9811,0.82142,0.25,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,1,0,18],[53,53,1.0,0.7991,0.21975,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,9,0,0,5,0,13]]},{"b":4,"e":1.0,"k":"flat","v":0.92857,"x":0.99554,"p":[[0,31,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,31,0.129,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[8,31,0.2581,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,31,0.3871,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,31,0.5161,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,31,0.6452,0.92857,0.16752,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[24,31,0.7742,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[28,31,0.9032,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[31,31,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"a8533492041d051d","q":"The number of all positive integers  $n$  such that  $n + s(n) = 2016$ , where  $s(n)$  is the sum of all digits of  $n$  is\n\n(A):  $1$  (B):  $2$  (C):  $3$  (D):  $4$  (E): None of the above.","t":[{"b":4,"e":1.0,"k":"flat","v":0.87944,"x":1.0,"p":[[0,23,0.0,0.87944,0.19924,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,0,0,23],[4,23,0.1739,0.87947,0.17536,0.71429,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,0,0,21],[8,23,0.3478,0.92857,0.17128,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,27],[12,23,0.5217,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[16,23,0.6957,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,89,0.0,0.82143,0.18557,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,2,0,15],[4,89,0.0449,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[8,89,0.0899,0.92411,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[12,89,0.1348,0.93304,0.17122,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,27],[16,89,0.1798,0.95534,0.11543,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[20,89,0.2247,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[24,89,0.2697,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[28,89,0.3146,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[32,89,0.3596,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,89,0.4045,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,89,0.4494,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,89,0.4944,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[48,89,0.5393,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,89,0.5843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,89,0.6292,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,89,0.6742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,89,0.7191,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,89,0.764,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,89,0.809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,89,0.8539,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,89,0.8989,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[84,89,0.9438,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,89,0.9888,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[89,89,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"3fbd6a6d8f187e15","q":"There are lamps in every field of  $n\\times n$  table. At start all the lamps are off. A move consists of chosing  $m$  consecutive fields in a row or a column and changing the status of that  $m$  lamps. Prove that you can reach a state in which all the lamps are on only if  $m$  divides  $n.$","t":[{"b":5,"e":0.2857,"k":"falling","v":0.06918,"x":0.74551,"p":[[0,31,0.0,0.57143,0.41649,0.10717,0.71429,1.0,0.0,1.0,8,12,0,8,0,1,0,0,4,0,0,0,0,0,2,0,0,3,0,0,2,0,12],[4,31,0.129,0.65624,0.37603,0.42857,0.78564,1.0,0.0,1.0,6,12,0,6,0,0,0,0,1,0,0,4,0,0,0,0,0,5,0,0,4,0,12],[8,31,0.2581,0.74551,0.35846,0.571,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,2,0,0,0,0,0,3,0,0,1,0,0,3,0,18],[12,31,0.3871,0.55801,0.39987,0.24999,0.57121,1.0,0.0,1.0,7,11,0,7,0,1,0,0,5,0,0,1,0,0,3,0,0,2,0,0,2,0,11],[16,31,0.5161,0.40844,0.37423,0.0,0.49979,0.71429,0.0,1.0,11,5,0,11,1,2,0,0,1,0,0,1,0,0,6,0,0,5,0,0,0,0,5],[20,31,0.6452,0.26777,0.36903,0.0,0.0,0.42857,0.0,1.0,17,5,0,17,0,4,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,5],[24,31,0.7742,0.20535,0.2647,0.0,0.07143,0.32143,0.0,1.0,16,1,0,16,0,3,0,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,1],[28,31,0.9032,0.06918,0.15093,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,3,0,0,0,0,1,2,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.23212,0.24153,0.0,0.14286,0.32143,0.0,0.85714,9,0,0,9,0,12,0,0,3,0,0,2,0,0,3,0,0,2,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.11607,"x":0.9375,"p":[[0,45,0.0,0.6964,0.34023,0.5354,0.71429,1.0,0.0,1.0,4,14,1,4,0,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,1,0,14],[4,45,0.0889,0.9375,0.18189,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[8,45,0.1778,0.85714,0.25754,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,1,0,22],[12,45,0.2667,0.86606,0.26713,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,23],[16,45,0.3556,0.71875,0.35081,0.53572,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,1,0,16],[20,45,0.4444,0.62054,0.39222,0.28571,0.71429,1.0,0.0,1.0,6,13,0,6,0,1,0,0,3,0,0,1,0,0,4,0,0,2,0,0,2,0,13],[24,45,0.5333,0.62946,0.39263,0.28571,0.71429,1.0,0.0,1.0,6,14,0,6,0,0,0,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,14],[28,45,0.6222,0.63391,0.3895,0.28571,0.78571,1.0,0.0,1.0,5,14,0,5,0,0,0,0,7,0,0,0,0,0,2,0,0,2,0,0,2,0,14],[32,45,0.7111,0.55354,0.40681,0.10714,0.57121,1.0,0.0,1.0,8,11,0,8,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,0,2,0,11],[36,45,0.8,0.56695,0.43956,0.0,0.78564,1.0,0.0,1.0,10,12,0,10,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,4,0,12],[40,45,0.8889,0.51338,0.391,0.10714,0.49979,0.89286,0.0,1.0,8,8,0,8,0,2,0,0,2,0,0,4,0,0,2,0,0,3,0,0,3,0,8],[44,45,0.9778,0.15177,0.29218,0.0,0.0,0.03572,0.0,1.0,24,1,0,24,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,1],[45,45,1.0,0.11607,0.23808,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,1]]}]},{"i":"39c1b1e12942342b","q":"[list=a]\n[*]Let  $a,b,c,d$  be real numbers with  $0\\leqslant a,b,c,d\\leqslant 1$ . Prove that  $$ ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\\leqslant \\frac{8}{27}. $$  [/*]\n[*]Find all quadruples  $(a,b,c,d)$  of real numbers with  $0\\leqslant a,b,c,d\\leqslant 1$  for which equality holds in the above inequality.\n[/list]","t":[{"b":0,"e":0.42857,"k":"flat","v":0.35266,"x":0.50001,"p":[[0,68,0.0,0.45089,0.25532,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,7,0,0,5,0,0,9,0,0,5,0,0,2,0,0,1,0,3],[4,68,0.0588,0.41963,0.19211,0.28571,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,4,0,0,8,0,0,12,0,0,5,0,0,1,0,0,1,0,1],[8,68,0.1176,0.46429,0.2369,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,5,0,0,6,0,0,8,0,0,8,0,0,2,0,0,0,0,3],[12,68,0.1765,0.46427,0.18897,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,14,0,0,6,0,0,1,0,0,2,0,1],[16,68,0.2353,0.50001,0.21128,0.39293,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,10,0,0,8,0,0,1,0,0,4,0,1],[20,68,0.2941,0.42409,0.1906,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,4,0,0,5,0,0,17,0,0,4,0,0,0,0,0,0,0,2],[24,68,0.3529,0.39286,0.14286,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,5,0,0,16,0,0,5,0,0,1,0,0,0,0,0],[28,68,0.4118,0.42857,0.22588,0.28571,0.42857,0.42858,0.14286,1.0,0,3,0,0,0,6,0,0,4,0,0,15,0,0,4,0,0,0,0,0,0,0,3],[32,68,0.4706,0.4107,0.16655,0.28571,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,5,0,0,5,0,0,14,0,0,6,0,0,1,0,0,1,0,0],[36,68,0.5294,0.39732,0.21349,0.24999,0.42857,0.42857,0.0,1.0,1,1,0,1,0,7,0,0,3,0,0,14,0,0,4,0,0,1,0,0,1,0,1],[40,68,0.5882,0.38839,0.12993,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,0,15,0,0,4,0,0,1,0,0,0,0,0],[44,68,0.6471,0.38839,0.14826,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,8,0,0,10,0,0,9,0,0,0,0,0,0,0,0],[48,68,0.7059,0.46429,0.18898,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,12,0,0,8,0,0,2,0,0,1,0,1],[52,68,0.7647,0.40625,0.15612,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,6,0,0,11,0,0,9,0,0,1,0,0,0,0,0],[56,68,0.8235,0.40625,0.17896,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,6,0,0,10,0,0,8,0,0,1,0,0,1,0,0],[60,68,0.8824,0.35266,0.11834,0.28571,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,4,0,0,12,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[64,68,0.9412,0.43304,0.15355,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,5,0,0,10,0,0,12,0,0,1,0,0,0,0,0],[68,68,1.0,0.39283,0.1336,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,4,0,0,7,0,0,14,0,0,7,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.3125,"x":0.49552,"p":[[0,47,0.0,0.45535,0.26591,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,10,0,0,6,0,0,2,0,0,4,0,0,3,0,2],[4,47,0.0851,0.39286,0.16751,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,9,0,0,13,0,0,5,0,0,0,0,0,0,0,1],[8,47,0.1702,0.43749,0.16727,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,4,0,0,9,0,0,12,0,0,2,0,0,0,0,0],[12,47,0.2553,0.49552,0.22583,0.42857,0.42857,0.57111,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,16,0,0,4,0,0,0,0,0,1,0,4],[16,47,0.3404,0.45536,0.22428,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,9,0,0,9,0,0,7,0,0,1,0,0,0,0,3],[20,47,0.4255,0.43749,0.16727,0.39286,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,15,0,0,7,0,0,1,0,0,0,0,1],[24,47,0.5106,0.44195,0.15302,0.39286,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,16,0,0,6,0,0,1,0,0,0,0,1],[28,47,0.5957,0.40625,0.18595,0.14286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,1,0,0,10,0,0,10,0,0,2,0,0,0,0,0],[32,47,0.6809,0.49106,0.20805,0.39286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,10,0,0,10,0,0,0,0,0,2,0,2],[36,47,0.766,0.39277,0.1787,0.28571,0.42857,0.46429,0.14,1.0,0,1,0,0,0,5,0,0,9,0,0,10,0,0,7,0,0,0,0,0,0,0,1],[40,47,0.8511,0.433,0.17669,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,5,0,0,9,0,0,11,0,0,1,0,0,1,0,0],[44,47,0.9362,0.3125,0.1357,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,8,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[47,47,1.0,0.37499,0.1587,0.24999,0.42857,0.46418,0.14286,0.57143,0,0,0,0,0,8,0,0,4,0,0,12,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"b187f36a3e7e77df","q":"let  $ABCD$  be a isosceles trapezium having an incircle with  $AB$  parallel to  $CD$ .\nlet  $CE$  be the perpendicular from  $C$  on  $AB$ prove that  $ CE^2 = AB. CD $","t":[{"b":3,"e":0.2857,"k":"flat","v":0.39732,"x":0.55357,"p":[[0,19,0.0,0.39732,0.06902,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.55357,0.1915,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,8,0,0,12,0,0,1,0,0,5,0,1],[8,19,0.4211,0.51786,0.1948,0.42857,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,16,0,0,9,0,0,0,0,0,1,0,3],[12,19,0.6316,0.45982,0.09268,0.42857,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,17,0,0,11,0,0,0,0,0,0,0,0],[16,19,0.8421,0.50892,0.10061,0.42857,0.57121,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,16,0,0,0,0,0,1,0,0],[19,19,1.0,0.48214,0.10564,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,18,0,0,11,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.39286,"x":0.47765,"p":[[0,35,0.0,0.39286,0.08748,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,3,0,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.45089,0.13415,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,25,0,0,1,0,0,1,0,0,2,0,0],[8,35,0.2286,0.47765,0.14109,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,18,0,0,9,0,0,1,0,0,0,0,1],[12,35,0.3429,0.42857,0.11293,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,21,0,0,5,0,0,1,0,0,0,0,0],[16,35,0.4571,0.45094,0.11355,0.42857,0.42857,0.42858,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,1],[20,35,0.5714,0.46427,0.11292,0.42857,0.42857,0.42858,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,26,0,0,3,0,0,0,0,0,2,0,0],[24,35,0.6857,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0],[28,35,0.8,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.40624,0.07237,0.42857,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,4,0,0,26,0,0,1,0,0,0,0,0,0,0,0],[35,35,1.0,0.40183,0.06624,0.42857,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"17897142c3292307","q":"The sequence $(a_{n})$ is defined by $a_{1}=1$ and\n\n$$\na_{n}=\\frac{1}{n}+\\frac{1}{a_{1} \\cdot \\ldots \\cdot a_{n-1}}\n$$\n\nShow that for all integers $m \\geqslant 3$, we have $a_{m} \\leqslant 1$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.98213,"x":1.0,"p":[[0,45,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,45,0.0889,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,45,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,45,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,45,0.3556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,45,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,45,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,45,0.6222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,45,0.7111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,45,0.9778,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[45,45,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":0.99554,"p":[[0,34,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,34,0.1176,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[8,34,0.2353,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,34,0.3529,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,34,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,34,0.5882,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,34,0.7059,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,34,0.8235,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,34,0.9412,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[34,34,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"4ec443ed98171332","q":"$2023$  balls are divided into several buckets such that no bucket contains more than  $99$  balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least  $100$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.82589,"x":0.85267,"p":[[0,16,0.0,0.83035,0.06621,0.85711,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,24,0,1],[4,16,0.25,0.83036,0.05576,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0],[8,16,0.5,0.82589,0.06902,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0],[12,16,0.75,0.83035,0.05576,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0],[16,16,1.0,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.70981,"x":0.84375,"p":[[0,22,0.0,0.84375,0.04164,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0],[4,22,0.1818,0.82589,0.06901,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0],[8,22,0.3636,0.82589,0.05906,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0],[12,22,0.5455,0.82589,0.05906,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0],[16,22,0.7273,0.80357,0.08564,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,22,0,0],[20,22,0.9091,0.70981,0.13594,0.71429,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,18,0,0,8,0,0],[22,22,1.0,0.74106,0.09742,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,11,0,0]]}]},{"i":"6c1539c3a043ddf9","q":"The sets  $A_1,A_2,...,A_n$  are finite. With  $d$  we denote the number of elements in  $\\bigcup_{i=1}^n A_i$  which are in odd number of the sets  $A_i$ . Prove that the number: $D(k)=d-\\sum_{i=1}^n|A_i|+2\\sum_{i<j}|A_i\\cap A_j |+...+(-1)^k2^{k-1}\\sum_{i_1<i_2<...<i_k}|A_{i_1}\\cap A_{i_2}\\cap ...\\cap A_{i_k}|$ is divisible by  $2^k$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.97321,"x":0.99554,"p":[[0,17,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,17,0.2353,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,17,0.4706,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,17,0.7059,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,17,0.9412,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[17,17,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":4,"e":1.0,"k":"flat","v":0.94643,"x":0.98661,"p":[[0,38,0.0,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,38,0.1053,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,38,0.2105,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,38,0.3158,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,38,0.4211,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[20,38,0.5263,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,38,0.6316,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,38,0.7368,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,38,0.8421,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[36,38,0.9474,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[38,38,1.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]}]},{"i":"6f7c0d3c3b4dca10","q":"$(USS 5)$  Given  $5$  points in the plane, no three of which are collinear, prove that we can choose  $4$  points among them that form a convex quadrilateral.","t":[{"b":0,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,23,0.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,23,0.1739,0.95088,0.16605,1.0,1.0,1.0,0.1429,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[8,23,0.3478,0.91964,0.19865,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,53,0.0,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[4,53,0.0755,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,53,0.1509,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,53,0.2264,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,53,0.3019,0.94642,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[20,53,0.3774,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[24,53,0.4528,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,53,0.5283,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,53,0.6038,0.95981,0.16461,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[36,53,0.6792,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,53,0.7547,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,53,0.8302,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,53,0.9057,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,53,0.9811,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[53,53,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"4ee4ae030e61c0e5","q":"The roots of the equation  $ax^2 - 4bx + 4c = 0$  with  $ a > 0$  belong to interval  $[2, 3]$ . Prove that:\na)  $a \\le  b \\le  c < a + b.$ b)  $\\frac{a}{a+c} + \\frac{b}{b+a} > \\frac{c}{b+c} .$","t":[{"b":1,"e":1.0,"k":"flat","v":0.94642,"x":1.0,"p":[[0,36,0.0,0.94642,0.09284,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[4,36,0.1111,0.94642,0.11152,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,23],[8,36,0.2222,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[12,36,0.3333,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,36,0.4444,0.95534,0.09745,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[20,36,0.5556,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[24,36,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.93303,"x":1.0,"p":[[0,12,0.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,12,0.3333,0.93303,0.11286,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,20],[8,12,0.6667,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"bd1a398ba522fd20","q":"There are 6 distinct values of  $x$  strictly between  $0$  and  $\\frac{\\pi}{2}$  that satisfy the equation \n\\[ \n  \\tan(15 x) = 15 \\tan(x) . \n\\]\nCall these 6 values  $r_1$ ,  $r_2$ ,  $r_3$ ,  $r_4$ ,  $r_5$ , and  $r_6$ .  What is the value of the sum\n\\[\n  \\frac{1}{\\tan^2 r_1} + \n   \\frac{1}{\\tan^2 r_2} + \n    \\frac{1}{\\tan^2 r_3} + \n     \\frac{1}{\\tan^2 r_4} + \n      \\frac{1}{\\tan^2 r_5} + \n       \\frac{1}{\\tan^2 r_6} \\, ?\n\\]","t":[{"b":1,"e":0.0,"k":"falling","v":0.18304,"x":1.0,"p":[[0,124,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,124,0.0323,0.94196,0.18509,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[8,124,0.0645,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[12,124,0.0968,0.94643,0.18123,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[16,124,0.129,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[20,124,0.1613,0.86161,0.29121,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,25],[24,124,0.1935,0.87946,0.26027,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,25],[28,124,0.2258,0.92857,0.19885,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,27],[32,124,0.2581,0.81696,0.30354,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,20],[36,124,0.2903,0.92857,0.17857,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,26],[40,124,0.3226,0.90625,0.25657,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,26],[44,124,0.3548,0.78572,0.36943,0.67857,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,23],[48,124,0.3871,0.81696,0.31387,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,22],[52,124,0.4194,0.81696,0.30771,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,22],[56,124,0.4516,0.77679,0.27879,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,4,0,0,1,0,17],[60,124,0.4839,0.65179,0.43439,0.10714,0.92857,1.0,0.0,1.0,8,16,0,8,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,16],[64,124,0.5161,0.70089,0.38193,0.39286,1.0,1.0,0.0,1.0,5,17,0,5,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,17],[68,124,0.5484,0.72768,0.39182,0.39286,1.0,1.0,0.0,1.0,5,20,0,5,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,20],[72,124,0.5806,0.67411,0.38834,0.28571,1.0,1.0,0.0,1.0,4,17,0,4,0,2,0,0,3,0,0,3,0,0,0,0,0,3,0,0,0,0,17],[76,124,0.6129,0.72321,0.35881,0.42857,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,19],[80,124,0.6452,0.63839,0.38793,0.28571,0.78571,1.0,0.0,1.0,4,15,0,4,0,2,0,0,4,0,0,4,0,0,0,0,0,2,0,0,1,0,15],[84,124,0.6774,0.50446,0.4103,0.14286,0.28571,1.0,0.0,1.0,6,11,0,6,0,5,0,0,6,0,0,1,0,0,0,0,0,2,0,0,1,0,11],[88,124,0.7097,0.49552,0.44605,0.0,0.35714,1.0,0.0,1.0,10,12,0,10,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,12],[92,124,0.7419,0.54018,0.40679,0.14286,0.57143,1.0,0.0,1.0,7,11,0,7,0,3,0,0,3,0,0,2,0,0,3,0,0,1,0,0,2,0,11],[96,124,0.7742,0.65625,0.37941,0.39286,0.85714,1.0,0.0,1.0,4,16,0,4,0,1,0,0,3,0,0,6,0,0,0,0,0,2,0,0,0,0,16],[100,124,0.8065,0.69196,0.37306,0.39286,1.0,1.0,0.0,1.0,3,17,0,3,0,2,0,0,3,0,0,4,0,0,0,0,0,2,0,0,1,0,17],[104,124,0.8387,0.72321,0.37276,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,4,0,0,0,0,0,1,0,0,2,0,0,2,0,18],[108,124,0.871,0.59821,0.4141,0.14286,0.78571,1.0,0.0,1.0,4,15,0,4,0,5,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,15],[112,124,0.9032,0.64286,0.42708,0.14286,1.0,1.0,0.0,1.0,6,17,0,6,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,17],[116,124,0.9355,0.52679,0.39518,0.14286,0.50001,0.89286,0.0,1.0,7,8,0,7,0,3,0,0,3,0,0,3,0,0,1,0,0,2,0,0,5,0,8],[120,124,0.9677,0.19196,0.21902,0.0,0.14286,0.28571,0.0,0.71429,13,0,0,13,0,6,0,0,9,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[124,124,1.0,0.18304,0.1931,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,9,0,0,8,0,0,2,0,0,1,0,0,0,0,0,1,0,0]]},{"b":4,"e":0.85714,"k":"falling","v":0.65623,"x":1.0,"p":[[0,232,0.0,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,232,0.0172,0.90179,0.22142,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,25],[8,232,0.0345,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[12,232,0.0517,0.94196,0.16311,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,28],[16,232,0.069,0.9375,0.18189,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,28],[20,232,0.0862,0.90179,0.21852,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,25],[24,232,0.1034,0.83036,0.2635,0.57143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,1,0,0,0,0,22],[28,232,0.1207,0.96429,0.12371,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[32,232,0.1379,0.92409,0.19882,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,27],[36,232,0.1552,0.92411,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[40,232,0.1724,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[44,232,0.1897,0.94643,0.1915,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[48,232,0.2069,0.94196,0.14664,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[52,232,0.2241,0.87053,0.24317,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,22],[56,232,0.2414,0.89286,0.23146,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,25],[60,232,0.2586,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[64,232,0.2759,0.9375,0.20183,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[68,232,0.2931,0.85714,0.29014,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,24],[72,232,0.3103,0.91072,0.21354,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,27],[76,232,0.3276,0.88839,0.2618,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,26],[80,232,0.3448,0.92411,0.20512,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[84,232,0.3621,0.86607,0.3008,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,25],[88,232,0.3793,0.94196,0.18851,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[92,232,0.3966,0.90179,0.25862,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,27],[96,232,0.4138,0.92857,0.24484,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[100,232,0.431,0.81695,0.34114,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,24],[104,232,0.4483,0.74107,0.40945,0.28571,1.0,1.0,0.0,1.0,5,22,0,5,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,22],[108,232,0.4655,0.88839,0.25935,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,26],[112,232,0.4828,0.85268,0.30823,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,25],[116,232,0.5,0.83929,0.29397,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,22],[120,232,0.5172,0.86607,0.28333,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,24],[124,232,0.5345,0.8125,0.35614,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[128,232,0.5517,0.88393,0.2976,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[132,232,0.569,0.83036,0.33776,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,24],[136,232,0.5862,0.73661,0.38649,0.28571,1.0,1.0,0.0,1.0,3,21,0,3,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,21],[140,232,0.6034,0.66518,0.4265,0.14286,1.0,1.0,0.0,1.0,7,18,0,7,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,18],[144,232,0.6207,0.74554,0.36023,0.39286,1.0,1.0,0.0,1.0,1,20,1,1,0,4,0,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,20],[148,232,0.6379,0.8125,0.33396,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[152,232,0.6552,0.75893,0.39518,0.60714,1.0,1.0,0.0,1.0,5,22,0,5,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,22],[156,232,0.6724,0.67411,0.406,0.28571,1.0,1.0,0.0,1.0,5,18,0,5,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,18],[160,232,0.6897,0.74554,0.37241,0.53571,1.0,1.0,0.0,1.0,3,20,0,3,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,20],[164,232,0.7069,0.75893,0.38206,0.5,1.0,1.0,0.0,1.0,4,22,0,4,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,22],[168,232,0.7241,0.6875,0.41869,0.14286,1.0,1.0,0.0,1.0,4,19,0,4,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,19],[172,232,0.7414,0.69643,0.41149,0.28571,1.0,1.0,0.0,1.0,6,20,0,6,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,20],[176,232,0.7586,0.77679,0.37447,0.64286,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,22],[180,232,0.7759,0.84821,0.30291,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,24],[184,232,0.7931,0.8125,0.34337,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,22],[188,232,0.8103,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[192,232,0.8276,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[196,232,0.8448,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[200,232,0.8621,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[204,232,0.8793,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[208,232,0.8966,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[212,232,0.9138,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[216,232,0.931,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[220,232,0.9483,0.96875,0.11142,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[224,232,0.9655,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[228,232,0.9828,0.65623,0.20473,0.57142,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,3,0,0,12,0,0,3,0,0,8,0,3],[232,232,1.0,0.65625,0.15916,0.57143,0.64286,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,7,0,0,9,0,0]]}]},{"i":"3a577ec8cc3078ea","q":"find all primes  $p$ , for which exist natural numbers, such that  $p=m^2+n^2$  and  $p|(m^3+n^3-4)$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.49552,"x":0.91964,"p":[[0,127,0.0,0.85714,0.24484,0.82132,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,22],[4,127,0.0315,0.64286,0.35893,0.28571,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,11,0,0,1,0,0,0,0,0,0,0,0,3,0,14],[8,127,0.063,0.80804,0.28707,0.82143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,5,0,0,1,0,0,0,0,0,1,0,0,6,0,18],[12,127,0.0945,0.76786,0.31491,0.39286,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,7,0,0,1,0,0,1,0,0,0,0,0,4,0,18],[16,127,0.126,0.82141,0.24745,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,5,0,17],[20,127,0.1575,0.84821,0.23941,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,11,0,16],[24,127,0.189,0.88839,0.20119,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,5,0,21],[28,127,0.2205,0.76339,0.28484,0.64286,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,3,0,0,0,0,0,3,0,0,8,0,13],[32,127,0.252,0.76337,0.29367,0.53539,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,3,0,0,3,0,0,2,0,0,1,0,0,6,0,15],[36,127,0.2835,0.79017,0.28344,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,1,0,0,0,0,0,3,0,0,6,0,16],[40,127,0.315,0.77669,0.29457,0.64286,0.85714,1.0,0.14,1.0,0,15,0,0,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,0,8,0,15],[44,127,0.3465,0.77678,0.28557,0.53572,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,1,0,0,7,0,15],[48,127,0.378,0.78125,0.2474,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,5,0,0,8,0,12],[52,127,0.4094,0.75,0.30305,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0,8,0,14],[56,127,0.4409,0.67411,0.34853,0.28571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,3,0,0,3,0,0,1,0,0,1,0,0,5,0,13],[60,127,0.4724,0.73212,0.26905,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,2,0,0,10,0,9],[64,127,0.5039,0.74997,0.27201,0.42859,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,0,0,0,7,0,13],[68,127,0.5354,0.83481,0.23721,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,0,4,0,18],[72,127,0.5669,0.67857,0.3312,0.39286,0.85714,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,4,0,0,5,0,0,0,0,0,1,0,0,6,0,12],[76,127,0.5984,0.63391,0.28107,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,3,0,0,7,0,0,2,0,0,3,0,0,9,0,5],[80,127,0.6299,0.54022,0.31688,0.28571,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,12,0,0,4,0,0,1,0,0,1,0,0,4,0,7],[84,127,0.6614,0.64285,0.3312,0.28571,0.78564,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,6,0,0,4,0,0,0,0,0,2,0,0,6,0,10],[88,127,0.6929,0.69197,0.30537,0.42857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,4,0,0,6,0,0,2,0,0,1,0,0,5,0,12],[92,127,0.7244,0.49552,0.29446,0.24999,0.42857,0.74996,0.14286,1.0,0,4,0,0,0,8,0,0,3,0,0,8,0,0,4,0,0,1,0,0,4,0,4],[96,127,0.7559,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[100,127,0.7874,0.82587,0.21649,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,9,0,14],[104,127,0.8189,0.88839,0.18117,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,17],[108,127,0.8504,0.89731,0.1439,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,18],[112,127,0.8819,0.88392,0.14035,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,11,0,15],[116,127,0.9134,0.86607,0.18536,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,15],[120,127,0.9449,0.84818,0.19216,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,15],[124,127,0.9764,0.85268,0.14054,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,10,0,12],[127,127,1.0,0.86605,0.15129,0.82132,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,10,0,14]]},{"b":6,"e":1.0,"k":"flat","v":0.80357,"x":0.99107,"p":[[0,53,0.0,0.88839,0.198,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,22],[4,53,0.0755,0.80357,0.31288,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,6,0,19],[8,53,0.1509,0.92411,0.2082,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,25],[12,53,0.2264,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[16,53,0.3019,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,53,0.3774,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,53,0.4528,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,53,0.5283,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,53,0.6038,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,53,0.6792,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,53,0.7547,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,53,0.8302,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[48,53,0.9057,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[52,53,0.9811,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[53,53,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"cbcd6667f33ce082","q":"$A$  and  $B$  are  $2\\times 2$  real valued matrices satisfying  $$ \\det A = \\det B = 1,\\quad \\text{tr}(A)>2,\\quad \\text{tr}(B)>2,\\quad \\text{tr}(ABA^{-1}B^{-1}) = 2 $$  Prove that  $A$  and  $B$  have a common eigenvector.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.72321,"x":0.91071,"p":[[0,60,0.0,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[4,60,0.0667,0.75,0.27664,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,3,0,0,0,0,0,0,0,0,3,0,0,6,0,0,10,0,9],[8,60,0.1333,0.79911,0.2392,0.82143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,9],[12,60,0.2,0.72321,0.32328,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,10],[16,60,0.2667,0.72768,0.30797,0.57143,0.85714,1.0,0.0,1.0,1,9,0,1,0,4,0,0,1,0,0,0,0,0,3,0,0,2,0,0,12,0,9],[20,60,0.3333,0.79464,0.26229,0.82143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,3,0,0,13,0,11],[24,60,0.4,0.79017,0.27195,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,12],[28,60,0.4667,0.73214,0.29179,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,3,0,0,11,0,9],[32,60,0.5333,0.83929,0.18123,0.85714,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,15,0,10],[36,60,0.6,0.84373,0.1204,0.71429,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,15,0,8],[40,60,0.6667,0.84821,0.12846,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,18,0,8],[44,60,0.7333,0.83036,0.15746,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,12,0,10],[48,60,0.8,0.88839,0.13236,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,15],[52,60,0.8667,0.84821,0.12846,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,20,0,7],[56,60,0.9333,0.83929,0.13243,0.82143,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,16,0,8],[60,60,1.0,0.85714,0.13832,0.82132,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,13,0,11]]},{"b":2,"e":0.85714,"k":"flat","v":0.83482,"x":0.94196,"p":[[0,54,0.0,0.87054,0.21829,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,18],[4,54,0.0741,0.86161,0.21572,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,19],[8,54,0.1481,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[12,54,0.2222,0.875,0.17405,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,9,0,17],[16,54,0.2963,0.875,0.16656,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,14],[20,54,0.3704,0.89732,0.1931,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,19],[24,54,0.4444,0.83929,0.16656,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,15,0,10],[28,54,0.5185,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[32,54,0.5926,0.87946,0.11904,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,12],[36,54,0.6667,0.90625,0.11633,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,17],[40,54,0.7407,0.89285,0.10102,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,12],[44,54,0.8148,0.90179,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[48,54,0.8889,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,12],[52,54,0.963,0.84373,0.12558,0.82132,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,16,0,8],[54,54,1.0,0.83482,0.12428,0.71429,0.85714,0.85714,0.5714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,16,0,7]]}]},{"i":"a396d73e9d4fcc86","q":"each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that  $3^k$  divides P.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.58929,"x":0.85268,"p":[[0,141,0.0,0.58929,0.31491,0.28571,0.71429,0.85714,0.0,1.0,4,5,4,4,0,0,0,0,5,0,0,2,0,0,3,0,0,9,0,0,4,0,5],[4,141,0.0284,0.85268,0.15355,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,3,0,15],[8,141,0.0567,0.80357,0.20124,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,13,0,0,2,0,13],[12,141,0.0851,0.8125,0.18013,0.71429,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,13,0,0,4,0,12],[16,141,0.1135,0.79018,0.19228,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,16,0,0,2,0,11],[20,141,0.1418,0.80804,0.16213,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,16,0,0,3,0,11],[24,141,0.1702,0.80804,0.18073,0.71429,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,12,0,0,4,0,12],[28,141,0.1986,0.79911,0.17807,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,15,0,0,3,0,11],[32,141,0.227,0.76786,0.20438,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,15,0,0,1,0,11],[36,141,0.2553,0.79464,0.15126,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,16,0,0,6,0,8],[40,141,0.2837,0.83036,0.15335,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,5,0,12],[44,141,0.3121,0.79464,0.17835,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,14,0,0,3,0,11],[48,141,0.3404,0.80804,0.18423,0.71429,0.78571,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,13,0,0,5,0,11],[52,141,0.3688,0.80802,0.18425,0.71429,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,0,3,0,13],[56,141,0.3972,0.80357,0.1915,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,2,0,12],[60,141,0.4255,0.77231,0.2572,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,1,0,0,2,0,0,9,0,0,5,0,12],[64,141,0.4539,0.77679,0.17835,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,17,0,0,3,0,9],[68,141,0.4823,0.83929,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,5,0,14],[72,141,0.5106,0.76786,0.22798,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,14,0,0,1,0,12],[76,141,0.539,0.77232,0.26693,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,7,0,0,2,0,15],[80,141,0.5674,0.85268,0.1838,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,4,0,17],[84,141,0.5957,0.77677,0.16729,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,0,3,0,9],[88,141,0.6241,0.70089,0.27049,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,9,0,0,1,0,0,6,0,0,4,0,10],[92,141,0.6525,0.69643,0.20438,0.57143,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,15,0,0,2,0,6],[96,141,0.6809,0.70982,0.1838,0.71429,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,19,0,0,0,0,6],[100,141,0.7092,0.70982,0.19061,0.71429,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,19,0,0,2,0,5],[104,141,0.7376,0.64732,0.19556,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,7,0,0,1,0,0,18,0,0,1,0,3],[108,141,0.766,0.71875,0.21275,0.64286,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,14,0,0,2,0,8],[112,141,0.7943,0.71875,0.21866,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,17,0,0,2,0,7],[116,141,0.8227,0.68304,0.21349,0.64286,0.71429,0.71429,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,18,0,0,0,0,6],[120,141,0.8511,0.67411,0.15663,0.67857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,21,0,0,0,0,3],[124,141,0.8794,0.66072,0.1915,0.42859,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,18,0,0,2,0,3],[128,141,0.9078,0.66071,0.20748,0.67857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0,19,0,0,2,0,3],[132,141,0.9362,0.62946,0.20159,0.53571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,5,0,0,1,0,0,20,0,0,2,0,1],[136,141,0.9645,0.625,0.18471,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,10,0,0,1,0,0,17,0,0,1,0,2],[140,141,0.9929,0.61607,0.19704,0.53571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,22,0,0,0,0,1],[141,141,1.0,0.68304,0.13709,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,24,0,0,0,0,2]]},{"b":5,"e":0.42857,"k":"flat","v":0.62946,"x":0.90625,"p":[[0,111,0.0,0.62946,0.25966,0.42857,0.71429,0.75,0.0,1.0,1,4,1,1,0,2,0,0,2,0,0,5,0,0,2,0,0,12,0,0,4,0,4],[4,111,0.036,0.87946,0.13415,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,6,0,16],[8,111,0.0721,0.85714,0.17128,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,5,0,16],[12,111,0.1081,0.78572,0.21129,0.71429,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,10,0,0,4,0,12],[16,111,0.1441,0.82589,0.19144,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,5,0,13],[20,111,0.1802,0.79018,0.16746,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,0,4,0,9],[24,111,0.2162,0.82143,0.21129,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,9,0,0,5,0,14],[28,111,0.2523,0.84375,0.18336,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,4,0,16],[32,111,0.2883,0.8192,0.24871,0.71429,0.96429,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,8,0,0,3,1,16],[36,111,0.3243,0.79018,0.24219,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,8,0,0,4,0,14],[40,111,0.3604,0.81695,0.17942,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,6,0,12],[44,111,0.3964,0.65625,0.28984,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,5,0,0,4,0,0,0,0,0,10,0,0,3,0,8],[48,111,0.4324,0.75893,0.2299,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,11,0,0,3,0,11],[52,111,0.4685,0.70536,0.27184,0.53572,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,4,0,0,3,0,0,1,0,0,11,0,0,2,0,10],[56,111,0.5045,0.79911,0.21683,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,10,0,0,4,0,13],[60,111,0.5405,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[64,111,0.5766,0.83929,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,6,0,14],[68,111,0.6126,0.82589,0.23887,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,3,0,0,7,0,16],[72,111,0.6486,0.82143,0.23958,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,7,0,0,3,0,17],[76,111,0.6847,0.83929,0.20748,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,7,0,0,5,0,16],[80,111,0.7207,0.85268,0.23003,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,3,0,19],[84,111,0.7568,0.83034,0.27069,0.82143,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,5,0,19],[88,111,0.7928,0.84152,0.24205,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,4,1,18],[92,111,0.8288,0.87499,0.1777,0.82143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,6,0,18],[96,111,0.8649,0.85714,0.16366,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,2,0,17],[100,111,0.9009,0.83034,0.24075,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,1,0,19],[104,111,0.9369,0.82143,0.27433,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,20],[108,111,0.973,0.66518,0.28259,0.42857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,4,0,0,9,0,0,2,0,0,1,0,11],[111,111,1.0,0.7098,0.2435,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,3,0,0,5,0,9]]}]},{"i":"ecd1fe782a1ac766","q":"There are  $100$  members in a ladies' club.Each lady has had tea (in private) with exactly  $56$  of her lady friends.The Board,consisting of the   $50$  most distinguished ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,with in each group,any lady has had tea with any other.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.51786,"x":0.74107,"p":[[0,30,0.0,0.57142,0.32733,0.28571,0.57143,0.85714,0.0,1.0,5,5,0,5,0,0,0,0,4,0,0,2,0,0,6,0,0,5,0,0,5,0,5],[4,30,0.1333,0.51786,0.30671,0.28571,0.42857,0.75,0.0,1.0,2,4,0,2,0,3,0,0,8,0,0,4,0,0,2,0,0,5,0,0,4,0,4],[8,30,0.2667,0.56697,0.29984,0.28571,0.64286,0.85714,0.0,1.0,3,2,0,3,0,2,0,0,4,0,0,3,0,0,4,0,0,6,0,0,8,0,2],[12,30,0.4,0.55802,0.28651,0.39286,0.57143,0.85714,0.0,1.0,2,2,0,2,0,3,0,0,3,0,0,5,0,0,6,0,0,3,0,0,8,0,2],[16,30,0.5333,0.61161,0.31991,0.39286,0.71429,0.85714,0.0,1.0,3,4,0,3,0,3,0,0,2,0,0,2,0,0,3,0,0,6,0,0,9,0,4],[20,30,0.6667,0.68304,0.30249,0.53571,0.85714,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,3,0,0,1,0,0,3,0,0,4,0,0,10,0,7],[24,30,0.8,0.74107,0.2126,0.57143,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,14,0,4],[28,30,0.9333,0.66071,0.25692,0.42857,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,2,0,0,5,0,0,2,0,0,6,0,0,14,0,1],[30,30,1.0,0.66518,0.31055,0.53572,0.85714,0.85714,0.0,1.0,3,5,0,3,0,2,0,0,0,0,0,3,0,0,4,0,0,3,0,0,12,0,5]]},{"b":3,"e":0.42857,"k":"falling","v":0.31688,"x":0.60265,"p":[[0,27,0.0,0.60265,0.27371,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,4,0,0,3,0,0,8,0,0,5,0,0,4,0,5],[4,27,0.1481,0.40625,0.22899,0.28571,0.28571,0.42858,0.14286,1.0,0,2,0,0,0,4,0,0,14,0,0,7,0,0,3,0,0,0,0,0,2,0,2],[8,27,0.2963,0.42411,0.25626,0.2857,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,7,0,0,11,0,0,3,0,0,4,0,0,2,0,0,4,0,1],[12,27,0.4444,0.43741,0.29876,0.14286,0.42857,0.57143,0.14,1.0,0,4,0,0,0,12,0,0,2,0,0,7,0,0,4,0,0,1,0,0,2,0,4],[16,27,0.5926,0.31688,0.23353,0.14286,0.21435,0.42857,0.14,1.0,0,1,0,0,0,16,0,0,5,0,0,6,0,0,2,0,0,0,0,0,2,0,1],[20,27,0.7407,0.33036,0.18707,0.24999,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,8,0,0,15,0,0,5,0,0,1,0,0,1,0,0,2,0,0],[24,27,0.8889,0.37945,0.18765,0.28571,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,7,0,0,13,0,0,2,0,0,1,0,0,2,0,0],[27,27,1.0,0.37946,0.22476,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,10,0,0,7,0,0,5,0,0,7,0,0,1,0,0,1,0,1]]}]},{"i":"86b68277e85f9fb0","q":"There are 100 members in a ladies' club. Each lady has had tea (in private) with exactly 56 of the other members of the club. The Board, consisting of the 50 most distinguished ladies, have all had tea with one another. Prove that the entire club may be split into two groups in such a way that, within each group, any lady has had tea with any other.","t":[{"b":1,"e":0.28571,"k":"flat","v":0.46429,"x":0.61598,"p":[[0,24,0.0,0.59821,0.29974,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,6,0,0,7,0,0,0,0,0,5,0,0,4,0,7],[4,24,0.1667,0.61598,0.31442,0.42857,0.64286,0.85714,0.0,1.0,1,7,0,1,0,4,0,0,1,0,0,9,0,0,1,0,0,2,0,0,7,0,7],[8,24,0.3333,0.55816,0.29081,0.28571,0.57121,0.85714,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,4,0,0,2,0,0,4,0,0,9,0,2],[12,24,0.5,0.54018,0.29609,0.28571,0.42857,0.85714,0.0,1.0,2,2,0,2,0,2,0,0,6,0,0,8,0,0,0,0,0,3,0,0,9,0,2],[16,24,0.6667,0.54017,0.30667,0.28571,0.42859,0.85714,0.0,1.0,1,5,0,1,0,5,0,0,4,0,0,7,0,0,2,0,0,4,0,0,4,0,5],[20,24,0.8333,0.46429,0.28121,0.28571,0.42857,0.60714,0.0,1.0,2,3,0,2,0,2,0,0,10,0,0,8,0,0,2,0,0,1,0,0,4,0,3],[24,24,1.0,0.48214,0.28738,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,7,0,0,9,0,0,1,0,0,3,0,0,4,0,3]]},{"b":7,"e":1.0,"k":"rising","v":0.5982,"x":1.0,"p":[[0,24,0.0,0.5982,0.33396,0.28571,0.4998,1.0,0.0,1.0,1,9,0,1,0,4,0,0,4,0,0,7,0,0,2,0,0,0,0,0,5,0,9],[4,24,0.1667,0.60714,0.2988,0.39286,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,4,0,0,7,0,0,1,0,0,2,0,0,9,0,5],[8,24,0.3333,0.66518,0.27342,0.42857,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,11,0,0,2,0,0,2,0,0,5,0,9],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"f3b469061fff4a5b","q":"$ n\\geq 2$  cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of  $ n$","t":[{"b":0,"e":0.85714,"k":"rising","v":0.29911,"x":0.83927,"p":[[0,59,0.0,0.58927,0.20123,0.42859,0.57143,0.57143,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,8,0,0,16,0,0,1,0,0,3,0,3],[4,59,0.0678,0.37054,0.2547,0.14286,0.42857,0.42857,0.0,1.0,5,1,0,5,0,4,0,0,3,0,0,16,0,0,0,0,0,0,0,0,3,0,1],[8,59,0.1356,0.39286,0.16366,0.42857,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,3,0,0,1,0,0,24,0,0,0,0,0,1,0,0,1,0,0],[12,59,0.2034,0.37054,0.20782,0.42857,0.42857,0.42857,0.0,1.0,5,1,0,5,0,2,0,0,0,0,0,23,0,0,0,0,0,1,0,0,0,0,1],[16,59,0.2712,0.40625,0.18249,0.42857,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,1,0,0,2,0,0,22,0,0,2,0,0,0,0,0,2,0,0],[20,59,0.339,0.375,0.17768,0.42857,0.42857,0.42857,0.0,0.85714,4,0,0,4,0,2,0,0,0,0,0,24,0,0,1,0,0,0,0,0,1,0,0],[24,59,0.4068,0.37499,0.21649,0.35714,0.42857,0.42857,0.0,1.0,4,1,0,4,0,4,0,0,0,0,0,21,0,0,1,0,0,0,0,0,1,0,1],[28,59,0.4746,0.29911,0.21237,0.10714,0.42857,0.42857,0.0,0.85714,8,0,0,8,0,4,0,0,0,0,0,19,0,0,0,0,0,0,0,0,1,0,0],[32,59,0.5424,0.41518,0.18681,0.42857,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,1,0,0,1,0,0,23,0,0,1,0,0,1,0,0,2,0,0],[36,59,0.6102,0.34822,0.19865,0.14289,0.42857,0.42857,0.0,1.0,4,1,0,4,0,5,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,1],[40,59,0.678,0.58482,0.25344,0.42857,0.4286,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,16,0,0,1,0,0,0,0,0,9,0,3],[44,59,0.7458,0.73651,0.24273,0.42857,0.85714,0.85714,0.14,1.0,0,7,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,2,0,0,13,0,7],[48,59,0.8136,0.69642,0.21944,0.42857,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,1,0,0,15,0,3],[52,59,0.8814,0.77679,0.20183,0.67857,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,17,0,6],[56,59,0.9492,0.83927,0.15046,0.85711,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,20,0,7],[59,59,1.0,0.79918,0.1281,0.82132,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,23,0,1]]},{"b":2,"e":0.0,"k":"falling","v":0.00446,"x":0.57588,"p":[[0,47,0.0,0.57588,0.19393,0.57143,0.57143,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,2,0,0,21,0,0,2,0,0,2,0,2],[4,47,0.0851,0.29464,0.25985,0.0,0.42857,0.42857,0.0,0.85714,10,0,0,10,0,4,0,0,1,0,0,14,0,0,0,0,0,0,0,0,3,0,0],[8,47,0.1702,0.33036,0.2683,0.10714,0.42857,0.42857,0.0,1.0,8,2,0,8,0,3,0,0,3,0,0,15,0,0,0,0,0,0,0,0,1,0,2],[12,47,0.2553,0.36161,0.23954,0.14286,0.42857,0.42857,0.0,1.0,4,1,0,4,0,7,0,0,0,0,0,17,0,0,0,0,0,2,0,0,1,0,1],[16,47,0.3404,0.40625,0.20858,0.42857,0.42857,0.42857,0.0,0.85714,4,0,0,4,0,1,0,0,1,0,0,22,0,0,1,0,0,0,0,0,3,0,0],[20,47,0.4255,0.34821,0.23941,0.14286,0.42857,0.42857,0.0,1.0,6,2,0,6,0,3,0,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,2],[24,47,0.5106,0.36598,0.17485,0.42857,0.42857,0.42857,0.0,0.85714,4,0,0,4,0,2,0,0,1,0,0,24,0,0,0,0,0,0,0,0,1,0,0],[28,47,0.5957,0.32143,0.20825,0.10714,0.42857,0.42857,0.0,0.85714,8,0,0,8,0,1,0,0,1,0,0,21,0,0,0,0,0,0,0,0,1,0,0],[32,47,0.6809,0.35259,0.26489,0.105,0.42857,0.42857,0.0,1.0,8,1,0,8,0,2,0,0,1,0,0,17,0,0,0,0,0,1,0,0,2,0,1],[36,47,0.766,0.30357,0.26667,0.0,0.42857,0.42857,0.0,1.0,11,2,0,11,0,1,0,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,2],[40,47,0.8511,0.19634,0.22801,0.0,0.07,0.42857,0.0,0.85714,16,0,0,16,0,3,0,0,1,0,0,11,0,0,0,0,0,0,0,0,1,0,0],[44,47,0.9362,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c10971e9af2e2684","q":"There are four basketball players  $A,B,C,D$ . Initially the ball is with  $A$ . The ball is always passed from one person to a different person. \nIn how many ways can the ball come back to  $A$  after  $\\textbf{seven}$  moves? (for example  $A\\rightarrow C\\rightarrow B\\rightarrow D\\rightarrow A\\rightarrow B\\rightarrow C\\rightarrow A$ , or  $A\\rightarrow D\\rightarrow A\\rightarrow D\\rightarrow C\\rightarrow A\\rightarrow B\\rightarrow A)$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,39,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,39,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,39,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,39,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,39,0.4103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,39,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,39,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,39,0.7179,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,39,0.8205,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,39,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[39,39,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,37,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,37,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"112870e7f5961c88","q":"$a,b,c \\geq-3/4$  and  $a+b+c=1$ . Show that:  $\\frac{a}{1+a^{2}}+\\frac{b}{1+b^{2}}+\\frac{c}{1+c^{2}}\\leq \\frac{9}{10}$","t":[{"b":3,"e":1.0,"k":"flat","v":0.82143,"x":1.0,"p":[[0,46,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,46,0.087,0.82143,0.36246,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[8,46,0.1739,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,46,0.2609,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,46,0.3478,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[20,46,0.4348,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.57143,"k":"falling","v":0.69201,"x":0.99554,"p":[[0,28,0.0,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,28,0.1429,0.9375,0.2257,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[8,28,0.2857,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[12,28,0.4286,0.92857,0.24484,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[16,28,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,28,0.7143,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[24,28,0.8571,0.7857,0.20204,0.57143,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,5,0,0,1,0,14],[28,28,1.0,0.69201,0.24246,0.53605,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,7,0,0,11,0,0,1,0,0,1,0,11]]}]},{"i":"2b3a061a547c7b4a","q":"$ABCD$  is a parallelogram of unit area and  $E, F, G, H$  are mid-points of the sides  $BC, CD, DA, AB$  respectively. The line segments  $AE,BF,CG$  and  $DH$  dissect the interior of  $ABCD$  into nine regions. Find the area of the central region.","t":[{"b":0,"e":1.0,"k":"flat","v":0.81249,"x":0.92857,"p":[[0,31,0.0,0.81249,0.21262,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,2,0,0,3,0,16],[4,31,0.129,0.87946,0.17536,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,1,0,21],[8,31,0.2581,0.90179,0.15335,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,21],[12,31,0.3871,0.92857,0.13832,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[16,31,0.5161,0.8482,0.17837,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,3,0,17],[20,31,0.6452,0.91517,0.14665,0.85711,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[24,31,0.7742,0.92411,0.14719,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,24],[28,31,0.9032,0.92856,0.15155,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,26],[31,31,1.0,0.92411,0.14279,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,24]]},{"b":4,"e":1.0,"k":"rising","v":0.70084,"x":0.95536,"p":[[0,48,0.0,0.70084,0.20631,0.57132,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,8,0,0,3,0,7],[4,48,0.0833,0.91071,0.15872,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,23],[8,48,0.1667,0.90176,0.16151,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,3,0,22],[12,48,0.25,0.93302,0.13359,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[16,48,0.3333,0.92411,0.14719,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,24],[20,48,0.4167,0.88393,0.16917,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,1,0,21],[24,48,0.5,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[28,48,0.5833,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[32,48,0.6667,0.92857,0.14725,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,25],[36,48,0.75,0.9107,0.1505,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,22],[40,48,0.8333,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[44,48,0.9167,0.89731,0.15663,0.82132,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,3,0,21],[48,48,1.0,0.91518,0.15916,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24]]}]},{"i":"783a89f7c52dd18b","q":"$f(x)$  is polynomial with integer coefficients, with module not exceeded  $5*10^6$ .  $f(x)=nx$  has integer root for  $n=1,2,...,20$ . Prove that  $f(0)=0$","t":[{"b":4,"e":0.14286,"k":"flat","v":0.16053,"x":0.22321,"p":[[0,66,0.0,0.17857,0.10101,0.14286,0.14286,0.28571,0.0,0.28571,5,0,3,5,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.22312,0.18191,0.14286,0.14286,0.28571,0.0,0.857,5,0,0,5,0,12,0,0,12,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[8,66,0.1212,0.16072,0.1171,0.10714,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,13,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.20536,0.12339,0.14286,0.2857,0.2857,0.0,0.57143,5,0,0,5,0,10,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,66,0.2424,0.17411,0.09932,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,18,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.22321,0.17835,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,13,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[24,66,0.3636,0.20983,0.14279,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[28,66,0.4242,0.17857,0.16751,0.14286,0.14286,0.14287,0.0,1.0,4,1,0,4,0,21,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,66,0.4848,0.1875,0.1729,0.14286,0.14286,0.2857,0.0,1.0,5,1,0,5,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,66,0.5455,0.20982,0.16746,0.14286,0.14286,0.2857,0.0,1.0,2,1,0,2,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,66,0.6061,0.17857,0.15152,0.14286,0.14286,0.2857,0.0,0.85714,5,0,0,5,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,66,0.6667,0.21429,0.17857,0.14286,0.14288,0.28571,0.0,1.0,3,1,0,3,0,17,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[48,66,0.7273,0.16053,0.08568,0.14286,0.14286,0.17857,0.0,0.28571,4,0,0,4,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.19196,0.1411,0.14286,0.14286,0.2857,0.0,0.85714,2,0,0,2,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[56,66,0.8485,0.22312,0.15129,0.14286,0.14286,0.28571,0.0,0.857,1,0,0,1,0,19,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[60,66,0.9091,0.16964,0.10374,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,17,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.19196,0.10479,0.14286,0.21428,0.28571,0.0,0.28571,5,0,0,5,0,11,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.16964,0.0974,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.03571,"x":0.17411,"p":[[0,40,0.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.28571,7,0,2,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.17411,0.17399,0.14286,0.14286,0.17857,0.0,1.0,6,1,0,6,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,40,0.2,0.12937,0.10325,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.17411,0.18117,0.10714,0.14286,0.2857,0.0,1.0,8,1,0,8,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,40,0.4,0.17411,0.22794,0.10714,0.14286,0.14286,0.0,1.0,8,2,0,8,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,40,0.5,0.125,0.11152,0.0,0.14286,0.17857,0.0,0.28571,12,0,0,12,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.15178,0.14258,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,11,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,40,0.7,0.12946,0.10926,0.0,0.14286,0.17857,0.0,0.28571,11,0,0,11,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.09375,0.08458,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d3b87a033b6e837b","q":"$ABCD$  is a cyclic quadrilateral with  $AC \\perp BD$ ;  $AC$  meets  $BD$  at  $E$ . Prove that \\[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \\]\r\nwhere  $R$  is the radius of the circumscribing circle.","t":[{"b":5,"e":1.0,"k":"rising","v":0.00446,"x":1.0,"p":[[0,60,0.0,0.25,0.42408,0.0,0.0,0.32143,0.0,1.0,23,7,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7],[4,60,0.0667,0.18304,0.38172,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[8,60,0.1333,0.10268,0.29066,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,60,0.2,0.19196,0.38896,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,60,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[24,60,0.4,0.12054,0.31966,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[28,60,0.4667,0.37491,0.46809,0.0,0.0,1.0,0.0,1.0,18,10,0,18,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,10],[32,60,0.5333,0.59375,0.47663,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,18],[36,60,0.6,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[40,60,0.6667,0.82142,0.36246,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[44,60,0.7333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,60,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,60,0.8667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.33482,"p":[[0,95,0.0,0.33482,0.46375,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,9],[4,95,0.0421,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,95,0.0842,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[12,95,0.1263,0.20982,0.39767,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[16,95,0.1684,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,95,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,95,0.2526,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,95,0.2947,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,95,0.3368,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,95,0.3789,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,95,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,95,0.4632,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,95,0.5053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,95,0.5474,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[56,95,0.5895,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,95,0.6316,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,95,0.6737,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[68,95,0.7158,0.0625,0.2257,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[72,95,0.7579,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,95,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,95,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,95,0.8842,0.05357,0.21053,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[88,95,0.9263,0.04911,0.19759,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[92,95,0.9684,0.04911,0.19759,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[95,95,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7698064ff9abca71","q":"Yesterday (=April 22, 2003) was Gittes birthday. She notices that her age equals the sum of the 4 digits of the year she was born in. \r\nHow old is she?","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,54,0.0,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[4,54,0.0741,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[8,54,0.1481,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,54,0.2222,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,54,0.2963,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,54,0.3704,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,54,0.4444,0.99106,0.03461,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,54,0.5185,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[32,54,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,54,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,54,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,54,0.8148,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,54,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,54,0.963,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[54,54,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.96874,"x":1.0,"p":[[0,38,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,38,0.1053,0.96874,0.06903,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,38,0.2105,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,38,0.3158,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,38,0.4211,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,38,0.5263,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,38,0.6316,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"0019e51928baeb83","q":"$ \\int_0^{\\pi^2/4} \\frac{dx}{1+\\sin\\sqrt x +\\cos\\sqrt x} $","t":[{"b":1,"e":0.4286,"k":"falling","v":0.49107,"x":0.89732,"p":[[0,17,0.0,0.89732,0.17941,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,24],[4,17,0.2353,0.84822,0.20805,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,0,1,0,20],[8,17,0.4706,0.76339,0.24383,0.57143,0.78571,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,0,0,0,0,0,16],[12,17,0.7059,0.62499,0.1915,0.57143,0.57143,0.57143,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,21,0,0,0,0,0,0,0,6],[16,17,0.9412,0.53125,0.11425,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,0,0,0,0,0,1],[17,17,1.0,0.49107,0.10062,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,10,0,0,18,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,23,0.0,0.91518,0.18161,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,26],[4,23,0.1739,0.94195,0.15513,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[8,23,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,23,0.5217,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"cb9741d273b6b691","q":"$ \\sin\\frac{\\pi }{4n}\\ge \\frac{\\sqrt 2 }{2n} ,\\quad \\forall n\\in\\mathbb{N} $","t":[{"b":1,"e":0.42857,"k":"falling","v":0.41518,"x":0.94643,"p":[[0,20,0.0,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[4,20,0.2,0.84375,0.25595,0.64286,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,1,0,0,0,0,23],[8,20,0.4,0.74554,0.28956,0.42857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,18],[12,20,0.6,0.5,0.22016,0.42857,0.42857,0.42857,0.28571,1.0,0,5,0,0,0,0,0,0,4,0,0,23,0,0,0,0,0,0,0,0,0,0,5],[16,20,0.8,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.41518,0.04164,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.78571,"x":1.0,"p":[[0,26,0.0,0.78571,0.27199,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,0,0,0,1,0,19],[4,26,0.1538,0.83482,0.26513,0.42857,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,23],[8,26,0.3077,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,26,0.4615,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8327199715e80ab9","q":"$ABC$  is an isosceles triangle with  $AB=AC$  and the angle in  $A$  is less than  $60^{\\circ}$ . Let  $D$  be a point on  $AC$  such that  $\\angle{DBC}=\\angle{BAC}$ .  $E$  is the intersection between the perpendicular bisector of  $BD$  and the line parallel to  $BC$  passing through  $A$ .  $F$  is a point on the line  $AC$  such that  $FA=2AC$  ( $A$  is between  $F$  and  $C$ ).\nShow that  $EB$  and  $AC$  are parallel and that the perpendicular from  $F$  to  $AB$ , the perpendicular from  $E$  to  $AC$  and  $BD$  are concurrent.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.1607,"x":0.44196,"p":[[0,159,0.0,0.36606,0.11256,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,16,0,0,12,0,0,2,0,0,1,0,0,0,0,0],[4,159,0.0252,0.33928,0.13716,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,15,0,0,8,0,0,3,0,0,1,0,0,0,0,0],[8,159,0.0503,0.33027,0.0908,0.28571,0.28571,0.42857,0.14,0.57143,0,0,0,0,0,2,0,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[12,159,0.0755,0.37946,0.16211,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,1,0,0,13,0,0,13,0,0,3,0,0,0,0,0,0,0,1],[16,159,0.1006,0.35268,0.11285,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,14,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[20,159,0.1258,0.33486,0.12683,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,18,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[24,159,0.1509,0.35714,0.07986,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[28,159,0.1761,0.4375,0.12846,0.42857,0.42857,0.4286,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,21,0,0,4,0,0,0,0,0,0,0,1],[32,159,0.2013,0.41964,0.08703,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,23,0,0,4,0,0,0,0,0,0,0,0],[36,159,0.2264,0.3884,0.10853,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,6,0,0,22,0,0,2,0,0,0,0,0,0,0,0],[40,159,0.2516,0.40177,0.14912,0.42857,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,0,0,0,3,0,0,20,0,0,6,0,0,0,0,0,0,0,0],[44,159,0.2767,0.43301,0.07558,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,23,0,0,5,0,0,0,0,0,0,0,0],[48,159,0.3019,0.43749,0.1181,0.42857,0.42857,0.4642,0.0,0.57143,1,0,0,1,0,1,0,0,1,0,0,21,0,0,8,0,0,0,0,0,0,0,0],[52,159,0.327,0.42857,0.10102,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,19,0,0,7,0,0,0,0,0,0,0,0],[56,159,0.3522,0.42411,0.07563,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[60,159,0.3774,0.4241,0.0756,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[64,159,0.4025,0.41517,0.09004,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,22,0,0,2,0,0,1,0,0,0,0,0],[68,159,0.4277,0.39732,0.07771,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,21,0,0,2,0,0,0,0,0,0,0,0],[72,159,0.4528,0.44195,0.06543,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0,0,0,0,0,0,0],[76,159,0.478,0.40625,0.10171,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,18,0,0,5,0,0,0,0,0,0,0,0],[80,159,0.5031,0.43751,0.09407,0.42857,0.42857,0.46431,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,18,0,0,8,0,0,0,0,0,0,0,0],[84,159,0.5283,0.40625,0.08828,0.42857,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,22,0,0,3,0,0,0,0,0,0,0,0],[88,159,0.5535,0.44196,0.12556,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,25,0,0,3,0,0,0,0,0,0,0,1],[92,159,0.5786,0.39731,0.10553,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,7,0,0,19,0,0,4,0,0,0,0,0,0,0,0],[96,159,0.6038,0.41071,0.06916,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,24,0,0,2,0,0,0,0,0,0,0,0],[100,159,0.6289,0.40178,0.07521,0.42857,0.42857,0.42858,0.14286,0.571,0,0,0,0,0,1,0,0,5,0,0,25,0,0,1,0,0,0,0,0,0,0,0],[104,159,0.6541,0.37947,0.12169,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,2,0,0,7,0,0,19,0,0,3,0,0,0,0,0,0,0,0],[108,159,0.6792,0.36161,0.12869,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,9,0,0,16,0,0,3,0,0,0,0,0,0,0,0],[112,159,0.7044,0.41072,0.06916,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,24,0,0,2,0,0,0,0,0,0,0,0],[116,159,0.7296,0.40177,0.1309,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,13,0,0,4,0,0,2,0,0,0,0,0],[120,159,0.7547,0.40177,0.10372,0.39286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,6,0,0,20,0,0,4,0,0,0,0,0,0,0,0],[124,159,0.7799,0.42858,0.09449,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,21,0,0,6,0,0,0,0,0,0,0,0],[128,159,0.805,0.39283,0.1129,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,9,0,0,16,0,0,5,0,0,0,0,0,0,0,0],[132,159,0.8302,0.41962,0.10058,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,2,0,0,3,0,0,22,0,0,5,0,0,0,0,0,0,0,0],[136,159,0.8553,0.42855,0.10098,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,19,0,0,7,0,0,0,0,0,0,0,0],[140,159,0.8805,0.41072,0.07784,0.42857,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,22,0,0,3,0,0,0,0,0,0,0,0],[144,159,0.9057,0.40179,0.12595,0.42857,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,2,0,0,4,0,0,20,0,0,5,0,0,0,0,0,0,0,0],[148,159,0.9308,0.33483,0.12168,0.28571,0.35729,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,11,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[152,159,0.956,0.41518,0.13997,0.42857,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,0,0,0,4,0,0,20,0,0,5,0,0,1,0,0,0,0,0],[156,159,0.9811,0.25446,0.17029,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,7,0,0,14,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[159,159,1.0,0.1607,0.1417,0.0,0.14286,0.2857,0.0,0.571,9,0,0,9,0,14,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.4286,"k":"flat","v":0.28125,"x":0.4598,"p":[[0,129,0.0,0.34375,0.12299,0.28571,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,2,0,0,15,0,0,11,0,0,3,0,0,0,0,0,0,0,0],[4,129,0.031,0.36607,0.09407,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,14,0,0,15,0,0,2,0,0,0,0,0,0,0,0],[8,129,0.062,0.38842,0.08917,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[12,129,0.093,0.41069,0.09935,0.28593,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,19,0,0,3,0,0,1,0,0,0,0,0],[16,129,0.124,0.4598,0.09929,0.42857,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,18,0,0,9,0,0,1,0,0,0,0,0],[20,129,0.155,0.45537,0.13091,0.42857,0.42857,0.46431,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,19,0,0,7,0,0,0,0,0,0,0,1],[24,129,0.186,0.41964,0.11259,0.28571,0.42857,0.46431,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,15,0,0,8,0,0,0,0,0,0,0,0],[28,129,0.2171,0.40179,0.08328,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0],[32,129,0.2481,0.41519,0.09688,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,20,0,0,5,0,0,0,0,0,0,0,0],[36,129,0.2791,0.39731,0.11141,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,8,0,0,17,0,0,5,0,0,0,0,0,0,0,0],[40,129,0.3101,0.44643,0.09943,0.42857,0.42857,0.46431,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,19,0,0,7,0,0,1,0,0,0,0,0],[44,129,0.3411,0.39286,0.08748,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,21,0,0,2,0,0,0,0,0,0,0,0],[48,129,0.3721,0.42857,0.07986,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,22,0,0,5,0,0,0,0,0,0,0,0],[52,129,0.4031,0.4107,0.11708,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,16,0,0,5,0,0,1,0,0,0,0,0],[56,129,0.4341,0.41965,0.09407,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,21,0,0,5,0,0,0,0,0,0,0,0],[60,129,0.4651,0.442,0.06542,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,0,1,0,0,0,0,0],[64,129,0.4961,0.43304,0.09771,0.42857,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,20,0,0,7,0,0,0,0,0,0,0,0],[68,129,0.5271,0.42414,0.08362,0.42857,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,21,0,0,5,0,0,0,0,0,0,0,0],[72,129,0.5581,0.41076,0.08565,0.39286,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,20,0,0,4,0,0,0,0,0,0,0,0],[76,129,0.5891,0.42411,0.12103,0.42857,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,20,0,0,5,0,0,1,0,0,0,0,0],[80,129,0.6202,0.40183,0.10376,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,9,0,0,17,0,0,5,0,0,0,0,0,0,0,0],[84,129,0.6512,0.39732,0.07771,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,21,0,0,2,0,0,0,0,0,0,0,0],[88,129,0.6822,0.40179,0.09062,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0],[92,129,0.7132,0.43302,0.09092,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,3,0,0,22,0,0,6,0,0,0,0,0,0,0,0],[96,129,0.7442,0.43304,0.08364,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,21,0,0,6,0,0,0,0,0,0,0,0],[100,129,0.7752,0.40177,0.13569,0.28571,0.42857,0.4642,0.0,0.57143,1,0,0,1,0,1,0,0,9,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[104,129,0.8062,0.38839,0.08171,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[108,129,0.8372,0.35714,0.11845,0.2857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[112,129,0.8682,0.36607,0.11809,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,9,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[116,129,0.8992,0.40177,0.12077,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,6,0,0,17,0,0,6,0,0,0,0,0,0,0,0],[120,129,0.9302,0.34375,0.11769,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,14,0,0,13,0,0,2,0,0,0,0,0,0,0,0],[124,129,0.9612,0.36606,0.11809,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,12,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[128,129,0.9922,0.3705,0.13761,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,2,0,0,12,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[129,129,1.0,0.28125,0.16164,0.14289,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,5,0,0,14,0,0,7,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"713547dc6af19ba1","q":"x,y,z positive real numbers such that  $x^2+y^2+z^2=25$ \r\nFind the min price of  $A=\\frac{xy}{z}+\\frac{yz}{x}+\\frac{zx}{y}$","t":[{"b":0,"e":1.0,"k":"flat","v":0.87946,"x":1.0,"p":[[0,15,0.0,0.87946,0.23987,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,1,0,24],[4,15,0.2667,0.91518,0.18851,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,25],[8,15,0.5333,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.28571,"k":"falling","v":0.56696,"x":0.91964,"p":[[0,26,0.0,0.91964,0.18536,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,25],[4,26,0.1538,0.84821,0.23128,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,8,0,0,2,0,19],[8,26,0.3077,0.73661,0.25781,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,11,0,0,3,0,11],[12,26,0.4615,0.67857,0.25254,0.60714,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,15,0,0,2,0,7],[16,26,0.6154,0.72767,0.26812,0.64286,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,8,0,0,6,0,10],[20,26,0.7692,0.64286,0.29233,0.28571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,0,0,6,0,7],[24,26,0.9231,0.58036,0.2765,0.28571,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,11,0,0,2,0,5],[26,26,1.0,0.56696,0.25874,0.28571,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,11,0,0,5,0,2]]}]},{"i":"241af8aea3ea06b7","q":"1-Let  $A$  and  $B$  be two diametrically opposite points on a circle with radius  $1$ . Points  $P_1,P_2,...,P_n$  are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of  $P_1,P_2,...,P_n$  from  $A$  and  $B$ , respectively. Show that at least one of the numbers  $a$  and  $b$  does not exceed  $\\sqrt{2}$","t":[{"b":5,"e":1.0,"k":"rising","v":0.54464,"x":0.98214,"p":[[0,19,0.0,0.54464,0.21558,0.42857,0.42857,0.46431,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,24,0,0,1,0,0,1,0,0,1,0,5],[4,19,0.2105,0.69643,0.24936,0.42857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,6,0,0,1,0,11],[8,19,0.4211,0.80804,0.24643,0.53571,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,4,0,0,0,0,19],[12,19,0.6316,0.88393,0.22428,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,25],[16,19,0.8421,0.89732,0.21498,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,26],[19,19,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"rising","v":0.5491,"x":0.85272,"p":[[0,13,0.0,0.58482,0.23517,0.42857,0.42857,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,21,0,0,1,0,0,3,0,0,0,0,7],[4,13,0.3077,0.5491,0.21162,0.42857,0.42857,0.60682,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,23,0,0,1,0,0,3,0,0,0,0,5],[8,13,0.6154,0.62946,0.2547,0.42857,0.42857,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,3,0,0,1,0,9],[12,13,0.9231,0.85272,0.24603,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,2,0,0,0,0,23],[13,13,1.0,0.85268,0.24087,0.67856,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,0,0,23]]}]},{"i":"c7c9fc05f4d04a80","q":"**8.** Find all integers  $a>1$  for which the least (integer) solution  $n$  of the congruence  $a^{n} \\equiv 1 \\pmod{p}$  differs from 6 (p is any prime number). **(N. 9)**","t":[{"b":4,"e":1.0,"k":"flat","v":0.89286,"x":0.94196,"p":[[0,32,0.0,0.92857,0.16366,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[4,32,0.125,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[8,32,0.25,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[12,32,0.375,0.89286,0.11845,0.82143,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,16],[16,32,0.5,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[20,32,0.625,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[24,32,0.75,0.90624,0.11075,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[28,32,0.875,0.93304,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22],[32,32,1.0,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15]]},{"b":6,"e":0.71429,"k":"flat","v":0.90179,"x":0.96875,"p":[[0,30,0.0,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[4,30,0.1333,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[8,30,0.2667,0.93304,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22],[12,30,0.4,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[16,30,0.5333,0.95522,0.09774,1.0,1.0,1.0,0.71,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[20,30,0.6667,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,30,0.8,0.92843,0.0948,0.85714,1.0,1.0,0.71,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[28,30,0.9333,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[30,30,1.0,0.90179,0.10374,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15]]}]},{"i":"d60ad9c6266b1a11","q":"$$ Problem 2: $$   Points  $D$  and  $E$  are taken on side  $CB$  of triangle  $ABC$ , with  $D$  between  $C$  and  $E$ ,\nsuch that  $\\angle BAE =\\angle CAD$ . If  $AC < AB$ , prove that  $AC.AE < AB.AD$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,28,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,28,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,23,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,23,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,23,0.3478,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ed5bd05a2cab4a75","q":"Determine the natural numbers  $m,n$  such as  $85^m-n^4=4$","t":[{"b":0,"e":0.71429,"k":"falling","v":0.72768,"x":0.94196,"p":[[0,98,0.0,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[4,98,0.0408,0.82589,0.12745,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,5,0,10],[8,98,0.0816,0.83036,0.1448,0.71429,0.78571,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,4,0,12],[12,98,0.1224,0.72768,0.14445,0.71429,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,22,0,0,2,0,4],[16,98,0.1633,0.77679,0.11811,0.71429,0.71429,0.71429,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7],[20,98,0.2041,0.82143,0.12372,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,6,0,9],[24,98,0.2449,0.79463,0.12848,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,3,0,8],[28,98,0.2857,0.80804,0.1411,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,3,0,10],[32,98,0.3265,0.8125,0.12078,0.71429,0.71429,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,6,0,8],[36,98,0.3673,0.75893,0.10374,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,3,0,4],[40,98,0.4082,0.78124,0.13358,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,1,0,8],[44,98,0.449,0.81696,0.11971,0.71429,0.71429,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,7,0,8],[48,98,0.4898,0.78571,0.11845,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,8,0,5],[52,98,0.5306,0.84819,0.1426,0.71429,0.857,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,3,0,14],[56,98,0.5714,0.83035,0.13092,0.71429,0.85707,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,7,0,10],[60,98,0.6122,0.75446,0.11425,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,1,0,5],[64,98,0.6531,0.87054,0.13533,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,3,0,16],[68,98,0.6939,0.76786,0.1171,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,4,0,5],[72,98,0.7347,0.82142,0.13835,0.71429,0.71429,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,3,0,11],[76,98,0.7755,0.86161,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,14],[80,98,0.8163,0.86161,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,14],[84,98,0.8571,0.80804,0.12682,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,3,0,9],[88,98,0.898,0.83928,0.13244,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,7,0,11],[92,98,0.9388,0.83927,0.15467,0.71429,0.71429,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,0,0,15],[96,98,0.9796,0.81696,0.12993,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,6,0,9],[98,98,1.0,0.78571,0.12877,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,1,0,8]]},{"b":7,"e":0.14286,"k":"falling","v":0.23214,"x":0.9241,"p":[[0,65,0.0,0.9241,0.12869,0.85711,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,23],[4,65,0.0615,0.83036,0.13092,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,11],[8,65,0.1231,0.80357,0.12753,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,2,0,9],[12,65,0.1846,0.80804,0.1411,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,17,0,0,5,0,9],[16,65,0.2462,0.83036,0.1357,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,2,0,12],[20,65,0.3077,0.79464,0.16342,0.71429,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,18,0,0,2,0,10],[24,65,0.3692,0.78571,0.18558,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,17,0,0,1,0,11],[28,65,0.4308,0.74105,0.15747,0.71429,0.71429,0.85704,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,20,0,0,5,0,4],[32,65,0.4923,0.79018,0.15561,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,2,0,9],[36,65,0.5538,0.57143,0.29014,0.28571,0.71429,0.75,0.0,1.0,1,4,0,1,0,1,0,0,11,0,0,1,0,0,0,0,0,10,0,0,4,0,4],[40,65,0.6154,0.67411,0.21793,0.67857,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,18,0,0,1,0,5],[44,65,0.6769,0.61161,0.24804,0.28571,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,0,0,0,3,0,0,14,0,0,3,0,3],[48,65,0.7385,0.59818,0.21852,0.39286,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,2,0,0,5,0,0,12,0,0,3,0,2],[52,65,0.8,0.65177,0.26472,0.49968,0.71429,0.85704,0.14286,1.0,0,6,0,0,0,2,0,0,6,0,0,0,0,0,3,0,0,12,0,0,3,0,6],[56,65,0.8615,0.49107,0.24984,0.2857,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,12,0,0,2,0,0,0,0,0,12,0,0,3,0,0],[60,65,0.9231,0.27231,0.17625,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,19,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[64,65,0.9846,0.23214,0.09278,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.24553,0.07349,0.14286,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9854a1f5999737fa","q":"$2$  darts are thrown randomly at a circular board with center  $O$ , such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked  $A$  and  $B$ . What is the probability that  $\\angle AOB$  is acute?","t":[{"b":4,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,46,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,46,0.087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,46,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,46,0.4348,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,46,0.5217,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,46,0.6087,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[32,46,0.6957,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[36,46,0.7826,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,46,0.8696,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[44,46,0.9565,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[46,46,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,27,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,0.99553,0.02486,1.0,1.0,1.0,0.8571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"7cd666109d779136","q":"$A B C D$ is a convex quadrilateral in which $A B$ is the longest side. Points $M$ and $N$ are located on sides $A B$ and $B C$ respectively, so that each of the segments $A N$ and $C M$ divides the quadrilateral into two parts of equal area. Prove that the segment $M N$ bisects the diagonal $B D$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.19186,"x":0.4464,"p":[[0,58,0.0,0.39731,0.23345,0.25002,0.42857,0.57111,0.0,0.857,4,0,0,4,0,4,0,0,3,0,0,12,0,0,3,0,0,5,0,0,1,0,0],[4,58,0.069,0.4464,0.22231,0.2857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,10,0,0,5,0,0,2,0,0,4,0,0],[8,58,0.1379,0.37499,0.1847,0.2857,0.28571,0.4642,0.14286,0.85714,0,0,0,0,0,6,0,0,12,0,0,6,0,0,5,0,0,2,0,0,1,0,0],[12,58,0.2069,0.32143,0.17857,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,11,0,0,9,0,0,8,0,0,2,0,0,1,0,0,1,0,0],[16,58,0.2759,0.37054,0.18509,0.25,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,7,0,0,4,0,0,16,0,0,1,0,0,2,0,0,1,0,0],[20,58,0.3448,0.36607,0.14699,0.28571,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,6,0,0,7,0,0,16,0,0,1,0,0,2,0,0,0,0,0],[24,58,0.4138,0.25437,0.14617,0.14286,0.21428,0.32143,0.0,0.71429,1,0,0,1,0,15,0,0,8,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[28,58,0.4828,0.22304,0.11824,0.14286,0.14286,0.28571,0.14,0.4286,0,0,0,0,0,21,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.19186,0.10479,0.14286,0.14286,0.14287,0.14,0.571,0,0,0,0,0,25,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,58,0.6207,0.27231,0.14443,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,13,0,0,8,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[40,58,0.6897,0.20536,0.12339,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,20,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.2142,0.12377,0.14286,0.14286,0.28571,0.0,0.4286,1,0,0,1,0,21,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.29459,0.16352,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,7,0,0,9,0,0,2,0,0,1,0,0,0,0,0],[52,58,0.8966,0.25882,0.16921,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,18,0,0,7,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[56,58,0.9655,0.2409,0.16928,0.14286,0.14286,0.32143,0.0,0.71429,2,0,0,2,0,18,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[58,58,1.0,0.26784,0.12239,0.14286,0.2857,0.32143,0.14286,0.571,0,0,0,0,0,13,0,0,11,0,0,7,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.28571,"x":0.47764,"p":[[0,90,0.0,0.44192,0.23784,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,9,0,0,4,0,0,7,0,0,4,0,0,3,0,0],[4,90,0.0444,0.41963,0.25984,0.24999,0.42857,0.60714,0.0,1.0,2,1,0,2,0,6,0,0,7,0,0,6,0,0,3,0,0,5,0,0,2,0,1],[8,90,0.0889,0.29899,0.2032,0.14286,0.21428,0.42858,0.0,0.85714,1,0,0,1,0,15,0,0,5,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[12,90,0.1333,0.38406,0.21552,0.25,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,7,0,0,7,0,0,10,0,0,2,0,0,3,0,0,2,0,0],[16,90,0.1778,0.30348,0.19487,0.14286,0.2857,0.42857,0.0,0.85714,1,0,0,1,0,13,0,0,6,0,0,9,0,0,0,0,0,2,0,0,1,0,0],[20,90,0.2222,0.29017,0.19717,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,11,0,0,7,0,0,6,0,0,4,0,0,0,0,0,1,0,0],[24,90,0.2667,0.37054,0.24186,0.14286,0.35714,0.46431,0.0,1.0,1,1,0,1,0,11,0,0,4,0,0,8,0,0,3,0,0,3,0,0,1,0,1],[28,90,0.3111,0.39283,0.23417,0.14286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,12,0,0,3,0,0,6,0,0,4,0,0,6,0,0,1,0,0],[32,90,0.3556,0.33929,0.20124,0.14286,0.28571,0.4286,0.0,0.85714,1,0,0,1,0,11,0,0,5,0,0,8,0,0,5,0,0,1,0,0,1,0,0],[36,90,0.4,0.28571,0.21724,0.14286,0.2857,0.42857,0.0,0.85714,4,0,0,4,0,11,0,0,7,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[40,90,0.4444,0.38838,0.17214,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,5,0,0,9,0,0,12,0,0,3,0,0,2,0,0,1,0,0],[44,90,0.4889,0.30801,0.17532,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,11,0,0,9,0,0,5,0,0,5,0,0,1,0,0,0,0,0],[48,90,0.5333,0.45526,0.24609,0.28571,0.42857,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,4,0,0,11,0,0,6,0,0,2,0,0,3,0,1],[52,90,0.5778,0.45755,0.22434,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,8,0,0,6,0,1,7,0,0,2,0,0,4,0,0],[56,90,0.6222,0.47764,0.25903,0.28571,0.42859,0.57143,0.0,1.0,3,1,0,3,0,1,0,0,6,0,0,8,0,0,7,0,0,1,0,0,5,0,1],[60,90,0.6667,0.47316,0.22426,0.42857,0.42859,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,1,0,0,12,0,0,9,0,0,2,0,0,2,0,1],[64,90,0.7111,0.45978,0.15037,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,12,0,0,9,0,0,2,0,0,1,0,0],[68,90,0.7556,0.41964,0.17474,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,11,0,0,9,0,0,7,0,0,2,0,0,1,0,0],[72,90,0.8,0.38839,0.20277,0.2857,0.42857,0.42858,0.0,0.85714,2,0,0,2,0,5,0,0,5,0,0,13,0,0,3,0,0,3,0,0,1,0,0],[76,90,0.8444,0.4375,0.22851,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,4,0,0,4,0,0,11,0,0,5,0,0,3,0,0,3,0,0],[80,90,0.8889,0.46427,0.17127,0.39286,0.42857,0.57141,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,14,0,0,6,0,0,1,0,0,3,0,0],[84,90,0.9333,0.45087,0.1561,0.39286,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,15,0,0,6,0,0,1,0,0,2,0,0],[88,90,0.9778,0.42857,0.15971,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,17,0,0,3,0,0,1,0,0,2,0,0],[90,90,1.0,0.3973,0.17026,0.28571,0.42857,0.4286,0.0,0.85714,2,0,0,2,0,2,0,0,5,0,0,18,0,0,3,0,0,1,0,0,1,0,0]]}]},{"i":"8bb74311ab084341","q":"$A$ set $S$ of $n-1$ natural numbers is given ( $n \\geq 3$ ). There exists at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.16518,"x":0.32142,"p":[[0,35,0.0,0.30795,0.24258,0.105,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,1,0,0,13,0,0,4,0,0,1,0,0,4,0,0,1,0,0],[4,35,0.1143,0.2366,0.21608,0.0,0.28571,0.32143,0.0,0.857,11,0,0,11,0,2,0,0,11,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[8,35,0.2286,0.22768,0.21975,0.0,0.28571,0.32143,0.0,0.85714,12,0,0,12,0,2,0,0,10,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[12,35,0.3429,0.26338,0.19918,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,11,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[16,35,0.4571,0.16518,0.18935,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,0,0,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[20,35,0.5714,0.27677,0.22849,0.0,0.28571,0.42857,0.0,1.0,9,1,0,9,0,1,0,0,12,0,0,7,0,0,1,0,0,1,0,0,0,0,1],[24,35,0.6857,0.30357,0.20124,0.2857,0.28571,0.28571,0.0,1.0,5,1,0,5,0,0,0,0,21,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[28,35,0.8,0.30804,0.13883,0.2857,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,4,0,0,15,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[32,35,0.9143,0.28125,0.07563,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.32142,0.07981,0.2857,0.28571,0.32143,0.143,0.571,0,0,0,0,0,1,0,0,23,0,0,7,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.14286,"x":0.35714,"p":[[0,55,0.0,0.35714,0.28793,0.14286,0.28571,0.71429,0.0,0.85714,7,0,0,7,0,3,0,0,11,0,0,1,0,0,1,0,0,6,0,0,3,0,0],[4,55,0.0727,0.29911,0.24836,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,2,0,0,12,0,0,5,0,0,2,0,0,1,0,0,1,0,1],[8,55,0.1455,0.22768,0.22263,0.0,0.28571,0.42857,0.0,0.71429,14,0,0,14,0,1,0,0,5,0,0,9,0,0,2,0,0,1,0,0,0,0,0],[12,55,0.2182,0.27679,0.26471,0.0,0.28571,0.42857,0.0,1.0,10,1,0,10,0,3,0,0,9,0,0,6,0,0,1,0,0,0,0,0,2,0,1],[16,55,0.2909,0.32588,0.263,0.14286,0.28571,0.4286,0.0,1.0,7,1,0,7,0,5,0,0,6,0,0,7,0,0,3,0,0,2,0,0,1,0,1],[20,55,0.3636,0.1874,0.2271,0.0,0.07,0.28571,0.0,0.71429,16,0,0,16,0,3,0,0,6,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[24,55,0.4364,0.18741,0.17657,0.0,0.2857,0.28571,0.0,0.4286,14,0,0,14,0,1,0,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.25,0.21724,0.0,0.2857,0.42857,0.0,0.85714,10,0,0,10,0,3,0,0,9,0,0,8,0,0,0,0,0,1,0,0,1,0,0],[32,55,0.5818,0.21874,0.22297,0.0,0.2857,0.28571,0.0,1.0,12,1,0,12,0,2,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[36,55,0.6545,0.16518,0.1992,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,2,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[40,55,0.7273,0.24105,0.2002,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,4,0,0,8,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[44,55,0.8,0.1607,0.18811,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,5,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[48,55,0.8727,0.14286,0.17496,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,1,0,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[52,55,0.9455,0.25445,0.17397,0.14286,0.28571,0.32143,0.0,0.57143,7,0,0,7,0,4,0,0,13,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[55,55,1.0,0.34366,0.11231,0.2857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,3,0,0,10,0,0,18,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4e6461d8484817e0","q":"We consider the equation  $x^2 + (a + b - 1)x + ab - a - b = 0$ , where  $a$  and  $b$  are positive integers with  $a \\leq b$ .\n\na) Show that the equation has  $2$  distinct real solutions.\n\nb) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and  $b < 2a.$","t":[{"b":5,"e":0.85714,"k":"flat","v":0.79909,"x":0.87053,"p":[[0,165,0.0,0.82585,0.09275,0.85711,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,26,0,1],[4,165,0.0242,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[8,165,0.0485,0.85714,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[12,165,0.0727,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[16,165,0.097,0.83481,0.09522,0.85714,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,28,0,1],[20,165,0.1212,0.85714,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[24,165,0.1455,0.83929,0.05923,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0],[28,165,0.1697,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[32,165,0.1939,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[36,165,0.2182,0.85267,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[40,165,0.2424,0.85714,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[44,165,0.2667,0.83927,0.09283,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,24,0,3],[48,165,0.2909,0.84374,0.07457,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,26,0,2],[52,165,0.3152,0.87053,0.07457,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,5],[56,165,0.3394,0.84373,0.08272,0.85714,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,27,0,2],[60,165,0.3636,0.84375,0.07457,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,29,0,1],[64,165,0.3879,0.84821,0.06121,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,1],[68,165,0.4121,0.87053,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5],[72,165,0.4364,0.84375,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,1],[76,165,0.4606,0.85267,0.10402,0.85714,0.85714,0.85714,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,26,0,4],[80,165,0.4848,0.85714,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[84,165,0.5091,0.84374,0.05486,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0],[88,165,0.5333,0.8482,0.07941,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,3],[92,165,0.5576,0.84375,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,1],[96,165,0.5818,0.8482,0.07092,0.85714,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,2],[100,165,0.6061,0.82587,0.06901,0.857,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0],[104,165,0.6303,0.83481,0.08827,0.85714,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,30,0,0],[108,165,0.6545,0.84821,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,1],[112,165,0.6788,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[116,165,0.703,0.85713,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[120,165,0.7273,0.83929,0.04725,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0],[124,165,0.7515,0.83479,0.08077,0.85714,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,27,0,1],[128,165,0.7758,0.82141,0.08752,0.85714,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,27,0,0],[132,165,0.8,0.83478,0.08835,0.85714,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,28,0,1],[136,165,0.8242,0.84374,0.05486,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0],[140,165,0.8485,0.81249,0.10974,0.85714,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,27,0,0],[144,165,0.8727,0.83036,0.05576,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0],[148,165,0.897,0.82141,0.08752,0.857,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,24,0,1],[152,165,0.9212,0.84374,0.08268,0.85714,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,29,0,1],[156,165,0.9455,0.84819,0.04979,0.85714,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0],[160,165,0.9697,0.83035,0.07523,0.85711,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,25,0,1],[164,165,0.9939,0.79909,0.10632,0.71429,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,23,0,0],[165,165,1.0,0.81249,0.0974,0.85708,0.85714,0.85714,0.42857,0.857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(GRE 1) Show that $\\frac{20}{60}<\\sin 20^{\\circ}<\\frac{21}{60}$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.35714,"x":0.97768,"p":[[0,15,0.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,15,0.2667,0.77679,0.33491,0.67857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,0,0,20],[8,15,0.5333,0.61607,0.43659,0.0,0.78571,1.0,0.0,1.0,10,15,0,10,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,15],[12,15,0.8,0.51339,0.42687,0.0,0.57143,1.0,0.0,1.0,11,11,0,11,0,0,0,0,3,0,0,0,0,0,3,0,0,4,0,0,0,0,11],[15,15,1.0,0.35714,0.46702,0.0,0.0,1.0,0.0,1.0,20,10,0,20,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,10]]},{"b":2,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,14,0.0,0.9375,0.17474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[4,14,0.2857,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[8,14,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,14,0.8571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[14,14,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"61c5bef300340591","q":"$120$  pirates distribute  $119$  gold pieces among themselves. Then the captain checks if any pirate has  $15$  or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with  $15$  or more gold pieces. Does this process end after a lot of checks?","t":[{"b":2,"e":0.0,"k":"flat","v":0.03571,"x":0.1875,"p":[[0,56,0.0,0.07143,0.24223,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,56,0.0714,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,56,0.1429,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,56,0.2143,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,56,0.2857,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[20,56,0.3571,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,56,0.4286,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,56,0.5,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,56,0.5714,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[36,56,0.6429,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[40,56,0.7143,0.16518,0.36089,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[44,56,0.7857,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,56,0.8571,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,56,0.9286,0.09821,0.2911,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[56,56,1.0,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":7,"e":0.0,"k":"flat","v":0.03125,"x":0.13393,"p":[[0,39,0.0,0.13393,0.32915,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,39,0.1026,0.10714,0.29014,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,39,0.2051,0.10268,0.29066,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,39,0.3077,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,39,0.4103,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,39,0.5128,0.13393,0.33108,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,39,0.6154,0.09821,0.2911,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[28,39,0.7179,0.03563,0.17491,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,39,0.8205,0.09821,0.2911,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[36,39,0.9231,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[39,39,1.0,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"108bc52faea7f687","q":"$ f(x)$  is a given polynomial whose degree at least 2. Define the following polynomial-sequence:  $ g_1(x)\\equal{}f(x), g_{n\\plus{}1}(x)\\equal{}f(g_n(x))$ , for all  $ n \\in N$ . Let  $ r_n$  be the average of  $ g_n(x)$ 's roots. If  $ r_{19}\\equal{}99$ , find  $ r_{99}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.97321,"x":0.99554,"p":[[0,8,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,8,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,8,1.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]},{"b":5,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,11,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,11,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,11,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[11,11,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"4d634eef4dd3e3e4","q":"12. (NET 1) Let $f, g$, and $a$ be polynomials with real coefficients, $f$ and $g$ in one variable and $a$ in two variables. Suppose  $$ f(x)-f(y)=a(x, y)(g(x)-g(y)) \\quad \\text { for all } x, y \\in \\mathbb{R} $$  Prove that there exists a polynomial $h$ with $f(x)=h(g(x))$ for all $x \\in \\mathbb{R}$.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.07143,"x":0.30804,"p":[[0,60,0.0,0.07143,0.22868,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,60,0.0667,0.23214,0.32684,0.0,0.14286,0.14286,0.0,1.0,11,4,0,11,0,15,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[8,60,0.1333,0.17857,0.27199,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[12,60,0.2,0.20536,0.30291,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,2],[16,60,0.2667,0.22321,0.30917,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,13,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,2],[20,60,0.3333,0.11607,0.18363,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,60,0.4,0.12947,0.2,0.0,0.14286,0.14286,0.0,0.85714,13,0,0,13,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[28,60,0.4667,0.17857,0.23958,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,15,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,1],[32,60,0.5333,0.17857,0.19233,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,23,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[36,60,0.6,0.23661,0.32656,0.0,0.14286,0.17857,0.0,1.0,12,3,0,12,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,3],[40,60,0.6667,0.20982,0.2575,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,16,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,0],[44,60,0.7333,0.25438,0.31693,0.0,0.14286,0.17857,0.0,1.0,9,2,0,9,0,15,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,2],[48,60,0.8,0.30804,0.34646,0.0,0.14286,0.60714,0.0,1.0,10,1,0,10,0,11,0,0,1,0,0,0,0,0,2,0,0,1,0,0,6,0,1],[52,60,0.8667,0.13838,0.1493,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,16,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[56,60,0.9333,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.10715,0.09449,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.1429,"k":"flat","v":0.0625,"x":0.35712,"p":[[0,52,0.0,0.0625,0.14698,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,52,0.0769,0.18303,0.2993,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,1],[8,52,0.1538,0.11598,0.17289,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,52,0.2308,0.16071,0.26426,0.0,0.14286,0.14286,0.0,1.0,13,2,0,13,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[16,52,0.3077,0.20982,0.30511,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,11,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,2],[20,52,0.3846,0.11161,0.21646,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[24,52,0.4615,0.27222,0.26091,0.14286,0.14286,0.42858,0.0,0.85714,6,0,0,6,0,14,0,0,3,0,0,2,0,0,2,0,0,3,0,0,2,0,0],[28,52,0.5385,0.29911,0.29957,0.14286,0.14286,0.57143,0.0,1.0,6,2,0,6,0,14,0,0,2,0,0,1,0,0,5,0,0,0,0,0,2,0,2],[32,52,0.6154,0.24105,0.26347,0.0,0.14286,0.32144,0.0,0.85714,9,0,0,9,0,13,0,0,2,0,0,1,0,0,3,0,0,2,0,0,2,0,0],[36,52,0.6923,0.35712,0.3154,0.14286,0.2143,0.57143,0.0,1.0,5,2,0,5,0,11,0,0,4,0,0,1,0,0,4,0,0,2,0,0,3,0,2],[40,52,0.7692,0.19642,0.2223,0.0,0.14286,0.1786,0.0,0.85714,10,0,0,10,0,14,0,0,1,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[44,52,0.8462,0.19642,0.24155,0.0,0.14286,0.14287,0.0,0.85714,10,0,0,10,0,15,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0],[48,52,0.9231,0.18304,0.19638,0.14286,0.14286,0.1429,0.0,0.85714,7,0,0,7,0,19,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[52,52,1.0,0.11152,0.11141,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,19,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e0f37edac2ad9674","q":"**Problem 1**\nLet  $f :  \\mathbb{R} \\rightarrow \\mathbb{R}$  be differentiable on  $\\mathbb{R}$ . Prove that there exists  $x \\in [0, 1]$  such that  $$ \\frac{4}{\\pi} ( f(1) - f(0) ) = (1+x^2) f'(x)      \\ . $$","t":[{"b":3,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,17,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,17,0.2353,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,17,0.4706,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,17,0.7059,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,17,0.9412,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[17,17,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]},{"b":5,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,46,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,46,0.087,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,46,0.1739,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,46,0.2609,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,46,0.4348,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,46,0.6957,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,46,0.7826,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"df1f31baa3d32adf","q":"$\\textbf{Problem C.1}$ There are two piles of coins, each containing  $2010$  pieces. Two players  $A$  and  $B$  play a game taking turns ( $A$  plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?","t":[{"b":4,"e":1.0,"k":"rising","v":0.51338,"x":0.95536,"p":[[0,105,0.0,0.51338,0.24185,0.39286,0.57121,0.57143,0.0,1.0,2,3,0,2,0,0,0,0,6,0,0,7,0,0,10,0,0,3,0,0,1,0,3],[4,105,0.0381,0.81696,0.30563,0.67857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,20],[8,105,0.0762,0.80801,0.18767,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,6,0,13],[12,105,0.1143,0.87495,0.17775,0.71421,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,3,0,20],[16,105,0.1524,0.83927,0.21945,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,1,0,0,4,0,18],[20,105,0.1905,0.90625,0.14987,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,21],[24,105,0.2286,0.83478,0.23454,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,4,0,18],[28,105,0.2667,0.80797,0.23859,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,4,0,16],[32,105,0.3048,0.87499,0.18474,0.82143,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,4,0,20],[36,105,0.3429,0.89731,0.17584,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,4,0,22],[40,105,0.381,0.8482,0.22572,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,18],[44,105,0.419,0.85713,0.19887,0.57143,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,2,0,20],[48,105,0.4571,0.87052,0.16508,0.82132,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,7,0,17],[52,105,0.4952,0.83481,0.18251,0.57143,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,6,0,15],[56,105,0.5333,0.89286,0.18557,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,4,0,22],[60,105,0.5714,0.84375,0.22406,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,6,0,17],[64,105,0.6095,0.81694,0.21794,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,1,0,0,6,0,15],[68,105,0.6476,0.91071,0.14617,0.85714,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,21],[72,105,0.6857,0.85712,0.2287,0.82132,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,5,0,19],[76,105,0.7238,0.82143,0.18898,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,4,0,15],[80,105,0.7619,0.85714,0.15972,0.82143,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,10,0,14],[84,105,0.8,0.85714,0.18898,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,5,0,18],[88,105,0.8381,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[92,105,0.8762,0.88838,0.21941,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,23],[96,105,0.9143,0.9375,0.13803,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,26],[100,105,0.9524,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[104,105,0.9905,0.94195,0.14226,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,27],[105,105,1.0,0.91963,0.15129,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24]]},{"b":5,"e":0.57143,"k":"rising","v":0.48214,"x":0.91518,"p":[[0,76,0.0,0.48214,0.26426,0.28571,0.42857,0.57143,0.14286,1.0,0,5,0,0,0,2,0,0,12,0,0,7,0,0,5,0,0,0,0,0,1,0,5],[4,76,0.0526,0.88392,0.2243,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,23],[8,76,0.1053,0.91518,0.15916,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,23],[12,76,0.1579,0.8571,0.18904,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,3,0,19],[16,76,0.2105,0.89731,0.16068,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,21],[20,76,0.2632,0.83482,0.2372,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,6,0,17],[24,76,0.3158,0.82143,0.26726,0.67857,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,6,0,17],[28,76,0.3684,0.91071,0.16656,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,24],[32,76,0.4211,0.82587,0.22516,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,2,0,0,4,0,17],[36,76,0.4737,0.87945,0.18251,0.82132,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,20],[40,76,0.5263,0.80351,0.21949,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,1,0,0,6,0,14],[44,76,0.5789,0.90621,0.1699,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,24],[48,76,0.6316,0.87495,0.1813,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,3,0,20],[52,76,0.6842,0.8303,0.1802,0.67857,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,4,0,15],[56,76,0.7368,0.8571,0.17864,0.67857,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,6,0,17],[60,76,0.7895,0.83927,0.18816,0.57143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,3,0,17],[64,76,0.8421,0.8214,0.21432,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,3,0,0,6,0,15],[68,76,0.8947,0.8348,0.21164,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,4,0,17],[72,76,0.9474,0.81695,0.20898,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,5,0,15],[76,76,1.0,0.80354,0.18475,0.57143,0.85707,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,0,6,0,12]]}]},{"i":"f4ca9e9c1ca05df9","q":"(a) Prove that for all $a, b, c, d \\in \\mathbb{R}$ with $a+b+c+d=0$,\n\n$$\n\\max (a, b)+\\max (a, c)+\\max (a, d)+\\max (b, c)+\\max (b, d)+\\max (c, d) \\geqslant 0\n$$\n\n(b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \\in \\mathbb{R}$ with $a+b+c+d=0$.\n\n#","t":[{"b":0,"e":0.85714,"k":"rising","v":0.67409,"x":0.98214,"p":[[0,122,0.0,0.67409,0.30563,0.53539,0.71429,1.0,0.0,1.0,1,9,0,1,0,3,0,0,3,0,0,1,0,0,4,0,0,6,0,0,5,0,9],[4,122,0.0328,0.95536,0.13092,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[8,122,0.0656,0.9375,0.14698,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[12,122,0.0984,0.94196,0.1551,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[16,122,0.1311,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,122,0.1639,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,122,0.1967,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,122,0.2295,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[32,122,0.2623,0.91071,0.15465,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,5,0,21],[36,122,0.2951,0.94643,0.14617,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[40,122,0.3279,0.92857,0.16751,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[44,122,0.3607,0.88839,0.21049,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,6,0,21],[48,122,0.3934,0.91071,0.18472,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,4,0,23],[52,122,0.4262,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[56,122,0.459,0.92857,0.17857,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[60,122,0.4918,0.91071,0.19805,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,23],[64,122,0.5246,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[68,122,0.5574,0.90625,0.19434,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,24],[72,122,0.5902,0.91509,0.20193,1.0,1.0,1.0,0.14,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,25],[76,122,0.623,0.87498,0.18816,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,9,0,17],[80,122,0.6557,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[84,122,0.6885,0.91071,0.16269,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,4,0,22],[88,122,0.7213,0.92857,0.14725,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[92,122,0.7541,0.91518,0.16698,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,23],[96,122,0.7869,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[100,122,0.8197,0.94643,0.13243,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[104,122,0.8525,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[108,122,0.8852,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[112,122,0.918,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[116,122,0.9508,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[120,122,0.9836,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[122,122,1.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]},{"b":4,"e":0.14286,"k":"falling","v":0.10697,"x":0.87947,"p":[[0,84,0.0,0.70089,0.3376,0.53571,0.85707,1.0,0.0,1.0,3,12,1,3,0,2,0,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,12],[4,84,0.0476,0.87947,0.24249,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[8,84,0.0952,0.83929,0.26184,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,4,0,19],[12,84,0.1429,0.83482,0.27225,0.82143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,18],[16,84,0.1905,0.83929,0.24679,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,18],[20,84,0.2381,0.62045,0.37572,0.24999,0.78571,0.89286,0.0,1.0,5,8,0,5,0,3,0,0,2,0,0,1,0,0,0,0,0,5,0,0,8,0,8],[24,84,0.2857,0.69641,0.32291,0.5354,0.85707,1.0,0.0,1.0,2,10,0,2,0,3,0,0,1,0,0,2,0,0,2,0,0,5,0,0,7,0,10],[28,84,0.3333,0.5982,0.38205,0.25,0.71429,0.89286,0.0,1.0,7,8,0,7,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,0,6,0,8],[32,84,0.381,0.66517,0.35103,0.28571,0.85707,1.0,0.0,1.0,4,9,0,4,0,1,0,0,4,0,0,0,0,0,1,0,0,5,0,0,8,0,9],[36,84,0.4286,0.60268,0.37582,0.25,0.71429,0.89286,0.0,1.0,5,8,0,5,0,3,0,0,3,0,0,0,0,0,2,0,0,4,0,0,7,0,8],[40,84,0.4762,0.62946,0.34784,0.28571,0.71429,0.89286,0.0,1.0,3,8,0,3,0,4,0,0,2,0,0,1,0,0,2,0,0,6,0,0,6,0,8],[44,84,0.5238,0.69643,0.34764,0.39285,0.85714,1.0,0.0,1.0,3,13,0,3,0,1,0,0,4,0,0,1,0,0,2,0,0,3,0,0,5,0,13],[48,84,0.5714,0.49096,0.32337,0.14286,0.57143,0.71429,0.0,1.0,6,1,0,6,0,3,0,0,3,0,0,1,0,0,4,0,0,9,0,0,5,0,1],[52,84,0.619,0.53561,0.34636,0.24999,0.57141,0.85714,0.0,1.0,4,5,0,4,0,4,0,0,4,0,0,2,0,0,4,0,0,3,0,0,6,0,5],[56,84,0.6667,0.68289,0.26662,0.57143,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,3,0,0,1,0,0,3,0,0,12,0,0,5,0,6],[60,84,0.7143,0.77232,0.1551,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,7,0,6],[64,84,0.7619,0.54462,0.35613,0.14286,0.64286,0.85714,0.0,1.0,6,5,0,6,0,3,0,0,1,0,0,2,0,0,4,0,0,6,0,0,5,0,5],[68,84,0.8095,0.52223,0.35832,0.14286,0.71429,0.85714,0.0,1.0,6,4,0,6,0,4,0,0,2,0,0,1,0,0,2,0,0,8,0,0,5,0,4],[72,84,0.8571,0.47321,0.33204,0.14286,0.42859,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,6,0,0,2,0,0,0,0,0,8,0,0,5,0,2],[76,84,0.9048,0.33473,0.30854,0.14286,0.2857,0.60714,0.0,1.0,6,1,0,6,0,9,0,0,8,0,0,0,0,0,1,0,0,3,0,0,4,0,1],[80,84,0.9524,0.30803,0.31966,0.0,0.2857,0.46429,0.0,1.0,11,2,0,11,0,3,0,0,9,0,0,1,0,0,1,0,0,3,0,0,2,0,2],[84,84,1.0,0.10697,0.10709,0.0,0.14143,0.14286,0.0,0.28571,14,0,0,14,0,12,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"974bdb095ba40973","q":"13. G4 (USA) Let $\\triangle A B C$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $A P, B P, C P$ meet the sides $B C, C A, A B$ in the points $A_{1}, B_{1}, C_{1}$ respectively. Prove that  $$ A_{1} B_{1} \\cdot B_{1} C_{1} \\cdot C_{1} A_{1} \\geq A_{1} B \\cdot B_{1} C \\cdot C_{1} A $$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.5892,"x":0.96875,"p":[[0,170,0.0,0.5892,0.31504,0.28571,0.57143,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,3,0,0,2,0,0,7,0,0,5,0,0,1,0,8],[4,170,0.0235,0.71875,0.27078,0.57143,0.71429,1.0,0.0,1.0,1,12,1,1,0,2,0,0,0,0,0,0,0,0,10,0,0,7,0,0,0,0,12],[8,170,0.0471,0.90625,0.2412,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[12,170,0.0706,0.95981,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[16,170,0.0941,0.93304,0.14719,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,26],[20,170,0.1176,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[24,170,0.1412,0.91964,0.16342,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,24],[28,170,0.1647,0.81696,0.27019,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,1,0,20],[32,170,0.1882,0.94196,0.1984,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[36,170,0.2118,0.91518,0.16311,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[40,170,0.2353,0.82589,0.21349,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,7,0,0,1,0,17],[44,170,0.2588,0.87944,0.22339,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,0,0,0,24],[48,170,0.2824,0.90625,0.16602,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,0,0,24],[52,170,0.3059,0.87946,0.20238,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,22],[56,170,0.3294,0.83482,0.2372,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,1,0,19],[60,170,0.3529,0.87497,0.21654,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,23],[64,170,0.3765,0.86159,0.20974,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,1,0,21],[68,170,0.4,0.8527,0.22719,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,0,0,20],[72,170,0.4235,0.83478,0.20555,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,1,0,18],[76,170,0.4471,0.81696,0.26058,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,3,0,0,0,0,20],[80,170,0.4706,0.83034,0.23267,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,6,0,0,0,0,19],[84,170,0.4941,0.80803,0.249,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,7,0,0,0,0,18],[88,170,0.5176,0.93303,0.13825,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[92,170,0.5412,0.88393,0.18707,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,0,0,22],[96,170,0.5647,0.83036,0.20025,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,0,1,0,17],[100,170,0.5882,0.85267,0.1731,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,2,0,17],[104,170,0.6118,0.85266,0.19721,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,1,0,19],[108,170,0.6353,0.81249,0.25365,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,2,0,0,1,0,19],[112,170,0.6588,0.83482,0.2372,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,8,0,0,0,0,19],[116,170,0.6824,0.77229,0.2522,0.57143,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,4,0,0,0,0,16],[120,170,0.7059,0.83927,0.19806,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,2,0,17],[124,170,0.7294,0.76782,0.20128,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,8,0,0,1,0,12],[128,170,0.7529,0.78125,0.23141,0.67857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,9,0,0,1,0,14],[132,170,0.7765,0.8214,0.22019,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,7,0,0,1,0,17],[136,170,0.8,0.79016,0.25502,0.67857,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,2,0,0,0,0,0,5,0,0,6,0,0,3,0,15],[140,170,0.8235,0.83036,0.22428,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,0,2,0,18],[144,170,0.8471,0.83482,0.19597,0.57143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,1,0,18],[148,170,0.8706,0.87945,0.17171,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,0,0,21],[152,170,0.8941,0.76783,0.21653,0.57143,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,0,1,0,13],[156,170,0.9176,0.62051,0.29798,0.5354,0.57143,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,2,0,0,1,0,0,9,0,0,6,0,0,1,0,8],[160,170,0.9412,0.77675,0.22572,0.57143,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,6,0,0,1,0,14],[164,170,0.9647,0.83927,0.18125,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,0,0,17],[168,170,0.9882,0.68746,0.19048,0.57143,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,11,0,0,1,0,6],[170,170,1.0,0.61604,0.10973,0.57143,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,15,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.59821,"x":1.0,"p":[[0,71,0.0,0.68747,0.32231,0.53539,0.57143,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,3,0,0,1,0,0,9,0,0,0,0,0,0,0,15],[4,71,0.0563,0.79018,0.2879,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,0,0,0,19],[8,71,0.1127,0.63393,0.41487,0.14286,0.78571,1.0,0.0,1.0,7,15,0,7,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,1,0,15],[12,71,0.169,0.59821,0.31831,0.53571,0.71429,0.71429,0.0,1.0,4,7,0,4,0,2,0,0,1,0,0,1,0,0,7,0,0,10,0,0,0,0,7],[16,71,0.2254,0.68304,0.31891,0.57143,0.71429,1.0,0.0,1.0,2,12,0,2,0,3,0,0,1,0,0,0,0,0,6,0,0,8,0,0,0,0,12],[20,71,0.2817,0.61152,0.35229,0.4286,0.57143,1.0,0.0,1.0,4,11,0,4,0,3,0,0,0,0,0,3,0,0,7,0,0,4,0,0,0,0,11],[24,71,0.338,0.64723,0.29678,0.42857,0.57143,1.0,0.0,1.0,1,10,0,1,0,2,0,0,3,0,0,3,0,0,8,0,0,4,0,0,1,0,10],[28,71,0.3944,0.71874,0.29985,0.57143,0.71429,1.0,0.0,1.0,2,13,0,2,0,1,0,0,1,0,0,2,0,0,5,0,0,7,0,0,1,0,13],[32,71,0.4507,0.61161,0.33926,0.28571,0.57143,1.0,0.0,1.0,1,11,0,1,0,5,0,0,3,0,0,4,0,0,4,0,0,3,0,0,1,0,11],[36,71,0.507,0.60714,0.40564,0.14286,0.71429,1.0,0.0,1.0,7,13,0,7,0,2,0,0,1,0,0,1,0,0,3,0,0,4,0,0,1,0,13],[40,71,0.5634,0.6829,0.31079,0.53572,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,5,0,0,1,0,0,5,0,0,6,0,0,1,0,12],[44,71,0.6197,0.68304,0.34577,0.4286,0.71429,1.0,0.0,1.0,2,15,0,2,0,3,0,0,2,0,0,2,0,0,5,0,0,3,0,0,0,0,15],[48,71,0.6761,0.63827,0.30941,0.42857,0.64286,1.0,0.0,1.0,2,9,0,2,0,1,0,0,4,0,0,3,0,0,6,0,0,4,0,0,3,0,9],[52,71,0.7324,0.61607,0.40317,0.14286,0.64286,1.0,0.0,1.0,5,15,0,5,0,4,0,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,15],[56,71,0.7887,0.85268,0.26119,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,3,0,0,0,0,23],[60,71,0.8451,0.81695,0.25813,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,6,0,0,0,0,19],[64,71,0.9014,0.80356,0.2896,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,4,0,0,0,0,20],[68,71,0.9577,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[71,71,1.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]}]},{"i":"d8c5146e1059dc7b","q":"(a) Prove the inequality  $$ \\frac{x^{2}}{(x-1)^{2}}+\\frac{y^{2}}{(y-1)^{2}}+\\frac{z^{2}}{(z-1)^{2}} \\geq 1 $$  for real numbers $x, y, z \\neq 1$ satisfying the condition $x y z=1$.  (b) Show that there are infinitely many triples of rational numbers $x, y, z$ for which this inequality turns into equality.","t":[{"b":5,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,91,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,91,0.044,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,91,0.0879,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,91,0.1319,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,91,0.1758,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,91,0.2198,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,91,0.2637,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,91,0.3077,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,91,0.3516,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,91,0.3956,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,91,0.4396,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,91,0.4835,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,91,0.5275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,91,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,91,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,91,0.6593,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,91,0.7033,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,91,0.7473,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,91,0.7912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,91,0.8352,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,91,0.8791,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,91,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,91,0.967,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[91,91,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,44,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,44,0.0909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,44,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,44,0.2727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,44,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,44,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,44,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,44,0.6364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d089cf51760a74f1","q":"11. (POL 3) The matrix  $$ \\left(\\begin{array}{ccc} a_{11} & \\ldots & a_{1 n} \\\\ \\vdots & \\ldots & \\vdots \\\\ a_{n 1} & \\ldots & a_{n n} \\end{array}\\right) $$  satisfies the inequality $\\sum_{j=1}^{n}\\left|a_{j 1} x_{1}+\\cdots+a_{j n} x_{n}\\right| \\leq M$ for each choice of numbers $x_{i}$ equal to $\\pm 1$. Show that  $$ \\left|a_{11}+a_{22}+\\cdots+a_{n n}\\right| \\leq M $$","t":[{"b":1,"e":0.0,"k":"falling","v":0.19196,"x":0.59375,"p":[[0,15,0.0,0.59375,0.39465,0.24999,0.71429,1.0,0.0,1.0,5,13,0,5,0,3,0,0,4,0,0,1,0,0,2,0,0,4,0,0,0,0,13],[4,15,0.2667,0.32589,0.32189,0.0,0.28571,0.57143,0.0,1.0,12,3,0,12,0,2,0,0,4,0,0,3,0,0,7,0,0,1,0,0,0,0,3],[8,15,0.5333,0.29911,0.31816,0.0,0.21429,0.57143,0.0,1.0,14,2,0,14,0,2,0,0,2,0,0,4,0,0,5,0,0,3,0,0,0,0,2],[12,15,0.8,0.19643,0.25442,0.0,0.0,0.42857,0.0,1.0,17,1,0,17,0,2,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,1],[15,15,1.0,0.19196,0.27107,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,3,0,0,3,0,0,3,0,0,2,0,0,2,0,0,0,0,1]]},{"b":3,"e":0.71429,"k":"falling","v":0.2633,"x":0.45981,"p":[[0,19,0.0,0.45981,0.38916,0.0,0.42857,0.78571,0.0,1.0,9,8,0,9,0,3,0,0,2,0,0,3,0,0,4,0,0,3,0,0,0,0,8],[4,19,0.2105,0.38392,0.3415,0.0,0.35714,0.60714,0.0,1.0,10,4,0,10,0,2,0,0,4,0,0,4,0,0,4,0,0,4,0,0,0,0,4],[8,19,0.4211,0.27232,0.30169,0.0,0.21428,0.42857,0.0,1.0,13,2,0,13,0,3,0,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,2],[12,19,0.6316,0.2633,0.26517,0.0,0.21428,0.42857,0.0,1.0,11,1,0,11,0,5,0,0,6,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[16,19,0.8421,0.3125,0.32623,0.0,0.28571,0.57143,0.0,1.0,12,3,0,12,0,3,0,0,5,0,0,3,0,0,3,0,0,3,0,0,0,0,3],[19,19,1.0,0.27231,0.30796,0.0,0.21429,0.57143,0.0,1.0,15,2,0,15,0,1,0,0,5,0,0,1,0,0,7,0,0,1,0,0,0,0,2]]}]},{"i":"9e14be3b3c2d30c9","q":"11. (IRE 1) ${ }^{\\mathrm{IMO1}}$ Let $n>1$ be an integer and let $f(x)=x^{n}+5 x^{n-1}+3$. Prove that there do not exist polynomials $g(x), h(x)$, each having integer coefficients and degree at least one, such that $f(x)=g(x) h(x)$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.50893,"p":[[0,50,0.0,0.50893,0.38455,0.14286,0.57143,0.89286,0.0,1.0,6,8,0,6,0,5,0,0,3,0,0,1,0,0,3,0,0,4,0,0,2,0,8],[4,50,0.08,0.38839,0.37667,0.0,0.2857,0.71429,0.0,1.0,10,6,0,10,0,5,0,0,3,0,0,1,0,0,4,0,0,3,0,0,0,0,6],[8,50,0.16,0.40402,0.4152,0.0,0.2857,0.85714,0.0,1.0,13,7,0,13,1,1,0,0,3,0,0,0,0,0,3,0,0,2,0,0,2,0,7],[12,50,0.24,0.36161,0.38793,0.0,0.14286,0.75,0.0,1.0,11,5,0,11,0,7,0,0,2,0,0,0,0,0,3,0,0,1,0,0,3,0,5],[16,50,0.32,0.27232,0.37519,0.0,0.0,0.57143,0.0,1.0,17,4,0,17,0,4,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,4],[20,50,0.4,0.19643,0.34022,0.0,0.0,0.28571,0.0,1.0,21,4,0,21,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,4],[24,50,0.48,0.24554,0.39486,0.0,0.0,0.35713,0.0,1.0,21,5,0,21,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,5],[28,50,0.56,0.22321,0.36759,0.0,0.0,0.28571,0.0,1.0,21,5,0,21,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[32,50,0.64,0.20982,0.3694,0.0,0.0,0.17857,0.0,1.0,22,5,0,22,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[36,50,0.72,0.16964,0.27994,0.0,0.0,0.2857,0.0,1.0,20,2,0,20,0,2,0,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,2],[40,50,0.8,0.12946,0.30169,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[44,50,0.88,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"falling","v":0.09375,"x":0.5223,"p":[[0,31,0.0,0.48659,0.40067,0.10714,0.42836,1.0,0.0,1.0,8,9,0,8,0,3,0,0,5,0,0,0,0,0,3,0,0,3,0,0,1,0,9],[4,31,0.129,0.33929,0.38091,0.0,0.21429,0.57143,0.0,1.0,14,6,0,14,0,2,0,0,4,0,0,0,0,0,6,0,0,0,0,0,0,0,6],[8,31,0.2581,0.4107,0.40049,0.0,0.28571,0.78571,0.0,1.0,11,8,0,11,0,3,0,0,4,0,0,0,0,0,5,0,0,1,0,0,0,0,8],[12,31,0.3871,0.43304,0.39202,0.10714,0.28571,0.89286,0.0,1.0,8,8,0,8,0,5,0,0,5,0,0,1,0,0,3,0,0,1,0,0,1,0,8],[16,31,0.5161,0.5223,0.38895,0.14286,0.57143,1.0,0.0,1.0,7,9,0,7,0,3,0,0,3,0,0,0,0,0,7,0,0,1,0,0,2,0,9],[20,31,0.6452,0.35714,0.33693,0.0,0.28571,0.57143,0.0,1.0,9,4,0,9,0,3,0,0,9,0,0,1,0,0,3,0,0,2,0,0,1,0,4],[24,31,0.7742,0.40625,0.39465,0.0,0.42857,0.64286,0.0,1.0,13,6,0,13,0,1,0,0,2,0,0,0,0,0,8,0,0,0,0,0,2,0,6],[28,31,0.9032,0.35714,0.43886,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,9],[31,31,1.0,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"f6e8dbdea0eaa474","q":"11. (NET 6) Consider a sequence of circles $K_{1}, K_{2}, K_{3}, K_{4}, \\ldots$ of radii $r_{1}, r_{2}, r_{3}, r_{4}, \\ldots$, respectively, situated inside a triangle $A B C$. The circle $K_{1}$ is tangent to $A B$ and $A C ; K_{2}$ is tangent to $K_{1}, B A$, and $B C ; K_{3}$ is tangent to $K_{2}, C A$, and $C B ; K_{4}$ is tangent to $K_{3}, A B$, and $A C$; etc. (a) Prove the relation  $$ r_{1} \\cot \\frac{1}{2} A+2 \\sqrt{r_{1} r_{2}}+r_{2} \\cot \\frac{1}{2} B=r\\left(\\cot \\frac{1}{2} A+\\cot \\frac{1}{2} B\\right), $$  where $r$ is the radius of the incircle of the triangle $A B C$. Deduce the existence of a $t_{1}$ such that  $$ r_{1}=r \\cot \\frac{1}{2} B \\cot \\frac{1}{2} C \\sin ^{2} t_{1} . $$  (b) Prove that the sequence of circles $K_{1}, K_{2}, \\ldots$ is periodic.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.44195,"x":0.91517,"p":[[0,91,0.0,0.44195,0.1726,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,15,0,0,5,0,0,7,0,0,4,0,0,1,0,0],[4,91,0.044,0.87946,0.16015,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,7,0,17],[8,91,0.0879,0.91517,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[12,91,0.1319,0.80355,0.23078,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,8,0,12],[16,91,0.1758,0.87051,0.1612,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,6,0,17],[20,91,0.2198,0.875,0.17033,0.85714,0.85714,1.0,0.143,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,14],[24,91,0.2637,0.90178,0.1357,0.85711,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,19],[28,91,0.3077,0.91516,0.13771,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[32,91,0.3516,0.81233,0.2127,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,7,0,0,5,0,14],[36,91,0.3956,0.88392,0.14032,0.82132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,8,0,16],[40,91,0.4396,0.85714,0.15152,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,10,0,13],[44,91,0.4835,0.84373,0.13057,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,14,0,9],[48,91,0.5275,0.80802,0.23039,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,9,0,12],[52,91,0.5714,0.87946,0.11356,0.82132,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,13],[56,91,0.6154,0.85266,0.15357,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,10,0,13],[60,91,0.6593,0.73655,0.18253,0.57132,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,7,0,6],[64,91,0.7033,0.72768,0.1868,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,7,0,0,9,0,5],[68,91,0.7473,0.71873,0.21573,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,1,0,0,6,0,0,12,0,0,5,0,6],[72,91,0.7912,0.66965,0.21257,0.53575,0.71414,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,7,0,0,5,0,5],[76,91,0.8352,0.71425,0.17497,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,7,0,0,10,0,3],[80,91,0.8791,0.69642,0.18472,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,10,0,0,6,0,4],[84,91,0.9231,0.70534,0.21111,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,7,0,0,11,0,3],[88,91,0.967,0.68746,0.20341,0.57142,0.71429,0.85704,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,11,0,0,7,0,3],[91,91,1.0,0.54012,0.16648,0.42857,0.571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,12,0,0,10,0,0,4,0,0,2,0,1]]},{"b":7,"e":0.857,"k":"rising","v":0.40625,"x":0.87499,"p":[[0,99,0.0,0.40625,0.17536,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,17,0,0,5,0,0,6,0,0,1,0,0,2,0,0],[4,99,0.0404,0.82142,0.20518,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,5,0,15],[8,99,0.0808,0.82141,0.15154,0.71429,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,9,0,10],[12,99,0.1212,0.87497,0.15467,0.857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,13,0,14],[16,99,0.1616,0.76782,0.23353,0.67857,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,12,0,8],[20,99,0.202,0.84374,0.13997,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,12,0,10],[24,99,0.2424,0.85267,0.17308,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,8,0,15],[28,99,0.2828,0.87499,0.14617,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,16],[32,99,0.3232,0.8482,0.16344,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,8,0,13],[36,99,0.3636,0.77677,0.19866,0.71429,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,10,0,0,5,0,10],[40,99,0.404,0.77677,0.16727,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,11,0,0,8,0,7],[44,99,0.4444,0.8482,0.15544,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,11,0,12],[48,99,0.4848,0.86161,0.16164,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,7,0,15],[52,99,0.5253,0.77229,0.20475,0.71429,0.78564,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,9,0,0,7,0,9],[56,99,0.5657,0.7857,0.22016,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,12,0,8],[60,99,0.6061,0.75888,0.20962,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,10,0,0,7,0,8],[64,99,0.6465,0.78124,0.18552,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,14,0,6],[68,99,0.6869,0.7098,0.2868,0.57142,0.71429,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,7,0,8],[72,99,0.7273,0.78574,0.22581,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,9,0,10],[76,99,0.7677,0.72767,0.28651,0.57143,0.85707,1.0,0.0,1.0,1,10,0,1,0,2,0,0,1,0,0,3,0,0,3,0,0,4,0,0,8,0,10],[80,99,0.8081,0.61601,0.36323,0.4996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C3 (CAN 5) Show that  $$ \\frac{1-s^{a}}{1-s} \\leq(1+s)^{a-1} $$  holds for every $1 \\neq s>0$ real and $0<a \\leq 1$ rational.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,61,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,61,0.1311,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,61,0.1967,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,61,0.2623,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,61,0.3279,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,61,0.3934,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,61,0.459,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,61,0.5246,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,61,0.6557,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,61,0.7869,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,61,0.918,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,61,0.9836,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,83,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,83,0.0482,0.0625,0.15542,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,83,0.0964,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,83,0.1446,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,83,0.1928,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,83,0.241,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,83,0.2892,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,83,0.3373,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,83,0.3855,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,83,0.4337,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,83,0.4819,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,83,0.5301,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,83,0.5783,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,83,0.6265,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,83,0.6747,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,83,0.7229,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,83,0.7711,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,83,0.8193,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,83,0.8675,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,83,0.9157,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,83,0.9639,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[83,83,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"827a3594cc76b2a1","q":"22. (VIE 3) ${ }^{\\mathrm{IMO} 4}$ Does there exist a function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$, such that $f(f(n))=$ $n+1987$ for every natural number $n$ ?","t":[{"b":4,"e":1.0,"k":"rising","v":0.62054,"x":0.99554,"p":[[0,38,0.0,0.62054,0.42948,0.0,0.85714,1.0,0.0,1.0,9,13,6,9,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,6,0,13],[4,38,0.1053,0.81694,0.27257,0.85714,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,10,0,15],[8,38,0.2105,0.84372,0.26334,0.82132,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,19],[12,38,0.3158,0.69196,0.39303,0.28571,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,16],[16,38,0.4211,0.72321,0.34799,0.42859,0.85714,1.0,0.0,1.0,3,14,0,3,0,1,0,0,3,0,0,2,0,0,1,0,0,1,0,0,7,0,14],[20,38,0.5263,0.65178,0.38949,0.28571,0.85707,1.0,0.0,1.0,5,14,0,5,0,0,0,0,6,0,0,1,0,0,1,0,0,1,0,0,4,0,14],[24,38,0.6316,0.64729,0.35172,0.28571,0.85714,1.0,0.0,1.0,4,9,0,4,0,0,0,0,6,0,0,0,0,0,3,0,0,2,0,0,8,0,9],[28,38,0.7368,0.79909,0.28317,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,7,0,16],[32,38,0.8421,0.83036,0.30397,0.85714,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,18],[36,38,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[38,38,1.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"rising","v":0.37054,"x":1.0,"p":[[0,63,0.0,0.7857,0.32538,0.82132,0.92857,1.0,0.0,1.0,3,16,3,3,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,8,0,16],[4,63,0.0635,0.79464,0.30501,0.85714,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,11,0,14],[8,63,0.127,0.59374,0.36089,0.2857,0.78571,0.85714,0.0,1.0,5,6,0,5,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,0,10,0,6],[12,63,0.1905,0.6741,0.36983,0.28571,0.85714,1.0,0.0,1.0,4,11,0,4,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,9,0,11],[16,63,0.254,0.53571,0.43448,0.0,0.85707,1.0,0.0,1.0,10,9,0,10,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,8,0,9],[20,63,0.3175,0.57588,0.39039,0.25,0.71429,1.0,0.0,1.0,7,9,0,7,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,0,5,0,9],[24,63,0.381,0.44642,0.4222,0.0,0.28571,0.85714,0.0,1.0,12,6,0,12,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,0,7,0,6],[28,63,0.4444,0.45536,0.40317,0.0,0.28571,0.85714,0.0,1.0,11,6,0,11,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,5,0,6],[32,63,0.5079,0.56248,0.37447,0.24999,0.57143,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,4,0,0,2,0,0,3,0,0,0,0,0,8,0,7],[36,63,0.5714,0.47766,0.39221,0.0,0.28571,0.85714,0.0,1.0,9,6,0,9,0,0,0,0,8,0,0,0,0,0,2,0,0,1,0,0,6,0,6],[40,63,0.6349,0.37054,0.41011,0.0,0.28571,0.85714,0.0,1.0,15,6,0,15,0,0,0,0,5,0,0,0,0,0,2,0,0,1,0,0,3,0,6],[44,63,0.6984,0.49553,0.37962,0.10714,0.57143,0.85714,0.0,1.0,8,3,0,8,0,2,0,0,5,0,0,0,0,0,2,0,0,2,0,0,10,0,3],[48,63,0.7619,0.39284,0.34441,0.0,0.28571,0.71429,0.0,1.0,9,3,0,9,0,2,0,0,8,0,0,0,0,0,4,0,0,3,0,0,3,0,3],[52,63,0.8254,0.46875,0.40913,0.0,0.5,0.85714,0.0,1.0,11,5,0,11,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,0,7,0,5],[56,63,0.8889,0.74107,0.2635,0.67857,0.85714,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,4,0,0,1,0,0,5,0,0,11,0,8],[60,63,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[63,63,1.0,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30]]}]},{"i":"d147a8a451c38d31","q":"21. (HKG 6) Let $f(x)$ be a monic polynomial of degree 1991 with integer coefficients. Define $g(x)=f^{2}(x)-9$. Show that the number of distinct integer solutions of $g(x)=0$ cannot exceed 1995.","t":[{"b":5,"e":0.1429,"k":"falling","v":0.17402,"x":0.75,"p":[[0,29,0.0,0.75,0.28121,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,4,0,0,5,0,0,3,0,0,2,0,15],[4,29,0.1379,0.65624,0.274,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,6,0,0,8,0,0,3,0,0,1,0,10],[8,29,0.2759,0.64731,0.3072,0.42857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,2,0,0,5,0,0,7,0,0,1,0,0,2,0,11],[12,29,0.4138,0.73214,0.31084,0.42859,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,0,0,0,1,0,17],[16,29,0.5517,0.74107,0.25614,0.57143,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,3,0,0,8,0,0,3,0,0,4,0,12],[20,29,0.6897,0.5357,0.32927,0.25,0.49979,0.89286,0.14286,1.0,0,8,0,0,0,8,0,0,4,0,0,4,0,0,6,0,0,0,0,0,2,0,8],[24,29,0.8276,0.46875,0.33737,0.14286,0.42857,0.85714,0.14286,1.0,0,5,0,0,0,14,0,0,0,0,0,6,0,0,1,0,0,2,0,0,4,0,5],[28,29,0.9655,0.17402,0.06906,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,26,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.1875,0.0974,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,25,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"falling","v":0.41509,"x":0.86606,"p":[[0,100,0.0,0.86606,0.18538,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,4,0,19],[4,100,0.04,0.8125,0.22711,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,2,0,17],[8,100,0.08,0.82141,0.25755,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,2,0,0,1,0,20],[12,100,0.12,0.72768,0.27283,0.53571,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,4,0,0,4,0,12],[16,100,0.16,0.68303,0.29609,0.4286,0.71429,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,0,0,0,5,0,0,6,0,0,4,0,0,1,0,12],[20,100,0.2,0.84375,0.25843,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,0,3,0,21],[24,100,0.24,0.72321,0.29437,0.42857,0.85707,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,1,0,0,5,0,0,3,0,0,3,0,0,4,0,13],[28,100,0.28,0.62945,0.34043,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,4,0,0,2,0,0,5,0,0,0,0,0,4,0,11],[32,100,0.32,0.80803,0.25904,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,4,0,0,4,0,17],[36,100,0.36,0.68304,0.36023,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,7,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,16],[40,100,0.4,0.71429,0.30514,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,4,0,0,4,0,0,3,0,0,2,0,0,3,0,14],[44,100,0.44,0.77231,0.28091,0.53539,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,6,0,0,3,0,0,2,0,0,2,0,17],[48,100,0.48,0.74554,0.29393,0.53572,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,4,0,0,2,0,0,1,0,0,5,0,0,4,0,14],[52,100,0.52,0.75,0.32341,0.42859,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,2,0,0,3,0,17],[56,100,0.56,0.6384,0.34253,0.28571,0.78571,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,3,0,0,5,0,0,1,0,0,1,0,0,5,0,11],[60,100,0.6,0.74554,0.28956,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,3,0,0,4,0,14],[64,100,0.64,0.41509,0.32222,0.14286,0.28571,0.60714,0.14,1.0,0,5,0,0,0,14,0,0,4,0,0,5,0,0,1,0,0,1,0,0,2,0,5],[68,100,0.68,0.46875,0.33926,0.14286,0.35714,0.75,0.14286,1.0,0,6,0,0,0,13,0,0,3,0,0,3,0,0,2,0,0,3,0,0,2,0,6],[72,100,0.72,0.8348,0.20553,0.71429,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,15],[76,100,0.76,0.76784,0.25693,0.57132,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,1,0,0,7,0,13],[80,100,0.8,0.79909,0.23922,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,5,0,0,6,0,14],[84,100,0.84,0.83482,0.20551,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,5,0,16],[88,100,0.88,0.75,0.25754,0.53572,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,4,0,0,4,0,13],[92,100,0.92,0.58033,0.27418,0.42857,0.57121,0.85714,0.14286,1.0,0,4,0,0,0,5,0,0,1,0,0,8,0,0,4,0,0,5,0,0,5,0,4],[96,100,0.96,0.66968,0.17648,0.53607,0.71429,0.85704,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,10,0,0,7,0,2],[100,100,1.0,0.66072,0.11709,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,21,0,0,2,0,0]]}]},{"i":"ed10e909686585b8","q":"24. (USA 5) A circle O with center $O$ on base $B C$ of an isosceles triangle $A B C$ is tangent to the equal sides $A B, A C$. If point $P$ on $A B$ and point $Q$ on $A C$ are selected such that $P B \\times C Q=(B C / 2)^{2}$, prove that line segment $P Q$ is tangent to circle O , and prove the converse.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.23634,"x":0.54012,"p":[[0,117,0.0,0.23634,0.18087,0.14286,0.14286,0.2857,0.14,1.0,0,1,0,0,0,22,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[4,117,0.0342,0.32137,0.25498,0.14286,0.14286,0.571,0.14286,1.0,0,2,0,0,0,20,0,0,0,0,0,2,0,0,8,0,0,0,0,0,0,0,2],[8,117,0.0684,0.28572,0.26244,0.14286,0.14286,0.32142,0.14286,1.0,0,3,0,0,0,22,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[12,117,0.1026,0.28097,0.26132,0.14286,0.14286,0.2857,0.14,1.0,0,3,0,0,0,22,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,3],[16,117,0.1368,0.36602,0.30497,0.14286,0.1429,0.571,0.0,1.0,1,4,0,1,0,17,0,0,1,0,0,2,0,0,6,0,0,1,0,0,0,0,4],[20,117,0.1709,0.33026,0.32431,0.14286,0.14286,0.42857,0.0,1.0,2,5,0,2,0,19,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,5],[24,117,0.2051,0.39727,0.31485,0.14286,0.2857,0.571,0.0,1.0,1,5,0,1,0,14,0,0,3,0,0,3,0,0,5,0,0,1,0,0,0,0,5],[28,117,0.2393,0.4374,0.34622,0.14286,0.42857,0.75,0.0,1.0,4,5,0,4,0,9,0,0,2,0,0,5,0,0,3,0,0,1,0,0,3,0,5],[32,117,0.2735,0.31249,0.28667,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,19,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,3],[36,117,0.3077,0.35714,0.31134,0.14286,0.14286,0.46429,0.0,1.0,1,4,0,1,0,18,0,0,0,0,0,5,0,0,2,0,0,1,0,0,1,0,4],[40,117,0.3419,0.33479,0.30008,0.14286,0.14286,0.42858,0.0,1.0,3,3,0,3,0,14,0,0,4,0,0,4,0,0,2,0,0,0,0,0,2,0,3],[44,117,0.3761,0.25882,0.27536,0.14286,0.14286,0.2857,0.0,1.0,5,2,0,5,0,18,0,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,2],[48,117,0.4103,0.26338,0.25024,0.14286,0.14286,0.1786,0.0,1.0,1,1,0,1,0,23,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,1],[52,117,0.4444,0.23643,0.20403,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,22,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[56,117,0.4786,0.25,0.2369,0.14286,0.14286,0.14287,0.14286,1.0,0,2,0,0,0,25,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[60,117,0.5128,0.40175,0.31222,0.14286,0.21429,0.60714,0.0,1.0,1,4,0,1,0,15,0,0,1,0,0,3,0,0,4,0,0,4,0,0,0,0,4],[64,117,0.547,0.29462,0.25234,0.14286,0.14286,0.35702,0.14286,0.85714,0,0,0,0,0,22,0,0,2,0,0,0,0,0,3,0,0,2,0,0,3,0,0],[68,117,0.5812,0.48661,0.35689,0.14286,0.42859,0.89286,0.14286,1.0,0,8,0,0,0,14,0,0,0,0,0,6,0,0,0,0,0,3,0,0,1,0,8],[72,117,0.6154,0.2991,0.25841,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,20,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,3],[76,117,0.6496,0.42856,0.35174,0.14286,0.2143,0.75,0.14286,1.0,0,7,0,0,0,16,0,0,3,0,0,2,0,0,2,0,0,1,0,0,1,0,7],[80,117,0.6838,0.28572,0.26486,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,23,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,3],[84,117,0.7179,0.30794,0.26994,0.14286,0.14286,0.42857,0.14,1.0,0,3,0,0,0,21,0,0,0,0,0,6,0,0,1,0,0,1,0,0,0,0,3],[88,117,0.7521,0.33033,0.29758,0.14286,0.14286,0.4642,0.0,1.0,1,4,0,1,0,19,0,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,4],[92,117,0.7863,0.32134,0.29456,0.14286,0.14288,0.32143,0.14,1.0,0,4,0,0,0,19,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,4],[96,117,0.8205,0.31248,0.28666,0.14286,0.14286,0.32143,0.14286,1.0,0,4,0,0,0,20,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,4],[100,117,0.8547,0.36606,0.3368,0.14286,0.14286,0.57111,0.14286,1.0,0,6,0,0,0,20,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,6],[104,117,0.8889,0.40625,0.34646,0.14286,0.14286,0.64282,0.14286,1.0,0,6,0,0,0,18,0,0,1,0,0,3,0,0,2,0,0,0,0,0,2,0,6],[108,117,0.9231,0.31239,0.28223,0.14286,0.14286,0.571,0.14,1.0,0,3,0,0,0,22,0,0,1,0,0,0,0,0,5,0,0,1,0,0,0,0,3],[112,117,0.9573,0.308,0.23444,0.14286,0.1429,0.42857,0.14286,1.0,0,1,0,0,0,18,0,0,4,0,0,3,0,0,4,0,0,1,0,0,1,0,1],[116,117,0.9915,0.48657,0.17805,0.39286,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,9,0,0,11,0,0,2,0,0,1,0,1],[117,117,1.0,0.54012,0.22226,0.39286,0.571,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,5,0,0,9,0,0,6,0,0,2,0,2]]},{"b":3,"e":0.0,"k":"falling","v":0.10027,"x":0.41294,"p":[[0,79,0.0,0.33913,0.23352,0.14286,0.2143,0.571,0.14,1.0,0,1,0,0,0,16,0,0,3,0,0,3,0,0,7,0,0,2,0,0,0,0,1],[4,79,0.0506,0.36161,0.32924,0.14286,0.14286,0.60714,0.0,1.0,1,5,0,1,0,18,0,0,3,0,0,1,0,0,1,0,0,3,0,0,0,0,5],[8,79,0.1013,0.28097,0.24881,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,19,0,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,2],[12,79,0.1519,0.29018,0.30196,0.14286,0.14286,0.2857,0.0,1.0,2,4,0,2,0,21,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,4],[16,79,0.2025,0.26778,0.28295,0.14286,0.14286,0.32142,0.0,1.0,4,3,0,4,0,19,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[20,79,0.2532,0.24554,0.26058,0.14286,0.14286,0.2857,0.0,1.0,5,2,0,5,0,18,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[24,79,0.3038,0.30804,0.25281,0.14286,0.14295,0.42857,0.0,1.0,2,2,0,2,0,15,0,0,5,0,0,3,0,0,4,0,0,1,0,0,0,0,2],[28,79,0.3544,0.27223,0.27051,0.14286,0.14286,0.32143,0.0,1.0,2,3,0,2,0,21,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[32,79,0.4051,0.26777,0.2013,0.14286,0.14286,0.32143,0.0,1.0,1,1,0,1,0,17,0,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[36,79,0.4557,0.41294,0.3163,0.14286,0.39279,0.71429,0.0,1.0,2,4,0,2,0,12,0,0,1,0,1,6,0,0,1,0,0,4,0,0,1,0,4],[40,79,0.5063,0.27667,0.24729,0.14286,0.14286,0.42858,0.0,1.0,3,1,0,3,0,18,0,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,1],[44,79,0.557,0.3124,0.28225,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,18,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,2],[48,79,0.6076,0.34374,0.33854,0.14286,0.14286,0.57111,0.0,1.0,2,5,0,2,0,19,0,0,2,0,0,0,0,0,2,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(BUL) Prove that the sum of distances from the center of the circumsphere of the regular tetrahedron to its four vertices is less than the sum of distances from any other point to the four vertices.  Second Day","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13384,"x":0.16964,"p":[[0,22,0.0,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.14732,0.06666,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.13384,0.07935,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.16518,0.08073,0.14286,0.14286,0.1786,0.0,0.28571,3,0,0,3,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.16518,0.06297,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.05357,"x":0.16072,"p":[[0,9,0.0,0.16072,0.08564,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,9,0.4444,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,21,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.09821,0.1357,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6d573fa2f60fc041","q":"0<a,b,c<1  ==> \\sqrt (abc) +  \\sqrt (1-a)(1-b)(1-c)  <1","t":[{"b":1,"e":0.14286,"k":"falling","v":0.17857,"x":0.35714,"p":[[0,14,0.0,0.35268,0.36594,0.14286,0.14286,0.5,0.0,1.0,3,7,0,3,0,19,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,7],[4,14,0.2857,0.27232,0.27976,0.14286,0.14286,0.2857,0.0,1.0,2,3,0,2,0,21,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,3],[8,14,0.5714,0.35714,0.34626,0.14286,0.14286,0.42858,0.0,1.0,2,6,0,2,0,18,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,6],[12,14,0.8571,0.17857,0.16751,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[14,14,1.0,0.1875,0.16536,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,27,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.14286,"k":"flat","v":0.10706,"x":0.22768,"p":[[0,23,0.0,0.22768,0.27166,0.14286,0.14286,0.14287,0.0,1.0,6,3,0,6,0,19,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[4,23,0.1739,0.22322,0.29867,0.14286,0.14286,0.14286,0.0,1.0,6,4,0,6,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,23,0.3478,0.14714,0.08365,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.20089,0.18851,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,21,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[16,23,0.6957,0.12054,0.08073,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.143,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b8e6d5d8d8d3e21","q":"12. (POL 1) If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that  $$ \\left(1+\\frac{a}{b}\\right)^{m}+\\left(1+\\frac{b}{a}\\right)^{m} \\geq 2^{m+1} $$","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.26339,"p":[[0,15,0.0,0.14732,0.27313,0.0,0.0,0.17857,0.0,1.0,21,2,0,21,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[4,15,0.2667,0.26339,0.29474,0.0,0.21428,0.42857,0.0,1.0,13,1,0,13,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,0,3,0,1],[8,15,0.5333,0.24998,0.29448,0.0,0.14286,0.4642,0.0,1.0,14,1,0,14,0,5,0,0,3,0,0,2,0,0,4,0,0,2,0,0,1,0,1],[12,15,0.8,0.07589,0.12869,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.02232,"x":0.22767,"p":[[0,26,0.0,0.22767,0.346,0.0,0.0,0.35714,0.0,1.0,20,3,0,20,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,3],[4,26,0.1538,0.19643,0.3004,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,0,2,0,1],[8,26,0.3077,0.13839,0.28679,0.0,0.0,0.07143,0.0,1.0,24,2,0,24,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[12,26,0.4615,0.10268,0.21199,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,26,0.6154,0.04018,0.1394,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,26,0.7692,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e80f3657c55cc9aa","q":"3. (HUN) Prove that if all the angles of a convex $n$-gon are equal and the lengths of consecutive edges $a_{1}, \\ldots, a_{n}$ satisfy $a_{1} \\geq a_{2} \\geq \\cdots \\geq a_{n}$, then $a_{1}=a_{2}=\\cdots=a_{n}$.  Second Day","t":[{"b":1,"e":1.0,"k":"flat","v":0.80357,"x":0.97321,"p":[[0,60,0.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[4,60,0.0667,0.85713,0.23692,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,21],[8,60,0.1333,0.81696,0.23483,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,9,0,0,1,0,17],[12,60,0.2,0.85268,0.20666,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,0,0,20],[16,60,0.2667,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[20,60,0.3333,0.86161,0.15765,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0,17],[24,60,0.4,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[28,60,0.4667,0.80357,0.28516,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,3,0,0,1,0,0,5,0,0,1,0,19],[32,60,0.5333,0.87499,0.2044,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,0,1,0,21],[36,60,0.6,0.87498,0.17407,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,1,0,20],[40,60,0.6667,0.93304,0.15146,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[44,60,0.7333,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[48,60,0.8,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[52,60,0.8667,0.90625,0.16602,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,23],[56,60,0.9333,0.95535,0.12078,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[60,60,1.0,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]},{"b":4,"e":0.85714,"k":"flat","v":0.71428,"x":0.96429,"p":[[0,59,0.0,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[4,59,0.0678,0.94197,0.12807,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[8,59,0.1356,0.84821,0.1954,0.71429,1.0,1.0,0.286,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,1,0,18],[12,59,0.2034,0.7811,0.20826,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,16,0,0,0,0,12],[16,59,0.2712,0.71428,0.20203,0.71429,0.71429,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,21,0,0,2,0,5],[20,59,0.339,0.73213,0.2044,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,16,0,0,1,0,8],[24,59,0.4068,0.73661,0.2055,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,17,0,0,0,0,9],[28,59,0.4746,0.77233,0.26451,0.57143,0.85714,1.0,0.143,1.0,0,16,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,7,0,0,0,0,16],[32,59,0.5424,0.77232,0.21387,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,14,0,0,2,0,11],[36,59,0.6102,0.73659,0.1927,0.67857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,14,0,0,2,0,8],[40,59,0.678,0.85714,0.22016,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,7,0,0,3,0,19],[44,59,0.7458,0.85714,0.16751,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,2,0,17],[48,59,0.8136,0.91517,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,18],[52,59,0.8814,0.78123,0.15147,0.71429,0.78564,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,10,0,6],[56,59,0.9492,0.83929,0.15047,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,8,0,12],[59,59,1.0,0.82588,0.16652,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,12,0,10]]}]},{"i":"bec50d60b51b1bca","q":"2500 chess kings have to be placed on a  $100 \\times 100$  chessboard so that**(i)** no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);**(ii)** each row and each column contains exactly 25 kings.\n\nFind the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)\n\n*Proposed by Sergei Berlov, Russia*","t":[{"b":5,"e":0.14286,"k":"falling","v":0.14286,"x":0.74107,"p":[[0,33,0.0,0.51339,0.36045,0.10714,0.5,0.85714,0.0,1.0,8,3,8,8,0,1,0,0,0,0,0,7,0,0,2,0,0,2,0,0,9,0,3],[4,33,0.1212,0.72768,0.27747,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,2,0,0,6,0,0,3,0,0,6,0,11],[8,33,0.2424,0.62944,0.24708,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,10,0,0,5,0,0,4,0,0,4,0,6],[12,33,0.3636,0.74107,0.2299,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,0,11,0,7],[16,33,0.4848,0.47768,0.27574,0.25,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,8,0,0,4,0,0,6,0,0,6,0,0,1,0,0,5,0,2],[20,33,0.6061,0.39286,0.23958,0.14286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,12,0,0,2,0,0,9,0,0,3,0,0,3,0,0,3,0,0],[24,33,0.7273,0.4375,0.22286,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,5,0,0,12,0,0,3,0,0,3,0,0,2,0,1],[28,33,0.8485,0.30795,0.19605,0.14286,0.21428,0.42857,0.14,0.71429,0,0,0,0,0,16,0,0,4,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[32,33,0.9697,0.16062,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[33,33,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.23652,"x":0.82143,"p":[[0,44,0.0,0.42411,0.35622,0.0,0.42857,0.75,0.0,1.0,11,1,9,11,0,1,0,0,1,0,0,4,0,0,4,0,0,3,0,0,7,0,1],[4,44,0.0909,0.82143,0.26,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,4,0,0,1,0,0,1,0,0,7,0,17],[8,44,0.1818,0.71427,0.25001,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,6,0,0,1,0,0,7,0,0,7,0,8],[12,44,0.2727,0.73659,0.25782,0.57132,0.78571,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,5,0,0,4,0,0,5,0,0,5,0,11],[16,44,0.3636,0.66513,0.25156,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,5,0,0,5,0,0,5,0,0,8,0,5],[20,44,0.4545,0.51785,0.32291,0.14286,0.42857,0.85714,0.14286,1.0,0,2,0,0,0,11,0,0,1,0,0,5,0,0,1,0,0,2,0,0,10,0,2],[24,44,0.5455,0.34822,0.26711,0.14286,0.14286,0.50002,0.0,0.85714,1,0,0,1,0,16,0,0,2,0,0,5,0,0,0,0,0,5,0,0,3,0,0],[28,44,0.6364,0.34819,0.25487,0.14286,0.2857,0.57111,0.0,0.85714,2,0,0,2,0,12,0,0,6,0,0,2,0,0,5,0,0,2,0,0,3,0,0],[32,44,0.7273,0.39267,0.26017,0.14286,0.28571,0.57143,0.14,1.0,0,1,0,0,0,12,0,0,6,0,0,3,0,0,4,0,0,4,0,0,2,0,1],[36,44,0.8182,0.28347,0.17538,0.14286,0.24999,0.32143,0.14286,0.71429,0,0,0,0,0,15,0,1,8,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[40,44,0.9091,0.33036,0.20341,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,12,0,0,9,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[44,44,1.0,0.23652,0.11079,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,14,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9b0aecd98dce3e87","q":"1. A1 (SLO) Let $a, b$, and $c$ be positive real numbers such that $a b c=1$. Prove that  $$ \\frac{a b}{a^{5}+b^{5}+a b}+\\frac{b c}{b^{5}+c^{5}+b c}+\\frac{c a}{c^{5}+a^{5}+c a} \\leq 1 $$  When does equality hold?","t":[{"b":1,"e":1.0,"k":"rising","v":0.73212,"x":1.0,"p":[[0,23,0.0,0.73212,0.29614,0.571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,1,0,0,3,0,0,7,0,0,1,0,0,2,0,15],[4,23,0.1739,0.88393,0.25364,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,26],[8,23,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.74552,"x":1.0,"p":[[0,34,0.0,0.74552,0.29394,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,0,0,0,3,0,0,9,0,0,0,0,0,0,0,17],[4,34,0.1176,0.78571,0.26726,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,0,0,0,18],[8,34,0.2353,0.78125,0.32338,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,21],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,34,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,34,0.9412,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[34,34,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"2c16764ab35dfb25","q":"3. (BUL 6) ${ }^{\\mathrm{IMO}}$ In the tetrahedron $S A B C$ the angle $B S C$ is a right angle, and the projection of the vertex $S$ to the plane $A B C$ is the intersection of the altitudes of the triangle $A B C$. Let $z$ be the radius of the inscribed circle of the triangle $A B C$. Prove that  $$ S A^{2}+S B^{2}+S C^{2} \\geq 18 z^{2} $$","t":[{"b":4,"e":0.57143,"k":"rising","v":0.20536,"x":0.7232,"p":[[0,110,0.0,0.20536,0.29001,0.0,0.14286,0.21432,0.0,1.0,15,2,0,15,0,9,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,2],[4,110,0.0364,0.63397,0.25486,0.42859,0.64286,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,8,0,0,5,0,0,6,0,0,5,0,5],[8,110,0.0727,0.49551,0.23952,0.39286,0.4286,0.60714,0.0,0.85714,1,0,0,1,0,4,0,0,3,0,0,9,0,0,7,0,0,2,0,0,6,0,0],[12,110,0.1091,0.58929,0.34209,0.28571,0.64286,0.89286,0.0,1.0,3,8,0,3,0,4,0,0,2,0,0,3,0,0,4,0,0,5,0,0,3,0,8],[16,110,0.1455,0.7232,0.2765,0.57143,0.85714,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,1,0,0,2,0,0,4,0,0,4,0,0,10,0,8],[20,110,0.1818,0.63839,0.3214,0.42857,0.64286,1.0,0.0,1.0,2,9,0,2,0,3,0,0,0,0,0,7,0,0,4,0,0,2,0,0,5,0,9],[24,110,0.2182,0.66964,0.25862,0.57142,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,2,0,0,3,0,0,7,0,0,6,0,0,6,0,6],[28,110,0.2545,0.6875,0.23538,0.53571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,5,0,0,9,0,5],[32,110,0.2909,0.71429,0.29014,0.53571,0.85714,1.0,0.0,1.0,1,10,0,1,0,2,0,0,1,0,0,4,0,0,3,0,0,4,0,0,7,0,10],[36,110,0.3273,0.57142,0.28121,0.42857,0.64286,0.85714,0.0,1.0,3,1,0,3,0,1,0,0,2,0,0,8,0,0,2,0,0,6,0,0,9,0,1],[40,110,0.3636,0.53563,0.27677,0.39286,0.42859,0.85714,0.0,1.0,2,1,0,2,0,2,0,0,4,0,0,10,0,0,0,0,0,5,0,0,8,0,1],[44,110,0.4,0.55797,0.17983,0.42857,0.5712,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,9,0,0,9,0,0,8,0,0,3,0,0],[48,110,0.4364,0.50443,0.24995,0.28571,0.4286,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,5,0,0,9,0,0,4,0,0,4,0,0,5,0,1],[52,110,0.4727,0.5714,0.23419,0.42857,0.57143,0.85711,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,7,0,0,6,0,0,4,0,0,9,0,0],[56,110,0.5091,0.59373,0.23449,0.42857,0.571,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,11,0,0,4,0,0,4,0,0,6,0,3],[60,110,0.5455,0.61606,0.20959,0.42857,0.57143,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,10,0,0,5,0,0,7,0,0,6,0,2],[64,110,0.5818,0.4999,0.25012,0.42857,0.42857,0.71429,0.0,1.0,2,1,0,2,0,3,0,0,2,0,0,11,0,0,3,0,0,7,0,0,3,0,1],[68,110,0.6182,0.50447,0.23686,0.42857,0.50001,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,2,0,0,9,0,0,3,0,0,11,0,0,2,0,0],[72,110,0.6545,0.48215,0.21943,0.39286,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,12,0,0,5,0,0,2,0,0,5,0,0],[76,110,0.6909,0.47768,0.26871,0.24999,0.5,0.71429,0.0,1.0,2,1,0,2,0,6,0,0,2,0,0,6,0,0,6,0,0,6,0,0,3,0,1],[80,110,0.7273,0.42856,0.22303,0.28571,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,3,0,0,4,0,0,10,0,0,6,0,0,5,0,0,1,0,0],[84,110,0.7636,0.59374,0.23987,0.42857,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,2,0,0,6,0,0,5,0,0,7,0,0,9,0,0],[88,110,0.8,0.50446,0.27889,0.28571,0.42859,0.71429,0.0,0.85714,3,0,0,3,0,3,0,0,3,0,0,8,0,0,2,0,0,6,0,0,7,0,0],[92,110,0.8364,0.41517,0.27048,0.14286,0.42857,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,1,0,0,12,0,0,3,0,0,4,0,0,2,0,1],[96,110,0.8727,0.52672,0.22988,0.42857,0.4998,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,10,0,0,7,0,0,2,0,0,7,0,0],[100,110,0.9091,0.60713,0.18558,0.42857,0.57143,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,10,0,0,6,0,0,6,0,0,8,0,0],[104,110,0.9455,0.57588,0.26118,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,5,0,0,6,0,0,4,0,0,9,0,1],[108,110,0.9818,0.60268,0.22227,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,9,0,0,2,0,0,8,0,0,9,0,0],[110,110,1.0,0.50445,0.31131,0.28571,0.57143,0.71429,0.0,0.85714,7,0,0,7,0,0,0,0,2,0,0,4,0,0,5,0,0,7,0,0,7,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.25445,"x":0.95534,"p":[[0,58,0.0,0.25445,0.23885,0.0,0.14286,0.42858,0.0,0.71429,10,0,0,10,0,8,0,0,2,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[4,58,0.069,0.76338,0.2615,0.57132,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,2,0,0,3,0,15],[8,58,0.1379,0.69641,0.28516,0.42859,0.78564,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,5,0,0,3,0,0,3,0,0,6,0,10],[12,58,0.2069,0.66072,0.24936,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,7,0,0,4,0,0,7,0,0,5,0,6],[16,58,0.2759,0.69192,0.2746,0.571,0.85707,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,0,0,0,4,0,0,8,0,0,0,0,0,10,0,7],[20,58,0.3448,0.85713,0.1713,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,4,0,17],[24,58,0.4138,0.85268,0.19061,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,5,0,17],[28,58,0.4828,0.91517,0.13767,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[32,58,0.5517,0.92409,0.15149,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,25],[36,58,0.6207,0.875,0.17035,0.71429,1.0,1.0,0.5714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,4,0,19],[40,58,0.6897,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[44,58,0.7586,0.91071,0.13243,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,20],[48,58,0.8276,0.91963,0.14701,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,23],[52,58,0.8966,0.91964,0.14258,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[56,58,0.9655,0.86607,0.15542,0.82143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,15],[58,58,1.0,0.95534,0.09745,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25]]}]},{"i":"633110cd7ea8edf4","q":"4. (HUN) Each of 17 students talked with every other student. They all talked about three different topics. Each pair of students talked about one topic. Prove that there are three students that talked about the same topic among themselves.","t":[{"b":0,"e":0.0,"k":"volatile","v":0.19643,"x":0.94196,"p":[[0,11,0.0,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,11,0.3636,0.87054,0.27975,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,26],[8,11,0.7273,0.30134,0.36234,0.0,0.14286,0.55357,0.0,1.0,15,4,0,15,0,4,0,0,0,0,0,4,0,1,0,0,0,4,0,0,0,0,4],[11,11,1.0,0.19643,0.31288,0.0,0.0,0.21429,0.0,1.0,18,3,0,18,0,6,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,3]]},{"b":2,"e":1.0,"k":"flat","v":0.87054,"x":1.0,"p":[[0,9,0.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,9,0.4444,0.87054,0.32996,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[9,9,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5daee43f20b6ad64","q":"4. (BUL 4) Suppose medians $m_{a}$ and $m_{b}$ of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled triangle. (b) The inequality  $$ 5\\left(a^{2}+b^{2}-c^{2}\\right) \\geq 8 a b $$  is valid, where $a, b$, and $c$ are side lengths of the given triangle.","t":[{"b":1,"e":0.57143,"k":"flat","v":0.89732,"x":0.96429,"p":[[0,28,0.0,0.96427,0.08754,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,28,0.1429,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[8,28,0.2857,0.94642,0.11714,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[12,28,0.4286,0.92409,0.13359,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[16,28,0.5714,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[20,28,0.7143,0.89732,0.16065,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,21],[24,28,0.8571,0.9241,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[28,28,1.0,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23]]},{"b":2,"e":1.0,"k":"flat","v":0.89284,"x":0.99107,"p":[[0,30,0.0,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[4,30,0.1333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,30,0.2667,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,30,0.4,0.91071,0.14617,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,0,0,23],[16,30,0.5333,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[20,30,0.6667,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[24,30,0.8,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[28,30,0.9333,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[30,30,1.0,0.89284,0.12372,0.85711,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,12,0,15]]}]},{"i":"1d2aac2424aeaa96","q":"37. (ROM 3) Let $A_{1}, A_{2}, \\ldots, A_{n+1}$ be positive integers such that $\\left(A_{i}, A_{n+1}\\right)$ $=1$ for every $i=1,2, \\ldots, n$. Show that the equation  $$ x_{1}^{A_{1}}+x_{2}^{A_{2}}+\\cdots+x_{n}^{A_{n}}=x_{n+1}^{A_{n+1}} $$  has an infinite set of solutions $\\left(x_{1}, x_{2}, \\ldots, x_{n+1}\\right)$ in positive integers.","t":[{"b":4,"e":0.1429,"k":"flat","v":0.32136,"x":0.67857,"p":[[0,41,0.0,0.48659,0.28762,0.2857,0.4998,0.71429,0.0,0.85714,5,0,1,5,0,2,0,0,2,0,0,7,0,0,3,0,0,8,0,0,5,0,0],[4,41,0.0976,0.57589,0.34715,0.28571,0.42857,1.0,0.0,1.0,1,12,0,1,0,4,0,0,5,0,0,9,0,0,1,0,0,0,0,0,0,0,12],[8,41,0.1951,0.59822,0.40317,0.25,0.71429,1.0,0.0,1.0,4,14,0,4,0,4,0,0,6,0,0,0,0,0,1,0,0,2,0,0,1,0,14],[12,41,0.2927,0.43293,0.35449,0.14286,0.28571,0.78571,0.0,1.0,2,8,0,2,0,9,0,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,8],[16,41,0.3902,0.50446,0.37113,0.14286,0.35714,1.0,0.0,1.0,3,9,0,3,0,6,0,0,7,0,0,3,0,0,1,0,0,1,0,0,2,0,9],[20,41,0.4878,0.58927,0.39407,0.25,0.64284,1.0,0.0,1.0,4,13,0,4,0,4,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,13],[24,41,0.5854,0.51785,0.3458,0.1429,0.42857,1.0,0.14286,1.0,0,9,0,0,0,9,0,0,5,0,0,5,0,0,2,0,0,1,0,0,1,0,9],[28,41,0.6829,0.67857,0.36246,0.28571,0.85714,1.0,0.0,1.0,1,15,0,1,0,5,0,0,3,0,0,3,0,0,1,0,0,1,0,0,3,0,15],[32,41,0.7805,0.58927,0.3973,0.14289,0.64286,1.0,0.0,1.0,3,13,0,3,0,6,0,0,5,0,0,0,0,0,2,0,0,1,0,0,2,0,13],[36,41,0.878,0.43294,0.3545,0.14286,0.28571,0.71429,0.0,1.0,3,7,0,3,0,9,0,0,8,0,0,0,0,0,2,0,0,3,0,0,0,0,7],[40,41,0.9756,0.32136,0.20508,0.14286,0.28571,0.571,0.0,0.71429,2,0,0,2,0,11,0,0,7,0,0,3,0,0,7,0,0,2,0,0,0,0,0],[41,41,1.0,0.33926,0.25187,0.14286,0.21431,0.46418,0.0,1.0,1,1,0,1,0,15,0,0,3,0,0,5,0,0,3,0,0,3,0,0,1,0,1]]},{"b":5,"e":0.4286,"k":"falling","v":0.24107,"x":0.62499,"p":[[0,55,0.0,0.50447,0.2879,0.42857,0.42857,0.71429,0.0,1.0,4,4,1,4,0,1,0,0,1,0,0,13,0,0,4,0,0,3,0,0,2,0,4],[4,55,0.0727,0.54464,0.38538,0.14286,0.5,1.0,0.0,1.0,2,10,0,2,0,9,0,0,4,0,0,1,0,0,1,0,0,2,0,0,3,0,10],[8,55,0.1455,0.57125,0.39467,0.14286,0.42859,1.0,0.0,1.0,1,13,0,1,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,2,0,13],[12,55,0.2182,0.55803,0.39182,0.24999,0.42857,1.0,0.0,1.0,3,13,0,3,0,5,0,0,7,0,0,2,0,0,1,0,0,1,0,0,0,0,13],[16,55,0.2909,0.57588,0.37709,0.14289,0.49979,1.0,0.14286,1.0,0,12,0,0,0,9,0,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,12],[20,55,0.3636,0.55802,0.36133,0.28571,0.57143,1.0,0.0,1.0,4,10,0,4,0,2,0,0,7,0,0,1,0,0,4,0,0,4,0,0,0,0,10],[24,55,0.4364,0.62499,0.36202,0.2857,0.71429,1.0,0.0,1.0,1,13,0,1,0,4,0,0,8,0,0,1,0,0,0,0,0,4,0,0,1,0,13],[28,55,0.5091,0.55803,0.37688,0.14289,0.57143,1.0,0.0,1.0,3,11,0,3,0,7,0,0,2,0,0,2,0,0,4,0,0,3,0,0,0,0,11],[32,55,0.5818,0.5,0.36422,0.14286,0.28571,1.0,0.0,1.0,1,9,0,1,0,8,0,0,9,0,0,2,0,0,0,0,0,1,0,0,2,0,9],[36,55,0.6545,0.48205,0.32693,0.2857,0.42857,0.74996,0.0,1.0,3,6,0,3,0,4,0,0,7,0,0,5,0,0,4,0,0,1,0,0,2,0,6],[40,55,0.7273,0.24107,0.18707,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,9,0,0,11,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[44,55,0.8,0.27888,0.25254,0.14286,0.2857,0.28571,0.0,1.0,5,2,0,5,1,8,0,0,12,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[48,55,0.8727,0.38828,0.22941,0.2857,0.28571,0.571,0.0,1.0,1,1,0,1,0,6,0,0,11,0,0,5,0,0,4,0,0,3,0,0,1,0,1],[52,55,0.9455,0.29009,0.20046,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,13,0,0,8,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[55,55,1.0,0.31249,0.19376,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,8,0,0,8,0,0,9,0,0,1,0,0,3,0,0,0,0,0]]}]},{"i":"7482ce238ee34a56","q":"1. A1 (USA) Let $a_{0}=1994$ and $a_{n+1}=\\frac{a_{n}^{2}}{a_{n}+1}$ for each nonnegative integer $n$. Prove that $1994-n$ is the greatest integer less than or equal to $a_{n}$, $0 \\leq n \\leq 998$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,38,0.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,38,0.1053,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,38,0.2105,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,38,0.3158,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.97768,"x":1.0,"p":[[0,36,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,36,0.1111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,36,0.2222,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,36,0.5556,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,36,0.6667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6231f21eea54f057","q":"15. (CUB 1) Prove that for all $n \\in \\mathbb{N}$ the following is true:  $$ 2^{n} \\prod_{k=1}^{n} \\sin \\frac{k \\pi}{2 n+1}=\\sqrt{2 n+1} $$","t":[{"b":4,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,27,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,58,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,58,0.069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,58,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,58,0.2069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,58,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,58,0.3448,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,58,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,58,0.4828,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,58,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,58,0.6207,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,58,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,58,0.7586,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,58,0.8276,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,58,0.8966,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,58,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[58,58,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a157bef2cd06e962","q":"23. (HUN 4) Prove that for an arbitrary pair of vectors $f$ and $g$ in the plane, the inequality  $$ a f^{2}+b f g+c g^{2} \\geq 0 $$  holds if and only if the following conditions are fulfilled: $a \\geq 0, c \\geq 0$, $4 a c \\geq b^{2}$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,19,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,19,0.2105,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,19,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,19,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,19,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.95088,"x":0.97768,"p":[[0,7,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,7,0.5714,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[7,7,1.0,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22]]}]},{"i":"65087d248651772f","q":"60. (VIE 4) Suppose $x_{0}, x_{1}, \\ldots, x_{n}$ are integers and $x_{0}>x_{1}>\\cdots>x_{n}$. Prove that at least one of the numbers $\\left|F\\left(x_{0}\\right)\\right|,\\left|F\\left(x_{1}\\right)\\right|,\\left|F\\left(x_{2}\\right)\\right|, \\ldots$, $\\left|F\\left(x_{n}\\right)\\right|$, where  $$ F(x)=x^{n}+a_{1} x^{n-1}+\\cdots+a_{n}, \\quad a_{i} \\in \\mathbb{R}, \\quad i=1, \\ldots, n $$  is greater than $\\frac{n!}{2^{n}}$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,56,0.0,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[4,56,0.0714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,56,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,56,0.3571,0.93749,0.19215,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[24,56,0.4286,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[28,56,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,56,0.5714,0.95982,0.12993,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[36,56,0.6429,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[40,56,0.7143,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[44,56,0.7857,0.92411,0.18205,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,26],[48,56,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.93748,"x":1.0,"p":[[0,30,0.0,0.93748,0.17108,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[4,30,0.1333,0.95982,0.12993,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[8,30,0.2667,0.94643,0.1915,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[12,30,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,30,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,30,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,30,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,30,0.9333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"7fb7e86719882bc3","q":"7. (FRA 5) We consider three distinct half-lines $O x, O y, O z$ in a plane. Prove the existence and uniqueness of three points $A \\in O x, B \\in O y$, $C \\in O z$ such that the perimeters of the triangles $O A B, O B C, O C A$ are all equal to a given number $2 p>0$.","t":[{"b":0,"e":0.571,"k":"falling","v":0.60712,"x":0.87946,"p":[[0,92,0.0,0.77677,0.20185,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,6,0,0,3,0,12],[4,92,0.0435,0.87946,0.17536,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,4,0,20],[8,92,0.087,0.8125,0.20341,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,2,0,16],[12,92,0.1304,0.81247,0.20344,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,2,0,16],[16,92,0.1739,0.79463,0.27186,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,0,0,0,6,0,0,5,0,0,0,0,18],[20,92,0.2174,0.83036,0.19704,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,2,0,17],[24,92,0.2609,0.7723,0.21684,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,7,0,0,3,0,12],[28,92,0.3043,0.87499,0.1777,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,3,0,20],[32,92,0.3478,0.76337,0.23586,0.57143,0.78564,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,13,0,0,1,0,0,2,0,14],[36,92,0.3913,0.7857,0.22589,0.57143,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,4,0,0,1,0,15],[40,92,0.4348,0.77677,0.18879,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,2,0,12],[44,92,0.4783,0.78124,0.202,0.57143,0.71429,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,1,0,14],[48,92,0.5217,0.83481,0.19271,0.57143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,3,0,17],[52,92,0.5652,0.73659,0.19921,0.57143,0.57143,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,2,0,0,1,0,11],[56,92,0.6087,0.77231,0.20474,0.57143,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,3,0,0,2,0,13],[60,92,0.6522,0.79016,0.23142,0.57143,0.9285,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,2,0,0,1,0,16],[64,92,0.6957,0.76786,0.20124,0.57143,0.71429,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,3,0,0,1,0,13],[68,92,0.7391,0.71875,0.19393,0.57143,0.57143,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,3,0,0,2,0,9],[72,92,0.7826,0.77229,0.21089,0.57143,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,4,0,0,4,0,12],[76,92,0.8261,0.72766,0.21831,0.57143,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,1,0,0,0,0,12],[80,92,0.8696,0.82575,0.18476,0.67536,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,1,0,16],[84,92,0.913,0.74106,0.18365,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,10,0,0,3,0,8],[88,92,0.9565,0.62054,0.13175,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,3,0,0,2,0,2],[92,92,1.0,0.60712,0.10715,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,5,0,0,1,0,1]]},{"b":7,"e":1.0,"k":"flat","v":0.71426,"x":0.90623,"p":[[0,86,0.0,0.76338,0.20081,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,4,0,0,2,0,12],[4,86,0.0465,0.78561,0.23445,0.57143,0.85714,1.0,0.14,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,4,0,0,0,0,16],[8,86,0.093,0.82142,0.20205,0.57143,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,1,0,17],[12,86,0.1395,0.74997,0.18901,0.57143,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,0,3,0,10],[16,86,0.186,0.71426,0.1856,0.57143,0.57143,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,2,0,0,3,0,8],[20,86,0.2326,0.72321,0.19541,0.57143,0.57143,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,2,0,0,1,0,10],[24,86,0.2791,0.84372,0.19682,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,1,0,19],[28,86,0.3256,0.77232,0.19186,0.57143,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,0,2,0,12],[32,86,0.3721,0.78122,0.23956,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,0,0,0,0,0,17],[36,86,0.4186,0.86607,0.17474,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,2,0,19],[40,86,0.4651,0.84821,0.18877,0.67857,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,0,0,19],[44,86,0.5116,0.80357,0.19805,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,2,0,15],[48,86,0.5581,0.84821,0.19865,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,2,0,19],[52,86,0.6047,0.81249,0.18015,0.67857,0.78571,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,2,0,14],[56,86,0.6512,0.8683,0.15579,0.82132,0.89286,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,8,1,15],[60,86,0.6977,0.83033,0.20962,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,2,0,0,0,0,19],[64,86,0.7442,0.90178,0.14914,0.82132,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,21],[68,86,0.7907,0.88838,0.15043,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,7,0,18],[72,86,0.8372,0.86158,0.16167,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,3,0,17],[76,86,0.8837,0.90623,0.1541,0.85711,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,22],[80,86,0.9302,0.83927,0.18125,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,3,0,16],[84,86,0.9767,0.87052,0.16117,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,3,0,18],[86,86,1.0,0.8973,0.17944,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,20]]}]},{"i":"b078cca1732197a6","q":"7. C2 (COL) In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x^{\\prime}$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x=x_{0}, x_{1}, \\ldots, x_{n}=x^{\\prime}$ for some integer $n \\geq 2$ such that $x_{i-1}$ and $x_{i}$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.72767,"x":0.94196,"p":[[0,17,0.0,0.86161,0.22156,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,8,0,0,0,0,21],[4,17,0.2353,0.94196,0.15093,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[8,17,0.4706,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20],[12,17,0.7059,0.80356,0.13244,0.71429,0.71429,1.0,0.714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,10],[16,17,0.9412,0.75446,0.09606,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,1,0,4],[17,17,1.0,0.72767,0.11495,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.88393,"x":1.0,"p":[[0,9,0.0,0.88393,0.19704,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,22],[4,9,0.4444,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[9,9,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a29be86fe6ba3e9f","q":"16. G2 (KOR) ${ }^{\\mathrm{IMO} 1}$ In acute triangle $A B C$ with circumcenter $O$ and altitude $A P, \\measuredangle C \\geq \\measuredangle B+30^{\\circ}$. Prove that $\\measuredangle A+\\measuredangle C O P<90^{\\circ}$.","t":[{"b":1,"e":0.42857,"k":"falling","v":0.45982,"x":0.67856,"p":[[0,107,0.0,0.61605,0.16535,0.42859,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,9,0,0,4,0,1],[4,107,0.0374,0.58483,0.20627,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,9,0,0,7,0,0,7,0,0,2,0,3],[8,107,0.0748,0.61157,0.20897,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,8,0,0,10,0,0,6,0,0,2,0,4],[12,107,0.1121,0.67856,0.23146,0.53539,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,7,0,0,8,0,0,6,0,0,2,0,8],[16,107,0.1495,0.54018,0.17029,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,13,0,0,6,0,0,10,0,0,1,0,0],[20,107,0.1869,0.58034,0.21998,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,8,0,0,7,0,0,9,0,0,1,0,3],[24,107,0.2243,0.55802,0.20935,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,8,0,0,6,0,0,11,0,0,0,0,2],[28,107,0.2617,0.59374,0.14773,0.42859,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,9,0,0,8,0,0,13,0,0,0,0,1],[32,107,0.2991,0.56698,0.20663,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,7,0,0,6,0,0,11,0,0,2,0,1],[36,107,0.3364,0.58036,0.2141,0.42859,0.57143,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,6,0,0,10,0,0,9,0,0,3,0,1],[40,107,0.3738,0.55801,0.2,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,9,0,0,7,0,0,9,0,0,2,0,1],[44,107,0.4112,0.53125,0.17582,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,11,0,0,9,0,0,6,0,0,1,0,1],[48,107,0.4486,0.57585,0.16935,0.42857,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,10,0,0,14,0,0,4,0,0,0,0,3],[52,107,0.486,0.53124,0.17214,0.42857,0.5712,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,12,0,0,12,0,0,4,0,0,1,0,1],[56,107,0.5234,0.52231,0.14987,0.42857,0.57121,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,12,0,0,13,0,0,2,0,0,1,0,1],[60,107,0.5607,0.52679,0.09055,0.42857,0.57121,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,16,0,0,3,0,0,0,0,0],[64,107,0.5981,0.5089,0.14697,0.42857,0.571,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,13,0,0,13,0,0,3,0,0,1,0,0],[68,107,0.6355,0.54015,0.14166,0.42857,0.57141,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,11,0,0,13,0,0,5,0,0,0,0,1],[72,107,0.6729,0.53569,0.14285,0.42857,0.57143,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,1,0,0,9,0,0,17,0,0,3,0,0,0,0,1],[76,107,0.7103,0.48212,0.09276,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,20,0,0,9,0,0,2,0,0,0,0,0],[80,107,0.7477,0.47317,0.1037,0.42857,0.42857,0.5711,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,18,0,0,9,0,0,2,0,0,0,0,0],[84,107,0.785,0.5,0.10101,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,11,0,0,17,0,0,1,0,0,0,0,0],[88,107,0.8224,0.50443,0.10702,0.42857,0.4998,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,14,0,0,13,0,0,3,0,0,0,0,0],[92,107,0.8598,0.49103,0.10674,0.42857,0.49979,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,14,0,0,15,0,0,1,0,0,0,0,0],[96,107,0.8972,0.49107,0.15542,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,14,0,0,13,0,0,1,0,0,0,0,1],[100,107,0.9346,0.49102,0.08698,0.42857,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,14,0,0,16,0,0,0,0,0,0,0,0],[104,107,0.972,0.51785,0.13716,0.42857,0.4286,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,17,0,0,9,0,0,4,0,0,0,0,1],[107,107,1.0,0.45982,0.13236,0.42857,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,2,0,0,14,0,0,14,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"falling","v":0.25,"x":0.66518,"p":[[0,69,0.0,0.62485,0.17028,0.53539,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,9,0,0,12,0,0,0,0,3],[4,69,0.058,0.60268,0.20119,0.42857,0.57143,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,1,0,0,10,0,0,5,0,0,11,0,0,1,0,3],[8,69,0.1159,0.57141,0.21429,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,9,0,0,8,0,0,8,0,0,2,0,2],[12,69,0.1739,0.66518,0.19434,0.57143,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,3,0,0,10,0,0,11,0,0,4,0,3],[16,69,0.2319,0.53121,0.25059,0.42857,0.571,0.71429,0.0,1.0,3,3,0,3,0,0,0,0,2,0,0,9,0,0,9,0,0,5,0,0,1,0,3],[20,69,0.2899,0.52679,0.18013,0.42857,0.50001,0.60714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,13,0,0,8,0,0,5,0,0,3,0,0],[24,69,0.3478,0.59372,0.20238,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,7,0,0,12,0,0,7,0,0,1,0,3],[28,69,0.4058,0.5714,0.21724,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,7,0,0,9,0,0,9,0,0,0,0,3],[32,69,0.4638,0.62052,0.17717,0.4286,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,10,0,0,1,0,3],[36,69,0.5217,0.60263,0.20434,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,7,0,0,12,0,0,4,0,0,2,0,4],[40,69,0.5797,0.56256,0.19878,0.42857,0.57143,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,8,0,0,12,0,0,3,0,0,3,0,2],[44,69,0.6377,0.62946,0.25719,0.42857,0.71429,0.74996,0.0,1.0,2,5,0,2,0,0,0,0,1,0,0,7,0,0,5,0,0,9,0,0,3,0,5],[48,69,0.6957,0.54909,0.2055,0.42857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,10,0,0,7,0,0,6,0,0,2,0,2],[52,69,0.7536,0.55805,0.24575,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,1,0,0,9,0,0,8,0,0,6,0,0,2,0,3],[56,69,0.8116,0.58482,0.23244,0.42857,0.57143,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,2,0,0,7,0,0,6,0,0,9,0,0,5,0,1],[60,69,0.8696,0.37044,0.27867,0.105,0.42857,0.57143,0.0,0.857,8,0,0,8,0,4,0,0,1,0,0,7,0,0,5,0,0,6,0,0,1,0,0],[64,69,0.9275,0.37502,0.25441,0.14286,0.42857,0.571,0.0,1.0,4,2,0,4,0,6,0,0,4,0,0,9,0,0,6,0,0,1,0,0,0,0,2],[68,69,0.9855,0.30356,0.2389,0.0,0.28571,0.4286,0.0,0.85714,9,0,0,9,0,2,0,0,7,0,0,7,0,0,5,0,0,1,0,0,1,0,0],[69,69,1.0,0.25,0.21724,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,4,0,0,7,0,0,8,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"dacf4eefefd6ae59","q":"7. (CAN 5) Let $a$ be a positive integer and let $\\left\\{a_{n}\\right\\}$ be defined by $a_{0}=0$ and  $$ a_{n+1}=\\left(a_{n}+1\\right) a+(a+1) a_{n}+2 \\sqrt{a(a+1) a_{n}\\left(a_{n}+1\\right)} \\quad(n=1,2 \\ldots) $$  Show that for each positive integer $n, a_{n}$ is a positive integer.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.2366,"x":0.7857,"p":[[0,47,0.0,0.43303,0.2435,0.28571,0.28571,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,19,0,0,3,0,0,1,0,0,3,0,0,4,0,1],[4,47,0.0851,0.72768,0.31615,0.53571,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,3,0,0,2,0,0,4,0,0,1,0,0,5,0,14],[8,47,0.1702,0.74999,0.29452,0.57132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,2,0,0,2,0,16],[12,47,0.2553,0.70089,0.3038,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,5,0,0,6,0,0,2,0,0,2,0,0,2,0,14],[16,47,0.3404,0.61607,0.35254,0.2857,0.64286,1.0,0.0,1.0,1,12,0,1,0,3,0,0,9,0,0,1,0,0,2,0,0,2,0,0,2,0,12],[20,47,0.4255,0.78124,0.29664,0.57132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,4,0,0,2,0,0,2,0,0,3,0,0,2,0,18],[24,47,0.5106,0.7857,0.28348,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,2,0,0,3,0,17],[28,47,0.5957,0.60267,0.30874,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,6,0,0,5,0,0,6,0,0,1,0,0,1,0,10],[32,47,0.6809,0.71429,0.31944,0.42859,0.78571,1.0,0.0,1.0,1,15,0,1,0,2,0,0,3,0,0,3,0,0,3,0,0,4,0,0,1,0,15],[36,47,0.766,0.43317,0.23812,0.28571,0.28786,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,13,0,0,5,0,0,3,0,0,4,0,0,1,0,2],[40,47,0.8511,0.38393,0.25862,0.2857,0.28571,0.42858,0.0,1.0,1,3,0,1,0,6,0,0,14,0,0,4,0,0,1,0,0,3,0,0,0,0,3],[44,47,0.9362,0.26786,0.16656,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,10,0,0,12,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[47,47,1.0,0.2366,0.12169,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.58927,"x":1.0,"p":[[0,70,0.0,0.58927,0.31693,0.28571,0.57121,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,12,0,0,2,0,0,4,0,0,0,0,0,5,0,8],[4,70,0.0571,0.92411,0.15561,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,4,0,24],[8,70,0.1143,0.91072,0.17034,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,23],[12,70,0.1714,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[16,70,0.2286,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[20,70,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,70,0.3429,0.95084,0.13189,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[28,70,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,70,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,70,0.5143,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[40,70,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,70,0.6286,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,70,0.6857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,70,0.7429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,70,0.8,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,70,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,70,0.9143,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[68,70,0.9714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[70,70,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"0226109f20fdceca","q":"8. (SWE 3) Let $P$ be a polynomial with real coefficients such that $P(x)>0$ if $x>0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x)=\\frac{Q(x)}{R(x)}$ if $x>0$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.33917,"x":0.4866,"p":[[0,28,0.0,0.46427,0.32536,0.28571,0.28571,0.71429,0.0,1.0,3,6,0,3,0,3,0,0,12,0,0,1,0,0,4,0,0,2,0,0,1,0,6],[4,28,0.1429,0.40178,0.28221,0.24999,0.5,0.57143,0.0,1.0,7,2,0,7,0,1,0,0,7,0,0,1,0,0,12,0,0,2,0,0,0,0,2],[8,28,0.2857,0.45534,0.34151,0.0,0.57143,0.57143,0.0,1.0,9,5,0,9,0,0,0,0,3,0,0,1,0,0,12,0,0,2,0,0,0,0,5],[12,28,0.4286,0.4866,0.31105,0.28571,0.57143,0.57143,0.0,1.0,5,5,0,5,0,1,0,0,6,0,0,1,0,0,13,0,0,0,0,0,1,0,5],[16,28,0.5714,0.33917,0.26909,0.0,0.35714,0.57143,0.0,0.85714,10,0,0,10,0,1,0,0,5,0,0,3,0,0,10,0,0,2,0,0,1,0,0],[20,28,0.7143,0.39732,0.28956,0.10714,0.50001,0.57143,0.0,1.0,8,2,0,8,0,1,0,0,5,0,0,2,0,0,12,0,0,2,0,0,0,0,2],[24,28,0.8571,0.46427,0.1821,0.42857,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,0,0,0,4,0,0,5,0,0,19,0,0,1,0,0,0,0,0],[28,28,1.0,0.39729,0.21347,0.2857,0.571,0.57143,0.0,0.57143,5,0,0,5,0,1,0,0,7,0,0,2,0,0,17,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.30803,"x":0.57133,"p":[[0,43,0.0,0.57133,0.31351,0.28571,0.57143,0.89286,0.0,1.0,1,8,1,1,0,3,0,0,7,0,0,2,0,0,8,0,0,1,0,0,2,0,8],[4,43,0.093,0.44196,0.24317,0.2857,0.57143,0.57143,0.0,1.0,4,1,0,4,0,2,0,0,6,0,0,1,0,0,15,0,0,3,0,0,0,0,1],[8,43,0.186,0.35266,0.2314,0.24999,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,1,0,0,9,0,0,1,0,0,13,0,0,1,0,0,0,0,0],[12,43,0.2791,0.45982,0.23619,0.28571,0.57143,0.57143,0.0,0.85714,5,0,0,5,0,1,0,0,3,0,0,0,0,0,20,0,0,2,0,0,1,0,0],[16,43,0.3721,0.36159,0.23414,0.2857,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,0,0,0,10,0,0,1,0,0,12,0,0,2,0,0,0,0,0],[20,43,0.4651,0.40177,0.2683,0.2857,0.42859,0.57143,0.0,0.85714,7,0,0,7,0,0,0,0,8,0,0,2,0,0,9,0,0,4,0,0,2,0,0],[24,43,0.5581,0.30803,0.26751,0.0,0.28571,0.57143,0.0,1.0,10,1,0,10,0,2,0,0,8,0,0,1,0,0,9,0,0,1,0,0,0,0,1],[28,43,0.6512,0.44194,0.30588,0.24999,0.57121,0.57143,0.0,1.0,7,2,0,7,0,1,0,0,5,0,0,1,0,0,12,0,0,1,0,0,3,0,2],[32,43,0.7442,0.38392,0.27533,0.24999,0.35714,0.57143,0.0,1.0,7,1,0,7,0,1,0,0,8,0,0,3,0,0,9,0,0,1,0,0,2,0,1],[36,43,0.8372,0.48213,0.23622,0.5354,0.57143,0.57143,0.0,1.0,5,1,0,5,0,0,0,0,2,0,0,1,0,0,21,0,0,2,0,0,0,0,1],[40,43,0.9302,0.41516,0.21828,0.28571,0.57121,0.57143,0.0,0.71429,5,0,0,5,0,0,0,0,7,0,0,3,0,0,15,0,0,2,0,0,0,0,0],[43,43,1.0,0.3571,0.23416,0.21427,0.42859,0.57143,0.0,0.57143,8,0,0,8,0,0,0,0,7,0,0,2,0,0,15,0,0,0,0,0,0,0,0]]}]},{"i":"e02a6e2a8184bd70","q":"6. (CZS 1) For a triangle $A B C$, let $k$ be its circumcircle with radius $r$. The bisectors of the inner angles $A, B$, and $C$ of the triangle intersect respectively the circle $k$ again at points $A^{\\prime}, B^{\\prime}$, and $C^{\\prime}$. Prove the inequality  $$ 16 Q^{3} \\geq 27 r^{4} P $$  where $Q$ and $P$ are the areas of the triangles $A^{\\prime} B^{\\prime} C^{\\prime}$ and $A B C$ respectively.","t":[{"b":3,"e":0.57143,"k":"falling","v":0.6116,"x":0.9107,"p":[[0,54,0.0,0.875,0.12242,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,14,0,12],[4,54,0.0741,0.9107,0.1462,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,21],[8,54,0.1481,0.87499,0.17407,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,8,0,17],[12,54,0.2222,0.85714,0.21429,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,0,8,0,17],[16,54,0.2963,0.84375,0.19019,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,14,0,12],[20,54,0.3704,0.76786,0.23351,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,12,0,8],[24,54,0.4444,0.8125,0.16146,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,12,0,8],[28,54,0.5185,0.79016,0.22012,0.67857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,0,7,0,12],[32,54,0.5926,0.77675,0.2313,0.67857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,0,12,0,9],[36,54,0.6667,0.82589,0.21049,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,8,0,0,6,0,14],[40,54,0.7407,0.85714,0.13832,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,16,0,10],[44,54,0.8148,0.82589,0.19475,0.85714,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,15,0,10],[48,54,0.8889,0.75891,0.18708,0.57143,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,3,0,0,14,0,5],[52,54,0.963,0.62051,0.26632,0.39286,0.71414,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,3,0,0,4,0,0,6,0,0,7,0,4],[54,54,1.0,0.6116,0.27254,0.42857,0.64286,0.85714,0.0,1.0,2,2,0,2,0,2,0,0,0,0,0,7,0,0,5,0,0,4,0,0,10,0,2]]},{"b":4,"e":1.0,"k":"flat","v":0.86158,"x":0.95089,"p":[[0,81,0.0,0.87946,0.13882,0.85711,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,10,0,15],[4,81,0.0494,0.86607,0.17105,0.85711,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,10,0,15],[8,81,0.0988,0.86159,0.16166,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,8,0,15],[12,81,0.1481,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,81,0.1975,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[20,81,0.2469,0.86607,0.20183,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,11,0,16],[24,81,0.2963,0.87945,0.18937,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,17],[28,81,0.3457,0.92857,0.14286,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,24],[32,81,0.3951,0.91071,0.17768,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,21],[36,81,0.4444,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[40,81,0.4938,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[44,81,0.5432,0.88392,0.16917,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,8,0,18],[48,81,0.5926,0.90177,0.11542,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,16],[52,81,0.642,0.92411,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,21],[56,81,0.6914,0.89732,0.22654,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,23],[60,81,0.7407,0.90624,0.17356,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,7,0,21],[64,81,0.7901,0.90625,0.12682,0.85714,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,17],[68,81,0.8395,0.92408,0.10711,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[72,81,0.8889,0.89286,0.13363,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,18],[76,81,0.9383,0.89732,0.1525,0.82132,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,4,0,20],[80,81,0.9877,0.86158,0.15357,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,6,0,15],[81,81,1.0,0.86161,0.18723,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,17]]}]},{"i":"40e6d42883d885ad","q":"9. (POL 1) Let $T_{k}=k-1$ for $k=1,2,3,4$ and  $$ T_{2 k-1}=T_{2 k-2}+2^{k-2}, \\quad T_{2 k}=T_{2 k-5}+2^{k} \\quad(k \\geq 3) $$  Show that for all $k$,  $$ 1+T_{2 n-1}=\\left[\\frac{12}{7} 2^{n-1}\\right] \\quad \\text { and } \\quad 1+T_{2 n}=\\left[\\frac{17}{7} 2^{n-1}\\right] $$  where $[x]$ denotes the greatest integer not exceeding $x$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.83482,"x":0.99554,"p":[[0,106,0.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[4,106,0.0377,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,106,0.0755,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,106,0.1132,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,106,0.1509,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,106,0.1887,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[24,106,0.2264,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[28,106,0.2642,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[32,106,0.3019,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,106,0.3396,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[40,106,0.3774,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,106,0.4151,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[48,106,0.4528,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,106,0.4906,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,106,0.5283,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[60,106,0.566,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,106,0.6038,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[68,106,0.6415,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[72,106,0.6792,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[76,106,0.717,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[80,106,0.7547,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[84,106,0.7925,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,106,0.8302,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[92,106,0.8679,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[96,106,0.9057,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[100,106,0.9434,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[104,106,0.9811,0.89286,0.21429,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,9,0,20],[106,106,1.0,0.83482,0.25028,0.53572,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,1,0,0,0,0,22]]},{"b":5,"e":0.85714,"k":"flat","v":0.94196,"x":1.0,"p":[[0,40,0.0,0.94196,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[4,40,0.1,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,40,0.2,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,40,0.3,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[16,40,0.4,0.96428,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[20,40,0.5,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,40,0.7,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,40,0.8,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,40,0.9,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[40,40,1.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25]]}]},{"i":"9c46846368509b09","q":"2. (HUN) Denote by $a, b, c$ the lengths of the sides of a triangle. Prove that  $$ a^{2}(b+c-a)+b^{2}(c+a-b)+c^{2}(a+b-c) \\leq 3 a b c $$","t":[{"b":2,"e":0.2857,"k":"volatile","v":0.12945,"x":0.96875,"p":[[0,22,0.0,0.86607,0.33108,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[4,22,0.1818,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,22,0.3636,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,22,0.5455,0.76339,0.40816,0.82143,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,22],[16,22,0.7273,0.85268,0.34531,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[20,22,0.9091,0.59821,0.47974,0.0,1.0,1.0,0.0,1.0,12,18,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,18],[22,22,1.0,0.12945,0.18678,0.0,0.0,0.28571,0.0,0.71429,20,0,0,20,0,0,0,0,10,0,0,0,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.8125,"x":1.0,"p":[[0,58,0.0,0.86161,0.33021,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[4,58,0.069,0.92857,0.24484,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[8,58,0.1379,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[12,58,0.2069,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,58,0.2759,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,58,0.3448,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,58,0.4138,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[28,58,0.4828,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,58,0.5517,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[36,58,0.6207,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[40,58,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,58,0.7586,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[48,58,0.8276,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,58,0.8966,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,58,0.9655,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[58,58,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"a48abbe7789d17d4","q":"19. (GBR 1) Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^{n}$ are a sequence $a_{m}, a_{m-1}, \\ldots, a_{1}=5\\left(0 \\leq a_{j}<10\\right)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_{i}$ is even, then $a_{i-1}$ is odd, and if $a_{i}$ is odd, then $a_{i-1}$ is even).","t":[{"b":1,"e":0.4286,"k":"falling","v":0.54016,"x":0.8124,"p":[[0,40,0.0,0.7455,0.28062,0.57132,0.857,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,3,0,0,5,0,13],[4,40,0.1,0.80354,0.25941,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,4,0,0,3,0,17],[8,40,0.2,0.8124,0.26132,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,1,0,0,5,0,0,3,0,0,4,0,17],[12,40,0.3,0.80808,0.23306,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,5,0,0,3,0,16],[16,40,0.4,0.78125,0.2575,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,6,0,0,5,0,14],[20,40,0.5,0.68748,0.26831,0.39286,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,8,0,0,1,0,0,1,0,0,9,0,0,5,0,8],[24,40,0.6,0.65172,0.27652,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,4,0,0,5,0,0,7,0,0,1,0,9],[28,40,0.7,0.77229,0.20161,0.57143,0.78571,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,7,0,0,6,0,10],[32,40,0.8,0.54016,0.24675,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,4,0,0,4,0,0,11,0,0,0,0,3],[36,40,0.9,0.60265,0.27137,0.28571,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,7,0,0,3,0,0,3,0,0,10,0,0,1,0,6],[40,40,1.0,0.59373,0.24251,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,5,0,0,3,0,0,11,0,0,3,0,3]]},{"b":4,"e":0.14286,"k":"falling","v":0.39286,"x":0.8125,"p":[[0,25,0.0,0.66516,0.30849,0.39286,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,5,0,0,2,0,0,5,0,0,3,0,0,3,0,11],[4,25,0.16,0.80355,0.24682,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,4,0,0,2,0,17],[8,25,0.32,0.8125,0.21852,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,6,0,14],[12,25,0.48,0.67842,0.25753,0.5354,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,3,0,0,5,0,0,9,0,0,1,0,9],[16,25,0.64,0.57586,0.27545,0.28571,0.571,0.85704,0.14286,1.0,0,6,0,0,0,1,0,0,9,0,0,5,0,0,5,0,0,3,0,0,3,0,6],[20,25,0.8,0.48203,0.27617,0.2857,0.42859,0.71429,0.0,1.0,1,3,1,1,0,5,0,0,7,0,0,4,0,0,6,0,0,4,0,0,2,0,3],[24,25,0.96,0.44631,0.21662,0.2857,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,3,0,0,10,0,0,5,0,0,3,0,0,10,0,0,0,0,0],[25,25,1.0,0.39286,0.27199,0.25,0.28571,0.60714,0.0,1.0,1,2,0,1,0,7,0,0,14,0,0,1,0,0,1,0,0,4,0,0,2,0,2]]}]},{"i":"99321ee759b638c7","q":"4. (GDR) In the interior of $\\triangle P_{1} P_{2} P_{3}$ a point $P$ is given. Let $Q_{1}, Q_{2}$, and $Q_{3}$ respectively be the intersections of $P P_{1}, P P_{2}$, and $P P_{3}$ with the opposing edges of $\\triangle P_{1} P_{2} P_{3}$. Prove that among the ratios $P P_{1} / P Q_{1}, P P_{2} / P Q_{2}$, and $P P_{3} / P Q_{3}$ there exists at least one not larger than 2 and at least one not smaller than 2.","t":[{"b":1,"e":1.0,"k":"flat","v":0.83035,"x":0.91071,"p":[[0,25,0.0,0.83035,0.19045,0.57143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,4,0,16],[4,25,0.16,0.91071,0.17405,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,24],[8,25,0.32,0.83481,0.17538,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,0,4,0,14],[12,25,0.48,0.83928,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,3,0,15],[16,25,0.64,0.87945,0.16796,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,2,0,20],[20,25,0.8,0.89284,0.16754,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,0,0,22],[24,25,0.96,0.84375,0.2,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,10,0,0,2,0,17],[25,25,1.0,0.88393,0.16917,0.71429,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,0,0,21]]},{"b":7,"e":1.0,"k":"rising","v":0.83033,"x":1.0,"p":[[0,25,0.0,0.83033,0.20028,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,1,0,18],[4,25,0.16,0.88837,0.18471,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,0,0,23],[8,25,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,25,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,25,0.64,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3bfbd8e243d97c39","q":"5. (SPA 4) In the triangle $A B C$, with $\\measuredangle A=60^{\\circ}$, a parallel $I F$ to $A C$ is drawn through the incenter $I$ of the triangle, where $F$ lies on the side $A B$. The point $P$ on the side $B C$ is such that $3 B P=B C$. Show that $\\measuredangle B F P=\\measuredangle B / 2$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.85714,"x":0.96875,"p":[[0,68,0.0,0.875,0.20045,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,8,2,17],[4,68,0.0588,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,68,0.1176,0.92633,0.15515,0.91074,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,1,23],[12,68,0.1765,0.92633,0.14451,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,1,0,6,0,22],[16,68,0.2353,0.85714,0.24484,0.85711,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,0,3,0,22],[20,68,0.2941,0.94642,0.11152,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[24,68,0.3529,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[28,68,0.4118,0.89283,0.1786,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,9,0,19],[32,68,0.4706,0.93303,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22],[36,68,0.5294,0.91736,0.1374,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,1,21],[40,68,0.5882,0.91963,0.11263,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[44,68,0.6471,0.91741,0.17045,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,6,1,22],[48,68,0.7059,0.90402,0.1532,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,7,1,19],[52,68,0.7647,0.87054,0.22968,0.85714,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,4,0,21],[56,68,0.8235,0.90168,0.23569,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[60,68,0.8824,0.90625,0.19103,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,21],[64,68,0.9412,0.90191,0.17002,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,1,0,9,1,18],[68,68,1.0,0.8638,0.219,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,1,18]]},{"b":6,"e":0.857,"k":"flat","v":0.66516,"x":0.96874,"p":[[0,93,0.0,0.90855,0.10136,0.85714,0.93,1.0,0.714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,11,0,16],[4,93,0.043,0.96874,0.06903,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,93,0.086,0.89508,0.19396,0.85711,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,1,0,5,0,21],[12,93,0.129,0.89734,0.14267,0.857,1.0,1.0,0.43,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,2,0,7,0,18],[16,93,0.172,0.89953,0.19956,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,1,21],[20,93,0.2151,0.89508,0.20825,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,2,1,0,2,0,0,1,0,24],[24,93,0.2581,0.8705,0.18341,0.857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,8,0,17],[28,93,0.3011,0.80578,0.26621,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,2,0,0,0,0,0,1,0,0,4,1,0,8,0,14],[32,93,0.3441,0.66516,0.35511,0.33918,0.82142,1.0,0.0,1.0,3,12,0,3,0,2,0,0,3,0,1,2,0,0,1,0,0,3,1,0,4,0,12],[36,93,0.3871,0.79685,0.26313,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,2,0,0,9,1,13],[40,93,0.4301,0.88835,0.19804,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,20],[44,93,0.4731,0.90848,0.16197,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,4,1,21],[48,93,0.5161,0.88616,0.20349,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,1,19],[52,93,0.5591,0.84149,0.23808,0.76786,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,3,1,0,6,0,17],[56,93,0.6022,0.74553,0.30615,0.58939,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,5,0,0,1,0,0,0,1,0,2,0,0,8,1,12],[60,93,0.6452,0.83033,0.20266,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,9,2,12],[64,93,0.6882,0.78569,0.29234,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,8,0,15],[68,93,0.7312,0.84374,0.23517,0.857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,10,0,15],[72,93,0.7742,0.7657,0.26244,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,1,2,0,0,0,0,0,5,0,0,12,0,9],[76,93,0.8172,0.76999,0.26371,0.71429,0.85714,1.0,0.14,1.0,0,9,0,0,0,2,0,0,3,0,0,1,0,0,0,1,0,2,1,0,12,1,9],[80,93,0.8602,0.82588,0.27372,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,6,0,18],[84,93,0.9032,0.69418,0.35139,0.62511,0.85714,0.85714,0.0,1.0,5,7,0,5,0,1,0,0,1,0,0,0,0,0,1,1,0,1,0,0,15,0,7],[88,93,0.9462,0.81918,0.19396,0.71429,0.85714,0.94643,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,13,1,8],[92,93,0.9892,0.8348,0.23517,0.857,0.85714,1.0,0.1429,1.0,0,14,0,0,0,2,0,0,0,0,1,1,0,0,0,0,0,2,1,0,11,0,14],[93,93,1.0,0.83258,0.18264,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,6,1,0,11,0,11]]}]},{"i":"3978c05b95d65bf7","q":"5. (GBR 1) Find, with proof, the point $P$ in the interior of an acute-angled triangle $A B C$ for which $B L^{2}+C M^{2}+A N^{2}$ is a minimum, where $L, M, N$ are the feet of the perpendiculars from $P$ to $B C, C A, A B$ respectively.","t":[{"b":4,"e":0.0,"k":"falling","v":0.2232,"x":1.0,"p":[[0,123,0.0,0.85268,0.33021,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[4,123,0.0325,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[8,123,0.065,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,123,0.0976,0.94196,0.16311,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,28],[16,123,0.1301,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[20,123,0.1626,0.94196,0.1551,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[24,123,0.1951,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,123,0.2276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,123,0.2602,0.93303,0.19557,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[36,123,0.2927,0.95981,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[40,123,0.3252,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[44,123,0.3577,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[48,123,0.3902,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[52,123,0.4228,0.91516,0.15513,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23],[56,123,0.4553,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[60,123,0.4878,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[64,123,0.5203,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[68,123,0.5528,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[72,123,0.5854,0.93304,0.19556,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[76,123,0.6179,0.94196,0.12807,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[80,123,0.6504,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[84,123,0.6829,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[88,123,0.7154,0.93749,0.19544,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[92,123,0.748,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[96,123,0.7805,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[100,123,0.813,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[104,123,0.8455,0.91518,0.21086,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,26],[108,123,0.878,0.79464,0.26949,0.57143,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,3,0,16],[112,123,0.9106,0.68742,0.2486,0.571,0.57143,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,0,0,0,16,0,0,1,0,0,3,0,9],[116,123,0.9431,0.74997,0.23422,0.57143,0.78571,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,5,0,11],[120,123,0.9756,0.71424,0.25508,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,0,0,0,14,0,0,0,0,0,5,0,10],[123,123,1.0,0.2232,0.27648,0.0,0.0,0.57143,0.0,0.85714,18,0,0,18,0,1,0,0,2,0,0,1,0,0,9,0,0,0,0,0,1,0,0]]},{"b":5,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,119,0.0,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[4,119,0.0336,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,119,0.0672,0.96875,0.11142,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[12,119,0.1008,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,119,0.1345,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,119,0.1681,0.90625,0.22759,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,25],[24,119,0.2017,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,119,0.2353,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[32,119,0.2689,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[36,119,0.3025,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[40,119,0.3361,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[44,119,0.3697,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[48,119,0.4034,0.94641,0.12247,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[52,119,0.437,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[56,119,0.4706,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[60,119,0.5042,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,119,0.5378,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[68,119,0.5714,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[72,119,0.605,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[76,119,0.6387,0.94643,0.15047,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[80,119,0.6723,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[84,119,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,119,0.7395,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[92,119,0.7731,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,119,0.8067,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[100,119,0.8403,0.94642,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[104,119,0.8739,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[108,119,0.9076,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,119,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,119,0.9748,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[119,119,1.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29]]}]},{"i":"e55bcd0605123552","q":"A  $ 9 \\times 9$  square is divided into unit squares. Is it possible to fill each unit square with a number  $ 1, 2,..., 9$  in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?","t":[{"b":0,"e":0.57143,"k":"flat","v":0.55797,"x":0.73208,"p":[[0,33,0.0,0.57588,0.29555,0.57132,0.57143,0.64286,0.0,1.0,1,7,0,1,0,6,0,0,0,0,0,0,0,0,17,0,0,0,0,0,1,0,7],[4,33,0.1212,0.63387,0.20185,0.57143,0.57143,0.57143,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,6],[8,33,0.2424,0.6473,0.17123,0.57143,0.57143,0.57143,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,0,0,0,6],[12,33,0.3636,0.57585,0.22156,0.57143,0.57143,0.57143,0.0,1.0,2,4,0,2,0,1,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,4],[16,33,0.4848,0.62491,0.20751,0.571,0.57143,0.57143,0.14286,1.0,0,6,0,0,0,2,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,6],[20,33,0.6061,0.58034,0.17105,0.57143,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,1,0,0,26,0,0,0,0,0,0,0,3],[24,33,0.7273,0.73208,0.23356,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,13],[28,33,0.8485,0.55797,0.19999,0.571,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,4,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,3],[32,33,0.9697,0.57141,0.31542,0.5713,0.57143,0.67857,0.0,1.0,3,8,0,3,0,4,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,8],[33,33,1.0,0.64282,0.23147,0.57143,0.57143,0.64286,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,0,0,0,22,0,0,0,0,0,1,0,7]]},{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.6339,"p":[[0,39,0.0,0.6339,0.25238,0.57143,0.57143,0.67857,0.0,1.0,1,8,0,1,0,2,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,8],[4,39,0.1026,0.54458,0.2034,0.57132,0.57143,0.57143,0.0,1.0,3,2,0,3,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,2],[8,39,0.2051,0.58926,0.20748,0.57143,0.57143,0.57143,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,4],[12,39,0.3077,0.55802,0.24317,0.57143,0.57143,0.57143,0.0,1.0,3,4,0,3,0,1,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,4],[16,39,0.4103,0.57589,0.3204,0.57143,0.57143,0.67857,0.0,1.0,5,8,0,5,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,8],[20,39,0.5128,0.47764,0.26391,0.571,0.57143,0.57143,0.0,1.0,6,2,0,6,0,1,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,2],[24,39,0.6154,0.58926,0.20748,0.57143,0.57143,0.57143,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,4],[28,39,0.7179,0.49107,0.25738,0.57143,0.57143,0.57143,0.0,1.0,6,2,0,6,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,2],[32,39,0.8205,0.51784,0.34022,0.14286,0.57143,0.57143,0.0,1.0,6,7,0,6,0,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,7],[36,39,0.9231,0.45088,0.38483,0.0,0.57143,0.57143,0.0,1.0,12,7,0,12,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,7],[39,39,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b0df477384d50661","q":"A *site* is any point  $(x, y)$  in the plane such that  $x$  and  $y$  are both positive integers less than or equal to 20.\n\nInitially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to  $\\sqrt{5}$ . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.\n\nFind the greatest  $K$  such that Amy can ensure that she places at least  $K$  red stones, no matter how Ben places his blue stones.\n\n*Proposed by Gurgen Asatryan, Armenia*","t":[{"b":0,"e":1.0,"k":"rising","v":0.19196,"x":0.73655,"p":[[0,38,0.0,0.19196,0.21011,0.0,0.0,0.42857,0.0,0.42857,17,0,17,17,0,1,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.49999,0.25253,0.39286,0.42857,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,4,0,0,10,0,0,4,0,0,6,0,0,2,0,2],[8,38,0.2105,0.47321,0.2911,0.25,0.5,0.71429,0.0,1.0,5,2,0,5,0,3,0,0,1,0,0,7,0,0,5,0,0,8,0,0,1,0,2],[12,38,0.3158,0.40172,0.27063,0.14286,0.42857,0.57111,0.0,1.0,6,1,0,6,0,4,0,0,1,0,0,8,0,0,8,0,0,3,0,0,1,0,1],[16,38,0.4211,0.62943,0.15917,0.5354,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,11,0,0,2,0,2],[20,38,0.5263,0.58928,0.15872,0.42859,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,9,0,0,7,0,0,3,0,1],[24,38,0.6316,0.63836,0.16746,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,12,0,0,5,0,1],[28,38,0.7368,0.70975,0.20044,0.571,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,8,0,0,4,0,7],[32,38,0.8421,0.6205,0.18424,0.42859,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,10,0,0,10,0,0,7,0,0,1,0,4],[36,38,0.9474,0.73655,0.20554,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,10,0,0,7,0,6],[38,38,1.0,0.66512,0.18076,0.571,0.57143,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,7,0,0,4,0,4]]},{"b":2,"e":0.42857,"k":"rising","v":0.19634,"x":0.72319,"p":[[0,63,0.0,0.19634,0.23353,0.0,0.0,0.42857,0.0,0.85714,17,0,17,17,0,2,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0],[4,63,0.0635,0.47766,0.24118,0.42857,0.42857,0.60714,0.0,1.0,2,1,0,2,0,3,0,0,2,0,0,13,0,0,4,0,0,4,0,0,3,0,1],[8,63,0.127,0.56691,0.21866,0.42857,0.5712,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,8,0,0,9,0,0,6,0,0,3,0,2],[12,63,0.1905,0.6161,0.19373,0.42857,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,9,0,0,3,0,3],[16,63,0.254,0.62046,0.21012,0.42859,0.571,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,10,0,0,8,0,0,6,0,0,3,0,4],[20,63,0.3175,0.51783,0.17767,0.42857,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,17,0,0,5,0,0,6,0,0,1,0,1],[24,63,0.381,0.66963,0.24857,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,9,0,0,4,0,0,6,0,0,3,0,8],[28,63,0.4444,0.57587,0.2004,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,14,0,0,7,0,0,5,0,0,2,0,3],[32,63,0.5079,0.62495,0.22799,0.42857,0.571,0.85704,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,13,0,0,4,0,0,4,0,0,6,0,4],[36,63,0.5714,0.61604,0.23808,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,11,0,0,4,0,0,8,0,0,2,0,5],[40,63,0.6349,0.6518,0.1986,0.42857,0.57143,0.85704,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,6,0,0,5,0,4],[44,63,0.6984,0.67856,0.20826,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,10,0,0,3,0,6],[48,63,0.7619,0.65622,0.18851,0.4286,0.64286,0.85704,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,7,0,0,6,0,3],[52,63,0.8254,0.72319,0.19867,0.57132,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,9,0,0,5,0,7],[56,63,0.8889,0.63394,0.18873,0.42857,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,6,0,0,8,0,0,4,0,3],[60,63,0.9524,0.63386,0.21412,0.5354,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,7,0,0,10,0,0,7,0,0,3,0,4],[63,63,1.0,0.54,0.16247,0.42857,0.42857,0.71107,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,20,0,0,3,0,0,6,0,0,2,0,1]]}]},{"i":"5f1386a1cc96bbe1","q":"A convex quadrangle $A B C D$ is inscribed in a circle with the centre $O$. The angles $\\angle A O B, \\angle B O C, \\angle C O D$ and $\\angle D O A$, taken in some order, are of the same size as the angles of quadrangle $A B C D$. Prove that $A B C D$ is a square.","t":[{"b":4,"e":0.57143,"k":"flat","v":0.31695,"x":0.56249,"p":[[0,40,0.0,0.35256,0.27895,0.14286,0.14288,0.57143,0.0,1.0,1,3,0,1,0,16,0,0,2,0,0,2,0,0,8,0,0,0,0,0,0,0,3],[4,40,0.1,0.52229,0.32656,0.14286,0.571,0.85704,0.14286,1.0,0,7,0,0,0,9,0,0,4,0,0,2,0,0,7,0,0,1,0,0,2,0,7],[8,40,0.2,0.56249,0.34981,0.14286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,10,0,0,1,0,0,3,0,0,6,0,0,1,0,0,1,0,10],[12,40,0.3,0.5223,0.31665,0.25,0.57143,0.60714,0.0,1.0,2,7,0,2,0,6,0,0,2,0,0,3,0,0,11,0,0,1,0,0,0,0,7],[16,40,0.4,0.39285,0.27893,0.14286,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,15,0,0,2,0,0,1,0,0,10,0,0,1,0,0,0,0,3],[20,40,0.5,0.44194,0.28201,0.14286,0.57121,0.57143,0.0,1.0,1,3,0,1,0,11,0,0,1,0,0,1,0,0,14,0,0,0,0,0,1,0,3],[24,40,0.6,0.31695,0.21348,0.14286,0.14288,0.57143,0.0,0.71429,1,0,0,1,0,16,0,0,2,0,0,3,0,0,8,0,0,2,0,0,0,0,0],[28,40,0.7,0.43302,0.27077,0.14286,0.4998,0.57143,0.0,1.0,1,3,0,1,0,10,0,0,2,0,0,3,0,0,12,0,0,1,0,0,0,0,3],[32,40,0.8,0.49105,0.29867,0.14286,0.571,0.57143,0.14286,1.0,0,6,0,0,0,9,0,0,3,0,0,3,0,0,11,0,0,0,0,0,0,0,6],[36,40,0.9,0.39284,0.20824,0.14286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,9,0,0,6,0,0,4,0,0,12,0,0,0,0,0,0,0,1],[40,40,1.0,0.41962,0.21408,0.14286,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,10,0,0,1,0,0,5,0,0,15,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.57143,"k":"flat","v":0.39727,"x":0.60268,"p":[[0,64,0.0,0.39727,0.24928,0.14286,0.35716,0.57143,0.14286,1.0,0,1,0,0,0,13,0,0,3,0,0,1,0,0,11,0,0,2,0,0,1,0,1],[4,64,0.0625,0.58479,0.31816,0.39286,0.57143,1.0,0.0,1.0,1,9,0,1,0,6,0,0,1,0,0,2,0,0,11,0,0,2,0,0,0,0,9],[8,64,0.125,0.5,0.34626,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,12,0,0,2,0,0,3,0,0,5,0,0,1,0,0,1,0,8],[12,64,0.1875,0.51328,0.35519,0.14286,0.57143,0.89286,0.0,1.0,2,8,0,2,0,10,0,0,1,0,0,0,0,0,9,0,0,1,0,0,1,0,8],[16,64,0.25,0.60268,0.30667,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,5,0,0,2,0,0,10,0,0,1,0,0,0,0,10],[20,64,0.3125,0.50881,0.31741,0.14286,0.57143,0.57143,0.0,1.0,1,7,0,1,0,8,0,0,3,0,0,1,0,0,12,0,0,0,0,0,0,0,7],[24,64,0.375,0.50882,0.26723,0.28571,0.57143,0.57143,0.14,1.0,0,4,0,0,0,7,0,0,3,0,0,2,0,0,14,0,0,1,0,0,1,0,4],[28,64,0.4375,0.46417,0.2501,0.14286,0.57143,0.57143,0.14,1.0,0,2,0,0,0,10,0,0,1,0,0,1,0,0,15,0,0,3,0,0,0,0,2],[32,64,0.5,0.46414,0.23429,0.24999,0.57121,0.57143,0.14,1.0,0,2,0,0,0,8,0,0,3,0,0,1,0,0,17,0,0,1,0,0,0,0,2],[36,64,0.5625,0.4687,0.24282,0.14286,0.57121,0.57143,0.14286,1.0,0,2,0,0,0,9,0,0,1,0,0,2,0,0,17,0,0,0,0,0,1,0,2],[40,64,0.625,0.482,0.2443,0.25,0.5712,0.57143,0.14,1.0,0,3,0,0,0,8,0,0,1,0,0,3,0,0,17,0,0,0,0,0,0,0,3],[44,64,0.6875,0.40621,0.19916,0.14286,0.571,0.57143,0.14286,0.57143,0,0,0,0,0,11,0,0,1,0,0,2,0,0,18,0,0,0,0,0,0,0,0],[48,64,0.75,0.49993,0.25252,0.28571,0.571,0.57143,0.0,1.0,1,3,0,1,0,5,0,0,3,0,0,3,0,0,15,0,0,1,0,0,1,0,3],[52,64,0.8125,0.42409,0.21274,0.14286,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,9,0,0,3,0,0,3,0,0,16,0,0,0,0,0,0,0,1],[56,64,0.875,0.40172,0.22423,0.14286,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,12,0,0,1,0,0,3,0,0,15,0,0,0,0,0,0,0,1],[60,64,0.9375,0.40595,0.2027,0.14286,0.57121,0.57143,0.0,0.57143,1,0,0,1,0,9,0,0,2,0,0,2,0,0,18,0,0,0,0,0,0,0,0],[64,64,1.0,0.40611,0.20245,0.14286,0.571,0.57143,0.14,0.57143,0,0,0,0,0,11,0,0,2,0,0,0,0,0,19,0,0,0,0,0,0,0,0]]}]},{"i":"19c184ede8335817","q":"Consider 70-digit numbers $n$, with the property that each of the digits $1,2,3, \\ldots, 7$ appears in the decimal expansion of $n$ ten times (and 8, 9, and 0 do not appear). Show that no number of this form can divide another number of this form.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.71875,"x":0.82588,"p":[[0,21,0.0,0.82588,0.20435,0.71429,0.85714,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,9,0,12],[4,21,0.1905,0.79017,0.18552,0.71429,0.78564,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,9,0,7],[8,21,0.381,0.72321,0.10062,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,9,0,0],[12,21,0.5714,0.77676,0.147,0.67857,0.857,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,12,0,5],[16,21,0.7619,0.71875,0.12103,0.57143,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,12,0,0,9,0,1],[20,21,0.9524,0.78125,0.10705,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,11,0,3],[21,21,1.0,0.72768,0.09006,0.71429,0.71429,0.75,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,19,0,0,8,0,0]]},{"b":5,"e":0.71429,"k":"falling","v":0.70089,"x":0.89732,"p":[[0,21,0.0,0.89732,0.11971,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,16],[4,21,0.1905,0.76338,0.19435,0.71429,0.71429,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,9,0,6],[8,21,0.381,0.70981,0.17308,0.71429,0.71429,0.85714,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,0,9,0,1],[12,21,0.5714,0.70089,0.11494,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,17,0,0,5,0,1],[16,21,0.7619,0.76338,0.11073,0.71429,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,11,0,2],[20,21,0.9524,0.74107,0.11538,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,8,0,2],[21,21,1.0,0.73212,0.09944,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,10,0,0]]}]},{"i":"9a1e8e78538c8fa4","q":"A circle  $\\omega$  cuts the sides  $BC,CA,AB$  of the triangle  $ABC$  at  $A_1$  and  $A_2$ ;  $B_1$  and  $B_2$ ;  $C_1$  and  $C_2$ , respectively. Let  $P$  be the center of  $\\omega$ .  $A'$  is the circumcenter of the triangle  $A_1A_2P$ ,  $B'$  is the circumcenter of the triangle  $B_1B_2P$ ,  $C'$  is the circumcenter of the triangle  $C_1C_2P$ . Prove that  $AA', BB'$  and  $CC'$  concur.","t":[{"b":0,"e":1.0,"k":"rising","v":0.30357,"x":0.84371,"p":[[0,61,0.0,0.30357,0.13716,0.2857,0.28571,0.28571,0.0,0.71429,2,0,1,2,0,1,0,0,25,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[4,61,0.0656,0.54017,0.35845,0.28571,0.28571,1.0,0.0,1.0,3,9,0,3,0,0,0,0,15,0,0,0,0,0,0,0,0,2,0,0,3,0,9],[8,61,0.1311,0.67857,0.33312,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,9,0,0,0,0,0,0,0,0,6,0,0,2,0,13],[12,61,0.1967,0.57364,0.33525,0.2857,0.71429,0.85714,0.0,1.0,3,6,1,3,0,1,0,0,9,0,0,1,0,0,1,0,0,6,0,0,4,1,6],[16,61,0.2623,0.57588,0.37369,0.28571,0.42836,1.0,0.0,1.0,2,12,0,2,0,3,0,0,11,0,0,0,0,0,2,0,0,0,0,0,2,0,12],[20,61,0.3279,0.59821,0.33586,0.28571,0.57144,1.0,0.0,1.0,1,9,0,1,0,1,0,0,12,0,0,2,0,0,0,0,0,2,0,0,5,0,9],[24,61,0.3934,0.53568,0.33693,0.2857,0.28571,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,14,0,0,0,0,0,2,0,0,3,0,0,2,0,8],[28,61,0.459,0.58927,0.34395,0.28571,0.64264,1.0,0.0,1.0,2,9,0,2,0,1,0,0,11,0,0,1,0,0,1,0,0,3,0,0,4,0,9],[32,61,0.5246,0.55804,0.34323,0.28571,0.35716,1.0,0.0,1.0,2,9,0,2,0,0,0,0,14,0,0,2,0,0,0,0,0,2,0,0,3,0,9],[36,61,0.5902,0.63829,0.32744,0.28571,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,10,0,0,0,0,0,1,0,0,7,0,0,1,0,11],[40,61,0.6557,0.65176,0.2878,0.28571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,8,0,0,3,0,0,1,0,0,5,0,0,7,0,7],[44,61,0.7213,0.65622,0.30064,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,8,0,0,0,0,0,3,0,0,6,0,0,4,0,9],[48,61,0.7869,0.7119,0.30852,0.28571,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,8,0,0,1,0,0,1,0,0,3,1,0,4,0,13],[52,61,0.8525,0.75892,0.25615,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,7,0,0,5,0,12],[56,61,0.918,0.70977,0.25627,0.57132,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,1,0,0,4,0,0,11,0,0,4,0,8],[60,61,0.9836,0.7812,0.18556,0.71429,0.78564,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,11,0,0,8,0,8],[61,61,1.0,0.84371,0.14885,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,9,0,12]]},{"b":3,"e":0.28571,"k":"flat","v":0.23659,"x":0.63391,"p":[[0,46,0.0,0.33482,0.12682,0.28571,0.28571,0.32143,0.0,0.71429,1,0,1,1,0,0,0,0,23,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[4,46,0.087,0.5803,0.30079,0.28571,0.571,0.857,0.0,1.0,1,6,0,1,0,1,0,0,10,0,0,2,0,0,3,0,0,5,0,0,4,0,6],[8,46,0.1739,0.62945,0.31106,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,8,0,0,1,0,0,1,0,0,8,0,0,2,0,9],[12,46,0.2609,0.57142,0.33312,0.28571,0.64286,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,10,0,0,0,0,0,2,0,0,5,0,0,4,0,7],[16,46,0.3478,0.63391,0.3008,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,11,0,0,0,0,0,1,0,0,7,0,0,4,0,8],[20,46,0.4348,0.51782,0.32877,0.24999,0.4998,0.857,0.0,1.0,1,5,0,1,0,7,0,0,7,0,0,1,0,0,2,0,0,5,0,0,4,0,5],[24,46,0.5217,0.50893,0.32721,0.28571,0.28571,0.75,0.0,1.0,2,7,0,2,0,2,0,0,13,0,0,1,0,0,2,0,0,4,0,0,1,0,7],[28,46,0.6087,0.57141,0.35174,0.2857,0.49979,1.0,0.0,1.0,1,10,0,1,0,4,0,0,10,0,0,1,0,0,1,0,0,3,0,0,2,0,10],[32,46,0.6957,0.33032,0.23803,0.14286,0.2857,0.571,0.0,1.0,1,1,0,1,0,14,0,0,6,0,0,2,0,0,5,0,0,3,0,0,0,0,1],[36,46,0.7826,0.27204,0.20949,0.14286,0.28571,0.28571,0.0,1.0,4,1,0,4,0,9,0,0,13,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[40,46,0.8696,0.24554,0.17215,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,13,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[44,46,0.9565,0.23659,0.11629,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,13,0,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[46,46,1.0,0.29461,0.15942,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,8,0,0,17,0,0,2,0,0,2,0,0,2,0,0,0,0,0]]}]},{"i":"e81b30da9656d18b","q":"5. (GDR 2) For every integer $d \\geq 1$, let $M_{d}$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A=M_{1}, B=M_{2} \\backslash\\{2\\}, C=M_{3}$. Prove that every $c \\in C$ may be written in a unique way as $c=a b$ with $a \\in A, b \\in B$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.8616,"x":0.95535,"p":[[0,97,0.0,0.8616,0.23003,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,21],[4,97,0.0412,0.95535,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[8,97,0.0825,0.89732,0.17582,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,7,0,20],[12,97,0.1237,0.93301,0.11842,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[16,97,0.1649,0.94195,0.10635,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[20,97,0.2062,0.88835,0.17036,0.82143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,3,0,21],[24,97,0.2474,0.90625,0.14555,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,20],[28,97,0.2887,0.91516,0.15097,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,22],[32,97,0.3299,0.90624,0.15818,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[36,97,0.3711,0.90178,0.16536,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,21],[40,97,0.4124,0.91963,0.13337,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[44,97,0.4536,0.93749,0.09408,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[48,97,0.4948,0.91071,0.14174,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[52,97,0.5361,0.92855,0.12376,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[56,97,0.5773,0.86604,0.15131,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,8,0,15],[60,97,0.6186,0.87052,0.18338,0.82132,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,5,0,19],[64,97,0.6598,0.93303,0.11837,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,21],[68,97,0.701,0.87052,0.16508,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,16],[72,97,0.7423,0.87051,0.17266,0.85711,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,8,0,17],[76,97,0.7835,0.89728,0.14833,0.85711,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,19],[80,97,0.8247,0.86606,0.17837,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,17],[84,97,0.866,0.86605,0.16344,0.857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,10,0,15],[88,97,0.9072,0.88837,0.17031,0.85711,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,7,0,19],[92,97,0.9485,0.91963,0.14261,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[96,97,0.9897,0.93301,0.11841,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[97,97,1.0,0.9241,0.15146,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,24]]},{"b":6,"e":0.857,"k":"flat","v":0.77678,"x":0.97321,"p":[[0,58,0.0,0.84369,0.20941,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,6,0,17],[4,58,0.069,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,58,0.1379,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,58,0.2069,0.94641,0.07787,0.85714,1.0,1.0,0.714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[16,58,0.2759,0.91964,0.14698,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,23],[20,58,0.3448,0.92857,0.12372,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,21],[24,58,0.4138,0.8973,0.15254,0.857,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,20],[28,58,0.4828,0.87944,0.15615,0.857,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,8,0,17],[32,58,0.5517,0.90624,0.15818,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,20],[36,58,0.6207,0.91068,0.13723,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,20],[40,58,0.6897,0.93749,0.12344,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[44,58,0.7586,0.87945,0.19599,0.82143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,20],[48,58,0.8276,0.91071,0.14174,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[52,58,0.8966,0.91963,0.12848,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[56,58,0.9655,0.84374,0.17987,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,6,0,15],[58,58,1.0,0.77678,0.15542,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,7,0,7]]}]},{"i":"fd1e36340e175aac","q":"6. (HUN 1) If $a_{i}(i=1,2, \\ldots, n)$ are distinct non-zero real numbers, prove that the equation  $$ \\frac{a_{1}}{a_{1}-x}+\\frac{a_{2}}{a_{2}-x}+\\cdots+\\frac{a_{n}}{a_{n}-x}=n $$  has at least $n-1$ real roots.","t":[{"b":1,"e":1.0,"k":"flat","v":0.65179,"x":0.92857,"p":[[0,12,0.0,0.76339,0.27341,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,1,0,0,0,0,18],[4,12,0.3333,0.81697,0.29066,0.64286,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,5,0,0,0,0,0,2,0,0,0,0,22],[8,12,0.6667,0.92857,0.18898,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,28],[12,12,1.0,0.65179,0.3071,0.42857,0.4286,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,13,0,0,0,0,0,2,0,0,0,0,13]]},{"b":7,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,40,0.0,0.90625,0.21902,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,27],[4,40,0.1,0.92857,0.18898,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,28],[8,40,0.2,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,40,0.3,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,40,0.6,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,40,0.7,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,40,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a5f00852c3dbcae0","q":"Consider a  $m\\times n$  rectangular board consisting of  $mn$  unit squares. Two of its unit squares are called *adjacent* if they have a common edge, and a *path* is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called *non-intersecting* if they don't share any common squares.\r\n\r\nEach unit square of the rectangular board can be colored black or white. We speak of a *coloring* of the board if all its  $mn$  unit squares are colored.\r\n\r\nLet  $N$  be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let  $M$  be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.\r\n\r\nProve that  $N^{2}\\geq M\\cdot 2^{mn}$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.07143,"x":0.30804,"p":[[0,19,0.0,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.13839,0.14054,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,20,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,19,0.4211,0.20982,0.21124,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,20,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0],[12,19,0.6316,0.30804,0.27919,0.14286,0.1429,0.42857,0.0,1.0,5,1,0,5,0,12,0,0,5,0,0,3,0,0,1,0,0,3,0,0,2,0,1],[16,19,0.8421,0.26786,0.22517,0.14286,0.21435,0.42857,0.0,1.0,6,1,0,6,0,10,0,0,5,0,0,8,0,0,1,0,0,1,0,0,0,0,1],[19,19,1.0,0.15179,0.18189,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,13,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.06688,"x":0.27232,"p":[[0,23,0.0,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.20982,0.27661,0.0,0.07143,0.28571,0.0,0.85714,16,0,0,16,0,4,0,0,5,0,0,1,0,0,3,0,0,0,0,0,3,0,0],[8,23,0.3478,0.06688,0.09432,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.27232,0.27049,0.0,0.21428,0.42857,0.0,0.85714,12,0,0,12,0,4,0,0,2,0,0,9,0,0,1,0,0,2,0,0,2,0,0],[16,23,0.6957,0.22767,0.23105,0.0,0.14286,0.42857,0.0,0.85714,11,0,0,11,0,8,0,0,3,0,0,5,0,0,4,0,0,0,0,0,1,0,0],[20,23,0.8696,0.15625,0.21237,0.0,0.14286,0.17857,0.0,0.85714,15,0,0,15,0,9,0,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[23,23,1.0,0.10706,0.11843,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9a1823a6b3f4a065","q":"6. (POL) Let $M, K$, and $L$ be points on $(A B),(B C)$, and $(C A)$, respectively. Prove that the area of at least one of the three triangles $\\triangle M A L$, $\\triangle K B M$, and $\\triangle L C K$ is less than or equal to one-fourth the area of $\\triangle A B C$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13839,"x":0.18518,"p":[[0,34,0.0,0.15178,0.11811,0.0,0.14286,0.28571,0.0,0.28571,10,0,0,10,0,10,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.16071,0.10564,0.14286,0.14286,0.28571,0.0,0.28571,7,0,0,7,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.18518,0.1024,0.14286,0.14288,0.28571,0.0,0.28571,5,0,0,5,0,12,0,1,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.15621,0.09856,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,1,15,0,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.13839,0.10999,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.14732,0.10999,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.1384,0.10403,0.0,0.14286,0.1786,0.0,0.28571,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.16071,0.1171,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,10,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.14286,0.10714,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.15625,0.10926,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,13,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.14732,"x":0.20089,"p":[[0,22,0.0,0.14732,0.12103,0.0,0.14286,0.28571,0.0,0.28571,11,0,0,11,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.20089,0.09354,0.14286,0.21428,0.28571,0.0,0.28571,3,0,0,3,0,13,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.19643,0.09942,0.14286,0.21428,0.28571,0.0,0.28571,4,0,0,4,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.17857,0.10101,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.16518,0.11355,0.10714,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.16518,0.12428,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,10,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.20089,0.1063,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,9,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ecfdab6040baeaa9","q":"A communications network consisting of some terminals is called a *$3$ -connector* if among any three terminals, some two of them can directly communicate with each other. A communications network contains a *windmill* with  $n$  blades if there exist  $n$  pairs of terminals  $\\{x_{1},y_{1}\\},\\{x_{2},y_{2}\\},\\ldots,\\{x_{n},y_{n}\\}$  such that each  $x_{i}$  can directly communicate with the corresponding  $y_{i}$  and there is a *hub* terminal that can directly communicate with each of the  $2n$  terminals  $x_{1}, y_{1},\\ldots,x_{n}, y_{n}$  . Determine the minimum value of  $f (n)$ , in terms of  $n$ , such that a  $3$  -connector with  $f (n)$  terminals always contains a windmill with  $n$  blades.","t":[{"b":2,"e":0.0,"k":"falling","v":0.00884,"x":0.45529,"p":[[0,46,0.0,0.1875,0.20025,0.0,0.14286,0.42857,0.0,0.71429,13,0,13,13,0,8,0,0,1,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[4,46,0.087,0.45529,0.31829,0.14286,0.571,0.71429,0.0,1.0,5,3,0,5,0,6,0,0,2,0,0,1,0,0,8,0,0,6,0,0,1,0,3],[8,46,0.1739,0.37499,0.31491,0.0,0.42857,0.57143,0.0,1.0,9,2,0,9,0,4,0,0,1,0,0,6,0,0,5,0,0,4,0,0,1,0,2],[12,46,0.2609,0.28125,0.2461,0.0,0.2857,0.42858,0.0,0.85714,9,0,0,9,0,6,0,0,4,0,0,7,0,0,3,0,0,2,0,0,1,0,0],[16,46,0.3478,0.12945,0.19349,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,8,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[20,46,0.4348,0.25893,0.27994,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,9,0,0,3,0,0,5,0,0,1,0,0,1,0,0,2,0,1],[24,46,0.5217,0.2857,0.29232,0.0,0.21428,0.57111,0.0,0.85714,14,0,0,14,0,2,0,0,1,0,0,6,0,0,5,0,0,2,0,0,2,0,0],[28,46,0.6087,0.08481,0.17441,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,5,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[32,46,0.6957,0.08927,0.17401,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[36,46,0.7826,0.04911,0.11071,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,46,0.8696,0.06696,0.14719,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,46,0.9565,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"rising","v":0.20088,"x":0.55354,"p":[[0,55,0.0,0.24107,0.19377,0.0,0.28571,0.42857,0.0,0.4286,10,0,10,10,0,6,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.41964,0.3213,0.0,0.57143,0.71429,0.0,1.0,9,1,0,9,0,3,0,0,0,0,0,3,0,0,7,0,0,7,0,0,2,0,1],[8,55,0.1455,0.37499,0.28064,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,6,0,0,2,0,0,8,0,0,3,0,0,5,0,0,1,0,1],[12,55,0.2182,0.40624,0.27224,0.14286,0.42857,0.71429,0.0,1.0,3,1,0,3,0,8,0,0,3,0,0,7,0,0,2,0,0,7,0,0,1,0,1],[16,55,0.2909,0.2857,0.2812,0.0,0.2857,0.42857,0.0,1.0,10,2,0,10,0,4,0,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[20,55,0.3636,0.20088,0.23105,0.0,0.14286,0.32143,0.0,0.71429,13,0,0,13,0,8,0,0,3,0,0,4,0,0,1,0,0,3,0,0,0,0,0],[24,55,0.4364,0.29017,0.24609,0.0,0.28571,0.57111,0.0,0.85714,9,0,0,9,0,4,0,0,8,0,0,2,0,0,7,0,0,1,0,0,1,0,0],[28,55,0.5091,0.3436,0.24426,0.14286,0.28571,0.57141,0.0,0.71429,6,0,0,6,0,5,0,0,6,0,0,5,0,0,5,0,0,5,0,0,0,0,0],[32,55,0.5818,0.46868,0.20586,0.28571,0.4998,0.60714,0.0,0.71429,1,0,0,1,0,4,0,0,4,0,0,7,0,0,8,0,0,8,0,0,0,0,0],[36,55,0.6545,0.55354,0.24418,0.42859,0.57143,0.71429,0.0,1.0,2,1,0,2,0,3,0,0,0,0,0,5,0,0,9,0,0,9,0,0,3,0,1],[40,55,0.7273,0.54011,0.16649,0.4286,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,0,0,0,3,0,0,6,0,0,14,0,0,7,0,0,1,0,0],[44,55,0.8,0.55346,0.17426,0.42857,0.57143,0.71429,0.14,0.85714,0,0,0,0,0,3,0,0,0,0,0,7,0,0,11,0,0,10,0,0,1,0,0],[48,55,0.8727,0.53124,0.18977,0.42859,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,1,0,0,6,0,0,11,0,0,9,0,0,1,0,0],[52,55,0.9455,0.52678,0.13092,0.42857,0.57141,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,11,0,0,11,0,0,7,0,0,0,0,0],[55,55,1.0,0.51782,0.11149,0.42857,0.4286,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,8,0,0,6,0,0,0,0,0]]}]},{"i":"9007aaae96b905ca","q":"Consider a table with one real number in each cell. In one step, one may switch the sign of the numbers in one row or one column simultaneously. Prove that one can obtain a table with non-negative sums in each row and each column.","t":[{"b":3,"e":0.57143,"k":"flat","v":0.64284,"x":0.83036,"p":[[0,28,0.0,0.64284,0.36246,0.53539,0.71429,1.0,0.0,1.0,6,12,0,6,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,1,0,12],[4,28,0.1429,0.83036,0.2299,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,1,0,18],[8,28,0.2857,0.70536,0.32328,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,4,0,0,1,0,0,1,0,0,6,0,0,2,0,0,4,0,13],[12,28,0.4286,0.83034,0.30187,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,21],[16,28,0.5714,0.76339,0.33429,0.71429,1.0,1.0,0.0,1.0,4,17,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,1,0,17],[20,28,0.7143,0.75446,0.32583,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,1,0,0,0,0,0,6,0,0,3,0,0,0,0,18],[24,28,0.8571,0.77232,0.2809,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,0,0,0,1,0,0,5,0,0,5,0,0,3,0,15],[28,28,1.0,0.67853,0.21726,0.57143,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,0,0,0,18,0,0,3,0,0,1,0,8]]},{"b":5,"e":0.71429,"k":"rising","v":0.625,"x":0.91964,"p":[[0,42,0.0,0.625,0.36025,0.28571,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,5,0,0,1,0,0,3,0,0,6,0,0,0,0,12],[4,42,0.0952,0.85268,0.21275,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,6,0,0,0,0,20],[8,42,0.1905,0.78571,0.29233,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,0,0,0,0,0,0,6,0,0,4,0,0,2,0,17],[12,42,0.2857,0.80356,0.25693,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,5,0,0,4,0,16],[16,42,0.381,0.83036,0.28446,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,20],[20,42,0.4762,0.75,0.29451,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,1,0,0,1,0,0,6,0,0,4,0,0,2,0,15],[24,42,0.5714,0.75892,0.22991,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,1,0,12],[28,42,0.6667,0.8482,0.2313,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,1,0,0,1,0,21],[32,42,0.7619,0.91964,0.12846,0.85711,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[36,42,0.8571,0.83482,0.13882,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,10,0,10],[40,42,0.9524,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[42,42,1.0,0.84375,0.13054,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,5,0,12]]}]},{"i":"6444c2c1b04d8d3c","q":"5. A5 (POL) Let $f(x)=\\frac{x^{2}+1}{2 x}$ for $x \\neq 0$. Define $f^{(0)}(x)=x$ and $f^{(n)}(x)=$ $f\\left(f^{(n-1)}(x)\\right)$ for all positive integers $n$ and $x \\neq 0$. Prove that for all nonnegative integers $n$ and $x \\neq-1,0$, or 1 ,  $$ \\frac{f^{(n)}(x)}{f^{(n+1)}(x)}=1+\\frac{1}{f\\left(\\left(\\frac{x+1}{x-1}\\right)^{2^{n}}\\right)} $$","t":[{"b":6,"e":0.42857,"k":"flat","v":0.31241,"x":0.5625,"p":[[0,24,0.0,0.48659,0.23381,0.42857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,2,0,0,13,0,0,2,0,0,7,0,0,2,0,1],[4,24,0.1667,0.5,0.17128,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,11,0,0,6,0,0,9,0,0,0,0,0],[8,24,0.3333,0.31241,0.19053,0.14286,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,10,0,0,1,0,0,16,0,0,1,0,0,0,0,0,1,0,0],[12,24,0.5,0.33483,0.16213,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,2,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[16,24,0.6667,0.50893,0.15126,0.42857,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,21,0,0,1,0,0,8,0,0,1,0,0],[20,24,0.8333,0.48659,0.18509,0.42857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,18,0,0,4,0,0,5,0,0,2,0,0],[24,24,1.0,0.5625,0.17473,0.42857,0.64286,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,12,0,0,2,0,0,16,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.26784,"x":0.50892,"p":[[0,24,0.0,0.50892,0.23402,0.42857,0.57143,0.71429,0.0,0.85714,3,0,0,3,0,1,0,0,3,0,0,7,0,0,5,0,0,12,0,0,1,0,0],[4,24,0.1667,0.47768,0.18766,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,15,0,0,4,0,0,6,0,0,0,0,1],[8,24,0.3333,0.27902,0.20389,0.14286,0.24999,0.42857,0.0,0.71429,5,0,0,5,0,10,0,1,3,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[12,24,0.5,0.28124,0.20353,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,6,0,0,5,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[16,24,0.6667,0.26784,0.19145,0.14286,0.2143,0.42857,0.0,0.714,5,0,0,5,0,11,0,0,3,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[20,24,0.8333,0.28125,0.20666,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,2,0,0,12,0,0,1,0,0,2,0,0,0,0,0],[24,24,1.0,0.28348,0.20083,0.14286,0.35714,0.42857,0.0,0.71429,5,0,0,5,1,9,0,0,1,0,0,14,0,0,0,0,0,2,0,0,0,0,0]]}]},{"i":"f465754c6dca306c","q":"Consider a circle  $S$ , and a point  $P$  outside it. The tangent lines from  $P$  meet  $S$  at  $A$  and  $B$ , respectively. Let  $M$  be the midpoint of  $AB$ . The perpendicular bisector of  $AM$  meets  $S$  in a point  $C$  lying inside the triangle  $ABP$ .  $AC$  intersects  $PM$  at  $G$ , and  $PM$  meets  $S$  in a point  $D$  lying outside the triangle  $ABP$ . If  $BD$  is parallel to  $AC$ , show that  $G$  is the centroid of the triangle  $ABP$ .\r\n\r\n*Arnoldo Aguilar (El Salvador)*","t":[{"b":5,"e":0.42857,"k":"flat","v":0.29465,"x":0.45087,"p":[[0,244,0.0,0.43308,0.08364,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,27,0,0,2,0,0,1,0,0,0,0,0],[4,244,0.0164,0.43304,0.06667,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0],[8,244,0.0328,0.41964,0.10677,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,1,0,0,25,0,0,4,0,0,0,0,0,0,0,0],[12,244,0.0492,0.42865,0.08746,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0],[16,244,0.0656,0.41516,0.07454,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,3,0,0,26,0,0,2,0,0,0,0,0,0,0,0],[20,244,0.082,0.39731,0.11699,0.42857,0.42857,0.42857,0.0,0.571,2,0,0,2,0,1,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0],[24,244,0.0984,0.43304,0.07562,0.42857,0.42857,0.42857,0.1429,0.71429,0,0,0,0,0,1,0,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0],[28,244,0.1148,0.41518,0.13534,0.42857,0.42857,0.42858,0.0,0.71429,2,0,0,2,0,1,0,0,0,0,0,25,0,0,3,0,0,1,0,0,0,0,0],[32,244,0.1311,0.41968,0.06119,0.42857,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,1,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[36,244,0.1475,0.4375,0.07087,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0,0,0,0],[40,244,0.1639,0.41963,0.1006,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,3,0,0,24,0,0,4,0,0,0,0,0,0,0,0],[44,244,0.1803,0.45087,0.06292,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0],[48,244,0.1967,0.43749,0.07934,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,1,0,0,25,0,0,5,0,0,0,0,0,0,0,0],[52,244,0.2131,0.4241,0.13591,0.42857,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,1,0,0,0,0,0,22,0,0,7,0,0,0,0,0,0,0,0],[56,244,0.2295,0.43302,0.10997,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,25,0,0,2,0,0,2,0,0,0,0,0],[60,244,0.2459,0.42411,0.07563,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[64,244,0.2623,0.44639,0.07777,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0],[68,244,0.2787,0.42411,0.11564,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,25,0,0,3,0,0,1,0,0,0,0,0],[72,244,0.2951,0.42411,0.12103,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,0,0,0,25,0,0,5,0,0,0,0,0,0,0,0],[76,244,0.3115,0.40625,0.13882,0.42857,0.42857,0.4286,0.0,0.57143,3,0,0,3,0,0,0,0,0,0,0,25,0,0,4,0,0,0,0,0,0,0,0],[80,244,0.3279,0.40626,0.09521,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0],[84,244,0.3443,0.41518,0.11495,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,0,0,0,27,0,0,3,0,0,0,0,0,0,0,0],[88,244,0.3607,0.44643,0.04724,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0],[92,244,0.377,0.44194,0.04156,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0],[96,244,0.3934,0.42858,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[100,244,0.4098,0.43302,0.06664,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0],[104,244,0.4262,0.40177,0.14031,0.42857,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,0,0,0,1,0,0,24,0,0,4,0,0,0,0,0,0,0,0],[108,244,0.4426,0.44643,0.08564,0.42857,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,26,0,0,4,0,0,1,0,0,0,0,0],[112,244,0.459,0.42857,0.08748,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0],[116,244,0.4754,0.42411,0.10402,0.42857,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,3,0,0,0,0,0,24,0,0,5,0,0,0,0,0,0,0,0],[120,244,0.4918,0.42856,0.06183,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0],[124,244,0.5082,0.433,0.08357,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,24,0,0,5,0,0,0,0,0,0,0,0],[128,244,0.5246,0.42409,0.08361,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0],[132,244,0.541,0.41515,0.09684,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,0,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[136,244,0.5574,0.41518,0.10326,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,1,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[140,244,0.5738,0.42411,0.09094,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,27,0,0,3,0,0,0,0,0,0,0,0],[144,244,0.5902,0.42411,0.10403,0.42857,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,1,0,0,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0],[148,244,0.6066,0.41072,0.08564,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,2,0,0,26,0,0,2,0,0,0,0,0,0,0,0],[152,244,0.623,0.44195,0.07454,0.42857,0.42857,0.42857,0.1429,0.57143,0,0,0,0,0,1,0,0,0,0,0,26,0,0,5,0,0,0,0,0,0,0,0],[156,244,0.6393,0.41965,0.11811,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,0,0,0,24,0,0,5,0,0,0,0,0,0,0,0],[160,244,0.6557,0.41963,0.087,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,1,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[164,244,0.6721,0.39731,0.12232,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,1,0,0,24,0,0,3,0,0,0,0,0,0,0,0],[168,244,0.6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the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.09819,"x":0.32588,"p":[[0,26,0.0,0.30357,0.13716,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[4,26,0.1538,0.32588,0.27486,0.10714,0.28571,0.42857,0.0,1.0,8,2,0,8,0,1,0,0,12,0,0,5,0,0,2,0,0,1,0,0,1,0,2],[8,26,0.3077,0.29454,0.2421,0.14214,0.28571,0.42857,0.0,1.0,7,1,0,7,0,4,0,0,12,0,0,4,0,0,1,0,0,3,0,0,0,0,1],[12,26,0.4615,0.29464,0.22851,0.21427,0.28571,0.42857,0.0,1.0,8,1,0,8,0,0,0,0,14,0,0,7,0,0,0,0,0,2,0,0,0,0,1],[16,26,0.6154,0.22768,0.19516,0.0,0.28571,0.32143,0.0,0.71429,11,0,0,11,0,2,0,0,11,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[20,26,0.7692,0.22758,0.24187,0.0,0.21429,0.42857,0.0,0.85714,14,0,0,14,0,2,0,0,6,0,0,6,0,0,2,0,0,1,0,0,1,0,0],[24,26,0.9231,0.09819,0.18006,0.0,0.0,0.14287,0.0,0.571,23,0,0,23,0,3,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[26,26,1.0,0.13393,0.2111,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.28571,"k":"falling","v":0.08927,"x":0.375,"p":[[0,21,0.0,0.375,0.24157,0.2857,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,2,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[4,21,0.1905,0.36607,0.32328,0.21427,0.28571,0.42858,0.0,1.0,8,5,0,8,0,0,0,0,13,0,0,4,0,0,1,0,0,1,0,0,0,0,5],[8,21,0.381,0.35268,0.2789,0.28571,0.28571,0.42857,0.0,1.0,7,3,0,7,0,0,0,0,12,0,0,8,0,0,0,0,0,2,0,0,0,0,3],[12,21,0.5714,0.24553,0.26781,0.0,0.2857,0.42857,0.0,0.85714,14,0,0,14,0,1,0,0,8,0,0,4,0,0,1,0,0,2,0,0,2,0,0],[16,21,0.7619,0.15616,0.17628,0.0,0.14143,0.28571,0.0,0.71429,15,0,0,15,0,4,0,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[20,21,0.9524,0.12051,0.21455,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[21,21,1.0,0.08927,0.14612,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"75521faf7fb535c1","q":"Consider on the coordinate plane all rectangles whose\n(i) vertices have integer coordinates;\n(ii) edges are parallel to coordinate axes;\n(iii) area is  $2^k$ , where  $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?","t":[{"b":5,"e":1.0,"k":"flat","v":0.08929,"x":0.57589,"p":[[0,42,0.0,0.57589,0.4798,0.0,0.92857,1.0,0.0,1.0,13,16,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,16],[4,42,0.0952,0.08929,0.22232,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,42,0.1905,0.12054,0.30328,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[12,42,0.2857,0.15179,0.33108,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,42,0.381,0.1875,0.35072,0.0,0.0,0.2857,0.0,1.0,23,4,0,23,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[20,42,0.4762,0.33036,0.39357,0.0,0.2143,0.71429,0.0,1.0,15,6,0,15,0,1,0,0,6,0,0,1,0,0,0,0,0,2,0,0,1,0,6],[24,42,0.5714,0.29018,0.35978,0.0,0.2857,0.28571,0.0,1.0,15,5,0,15,0,0,0,0,10,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[28,42,0.6667,0.27679,0.31326,0.0,0.28571,0.28571,0.0,1.0,13,3,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,3,0,0,0,0,3],[32,42,0.7619,0.28125,0.36681,0.0,0.21428,0.28571,0.0,1.0,15,6,0,15,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[36,42,0.8571,0.41071,0.3989,0.0,0.28571,1.0,0.0,1.0,10,9,0,10,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[40,42,0.9524,0.44195,0.36658,0.2857,0.28571,0.78571,0.0,1.0,7,8,0,7,0,0,0,0,13,0,0,0,0,0,3,0,0,1,0,0,0,0,8],[42,42,1.0,0.51786,0.35129,0.28571,0.28571,1.0,0.0,1.0,3,9,0,3,0,1,0,0,14,0,0,0,0,0,1,0,0,4,0,0,0,0,9]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.76786,"p":[[0,51,0.0,0.76786,0.38424,0.71429,1.0,1.0,0.0,1.0,6,21,1,6,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,21],[4,51,0.0784,0.39731,0.43993,0.0,0.2857,1.0,0.0,1.0,15,10,0,15,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,10],[8,51,0.1569,0.36161,0.47243,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11],[12,51,0.2353,0.12944,0.29089,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,2],[16,51,0.3137,0.14286,0.3312,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[20,51,0.3922,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,51,0.4706,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,51,0.549,0.06696,0.1988,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[32,51,0.6275,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,51,0.7843,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.25446,0.40522,0.0,0.0,0.28571,0.0,1.0,21,7,0,21,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[48,51,0.9412,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[51,51,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"234e38e7a14e46de","q":"9. II 3 (CUB 3) Let $x, y, z$ be real numbers each of whose absolute value is different from $1 / \\sqrt{3}$ such that $x+y+z=x y z$. Prove that  $$ \\frac{3 x-x^{3}}{1-3 x^{2}}+\\frac{3 y-y^{3}}{1-3 y^{2}}+\\frac{3 z-z^{3}}{1-3 z^{2}}=\\frac{3 x-x^{3}}{1-3 x^{2}} \\cdot \\frac{3 y-y^{3}}{1-3 y^{2}} \\cdot \\frac{3 z-z^{3}}{1-3 z^{2}} $$","t":[{"b":1,"e":0.857,"k":"flat","v":0.9375,"x":0.98661,"p":[[0,26,0.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,26,0.1538,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,26,0.3077,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,26,0.4615,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,26,0.6154,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[20,26,0.7692,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,26,0.9231,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[26,26,1.0,0.94196,0.07874,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20]]},{"b":5,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,29,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,29,0.1379,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,29,0.2759,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,29,0.4138,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"45e61283ea1b5dac","q":"Andile and Zandre play a game on a  $2017 \\times 2017$  board. At the beginning, Andile declares some of the squares *forbidden*, meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game.\n\nWhat is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)","t":[{"b":3,"e":0.42857,"k":"rising","v":0.17411,"x":0.48218,"p":[[0,20,0.0,0.17411,0.19144,0.0,0.07143,0.28571,0.0,0.57143,16,0,7,16,0,1,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[4,20,0.2,0.4242,0.19719,0.42857,0.42857,0.42895,0.0,1.0,2,1,0,2,0,2,0,0,3,0,0,19,0,0,3,0,0,1,0,0,1,0,1],[8,20,0.4,0.43304,0.18029,0.42857,0.42857,0.42858,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,22,0,0,2,0,0,0,0,0,0,0,2],[12,20,0.6,0.45532,0.17287,0.42857,0.42857,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,3,0,0,22,0,0,4,0,0,0,0,0,0,0,2],[16,20,0.8,0.42862,0.12372,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,1],[20,20,1.0,0.48218,0.17766,0.42857,0.42857,0.4286,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,23,0,0,3,0,0,0,0,0,0,0,3]]},{"b":5,"e":1.0,"k":"rising","v":0.21875,"x":0.51788,"p":[[0,13,0.0,0.21875,0.19556,0.0,0.28571,0.42857,0.0,0.71429,12,0,5,12,0,2,0,0,9,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[4,13,0.3077,0.42407,0.16551,0.42857,0.42857,0.42858,0.0,1.0,2,1,0,2,0,0,0,0,4,0,0,20,0,0,5,0,0,0,0,0,0,0,1],[8,13,0.6154,0.45088,0.16408,0.42857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,20,0,0,5,0,0,0,0,0,1,0,1],[12,13,0.9231,0.51788,0.18811,0.42857,0.42857,0.46525,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,24,0,0,4,0,0,0,0,0,0,0,4],[13,13,1.0,0.45536,0.14914,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,26,0,0,1,0,0,0,0,0,0,0,2]]}]},{"i":"636b4f2b19ea4389","q":"Consider a right-angled triangle  $ABC$  with the hypothenuse  $AB=1$ . The bisector of  $\\angle{ACB}$  cuts the medians  $BE$  and  $AF$  at  $P$  and  $M$ , respectively. If  ${AF}\\cap{BE}=\\{P\\}$ , determine the maximum value of the area of  $\\triangle{MNP}$ .","t":[{"b":0,"e":0.42857,"k":"falling","v":0.07589,"x":0.88393,"p":[[0,131,0.0,0.88393,0.2911,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[4,131,0.0305,0.75,0.35714,0.42857,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,21],[8,131,0.0611,0.80357,0.32488,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,22],[12,131,0.0916,0.78571,0.31944,0.39286,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,22],[16,131,0.1221,0.39286,0.34069,0.24999,0.28571,0.42857,0.0,1.0,5,7,0,5,0,3,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,7],[20,131,0.1527,0.36165,0.25251,0.2857,0.28571,0.28571,0.0,1.0,1,4,0,1,0,3,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[24,131,0.1832,0.38839,0.27947,0.28571,0.28571,0.42857,0.0,1.0,2,5,0,2,0,2,0,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,5],[28,131,0.2137,0.25893,0.2822,0.0,0.2857,0.28571,0.0,1.0,11,3,0,11,0,3,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[32,131,0.2443,0.22768,0.28316,0.0,0.21428,0.28571,0.0,1.0,13,3,0,13,0,3,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[36,131,0.2748,0.16518,0.13415,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,6,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,131,0.3053,0.22322,0.19865,0.0,0.28571,0.28571,0.0,1.0,9,1,0,9,0,3,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[44,131,0.3359,0.16964,0.14032,0.0,0.14288,0.28571,0.0,0.42857,11,0,0,11,0,6,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,131,0.3664,0.15178,0.14698,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,4,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,131,0.3969,0.16964,0.14032,0.0,0.2857,0.28571,0.0,0.42857,12,0,0,12,0,3,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,131,0.4275,0.17857,0.19885,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,6,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[60,131,0.458,0.16964,0.14914,0.0,0.2857,0.28571,0.0,0.4286,13,0,0,13,0,2,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[64,131,0.4885,0.15625,0.13054,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,131,0.5191,0.16965,0.20652,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,2,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[72,131,0.5496,0.17857,0.20203,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,4,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[76,131,0.5802,0.17411,0.20743,0.0,0.21428,0.28571,0.0,1.0,15,1,0,15,0,1,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[80,131,0.6107,0.19196,0.249,0.0,0.14286,0.28571,0.0,1.0,14,2,0,14,0,4,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[84,131,0.6412,0.13393,0.12339,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,8,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,131,0.6718,0.08928,0.13243,0.0,0.0,0.17857,0.0,0.42857,21,0,0,21,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,131,0.7023,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,131,0.7328,0.11611,0.14923,0.0,0.0,0.2857,0.0,0.43,19,0,0,19,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[100,131,0.7634,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,131,0.7939,0.09821,0.1357,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[108,131,0.8244,0.10714,0.15152,0.0,0.0,0.17857,0.0,0.57143,19,0,0,19,0,5,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[112,131,0.855,0.125,0.14617,0.0,0.07143,0.2857,0.0,0.57143,16,0,0,16,0,6,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[116,131,0.8855,0.08482,0.19186,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[120,131,0.916,0.17857,0.26964,0.0,0.0,0.2857,0.0,1.0,18,2,0,18,0,1,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[124,131,0.9466,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[128,131,0.9771,0.17411,0.28735,0.0,0.0,0.2857,0.0,1.0,17,3,0,17,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[131,131,1.0,0.14277,0.15567,0.0,0.07,0.28571,0.0,0.42857,16,0,0,16,0,3,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.79911,"x":1.0,"p":[[0,145,0.0,0.79911,0.3423,0.67857,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,23],[4,145,0.0276,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,145,0.0552,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[12,145,0.0828,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,145,0.1103,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,145,0.1379,0.94643,0.17035,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[24,145,0.1655,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[28,145,0.1931,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[32,145,0.2207,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,145,0.2483,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[40,145,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,145,0.3034,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0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up-right path from lattice points  $P$  and  $Q$  on the  $xy$ -plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from  $(0,0)$  to  $(10,3)$  does not intersect the graph of  $y=x^2+0.5$  can be written as  $\\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .\n\n*Proposed by **HrishiP***","t":[{"b":0,"e":1.0,"k":"flat","v":0.87946,"x":1.0,"p":[[0,91,0.0,0.87946,0.2372,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,4,0,23],[4,91,0.044,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,91,0.0879,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,91,0.1319,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[16,91,0.1758,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,91,0.2198,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,91,0.2637,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,91,0.3077,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,91,0.3516,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[36,91,0.3956,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,91,0.4396,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,91,0.4835,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[48,91,0.5275,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[52,91,0.5714,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[56,91,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,91,0.6593,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,91,0.7033,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,91,0.7473,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[72,91,0.7912,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,91,0.8352,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[80,91,0.8791,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,91,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,91,0.967,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[91,91,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,63,0.0,0.89732,0.24804,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,26],[4,63,0.0635,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,63,0.127,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[12,63,0.1905,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,63,0.254,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,63,0.3175,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,63,0.381,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,63,0.4444,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[32,63,0.5079,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,63,0.5714,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[40,63,0.6349,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[44,63,0.6984,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,63,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,63,0.8254,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,63,0.8889,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[60,63,0.9524,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[63,63,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"d092240467f67fd8","q":"Define real number  $y$  as the fractional part of real number  $x$  such that  $0\\leq y<1$  and  $x-y$  is integer. Denote this by  $<x>$ .\n\nFor real number  $a$ , define an infinite sequence  $\\{a_n\\}\\ (n=1,\\ 2,\\ 3,\\ \\cdots)$  inductively as follows.\n\n(i)  $a_1=<a>$ (ii) If  $a\\n\\neq 0$ , then  $a_{n+1}=\\left<\\frac{1}{a_n}\\right>$ ,\n\nif  $a_n=0$ , then  $a_{n+1}=0$ .\n\n(1) For  $a=\\sqrt{2}$ , find  $a_n$ .\n\n(2) For any natural number  $n$ , find real number  $a\\geq \\frac 13$  such that  $a_n=a$ .\n\n(3) Let  $a$  be a rational number. When we express  $a=\\frac{p}{q}$  with integer  $p$ , natural number  $q$ , prove that  $a_n=0$  for any natural number  $n\\geq q$ .\n\n*2011 Tokyo University entrance exam/Science, Problem 2*","t":[{"b":1,"e":0.71429,"k":"falling","v":0.70089,"x":0.96429,"p":[[0,12,0.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[4,12,0.3333,0.82128,0.17138,0.71429,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,3,0,13],[8,12,0.6667,0.71874,0.02486,0.71429,0.71429,0.71429,0.714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0],[12,12,1.0,0.70089,0.07457,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,29,0,0,1,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.80357,"x":0.98661,"p":[[0,15,0.0,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[4,15,0.2667,0.83929,0.16269,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,1,0,15],[8,15,0.5333,0.80357,0.14174,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,2,0,10],[12,15,0.8,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[15,15,1.0,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28]]}]},{"i":"017ebee7e8a1d0ed","q":"An ordered triple  $(p, q, r)$  of prime numbers is called *parcera* if  $p$  divides  $q^2-4$ ,  $q$  divides  $r^2-4$  and  $r$  divides  $p^2-4$ . Find all parcera triples.","t":[{"b":3,"e":0.42857,"k":"falling","v":0.53572,"x":0.8348,"p":[[0,84,0.0,0.8348,0.21165,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,1,0,18],[4,84,0.0476,0.77232,0.23381,0.57143,0.78564,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,6,0,0,2,0,14],[8,84,0.0952,0.72768,0.21535,0.64286,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,12,0,0,4,0,8],[12,84,0.1429,0.72767,0.25595,0.42857,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,4,0,0,5,0,11],[16,84,0.1905,0.73213,0.22799,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,7,0,0,4,0,10],[20,84,0.2381,0.75445,0.23753,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,8,0,0,3,0,12],[24,84,0.2857,0.74107,0.23266,0.42859,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,8,0,0,5,0,10],[28,84,0.3333,0.73214,0.23077,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,10,0,0,3,0,10],[32,84,0.381,0.69643,0.25692,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,11,0,0,0,0,0,4,0,0,6,0,9],[36,84,0.4286,0.71428,0.25505,0.42857,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,9,0,0,2,0,0,3,0,0,6,0,10],[40,84,0.4762,0.64286,0.26245,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,11,0,0,0,0,0,8,0,0,0,0,9],[44,84,0.5238,0.69642,0.25692,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,9,0,0,0,0,0,6,0,0,5,0,9],[48,84,0.5714,0.73661,0.23449,0.53571,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,5,0,0,2,0,12],[52,84,0.619,0.76785,0.24157,0.57143,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,4,0,0,3,0,14],[56,84,0.6667,0.73659,0.24252,0.57132,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,2,0,12],[60,84,0.7143,0.7232,0.23129,0.53539,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,6,0,0,7,0,8],[64,84,0.7619,0.75,0.22868,0.53571,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,8,0,0,2,0,12],[68,84,0.8095,0.75446,0.24804,0.53571,0.78571,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,6,0,0,3,0,13],[72,84,0.8571,0.80804,0.20705,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,11,0,0,1,0,15],[76,84,0.9048,0.79911,0.20159,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,4,0,13],[80,84,0.9524,0.64286,0.22588,0.42857,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,9,0,0,2,0,6],[84,84,1.0,0.53572,0.18211,0.42857,0.42857,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,22,0,0,2,0,0,5,0,0,0,0,3]]},{"b":5,"e":0.2857,"k":"falling","v":0.58482,"x":0.86607,"p":[[0,83,0.0,0.86607,0.19541,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,4,0,19],[4,83,0.0482,0.85267,0.1838,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,5,0,16],[8,83,0.0964,0.82143,0.22304,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,6,0,0,6,0,15],[12,83,0.1446,0.84374,0.18681,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,9,0,14],[16,83,0.1928,0.75893,0.2299,0.64286,0.78571,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,8,0,0,5,0,11],[20,83,0.241,0.71875,0.23278,0.42859,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,6,0,0,5,0,9],[24,83,0.2892,0.71428,0.22015,0.42857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,10,0,0,4,0,8],[28,83,0.3373,0.70982,0.23003,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,8,0,0,5,0,8],[32,83,0.3855,0.65179,0.24468,0.42857,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,11,0,0,0,0,0,7,0,0,5,0,6],[36,83,0.4337,0.6384,0.24996,0.42857,0.57144,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,14,0,0,0,0,0,7,0,0,1,0,8],[40,83,0.4819,0.66071,0.26184,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,9,0,0,2,0,0,6,0,0,2,0,9],[44,83,0.5301,0.6875,0.23538,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,6,0,0,4,0,8],[48,83,0.5783,0.65179,0.24984,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,4,0,0,8,0,0,3,0,0,8,0,0,1,0,8],[52,83,0.6265,0.71429,0.26,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,5,0,0,4,0,11],[56,83,0.6747,0.78571,0.23419,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,7,0,0,7,0,12],[60,83,0.7229,0.71875,0.2461,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,6,0,0,6,0,9],[64,83,0.7711,0.7366,0.23175,0.42857,0.85707,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,4,0,0,8,0,9],[68,83,0.8193,0.70982,0.26603,0.42857,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,9,0,0,0,0,0,5,0,0,4,0,11],[72,83,0.8675,0.77679,0.23673,0.53572,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,4,0,0,4,0,14],[76,83,0.9157,0.76339,0.21902,0.57143,0.78564,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,7,0,0,5,0,11],[80,83,0.9639,0.70534,0.22851,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,6,0,0,5,0,8],[83,83,1.0,0.58482,0.25345,0.42857,0.42857,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,14,0,0,0,0,0,5,0,0,2,0,6]]}]},{"i":"55677effdbc5f74b","q":"Define the sequence  $a_n$  by the rule   $$ a_{n+1} =\\left  \\lfloor \\frac{a_n} 2 \\right \\rfloor + \\left \\lfloor \\frac{a_n}3 \\right \\rfloor $$   for  $n \\in \\{1, 2, 3, 4, 5, 6, 7\\}$ , where  $\\lfloor x \\rfloor$  denotes the greatest integer not greater than  $x$ . If  $a_8=8$ , how many possible values are there for  $a_1$  given that it is a positive integer?\n\n*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 10)*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.14277,"x":0.23661,"p":[[0,123,0.0,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,123,0.0325,0.16964,0.10971,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,123,0.065,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,123,0.0976,0.17858,0.13832,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,123,0.1301,0.20982,0.16745,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[20,123,0.1626,0.16518,0.11355,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,123,0.1951,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,123,0.2276,0.16955,0.12597,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,29,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[32,123,0.2602,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,123,0.2927,0.16964,0.10374,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,123,0.3252,0.165,0.10783,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,123,0.3577,0.19643,0.14174,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[48,123,0.3902,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,123,0.4228,0.19196,0.12682,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[56,123,0.4553,0.17402,0.12747,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[60,123,0.4878,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,123,0.5203,0.19643,0.14174,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[68,123,0.5528,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,123,0.5854,0.18268,0.12504,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,28,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[76,123,0.6179,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,123,0.6504,0.16518,0.10779,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[84,123,0.6829,0.1784,0.1429,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[88,123,0.7154,0.17848,0.13834,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[92,123,0.748,0.16964,0.12595,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,29,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[96,123,0.7805,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,123,0.813,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[104,123,0.8455,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[108,123,0.878,0.16519,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[112,123,0.9106,0.23661,0.20079,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0],[116,123,0.9431,0.20089,0.13296,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,25,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[120,123,0.9756,0.19625,0.14181,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[123,123,1.0,0.17411,0.05905,0.14286,0.14286,0.14287,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.14268,"x":0.19643,"p":[[0,179,0.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,179,0.0223,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,179,0.0447,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,179,0.067,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,179,0.0894,0.16946,0.10379,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,179,0.1117,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,179,0.1341,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,179,0.1564,0.15616,0.05489,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,179,0.1788,0.17857,0.13832,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[36,179,0.2011,0.16956,0.10376,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,179,0.2235,0.17857,0.13832,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[44,179,0.2458,0.19634,0.1462,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,27,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[48,179,0.2682,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,179,0.2905,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,179,0.3128,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,179,0.3352,0.17839,0.12377,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[64,179,0.3575,0.16062,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[68,179,0.3799,0.15616,0.10327,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,179,0.4022,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,179,0.4246,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,179,0.4469,0.1875,0.14032,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[84,179,0.4693,0.15616,0.07459,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,179,0.4916,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,179,0.514,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[96,179,0.5363,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,179,0.5587,0.18304,0.1394,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[104,179,0.581,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[108,179,0.6034,0.18277,0.13948,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[112,179,0.6257,0.17839,0.13837,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[116,179,0.648,0.17848,0.13834,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[120,179,0.6704,0.16955,0.10974,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[124,179,0.6927,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,179,0.7151,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,179,0.7374,0.16054,0.09946,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[136,179,0.7598,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,179,0.7821,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,179,0.8045,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[148,179,0.8268,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,179,0.8492,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[156,179,0.8715,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[160,179,0.8939,0.19643,0.16656,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[164,179,0.9162,0.16964,0.10374,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[168,179,0.9385,0.16964,0.10374,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[172,179,0.9609,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[176,179,0.9832,0.16955,0.10376,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[179,179,1.0,0.165,0.10174,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"f5a590eebfde6ac5","q":"Denote by  $S_n$  the group of permutations of the sequence  $(1,2,\\dots,n).$  Suppose that  $G$  is a subgroup of  $S_n,$  such that for every  $\\pi\\in G\\setminus\\{e\\}$  there exists a unique  $k\\in \\{1,2,\\dots,n\\}$  for which  $\\pi(k)=k.$  (Here  $e$  is the unit element of the group  $S_n.$ ) Show that this  $k$  is the same for all  $\\pi \\in G\\setminus \\{e\\}.$","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.67409,"p":[[0,61,0.0,0.53124,0.45208,0.0,0.57121,1.0,0.0,1.0,11,14,6,11,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,14],[4,61,0.0656,0.66518,0.425,0.21429,1.0,1.0,0.0,1.0,8,17,0,8,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,17],[8,61,0.1311,0.67409,0.37327,0.49968,0.78571,1.0,0.0,1.0,5,14,0,5,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,0,2,0,14],[12,61,0.1967,0.61159,0.40913,0.21429,0.78564,1.0,0.0,1.0,8,13,0,8,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,3,0,13],[16,61,0.2623,0.6607,0.43117,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,17],[20,61,0.3279,0.50447,0.41185,0.0,0.57143,0.89286,0.0,1.0,10,8,0,10,0,1,0,0,3,0,0,0,0,0,5,0,0,0,0,0,5,0,8],[24,61,0.3934,0.29464,0.38289,0.0,0.0,0.71429,0.0,1.0,17,5,0,17,0,1,0,0,5,0,0,0,0,0,0,0,0,4,0,0,0,0,5],[28,61,0.459,0.37054,0.40855,0.0,0.28571,0.75,0.0,1.0,15,7,0,15,0,0,0,0,3,0,0,4,0,0,0,0,0,2,0,0,1,0,7],[32,61,0.5246,0.48213,0.37922,0.0,0.42859,0.85714,0.0,1.0,9,5,0,9,0,1,0,0,3,0,0,4,0,0,1,0,0,4,0,0,5,0,5],[36,61,0.5902,0.25445,0.3326,0.0,0.07143,0.46418,0.0,1.0,16,3,0,16,0,3,0,0,4,0,0,1,0,0,4,0,0,0,0,0,1,0,3],[40,61,0.6557,0.26339,0.37476,0.0,0.0,0.57143,0.0,1.0,18,5,0,18,0,3,0,0,2,0,0,0,0,0,3,0,0,1,0,0,0,0,5],[44,61,0.7213,0.27231,0.34875,0.0,0.0,0.60714,0.0,1.0,19,2,0,19,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,0,0,2],[48,61,0.7869,0.26339,0.34462,0.0,0.07143,0.42858,0.0,1.0,16,4,0,16,0,3,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,4],[52,61,0.8525,0.3125,0.35072,0.0,0.21428,0.5,0.0,1.0,14,2,0,14,0,2,0,0,4,0,0,4,0,0,0,0,0,2,0,0,4,0,2],[56,61,0.918,0.11607,0.28446,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[60,61,0.9836,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.66964,"p":[[0,41,0.0,0.66964,0.39518,0.5,0.78571,1.0,0.0,1.0,7,15,2,7,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,1,0,15],[4,41,0.0976,0.66518,0.43244,0.14289,1.0,1.0,0.0,1.0,6,19,0,6,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,19],[8,41,0.1951,0.54911,0.39141,0.28571,0.50001,1.0,0.0,1.0,7,11,0,7,0,0,0,0,5,0,0,4,0,0,2,0,0,2,0,0,1,0,11],[12,41,0.2927,0.60714,0.37796,0.28571,0.71429,1.0,0.0,1.0,6,11,0,6,0,0,0,0,4,0,0,2,0,0,2,0,0,5,0,0,2,0,11],[16,41,0.3902,0.51786,0.39082,0.0,0.71429,0.85714,0.0,1.0,9,7,0,9,0,0,0,0,4,0,0,2,0,0,0,0,0,7,0,0,3,0,7],[20,41,0.4878,0.35713,0.38132,0.0,0.28571,0.71429,0.0,1.0,13,6,0,13,0,1,0,0,6,0,0,2,0,0,1,0,0,3,0,0,0,0,6],[24,41,0.5854,0.21875,0.28118,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,1,0,0,8,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[28,41,0.6829,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.07589,0.17122,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac5a5088ee43a545","q":"Consider a polynomial  $P(x) = ax^2 + bx + c$  with  $a > 0$  that has two real roots  $x_1, x_2$ . Prove that the absolute values of both roots are less than or equal to  $1$  if and only if  $a + b + c \\ge 0, a -b + c \\ge 0$ , and  $a - c \\ge 0$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.82589,"x":0.86607,"p":[[0,9,0.0,0.86607,0.13333,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,15],[4,9,0.4444,0.86607,0.12846,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,6,0,14],[8,9,0.8889,0.83036,0.13092,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,11],[9,9,1.0,0.82589,0.13236,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,11]]},{"b":3,"e":0.85714,"k":"flat","v":0.83937,"x":0.91964,"p":[[0,42,0.0,0.87946,0.12428,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,7,0,15],[4,42,0.0952,0.91964,0.11811,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[8,42,0.1905,0.87054,0.13054,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,5,0,15],[12,42,0.2857,0.83937,0.12754,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,6,0,11],[16,42,0.381,0.91071,0.13716,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[20,42,0.4762,0.85713,0.12878,0.71429,0.85714,1.0,0.714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,6,0,13],[24,42,0.5714,0.87946,0.12931,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,5,0,16],[28,42,0.6667,0.85254,0.12634,0.71429,0.85714,1.0,0.71,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,7,0,12],[32,42,0.7619,0.84375,0.13533,0.71429,0.78571,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,3,0,13],[36,42,0.8571,0.85268,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,13],[40,42,0.9524,0.86607,0.13333,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,15],[42,42,1.0,0.84821,0.12846,0.71429,0.85707,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,12]]}]},{"i":"99b1a1fce8d25c34","q":"Determine all functions  $f : \\mathbb{R} \\to \\mathbb{R}$  that satisfy the following two properties.\n(i) The Riemann integral  $\\int_a^b f(t) \\mathrm dt$  exists for all real numbers  $a < b$ .\n(ii) For every real number  $x$  and every integer  $n \\ge 1$  we have\n\\[ f(x) = \\frac{n}{2} \\int_{x-\\frac{1}{n}}^{x+\\frac{1}{n}} f(t) \\mathrm dt. \\]","t":[{"b":5,"e":1.0,"k":"rising","v":0.45982,"x":0.89277,"p":[[0,88,0.0,0.45982,0.43262,0.0,0.35714,1.0,0.0,1.0,10,11,0,10,0,5,0,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,11],[4,88,0.0455,0.73212,0.30672,0.5354,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,1,0,0,3,0,0,3,0,0,3,0,0,4,0,14],[8,88,0.0909,0.72767,0.34508,0.42857,0.92857,1.0,0.0,1.0,2,16,0,2,0,3,0,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,16],[12,88,0.1364,0.82587,0.25937,0.82132,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,8,0,16],[16,88,0.1818,0.69633,0.31708,0.53572,0.857,1.0,0.0,1.0,2,10,0,2,0,2,0,0,2,0,0,2,0,0,3,0,0,4,0,0,7,0,10],[20,88,0.2273,0.73659,0.31966,0.57143,0.78571,1.0,0.0,1.0,3,15,0,3,0,0,0,0,2,0,0,1,0,0,3,0,0,7,0,0,1,0,15],[24,88,0.2727,0.81473,0.33065,0.85714,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,1,20],[28,88,0.3182,0.77229,0.31108,0.57132,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,18],[32,88,0.3636,0.7098,0.3164,0.57132,0.71429,1.0,0.0,1.0,3,13,0,3,0,0,0,0,2,0,0,2,0,0,3,0,0,8,0,0,1,0,13],[36,88,0.4091,0.73214,0.36202,0.5,1.0,1.0,0.0,1.0,2,18,0,2,0,4,0,0,2,0,0,0,0,0,1,0,0,4,0,0,1,0,18],[40,88,0.4545,0.70522,0.31326,0.53572,0.71429,1.0,0.0,1.0,3,13,0,3,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,0,0,13],[44,88,0.5,0.76784,0.33835,0.67857,1.0,1.0,0.0,1.0,2,18,0,2,0,3,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,18],[48,88,0.5455,0.74988,0.30113,0.571,0.85714,1.0,0.14,1.0,0,14,0,0,0,4,0,0,1,0,0,2,0,0,2,0,0,4,0,0,5,0,14],[52,88,0.5909,0.69196,0.35012,0.42859,0.78564,1.0,0.0,1.0,3,14,0,3,0,2,0,0,2,0,0,2,0,0,2,0,0,5,0,0,2,0,14],[56,88,0.6364,0.79016,0.27891,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,2,0,17],[60,88,0.6818,0.86607,0.24468,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,3,0,21],[64,88,0.7273,0.79464,0.26471,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,5,0,0,2,0,17],[68,88,0.7727,0.79908,0.30277,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,2,0,0,0,0,0,2,0,0,4,0,0,2,0,19],[72,88,0.8182,0.79888,0.29227,0.67536,1.0,1.0,0.14,1.0,0,20,0,0,0,2,0,0,3,0,0,1,0,0,2,0,0,4,0,0,0,0,20],[76,88,0.8636,0.875,0.17768,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,5,0,18],[80,88,0.9091,0.8616,0.26842,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,0,3,0,23],[84,88,0.9545,0.75,0.37458,0.60714,1.0,1.0,0.0,1.0,3,20,0,3,0,3,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,20],[88,88,1.0,0.89277,0.21755,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,23]]},{"b":6,"e":1.0,"k":"rising","v":0.25446,"x":0.96875,"p":[[0,49,0.0,0.25446,0.35307,0.0,0.14286,0.32143,0.0,1.0,15,5,0,15,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,5],[4,49,0.0816,0.83482,0.29038,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,1,0,22],[8,49,0.1633,0.93302,0.15149,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,26],[12,49,0.2449,0.89286,0.24744,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,23],[16,49,0.3265,0.8704,0.21247,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,2,0,21],[20,49,0.4082,0.86383,0.22753,0.82132,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,1,21],[24,49,0.4898,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[28,49,0.5714,0.85714,0.24223,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,22],[32,49,0.6531,0.85714,0.28794,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,25],[36,49,0.7347,0.82143,0.27894,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,0,2,0,20],[40,49,0.8163,0.89732,0.18977,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,0,0,24],[44,49,0.898,0.86607,0.24727,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[48,49,0.9796,0.9374,0.19578,1.0,1.0,1.0,0.14,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[49,49,1.0,0.89286,0.22304,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,24]]}]},{"i":"d238aad32ff336e0","q":"Determine all real number  $(x,y)$  pairs that satisfy the equation.   $$ 2x^2+y^2+7=2(x+1)(y+1) $$","t":[{"b":1,"e":1.0,"k":"flat","v":0.91071,"x":0.99107,"p":[[0,9,0.0,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[4,9,0.4444,0.91071,0.1915,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,25],[8,9,0.8889,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[9,9,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.94196,"x":0.98661,"p":[[0,10,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,10,0.4,0.94196,0.17807,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[8,10,0.8,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[10,10,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"323ee3ce3321bff6","q":"Determine all pairs of prime numbers  $p$  and  $q$  greater than  $1$  and less than  $100$ , such that the following five numbers:   $$ p+6,p+10,q+4,q+10,p+q+1, $$   are all prime numbers.","t":[{"b":4,"e":1.0,"k":"flat","v":0.89732,"x":1.0,"p":[[0,68,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,68,0.0588,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,68,0.1176,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,68,0.1765,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,68,0.2353,0.89732,0.26543,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,25],[20,68,0.2941,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[24,68,0.3529,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,68,0.4118,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,68,0.4706,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,68,0.5294,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,68,0.5882,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,68,0.6471,0.95089,0.11071,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[48,68,0.7059,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[52,68,0.7647,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[56,68,0.8235,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,68,0.8824,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[64,68,0.9412,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[68,68,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,51,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,51,0.0784,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,51,0.1569,0.92857,0.14286,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[12,51,0.2353,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[16,51,0.3137,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,51,0.3922,0.92857,0.12372,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,21],[24,51,0.4706,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[28,51,0.549,0.95534,0.09067,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[32,51,0.6275,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,51,0.7059,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,51,0.7843,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,51,0.8627,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[48,51,0.9412,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[51,51,1.0,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19]]}]},{"i":"cb8762801ec26f7c","q":"Determine all integers $n \\geqslant 2$ satisfying the following property: for all integers $a_{1}, a_{2}, \\ldots, a_{n}$ whose sum is not divisible by $n$, there exists an index $i$ such that none of the numbers\n\n$$\na_{i}, a_{i}+a_{i+1}, \\ldots, a_{i}+\\cdots+a_{i+n-1}\n$$\n\nis divisible by $n$ (for $i>n$, we set $a_{i}=a_{i-n}$ ).","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.13839,"p":[[0,45,0.0,0.13839,0.13592,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.10714,0.24744,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[8,45,0.1778,0.06701,0.175,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,45,0.2667,0.10713,0.2422,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[16,45,0.3556,0.0625,0.17474,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[20,45,0.4444,0.05357,0.15047,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,45,0.5333,0.06697,0.13825,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,45,0.7111,0.03572,0.09449,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.02231,0.10163,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,45,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.20536,"p":[[0,55,0.0,0.14284,0.14282,0.0,0.14286,0.2857,0.0,0.571,13,0,0,13,0,8,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,55,0.0727,0.1607,0.26424,0.0,0.0,0.32142,0.0,0.85714,22,0,0,22,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,0,1,0,0],[8,55,0.1455,0.05348,0.15041,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,55,0.2182,0.0625,0.13333,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,55,0.2909,0.20536,0.28333,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,0,0,0,8,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[20,55,0.3636,0.07589,0.20511,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[24,55,0.4364,0.06697,0.15966,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,55,0.5091,0.12051,0.23171,0.0,0.0,0.1786,0.0,1.0,23,1,0,23,0,1,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[32,55,0.5818,0.06687,0.12864,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.0625,0.13333,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,55,0.8,0.04464,0.15335,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,55,0.8727,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b9a069ccc232a04c","q":"Consider all 1000-element subsets of the set  $\\{1,2,3,\\dots,2015\\}$ . From each such subset choose the least element. The arithmetic mean of all of these least elements is  $\\tfrac{p}{q}$ , where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,47,0.0,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[4,47,0.0851,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,47,0.1702,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,47,0.2553,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,47,0.3404,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[20,47,0.4255,0.93304,0.15561,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,26],[24,47,0.5106,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,47,0.5957,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[32,47,0.6809,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,47,0.766,0.95982,0.1394,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[47,47,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.875,"x":0.97321,"p":[[0,36,0.0,0.92857,0.16366,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,26],[4,36,0.1111,0.89732,0.22084,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,25],[8,36,0.2222,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,36,0.3333,0.90179,0.16917,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,2,0,22],[16,36,0.4444,0.87946,0.20238,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,0,0,22],[20,36,0.5556,0.88839,0.21939,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,0,0,24],[24,36,0.6667,0.90179,0.19704,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,1,0,24],[28,36,0.7778,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,36,0.8889,0.875,0.21943,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,6,0,0,1,0,22],[36,36,1.0,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29]]}]},{"i":"a90201c45cfb1adb","q":"Determine all positive integer numbers  $n$  satisfying the following condition:\n the sum of the squares of any  $n$  prime numbers greater than  $3$  is divisible by  $n$ .","t":[{"b":0,"e":0.57143,"k":"flat","v":0.70982,"x":0.9241,"p":[[0,34,0.0,0.81694,0.17944,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,8,0,12],[4,34,0.1176,0.86607,0.1234,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,19,0,9],[8,34,0.2353,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[12,34,0.3529,0.9241,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21],[16,34,0.4706,0.82588,0.22229,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,9,0,14],[20,34,0.5882,0.79464,0.2111,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,7,0,12],[24,34,0.7059,0.77679,0.19212,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,4,0,0,7,0,10],[28,34,0.8235,0.83927,0.17037,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,4,0,15],[32,34,0.9412,0.70982,0.17307,0.57143,0.57143,0.85704,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,6,0,0,2,0,7],[34,34,1.0,0.74997,0.20205,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,5,0,0,0,0,12]]},{"b":7,"e":0.857,"k":"flat","v":0.82143,"x":0.91071,"p":[[0,24,0.0,0.82143,0.17857,0.71429,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,6,0,13],[4,24,0.1667,0.83927,0.17407,0.85711,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,14,0,11],[8,24,0.3333,0.89286,0.10102,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,12],[12,24,0.5,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[16,24,0.6667,0.88838,0.08557,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,9],[20,24,0.8333,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[24,24,1.0,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3]]}]},{"i":"81e8f93d7744ff82","q":"Consider the  $2015$  integers  $n$ , from  $ 1$  to  $2015$ . Determine for how many values \u200b\u200bof  $n$  it is verified that the number  $n^3 + 3^n$  is a multiple of  $5$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.69642,"x":0.72321,"p":[[0,71,0.0,0.70534,0.04975,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0],[4,71,0.0563,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[8,71,0.1127,0.70969,0.02484,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[12,71,0.169,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[16,71,0.2254,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[20,71,0.2817,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[24,71,0.338,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[28,71,0.3944,0.71861,0.02487,0.71429,0.71429,0.71429,0.71,0.857,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0],[32,71,0.4507,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[36,71,0.507,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[40,71,0.5634,0.72321,0.03458,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0],[44,71,0.6197,0.71875,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0],[48,71,0.6761,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[52,71,0.7324,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[56,71,0.7887,0.72321,0.03458,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0],[60,71,0.8451,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[64,71,0.9014,0.69642,0.04728,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[68,71,0.9577,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[71,71,1.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.69642,"x":0.73214,"p":[[0,95,0.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[4,95,0.0421,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[8,95,0.0842,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[12,95,0.1263,0.72321,0.03456,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0],[16,95,0.1684,0.71429,0.03571,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0],[20,95,0.2105,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[24,95,0.2526,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[28,95,0.2947,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[32,95,0.3368,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,95,0.3789,0.71429,0.03571,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0],[40,95,0.4211,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[44,95,0.4632,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[48,95,0.5053,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[52,95,0.5474,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[56,95,0.5895,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[60,95,0.6316,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[64,95,0.6737,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[68,95,0.7158,0.70969,0.02484,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[72,95,0.7579,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[76,95,0.8,0.69642,0.06918,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,1,0,0],[80,95,0.8421,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[84,95,0.8842,0.71875,0.02486,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0],[88,95,0.9263,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[92,95,0.9684,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[95,95,1.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0]]}]},{"i":"5ccffa6174dc562a","q":"Define a sequence $\\left\\langle a_{n}\\right\\rangle_{n=1}^{\\infty}$ as follows:\n\n$$\na_{n}= \\begin{cases}0, & \\text { if the number of positive divisors of } n \\text { is odd } \\\\ 1, & \\text { if the number of positive divisors of } n \\text { is even }\\end{cases}\n$$\n\n(The positive divisors of $n$ include 1 as well as $n$.) Let $x=0 . a_{1} a_{2} a_{3} \\ldots$ be the real number whose decimal expansion contains $a_{n}$ in the $n$-th place, $n \\geq 1$. Determine, with proof, whether $x$ is rational or irrational.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.63838,"x":0.80803,"p":[[0,54,0.0,0.67409,0.24804,0.42857,0.57143,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,8,0,0,8,0,0,3,0,0,1,0,10],[4,54,0.0741,0.75446,0.2237,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,5,0,0,3,0,12],[8,54,0.1481,0.73214,0.23623,0.42859,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,4,0,0,4,0,11],[12,54,0.2222,0.64731,0.21424,0.42857,0.57143,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,7,0,0,7,0,0,0,0,7],[16,54,0.2963,0.74553,0.19799,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,5,0,0,7,0,8],[20,54,0.3704,0.65625,0.20781,0.42857,0.57143,0.74996,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,7,0,0,2,0,6],[24,54,0.4444,0.70088,0.2212,0.53539,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,6,0,0,2,0,9],[28,54,0.5185,0.63838,0.18893,0.42857,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,10,0,0,6,0,0,3,0,4],[32,54,0.5926,0.79463,0.20498,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,4,0,0,5,0,13],[36,54,0.6667,0.65177,0.20497,0.42857,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,8,0,0,5,0,0,4,0,5],[40,54,0.7407,0.6875,0.21852,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,4,0,0,1,0,9],[44,54,0.8148,0.79019,0.20815,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,7,0,12],[48,54,0.8889,0.80803,0.21011,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,6,0,14],[52,54,0.963,0.7232,0.21707,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,6,0,0,0,0,11],[54,54,1.0,0.75892,0.20342,0.57143,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,3,0,0,6,0,10]]},{"b":5,"e":0.42857,"k":"flat","v":0.47768,"x":0.71429,"p":[[0,59,0.0,0.66964,0.23538,0.42857,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,10,0,0,6,0,0,4,0,0,3,0,8],[4,59,0.0678,0.71429,0.23958,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,4,0,0,6,0,9],[8,59,0.1356,0.47768,0.16602,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,16,0,0,2,0,0,5,0,0,2,0,0],[12,59,0.2034,0.61607,0.15335,0.57142,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,14,0,0,7,0,0,2,0,2],[16,59,0.2712,0.51784,0.12242,0.42857,0.57121,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,13,0,0,13,0,0,3,0,0,1,0,0],[20,59,0.339,0.59819,0.11539,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,0,11,0,0,1,0,0],[24,59,0.4068,0.58481,0.16888,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,7,0,0,8,0,0,3,0,1],[28,59,0.4746,0.55802,0.15303,0.42857,0.57141,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,11,0,0,10,0,0,6,0,0,3,0,0],[32,59,0.5424,0.55801,0.17627,0.42857,0.571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,12,0,0,6,0,0,8,0,0,2,0,1],[36,59,0.6102,0.62051,0.20705,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,12,0,0,9,0,0,5,0,0,0,0,6],[40,59,0.678,0.61159,0.16457,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,9,0,0,7,0,0,11,0,0,3,0,1],[44,59,0.7458,0.55357,0.12753,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,14,0,0,9,0,0,8,0,0,1,0,0],[48,59,0.8136,0.49554,0.15561,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,17,0,0,5,0,0,5,0,0,0,0,1],[52,59,0.8814,0.56696,0.2004,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,11,0,0,6,0,0,6,0,0,3,0,2],[56,59,0.9492,0.60268,0.1665,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,10,0,0,5,0,0,5,0,1],[59,59,1.0,0.57589,0.19061,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,14,0,0,8,0,0,4,0,0,2,0,3]]}]},{"i":"8564bb9cd3866384","q":"Define a sequence $\\left(n_{k}\\right)_{k \\geq 0}$ by $n_{0}=n_{1}=1$, and $n_{2 k}=n_{k}+n_{k-1}$ and $n_{2 k+1}=n_{k}$ for $k \\geq 1$. Let further $q_{k}=n_{k} / n_{k-1}$ for each $k \\geq 1$. Show that every positive rational number is present exactly once in the sequence $\\left(q_{k}\\right)_{k \\geq 1}$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.96428,"x":1.0,"p":[[0,25,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,25,0.16,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,25,0.32,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,25,0.48,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,25,0.64,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,33,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,33,0.1212,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,33,0.2424,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,33,0.3636,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,33,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,33,0.6061,0.98214,0.04725,1.0,1.0,1.0,0.8571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,33,0.9697,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[33,33,1.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"05669775b0b12fb4","q":"Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?","t":[{"b":0,"e":0.28571,"k":"flat","v":0.54464,"x":0.85267,"p":[[0,13,0.0,0.54464,0.33396,0.2857,0.64286,0.85714,0.0,1.0,2,4,0,2,0,4,0,0,9,0,0,0,0,0,1,0,0,4,0,0,8,0,4],[4,13,0.3077,0.84821,0.20806,0.85711,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,13],[8,13,0.6154,0.85267,0.11564,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,14,0,9],[12,13,0.9231,0.68749,0.27994,0.28571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,8,0,0,8,0,7],[13,13,1.0,0.6741,0.27254,0.28571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,8,0,0,0,0,0,1,0,0,7,0,0,10,0,5]]},{"b":3,"e":0.71429,"k":"rising","v":0.56695,"x":0.884,"p":[[0,14,0.0,0.56695,0.31234,0.28571,0.71429,0.85714,0.0,1.0,2,4,0,2,0,3,0,0,7,0,0,1,0,0,1,0,0,9,0,0,5,0,4],[4,14,0.2857,0.83929,0.14617,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,13,0,9],[8,14,0.5714,0.87053,0.12037,0.82132,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,12],[12,14,0.8571,0.884,0.1208,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,11,0,14],[14,14,1.0,0.84375,0.10926,0.71429,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,16,0,7]]}]},{"i":"74aabd6fe63156ed","q":"Assume we can fill a table  $n\\times n$  with all numbers  $1,2,\\ldots,n^2-1,n^2$  in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers  $n$ .","t":[{"b":6,"e":0.28571,"k":"flat","v":0.21428,"x":0.35268,"p":[[0,75,0.0,0.21428,0.11845,0.14286,0.28571,0.28571,0.0,0.42857,6,0,6,6,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.35268,0.18893,0.2857,0.28571,0.32164,0.14286,1.0,0,1,0,0,0,4,0,0,20,0,0,2,0,0,4,0,0,0,0,0,1,0,1],[8,75,0.1067,0.2634,0.08073,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,7,0,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,75,0.16,0.27232,0.1488,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,9,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,75,0.2133,0.24998,0.09444,0.14286,0.28571,0.28571,0.0,0.571,1,0,0,1,0,8,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,75,0.2667,0.33482,0.16602,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,4,0,0,22,0,0,1,0,0,2,0,0,2,0,0,1,0,0],[24,75,0.32,0.33035,0.20023,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,19,0,0,1,0,0,2,0,0,1,0,0,1,0,1],[28,75,0.3733,0.28125,0.07563,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,75,0.4267,0.29018,0.05629,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,75,0.48,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.28571,0.07143,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,75,0.6933,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,75,0.7467,0.27233,0.04164,0.28571,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.27678,0.03457,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.27232,0.04164,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"rising","v":0.21428,"x":0.45536,"p":[[0,21,0.0,0.21428,0.10101,0.14286,0.28571,0.28571,0.0,0.28571,4,0,3,4,0,8,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.27678,0.1234,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,9,0,0,19,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,21,0.381,0.29911,0.13997,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,5,0,0,24,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[12,21,0.5714,0.29463,0.10058,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,25,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,21,0.7619,0.3125,0.12595,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,27,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[20,21,0.9524,0.33928,0.21943,0.2857,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,4,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[21,21,1.0,0.45536,0.31831,0.28571,0.28571,0.57143,0.14286,1.0,0,8,0,0,0,3,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,8]]}]},{"i":"f72bcb8753f70b22","q":"Beto plays the following game with his computer: initially the computer randomly picks  $30$  integers from  $1$  to  $2015$ , and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer  $k$  and some if the numbers written on the chalkboard, and subtracts  $k$  from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all  $30$  numbers to  $0$ , in which case the game ends. Find the minimal number  $n$  such that, regardless of which numbers the computer chooses, Beto can end the game in at most  $n$  turns.","t":[{"b":1,"e":0.42857,"k":"rising","v":0.52679,"x":0.89282,"p":[[0,20,0.0,0.52679,0.32818,0.25,0.57143,0.71429,0.0,1.0,7,4,5,7,0,1,0,0,1,0,0,0,0,0,8,0,0,11,0,0,0,0,4],[4,20,0.2,0.89282,0.2202,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,24],[8,20,0.4,0.8125,0.25614,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,2,0,18],[12,20,0.6,0.73213,0.27375,0.4286,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,8,0,0,5,0,0,1,0,0,4,0,13],[16,20,0.8,0.75893,0.30396,0.4286,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,7,0,0,2,0,0,1,0,0,4,0,16],[20,20,1.0,0.7991,0.21087,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,5,0,0,8,0,12]]},{"b":7,"e":1.0,"k":"rising","v":0.45534,"x":0.95089,"p":[[0,31,0.0,0.45534,0.38372,0.0,0.57143,0.71429,0.0,1.0,12,6,6,12,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,0,0,0,6],[4,31,0.129,0.87945,0.18597,0.82143,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,3,0,21],[8,31,0.2581,0.87499,0.21055,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,0,0,23],[12,31,0.3871,0.85268,0.22724,0.67857,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,0,0,22],[16,31,0.5161,0.87052,0.20936,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,22],[20,31,0.6452,0.81246,0.22433,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,1,0,0,1,0,18],[24,31,0.7742,0.81696,0.24546,0.67836,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,4,0,0,1,0,19],[28,31,0.9032,0.94194,0.12815,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[31,31,1.0,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27]]}]},{"i":"8d1a14ac240f0336","q":"Are there different integers  $a,b,c,d,e,f$  such that they are the  $6$  roots of  $$ (x+a)(x^2+bx+c)(x^3+dx^2+ex+f)=0? $$","t":[{"b":2,"e":0.571,"k":"flat","v":0.40179,"x":0.86607,"p":[[0,196,0.0,0.41963,0.28779,0.28571,0.42857,0.57143,0.0,1.0,7,2,0,7,0,0,0,0,4,0,0,10,0,0,5,0,0,2,0,0,2,0,2],[4,196,0.0204,0.86607,0.20497,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,19],[8,196,0.0408,0.78572,0.28121,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,0,0,0,2,0,0,2,0,0,5,0,0,5,0,15],[12,196,0.0612,0.75444,0.25812,0.57143,0.857,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,7,0,10],[16,196,0.0816,0.77232,0.21086,0.67857,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,5,0,0,11,0,8],[20,196,0.102,0.79462,0.24985,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,5,0,0,8,0,13],[24,196,0.1224,0.81247,0.20655,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,9,0,11],[28,196,0.1429,0.75892,0.27534,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,7,0,12],[32,196,0.1633,0.80355,0.19479,0.71429,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,12,0,9],[36,196,0.1837,0.71428,0.26486,0.53572,0.857,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,1,0,0,5,0,0,2,0,0,5,0,0,10,0,7],[40,196,0.2041,0.66071,0.3067,0.4286,0.71429,0.85714,0.0,1.0,3,6,0,3,0,1,0,0,1,0,0,4,0,0,3,0,0,5,0,0,9,0,6],[44,196,0.2245,0.74106,0.26592,0.57143,0.78571,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,6,0,10],[48,196,0.2449,0.80357,0.24157,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,4,0,0,7,0,14],[52,196,0.2653,0.76336,0.21013,0.57143,0.85707,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,5,0,0,10,0,8],[56,196,0.2857,0.73213,0.2918,0.57143,0.78571,1.0,0.0,1.0,2,12,0,2,0,0,0,0,3,0,0,0,0,0,5,0,0,6,0,0,4,0,12],[60,196,0.3061,0.62498,0.27837,0.42859,0.57143,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,2,0,0,7,0,0,7,0,0,4,0,0,3,0,7],[64,196,0.3265,0.61607,0.24074,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,9,0,0,7,0,0,3,0,0,3,0,6],[68,196,0.3469,0.69643,0.26183,0.57142,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,0,8,0,7],[72,196,0.3673,0.68301,0.26181,0.57143,0.71429,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,0,0,0,2,0,0,7,0,0,7,0,0,8,0,5],[76,196,0.3878,0.67852,0.27665,0.5354,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,6,0,0,1,0,0,6,0,0,6,0,0,2,0,10],[80,196,0.4082,0.60268,0.34761,0.28571,0.64286,0.89286,0.0,1.0,4,8,0,4,0,2,0,0,3,0,0,2,0,0,5,0,0,3,0,0,5,0,8],[84,196,0.4286,0.72765,0.21831,0.571,0.78564,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,10,0,6],[88,196,0.449,0.60266,0.25935,0.4286,0.71429,0.74996,0.0,1.0,2,2,0,2,0,1,0,0,3,0,0,3,0,0,6,0,0,9,0,0,6,0,2],[92,196,0.4694,0.6741,0.26543,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,7,0,0,6,0,7],[96,196,0.4898,0.65177,0.30291,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,3,0,0,4,0,0,4,0,0,2,0,0,7,0,8],[100,196,0.5102,0.74552,0.24153,0.67857,0.78564,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,9,0,7],[104,196,0.5306,0.66963,0.29329,0.5354,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,2,0,0,3,0,0,5,0,0,6,0,0,5,0,8],[108,196,0.551,0.63833,0.29879,0.5354,0.57143,1.0,0.0,1.0,2,9,0,2,0,1,0,0,3,0,0,2,0,0,10,0,0,3,0,0,2,0,9],[112,196,0.5714,0.6964,0.228,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,9,0,0,4,0,7],[116,196,0.5918,0.65624,0.23653,0.4286,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,6,0,0,7,0,0,6,0,0,4,0,6],[120,196,0.6122,0.50442,0.32923,0.24999,0.57121,0.85704,0.0,1.0,5,3,0,5,0,3,0,0,4,0,0,1,0,0,8,0,0,2,0,0,6,0,3],[124,196,0.6327,0.62945,0.28763,0.39286,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,6,0,0,3,0,0,4,0,0,5,0,0,6,0,6],[128,196,0.6531,0.57142,0.24483,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,3,0,0,5,0,0,9,0,0,5,0,0,5,0,2],[132,196,0.6735,0.6161,0.24853,0.42964,0.57143,0.74996,0.0,1.0,2,4,0,2,0,0,0,0,1,0,0,6,0,0,9,0,0,6,0,0,4,0,4],[136,196,0.6939,0.66513,0.27109,0.571,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,3,0,0,2,0,0,5,0,0,9,0,0,5,0,6],[140,196,0.7143,0.62047,0.23314,0.571,0.57143,0.74999,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,3,0,0,11,0,0,6,0,0,5,0,3],[144,196,0.7347,0.64729,0.24996,0.4286,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,6,0,0,4,0,6],[148,196,0.7551,0.64722,0.29463,0.4286,0.71429,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,1,0,0,5,0,0,2,0,0,8,0,0,6,0,6],[152,196,0.7755,0.65623,0.21975,0.4286,0.71429,0.85704,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,8,0,0,4,0,5],[156,196,0.7959,0.55347,0.24694,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,2,0,0,8,0,0,7,0,0,8,0,0,1,0,3],[160,196,0.8163,0.51783,0.19149,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,10,0,0,8,0,0,7,0,0,2,0,0],[164,196,0.8367,0.54464,0.22142,0.42857,0.57143,0.60714,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,10,0,0,9,0,0,2,0,0,4,0,2],[168,196,0.8571,0.45533,0.25861,0.28571,0.4286,0.60714,0.0,1.0,4,1,0,4,0,1,0,0,6,0,0,7,0,0,6,0,0,5,0,0,2,0,1],[172,196,0.8776,0.53569,0.18898,0.42857,0.571,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,10,0,0,8,0,0,6,0,0,2,0,1],[176,196,0.898,0.50432,0.23966,0.39286,0.571,0.60714,0.0,1.0,2,1,0,2,0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0,0,4,0,0,2,0,0,5,0,0,10,0,9],[160,253,0.6324,0.71875,0.24868,0.57143,0.85707,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,5,0,0,11,0,6],[164,253,0.6482,0.70535,0.26709,0.571,0.71429,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,0,0,0,2,0,0,5,0,0,7,0,0,9,0,6],[168,253,0.664,0.76339,0.24119,0.71429,0.85707,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,9,0,0,8,0,9],[172,253,0.6798,0.74996,0.2287,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,8,0,0,5,0,10],[176,253,0.6957,0.71875,0.23551,0.53572,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,8,0,0,6,0,8],[180,253,0.7115,0.66516,0.26392,0.42857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,3,0,0,6,0,0,4,0,0,6,0,0,5,0,7],[184,253,0.7273,0.67408,0.27947,0.57143,0.71429,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,1,0,0,1,0,0,4,0,0,8,0,0,10,0,4],[188,253,0.7431,0.70979,0.25377,0.57132,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,0,7,0,7],[192,253,0.7589,0.68303,0.23618,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,5,0,0,6,0,0,8,0,0,5,0,6],[196,253,0.7747,0.73659,0.26752,0.5713,0.78564,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,5,0,0,4,0,12],[200,253,0.7905,0.72318,0.22852,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,7,0,0,7,0,7],[204,253,0.8063,0.70088,0.28203,0.57143,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,3,0,0,1,0,0,5,0,0,6,0,0,7,0,8],[208,253,0.8221,0.71872,0.23552,0.57142,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,3,0,0,6,0,0,7,0,0,7,0,7],[212,253,0.8379,0.66516,0.26632,0.42857,0.71429,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,0,0,0,6,0,0,2,0,0,7,0,0,11,0,3],[216,253,0.8538,0.73659,0.1825,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,11,0,0,6,0,6],[220,253,0.8696,0.69642,0.23623,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,9,0,0,6,0,6],[224,253,0.8854,0.6696,0.29761,0.4286,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,5,0,0,2,0,0,5,0,0,5,0,0,3,0,10],[228,253,0.9012,0.65625,0.25965,0.53572,0.71429,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,1,0,0,4,0,0,5,0,0,7,0,0,8,0,4],[232,253,0.917,0.60711,0.28122,0.39286,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,5,0,0,3,0,0,4,0,0,8,0,0,4,0,5],[236,253,0.9328,0.75445,0.23482,0.67857,0.85707,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,12,0,7],[240,253,0.9486,0.66514,0.27574,0.5354,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,3,0,0,3,0,0,4,0,0,8,0,0,6,0,6],[244,253,0.9644,0.70531,0.21708,0.57132,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,3,0,0,6,0,0,6,0,0,12,0,3],[248,253,0.9802,0.64727,0.21426,0.571,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,3,0,0,9,0,0,8,0,0,7,0,2],[252,253,0.996,0.61159,0.25059,0.42859,0.57143,0.85704,0.0,1.0,1,3,0,1,0,1,0,0,4,0,0,3,0,0,8,0,0,6,0,0,6,0,3],[253,253,1.0,0.54015,0.18115,0.42857,0.57121,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,10,0,0,11,0,0,2,0,0,5,0,0]]}]},{"i":"081e8f527520d66b","q":"Determine the greatest positive integer \\(n\\) for which there exists a sequence of distinct positive integers \\(s_1\\), \\(s_2\\), \\(\\ldots\\), \\(s_n\\) satisfying \\[s_1^{s_2}=s_2^{s_3}=\\cdots=s_{n-1}^{s_n}.\\]\n\n*Proposed by Holden Mui*","t":[{"b":2,"e":0.28571,"k":"flat","v":0.1875,"x":0.558,"p":[[0,61,0.0,0.1875,0.20652,0.0,0.14286,0.28571,0.0,0.71429,14,0,12,14,0,5,0,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,61,0.0656,0.53124,0.30563,0.28571,0.57143,0.75,0.0,1.0,3,3,0,3,0,4,0,0,2,0,0,5,0,0,5,0,0,5,0,0,5,0,3],[8,61,0.1311,0.37944,0.35464,0.10714,0.28571,0.57143,0.0,1.0,8,5,0,8,0,5,0,0,7,0,0,1,0,0,4,0,0,0,0,0,2,0,5],[12,61,0.1967,0.558,0.30797,0.28571,0.57143,0.85704,0.0,1.0,4,3,0,4,0,1,0,0,4,0,0,3,0,0,5,0,0,6,0,0,6,0,3],[16,61,0.2623,0.46875,0.32189,0.14289,0.42859,0.71429,0.0,1.0,5,3,0,5,0,4,0,0,3,0,0,5,0,0,5,0,0,3,0,0,4,0,3],[20,61,0.3279,0.46428,0.35714,0.14286,0.42857,0.75,0.0,1.0,7,4,0,7,0,4,0,0,2,0,0,5,0,0,1,0,0,5,0,0,4,0,4],[24,61,0.3934,0.50893,0.37276,0.14286,0.5,0.85714,0.0,1.0,6,6,0,6,0,4,0,0,3,0,0,3,0,0,2,0,0,3,0,0,5,0,6],[28,61,0.459,0.46428,0.31944,0.24999,0.42857,0.71429,0.0,1.0,5,2,0,5,0,3,0,0,6,0,0,4,0,0,2,0,0,5,0,0,5,0,2],[32,61,0.5246,0.37052,0.34042,0.14286,0.28571,0.60714,0.0,1.0,7,3,0,7,0,7,0,0,6,0,0,2,0,0,2,0,0,1,0,0,4,0,3],[36,61,0.5902,0.31686,0.2896,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,6,0,0,4,0,0,3,0,0,5,0,0,3,0,0,1,0,1],[40,61,0.6557,0.4107,0.33834,0.10714,0.35714,0.71429,0.0,1.0,8,2,0,8,0,4,0,0,4,0,0,2,0,0,4,0,0,4,0,0,4,0,2],[44,61,0.7213,0.25893,0.27994,0.0,0.14286,0.42858,0.0,0.85714,13,0,0,13,0,5,0,0,2,0,0,5,0,0,3,0,0,2,0,0,2,0,0],[48,61,0.7869,0.34363,0.32317,0.0,0.28571,0.57111,0.0,1.0,10,3,0,10,0,4,0,0,3,0,0,5,0,0,5,0,0,1,0,0,1,0,3],[52,61,0.8525,0.24097,0.26351,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,10,0,0,7,0,0,1,0,0,2,0,0,0,0,0,2,0,1],[56,61,0.918,0.25446,0.29175,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,9,0,0,3,0,0,3,0,0,1,0,0,2,0,0,2,0,1],[60,61,0.9836,0.1875,0.2299,0.0,0.14286,0.2857,0.0,0.85714,13,0,0,13,0,9,0,0,4,0,0,3,0,0,0,0,0,2,0,0,1,0,0],[61,61,1.0,0.23214,0.14617,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,11,0,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.10268,"x":0.61157,"p":[[0,79,0.0,0.18303,0.20277,0.0,0.07143,0.32143,0.0,0.71429,16,0,16,16,0,1,0,0,7,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[4,79,0.0506,0.45981,0.32874,0.14286,0.35714,0.71429,0.0,1.0,3,4,0,3,0,7,0,0,6,0,0,2,0,0,3,0,0,4,0,0,3,0,4],[8,79,0.1013,0.60713,0.30723,0.28571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,4,0,0,1,0,0,3,0,0,9,0,0,5,0,5],[12,79,0.1519,0.56248,0.30916,0.28571,0.64286,0.71429,0.0,1.0,4,4,0,4,0,1,0,0,5,0,0,0,0,0,6,0,0,9,0,0,3,0,4],[16,79,0.2025,0.40625,0.32754,0.10714,0.35714,0.71429,0.0,1.0,8,1,0,8,0,3,0,0,5,0,0,3,0,0,3,0,0,4,0,0,5,0,1],[20,79,0.2532,0.5491,0.30745,0.28571,0.57143,0.85704,0.0,1.0,3,3,0,3,0,3,0,0,3,0,0,5,0,0,3,0,0,6,0,0,6,0,3],[24,79,0.3038,0.60714,0.3481,0.42857,0.64286,1.0,0.0,1.0,4,9,0,4,0,1,0,0,2,0,0,8,0,0,1,0,0,2,0,0,5,0,9],[28,79,0.3544,0.50893,0.35344,0.14286,0.57143,0.85714,0.0,1.0,5,5,0,5,0,5,0,0,2,0,0,3,0,0,3,0,0,5,0,0,4,0,5],[32,79,0.4051,0.47317,0.29973,0.28571,0.42857,0.71429,0.0,1.0,5,3,0,5,0,1,0,0,5,0,0,7,0,0,4,0,0,5,0,0,2,0,3],[36,79,0.4557,0.54464,0.34522,0.25001,0.42859,0.85714,0.0,1.0,3,7,0,3,0,5,0,0,1,0,0,9,0,0,0,0,0,3,0,0,4,0,7],[40,79,0.5063,0.41962,0.35702,0.0,0.35714,0.71429,0.0,1.0,9,3,0,9,0,3,0,0,4,0,0,2,0,0,3,0,0,4,0,0,4,0,3],[44,79,0.557,0.52231,0.28483,0.28571,0.57143,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,3,0,0,5,0,0,8,0,0,3,0,0,4,0,3],[48,79,0.6076,0.61157,0.33926,0.39286,0.71429,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,0,6,0,7],[52,79,0.6582,0.50445,0.36593,0.10717,0.57121,0.85714,0.0,1.0,8,3,0,8,0,1,0,0,3,0,0,3,0,0,3,0,0,2,0,0,9,0,3],[56,79,0.7089,0.46425,0.29449,0.28571,0.4286,0.71429,0.0,1.0,5,3,0,5,0,1,0,0,6,0,0,5,0,0,6,0,0,5,0,0,1,0,3],[60,79,0.7595,0.46428,0.34255,0.14286,0.5,0.74996,0.0,1.0,6,2,0,6,0,5,0,0,3,0,0,2,0,0,3,0,0,5,0,0,6,0,2],[64,79,0.8101,0.32142,0.30092,0.10714,0.21428,0.57111,0.0,1.0,8,2,0,8,0,8,0,0,3,0,0,4,0,0,4,0,0,2,0,0,1,0,2],[68,79,0.8608,0.41517,0.31815,0.14286,0.42859,0.71429,0.0,1.0,6,2,0,6,0,6,0,0,3,0,0,3,0,0,4,0,0,6,0,0,2,0,2],[72,79,0.9114,0.43295,0.34723,0.14214,0.42857,0.75,0.0,1.0,7,3,0,7,0,4,0,0,4,0,0,5,0,0,1,0,0,3,0,0,5,0,3],[76,79,0.962,0.21426,0.21124,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,7,0,0,7,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[79,79,1.0,0.10268,0.16065,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"9fdf5defac0b729e","q":"Determine the lowest positive integer n such that following statement is true:\r\nIf polynomial with integer coefficients gets value 2 for n different integers, \r\nthen it can't take value 4 for any integer.","t":[{"b":1,"e":1.0,"k":"flat","v":0.97321,"x":0.99107,"p":[[0,32,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,32,0.125,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[8,32,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,32,0.375,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,32,0.5,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,32,0.625,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,32,0.75,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,32,0.875,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,32,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,29,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,29,0.1379,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,29,0.2759,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,29,0.4138,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,29,0.5517,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,29,0.6897,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,29,0.8276,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"476d266b4c763bff","q":"Determine all non-negative integral pairs  $ (x, y)$  for which\r\n\\[ (xy \\minus{} 7)^2 \\equal{} x^2 \\plus{} y^2.\\]","t":[{"b":1,"e":0.14286,"k":"flat","v":0.14732,"x":0.16955,"p":[[0,15,0.0,0.16929,0.16152,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,15,0.2667,0.16955,0.14915,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,15,0.5333,0.16072,0.06916,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.16509,0.1243,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.12054,"x":0.17384,"p":[[0,17,0.0,0.16955,0.16537,0.14286,0.14286,0.14286,0.0,0.85714,3,0,0,3,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,17,0.2353,0.14268,0.06187,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.17384,0.22798,0.14,0.14286,0.14286,0.0,1.0,7,2,0,7,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[16,17,0.9412,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.16063,0.06919,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"961a7b8e3c0c13a5","q":"Circles of radii  $r_1, r_2$  and  $r_3$  are externally touching each other at points  $A, B$ , and   $C$ . If the triangle  $ABC$  has perimeter equal to  $p$ , prove that    $$ \\frac{1}{r_1}+\\frac{1}{r_2}+\\frac{1}{r_3}\\ge \\frac{9}{p}. $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.35267,"x":0.79911,"p":[[0,56,0.0,0.38169,0.18,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,7,0,1,12,0,0,2,0,0,4,0,0,0,0,0],[4,56,0.0714,0.47322,0.25614,0.28571,0.42857,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,6,0,0,9,0,0,3,0,0,6,0,0,0,0,3],[8,56,0.1429,0.44643,0.23077,0.28571,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,6,0,0,10,0,0,2,0,0,7,0,0,2,0,0],[12,56,0.2143,0.46428,0.27664,0.28571,0.42857,0.5,0.0,1.0,2,4,0,2,0,4,0,0,3,0,0,15,0,0,0,0,0,3,0,0,1,0,4],[16,56,0.2857,0.42857,0.22588,0.28571,0.42857,0.4286,0.0,1.0,1,2,0,1,0,4,0,0,6,0,0,14,0,0,1,0,0,4,0,0,0,0,2],[20,56,0.3571,0.48214,0.26905,0.28571,0.42857,0.60714,0.0,1.0,2,4,0,2,0,2,0,0,7,0,0,8,0,0,5,0,0,4,0,0,0,0,4],[24,56,0.4286,0.54017,0.24675,0.42857,0.4286,0.60714,0.0,1.0,1,4,0,1,0,2,0,0,1,0,0,13,0,0,7,0,0,2,0,0,2,0,4],[28,56,0.5,0.52677,0.20024,0.42857,0.42857,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,12,0,0,6,0,0,4,0,0,2,0,2],[32,56,0.5714,0.46428,0.22867,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,12,0,0,3,0,0,4,0,0,1,0,2],[36,56,0.6429,0.45534,0.26107,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,6,0,0,7,0,0,9,0,0,3,0,0,2,0,0,2,0,3],[40,56,0.7143,0.55133,0.28416,0.41068,0.57121,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,3,0,1,7,0,0,4,0,0,7,0,0,1,0,5],[44,56,0.7857,0.49098,0.24997,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,10,0,0,6,0,0,7,0,0,2,0,0,2,0,3],[48,56,0.8571,0.39732,0.25688,0.28571,0.28571,0.46429,0.0,1.0,3,2,0,3,0,3,0,0,11,0,0,7,0,0,2,0,0,3,0,0,1,0,2],[52,56,0.9286,0.35267,0.2201,0.24999,0.28571,0.42858,0.0,1.0,2,1,0,2,0,6,0,0,12,0,0,5,0,0,3,0,0,3,0,0,0,0,1],[56,56,1.0,0.79911,0.23381,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,5,0,0,4,0,15]]},{"b":5,"e":0.14286,"k":"flat","v":0.22321,"x":0.45089,"p":[[0,82,0.0,0.39731,0.16261,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,5,0,0,6,0,0,15,0,0,4,0,0,1,0,0,1,0,0],[4,82,0.0488,0.37054,0.24964,0.14286,0.35714,0.42858,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,9,0,0,1,0,0,4,0,0,1,0,1],[8,82,0.0976,0.39723,0.24162,0.2857,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,4,0,0,8,0,0,8,0,0,3,0,0,3,0,0,3,0,0],[12,82,0.1463,0.43302,0.24867,0.28571,0.42857,0.57143,0.0,1.0,3,2,0,3,0,2,0,0,7,0,0,10,0,0,3,0,0,5,0,0,0,0,2],[16,82,0.1951,0.30357,0.21651,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,5,0,0,9,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[20,82,0.2439,0.37946,0.21312,0.28571,0.42857,0.42857,0.0,1.0,3,1,0,3,0,2,0,0,9,0,0,13,0,0,2,0,0,1,0,0,1,0,1],[24,82,0.2927,0.45089,0.22898,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,9,0,0,10,0,0,0,0,0,6,0,0,2,0,1],[28,82,0.3415,0.25893,0.27302,0.0,0.21428,0.28571,0.0,1.0,9,2,0,9,0,7,0,0,10,0,0,1,0,0,2,0,0,0,0,0,1,0,2],[32,82,0.3902,0.24545,0.16463,0.14286,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,5,0,0,16,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[36,82,0.439,0.22321,0.19212,0.10714,0.21428,0.28571,0.0,0.71429,8,0,0,8,0,8,0,0,11,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[40,82,0.4878,0.22321,0.15542,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,8,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[44,82,0.5366,0.25893,0.2126,0.14286,0.2857,0.28571,0.0,1.0,4,1,0,4,0,11,0,0,12,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[48,82,0.5854,0.25446,0.12234,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,13,0,0,15,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[52,82,0.6341,0.32143,0.25505,0.24999,0.28571,0.28571,0.0,1.0,5,2,0,5,0,3,0,0,17,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[56,82,0.6829,0.27232,0.16506,0.14286,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,17,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[60,82,0.7317,0.33025,0.20965,0.25,0.28571,0.42857,0.0,1.0,2,1,0,2,0,6,0,0,14,0,0,5,0,0,3,0,0,0,0,0,1,0,1],[64,82,0.7805,0.27232,0.15303,0.14286,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,7,0,0,20,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[68,82,0.8293,0.23661,0.12682,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,7,0,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[72,82,0.878,0.29018,0.17672,0.24999,0.28571,0.28571,0.0,1.0,2,1,0,2,0,6,0,0,19,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[76,82,0.9268,0.25446,0.08552,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.25893,0.10972,0.25,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,6,0,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[82,82,1.0,0.28125,0.15355,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,6,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"fe00a03fd7989fe2","q":"Consider the set \n\\[\\{0, 1\\}^n = \\{X = (x_1, x_2,\\dots , x_n) : x_i \\in \\{0, 1\\}, 1 \\leq i \\leq n\\}.\\]\nWe say that  $X > Y$  if  $X \\neq Y$  and the following  $n$  inequalities are satisfy \n\\[x_1 \\geq y_1, x_1 + x_2 \\geq y_1 + y_2,\\dots , x_1 + x_2 + \\cdots + x_n \\geq y_1 + y_2 + \\cdots + y_n.\\]\nWe define a chain of length  $k$  as a subset  ${Z_1,\\dots , Z_k} \\subseteq \\{0, 1\\}^n$  of distinct elements such that  $Z_1 > Z_2 > \\cdots > Z_k.$  \nDetermine the lenght of longest chain in  $\\{0,1\\}^n$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.35714,"x":0.95536,"p":[[0,38,0.0,0.35714,0.41342,0.0,0.07143,0.75,0.0,1.0,16,4,6,16,0,3,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,4],[4,38,0.1053,0.85268,0.3204,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,26],[8,38,0.2105,0.87054,0.29957,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,24],[12,38,0.3158,0.86161,0.28901,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,24],[16,38,0.4211,0.82143,0.3677,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[20,38,0.5263,0.83036,0.2976,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,21],[24,38,0.6316,0.95536,0.12078,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[28,38,0.7368,0.8125,0.24598,0.42859,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,6,0,17],[32,38,0.8421,0.72768,0.26572,0.42857,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,0,0,0,10,0,0,1,0,0,5,0,0,2,0,13],[36,38,0.9474,0.48213,0.11151,0.42857,0.42857,0.4642,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,23,0,0,3,0,0,5,0,0,0,0,0],[38,38,1.0,0.49107,0.15947,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,25,0,0,1,0,0,3,0,0,0,0,2]]},{"b":6,"e":0.4286,"k":"rising","v":0.32589,"x":0.97767,"p":[[0,56,0.0,0.32589,0.40126,0.0,0.14286,0.85714,0.0,1.0,14,5,10,14,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,5],[4,56,0.0714,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,56,0.1429,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[12,56,0.2143,0.78571,0.37627,0.7499,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,23],[16,56,0.2857,0.85714,0.26726,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,23],[20,56,0.3571,0.85268,0.31029,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,24],[24,56,0.4286,0.9375,0.19212,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[28,56,0.5,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[32,56,0.5714,0.93304,0.15146,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,25],[36,56,0.6429,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[40,56,0.7143,0.93304,0.16745,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[44,56,0.7857,0.94196,0.16312,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[48,56,0.8571,0.90179,0.20025,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,25],[52,56,0.9286,0.91964,0.18877,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,27],[56,56,1.0,0.74107,0.33776,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,5,0,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,18]]}]},{"i":"af95fb1ac1a912c3","q":"Determine all integers  $n \\geq 2$ , satisfying  $$ n=a^2+b^2, $$  where  $a$  is the smallest divisor of  $n$  different from  $1$  and  $b$  is an arbitrary divisor of  $n$ .\n*Proposed by Walther Janous*","t":[{"b":5,"e":1.0,"k":"flat","v":0.87945,"x":1.0,"p":[[0,18,0.0,0.87945,0.15615,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,18],[4,18,0.2222,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[8,18,0.4444,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,18,0.6667,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[16,18,0.8889,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.76786,"x":0.91517,"p":[[0,41,0.0,0.89284,0.14728,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,20],[4,41,0.0976,0.91517,0.16312,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[8,41,0.1951,0.82141,0.16754,0.71429,0.71429,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,1,0,14],[12,41,0.2927,0.82143,0.21724,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,0,1,0,18],[16,41,0.3902,0.76786,0.22798,0.71429,0.71429,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,0,1,0,12],[20,41,0.4878,0.83036,0.17655,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,3,0,15],[24,41,0.5854,0.82143,0.20516,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,3,0,16],[28,41,0.6829,0.81696,0.20277,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,2,0,16],[32,41,0.7805,0.81693,0.22086,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,0,3,0,17],[36,41,0.878,0.86607,0.20183,0.67857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,2,0,21],[40,41,0.9756,0.85713,0.20205,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,4,0,19],[41,41,1.0,0.85713,0.23147,0.71429,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,3,0,20]]}]},{"i":"6d6ea36e2e5c2e17","q":"Consider two fixed points  $B,C$  on a circle  $w$ . Find the locus of the incenters of all triangles  $ABC$  when point  $A$  describes  $w$ .","t":[{"b":0,"e":0.4286,"k":"falling","v":0.62057,"x":0.99554,"p":[[0,44,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,44,0.0909,0.89284,0.2287,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,25],[8,44,0.1818,0.85714,0.25,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,1,0,22],[12,44,0.2727,0.76786,0.29613,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,1,0,17],[16,44,0.3636,0.89732,0.17582,0.82132,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,1,0,23],[20,44,0.4545,0.79911,0.2809,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,0,0,0,8,0,0,1,0,0,1,0,19],[24,44,0.5455,0.8125,0.26108,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,7,0,0,2,0,0,1,0,19],[28,44,0.6364,0.74107,0.33776,0.57143,0.92857,1.0,0.0,1.0,3,16,0,3,0,2,0,0,0,0,0,1,0,0,3,0,0,5,0,0,2,0,16],[32,44,0.7273,0.78111,0.28121,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,0,0,0,5,0,0,4,0,0,3,0,16],[36,44,0.8182,0.89284,0.20205,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,23],[40,44,0.9091,0.89283,0.14289,0.857,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,18],[44,44,1.0,0.62057,0.23034,0.42859,0.57121,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,14,0,0,5,0,0,3,0,0,3,0,6]]},{"b":3,"e":0.14286,"k":"falling","v":0.28125,"x":0.99107,"p":[[0,87,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,87,0.046,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,87,0.092,0.92857,0.20516,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[12,87,0.1379,0.92857,0.16366,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,26],[16,87,0.1839,0.93303,0.19557,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[20,87,0.2299,0.89284,0.2287,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,1,0,24],[24,87,0.2759,0.85714,0.25754,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,22],[28,87,0.3218,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,87,0.3678,0.91518,0.19516,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[36,87,0.4138,0.80356,0.27608,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,18],[40,87,0.4598,0.80802,0.27806,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,0,2,0,19],[44,87,0.5057,0.7812,0.30093,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,1,0,18],[48,87,0.5517,0.82593,0.26416,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,0,2,0,20],[52,87,0.5977,0.75445,0.34854,0.57143,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,0,1,0,18],[56,87,0.6437,0.69196,0.35013,0.42857,0.78571,1.0,0.0,1.0,3,14,0,3,0,2,0,0,2,0,0,2,0,0,2,0,0,5,0,0,2,0,14],[60,87,0.6897,0.80355,0.25693,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,0,3,0,16],[64,87,0.7356,0.73214,0.37415,0.39286,1.0,1.0,0.0,1.0,3,18,0,3,0,3,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,18],[68,87,0.7816,0.71874,0.31235,0.57132,0.78571,1.0,0.0,1.0,1,13,0,1,0,3,0,0,2,0,0,1,0,0,3,0,0,6,0,0,3,0,13],[72,87,0.8276,0.70982,0.37878,0.53569,0.92857,1.0,0.0,1.0,5,16,1,5,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,16],[76,87,0.8736,0.78561,0.32751,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,4,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,20],[80,87,0.9195,0.52233,0.37899,0.14286,0.42857,0.89286,0.0,1.0,2,8,0,2,0,11,0,0,1,0,0,3,0,0,0,0,0,3,0,0,4,0,8],[84,87,0.9655,0.37491,0.25947,0.14286,0.28571,0.42857,0.14,1.0,0,3,0,0,0,12,0,0,5,0,0,9,0,0,1,0,0,2,0,0,0,0,3],[87,87,1.0,0.28125,0.2461,0.14286,0.14286,0.42857,0.0,1.0,1,2,0,1,0,19,0,0,3,0,0,6,0,0,0,0,0,0,0,0,1,0,2]]}]},{"i":"8ab62a3c54343eb6","q":"Determine all positive integers  $n \\ge 2$  which have a positive divisor  $m | n$  satisfying   $$ n = d^3 + m^3. $$  where  $d$  is the smallest divisor of  $n$  which is greater than  $1$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.84821,"x":1.0,"p":[[0,37,0.0,0.84821,0.17105,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,6,0,15],[4,37,0.1081,0.92857,0.15152,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,25],[8,37,0.2162,0.91964,0.16342,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,24],[12,37,0.3243,0.92411,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,26],[16,37,0.4324,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[20,37,0.5405,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,26],[24,37,0.6486,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[37,37,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":7,"e":0.85714,"k":"flat","v":0.72767,"x":0.85268,"p":[[0,71,0.0,0.85268,0.18723,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,2,0,18],[4,71,0.0563,0.7857,0.19234,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,9,0,9],[8,71,0.1127,0.75445,0.15252,0.67857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,0,8,0,5],[12,71,0.169,0.72767,0.16116,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,9,0,0,6,0,5],[16,71,0.2254,0.78571,0.18211,0.67836,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,6,0,10],[20,71,0.2817,0.76786,0.15465,0.67857,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,0,6,0,7],[24,71,0.338,0.79911,0.16311,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,11,0,8],[28,71,0.3944,0.77677,0.12342,0.71429,0.78571,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,13,0,3],[32,71,0.4507,0.76786,0.14617,0.67857,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,10,0,5],[36,71,0.507,0.76338,0.13653,0.71429,0.71429,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,10,0,4],[40,71,0.5634,0.79018,0.12869,0.71429,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,17,0,3],[44,71,0.6197,0.82589,0.14167,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,8,0,10],[48,71,0.6761,0.80357,0.14173,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,13,0,6],[52,71,0.7324,0.79018,0.14279,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,13,0,5],[56,71,0.7887,0.8125,0.11538,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,17,0,4],[60,71,0.8451,0.80802,0.11074,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,15,0,4],[64,71,0.9014,0.79463,0.1068,0.71429,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,20,0,1],[68,71,0.9577,0.83482,0.08828,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,22,0,3],[71,71,1.0,0.83036,0.14032,0.82143,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,17,0,7]]}]},{"i":"3c5a32cd62ec779b","q":"Do there exist 12 rectangular parallelepipeds   $P_1,\\,P_2,\\ldots,P_{12}$  with edges parallel to coordinate axes  $OX,\\,OY,\\,OZ$  such that  $P_i$  and  $P_j$  have a common point iff  $i\\ne j\\pm 1$  modulo 12?","t":[{"b":5,"e":0.1429,"k":"flat","v":0.10714,"x":0.29911,"p":[[0,39,0.0,0.10714,0.23958,0.0,0.0,0.14286,0.0,1.0,22,1,3,22,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[4,39,0.1026,0.23214,0.18814,0.14286,0.14286,0.32143,0.0,0.57143,4,0,0,4,0,18,0,0,2,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[8,39,0.2051,0.20982,0.20511,0.14286,0.14286,0.1786,0.0,1.0,5,1,0,5,0,19,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[12,39,0.3077,0.28121,0.20661,0.14286,0.14286,0.571,0.0,0.57143,3,0,0,3,0,16,0,0,1,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[16,39,0.4103,0.23204,0.21055,0.105,0.14286,0.42857,0.0,0.71429,8,0,0,8,0,12,0,0,2,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[20,39,0.5128,0.20972,0.20199,0.14214,0.14286,0.21429,0.0,0.71429,7,0,0,7,0,17,0,0,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[24,39,0.6154,0.29911,0.20316,0.14286,0.14286,0.4286,0.0,0.71429,1,0,0,1,0,17,0,0,1,0,0,6,0,0,5,0,0,2,0,0,0,0,0],[28,39,0.7179,0.2991,0.21533,0.14286,0.1429,0.4642,0.0,0.71429,3,0,0,3,0,14,0,0,2,0,0,5,0,0,6,0,0,2,0,0,0,0,0],[32,39,0.8205,0.24107,0.20025,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,18,0,0,0,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[36,39,0.9231,0.18741,0.17657,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,20,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[39,39,1.0,0.24107,0.16536,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,19,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"rising","v":0.0625,"x":0.49543,"p":[[0,45,0.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.42857,20,0,3,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.43297,0.26599,0.14286,0.571,0.57143,0.14286,1.0,0,2,0,0,0,13,0,0,0,0,0,1,0,0,14,0,0,1,0,0,1,0,2],[8,45,0.1778,0.2767,0.22575,0.14286,0.14286,0.57143,0.0,0.71429,2,0,0,2,0,20,0,0,0,0,0,1,0,0,6,0,0,3,0,0,0,0,0],[12,45,0.2667,0.31685,0.24679,0.14286,0.14286,0.5711,0.0,1.0,2,1,0,2,0,16,0,0,2,0,0,2,0,0,7,0,0,2,0,0,0,0,1],[16,45,0.3556,0.21871,0.17483,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,25,0,0,0,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[20,45,0.4444,0.32121,0.23429,0.14286,0.14286,0.57143,0.0,0.85714,1,0,0,1,0,18,0,0,0,0,0,1,0,0,10,0,0,1,0,0,1,0,0],[24,45,0.5333,0.33471,0.27111,0.14286,0.14286,0.57111,0.14,1.0,0,2,0,0,0,19,0,0,2,0,0,1,0,0,6,0,0,1,0,0,1,0,2],[28,45,0.6222,0.36147,0.26963,0.14286,0.14288,0.57111,0.14,1.0,0,2,0,0,0,17,0,0,2,0,0,1,0,0,7,0,0,3,0,0,0,0,2],[32,45,0.7111,0.30802,0.27688,0.14286,0.14286,0.57111,0.14286,1.0,0,2,0,0,0,23,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,2],[36,45,0.8,0.35268,0.32534,0.14286,0.14286,0.57143,0.0,1.0,2,5,0,2,0,17,0,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,5],[40,45,0.8889,0.37942,0.33042,0.14286,0.14286,0.57143,0.0,1.0,4,4,0,4,0,13,0,0,1,0,0,2,0,0,5,0,0,2,0,0,1,0,4],[44,45,0.9778,0.49543,0.34262,0.14286,0.42857,0.75,0.0,1.0,1,7,0,1,0,11,0,0,1,0,0,4,0,0,4,0,0,3,0,0,1,0,7],[45,45,1.0,0.48214,0.33834,0.14286,0.5,0.85714,0.0,1.0,1,5,0,1,0,12,0,0,1,0,0,2,0,0,6,0,0,1,0,0,4,0,5]]}]},{"i":"6f1fa0dcaa28ab96","q":"Each integer is colored with one of two colors, red or blue. It is known that, for every finite set  $A$  of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set  $A$  is at most  $1000$ . Prove that there exists a set of  $2000$  consecutive integers in which there are exactly  $1000$  red numbers and  $1000$  numbers blue.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.80357,"x":0.91517,"p":[[0,24,0.0,0.91517,0.16312,0.85711,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[4,24,0.1667,0.84362,0.22414,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,10,0,0,0,0,19],[8,24,0.3333,0.81696,0.2506,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,4,0,0,1,0,19],[12,24,0.5,0.90179,0.15746,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,1,0,22],[16,24,0.6667,0.85714,0.20825,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,2,0,20],[20,24,0.8333,0.84375,0.20935,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,1,0,19],[24,24,1.0,0.80357,0.22799,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,9,0,0,1,0,16]]},{"b":3,"e":0.28571,"k":"falling","v":0.42411,"x":0.89286,"p":[[0,34,0.0,0.89286,0.21129,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,24],[4,34,0.1176,0.85268,0.23279,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,6,0,0,2,0,20],[8,34,0.2353,0.8213,0.25006,0.71321,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,5,0,0,1,0,19],[12,34,0.3529,0.875,0.20124,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,2,0,21],[16,34,0.4706,0.77678,0.25985,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,3,0,0,2,0,16],[20,34,0.5882,0.84821,0.2257,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,0,1,0,20],[24,34,0.7059,0.78571,0.23958,0.67857,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,8,0,0,0,0,16],[28,34,0.8235,0.65625,0.33093,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,8,0,0,4,0,0,1,0,0,3,0,0,0,0,14],[32,34,0.9412,0.42411,0.22442,0.28571,0.42857,0.60714,0.0,1.0,2,1,0,2,0,1,0,0,12,0,0,8,0,0,1,0,0,7,0,0,0,0,1],[34,34,1.0,0.49107,0.22851,0.28571,0.42857,0.71429,0.0,0.71429,2,0,0,2,0,2,0,0,5,0,0,8,0,0,1,0,0,14,0,0,0,0,0]]}]},{"i":"3be007047b8f6075","q":"Consider the numbers  $ a_n=1-\\binom{n}{3} +\\binom{n}{6} -\\cdots, b_n= -\\binom{n}{1} +\\binom{n}{4}-\\binom{n}{7} +\\cdots $  and  $ c_n=\\binom{n}{2} -\\binom{n}{5} +\\binom{n}{8} -\\cdots , $  for a natural number  $ n\\ge 2. $  Prove that  $$  a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}.  $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.11607,"x":0.27232,"p":[[0,61,0.0,0.15179,0.21998,0.0,0.07143,0.2857,0.0,0.85714,16,0,2,16,0,7,0,0,6,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[4,61,0.0656,0.19643,0.1948,0.14286,0.14286,0.2857,0.0,1.0,5,1,0,5,0,18,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,61,0.1311,0.21875,0.1988,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,13,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[12,61,0.1967,0.15625,0.19019,0.0,0.14286,0.14287,0.0,0.85714,11,0,0,11,0,14,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,61,0.2623,0.25446,0.23887,0.14286,0.14286,0.28571,0.0,1.0,2,2,0,2,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[20,61,0.3279,0.26339,0.28818,0.10714,0.2143,0.28571,0.0,1.0,8,3,0,8,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[24,61,0.3934,0.23205,0.14179,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,9,0,0,16,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,61,0.459,0.25446,0.19144,0.14286,0.2857,0.28571,0.0,1.0,4,1,0,4,0,10,0,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[32,61,0.5246,0.21875,0.09439,0.14286,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.27232,0.17627,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,15,0,0,10,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[40,61,0.6557,0.2008,0.16315,0.14286,0.14286,0.2857,0.0,0.85714,4,0,0,4,0,17,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[44,61,0.7213,0.16518,0.15612,0.14286,0.14286,0.17857,0.0,0.85714,7,0,0,7,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,61,0.7869,0.16071,0.11152,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,15,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.16071,0.12752,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,14,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,61,0.918,0.1384,0.08364,0.14286,0.14286,0.14287,0.0,0.28571,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,61,0.9836,0.14277,0.07986,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.05357,"x":0.20527,"p":[[0,50,0.0,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.28571,13,0,2,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.20527,0.22288,0.14214,0.14286,0.17857,0.0,0.85714,7,0,0,7,0,17,0,0,3,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[8,50,0.16,0.19195,0.21008,0.0,0.14286,0.2857,0.0,0.85714,9,0,0,9,0,14,0,0,5,0,0,0,0,0,2,0,0,1,0,0,1,0,0],[12,50,0.24,0.125,0.08564,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.16964,0.16917,0.14286,0.14286,0.14287,0.0,0.85714,7,0,0,7,0,18,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[20,50,0.4,0.11607,0.10374,0.0,0.14286,0.14287,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.10714,0.10714,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,12,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.125,0.10564,0.0,0.14286,0.1429,0.0,0.28571,11,0,0,11,0,14,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.10714,0.11294,0.0,0.14286,0.14287,0.0,0.28571,15,0,0,15,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.10714,0.10714,0.0,0.14286,0.14287,0.0,0.28571,14,0,0,14,0,12,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.09375,0.10479,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.09366,0.09847,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"af3cba9e69fbcfb9","q":"Do there exist prime numbers  $p$  and  $q$  such that  $p^2(p^3-1)=q(q+1)$  ?","t":[{"b":1,"e":0.1667,"k":"flat","v":0.04911,"x":0.19196,"p":[[0,64,0.0,0.18295,0.18295,0.105,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,17,0,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[4,64,0.0625,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,9,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,64,0.1875,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.17857,0.15972,0.14286,0.14286,0.17857,0.0,0.57143,7,0,0,7,0,17,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[20,64,0.3125,0.16517,0.16012,0.10714,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,18,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[24,64,0.375,0.17857,0.12877,0.14286,0.14286,0.17857,0.0,0.57143,4,0,0,4,0,20,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,64,0.4375,0.19196,0.1411,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,64,0.5,0.15625,0.12037,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,20,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,64,0.5625,0.11607,0.1448,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,14,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[40,64,0.625,0.12946,0.13533,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,17,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,64,0.6875,0.14277,0.17496,0.0,0.14286,0.17857,0.0,0.71429,14,0,0,14,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[48,64,0.75,0.08036,0.12339,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,64,0.8125,0.15607,0.16116,0.0,0.14286,0.17857,0.0,0.71429,10,0,0,10,0,14,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[56,64,0.875,0.13542,0.11487,0.0,0.14286,0.14287,0.0,0.42857,10,0,0,10,0,15,0,0,5,1,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.1436,0.12379,0.0,0.14286,0.14882,0.0,0.57143,9,0,0,9,0,15,1,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,64,1.0,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.10714,"x":0.26339,"p":[[0,42,0.0,0.13839,0.17307,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,12,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,42,0.0952,0.26328,0.19598,0.14286,0.14286,0.35704,0.0,0.71429,1,0,0,1,0,20,0,0,3,0,0,0,0,0,7,0,0,1,0,0,0,0,0],[8,42,0.1905,0.26339,0.17169,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,15,0,0,6,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[12,42,0.2857,0.16072,0.10564,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,22,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,42,0.381,0.21875,0.17852,0.14286,0.14286,0.17857,0.0,0.57143,3,0,0,3,0,21,0,0,2,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[20,42,0.4762,0.16518,0.12931,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,21,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,42,0.5714,0.21429,0.14286,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,22,0,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[28,42,0.6667,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.24554,0.18293,0.14286,0.14286,0.32142,0.0,0.57143,1,0,0,1,0,22,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[36,42,0.8571,0.16518,0.14773,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,21,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[40,42,0.9524,0.20536,0.18189,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,20,0,0,0,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[42,42,1.0,0.14732,0.12103,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,20,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"a81e68ccce7e6ce5","q":"El Chapul\u00edn observed that the number  $2014$   has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus  $4$ . A number of eight divisors with this unusual property is called the *red* number . How many *red* numbers  smaller than  $2014$  exist?","t":[{"b":1,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,93,0.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,93,0.043,0.91964,0.24206,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[8,93,0.086,0.91964,0.24206,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[12,93,0.129,0.91518,0.19516,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,23],[16,93,0.172,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[20,93,0.2151,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,93,0.2581,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[28,93,0.3011,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[32,93,0.3441,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,93,0.3871,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,93,0.4301,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,93,0.4731,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,93,0.5161,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,93,0.5591,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,93,0.6022,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[60,93,0.6452,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,93,0.6882,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,93,0.7312,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,93,0.7742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,93,0.8172,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[80,93,0.8602,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[84,93,0.9032,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[88,93,0.9462,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[92,93,0.9892,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[93,93,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":3,"e":1.0,"k":"flat","v":0.88393,"x":0.99554,"p":[[0,150,0.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[4,150,0.0267,0.88393,0.29329,1.0,1.0,1.0,0.0,1.0,3,26,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,26],[8,150,0.0533,0.92857,0.19233,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[12,150,0.08,0.95089,0.11632,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[16,150,0.1067,0.91071,0.24679,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[20,150,0.1333,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[24,150,0.16,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[28,150,0.1867,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[32,150,0.2133,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,150,0.24,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,150,0.2667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,150,0.2933,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,150,0.32,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,150,0.3467,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[56,150,0.3733,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[60,150,0.4,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,150,0.4267,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,150,0.4533,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[72,150,0.48,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[76,150,0.5067,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[80,150,0.5333,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[84,150,0.56,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,150,0.5867,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[92,150,0.6133,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[96,150,0.64,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[100,150,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,150,0.6933,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[108,150,0.72,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[112,150,0.7467,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[116,150,0.7733,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[120,150,0.8,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[124,150,0.8267,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[128,150,0.8533,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[132,150,0.88,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[136,150,0.9067,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[140,150,0.9333,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[144,150,0.96,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[148,150,0.9867,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[150,150,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]}]},{"i":"6445fdd42b6d184b","q":"Find all continuous functions  $f:\\left[0,1\\right]\\rightarrow[0,\\infty)$  such that: $\\int_{0}^{1}f\\left(x\\right)dx\\cdotp\\int_{0}^{1}f^{2}\\left(x\\right)dx\\cdotp...\\cdotp\\int_{0}^{1}f^{2020}\\left(x\\right)dx=\\left(\\int_{0}^{1}f^{2021}\\left(x\\right)dx\\right)^{1010}$","t":[{"b":0,"e":1.0,"k":"flat","v":0.9241,"x":1.0,"p":[[0,22,0.0,0.9241,0.17852,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,26],[4,22,0.1818,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,22,0.3636,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,22,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,34,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,34,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,34,0.2353,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,34,0.5882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,34,0.7059,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[34,34,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8b29b97e9fa57dd1","q":"Find a method by which one can compute the coefficients of  $P(x) = x^6 + a_1x^5 + \\cdots+  a_6$  from the roots of  $P(x) = 0$  by performing not more than  $15$  additions and  $15$  multiplications.","t":[{"b":3,"e":1.0,"k":"falling","v":0.59366,"x":0.96428,"p":[[0,65,0.0,0.95089,0.14555,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[4,65,0.0615,0.96428,0.11294,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[8,65,0.1231,0.90177,0.22144,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,25],[12,65,0.1846,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[16,65,0.2462,0.92857,0.15972,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[20,65,0.3077,0.90624,0.17719,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,24],[24,65,0.3692,0.95536,0.1357,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[28,65,0.4308,0.92408,0.1821,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,26],[32,65,0.4923,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[36,65,0.5538,0.87945,0.23989,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,23],[40,65,0.6154,0.90622,0.18078,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,24],[44,65,0.6769,0.83035,0.2299,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,4,0,17],[48,65,0.7385,0.87946,0.23448,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[52,65,0.8,0.79909,0.2809,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,0,4,0,17],[56,65,0.8615,0.74995,0.29454,0.5354,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,4,0,0,3,0,0,5,0,0,1,0,0,1,0,17],[60,65,0.9231,0.6473,0.25251,0.4286,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,6,0,0,3,0,0,8,0,0,4,0,6],[64,65,0.9846,0.59366,0.26513,0.42857,0.571,0.857,0.0,1.0,1,4,0,1,0,1,0,0,5,0,0,5,0,0,6,0,0,5,0,0,5,0,4],[65,65,1.0,0.60698,0.2552,0.42859,0.5712,0.857,0.14,1.0,0,5,0,0,0,2,0,0,3,0,0,7,0,0,8,0,0,2,0,0,5,0,5]]},{"b":4,"e":1.0,"k":"flat","v":0.92409,"x":0.99554,"p":[[0,25,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,25,0.16,0.92409,0.13828,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,24],[8,25,0.32,0.94641,0.15468,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,28],[12,25,0.48,0.92856,0.15975,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[16,25,0.64,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,25,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,25,0.96,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[25,25,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"3090f3153264c971","q":"Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16?\r\n\r\n*Bulgaria*","t":[{"b":3,"e":0.0,"k":"flat","v":0.11606,"x":0.60267,"p":[[0,243,0.0,0.26784,0.33644,0.0,0.0,0.57111,0.0,1.0,17,2,7,17,0,2,0,0,1,0,0,3,0,0,3,0,0,3,0,0,1,0,2],[4,243,0.0165,0.60267,0.37241,0.35714,0.71429,0.89286,0.0,1.0,7,8,2,7,0,1,0,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,8],[8,243,0.0329,0.33036,0.37532,0.0,0.14286,0.57143,0.0,1.0,15,4,0,15,0,2,0,0,0,0,0,6,0,0,2,0,0,0,0,0,3,0,4],[12,243,0.0494,0.32588,0.35755,0.0,0.21428,0.60714,0.0,1.0,15,3,1,15,0,1,0,0,2,0,0,3,0,0,3,0,0,4,0,0,1,0,3],[16,243,0.0658,0.33034,0.33203,0.0,0.35714,0.57143,0.0,1.0,14,1,1,14,0,1,0,0,1,0,0,4,0,0,6,0,0,2,0,0,3,0,1],[20,243,0.0823,0.3482,0.38785,0.0,0.21429,0.75,0.0,1.0,15,4,1,15,0,1,0,0,2,0,0,4,0,0,1,0,0,1,0,0,4,0,4],[24,243,0.0988,0.39286,0.3977,0.0,0.42857,0.75,0.0,1.0,15,4,0,15,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,4],[28,243,0.1152,0.40179,0.39679,0.0,0.42857,0.75,0.0,1.0,14,5,1,14,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,3,0,5],[32,243,0.1317,0.27678,0.33491,0.0,0.14286,0.46429,0.0,1.0,14,2,0,14,0,5,0,0,3,0,0,2,0,0,1,0,0,3,0,0,2,0,2],[36,243,0.1481,0.35713,0.42106,0.0,0.0,0.71429,0.0,1.0,17,7,0,17,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,7],[40,243,0.1646,0.41954,0.35888,0.105,0.42857,0.71429,0.0,1.0,8,5,0,8,0,5,0,0,1,0,0,6,0,0,3,0,0,2,0,0,2,0,5],[44,243,0.1811,0.43303,0.39201,0.0,0.35714,0.85704,0.0,1.0,10,6,0,10,0,3,0,0,3,0,0,3,0,0,1,0,0,3,0,0,3,0,6],[48,243,0.1975,0.34372,0.32703,0.0,0.28571,0.57143,0.0,1.0,12,2,0,12,0,2,0,0,3,0,0,2,0,0,7,0,0,3,0,0,1,0,2],[52,243,0.214,0.24106,0.32622,0.0,0.0,0.57111,0.0,1.0,19,1,0,19,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,0,2,0,1],[56,243,0.2305,0.25,0.31744,0.0,0.07143,0.46431,0.0,1.0,16,2,0,16,0,4,0,0,1,0,0,3,0,0,3,0,0,3,0,0,0,0,2],[60,243,0.2469,0.29463,0.31528,0.0,0.21428,0.57111,0.0,0.85714,14,0,0,14,0,2,0,0,3,0,0,4,0,0,3,0,0,2,0,0,4,0,0],[64,243,0.2634,0.26338,0.29037,0.0,0.14286,0.4286,0.0,1.0,14,1,0,14,0,3,0,0,2,0,0,7,0,0,2,0,0,2,0,0,1,0,1],[68,243,0.2798,0.31694,0.3724,0.0,0.07143,0.60714,0.0,1.0,16,4,0,16,0,2,0,0,1,0,0,1,0,0,4,0,0,4,0,0,0,0,4],[72,243,0.2963,0.2366,0.34368,0.0,0.0,0.57143,0.0,1.0,19,1,0,19,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,1],[76,243,0.3128,0.26331,0.32365,0.0,0.14143,0.46429,0.0,1.0,15,2,0,15,0,4,0,0,3,0,0,2,0,0,1,0,0,5,0,0,0,0,2],[80,243,0.3292,0.25446,0.29824,0.0,0.14286,0.46429,0.0,1.0,15,1,1,15,0,4,0,0,1,0,0,4,0,0,3,0,0,4,0,0,0,0,1],[84,243,0.3457,0.37946,0.38067,0.0,0.28571,0.71429,0.0,1.0,13,5,0,13,0,1,0,0,3,0,0,3,0,0,2,0,0,4,0,0,1,0,5],[88,243,0.3621,0.27232,0.31816,0.0,0.14286,0.42857,0.0,1.0,13,3,0,13,0,5,0,0,3,0,0,5,0,0,1,0,0,2,0,0,0,0,3],[92,243,0.3786,0.24106,0.32029,0.0,0.14286,0.42857,0.0,1.0,15,2,0,15,0,6,0,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,2],[96,243,0.3951,0.26338,0.33332,0.0,0.0,0.4642,0.0,1.0,17,3,0,17,0,2,0,0,0,0,0,5,0,0,4,0,0,1,0,0,0,0,3],[100,243,0.4115,0.22766,0.33475,0.0,0.0,0.42857,0.0,1.0,19,3,0,19,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,3],[104,243,0.428,0.28572,0.34442,0.0,0.14286,0.57143,0.0,1.0,15,3,0,15,0,4,0,0,1,0,0,3,0,0,2,0,0,4,0,0,0,0,3],[108,243,0.4444,0.24552,0.32581,0.0,0.0,0.57143,0.0,1.0,18,1,0,18,0,3,0,0,0,0,0,1,0,0,4,0,0,4,0,0,1,0,1],[112,243,0.4609,0.30356,0.36024,0.0,0.0,0.57143,0.0,1.0,17,2,0,17,0,0,0,0,3,0,0,0,0,0,5,0,0,2,0,0,3,0,2],[116,243,0.4774,0.25893,0.34151,0.0,0.0,0.57143,0.0,1.0,17,3,0,17,0,3,0,0,1,0,0,2,0,0,5,0,0,0,0,0,1,0,3],[120,243,0.4938,0.20536,0.29001,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,3,0,0,3,0,0,2,0,0,1,0,0,4,0,0,0,0,1],[124,243,0.5103,0.375,0.38589,0.0,0.21429,0.74996,0.0,1.0,13,3,0,13,0,3,0,0,2,0,0,1,0,0,2,0,0,3,0,0,5,0,3],[128,243,0.5267,0.2633,0.35378,0.0,0.07,0.42858,0.0,1.0,16,4,0,16,0,5,0,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,4],[132,243,0.5432,0.29463,0.35343,0.0,0.0,0.57143,0.0,1.0,17,3,0,17,0,1,0,0,0,0,0,4,0,0,3,0,0,4,0,0,0,0,3],[136,243,0.5597,0.35267,0.37793,0.0,0.2857,0.71429,0.0,1.0,14,4,0,14,0,1,0,0,3,0,0,4,0,0,1,0,0,2,0,0,3,0,4],[140,243,0.5761,0.3482,0.39275,0.0,0.14286,0.60714,0.0,1.0,15,5,0,15,0,2,0,0,2,0,0,0,0,0,5,0,0,1,0,0,2,0,5],[144,243,0.5926,0.36606,0.39275,0.0,0.2857,0.71429,0.0,1.0,15,5,0,15,0,0,0,0,3,0,0,1,0,0,4,0,0,2,0,0,2,0,5],[148,243,0.6091,0.32586,0.38336,0.0,0.14286,0.71429,0.0,1.0,15,4,0,15,0,4,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,4],[152,243,0.6255,0.31249,0.35433,0.0,0.14288,0.57143,0.0,1.0,15,3,0,15,0,2,0,0,2,0,0,2,0,0,4,0,0,3,0,0,1,0,3],[156,243,0.642,0.34375,0.38937,0.0,0.14286,0.60714,0.0,1.0,13,6,0,13,0,5,0,0,1,0,0,3,0,0,2,0,0,1,0,0,1,0,6],[160,243,0.6584,0.3125,0.38038,0.0,0.0,0.60714,0.0,1.0,17,3,0,17,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,3],[164,243,0.6749,0.29016,0.36678,0.0,0.07143,0.71429,0.0,1.0,16,2,0,16,0,5,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,2],[168,243,0.6914,0.19187,0.3474,0.0,0.0,0.14286,0.0,1.0,22,3,0,22,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,3],[172,243,0.7078,0.30802,0.38813,0.0,0.07143,0.57111,0.0,1.0,16,6,0,16,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,6],[176,243,0.7243,0.26337,0.33899,0.0,0.0,0.57111,0.0,1.0,17,1,0,17,0,3,0,0,1,0,0,2,0,0,2,0,0,3,0,0,3,0,1],[180,243,0.7407,0.16964,0.2976,0.0,0.0,0.21429,0.0,1.0,22,2,0,22,0,2,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,2],[184,243,0.7572,0.19196,0.29581,0.0,0.0,0.32142,0.0,1.0,18,2,0,18,0,5,0,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[188,243,0.7737,0.16964,0.29544,0.0,0.0,0.2857,0.0,1.0,21,2,0,21,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,2],[192,243,0.7901,0.32141,0.33881,0.0,0.21428,0.57143,0.0,1.0,14,2,0,14,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,0,1,0,2],[196,243,0.8066,0.19643,0.30671,0.0,0.0,0.42857,0.0,1.0,20,2,0,20,0,3,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,2],[200,243,0.823,0.24107,0.2889,0.0,0.14286,0.42858,0.0,1.0,14,1,0,14,0,5,0,0,4,0,0,2,0,0,3,0,0,2,0,0,1,0,1],[204,243,0.8395,0.3125,0.32622,0.0,0.14286,0.57143,0.0,1.0,12,2,0,12,0,5,0,0,1,0,0,5,0,0,2,0,0,4,0,0,1,0,2],[208,243,0.856,0.12938,0.22119,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,7,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[212,243,0.8724,0.17854,0.24218,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,6,0,0,1,0,0,2,0,0,4,0,0,2,0,0,0,0,0],[216,243,0.8889,0.16963,0.26105,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,1],[220,243,0.9053,0.14286,0.23146,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,7,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0],[224,243,0.9218,0.12947,0.24317,0.0,0.0,0.14287,0.0,0.85714,23,0,0,23,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,0],[228,243,0.9383,0.16961,0.27061,0.0,0.0,0.21432,0.0,1.0,20,1,0,20,0,4,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,1],[232,243,0.9547,0.19631,0.29611,0.0,0.0,0.32142,0.0,1.0,18,2,0,18,0,5,0,0,1,0,0,2,0,0,3,0,0,1,0,0,0,0,2],[236,243,0.9712,0.12052,0.21159,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[240,243,0.9877,0.11606,0.17287,0.0,0.0,0.1786,0.0,0.571,20,0,0,20,0,4,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[243,243,1.0,0.12946,0.2212,0.0,0.0,0.14287,0.0,0.71429,21,0,0,21,0,4,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"rising","v":0.16964,"x":0.45089,"p":[[0,78,0.0,0.16964,0.30813,0.0,0.0,0.17857,0.0,1.0,22,2,7,22,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[4,78,0.0513,0.42411,0.37199,0.0,0.42857,0.71429,0.0,1.0,11,4,7,11,0,2,0,0,1,0,0,3,0,0,3,0,0,6,0,0,2,0,4],[8,78,0.1026,0.27231,0.35238,0.0,0.0,0.46418,0.0,1.0,17,3,2,17,0,2,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,3],[12,78,0.1538,0.38391,0.34705,0.0,0.42857,0.71429,0.0,1.0,11,3,1,11,0,2,0,0,1,0,0,7,0,0,2,0,0,4,0,0,2,0,3],[16,78,0.2051,0.37499,0.41148,0.0,0.14286,0.71429,0.0,1.0,15,7,0,15,0,2,0,0,0,0,0,2,0,0,3,0,0,3,0,0,0,0,7],[20,78,0.2564,0.33482,0.37899,0.0,0.14286,0.71429,0.0,1.0,15,3,0,15,0,3,0,0,0,0,0,3,0,0,1,0,0,4,0,0,3,0,3],[24,78,0.3077,0.38393,0.33395,0.0,0.4286,0.71429,0.0,1.0,11,1,0,11,0,3,0,0,0,0,0,3,0,0,5,0,0,7,0,0,2,0,1],[28,78,0.359,0.26338,0.30744,0.0,0.14286,0.4642,0.0,1.0,15,1,0,15,0,4,0,0,0,0,0,5,0,0,3,0,0,3,0,0,1,0,1],[32,78,0.4103,0.31692,0.30455,0.0,0.2857,0.57111,0.0,1.0,10,2,0,10,0,5,0,0,5,0,0,1,0,0,6,0,0,3,0,0,0,0,2],[36,78,0.4615,0.24107,0.28446,0.0,0.14286,0.42857,0.0,1.0,14,2,0,14,0,4,0,0,3,0,0,7,0,0,1,0,0,1,0,0,0,0,2],[40,78,0.5128,0.3125,0.36846,0.0,0.14286,0.60714,0.0,1.0,15,4,0,15,0,3,0,0,1,0,0,4,0,0,1,0,0,3,0,0,1,0,4],[44,78,0.5641,0.22768,0.26452,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,4,0,0,3,0,0,3,0,0,3,0,0,4,0,0,0,0,0],[48,78,0.6154,0.3616,0.31335,0.0,0.42857,0.57143,0.0,1.0,9,2,0,9,0,5,0,0,1,0,0,5,0,0,6,0,0,3,0,0,1,0,2],[52,78,0.6667,0.45089,0.30327,0.24999,0.42857,0.71429,0.0,1.0,6,2,0,6,0,2,0,0,3,0,0,7,0,0,5,0,0,4,0,0,3,0,2],[56,78,0.7179,0.37938,0.27812,0.14214,0.42857,0.57143,0.0,1.0,7,1,0,7,0,4,0,0,3,0,0,4,0,0,10,0,0,2,0,0,1,0,1],[60,78,0.7692,0.37499,0.34579,0.0,0.28571,0.71429,0.0,1.0,9,3,0,9,0,6,0,0,3,0,0,1,0,0,3,0,0,6,0,0,1,0,3],[64,78,0.8205,0.37945,0.26391,0.14286,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,6,0,0,4,0,0,5,0,0,6,0,0,4,0,0,2,0,0],[68,78,0.8718,0.31695,0.27602,0.14286,0.14286,0.57111,0.0,0.85714,7,0,0,7,0,10,0,0,1,0,0,5,0,0,2,0,0,6,0,0,1,0,0],[72,78,0.9231,0.40177,0.2911,0.14286,0.42859,0.71429,0.0,1.0,6,1,0,6,0,6,0,0,2,0,0,3,0,0,6,0,0,8,0,0,0,0,1],[76,78,0.9744,0.38829,0.29723,0.14286,0.42859,0.57143,0.0,1.0,4,2,0,4,0,11,0,0,0,0,0,3,0,0,8,0,0,3,0,0,1,0,2],[78,78,1.0,0.42408,0.2823,0.14286,0.4998,0.57143,0.0,1.0,2,2,0,2,0,10,0,0,2,0,0,2,0,0,11,0,0,1,0,0,2,0,2]]}]},{"i":"05e2b7e8c993a3f7","q":"Find all functions  $f \\colon \\mathbb{R_{+}}\\to \\mathbb{R_{+}}$  satisfying : \\[f ( f (x)-x) = 2x\\] for all  $x > 0$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.54911,"x":0.65177,"p":[[0,88,0.0,0.54911,0.14335,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,25,0,0,0,0,0,1,0,1],[4,88,0.0455,0.61161,0.12492,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,25,0,0,2,0,0,3,0,1],[8,88,0.0909,0.58036,0.18877,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,2,0,0,20,0,0,2,0,0,4,0,1],[12,88,0.1364,0.59375,0.17169,0.57143,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,1,0,0,21,0,0,3,0,0,2,0,2],[16,88,0.1818,0.63393,0.13333,0.57143,0.57143,0.57143,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,4,0,2],[20,88,0.2273,0.62945,0.12808,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,1,0,0,5,0,1],[24,88,0.2727,0.62946,0.1063,0.57143,0.57143,0.60714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,3,0,0,5,0,0],[28,88,0.3182,0.59374,0.09523,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,1,0,0,1,0,1],[32,88,0.3636,0.58481,0.10926,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,1,0,0,1,0,1],[36,88,0.4091,0.60267,0.12234,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,1,0,0,1,0,2],[40,88,0.4545,0.5625,0.10062,0.57143,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,27,0,0,1,0,0,1,0,0],[44,88,0.5,0.58479,0.09007,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0,0,2,0,0],[48,88,0.5455,0.61161,0.11971,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,2,0,0,3,0,1],[52,88,0.5909,0.61606,0.12079,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,0,4,0,1],[56,88,0.6364,0.62054,0.10479,0.57143,0.57143,0.57143,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,1,0,0,5,0,0],[60,88,0.6818,0.56249,0.07936,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,0,0,0,1,0,0],[64,88,0.7273,0.60714,0.10714,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,2,0,0,2,0,1],[68,88,0.7727,0.59817,0.12079,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,25,0,0,0,0,0,3,0,1],[72,88,0.8182,0.57142,0.07143,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,1,0,0,1,0,0],[76,88,0.8636,0.57143,0.09449,0.57143,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,29,0,0,1,0,0,1,0,0],[80,88,0.9091,0.63838,0.1287,0.57143,0.57143,0.64286,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,0,0,0,8,0,0],[84,88,0.9545,0.59375,0.11355,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,25,0,0,1,0,0,2,0,1],[88,88,1.0,0.65177,0.16728,0.57143,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,19,0,0,0,0,0,11,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.55356,"x":0.69196,"p":[[0,40,0.0,0.57588,0.13115,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,1,0,0,22,0,0,5,0,0,0,0,1],[4,40,0.1,0.63391,0.14258,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,2,0,0,2,0,3],[8,40,0.2,0.56696,0.11564,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,1,0,0,23,0,0,4,0,0,1,0,0],[12,40,0.3,0.55356,0.08564,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,2,0,0,26,0,0,2,0,0,0,0,0],[16,40,0.4,0.58479,0.08268,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,0,3,0,0,1,0,0],[20,40,0.5,0.59375,0.09523,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,1,0,0,3,0,0],[24,40,0.6,0.56695,0.16935,0.57143,0.57143,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,1,0,0,26,0,0,1,0,0,0,0,2],[28,40,0.7,0.68304,0.1461,0.57143,0.57143,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,11,0,1],[32,40,0.8,0.68304,0.1461,0.57143,0.57143,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,3,0,0,10,0,1],[36,40,0.9,0.69196,0.14772,0.57143,0.57143,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,14,0,0],[40,40,1.0,0.6607,0.13244,0.57143,0.57143,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0]]}]},{"i":"20566be70e7d2621","q":"Find all functions  $f :[0, +\\infty) \\rightarrow [0, +\\infty)$  for which  $f(f(x)+f(y)) = xy f (x+y)$  \n\nfor every two non-negative real numbers  $x$  and  $y$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.35261,"x":0.88839,"p":[[0,117,0.0,0.37052,0.28762,0.14286,0.28571,0.57143,0.0,1.0,7,2,0,7,0,2,0,0,10,0,0,1,0,0,6,0,0,4,0,0,0,0,2],[4,117,0.0342,0.83482,0.16409,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,9,0,12],[8,117,0.0684,0.88839,0.20743,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,20],[12,117,0.1026,0.78125,0.27196,0.71429,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,9,0,12],[16,117,0.1368,0.76784,0.2594,0.57143,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,10,0,10],[20,117,0.1709,0.77229,0.24709,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,6,0,12],[24,117,0.2051,0.85268,0.2004,0.85714,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,13,0,13],[28,117,0.2393,0.85268,0.15765,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,11,0,12],[32,117,0.2735,0.70532,0.30293,0.571,0.78571,1.0,0.0,1.0,2,11,0,2,0,1,0,0,2,0,0,1,0,0,7,0,0,3,0,0,5,0,11],[36,117,0.3077,0.7857,0.2113,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,8],[40,117,0.3419,0.7991,0.2621,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,0,7,0,15],[44,117,0.3761,0.82587,0.17766,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,12,0,11],[48,117,0.4103,0.73213,0.2918,0.57143,0.85714,1.0,0.0,1.0,3,9,0,3,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,9,0,9],[52,117,0.4444,0.73214,0.30462,0.57143,0.85714,1.0,0.0,1.0,3,10,0,3,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,10,0,10],[56,117,0.4786,0.79014,0.22015,0.57143,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,2,0,0,8,0,12],[60,117,0.5128,0.66506,0.28724,0.571,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,2,0,0,2,0,0,7,0,0,5,0,0,6,0,7],[64,117,0.547,0.71874,0.29555,0.67857,0.85707,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,3,0,0,0,0,0,2,0,0,6,0,0,10,0,8],[68,117,0.5812,0.69642,0.29613,0.57132,0.71429,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,1,0,0,3,0,0,3,0,0,7,0,0,7,0,8],[72,117,0.6154,0.66517,0.35103,0.53572,0.78571,1.0,0.0,1.0,5,10,0,5,0,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,6,0,10],[76,117,0.6496,0.75445,0.29067,0.67857,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,12,0,10],[80,117,0.6838,0.72763,0.2458,0.57132,0.85707,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,2,0,0,6,0,0,4,0,0,10,0,7],[84,117,0.7179,0.7366,0.27919,0.57143,0.85714,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,1,0,0,0,0,0,5,0,0,4,0,0,11,0,8],[88,117,0.7521,0.69639,0.26185,0.571,0.71429,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,4,0,0,2,0,0,7,0,0,4,0,0,5,0,9],[92,117,0.7863,0.63839,0.2923,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,1,0,0,8,0,0,7,0,0,3,0,7],[96,117,0.8205,0.73659,0.28596,0.57143,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,0,0,0,6,0,0,4,0,0,8,0,10],[100,117,0.8547,0.70536,0.29653,0.42857,0.85714,1.0,0.0,1.0,2,9,0,2,0,0,0,0,3,0,0,4,0,0,1,0,0,5,0,0,8,0,9],[104,117,0.8889,0.57142,0.34442,0.28571,0.57143,0.85714,0.0,1.0,6,5,0,6,0,0,0,0,3,0,0,2,0,0,6,0,0,3,0,0,7,0,5],[108,117,0.9231,0.5714,0.33503,0.35716,0.57143,0.85714,0.0,1.0,5,3,0,5,0,3,0,0,0,0,0,1,0,0,9,0,0,1,0,0,10,0,3],[112,117,0.9573,0.50889,0.30291,0.28571,0.57121,0.71429,0.0,1.0,5,4,0,5,0,0,0,0,5,0,0,3,0,0,10,0,0,3,0,0,2,0,4],[116,117,0.9915,0.51334,0.29201,0.28571,0.57121,0.71429,0.0,1.0,3,3,0,3,0,3,0,0,5,0,0,2,0,0,7,0,0,7,0,0,2,0,3],[117,117,1.0,0.35261,0.2368,0.24999,0.28571,0.571,0.0,1.0,6,1,0,6,0,2,0,0,9,0,0,4,0,0,10,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.714,"k":"flat","v":0.23659,"x":0.86605,"p":[[0,235,0.0,0.31695,0.30458,0.0,0.28571,0.57143,0.0,1.0,13,1,0,13,0,0,0,0,6,0,0,0,0,0,9,0,0,2,0,0,1,0,1],[4,235,0.017,0.84821,0.1234,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,11,0,10],[8,235,0.034,0.78567,0.23149,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,1,0,0,7,0,13],[12,235,0.0511,0.78571,0.29014,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,9,0,14],[16,235,0.0681,0.74552,0.28957,0.57143,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,3,0,0,0,0,0,4,0,0,4,0,0,8,0,11],[20,235,0.0851,0.80804,0.27342,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,14],[24,235,0.1021,0.74542,0.26685,0.57132,0.85714,1.0,0.14,1.0,0,11,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,11],[28,235,0.1191,0.86605,0.15544,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,15,0,12],[32,235,0.1362,0.78124,0.25751,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,0,0,0,5,0,0,4,0,0,8,0,12],[36,235,0.1532,0.83482,0.23176,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,7,0,16],[40,235,0.1702,0.80357,0.22517,0.82143,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,15,0,9],[44,235,0.1872,0.74553,0.30248,0.57143,0.85714,1.0,0.0,1.0,3,12,0,3,0,0,0,0,1,0,0,0,0,0,6,0,0,3,0,0,7,0,12],[48,235,0.2043,0.70087,0.25345,0.57143,0.71429,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,1,0,0,1,0,0,8,0,0,5,0,0,10,0,5],[52,235,0.2213,0.82589,0.20433,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,12,0,11],[56,235,0.2383,0.74106,0.29111,0.57143,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,6,0,11],[60,235,0.2553,0.73661,0.21461,0.57143,0.78571,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,7,0,0,10,0,6],[64,235,0.2723,0.69193,0.25782,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,2,0,0,7,0,0,4,0,0,6,0,8],[68,235,0.2894,0.62053,0.37561,0.28571,0.78571,1.0,0.0,1.0,5,10,0,5,0,1,0,0,5,0,0,0,0,0,3,0,0,2,0,0,6,0,10],[72,235,0.3064,0.6339,0.37105,0.28571,0.85714,1.0,0.0,1.0,6,9,0,6,0,0,0,0,3,0,0,0,0,0,5,0,0,1,0,0,8,0,9],[76,235,0.3234,0.65177,0.37955,0.49968,0.78571,1.0,0.0,1.0,6,12,0,6,0,1,0,0,1,0,0,0,0,0,5,0,0,3,0,0,4,0,12],[80,235,0.3404,0.6473,0.37112,0.53539,0.857,0.89286,0.0,1.0,7,8,0,7,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,9,0,8],[84,235,0.3574,0.65622,0.30901,0.5354,0.71429,1.0,0.0,1.0,3,9,0,3,0,0,0,0,3,0,0,2,0,0,6,0,0,6,0,0,3,0,9],[88,235,0.3745,0.77231,0.26931,0.57143,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,10,0,11],[92,235,0.3915,0.78569,0.22018,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,4,0,0,10,0,10],[96,235,0.4085,0.70089,0.28428,0.57143,0.85707,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,2,0,0,1,0,0,4,0,0,5,0,0,11,0,6],[100,235,0.4255,0.73214,0.24679,0.57143,0.85707,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,1,0,0,8,0,0,3,0,0,9,0,8],[104,235,0.4426,0.60713,0.28572,0.53539,0.57143,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,4,0,0,1,0,0,9,0,0,5,0,0,6,0,4],[108,235,0.4596,0.76786,0.24157,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,4,0,0,8,0,11],[112,235,0.4766,0.71874,0.26842,0.57142,0.85714,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,10,0,7],[116,235,0.4936,0.66069,0.32879,0.49968,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,4,0,0,0,0,0,5,0,0,4,0,0,6,0,9],[120,235,0.5106,0.71875,0.29984,0.57143,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,2,0,0,1,0,0,6,0,0,0,0,0,11,0,9],[124,235,0.5277,0.6741,0.31589,0.57143,0.85707,0.85714,0.0,1.0,4,6,0,4,0,0,0,0,2,0,0,0,0,0,6,0,0,3,0,0,11,0,6],[128,235,0.5447,0.66069,0.30879,0.571,0.71429,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,2,0,0,1,0,0,5,0,0,8,0,0,4,0,8],[132,235,0.5617,0.64728,0.32731,0.53539,0.71429,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,1,0,0,2,0,0,6,0,0,3,0,0,8,0,7],[136,235,0.5787,0.68747,0.28447,0.57132,0.78571,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,2,0,0,8,0,0,2,0,0,9,0,7],[140,235,0.5957,0.61158,0.29502,0.42857,0.64286,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,4,0,0,3,0,0,6,0,0,4,0,0,8,0,4],[144,235,0.6128,0.69196,0.22335,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,0,0,0,10,0,0,9,0,0,4,0,6],[148,235,0.6298,0.66071,0.30878,0.57143,0.78571,0.85714,0.0,1.0,4,4,0,4,0,0,0,0,2,0,0,1,0,0,4,0,0,5,0,0,12,0,4],[152,235,0.6468,0.69643,0.30252,0.57143,0.85714,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,1,0,0,3,0,0,6,0,0,1,0,0,10,0,8],[156,235,0.6638,0.69197,0.29904,0.53572,0.85714,0.85714,0.0,1.0,3,6,0,3,0,0,0,0,2,0,0,3,0,0,1,0,0,6,0,0,11,0,6],[160,235,0.6809,0.54908,0.31157,0.39286,0.57143,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,3,0,0,5,0,0,8,0,0,2,0,0,4,0,5],[164,235,0.6979,0.57587,0.31438,0.39286,0.64286,0.85714,0.0,1.0,4,4,0,4,0,1,0,0,3,0,0,4,0,0,4,0,0,6,0,0,6,0,4],[168,235,0.7149,0.59372,0.2372,0.53539,0.57143,0.75,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,2,0,0,12,0,0,4,0,0,6,0,2],[172,235,0.7319,0.61605,0.32031,0.39286,0.71429,0.85714,0.0,1.0,4,5,0,4,0,0,0,0,4,0,0,2,0,0,4,0,0,5,0,0,8,0,5],[176,235,0.7489,0.64728,0.28791,0.571,0.71429,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,2,0,0,2,0,0,8,0,0,4,0,0,8,0,5],[180,235,0.766,0.66958,0.26352,0.571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,3,0,0,0,0,0,8,0,0,7,0,0,7,0,5],[184,235,0.783,0.65622,0.27862,0.42859,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,3,0,0,3,0,0,7,0,0,8,0,5],[188,235,0.8,0.70532,0.21411,0.5713,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,3,0,0,10,0,5],[192,235,0.817,0.75445,0.23484,0.57143,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,10,0,8],[196,235,0.834,0.71429,0.23145,0.57143,0.78571,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,4,0,0,10,0,6],[200,235,0.8511,0.67857,0.31135,0.57143,0.78571,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,2,0,0,1,0,0,4,0,0,5,0,0,9,0,7],[204,235,0.8681,0.65172,0.23943,0.571,0.64286,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,3,0,0,9,0,0,5,0,0,7,0,4],[208,235,0.8851,0.55354,0.30878,0.28571,0.57143,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,6,0,0,2,0,0,6,0,0,4,0,0,4,0,5],[212,235,0.9021,0.64725,0.2812,0.571,0.57143,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,1,0,0,2,0,0,11,0,0,2,0,0,8,0,5],[216,235,0.9191,0.65177,0.29868,0.53539,0.71429,0.85714,0.0,1.0,3,6,0,3,0,0,0,0,3,0,0,2,0,0,5,0,0,6,0,0,7,0,6],[220,235,0.9362,0.7098,0.24611,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,0,0,0,6,0,0,7,0,0,10,0,5],[224,235,0.9532,0.54462,0.3203,0.2857,0.57141,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,7,0,0,2,0,0,5,0,0,2,0,0,7,0,4],[228,235,0.9702,0.53124,0.29501,0.28571,0.57143,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,5,0,0,2,0,0,6,0,0,6,0,0,4,0,3],[232,235,0.9872,0.53121,0.28173,0.28571,0.57143,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,4,0,0,2,0,0,8,0,0,7,0,0,5,0,1],[235,235,1.0,0.23659,0.2219,0.0,0.28571,0.32143,0.0,0.57143,13,0,0,13,0,0,0,0,11,0,0,1,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"d4cc0f64aa392ea1","q":"Find all natural numbers  $N$  consisting of exactly  $1112$  digits (in decimal notation) such that:\n(a) The sum of the digits of  $N$  is divisible by  $2000$ ;\n(b) The sum of the digits of  $N+1$  is divisible by  $2000$ ;\n(c)  $1$  is a digit of  $N$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,40,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,40,0.1,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,40,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,40,0.3,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,40,0.7,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,40,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.71429,"k":"falling","v":0.77679,"x":1.0,"p":[[0,42,0.0,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[4,42,0.0952,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,42,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,42,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,42,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,42,0.8571,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[40,42,0.9524,0.79464,0.12846,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,9],[42,42,1.0,0.77679,0.11811,0.71429,0.71429,0.71429,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7]]}]},{"i":"910a2dfb97ce9dea","q":"Find all integer solutions  $(x,y,z)$  of the equation  $xy+yz+zx-xyz = 2$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,68,0.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[4,68,0.0588,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,68,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,68,0.1765,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,68,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,68,0.2941,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,68,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,68,0.4118,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,68,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,68,0.5294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,68,0.5882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,68,0.6471,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,68,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,68,0.7647,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,68,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,68,0.8824,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.857,"x":1.0,"p":[[0,71,0.0,0.857,0.23428,0.82132,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,5,0,19],[4,71,0.0563,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,71,0.1127,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,71,0.169,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,71,0.2254,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,71,0.2817,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,71,0.338,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,71,0.3944,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,71,0.4507,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,71,0.507,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,71,0.5634,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,71,0.6197,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,71,0.6761,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,71,0.7324,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,71,0.7887,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,71,0.8451,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[64,71,0.9014,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[68,71,0.9577,0.94642,0.07785,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[71,71,1.0,0.89286,0.11845,0.82143,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,16]]}]},{"i":"125d1bbd477290cd","q":"Elbert and Yaiza each draw  $10$  cards from a  $20$ -card deck with cards numbered  $1,2,3,\\dots,20$ . Then, starting with the player with the card numbered  $1$ , the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.\n\nGiven that Yaiza lost and  $5$  cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player\u2019s hand does not 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1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[228,493,0.4625,0.91518,0.21387,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[232,493,0.4706,0.90179,0.2299,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,27],[236,493,0.4787,0.91964,0.22851,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[240,493,0.4868,0.85268,0.29121,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,25],[244,493,0.4949,0.88391,0.2381,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,25],[248,493,0.503,0.88839,0.25187,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,26],[252,493,0.5112,0.93304,0.19557,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[256,493,0.5193,0.83929,0.2896,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,24],[260,493,0.5274,0.87499,0.24158,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,25],[264,493,0.5355,0.91964,0.18877,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,26],[268,493,0.5436,0.82142,0.30723,0.78561,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,23],[272,493,0.5517,0.88393,0.26351,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,26],[276,493,0.5598,0.89286,0.23145,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,26],[280,493,0.568,0.94196,0.18851,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[284,493,0.5761,0.92857,0.19885,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[288,493,0.5842,0.85714,0.28347,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,25],[292,493,0.5923,0.90179,0.20958,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,25],[296,493,0.6004,0.91964,0.21998,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[300,493,0.6085,0.94196,0.19186,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[304,493,0.6166,0.88839,0.23072,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,25],[308,493,0.6247,0.90179,0.26592,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[312,493,0.6329,0.88838,0.23349,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,25],[316,493,0.641,0.86606,0.2695,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,25],[320,493,0.6491,0.87054,0.2633,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,24],[324,493,0.6572,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[328,493,0.6653,0.96428,0.12878,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[332,493,0.6734,0.85716,0.24482,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,22],[336,493,0.6815,0.76786,0.33264,0.39286,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,21],[340,493,0.6897,0.78125,0.3174,0.42859,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,21],[344,493,0.6978,0.72768,0.28203,0.42859,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,7,0,0,3,0,0,4,0,0,0,0,15],[348,493,0.7059,0.72768,0.33189,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,19],[352,493,0.714,0.74991,0.32749,0.42857,1.0,1.0,0.14,1.0,0,19,0,0,0,2,0,0,5,0,0,4,0,0,0,0,0,1,0,0,1,0,19],[356,493,0.7221,0.79456,0.31749,0.4286,1.0,1.0,0.14,1.0,0,21,0,0,0,3,0,0,2,0,0,4,0,0,0,0,0,0,0,0,2,0,21],[360,493,0.7302,0.85714,0.26487,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,24],[364,493,0.7383,0.80803,0.31055,0.64286,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,22],[368,493,0.7465,0.75893,0.30813,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,6,0,0,5,0,0,1,0,0,0,0,0,1,0,19],[372,493,0.7546,0.73214,0.3458,0.39286,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,6,0,0,4,0,0,0,0,0,0,0,0,1,0,19],[376,493,0.7627,0.70982,0.35081,0.28571,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,7,0,0,4,0,0,0,0,0,0,0,0,1,0,18],[380,493,0.7708,0.74999,0.32537,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,5,0,0,4,0,0,0,0,0,0,0,0,3,0,18],[384,493,0.7789,0.7768,0.31527,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,21],[388,493,0.787,0.70982,0.35978,0.28571,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,19],[392,493,0.7951,0.875,0.24419,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,24],[396,493,0.8032,0.85714,0.28572,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,24],[400,493,0.8114,0.875,0.26184,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[404,493,0.8195,0.87054,0.26088,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,25],[408,493,0.8276,0.82143,0.29234,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,22],[412,493,0.8357,0.69643,0.31693,0.42857,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,8,0,0,1,0,0,1,0,0,0,0,16],[416,493,0.8438,0.73661,0.32362,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,19],[420,493,0.8519,0.74553,0.32289,0.39286,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,7,0,0,3,0,0,0,0,0,1,0,0,2,0,18],[424,493,0.86,0.79018,0.34066,0.42857,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,23],[428,493,0.8682,0.79019,0.30718,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,5,0,0,3,0,0,0,0,0,2,0,0,0,0,21],[432,493,0.8763,0.91071,0.22798,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,27],[436,493,0.8844,0.76786,0.30878,0.42857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,20],[440,493,0.8925,0.80357,0.27375,0.53572,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,1,0,0,1,0,20],[444,493,0.9006,0.78572,0.32142,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,5,0,0,2,0,0,0,0,0,1,0,0,1,0,21],[448,493,0.9087,0.76786,0.30252,0.42857,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,3,0,0,7,0,0,0,0,0,1,0,0,1,0,19],[452,493,0.9168,0.71429,0.32537,0.39286,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,7,0,0,4,0,0,2,0,0,0,0,0,1,0,17],[456,493,0.9249,0.73214,0.32488,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,4,0,0,6,0,0,1,0,0,0,0,0,1,0,18],[460,493,0.9331,0.75893,0.31225,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,7,0,0,4,0,0,0,0,0,1,0,0,1,0,19],[464,493,0.9412,0.80802,0.30643,0.53539,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,22],[468,493,0.9493,0.70536,0.32328,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,5,0,0,7,0,0,1,0,0,1,0,0,1,0,16],[472,493,0.9574,0.65178,0.33869,0.28571,0.57121,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,15],[476,493,0.9655,0.80804,0.29366,0.53571,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,22],[480,493,0.9736,0.82142,0.28571,0.64286,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,5,0,0,3,0,0,0,0,0,1,0,0,1,0,22],[484,493,0.9817,0.72326,0.33866,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,5,0,0,5,0,0,1,0,0,0,0,0,1,0,18],[488,493,0.9899,0.68304,0.32288,0.42857,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,4,0,0,9,0,0,0,0,0,1,0,0,1,0,15],[492,493,0.998,0.42848,0.29023,0.2857,0.28571,0.42858,0.14,1.0,0,6,0,0,0,6,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,6],[493,493,1.0,0.32589,0.11425,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,15,0,0,10,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"685dbbacef20d636","q":"Find all natural numbers  $n \\geqslant  2$  with the property that there are two permutations  $(a_1, a_2,\\ldots, a_n) $  and  $(b_1, b_2,\\ldots, b_n)$  of the numbers  $1, 2,\\ldots, n$  such that  $(a_1 + b_1, a_2 +b_2,\\ldots, a_n + b_n)$  are consecutive natural numbers.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.98214,"p":[[0,48,0.0,0.88839,0.1504,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,12,0,16],[4,48,0.0833,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[8,48,0.1667,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,48,0.25,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[16,48,0.3333,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,48,0.4167,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[24,48,0.5,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[28,48,0.5833,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,48,0.6667,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[36,48,0.75,0.92857,0.11293,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,19],[40,48,0.8333,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[44,48,0.9167,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[48,48,1.0,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19]]},{"b":5,"e":1.0,"k":"flat","v":0.88391,"x":0.96875,"p":[[0,44,0.0,0.88391,0.14917,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,11,0,16],[4,44,0.0909,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,44,0.1818,0.93304,0.11837,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,21],[12,44,0.2727,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[16,44,0.3636,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[20,44,0.4545,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,44,0.5455,0.91964,0.11812,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0,18],[28,44,0.6364,0.90625,0.14987,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,9,0,19],[32,44,0.7273,0.95536,0.10971,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[36,44,0.8182,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[40,44,0.9091,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[44,44,1.0,0.90625,0.1411,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,20]]}]},{"i":"185c22f4c10b02bb","q":"Find all pairs  $(x,y)$  of integers such that  $y^3-1=x^4+x^2$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.28572,"x":0.64732,"p":[[0,99,0.0,0.45533,0.32426,0.0,0.4998,0.71429,0.0,1.0,9,3,0,9,0,0,0,0,0,0,0,7,0,0,5,0,0,8,0,0,0,0,3],[4,99,0.0404,0.46874,0.28623,0.25,0.57143,0.71429,0.0,1.0,7,1,0,7,0,1,0,0,1,0,0,2,0,0,11,0,0,9,0,0,0,0,1],[8,99,0.0808,0.51785,0.22232,0.42857,0.57143,0.71429,0.0,0.71429,4,0,0,4,0,0,0,0,0,0,0,7,0,0,10,0,0,11,0,0,0,0,0],[12,99,0.1212,0.39286,0.32341,0.0,0.50001,0.71429,0.0,1.0,12,1,0,12,0,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,0,0,1],[16,99,0.1616,0.45088,0.27918,0.2857,0.57121,0.71429,0.0,0.71429,7,0,0,7,0,0,0,0,4,0,0,4,0,0,4,0,0,13,0,0,0,0,0],[20,99,0.202,0.37499,0.26425,0.0,0.42857,0.57143,0.0,0.71429,9,0,0,9,0,1,0,0,1,0,0,8,0,0,8,0,0,5,0,0,0,0,0],[24,99,0.2424,0.37944,0.30221,0.0,0.49979,0.71429,0.0,0.71429,11,0,0,11,0,1,0,0,1,0,0,3,0,0,7,0,0,9,0,0,0,0,0],[28,99,0.2828,0.34821,0.32328,0.0,0.42857,0.57143,0.0,1.0,13,1,0,13,0,1,0,0,1,0,0,3,0,0,7,0,0,5,0,0,1,0,1],[32,99,0.3232,0.35267,0.31335,0.0,0.42857,0.71429,0.0,0.71429,13,0,0,13,0,1,0,0,0,0,0,3,0,0,6,0,0,9,0,0,0,0,0],[36,99,0.3636,0.32588,0.295,0.0,0.42857,0.60714,0.0,0.71429,13,0,0,13,0,0,0,0,2,0,0,7,0,0,2,0,0,8,0,0,0,0,0],[40,99,0.404,0.31247,0.29108,0.0,0.35714,0.57143,0.0,0.71429,13,0,0,13,0,2,0,0,1,0,0,3,0,0,8,0,0,5,0,0,0,0,0],[44,99,0.4444,0.34374,0.29419,0.0,0.42857,0.60714,0.0,0.71429,12,0,0,12,0,1,0,0,0,0,0,8,0,0,3,0,0,8,0,0,0,0,0],[48,99,0.4848,0.28572,0.31744,0.0,0.0,0.71429,0.0,0.71429,17,0,0,17,0,0,0,0,0,0,0,5,0,0,1,0,0,9,0,0,0,0,0],[52,99,0.5253,0.45089,0.31361,0.10714,0.5,0.71429,0.0,1.0,8,3,0,8,0,1,0,0,0,0,0,7,0,0,7,0,0,6,0,0,0,0,3],[56,99,0.5657,0.3214,0.31337,0.0,0.42859,0.57143,0.0,0.85714,15,0,0,15,0,0,0,0,0,0,0,3,0,0,8,0,0,5,0,0,1,0,0],[60,99,0.6061,0.32141,0.29232,0.0,0.28571,0.57143,0.0,0.71429,13,0,0,13,0,0,0,0,4,0,0,2,0,0,7,0,0,6,0,0,0,0,0],[64,99,0.6465,0.64732,0.13825,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,16,0,0,0,0,2],[68,99,0.6869,0.61152,0.15253,0.57075,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,12,0,0,11,0,0,0,0,2],[72,99,0.7273,0.60712,0.13833,0.5354,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,11,0,0,11,0,0,1,0,1],[76,99,0.7677,0.57589,0.12103,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,0,12,0,0,0,0,0],[80,99,0.8081,0.5781,0.18242,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,8,0,1,8,0,0,11,0,0,1,0,1],[84,99,0.8485,0.59372,0.14334,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,10,0,0,10,0,0,1,0,1],[88,99,0.8889,0.60266,0.17762,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,11,0,0,6,0,0,10,0,0,2,0,2],[92,99,0.9293,0.57139,0.15152,0.42857,0.5714,0.60714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,12,0,0,6,0,0,0,0,2],[96,99,0.9697,0.51784,0.13243,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,16,0,0,8,0,0,7,0,0,0,0,0],[99,99,1.0,0.54014,0.14609,0.42857,0.4998,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,0,4,0,0,3,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.37497,"x":0.62052,"p":[[0,66,0.0,0.47321,0.31224,0.39286,0.42859,0.60714,0.0,1.0,7,4,0,7,0,0,0,0,1,0,0,9,0,0,7,0,0,3,0,0,1,0,4],[4,66,0.0606,0.37497,0.29611,0.0,0.42859,0.60714,0.0,0.71429,11,0,0,11,0,1,0,0,0,0,0,5,0,0,7,0,0,8,0,0,0,0,0],[8,66,0.1212,0.44195,0.30797,0.0,0.57143,0.71429,0.0,1.0,9,1,0,9,0,0,0,0,2,0,0,3,0,0,6,0,0,11,0,0,0,0,1],[12,66,0.1818,0.5357,0.17128,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,12,0,0,7,0,0,11,0,0,0,0,0],[16,66,0.2424,0.61158,0.14391,0.53539,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,15,0,0,0,0,1],[20,66,0.303,0.62052,0.14111,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,17,0,0,0,0,1],[24,66,0.3636,0.59816,0.12596,0.571,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,14,0,0,10,0,0,0,0,1],[28,66,0.4242,0.56249,0.12846,0.42857,0.57141,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,9,0,0,1,0,0],[32,66,0.4848,0.58033,0.1234,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,8,0,0,13,0,0,0,0,0],[36,66,0.5455,0.60711,0.11846,0.5354,0.64286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,16,0,0,0,0,0],[40,66,0.6061,0.58926,0.14617,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,13,0,0,0,0,1],[44,66,0.6667,0.55802,0.13054,0.42857,0.57141,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,9,0,0,1,0,0],[48,66,0.7273,0.52679,0.12078,0.42857,0.42857,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,8,0,0,0,0,0],[52,66,0.7879,0.50447,0.11285,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,5,0,0,6,0,0,0,0,0],[56,66,0.8485,0.51338,0.11768,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,20,0,0,5,0,0,7,0,0,0,0,0],[60,66,0.9091,0.47764,0.08454,0.42857,0.42857,0.571,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,7,0,0,2,0,0,0,0,0],[64,66,0.9697,0.48659,0.10011,0.42857,0.42857,0.57111,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,5,0,0,4,0,0,0,0,0],[66,66,1.0,0.45536,0.07523,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0]]}]},{"i":"6b5b6dfb9144354b","q":"Find all polynomials  $P(x, y)$  with real coefficients which for all real numbers  $x$  and  $y$  satisfy  $P(x + y, x - y) = 2P(x, y)$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.68749,"x":1.0,"p":[[0,59,0.0,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,59,0.0678,0.88393,0.18013,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,20],[8,59,0.1356,0.86161,0.20666,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,0,6,0,19],[12,59,0.2034,0.68749,0.23539,0.42857,0.57143,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,3,0,0,3,0,9],[16,59,0.2712,0.7366,0.21757,0.57143,0.78571,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,2,0,0,7,0,9],[20,59,0.339,0.79463,0.20185,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,4,0,13],[24,59,0.4068,0.75,0.19562,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,6,0,0,7,0,8],[28,59,0.4746,0.76786,0.20124,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,3,0,0,7,0,10],[32,59,0.5424,0.75,0.19562,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,5,0,0,7,0,8],[36,59,0.6102,0.70536,0.23402,0.53572,0.78571,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,1,0,0,9,0,7],[40,59,0.678,0.70533,0.22851,0.57132,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,5,0,0,5,0,8],[44,59,0.7458,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,59,0.8136,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,59,0.8814,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,59,0.9492,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[59,59,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.72768,"x":0.94196,"p":[[0,31,0.0,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,31,0.129,0.72768,0.21535,0.53571,0.78571,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,4,0,0,9,0,7],[8,31,0.2581,0.80356,0.2075,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,7,0,13],[12,31,0.3871,0.75446,0.2237,0.57143,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,4,0,0,7,0,10],[16,31,0.5161,0.80356,0.19151,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,9,0,11],[20,31,0.6452,0.86607,0.17474,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,7,0,17],[24,31,0.7742,0.83035,0.16146,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,10,0,11],[28,31,0.9032,0.88392,0.1357,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,15],[31,31,1.0,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,17]]}]},{"i":"f4ca9b8a64b8d46a","q":"Find all functions  $f:\\mathbb{R} \\to \\mathbb{R}$ , such that for any  $x, y \\in \\mathbb{R}$  holds the following:  $$ f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x) $$  *Proposed by Mykhailo Shtandenko*","t":[{"b":1,"e":1.0,"k":"rising","v":0.70982,"x":1.0,"p":[[0,167,0.0,0.70982,0.2461,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,6,0,0,8,0,7],[4,167,0.024,0.92411,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[8,167,0.0479,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[12,167,0.0719,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[16,167,0.0958,0.8616,0.15765,0.82132,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,11,0,13],[20,167,0.1198,0.84375,0.17261,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,11],[24,167,0.1437,0.88839,0.12745,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,13,0,14],[28,167,0.1677,0.91071,0.1171,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,14,0,16],[32,167,0.1916,0.89286,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[36,167,0.2156,0.91071,0.09943,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,16],[40,167,0.2395,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[44,167,0.2635,0.89286,0.11294,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,15],[48,167,0.2874,0.88839,0.11701,0.82143,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,15],[52,167,0.3114,0.88392,0.10374,0.85711,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,14,0,12],[56,167,0.3353,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[60,167,0.3593,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[64,167,0.3832,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[68,167,0.4072,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[72,167,0.4311,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[76,167,0.4551,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[80,167,0.479,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[84,167,0.503,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[88,167,0.5269,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[92,167,0.5509,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[96,167,0.5749,0.93748,0.10067,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[100,167,0.5988,0.93303,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[104,167,0.6228,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[108,167,0.6467,0.93081,0.0936,0.85714,1.0,1.0,0.643,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,11,0,19],[112,167,0.6707,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[116,167,0.6946,0.92411,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[120,167,0.7186,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[124,167,0.7425,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[128,167,0.7665,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[132,167,0.7904,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[136,167,0.8144,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[140,167,0.8383,0.96427,0.08755,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[144,167,0.8623,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[148,167,0.8862,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[152,167,0.9102,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[156,167,0.9341,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[160,167,0.9581,0.96427,0.10719,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[164,167,0.982,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[167,167,1.0,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]},{"b":6,"e":0.85714,"k":"rising","v":0.71875,"x":0.96429,"p":[[0,445,0.0,0.71875,0.24868,0.53571,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,12,0,0,0,0,11],[4,445,0.009,0.83929,0.23077,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,6,0,16],[8,445,0.018,0.8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,0,0,0,0,3,0,0,14,0,15],[372,445,0.836,0.90165,0.10995,0.85714,0.92857,1.0,0.71,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,16],[376,445,0.8449,0.90625,0.10479,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15],[380,445,0.8539,0.88839,0.1461,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,12,0,15],[384,445,0.8629,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[388,445,0.8719,0.89732,0.12492,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,13,0,15],[392,445,0.8809,0.87946,0.13415,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,14,0,13],[396,445,0.8899,0.91517,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[400,445,0.8989,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[404,445,0.9079,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[408,445,0.9169,0.92411,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[412,445,0.9258,0.90178,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[416,445,0.9348,0.87946,0.09523,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,10],[420,445,0.9438,0.91071,0.09942,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,15],[424,445,0.9528,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[428,445,0.9618,0.89286,0.14725,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,12,0,16],[432,445,0.9708,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[436,445,0.9798,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[440,445,0.9888,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[444,445,0.9978,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[445,445,1.0,0.92411,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20]]}]},{"i":"63056a931f9c5ca2","q":"Find all positive integers  $abcd=a^{a+b+c+d} - a^{-a+b-c+d} + a$ , where  $abcd$  is a four-digit number","t":[{"b":1,"e":0.85714,"k":"flat","v":0.85268,"x":0.95536,"p":[[0,95,0.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[4,95,0.0421,0.85714,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,5,0,14],[8,95,0.0842,0.88839,0.11701,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,14],[12,95,0.1263,0.85268,0.10999,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,13,0,9],[16,95,0.1684,0.87498,0.12246,0.82132,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,13],[20,95,0.2105,0.91071,0.10564,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[24,95,0.2526,0.86607,0.11811,0.82132,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,13,0,11],[28,95,0.2947,0.87946,0.15198,0.85711,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,9,0,16],[32,95,0.3368,0.87052,0.12556,0.857,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,13,0,12],[36,95,0.3789,0.92397,0.09469,0.85714,1.0,1.0,0.71,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[40,95,0.4211,0.86607,0.11811,0.82143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,13,0,11],[44,95,0.4632,0.88393,0.18363,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,15],[48,95,0.5053,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[52,95,0.5474,0.89283,0.13367,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,15],[56,95,0.5895,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[60,95,0.6316,0.89283,0.10719,0.85714,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[64,95,0.6737,0.87052,0.1093,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,19,0,9],[68,95,0.7158,0.88838,0.09268,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,11],[72,95,0.7579,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[76,95,0.8,0.87946,0.12428,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,16,0,12],[80,95,0.8421,0.89732,0.11425,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,15],[84,95,0.8842,0.86607,0.11259,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,15,0,10],[88,95,0.9263,0.89286,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[92,95,0.9684,0.93303,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[95,95,1.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22]]},{"b":5,"e":0.71429,"k":"falling","v":0.69195,"x":0.91071,"p":[[0,125,0.0,0.89732,0.16065,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,5,0,20],[4,125,0.032,0.91071,0.09943,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,16],[8,125,0.064,0.83927,0.21354,0.857,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,13,0,12],[12,125,0.096,0.875,0.12242,0.82132,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,13],[16,125,0.128,0.87053,0.10326,0.85711,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,15,0,10],[20,125,0.16,0.84375,0.19019,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,12],[24,125,0.192,0.875,0.12242,0.82132,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,13],[28,125,0.224,0.82143,0.11294,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,13,0,6],[32,125,0.256,0.76339,0.19434,0.71429,0.85707,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,13,0,4],[36,125,0.288,0.79463,0.07935,0.71429,0.857,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,16,0,1],[40,125,0.32,0.79911,0.10012,0.71429,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,14,0,3],[44,125,0.352,0.78125,0.15561,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,20,0,0],[48,125,0.384,0.79451,0.08715,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,17,0,1],[52,125,0.416,0.73214,0.22517,0.71429,0.71429,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,10,0,4],[56,125,0.448,0.76783,0.09281,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,12,0,1],[60,125,0.48,0.78571,0.09449,0.71429,0.85707,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,16,0,1],[64,125,0.512,0.76339,0.16982,0.71429,0.78571,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,14,0,2],[68,125,0.544,0.73657,0.16796,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,9,0,2],[72,125,0.576,0.76785,0.15872,0.71429,0.78564,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,15,0,1],[76,125,0.608,0.74998,0.10103,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,12,0,0],[80,125,0.64,0.75893,0.10972,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,10,0,2],[84,125,0.672,0.76339,0.15815,0.71429,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,14,0,1],[88,125,0.704,0.78557,0.11302,0.71429,0.85707,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,16,0,2],[92,125,0.736,0.77664,0.10688,0.71429,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,10,0,3],[96,125,0.768,0.76786,0.09279,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,12,0,1],[100,125,0.8,0.75892,0.07523,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,11,0,0],[104,125,0.832,0.72768,0.16115,0.71429,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,9,0,1],[108,125,0.864,0.75445,0.16459,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,11,0,2],[112,125,0.896,0.76784,0.11154,0.71429,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,9,0,3],[116,125,0.928,0.75891,0.09742,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,11,0,1],[120,125,0.96,0.73214,0.12242,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,0,0,10,0,1],[124,125,0.992,0.70982,0.145,0.71429,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,6,0,0],[125,125,1.0,0.69195,0.19921,0.71429,0.71429,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,6,0,1]]}]},{"i":"98a815bddf9e06d6","q":"Find all polynomyals  $P(x)$  with real coefficients which satisfy the following equality for all real numbers  $x$ : \\[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \\]","t":[{"b":5,"e":0.71429,"k":"rising","v":0.62945,"x":0.99107,"p":[[0,184,0.0,0.62945,0.39907,0.25001,0.85714,1.0,0.0,1.0,7,10,0,7,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0,9,0,10],[4,184,0.0217,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,184,0.0435,0.90178,0.20025,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,21],[12,184,0.0652,0.84375,0.26573,0.85714,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,9,0,17],[16,184,0.087,0.91071,0.1915,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,21],[20,184,0.1087,0.8125,0.31428,0.82143,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,19],[24,184,0.1304,0.87054,0.23517,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,21],[28,184,0.1522,0.8482,0.22287,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,9,0,16],[32,184,0.1739,0.82589,0.29824,0.85714,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,5,0,20],[36,184,0.1957,0.87052,0.24319,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,7,0,20],[40,184,0.2174,0.85268,0.30196,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,23],[44,184,0.2391,0.83928,0.26666,0.82132,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,3,0,0,4,0,20],[48,184,0.2609,0.85267,0.27079,0.85714,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,8,0,19],[52,184,0.2826,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[56,184,0.3043,0.78125,0.3369,0.82132,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,7,0,17],[60,184,0.3261,0.9375,0.11811,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[64,184,0.3478,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,184,0.3696,0.74107,0.31831,0.57143,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,11,0,11],[72,184,0.3913,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[76,184,0.413,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[80,184,0.4348,0.87946,0.08828,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,9],[84,184,0.4565,0.90178,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[88,184,0.4783,0.89285,0.08748,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11],[92,184,0.5,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[96,184,0.5217,0.91517,0.08646,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[100,184,0.5435,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[104,184,0.5652,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[108,184,0.587,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[112,184,0.6087,0.90625,0.09852,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,14],[116,184,0.6304,0.89285,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[120,184,0.6522,0.88839,0.08552,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,19,0,10],[124,184,0.6739,0.89731,0.08918,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12],[128,184,0.6957,0.91069,0.10569,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,16],[132,184,0.7174,0.86607,0.23673,0.85714,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,15],[136,184,0.7391,0.87054,0.23517,0.85714,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,15],[140,184,0.7609,0.90179,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[144,184,0.7826,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[148,184,0.8043,0.86607,0.17835,0.85714,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,11],[152,184,0.8261,0.90624,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[156,184,0.8478,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[160,184,0.8696,0.89732,0.10249,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14],[164,184,0.8913,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[168,184,0.913,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[172,184,0.9348,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[176,184,0.9565,0.86607,0.19212,0.85714,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,17,0,12],[180,184,0.9783,0.90179,0.17655,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,16],[184,184,1.0,0.82143,0.18558,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,11,0,9]]},{"b":6,"e":1.0,"k":"rising","v":0.62946,"x":1.0,"p":[[0,88,0.0,0.62946,0.37434,0.28571,0.78571,1.0,0.0,1.0,6,9,0,6,0,0,0,0,4,0,0,0,0,0,2,0,0,4,0,0,7,0,9],[4,88,0.0455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,88,0.0909,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,88,0.1364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,88,0.1818,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,88,0.2273,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,88,0.2727,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,88,0.3182,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,88,0.3636,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,88,0.4091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,88,0.4545,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,88,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,88,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,88,0.5909,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,88,0.6364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,88,0.6818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,88,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,88,0.7727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,88,0.8182,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,88,0.8636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,88,0.9091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,88,0.9545,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[88,88,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"46ce3222707e1f38","q":"Find all positive integers  $n$  for which there exists a set of exactly  $n$  distinct positive integers, none of which exceed  $n^2$ , whose reciprocals add up to  $1$ .","t":[{"b":2,"e":0.85714,"k":"rising","v":0.16518,"x":0.82588,"p":[[0,65,0.0,0.16518,0.1017,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,65,0.0615,0.741,0.23811,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,0,4,0,10],[8,65,0.1231,0.79907,0.21091,0.67857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,8,0,0,2,0,14],[12,65,0.1846,0.82588,0.20121,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,1,0,17],[16,65,0.2462,0.72319,0.17837,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,12,0,0,2,0,7],[20,65,0.3077,0.75892,0.16147,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,15,0,0,2,0,8],[24,65,0.3692,0.7366,0.25028,0.57143,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,4,0,0,1,0,13],[28,65,0.4308,0.67408,0.16842,0.57143,0.64286,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,9,0,0,3,0,4],[32,65,0.4923,0.683,0.1812,0.57132,0.71429,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,9,0,0,3,0,5],[36,65,0.5538,0.70081,0.17633,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,9,0,0,4,0,5],[40,65,0.6154,0.67837,0.20826,0.571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,9,0,0,4,0,5],[44,65,0.6769,0.77677,0.17106,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,13,0,0,2,0,10],[48,65,0.7385,0.72765,0.21831,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,10,0,0,8,0,0,1,0,10],[52,65,0.8,0.7857,0.20204,0.67857,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,10,0,0,1,0,13],[56,65,0.8615,0.75,0.15972,0.57143,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,14,0,0,1,0,8],[60,65,0.9231,0.80355,0.15048,0.71429,0.71429,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,3,0,10],[64,65,0.9846,0.79464,0.20806,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,11,0,0,2,0,13],[65,65,1.0,0.79463,0.15544,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,3,0,10]]},{"b":5,"e":0.71429,"k":"rising","v":0.22759,"x":0.78122,"p":[[0,114,0.0,0.22759,0.17449,0.14286,0.14286,0.2857,0.0,0.71429,1,0,0,1,0,21,0,0,6,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[4,114,0.0351,0.78122,0.20823,0.57143,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,0,2,0,13],[8,114,0.0702,0.72764,0.21832,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,10,0,0,8,0,0,1,0,10],[12,114,0.1053,0.73658,0.16411,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,13,0,0,4,0,6],[16,114,0.1404,0.66961,0.18708,0.57143,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,11,0,0,1,0,5],[20,114,0.1754,0.76786,0.19149,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,14,0,0,0,0,11],[24,114,0.2105,0.75446,0.1684,0.57143,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,0,4,0,8],[28,114,0.2456,0.69643,0.23891,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,2,0,0,7,0,0,10,0,0,2,0,8],[32,114,0.2807,0.70086,0.21537,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,0,0,0,7,0,0,12,0,0,5,0,5],[36,114,0.3158,0.70085,0.21832,0.571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,10,0,0,0,0,9],[40,114,0.3509,0.61159,0.27019,0.42857,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,5,0,0,1,0,0,4,0,0,5,0,0,8,0,0,5,0,4],[44,114,0.386,0.76338,0.2008,0.67857,0.71429,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,13,0,0,0,0,11],[48,114,0.4211,0.62052,0.23854,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,1,0,0,9,0,0,12,0,0,1,0,4],[52,114,0.4561,0.6339,0.1819,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,13,0,0,2,0,2],[56,114,0.4912,0.65624,0.19187,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,15,0,0,0,0,4],[60,114,0.5263,0.62496,0.22233,0.571,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,2,0,0,10,0,0,10,0,0,3,0,3],[64,114,0.5614,0.74552,0.20435,0.67857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,13,0,0,2,0,9],[68,114,0.5965,0.72768,0.10926,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0,1,0,3],[72,114,0.6316,0.60268,0.20743,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,4,0,0,4,0,0,17,0,0,0,0,2],[76,114,0.6667,0.64286,0.26486,0.42857,0.71429,0.75,0.0,1.0,1,6,0,1,0,2,0,0,1,0,0,6,0,0,2,0,0,12,0,0,2,0,6],[80,114,0.7018,0.67408,0.20278,0.57143,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,14,0,0,1,0,5],[84,114,0.7368,0.70536,0.11812,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,24,0,0,1,0,2],[88,114,0.7719,0.73214,0.09942,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,25,0,0,1,0,3],[92,114,0.807,0.73647,0.11358,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,0,0,4],[96,114,0.8421,0.74551,0.14613,0.71429,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,23,0,0,0,0,6],[100,114,0.8772,0.72768,0.10326,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,24,0,0,3,0,2],[104,114,0.9123,0.71874,0.15766,0.71429,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,22,0,0,0,0,5],[108,114,0.9474,0.71429,0.11294,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,26,0,0,1,0,2],[112,114,0.9825,0.71427,0.15569,0.71429,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,21,0,0,3,0,3],[114,114,1.0,0.74107,0.07523,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,2,0,2]]}]},{"i":"e035e37e537f26f8","q":"Find the number of pairs of integers  $x$  and  $y$  such that  $x^2 + xy + y^2 = 28$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.64732,"x":0.74554,"p":[[0,32,0.0,0.64732,0.19556,0.5357,0.71429,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,12,0,0,7,0,1],[4,32,0.125,0.73658,0.0883,0.71429,0.71429,0.74996,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,8,0,0],[8,32,0.25,0.70533,0.0794,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,0,3,0,0],[12,32,0.375,0.72768,0.09006,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,22,0,0,7,0,0],[16,32,0.5,0.74554,0.05906,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0],[20,32,0.625,0.74107,0.05575,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0],[24,32,0.75,0.73659,0.06295,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,6,0,0],[28,32,0.875,0.74106,0.06625,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,7,0,0],[32,32,1.0,0.72321,0.0497,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.66964,"x":0.74996,"p":[[0,28,0.0,0.66964,0.18363,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,18,0,0,7,0,0],[4,28,0.1429,0.74104,0.07528,0.71429,0.71429,0.74996,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,0,8,0,0],[8,28,0.2857,0.74996,0.08753,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,11,0,0],[12,28,0.4286,0.71862,0.08365,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,24,0,0,5,0,0],[16,28,0.5714,0.74107,0.12078,0.71429,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,19,0,0,11,0,0],[20,28,0.7143,0.73214,0.08564,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,7,0,0],[24,28,0.8571,0.74107,0.06622,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,7,0,0],[28,28,1.0,0.74553,0.08552,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,10,0,0]]}]},{"i":"b4b93c63a8a39d72","q":"Find the maximum constant  $C$  such that, whenever  $\\{a_n \\}_{n=1}^{\\infty}$  is a sequence of positive real numbers satisfying  $a_{n+1}-a_n=a_n(a_n+1)(a_n+2)$ ,  we have  $$ \\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C. $$","t":[{"b":1,"e":0.85714,"k":"flat","v":0.71429,"x":0.84821,"p":[[0,74,0.0,0.80357,0.15047,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,3,0,10],[4,74,0.0541,0.71429,0.15152,0.71429,0.71429,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,17,0,0,6,0,2],[8,74,0.1081,0.79464,0.12846,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,12,0,5],[12,74,0.1622,0.78571,0.10102,0.71429,0.78571,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,14,0,2],[16,74,0.2162,0.82143,0.12877,0.71429,0.85714,0.85714,0.5714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,13,0,7],[20,74,0.2703,0.79018,0.1988,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,13,0,6],[24,74,0.3243,0.80357,0.10564,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,13,0,4],[28,74,0.3784,0.75446,0.22084,0.71429,0.78571,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,12,0,4],[32,74,0.4324,0.79463,0.11261,0.71429,0.78571,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,12,0,4],[36,74,0.4865,0.84821,0.11259,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,15,0,8],[40,74,0.5405,0.82143,0.13363,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,11,0,8],[44,74,0.5946,0.77679,0.16342,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,11,0,0,15,0,3],[48,74,0.6486,0.76786,0.13716,0.71429,0.78569,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,13,0,3],[52,74,0.7027,0.81696,0.17582,0.71429,0.85714,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,16,0,6],[56,74,0.7568,0.79911,0.09354,0.71429,0.78571,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,13,0,3],[60,74,0.8108,0.82589,0.11701,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,15,0,6],[64,74,0.8649,0.84375,0.10326,0.71429,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,15,0,7],[68,74,0.9189,0.81696,0.10853,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,17,0,4],[72,74,0.973,0.81696,0.10853,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,17,0,4],[74,74,1.0,0.82143,0.10714,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,15,0,5]]},{"b":6,"e":0.85714,"k":"flat","v":0.65179,"x":0.85268,"p":[[0,71,0.0,0.72768,0.16115,0.71429,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,17,0,0,3,0,5],[4,71,0.0563,0.76339,0.09852,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,9,0,2],[8,71,0.1127,0.65179,0.21706,0.67857,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,0,0,0,1,0,0,4,0,0,18,0,0,6,0,0],[12,71,0.169,0.70536,0.15542,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,0,4,0,1],[16,71,0.2254,0.66518,0.21902,0.57143,0.71429,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,0,0,0,2,0,0,6,0,0,16,0,0,3,0,3],[20,71,0.2817,0.74107,0.09062,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,24,0,0,6,0,1],[24,71,0.338,0.71875,0.10999,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,19,0,0,6,0,1],[28,71,0.3944,0.72321,0.18536,0.71429,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,19,0,0,6,0,3],[32,71,0.4507,0.70982,0.20666,0.71429,0.71429,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,16,0,0,7,0,3],[36,71,0.507,0.80804,0.09182,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,18,0,2],[40,71,0.5634,0.73213,0.15873,0.71429,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,16,0,0,9,0,2],[44,71,0.6197,0.78125,0.10705,0.71429,0.78571,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,14,0,2],[48,71,0.6761,0.85268,0.11564,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,14,0,9],[52,71,0.7324,0.80357,0.13243,0.71429,0.78571,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,9,0,7],[56,71,0.7887,0.77679,0.16342,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,11,0,0,13,0,4],[60,71,0.8451,0.8125,0.12078,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,15,0,5],[64,71,0.9014,0.81696,0.12492,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,14,0,6],[68,71,0.9577,0.83482,0.08828,0.71429,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,19,0,4],[71,71,1.0,0.77679,0.16342,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,15,0,4]]}]},{"i":"025cf10f56630e42","q":"Find all positive integers  $n$  such that there are positive integers  $a_1,\\cdots,a_n, b_1,\\cdots,b_n$  that satisfy\n\n \\[(a_1^2+\\cdots+a_n^2)(b_1^2+\\cdots+b_n^2)-(a_1b_1+\\cdots+a_nb_n)^2=n\\]","t":[{"b":3,"e":0.57143,"k":"flat","v":0.62944,"x":0.81696,"p":[[0,112,0.0,0.66964,0.26592,0.42857,0.71429,1.0,0.0,1.0,1,10,1,1,0,0,0,0,1,0,0,9,0,0,4,0,0,7,0,0,0,0,10],[4,112,0.0357,0.71427,0.20517,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,11,0,0,1,0,8],[8,112,0.0714,0.73214,0.20124,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,8,0,0,6,0,7],[12,112,0.1071,0.75445,0.19312,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,10,0,0,2,0,10],[16,112,0.1429,0.81696,0.20277,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,3,0,15],[20,112,0.1786,0.69196,0.18249,0.57143,0.71429,0.85704,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,11,0,0,5,0,4],[24,112,0.2143,0.70536,0.2257,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,8,0,0,1,0,9],[28,112,0.25,0.7232,0.18878,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,9,0,0,7,0,5],[32,112,0.2857,0.70088,0.20001,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,10,0,0,2,0,7],[36,112,0.3214,0.73659,0.20859,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,5,0,0,5,0,9],[40,112,0.3571,0.65625,0.20781,0.53571,0.71429,0.71429,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,7,0,0,7,0,0,10,0,0,2,0,5],[44,112,0.3929,0.79462,0.19544,0.57143,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,3,0,13],[48,112,0.4286,0.74554,0.19475,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,10,0,0,1,0,10],[52,112,0.4643,0.77677,0.21999,0.57143,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,6,0,0,1,0,14],[56,112,0.5,0.76339,0.18766,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,6,0,0,6,0,9],[60,112,0.5357,0.75893,0.22142,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,9,0,0,1,0,12],[64,112,0.5714,0.73213,0.20125,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,7,0,0,3,0,9],[68,112,0.6071,0.73661,0.18595,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,10,0,0,1,0,9],[72,112,0.6429,0.75892,0.2096,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,8,0,0,1,0,12],[76,112,0.6786,0.72768,0.17261,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,12,0,0,4,0,6],[80,112,0.7143,0.70088,0.20628,0.57143,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,3,0,0,3,0,8],[84,112,0.75,0.70982,0.18029,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,8,0,0,4,0,6],[88,112,0.7857,0.73652,0.22922,0.57143,0.71429,1.0,0.14,1.0,0,10,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,6,0,0,4,0,10],[92,112,0.8214,0.63837,0.19556,0.57143,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,4,0,0,16,0,0,5,0,0,1,0,5],[96,112,0.8571,0.71429,0.18211,0.57143,0.64286,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,7,0,0,1,0,8],[100,112,0.8929,0.62944,0.13297,0.57143,0.57143,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,5,0,0,0,0,3],[104,112,0.9286,0.76338,0.20081,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,4,0,0,2,0,12],[108,112,0.9643,0.70088,0.13535,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,12,0,0,4,0,3],[112,112,1.0,0.6607,0.18122,0.57143,0.57143,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,18,0,0,4,0,0,4,0,4]]},{"b":4,"e":0.42857,"k":"flat","v":0.45536,"x":0.79911,"p":[[0,83,0.0,0.62946,0.27166,0.42857,0.64286,0.75,0.0,1.0,2,7,2,2,0,0,0,0,2,0,0,6,0,0,6,0,0,8,0,0,1,0,7],[4,83,0.0482,0.79911,0.21087,0.71429,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,0,0,0,15],[8,83,0.0964,0.71874,0.22157,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,11,0,0,1,0,9],[12,83,0.1446,0.68747,0.18365,0.57143,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,10,0,0,3,0,5],[16,83,0.1928,0.69644,0.2335,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,6,0,0,4,0,8],[20,83,0.241,0.67409,0.22371,0.53539,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,6,0,0,6,0,0,7,0,0,6,0,5],[24,83,0.2892,0.69197,0.22047,0.57143,0.64286,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,6,0,0,9,0,0,5,0,0,3,0,8],[28,83,0.3373,0.66513,0.22479,0.42859,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,7,0,0,8,0,0,4,0,0,5,0,6],[32,83,0.3855,0.63392,0.18188,0.5354,0.57143,0.71429,0.286,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,10,0,0,8,0,0,3,0,3],[36,83,0.4337,0.68304,0.24932,0.42857,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,9,0,0,5,0,0,3,0,0,4,0,9],[40,83,0.4819,0.71873,0.21573,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,6,0,0,5,0,8],[44,83,0.5301,0.67407,0.17584,0.57143,0.57143,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,5,0,0,5,0,4],[48,83,0.5783,0.68747,0.2096,0.57132,0.64286,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,6,0,0,3,0,7],[52,83,0.6265,0.66076,0.25688,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,10,0,0,5,0,0,4,0,0,2,0,9],[56,83,0.6747,0.60714,0.18898,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,9,0,0,5,0,0,11,0,0,4,0,1],[60,83,0.7229,0.58481,0.23517,0.42857,0.57121,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,11,0,0,5,0,0,6,0,0,1,0,5],[64,83,0.7711,0.57588,0.21572,0.42857,0.4998,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,11,0,0,3,0,0,5,0,0,7,0,1],[68,83,0.8193,0.57589,0.21572,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,8,0,0,7,0,0,6,0,0,4,0,2],[72,83,0.8675,0.51339,0.21683,0.42857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,14,0,0,2,0,0,3,0,0,5,0,1],[76,83,0.9157,0.49106,0.15946,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,14,0,0,6,0,0,4,0,0,2,0,0],[80,83,0.9639,0.45536,0.15335,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,16,0,0,6,0,0,1,0,0,2,0,0],[83,83,1.0,0.5134,0.1984,0.42857,0.42859,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,14,0,0,5,0,0,3,0,0,5,0,0]]}]},{"i":"0a62915c21455c8f","q":"Find the smallest positive integer  $k$  which is representable in the form  $k=19^n-5^m$  for some positive integers  $m$  and  $n$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.78125,"x":0.94643,"p":[[0,34,0.0,0.84822,0.27418,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,22],[4,34,0.1176,0.94643,0.11152,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,23],[8,34,0.2353,0.91964,0.18536,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,5,0,24],[12,34,0.3529,0.90179,0.21558,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,22],[16,34,0.4706,0.93304,0.17491,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,5,0,25],[20,34,0.5882,0.78125,0.29877,0.64286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,0,7,0,16],[24,34,0.7059,0.91072,0.18123,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,7,0,22],[28,34,0.8235,0.92411,0.21124,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,27],[32,34,0.9412,0.83482,0.30328,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,22],[34,34,1.0,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16]]},{"b":7,"e":0.71429,"k":"falling","v":0.53124,"x":0.97321,"p":[[0,38,0.0,0.85705,0.27687,0.85714,1.0,1.0,0.14,1.0,0,22,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,5,0,22],[4,38,0.1053,0.83482,0.2699,0.75,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,2,0,22],[8,38,0.2105,0.91518,0.19186,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,25],[12,38,0.3158,0.88839,0.23619,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,5,0,23],[16,38,0.4211,0.90625,0.19103,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,5,0,23],[20,38,0.5263,0.89286,0.25,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,25],[24,38,0.6316,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[28,38,0.7368,0.93303,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,25],[32,38,0.8421,0.67411,0.35934,0.39286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,7,0,0,1,0,0,5,0,0,1,0,0,0,0,0,3,0,15],[36,38,0.9474,0.66504,0.38065,0.14286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,10,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,15],[38,38,1.0,0.53124,0.39323,0.14286,0.42836,1.0,0.14286,1.0,0,10,0,0,0,15,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,10]]}]},{"i":"42592bd0aaccc2cd","q":"Find three consecutive odd numbers  $a,b,c$  such that  $a^2+b^2+c^2$  is a four digit number with four equal digits.","t":[{"b":0,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,18,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.857,"k":"flat","v":0.98214,"x":1.0,"p":[[0,63,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,63,0.0635,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,63,0.127,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,63,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,63,0.254,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,63,0.3175,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,63,0.381,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,63,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,63,0.5079,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,63,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,63,0.6349,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,63,0.6984,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,63,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,63,0.8254,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,63,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,63,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[63,63,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c9dd7c7237487b26","q":"Find the number of primes  $p$  between  $100$  and  $200$  for which  $x^{11}+y^{16}\\equiv 2013\\pmod p$  has a solution in integers  $x$  and  $y$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.82142,"x":0.98214,"p":[[0,51,0.0,0.85268,0.20355,0.57143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,21],[4,51,0.0784,0.91517,0.15513,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23],[8,51,0.1569,0.88839,0.17762,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,2,0,22],[12,51,0.2353,0.91071,0.15465,0.85714,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,23],[16,51,0.3137,0.88392,0.19047,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,2,0,22],[20,51,0.3922,0.875,0.19805,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,1,0,22],[24,51,0.4706,0.87946,0.19269,0.78571,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,2,0,22],[28,51,0.549,0.82143,0.20203,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,4,0,16],[32,51,0.6275,0.84374,0.21239,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,1,0,20],[36,51,0.7059,0.82142,0.2113,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,1,0,0,1,0,18],[40,51,0.7843,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,51,0.8627,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,51,0.9412,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[51,51,1.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]},{"b":3,"e":1.0,"k":"rising","v":0.82143,"x":0.98214,"p":[[0,48,0.0,0.82143,0.21724,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,19],[4,48,0.0833,0.86607,0.18536,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,4,0,19],[8,48,0.1667,0.89284,0.18901,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,2,0,23],[12,48,0.25,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[16,48,0.3333,0.875,0.17768,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,5,0,19],[20,48,0.4167,0.92856,0.15155,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,26],[24,48,0.5,0.84374,0.19353,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,5,0,17],[28,48,0.5833,0.86607,0.18877,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0,7,0,18],[32,48,0.6667,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[36,48,0.75,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,48,0.8333,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,48,0.9167,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[48,48,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"0767b3df02d43fcb","q":"Find the smallest term of the sequence  $a_1, a_2, a_3, \\ldots$  defined by  $a_1=2014^{2015^{2016}}$  and  $$ a_{n+1}= \n\\begin{cases}\n\\frac{a_n}{2} & \\text{ if } a_n \\text{ is even} \na_n + 7 & \\text{ if } a_n \\text{ is odd} \n\\end{cases} $$","t":[{"b":5,"e":0.42857,"k":"flat","v":0.29454,"x":0.63839,"p":[[0,128,0.0,0.29454,0.23678,0.14286,0.21429,0.42857,0.0,0.85714,5,0,2,5,0,11,0,0,4,0,0,5,0,0,5,0,0,0,0,0,2,0,0],[4,128,0.0312,0.63839,0.27429,0.42857,0.64286,0.85714,0.0,1.0,1,4,1,1,0,1,0,0,5,0,0,2,0,0,7,0,0,2,0,0,10,0,4],[8,128,0.0625,0.625,0.25692,0.42857,0.64286,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,11,0,0,2,0,0,2,0,0,12,0,2],[12,128,0.0938,0.58481,0.25092,0.42857,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,8,0,0,4,0,0,2,0,0,12,0,0],[16,128,0.125,0.61607,0.26107,0.42857,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,4,0,0,4,0,0,4,0,0,12,0,1],[20,128,0.1562,0.60714,0.24999,0.42857,0.57143,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,9,0,0,3,0,0,2,0,0,12,0,1],[24,128,0.1875,0.61607,0.25111,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,8,0,0,2,0,0,3,0,0,14,0,0],[28,128,0.2188,0.58929,0.23623,0.42857,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,7,0,0,6,0,0,4,0,0,10,0,0],[32,128,0.25,0.59821,0.26592,0.39286,0.57143,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,6,0,0,3,0,0,1,0,0,13,0,1],[36,128,0.2812,0.58036,0.23941,0.39286,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,6,0,0,4,0,0,3,0,0,11,0,0],[40,128,0.3125,0.54909,0.25028,0.42857,0.4998,0.75,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,9,0,0,5,0,0,3,0,0,6,0,2],[44,128,0.3438,0.51339,0.25966,0.42857,0.42857,0.75,0.0,0.85714,3,0,0,3,0,1,0,0,2,0,0,12,0,0,4,0,0,2,0,0,8,0,0],[48,128,0.375,0.54018,0.33069,0.28571,0.57143,0.85714,0.0,1.0,3,2,0,3,0,4,0,0,5,0,0,4,0,0,0,0,0,3,0,0,11,0,2],[52,128,0.4062,0.54015,0.19799,0.42857,0.4286,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,15,0,0,6,0,0,2,0,0,5,0,1],[56,128,0.4375,0.57588,0.22156,0.42857,0.4998,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,11,0,0,2,0,0,5,0,0,9,0,0],[60,128,0.4688,0.4732,0.31019,0.25,0.42857,0.75,0.0,1.0,4,2,0,4,0,4,0,0,3,0,0,9,0,0,1,0,0,3,0,0,6,0,2],[64,128,0.5,0.58482,0.27747,0.42857,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,4,0,0,6,0,0,5,0,0,1,0,0,12,0,1],[68,128,0.5312,0.51339,0.24964,0.39286,0.42857,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,13,0,0,2,0,0,0,0,0,8,0,1],[72,128,0.5625,0.49107,0.25739,0.28571,0.42857,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,10,0,0,7,0,0,4,0,0,2,0,0,6,0,1],[76,128,0.5938,0.50892,0.26229,0.28571,0.42857,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,6,0,0,11,0,0,3,0,0,2,0,0,4,0,3],[80,128,0.625,0.47313,0.27078,0.28571,0.42857,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,5,0,0,8,0,0,4,0,0,4,0,0,3,0,2],[84,128,0.6562,0.57143,0.27664,0.42857,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,4,0,0,8,0,0,3,0,0,2,0,0,11,0,1],[88,128,0.6875,0.48212,0.25442,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,5,0,0,11,0,0,5,0,0,0,0,0,6,0,1],[92,128,0.7188,0.45089,0.24513,0.28571,0.42857,0.60714,0.0,0.85714,2,0,0,2,0,1,0,0,10,0,0,10,0,0,1,0,0,2,0,0,6,0,0],[96,128,0.75,0.4107,0.16655,0.39286,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,1,0,0,5,0,0,18,0,0,4,0,0,1,0,0,1,0,0],[100,128,0.7812,0.40624,0.22045,0.28571,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,2,0,0,10,0,0,12,0,0,1,0,0,1,0,0,4,0,0],[104,128,0.8125,0.3839,0.19701,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,3,0,0,13,0,0,9,0,0,3,0,0,0,0,0,3,0,0],[108,128,0.8438,0.43304,0.23002,0.28571,0.42857,0.57143,0.0,1.0,2,2,0,2,0,0,0,0,12,0,0,9,0,0,4,0,0,2,0,0,1,0,2],[112,128,0.875,0.45089,0.19269,0.28571,0.42857,0.42858,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,15,0,0,1,0,0,1,0,0,3,0,1],[116,128,0.9062,0.41518,0.15714,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,9,0,0,17,0,0,3,0,0,0,0,0,2,0,0],[120,128,0.9375,0.375,0.15047,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,2,0,0,11,0,0,14,0,0,3,0,0,0,0,0,1,0,0],[124,128,0.9688,0.39732,0.14167,0.28571,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,18,0,0,0,0,0,0,0,0,2,0,0],[128,128,1.0,0.35268,0.14279,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,19,0,0,10,0,0,0,0,0,1,0,0,1,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.38839,"x":0.94196,"p":[[0,86,0.0,0.38839,0.28846,0.14286,0.42857,0.57143,0.0,1.0,3,2,3,3,0,10,0,0,2,0,0,7,0,0,5,0,0,0,0,0,3,0,2],[4,86,0.0465,0.67856,0.28122,0.42857,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0,3,0,0,8,0,8],[8,86,0.093,0.73661,0.27225,0.57143,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,2,0,0,4,0,0,3,0,0,10,0,9],[12,86,0.1395,0.69643,0.28516,0.42857,0.78571,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,3,0,0,5,0,0,0,0,0,6,0,0,8,0,8],[16,86,0.186,0.76339,0.27342,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,5,0,0,3,0,0,1,0,0,9,0,12],[20,86,0.2326,0.79911,0.2392,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,3,0,0,12,0,11],[24,86,0.2791,0.75,0.26726,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,5,0,0,3,0,0,2,0,0,10,0,10],[28,86,0.3256,0.72321,0.28107,0.42859,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,2,0,0,7,0,0,2,0,0,2,0,0,7,0,11],[32,86,0.3721,0.74554,0.28735,0.42857,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,7,0,0,2,0,0,3,0,0,4,0,14],[36,86,0.4186,0.75893,0.25111,0.42859,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,2,0,0,4,0,14],[40,86,0.4651,0.86607,0.21706,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,7,0,19],[44,86,0.5116,0.77232,0.25966,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,0,10,0,11],[48,86,0.5581,0.8616,0.2461,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,19],[52,86,0.6047,0.81697,0.23483,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,0,0,0,7,0,16],[56,86,0.6512,0.77679,0.31326,0.53571,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,0,0,0,5,0,0,1,0,0,1,0,0,5,0,17],[60,86,0.6977,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[64,86,0.7442,0.92857,0.15972,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,22],[68,86,0.7907,0.87946,0.11904,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,17,0,11],[72,86,0.8372,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[76,86,0.8837,0.8616,0.07563,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,23,0,5],[80,86,0.9302,0.83482,0.08828,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,22,0,3],[84,86,0.9767,0.84821,0.07936,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,22,0,4],[86,86,1.0,0.83482,0.0724,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,26,0,1]]}]},{"i":"1f30d7ed2e2bcb4b","q":"For  $a_1 = 3$ , define the sequence  $a_1, a_2, a_3, \\ldots$  for  $n \\geq 1$  as   $$ na_{n+1}=2(n+1)a_n-n-2. $$  Prove that for any odd prime  $p$ , there exist positive integer  $m,$  such that  $p|a_m$  and  $p|a_{m+1}.$","t":[{"b":0,"e":1.0,"k":"flat","v":0.88839,"x":0.99107,"p":[[0,46,0.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[4,46,0.087,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,46,0.1739,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,46,0.2609,0.90624,0.20397,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,25],[16,46,0.3478,0.89732,0.26302,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[20,46,0.4348,0.92857,0.16751,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[24,46,0.5217,0.88839,0.23347,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,24],[28,46,0.6087,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,46,0.6957,0.92857,0.16751,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[36,46,0.7826,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,46,0.8696,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,46,0.9565,0.94642,0.11156,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[46,46,1.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26]]},{"b":5,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,94,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,94,0.0426,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,94,0.0851,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,94,0.1277,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,94,0.1702,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,94,0.2128,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,94,0.2553,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,94,0.2979,0.9375,0.20183,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[32,94,0.3404,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,94,0.383,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[40,94,0.4255,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,94,0.4681,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,94,0.5106,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,94,0.5532,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,94,0.5957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,94,0.6383,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[64,94,0.6809,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[68,94,0.7234,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,94,0.766,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,94,0.8085,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[80,94,0.8511,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,94,0.8936,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,94,0.9362,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[92,94,0.9787,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[94,94,1.0,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"a187ba0afef7ae13","q":"Find the sum of all positive integers  $n$  such that \\[ \\frac{2n+1}{n(n-1)} \\] has a terminating decimal representation.\n\n*Proposed by Evan Chen*","t":[{"b":2,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,92,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,92,0.0435,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,92,0.087,0.97754,0.08868,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,92,0.1304,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,92,0.1739,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,92,0.2174,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,92,0.2609,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,92,0.3043,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,92,0.3478,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,92,0.3913,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,92,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,92,0.4783,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,92,0.5217,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,92,0.5652,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,92,0.6087,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,92,0.6522,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,92,0.6957,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[68,92,0.7391,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,92,0.7826,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,92,0.8261,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,92,0.8696,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,92,0.913,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[88,92,0.9565,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[92,92,1.0,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23]]},{"b":4,"e":0.85714,"k":"flat","v":0.95089,"x":1.0,"p":[[0,105,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,105,0.0381,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,105,0.0762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,105,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,105,0.1524,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,105,0.1905,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,105,0.2286,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,105,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,105,0.3048,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,105,0.3429,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[40,105,0.381,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[44,105,0.419,0.95981,0.08174,1.0,1.0,1.0,0.714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[48,105,0.4571,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,105,0.4952,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,105,0.5333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,105,0.5714,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,105,0.6095,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,105,0.6476,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,105,0.6857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,105,0.7238,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,105,0.7619,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,105,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,105,0.8381,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[92,105,0.8762,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,105,0.9143,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[100,105,0.9524,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[104,105,0.9905,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[105,105,1.0,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]}]},{"i":"fa0af85476ee4ba8","q":"Find all possible  $\\{ x_1,x_2,...x_n \\}$  permutations of  $ \\{1,2,...,n \\}$  so that  when  $1\\le i \\le n-2 $  then we have   $x_i < x_{i+2}$  and when  $1 \\le i \\le n-3$  then we have   $x_i < x_{i+3}$  . Here  $n \\ge 4$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.2231,"x":0.45086,"p":[[0,22,0.0,0.28562,0.29885,0.0,0.2857,0.28571,0.0,1.0,10,3,0,10,0,4,0,0,11,0,0,0,0,0,3,0,0,1,0,0,0,0,3],[4,22,0.1818,0.45086,0.34275,0.14286,0.28571,0.75,0.0,1.0,1,7,0,1,0,10,0,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,7],[8,22,0.3636,0.35268,0.26722,0.14286,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,11,0,0,14,0,0,1,0,0,0,0,0,1,0,0,3,0,2],[12,22,0.5455,0.26784,0.23074,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,21,0,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[16,22,0.7273,0.2231,0.20496,0.14286,0.14286,0.14287,0.0,0.85714,3,0,0,3,0,22,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,0],[20,22,0.9091,0.33469,0.21905,0.14286,0.28571,0.571,0.0,0.857,1,0,0,1,0,12,0,0,8,0,0,1,0,0,7,0,0,2,0,0,1,0,0],[22,22,1.0,0.27668,0.21709,0.14286,0.14286,0.32143,0.0,0.85714,1,0,0,1,0,19,0,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.03125,"x":0.41516,"p":[[0,55,0.0,0.36161,0.41647,0.0,0.21428,1.0,0.0,1.0,13,9,0,13,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[4,55,0.0727,0.41516,0.29311,0.24999,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,8,0,0,14,0,0,0,0,0,3,0,0,1,0,0,2,0,4],[8,55,0.1455,0.375,0.26904,0.2857,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,7,0,0,18,0,0,1,0,0,0,0,0,2,0,0,0,0,4],[12,55,0.2182,0.32134,0.23964,0.14286,0.28571,0.28571,0.14,1.0,0,2,0,0,0,13,0,0,12,0,0,1,0,0,3,0,0,0,0,0,1,0,2],[16,55,0.2909,0.33928,0.29397,0.14286,0.2857,0.28571,0.0,1.0,2,4,0,2,0,11,0,0,12,0,0,1,0,0,1,0,0,0,0,0,1,0,4],[20,55,0.3636,0.30801,0.24248,0.14286,0.2857,0.28571,0.0,1.0,3,2,0,3,0,9,0,0,13,0,0,1,0,0,3,0,0,1,0,0,0,0,2],[24,55,0.4364,0.37491,0.28746,0.14286,0.28571,0.57143,0.0,1.0,2,4,0,2,0,8,0,0,12,0,0,1,0,0,4,0,0,1,0,0,0,0,4],[28,55,0.5091,0.38828,0.33931,0.14286,0.28571,0.57111,0.0,1.0,4,6,0,4,0,9,0,0,8,0,0,1,0,0,3,0,0,1,0,0,0,0,6],[32,55,0.5818,0.29009,0.20979,0.14286,0.2857,0.28571,0.0,1.0,1,1,0,1,0,12,0,0,14,0,0,0,0,0,3,0,0,0,0,0,1,0,1],[36,55,0.6545,0.19643,0.23076,0.0,0.14288,0.28571,0.0,1.0,13,1,0,13,0,5,0,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[40,55,0.7273,0.1607,0.17764,0.0,0.14286,0.2857,0.0,0.57143,14,0,0,14,0,6,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[44,55,0.8,0.23661,0.27574,0.0,0.14286,0.28571,0.0,1.0,11,2,0,11,0,8,0,0,7,0,0,0,0,0,3,0,0,1,0,0,0,0,2],[48,55,0.8727,0.14723,0.24611,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[52,55,0.9455,0.14286,0.25754,0.0,0.0,0.17857,0.0,1.0,22,1,0,22,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[55,55,1.0,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5c892254639ba932","q":"Find all positive integers such that they have  $6$  divisors (without  $1$  and the number itself) and the sum of the divisors is  $14133$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.35715,"x":0.83929,"p":[[0,56,0.0,0.75008,0.26249,0.57132,0.85714,1.0,0.0,1.0,1,11,1,1,0,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,8,0,11],[4,56,0.0714,0.75892,0.27534,0.71429,0.85714,0.85714,0.0,1.0,2,7,1,2,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,16,0,7],[8,56,0.1429,0.83929,0.18123,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,8],[12,56,0.2143,0.70535,0.31326,0.71429,0.85714,0.85714,0.0,1.0,3,5,2,3,0,1,0,0,3,0,0,0,0,0,0,0,0,4,0,0,16,0,5],[16,56,0.2857,0.78112,0.24743,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,0,13,0,9],[20,56,0.3571,0.78125,0.2082,0.82143,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,2,0,0,19,0,5],[24,56,0.4286,0.70536,0.32328,0.42857,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,5,0,0,4,0,0,0,0,0,1,0,0,9,0,11],[28,56,0.5,0.59821,0.32623,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,5,0,0,7,0,0,3,0,0,0,0,0,2,0,0,9,0,6],[32,56,0.5714,0.67411,0.28845,0.42857,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,3,0,0,6,0,0,0,0,0,1,0,0,14,0,5],[36,56,0.6429,0.68748,0.28446,0.53539,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,5,0,0,1,0,0,2,0,0,7,0,0,8,0,7],[40,56,0.7143,0.47759,0.25167,0.28571,0.42857,0.60712,0.14,1.0,0,3,0,0,0,2,0,0,11,0,0,10,0,0,1,0,0,2,0,0,3,0,3],[44,56,0.7857,0.54465,0.3163,0.28571,0.42857,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,8,0,0,7,0,0,0,0,0,2,0,0,5,0,6],[48,56,0.8571,0.48219,0.27836,0.28571,0.42857,0.50107,0.0,1.0,1,5,0,1,0,4,0,0,4,0,0,15,0,0,0,0,0,2,0,0,1,0,5],[52,56,0.9286,0.41964,0.20183,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,16,0,0,1,0,0,1,0,0,2,0,1],[56,56,1.0,0.35715,0.10714,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,6,0,0,21,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.67411,"x":0.84165,"p":[[0,134,0.0,0.71875,0.25376,0.57143,0.85714,0.85714,0.0,1.0,1,5,1,1,0,1,0,0,1,0,0,4,0,0,2,0,0,5,0,0,13,0,5],[4,134,0.0299,0.77677,0.28108,0.82143,0.85714,1.0,0.0,1.0,1,10,1,1,0,3,0,0,0,0,0,0,0,0,3,0,0,1,0,0,14,0,10],[8,134,0.0597,0.77232,0.26453,0.71429,0.85714,1.0,0.0,1.0,1,10,1,1,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,0,11,0,10],[12,134,0.0896,0.71875,0.33973,0.64286,0.85714,1.0,0.0,1.0,3,10,2,3,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,0,12,0,10],[16,134,0.1194,0.83482,0.16409,0.85714,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,21,0,6],[20,134,0.1493,0.8125,0.25364,0.85714,0.85714,1.0,0.0,1.0,2,10,1,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,16,0,10],[24,134,0.1791,0.82589,0.23347,0.85714,0.85714,1.0,0.0,1.0,1,9,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,19,0,9],[28,134,0.209,0.81697,0.21199,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,17,0,8],[32,134,0.2388,0.80357,0.23351,0.82132,0.85714,0.89286,0.0,1.0,1,8,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,16,0,8],[36,134,0.2687,0.78125,0.27661,0.71429,0.85714,1.0,0.0,1.0,2,10,2,2,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,0,12,0,10],[40,134,0.2985,0.79463,0.20807,0.82143,0.85714,0.85714,0.0,1.0,1,4,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,20,0,4],[44,134,0.3284,0.79911,0.1984,0.82143,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,0,18,0,6],[48,134,0.3582,0.75893,0.22711,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,6,0,0,17,0,4],[52,134,0.3881,0.76772,0.18817,0.71429,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,16,0,4],[56,134,0.4179,0.76785,0.24419,0.85711,0.85714,0.85714,0.0,1.0,2,2,1,2,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,23,0,2],[60,134,0.4478,0.69196,0.27919,0.39286,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,7,0,0,1,0,0,0,0,0,4,0,0,15,0,4],[64,134,0.4776,0.7366,0.23987,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,4,0,0,17,0,4],[68,134,0.5075,0.69643,0.3004,0.53572,0.85714,0.85714,0.0,1.0,2,4,2,2,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,0,17,0,4],[72,134,0.5373,0.71428,0.27894,0.74989,0.85714,0.85714,0.0,1.0,1,2,1,1,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,22,0,2],[76,134,0.5672,0.79017,0.20198,0.85711,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,20,0,5],[80,134,0.597,0.84165,0.09051,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,1,0,24,0,3],[84,134,0.6269,0.81696,0.12492,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,19,0,4],[88,134,0.6567,0.80356,0.10564,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,21,0,1],[92,134,0.6866,0.82142,0.10101,0.82132,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,22,0,2],[96,134,0.7164,0.83929,0.11151,0.85714,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,20,0,5],[100,134,0.7463,0.79911,0.15916,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,19,0,4],[104,134,0.7761,0.82143,0.13363,0.85714,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,23,0,3],[108,134,0.806,0.78125,0.1636,0.71429,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,20,0,2],[112,134,0.8358,0.81696,0.17582,0.85714,0.85714,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,22,0,4],[116,134,0.8657,0.82589,0.11143,0.85711,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,22,0,3],[120,134,0.8955,0.80357,0.13716,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,21,0,2],[124,134,0.9254,0.83035,0.10374,0.85711,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,22,0,3],[128,134,0.9552,0.83928,0.11152,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,23,0,4],[132,134,0.9851,0.79464,0.16342,0.85711,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,23,0,2],[134,134,1.0,0.67411,0.21498,0.42857,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,11,0,0,1,0,0,4,0,0,13,0,2]]}]},{"i":"6162472ceb341b41","q":"Find the unique integer  $a > 1$  that satisfies\n\\[ \\int_{a}^{a^2} \\left(\\frac{1}{\\ln x} - \\frac{2}{(\\ln x)^3}\\right) dx = \\frac{a}{\\ln a}. \\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.80357,"x":0.98214,"p":[[0,45,0.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[4,45,0.0889,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,45,0.1778,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,45,0.2667,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[16,45,0.3556,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[20,45,0.4444,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[24,45,0.5333,0.89732,0.11971,0.82143,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,17],[28,45,0.6222,0.90625,0.19434,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,22],[32,45,0.7111,0.91964,0.11811,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[36,45,0.8,0.82143,0.12877,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,4,0,10],[40,45,0.8889,0.85714,0.12372,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,8,0,12],[44,45,0.9778,0.80357,0.13243,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,3,0,9],[45,45,1.0,0.84821,0.1234,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,8,0,11]]},{"b":3,"e":0.71429,"k":"flat","v":0.85268,"x":0.95534,"p":[[0,63,0.0,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[4,63,0.0635,0.9241,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[8,63,0.127,0.90179,0.12078,0.82143,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,6,0,18],[12,63,0.1905,0.90625,0.12169,0.82143,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,19],[16,63,0.254,0.95534,0.09743,1.0,1.0,1.0,0.714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[20,63,0.3175,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[24,63,0.381,0.88839,0.12745,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,17],[28,63,0.4444,0.88839,0.12234,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,16],[32,63,0.5079,0.88392,0.13092,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,17],[36,63,0.5714,0.90624,0.11071,0.85711,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,17],[40,63,0.6349,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[44,63,0.6984,0.90178,0.12595,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,19],[48,63,0.7619,0.88393,0.13092,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,17],[52,63,0.8254,0.91071,0.12242,0.82143,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[56,63,0.8889,0.90178,0.12595,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,19],[60,63,0.9524,0.85268,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,13],[63,63,1.0,0.8616,0.12619,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,7,0,13]]}]},{"i":"8d144b6f7ba31b42","q":"For a real number $x$, let $\\lfloor x\\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that\n\n$$\n\\left\\lfloor\\frac{(n-1)!}{n(n+1)}\\right\\rfloor\n$$\n\nis even for every positive integer $n$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.20087,"x":0.91071,"p":[[0,92,0.0,0.50445,0.38296,0.14286,0.42859,1.0,0.0,1.0,2,9,0,2,0,12,0,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,9],[4,92,0.0435,0.87052,0.24319,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,23],[8,92,0.087,0.86606,0.26472,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,21],[12,92,0.1304,0.86607,0.27185,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,23],[16,92,0.1739,0.87499,0.24419,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,21],[20,92,0.2174,0.84375,0.27747,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,21],[24,92,0.2609,0.80355,0.30253,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,20],[28,92,0.3043,0.85714,0.26,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,4,0,22],[32,92,0.3478,0.87497,0.25942,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,23],[36,92,0.3913,0.91071,0.23351,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[40,92,0.4348,0.84375,0.29957,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,23],[44,92,0.4783,0.64286,0.33881,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,6,0,0,2,0,0,1,0,0,2,0,0,7,0,0,3,0,10],[48,92,0.5217,0.89731,0.215,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,3,0,24],[52,92,0.5652,0.73661,0.32754,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,1,0,0,1,0,0,2,0,0,6,0,0,1,0,16],[56,92,0.6087,0.66964,0.37191,0.24999,0.85714,1.0,0.0,1.0,1,15,0,1,0,7,0,0,1,0,0,3,0,0,0,0,0,3,0,0,2,0,15],[60,92,0.6522,0.70087,0.3666,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,6,0,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,18],[64,92,0.6957,0.60268,0.38916,0.14286,0.71429,1.0,0.0,1.0,2,13,0,2,0,9,0,0,1,0,0,0,0,0,2,0,0,5,0,0,0,0,13],[68,92,0.7391,0.54463,0.37701,0.14286,0.64286,1.0,0.0,1.0,3,9,0,3,0,9,0,0,1,0,0,0,0,0,3,0,0,6,0,0,1,0,9],[72,92,0.7826,0.50445,0.35711,0.14286,0.42859,0.78571,0.14286,1.0,0,8,0,0,0,14,0,0,0,0,0,3,0,0,1,0,0,6,0,0,0,0,8],[76,92,0.8261,0.50893,0.40079,0.14286,0.42859,1.0,0.0,1.0,4,10,0,4,0,10,0,0,1,0,0,2,0,0,0,0,0,4,0,0,1,0,10],[80,92,0.8696,0.42848,0.38472,0.14286,0.21431,0.74996,0.0,1.0,6,7,0,6,0,10,0,0,1,0,0,3,0,0,0,0,0,4,0,0,1,0,7],[84,92,0.913,0.49545,0.37463,0.14286,0.50001,0.89286,0.0,1.0,3,8,0,3,0,11,0,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,8],[88,92,0.9565,0.51339,0.36919,0.14286,0.42859,1.0,0.0,1.0,2,9,0,2,0,10,0,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,9],[92,92,1.0,0.20087,0.25466,0.10714,0.14286,0.14286,0.0,1.0,8,2,0,8,0,19,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2]]},{"b":5,"e":0.14286,"k":"falling","v":0.09822,"x":0.85268,"p":[[0,89,0.0,0.54015,0.3209,0.2857,0.4998,0.85714,0.0,1.0,1,6,0,1,0,6,0,0,4,0,0,5,0,0,4,0,0,2,0,0,4,0,6],[4,89,0.0449,0.85268,0.2912,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,23],[8,89,0.0899,0.80356,0.32879,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,3,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,20],[12,89,0.1348,0.83928,0.23891,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,6,0,0,4,0,18],[16,89,0.1798,0.77232,0.31106,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,4,0,0,5,0,16],[20,89,0.2247,0.76339,0.32656,0.42859,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,0,0,0,5,0,0,1,0,0,2,0,0,1,0,19],[24,89,0.2697,0.73214,0.34209,0.53571,1.0,1.0,0.0,1.0,1,17,0,1,0,5,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,17],[28,89,0.3146,0.82142,0.30514,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,21],[32,89,0.3596,0.69197,0.36615,0.35716,0.85714,1.0,0.0,1.0,2,14,0,2,0,6,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,14],[36,89,0.4045,0.70088,0.29957,0.57143,0.78564,1.0,0.0,1.0,2,9,0,2,0,2,0,0,0,0,0,3,0,0,4,0,0,5,0,0,7,0,9],[40,89,0.4494,0.58036,0.40079,0.14286,0.71429,1.0,0.0,1.0,3,13,0,3,0,7,0,0,4,0,0,1,0,0,0,0,0,3,0,0,1,0,13],[44,89,0.4944,0.65179,0.3862,0.14286,0.85714,1.0,0.0,1.0,2,15,0,2,0,7,0,0,0,0,0,4,0,0,0,0,0,2,0,0,2,0,15],[48,89,0.5393,0.60268,0.37582,0.24999,0.71429,1.0,0.0,1.0,3,11,0,3,0,5,0,0,4,0,0,1,0,0,0,0,0,6,0,0,2,0,11],[52,89,0.5843,0.46866,0.37843,0.14286,0.28574,0.85714,0.0,1.0,2,7,0,2,0,14,0,0,0,0,0,3,0,0,1,0,0,1,0,0,4,0,7],[56,89,0.6292,0.59375,0.42724,0.14286,0.85714,1.0,0.0,1.0,5,15,0,5,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,15],[60,89,0.6742,0.40167,0.37537,0.14286,0.21428,0.75,0.0,1.0,6,6,0,6,0,10,0,0,4,0,0,0,0,0,2,0,0,2,0,0,2,0,6],[64,89,0.7191,0.51786,0.41149,0.14286,0.42857,1.0,0.0,1.0,3,10,0,3,0,13,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,10],[68,89,0.764,0.53124,0.38998,0.14286,0.42857,1.0,0.0,1.0,2,11,0,2,0,10,0,0,3,0,0,2,0,0,1,0,0,2,0,0,1,0,11],[72,89,0.809,0.30348,0.2807,0.14286,0.14286,0.42857,0.0,1.0,2,3,0,2,0,17,0,0,4,0,0,4,0,0,1,0,0,0,0,0,1,0,3],[76,89,0.8539,0.40176,0.33203,0.14286,0.21431,0.57143,0.0,1.0,3,5,0,3,0,13,0,0,1,0,0,2,0,0,6,0,0,2,0,0,0,0,5],[80,89,0.8989,0.11161,0.09268,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.14286,0.13363,0.10714,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,20,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[88,89,0.9888,0.11607,0.12079,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cc10e3746bc58643","q":"For a positive integer \\( n \\), let \\( \\tau(n) \\) denote the number of positive divisors of \\( n \\). Determine all positive integers \\( K \\) such that the equation\n\\[ \\tau(x) = \\tau(y) = \\tau(z) = \\tau(2x + 3y + 3z) = K \\]  holds for some positive integers  $x,y,z$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.49991,"x":0.73219,"p":[[0,61,0.0,0.5759,0.1906,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,4,0,0,10,0,0,3,0,0,13,0,0,0,0,2],[4,61,0.0656,0.73219,0.27831,0.4286,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,7,0,0,2,0,0,4,0,0,2,0,14],[8,61,0.1311,0.58033,0.26711,0.28571,0.71414,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,10,0,0,3,0,0,1,0,0,11,0,0,1,0,5],[12,61,0.1967,0.68748,0.16146,0.57143,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,16,0,0,1,0,4],[16,61,0.2623,0.64271,0.19586,0.57132,0.64286,0.71429,0.14,1.0,0,4,0,0,0,1,0,0,1,0,0,4,0,0,10,0,0,11,0,0,1,0,4],[20,61,0.3279,0.63393,0.20806,0.42857,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,8,0,0,2,0,0,14,0,0,1,0,4],[24,61,0.3934,0.71427,0.18559,0.67857,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,17,0,0,0,0,7],[28,61,0.459,0.65177,0.21998,0.42857,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,9,0,0,3,0,0,11,0,0,1,0,6],[32,61,0.5246,0.66964,0.19045,0.53572,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,13,0,0,3,0,4],[36,61,0.5902,0.65163,0.24204,0.42859,0.64071,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,6,0,0,6,0,0,8,0,0,0,0,8],[40,61,0.6557,0.56254,0.21407,0.42857,0.57141,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,9,0,0,7,0,0,7,0,0,1,0,3],[44,61,0.7213,0.58035,0.20497,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,9,0,0,3,0,0,15,0,0,1,0,1],[48,61,0.7869,0.53571,0.17496,0.42857,0.4998,0.71429,0.1429,0.71429,0,0,0,0,0,1,0,0,4,0,0,11,0,0,2,0,0,14,0,0,0,0,0],[52,61,0.8525,0.49991,0.19901,0.28571,0.4286,0.71429,0.14,1.0,0,1,0,0,0,1,0,0,9,0,0,7,0,0,5,0,0,9,0,0,0,0,1],[56,61,0.918,0.66964,0.10374,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,26,0,0,0,0,0],[60,61,0.9836,0.64286,0.10714,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,21,0,0,0,0,0],[61,61,1.0,0.64719,0.11829,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,23,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.52231,"x":0.92411,"p":[[0,186,0.0,0.62505,0.15866,0.57143,0.71429,0.71429,0.1429,0.71429,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,23,0,0,0,0,0],[4,186,0.0215,0.92411,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[8,186,0.043,0.90178,0.21261,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,23],[12,186,0.0645,0.7857,0.30094,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,19],[16,186,0.086,0.84826,0.20487,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,4,0,18],[20,186,0.1075,0.68748,0.31224,0.4286,0.78564,1.0,0.0,1.0,1,12,1,1,0,1,0,0,5,0,0,3,0,0,4,0,0,2,0,0,4,0,12],[24,186,0.129,0.71871,0.29772,0.53539,0.78564,1.0,0.0,1.0,1,13,0,1,0,1,0,0,3,0,0,3,0,0,4,0,0,4,0,0,3,0,13],[28,186,0.1505,0.80802,0.24644,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,4,0,0,6,0,15],[32,186,0.172,0.74991,0.29469,0.57143,0.85707,1.0,0.14,1.0,0,15,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,6,0,0,2,0,15],[36,186,0.1935,0.83022,0.21859,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,8,0,0,2,0,17],[40,186,0.2151,0.73213,0.2442,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,4,0,0,4,0,11],[44,186,0.2366,0.79463,0.25739,0.57143,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,0,2,0,17],[48,186,0.2581,0.70088,0.26332,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,6,0,0,2,0,11],[52,186,0.2796,0.75891,0.25615,0.57143,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,8,0,0,1,0,14],[56,186,0.3011,0.67397,0.27018,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,7,0,0,2,0,0,6,0,0,2,0,10],[60,186,0.3226,0.74085,0.31648,0.53572,0.92857,1.0,0.0,1.0,2,16,1,2,0,1,0,0,1,0,0,4,0,0,2,0,0,5,0,0,1,0,16],[64,186,0.3441,0.63391,0.30292,0.39286,0.64286,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,6,0,0,6,0,0,2,0,0,4,0,0,2,0,10],[68,186,0.3656,0.68304,0.26422,0.53572,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,4,0,0,2,0,0,12,0,0,2,0,8],[72,186,0.3871,0.70982,0.28004,0.53572,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,5,0,0,2,0,0,4,0,0,6,0,0,2,0,12],[76,186,0.4086,0.67409,0.22371,0.5354,0.71429,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,11,0,0,1,0,7],[80,186,0.4301,0.73213,0.24158,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,0,0,0,11,0,0,4,0,0,3,0,11],[84,186,0.4516,0.75893,0.25862,0.53572,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,5,0,0,3,0,14],[88,186,0.4731,0.74107,0.24337,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,8,0,0,3,0,11],[92,186,0.4946,0.82589,0.22794,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,0,6,0,15],[96,186,0.5161,0.82142,0.22868,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,0,3,0,17],[100,186,0.5376,0.72326,0.24722,0.53607,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,8,0,0,2,0,11],[104,186,0.5591,0.65622,0.27633,0.39286,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,8,0,0,1,0,0,8,0,0,4,0,0,1,0,10],[108,186,0.5806,0.71873,0.27546,0.42857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,4,0,0,2,0,13],[112,186,0.6022,0.73659,0.23721,0.57132,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,7,0,0,3,0,11],[116,186,0.6237,0.83034,0.23267,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,7,0,0,0,0,19],[120,186,0.6452,0.6205,0.25408,0.42857,0.57143,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,7,0,0,6,0,0,6,0,0,1,0,7],[124,186,0.6667,0.68301,0.28513,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,5,0,0,7,0,0,1,0,0,2,0,12],[128,186,0.6882,0.74552,0.25689,0.57143,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,4,0,0,1,0,14],[132,186,0.7097,0.66518,0.26149,0.53569,0.64286,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,4,0,0,8,0,0,6,0,0,1,0,9],[136,186,0.7312,0.67856,0.30515,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,4,0,0,6,0,0,3,0,0,3,0,0,1,0,13],[140,186,0.7527,0.58927,0.26184,0.39286,0.57143,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,7,0,0,4,0,0,9,0,0,4,0,0,0,0,7],[144,186,0.7742,0.58479,0.24836,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,6,0,0,7,0,0,5,0,0,2,0,5],[148,186,0.7957,0.52231,0.26871,0.2857,0.4998,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,8,0,0,4,0,0,3,0,0,9,0,0,0,0,4],[152,186,0.8172,0.55801,0.23244,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,5,0,0,10,0,0,6,0,0,0,0,4],[156,186,0.8387,0.70982,0.27313,0.53572,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,3,0,0,7,0,0,3,0,0,1,0,13],[160,186,0.8602,0.77232,0.21086,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,7,0,0,3,0,12],[164,186,0.8817,0.72766,0.25844,0.57132,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,1,0,0,3,0,0,5,0,0,8,0,9],[168,186,0.9032,0.80356,0.17407,0.67857,0.78571,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,4,0,12],[172,186,0.9247,0.82589,0.22794,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,2,0,18],[176,186,0.9462,0.79463,0.19214,0.57143,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,6,0,0,3,0,13],[180,186,0.9677,0.79464,0.18536,0.57143,0.85714,1.0,0.5714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,0,5,0,12],[184,186,0.9892,0.87946,0.11356,0.82132,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,13],[186,186,1.0,0.79018,0.11837,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,3,0,7]]}]},{"i":"d0c5e02844506d87","q":"Find the maximum value of the real constant  $C$  such that  $x^{2}+y^{2}+1\\geq C(x+y)$ , and  $ x^{2}+y^{2}+xy+1\\geq  C(x+y)$  for all reals  $x,y$ .","t":[{"b":3,"e":0.57143,"k":"falling","v":0.64283,"x":0.83927,"p":[[0,12,0.0,0.83927,0.19806,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,0,0,19],[4,12,0.3333,0.80356,0.23078,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,1,0,0,0,0,18],[8,12,0.6667,0.8125,0.21261,0.57143,1.0,1.0,0.5714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,18],[12,12,1.0,0.64283,0.17497,0.57143,0.57143,0.57143,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,0,0,0,0,0,6]]},{"b":4,"e":0.85714,"k":"flat","v":0.79462,"x":0.85268,"p":[[0,13,0.0,0.83036,0.22142,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,0,0,20],[4,13,0.3077,0.79462,0.20809,0.57143,0.85714,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,0,0,16],[8,13,0.6154,0.84819,0.19869,0.57143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,0,0,20],[12,13,0.9231,0.79909,0.21684,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,0,0,17],[13,13,1.0,0.85268,0.19719,0.57143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,1,0,20]]}]},{"i":"e9b7f8a2a3d76e92","q":"Find the minimal positive integer  $m$ , so that there exist positive integers  $n>k>1$ , which satisfy $11...1=11...1.m$ , where the first number has  $n$  digits  $1$ , and the second has  $k$  digits  $1$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,16,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,16,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,16,0.5,0.95982,0.13939,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[12,16,0.75,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.99107,"x":0.99554,"p":[[0,7,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,7,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[7,7,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"26ec4a4a9f9c7661","q":"Find the perimeter of a triangle whose altitudes are  $3,4,$  and  $6$ . $ \\textbf{(A)}\\ 12\\sqrt\\frac35 \\qquad \\textbf{(B)}\\ 16\\sqrt\\frac35 \\qquad \\textbf{(C)}\\ 20\\sqrt\\frac35 \\qquad \\textbf{(D)}\\ 24\\sqrt\\frac35 \\qquad \\textbf{(E)}\\ \\text{None}$","t":[{"b":5,"e":0.85714,"k":"falling","v":0.58928,"x":0.8125,"p":[[0,31,0.0,0.75,0.29451,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,6,0,0,2,0,0,2,0,0,1,0,17],[4,31,0.129,0.71427,0.32733,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,5,0,0,5,0,0,2,0,0,0,0,0,1,0,17],[8,31,0.2581,0.8125,0.2683,0.64286,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,2,0,0,3,0,19],[12,31,0.3871,0.80804,0.26392,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,5,0,0,1,0,0,4,0,0,1,0,19],[16,31,0.5161,0.79018,0.25501,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,5,0,0,1,0,17],[20,31,0.6452,0.77232,0.31715,0.53571,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,3,0,0,2,0,0,0,0,0,4,0,18],[24,31,0.7742,0.60714,0.28571,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,2,0,0,7,0,0,6,0,0,1,0,0,6,0,6],[28,31,0.9032,0.62054,0.28484,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,6,0,0,5,0,0,3,0,0,3,0,8],[31,31,1.0,0.58928,0.25442,0.42857,0.57143,0.74996,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,6,0,0,7,0,0,5,0,0,4,0,4]]},{"b":6,"e":1.0,"k":"flat","v":0.83036,"x":1.0,"p":[[0,30,0.0,0.85714,0.26486,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,23],[4,30,0.1333,0.83036,0.29974,0.85714,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,24],[8,30,0.2667,0.95089,0.19102,1.0,1.0,1.0,0.1429,1.0,0,30,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,30,0.4,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,30,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,30,0.6667,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"fd03a2a27c6e161b","q":"For a real number $x$, we denote $\\lfloor x\\rfloor$ as the greatest integer less than or equal to $x$ (for example, $\\lfloor 2.7\\rfloor=2, \\lfloor\\pi\\rfloor=3$ and $\\lfloor-1.5\\rfloor=-2$).\nLet $a, b$ be two real numbers such that\n\n$$\na+\\lfloor a\\rfloor=b+\\lfloor b\\rfloor .\n$$\n\nShow that $a=b$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.03571,"x":0.12498,"p":[[0,35,0.0,0.03571,0.14286,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,35,0.1143,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,35,0.2286,0.12498,0.29176,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[12,35,0.3429,0.07589,0.22011,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[16,35,0.4571,0.05804,0.20159,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[20,35,0.5714,0.06696,0.22862,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[24,35,0.6857,0.11607,0.27534,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[28,35,0.8,0.04463,0.1259,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,35,0.9143,0.06695,0.17118,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[35,35,1.0,0.09821,0.25111,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.11159,"p":[[0,30,0.0,0.11159,0.28059,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[4,30,0.1333,0.08036,0.25238,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[8,30,0.2667,0.05804,0.20159,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,30,0.4,0.05804,0.19516,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,30,0.5333,0.04018,0.12993,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,30,0.6667,0.08481,0.21083,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[24,30,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.08036,0.24727,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[30,30,1.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"a61eecebf2ef20d6","q":"For a complex number  $z=1+2\\sqrt{6}i$  and natural number  $n=1,\\ 2,\\ 3,\\ \\cdots$ , express the complex number  $z^n$  in using real numbers  $a_n,\\ b_n$  as  $z^n=a_n+b_ni$ .\nAnswer the following questions.\n\n(1) Show that  $a_n^2+b_n^2=5^{2n}\\ (n=1,\\ 2,\\ 3,\\ \\cdots).$ (2) Find the constants  $p,\\ q$  such that  $a_{n+2}=pa_{n+1}+qa_n$  holds for all  $n$ .\n\n(3) Show that  $a_n$  is not a multiple of  $5$  for any  $n$ .\n\n(4) Show that  $z^n\\ (n=1,\\ 2,\\ 3,\\ \\cdots)$  is not a real number.","t":[{"b":1,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,49,0.0,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[4,49,0.0816,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,49,0.1633,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22],[12,49,0.2449,0.91964,0.13333,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,23],[16,49,0.3265,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,49,0.4082,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,49,0.4898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,49,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,49,0.6531,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,49,0.7347,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,49,0.8163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,49,0.898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,49,0.9796,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[49,49,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,36,0.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[4,36,0.1111,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,36,0.2222,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"578bfbfbb2a9281c","q":"Find all positive real numbers  $\\lambda$  such that for all integers  $n\\geq 2$  and all positive real numbers  $a_1,a_2,\\cdots,a_n$  with  $a_1+a_2+\\cdots+a_n=n$ , the following inequality holds: $\\sum_{i=1}^n\\frac{1}{a_i}-\\lambda\\prod_{i=1}^{n}\\frac{1}{a_i}\\leq n-\\lambda$ .","t":[{"b":3,"e":0.28571,"k":"flat","v":0.32365,"x":0.7232,"p":[[0,111,0.0,0.32365,0.11843,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,19,0,1,8,0,0,2,0,0,0,0,0,0,0,0],[4,111,0.036,0.67856,0.18899,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,11,0,0,9,0,0,6,0,3],[8,111,0.0721,0.683,0.20436,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,2,0,0,7,0,0,12,0,0,6,0,3],[12,111,0.1081,0.65175,0.21707,0.57132,0.64286,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,2,0,0,11,0,0,7,0,0,6,0,3],[16,111,0.1441,0.7232,0.15127,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,5,0,0,14,0,1],[20,111,0.1802,0.63839,0.18893,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,2,0,0,10,0,0,12,0,0,6,0,0],[24,111,0.2162,0.66963,0.21853,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,1,0,0,8,0,0,6,0,0,12,0,1],[28,111,0.2523,0.66963,0.18364,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,0,0,0,8,0,0,15,0,0,7,0,0],[32,111,0.2883,0.69864,0.15125,0.57143,0.71429,0.80354,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,10,1,0,5,0,3],[36,111,0.3243,0.58479,0.25345,0.53539,0.57143,0.75,0.0,0.85714,3,0,0,3,0,0,0,0,3,0,0,2,0,0,9,0,0,7,0,0,8,0,0],[40,111,0.3604,0.63839,0.18552,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,3,0,0,9,0,0,13,0,0,4,0,1],[44,111,0.3964,0.66963,0.19704,0.57143,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,4,0,0,9,0,0,7,0,0,10,0,1],[48,111,0.4324,0.63393,0.20183,0.57143,0.64286,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,3,0,0,11,0,0,9,0,0,6,0,1],[52,111,0.4685,0.65398,0.16084,0.5713,0.71429,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,1,10,0,0,9,0,0,8,0,0],[56,111,0.5045,0.66513,0.15818,0.571,0.64286,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,7,0,0,8,0,1],[60,111,0.5405,0.61607,0.16536,0.57143,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,11,0,0,4,0,0],[64,111,0.5766,0.65179,0.19212,0.57143,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,2,0,0,12,0,0,6,0,0,10,0,0],[68,111,0.6126,0.68748,0.14914,0.57143,0.71429,0.85704,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,11,0,0,8,0,1],[72,111,0.6486,0.64729,0.18206,0.57143,0.57143,0.85704,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,0,0,0,16,0,0,5,0,0,9,0,0],[76,111,0.6847,0.63838,0.2172,0.57143,0.64286,0.85714,0.0,0.85714,2,0,0,2,0,0,0,0,0,0,0,3,0,0,11,0,0,6,0,0,10,0,0],[80,111,0.7207,0.63838,0.1988,0.57143,0.64286,0.85704,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,2,0,0,11,0,0,7,0,0,9,0,0],[84,111,0.7568,0.61164,0.20893,0.42964,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,4,0,0,8,0,0,6,0,0,9,0,0],[88,111,0.7928,0.54686,0.21915,0.41068,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,1,3,0,0,10,0,0,7,0,0,3,0,1],[92,111,0.8288,0.5625,0.26711,0.39286,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,5,0,0,5,0,0,5,0,0,8,0,1],[96,111,0.8649,0.53571,0.21724,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,4,0,0,6,0,0,10,0,0,6,0,0,4,0,0],[100,111,0.9009,0.59372,0.15613,0.571,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,14,0,0,7,0,0,4,0,0],[104,111,0.9369,0.61607,0.20652,0.42857,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,6,0,0,3,0,0,6,0,0,10,0,0,6,0,1],[108,111,0.973,0.49106,0.21409,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,9,0,0,5,0,0,3,0,0,5,0,0],[111,111,1.0,0.32589,0.23753,0.14286,0.28571,0.42858,0.0,0.85714,5,0,0,5,0,6,0,0,8,0,0,7,0,0,2,0,0,2,0,0,2,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.30357,"x":0.69641,"p":[[0,80,0.0,0.30357,0.07784,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,80,0.05,0.69641,0.17406,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,7,0,0,8,0,3],[8,80,0.1,0.61158,0.21793,0.571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,4,0,0,12,0,0,6,0,0,5,0,2],[12,80,0.15,0.68745,0.16921,0.57142,0.64286,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,7,0,0,5,0,4],[16,80,0.2,0.50888,0.19539,0.42857,0.57143,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,0,0,0,7,0,0,15,0,0,5,0,0,1,0,0],[20,80,0.25,0.58032,0.19864,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,7,0,0,10,0,0,5,0,0,5,0,1],[24,80,0.3,0.64061,0.16316,0.57143,0.57143,0.71429,0.28571,0.92857,0,0,0,0,0,0,0,0,2,0,0,3,0,0,12,0,0,8,0,0,6,1,0],[28,80,0.35,0.55356,0.22232,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,7,0,0,7,0,0,6,0,0,6,0,0],[32,80,0.4,0.52676,0.24598,0.42857,0.57121,0.71429,0.0,0.85714,3,0,0,3,0,0,0,0,4,0,0,7,0,0,6,0,0,7,0,0,5,0,0],[36,80,0.45,0.58928,0.22799,0.39286,0.57143,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,3,0,0,8,0,0,5,0,0,6,0,2],[40,80,0.5,0.61384,0.19468,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,1,1,0,0,4,0,0,12,0,0,6,0,0,6,0,1],[44,80,0.55,0.54462,0.20958,0.42857,0.57143,0.60714,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,9,0,0,11,0,0,3,0,0,4,0,1],[48,80,0.6,0.60045,0.15553,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,12,0,0,9,1,0,3,0,0],[52,80,0.65,0.54466,0.23263,0.42859,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,3,0,0,6,0,0,7,0,0,10,0,0,2,0,1],[56,80,0.7,0.48883,0.22095,0.42857,0.4998,0.71429,0.0,0.92857,3,0,0,3,0,0,0,0,4,0,0,9,0,0,7,0,0,8,0,0,0,1,0],[60,80,0.75,0.58479,0.19598,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,1,7,0,0,7,1,0,6,0,0,5,0,1],[64,80,0.8,0.54016,0.20743,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,3,0,0,8,0,0,9,0,0,7,0,0,2,0,1],[68,80,0.85,0.51338,0.20472,0.42857,0.57143,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,1,0,0,10,0,0,11,0,0,4,0,0,3,0,0],[72,80,0.9,0.56687,0.19738,0.42857,0.57143,0.71429,0.14,1.0,0,1,0,0,0,1,0,0,4,0,0,7,0,0,8,0,0,8,0,0,3,0,1],[76,80,0.95,0.42415,0.17308,0.42857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,3,0,0,15,0,0,9,0,0,0,0,0,1,0,0],[80,80,1.0,0.41071,0.11152,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,1,0,0,9,0,0,15,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"0c29a9b69a90b9d6","q":"For how many integers  $1\\leq k\\leq 2013$  does the decimal representation of  $k^k$  end with a  $1$ ?","t":[{"b":3,"e":0.85714,"k":"flat","v":0.875,"x":0.97768,"p":[[0,25,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,25,0.16,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[8,25,0.32,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[12,25,0.48,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[16,25,0.64,0.92857,0.12877,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,22],[20,25,0.8,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[24,25,0.96,0.875,0.16269,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,6,0,17],[25,25,1.0,0.90625,0.15407,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,7,0,20]]},{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":0.98661,"p":[[0,31,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,31,0.129,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[8,31,0.2581,0.91518,0.15093,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,21],[12,31,0.3871,0.94642,0.11156,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[16,31,0.5161,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[20,31,0.6452,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,31,0.7742,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[28,31,0.9032,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[31,31,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"45d637b4a04bab3f","q":"For each pair of positive integers  $m$  and  $n$ , we define  $f_m(n)$  as follows:    $$  f_m(n) = \\gcd(n, d_1) + \\gcd(n, d_2) + \\cdots + \\gcd(n, d_k),  $$    \nwhere  $1 = d_1 < d_2 < \\cdots < d_k = m$  are all the positive divisors of  $m$ . For example,   $f_4(6) = \\gcd(6,1) + \\gcd(6,2) + \\gcd(6,4) = 5$ .   $a)\\:$  Find all positive integers  $n$  such that  $f_{2017}(n) = f_n(2017)$ . $b)\\:$  Find all positive integers  $n$  such that  $f_6(n) = f_n(6)$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.91063,"x":0.99554,"p":[[0,85,0.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,85,0.0471,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[8,85,0.0941,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,85,0.1412,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,85,0.1882,0.91063,0.1885,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,23],[20,85,0.2353,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[24,85,0.2824,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[28,85,0.3294,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,85,0.3765,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,85,0.4235,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,85,0.4706,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[44,85,0.5176,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[48,85,0.5647,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[52,85,0.6118,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,85,0.6588,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,85,0.7059,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[64,85,0.7529,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,85,0.8,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[72,85,0.8471,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[76,85,0.8941,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,85,0.9412,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,85,0.9882,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[85,85,1.0,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27]]},{"b":5,"e":0.71429,"k":"falling","v":0.70981,"x":0.98214,"p":[[0,113,0.0,0.92857,0.13363,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22],[4,113,0.0354,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,113,0.0708,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,113,0.1062,0.9375,0.14698,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[16,113,0.1416,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[20,113,0.177,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[24,113,0.2124,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,113,0.2478,0.85714,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[32,113,0.2832,0.82143,0.15152,0.85714,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,21,0,4],[36,113,0.3186,0.875,0.09943,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,16,0,10],[40,113,0.354,0.90179,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,12],[44,113,0.3894,0.85714,0.08748,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,20,0,6],[48,113,0.4248,0.86607,0.07936,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,6],[52,113,0.4602,0.85713,0.08748,0.857,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,20,0,6],[56,113,0.4956,0.87946,0.08828,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,9],[60,113,0.531,0.875,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,6],[64,113,0.5664,0.86607,0.10062,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,8],[68,113,0.6018,0.84821,0.08702,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,20,0,5],[72,113,0.6372,0.83035,0.08328,0.85708,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,23,0,2],[76,113,0.6726,0.85714,0.07143,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4],[80,113,0.708,0.85254,0.07588,0.85714,0.85714,0.85714,0.71,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,23,0,4],[84,113,0.7434,0.84375,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,21,0,4],[88,113,0.7788,0.85714,0.10102,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,19,0,7],[92,113,0.8142,0.83927,0.11713,0.71429,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,16,0,7],[96,113,0.8496,0.83929,0.09279,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,21,0,4],[100,113,0.885,0.84375,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,2],[104,113,0.9204,0.82588,0.10555,0.857,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,22,0,3],[108,113,0.9558,0.80357,0.09942,0.71429,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,15,0,3],[112,113,0.9912,0.75445,0.1025,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,11,0,1],[113,113,1.0,0.70981,0.09096,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,0,6,0,0]]}]},{"i":"f48837af28455e14","q":"For each prime $p$, there is a kingdom of $p$-Landia consisting of $p$ islands numbered $1,2, \\ldots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $\\left(n^{2}-m+1\\right)\\left(m^{2}-n+1\\right)$. The bridges may pass over each other, but cannot cross. Prove that for infinitely many $p$ there are two islands in $p$-Landia not connected by a chain of bridges. (Denmark)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.21427,"x":0.33483,"p":[[0,33,0.0,0.30569,0.18839,0.14286,0.28571,0.46418,0.0,0.71429,2,0,0,2,0,10,0,0,10,0,1,1,0,0,7,0,0,1,0,0,0,0,0],[4,33,0.1212,0.3125,0.23538,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,3,0,0,14,0,0,2,0,0,4,0,0,2,0,0,0,0,1],[8,33,0.2424,0.33483,0.22191,0.24999,0.28571,0.42857,0.0,1.0,3,1,0,3,0,5,0,0,14,0,0,4,0,0,2,0,0,3,0,0,0,0,1],[12,33,0.3636,0.29909,0.21828,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,11,0,0,5,0,0,2,0,0,0,0,0,1,0,1],[16,33,0.4848,0.26782,0.14608,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,9,0,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[20,33,0.6061,0.21427,0.12873,0.14286,0.2143,0.28571,0.0,0.571,4,0,0,4,0,12,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.26999,0.22144,0.14286,0.2857,0.32143,0.0,0.85714,6,0,0,6,0,9,0,0,9,0,0,1,0,1,4,0,0,1,0,0,1,0,0],[28,33,0.8485,0.27223,0.14005,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,9,0,0,18,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[32,33,0.9697,0.22321,0.17105,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,14,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[33,33,1.0,0.24553,0.17215,0.14286,0.2857,0.28571,0.0,1.0,3,1,0,3,0,10,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]},{"b":3,"e":0.2857,"k":"flat","v":0.18304,"x":0.29462,"p":[[0,25,0.0,0.21866,0.22014,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,9,0,0,7,0,0,0,0,0,4,0,0,2,0,0,0,0,0],[4,25,0.16,0.29462,0.27647,0.10714,0.28571,0.42857,0.0,1.0,8,2,0,8,0,6,0,0,9,0,0,2,0,0,3,0,0,2,0,0,0,0,2],[8,25,0.32,0.28571,0.20516,0.14286,0.28571,0.32143,0.0,1.0,5,1,0,5,0,5,0,0,14,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[12,25,0.48,0.18304,0.14827,0.0,0.14288,0.28571,0.0,0.57143,9,0,0,9,0,9,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,25,0.64,0.24999,0.18208,0.14286,0.2143,0.28571,0.0,0.71429,4,0,0,4,0,12,0,0,9,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[20,25,0.8,0.20982,0.23686,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,10,0,0,8,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[24,25,0.96,0.24107,0.15745,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,7,0,0,16,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[25,25,1.0,0.22766,0.19513,0.14286,0.21428,0.28571,0.0,1.0,6,1,0,6,0,10,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,1]]}]},{"i":"3c56a81aa175c796","q":"For each positive integer  $ k$ , find the smallest number  $ n_{k}$  for which there exist real  $ n_{k}\\times n_{k}$  matrices  $ A_{1}, A_{2}, \\ldots, A_{k}$  such that all of the following conditions hold:\r\n\r\n(1)  $ A_{1}^{2}= A_{2}^{2}= \\ldots = A_{k}^{2}= 0$ ,\r\n\r\n(2)  $ A_{i}A_{j}= A_{j}A_{i}$  for all  $ 1 \\le i, j \\le k$ , and \r\n\r\n(3)  $ A_{1}A_{2}\\ldots A_{k}\\ne 0$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.10259,"x":0.57589,"p":[[0,67,0.0,0.57586,0.26361,0.42857,0.42859,0.74996,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,14,0,0,4,0,0,2,0,0,1,0,7],[4,67,0.0597,0.50437,0.33888,0.25,0.42857,0.85704,0.0,1.0,4,6,0,4,0,4,0,0,2,0,0,10,0,0,0,0,0,3,0,0,3,0,6],[8,67,0.1194,0.49553,0.35171,0.14286,0.42857,0.857,0.0,1.0,2,7,0,2,0,10,0,0,0,0,0,7,0,0,1,0,0,3,0,0,2,0,7],[12,67,0.1791,0.51336,0.30899,0.35715,0.4286,0.71429,0.0,1.0,2,6,0,2,0,6,0,0,0,0,0,9,0,0,6,0,0,2,0,0,1,0,6],[16,67,0.2388,0.39732,0.33261,0.14286,0.42857,0.57143,0.0,1.0,4,5,0,4,0,11,0,0,0,0,0,8,0,0,2,0,0,1,0,0,1,0,5],[20,67,0.2985,0.4017,0.34897,0.14286,0.42857,0.42858,0.0,1.0,4,7,0,4,0,11,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,7],[24,67,0.3582,0.45536,0.3203,0.14286,0.42857,0.57143,0.0,1.0,3,6,0,3,0,7,0,0,0,0,0,13,0,0,2,0,0,0,0,0,1,0,6],[28,67,0.4179,0.54464,0.35254,0.14286,0.64284,0.78571,0.0,1.0,3,8,0,3,0,7,0,0,0,0,0,5,0,0,1,0,0,8,0,0,0,0,8],[32,67,0.4776,0.57589,0.31234,0.42857,0.5,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,0,0,0,10,0,0,4,0,0,2,0,0,2,0,8],[36,67,0.5373,0.37938,0.29155,0.14286,0.42857,0.4286,0.0,1.0,3,3,0,3,0,11,0,0,0,0,0,11,0,0,1,0,0,2,0,0,1,0,3],[40,67,0.597,0.45535,0.33775,0.14286,0.42857,0.57143,0.0,1.0,4,7,0,4,0,7,0,0,0,0,0,10,0,0,4,0,0,0,0,0,0,0,7],[44,67,0.6567,0.35713,0.31541,0.10714,0.42857,0.42858,0.0,1.0,8,4,0,8,0,5,0,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,4],[48,67,0.7164,0.36606,0.29,0.14286,0.35714,0.4642,0.0,1.0,3,3,0,3,0,12,0,0,1,0,0,8,0,0,2,0,0,3,0,0,0,0,3],[52,67,0.7761,0.2009,0.19516,0.14286,0.14286,0.1429,0.0,1.0,3,1,0,3,0,24,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[56,67,0.8358,0.20054,0.22557,0.14286,0.14286,0.14286,0.0,1.0,3,2,0,3,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[60,67,0.8955,0.27233,0.32015,0.10714,0.14286,0.42857,0.0,1.0,8,3,0,8,0,15,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,3],[64,67,0.9552,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.32143,"x":0.59374,"p":[[0,71,0.0,0.54017,0.27603,0.42857,0.42859,0.71429,0.0,1.0,3,5,3,3,0,0,0,0,2,0,0,12,0,0,5,0,0,4,0,0,1,0,5],[4,71,0.0563,0.49553,0.33499,0.14286,0.42857,0.74996,0.0,1.0,4,6,0,4,0,5,0,0,0,0,0,11,0,0,1,0,0,3,0,0,2,0,6],[8,71,0.1127,0.57588,0.33213,0.42857,0.42859,1.0,0.0,1.0,1,9,0,1,0,6,0,0,0,0,0,10,0,0,3,0,0,0,0,0,3,0,9],[12,71,0.169,0.49107,0.33108,0.25001,0.42857,0.85714,0.0,1.0,3,7,0,3,0,5,0,0,1,0,0,14,0,0,0,0,0,0,0,0,2,0,7],[16,71,0.2254,0.5625,0.33108,0.42857,0.42857,1.0,0.0,1.0,4,9,0,4,0,1,0,0,0,0,0,14,0,0,0,0,0,4,0,0,0,0,9],[20,71,0.2817,0.49552,0.32337,0.35714,0.42857,0.71429,0.0,1.0,4,6,0,4,0,4,0,0,0,0,0,12,0,0,2,0,0,3,0,0,1,0,6],[24,71,0.338,0.45089,0.35734,0.14286,0.42857,0.71429,0.0,1.0,6,6,0,6,0,7,0,0,0,0,0,5,0,0,3,0,0,5,0,0,0,0,6],[28,71,0.3944,0.53124,0.35934,0.14289,0.5712,1.0,0.0,1.0,5,9,0,5,0,4,0,0,0,0,0,6,0,0,6,0,0,2,0,0,0,0,9],[32,71,0.4507,0.49554,0.29877,0.35715,0.42857,0.71429,0.0,1.0,2,5,0,2,0,6,0,0,0,0,0,12,0,0,1,0,0,6,0,0,0,0,5],[36,71,0.507,0.43749,0.3443,0.14286,0.42857,0.71429,0.0,1.0,5,5,0,5,0,8,0,0,0,0,0,7,0,0,3,0,0,2,0,0,2,0,5],[40,71,0.5634,0.47321,0.33963,0.14289,0.42857,0.64286,0.0,1.0,5,7,0,5,0,4,0,0,0,0,0,13,0,0,2,0,0,0,0,0,1,0,7],[44,71,0.6197,0.34822,0.32525,0.0,0.42857,0.4286,0.0,1.0,11,4,0,11,0,2,0,0,0,0,0,13,0,0,1,0,0,1,0,0,0,0,4],[48,71,0.6761,0.40625,0.33141,0.0,0.42857,0.60714,0.0,1.0,9,3,0,9,0,2,0,0,0,0,0,12,0,0,1,0,0,2,0,0,3,0,3],[52,71,0.7324,0.32143,0.29233,0.0,0.42857,0.4643,0.0,1.0,10,1,0,10,0,5,0,0,0,0,0,9,0,0,4,0,0,1,0,0,2,0,1],[56,71,0.7887,0.41071,0.3531,0.14286,0.42857,0.71429,0.0,1.0,7,6,0,7,0,6,0,0,1,0,0,9,0,0,0,0,0,3,0,0,0,0,6],[60,71,0.8451,0.33928,0.28737,0.14286,0.42857,0.42857,0.0,1.0,7,2,0,7,0,7,0,0,1,0,0,11,0,0,1,0,0,2,0,0,1,0,2],[64,71,0.9014,0.40624,0.36788,0.14286,0.28571,0.71429,0.0,1.0,7,6,0,7,0,9,0,0,0,0,0,4,0,0,2,0,0,4,0,0,0,0,6],[68,71,0.9577,0.59374,0.23176,0.42857,0.42859,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,19,0,0,3,0,0,3,0,0,0,0,7],[71,71,1.0,0.58032,0.20497,0.42857,0.42859,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,17,0,0,6,0,0,4,0,0,0,0,5]]}]},{"i":"21b742a245b22ead","q":"Five distinct  $2$ -digit numbers are in a geometric progression. Find the middle term.","t":[{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":0.99107,"p":[[0,66,0.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[4,66,0.0606,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,66,0.1212,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,66,0.1818,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,66,0.2424,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,66,0.303,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,66,0.3636,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,66,0.4242,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,66,0.4848,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,66,0.5455,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,66,0.6061,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[44,66,0.6667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,66,0.7273,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,66,0.7879,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,66,0.8485,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[60,66,0.9091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,66,0.9697,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[66,66,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.95089,"x":0.98661,"p":[[0,58,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,58,0.069,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,58,0.1379,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,58,0.2069,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[16,58,0.2759,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[20,58,0.3448,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[24,58,0.4138,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,58,0.4828,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,58,0.5517,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[36,58,0.6207,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,58,0.6897,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[44,58,0.7586,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[48,58,0.8276,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[52,58,0.8966,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,58,0.9655,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[58,58,1.0,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"720d8020c4e70bbf","q":"Find the sum of all positive integers  $n$  such that  $\\frac{2020}{n^3 + n}$  is an integer.","t":[{"b":5,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,16,0.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[4,16,0.25,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[8,16,0.5,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[12,16,0.75,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,60,0.0,0.92857,0.10715,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[4,60,0.0667,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[8,60,0.1333,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,60,0.2,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,60,0.2667,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,60,0.3333,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,60,0.4,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,60,0.4667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,60,0.5333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,60,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,60,0.7333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,60,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,60,0.8667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e7d5a30e78fec9b4","q":"For every integer $k \\geq 2$, prove that $2^{3 k}$ divides the number  $$ \\left(\\begin{array}{c} 2^{k+1} \\\\ 2^{k} \\end{array}\\right)-\\left(\\begin{array}{c} 2^{k} \\\\ 2^{k-1} \\end{array}\\right) $$  but $2^{3 k+1}$ does not.  (Poland)","t":[{"b":6,"e":0.2857,"k":"rising","v":0.24107,"x":0.66516,"p":[[0,118,0.0,0.24107,0.24856,0.0,0.14286,0.42857,0.0,0.85714,11,0,0,11,0,6,0,0,6,0,0,5,0,0,1,0,0,1,0,0,2,0,0],[4,118,0.0339,0.54461,0.27764,0.28571,0.571,0.85704,0.0,1.0,1,2,1,1,0,3,0,0,6,0,0,5,0,0,4,0,0,4,0,0,7,0,2],[8,118,0.0678,0.558,0.23516,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,7,0,0,9,0,0,3,0,0,5,0,2],[12,118,0.1017,0.61159,0.22371,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,9,0,0,7,0,0,4,0,0,6,0,3],[16,118,0.1356,0.65177,0.24207,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,6,0,0,3,0,0,7,0,0,6,0,5],[20,118,0.1695,0.55354,0.27141,0.28571,0.57121,0.71429,0.0,1.0,1,5,1,1,0,0,0,0,10,0,0,3,0,0,7,0,0,4,0,0,2,0,5],[24,118,0.2034,0.61159,0.25812,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,8,0,0,5,0,0,4,0,0,5,0,5],[28,118,0.2373,0.63393,0.30501,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,3,0,0,7,0,0,1,0,0,2,0,0,8,0,7],[32,118,0.2712,0.63839,0.25501,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,8,0,0,2,0,0,4,0,0,12,0,2],[36,118,0.3051,0.53119,0.24019,0.39286,0.49979,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,8,0,0,6,0,0,3,0,0,6,0,1],[40,118,0.339,0.558,0.23516,0.42857,0.49979,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,10,0,0,4,0,0,5,0,0,5,0,2],[44,118,0.3729,0.47768,0.26392,0.28571,0.42857,0.60714,0.0,1.0,1,3,0,1,0,4,0,0,6,0,0,9,0,0,4,0,0,3,0,0,2,0,3],[48,118,0.4068,0.52679,0.21852,0.42857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,13,0,0,2,0,0,6,0,0,4,0,1],[52,118,0.4407,0.59372,0.19269,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,8,0,0,10,0,0,5,0,0,4,0,2],[56,118,0.4746,0.46875,0.19638,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,15,0,0,5,0,0,2,0,0,2,0,1],[60,118,0.5085,0.5848,0.27282,0.42857,0.57143,0.857,0.0,1.0,2,3,0,2,0,1,0,0,3,0,0,7,0,0,4,0,0,6,0,0,6,0,3],[64,118,0.5424,0.57141,0.21724,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,5,0,0,9,0,0,6,0,0,5,0,1],[68,118,0.5763,0.66516,0.25155,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,7,0,0,2,0,0,7,0,0,6,0,6],[72,118,0.6102,0.54909,0.23448,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,6,0,0,8,0,0,6,0,0,3,0,2],[76,118,0.6441,0.49104,0.20182,0.39286,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,10,0,0,8,0,0,3,0,0,2,0,1],[80,118,0.678,0.53568,0.20824,0.42857,0.49979,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,10,0,0,5,0,0,6,0,0,5,0,0],[84,118,0.7119,0.54014,0.18808,0.42857,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,8,0,0,12,0,0,3,0,0,3,0,1],[88,118,0.7458,0.55356,0.21943,0.42857,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,13,0,0,4,0,0,4,0,0,3,0,3],[92,118,0.7797,0.54909,0.22334,0.42857,0.49979,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,11,0,0,6,0,0,4,0,0,4,0,2],[96,118,0.8136,0.52674,0.17654,0.42857,0.571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,10,0,0,11,0,0,4,0,0,0,0,2],[100,118,0.8475,0.47766,0.17353,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,15,0,0,9,0,0,1,0,0,1,0,1],[104,118,0.8814,0.49103,0.19863,0.42857,0.49979,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,9,0,0,12,0,0,1,0,0,2,0,1],[108,118,0.9153,0.46427,0.16366,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,14,0,0,4,0,0,5,0,0,1,0,0],[112,118,0.9492,0.45978,0.11139,0.42857,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,17,0,0,10,0,0,1,0,0,0,0,0],[116,118,0.9831,0.41069,0.12749,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,8,0,0,15,0,0,6,0,0,1,0,0,0,0,0],[118,118,1.0,0.44637,0.11146,0.39286,0.42857,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,12,0,0,12,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.20982,"x":0.66516,"p":[[0,94,0.0,0.20982,0.20511,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,7,0,0,7,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[4,94,0.0426,0.57143,0.22868,0.42857,0.57143,0.75,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,7,0,0,5,0,0,5,0,0,7,0,1],[8,94,0.0851,0.58479,0.22968,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,5,0,0,8,0,0,6,0,0,5,0,2],[12,94,0.1277,0.53125,0.24804,0.39286,0.42857,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,10,0,0,4,0,0,1,0,0,8,0,1],[16,94,0.1702,0.6205,0.25905,0.42857,0.64286,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,8,0,0,3,0,0,5,0,0,7,0,4],[20,94,0.2128,0.49554,0.23686,0.39286,0.42859,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,10,0,0,5,0,0,6,0,0,1,0,2],[24,94,0.2553,0.60714,0.25754,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,3,0,0,6,0,0,8,0,0,3,0,0,6,0,4],[28,94,0.2979,0.60267,0.22512,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,9,0,0,4,0,0,8,0,0,4,0,3],[32,94,0.3404,0.66516,0.23313,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,4,0,0,9,0,4],[36,94,0.383,0.56694,0.2461,0.42857,0.571,0.75,0.0,1.0,1,3,0,1,0,0,0,0,5,0,0,9,0,0,6,0,0,3,0,0,5,0,3],[40,94,0.4255,0.4598,0.22512,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,11,0,0,8,0,0,4,0,0,2,0,0,5,0,0],[44,94,0.4681,0.57588,0.22155,0.42857,0.57121,0.75,0.1429,1.0,0,1,0,0,0,1,0,0,4,0,0,10,0,0,4,0,0,5,0,0,7,0,1],[48,94,0.5106,0.54013,0.24674,0.39286,0.4998,0.74996,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,8,0,0,5,0,0,3,0,0,7,0,1],[52,94,0.5532,0.56694,0.26362,0.42857,0.49979,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,5,0,0,9,0,0,3,0,0,4,0,0,6,0,3],[56,94,0.5957,0.62054,0.20395,0.42857,0.57143,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,7,0,0,6,0,2],[60,94,0.6383,0.58482,0.26332,0.28571,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,9,0,0,5,0,0,1,0,0,5,0,0,9,0,2],[64,94,0.6809,0.62944,0.20159,0.5354,0.71429,0.71429,0.0,1.0,1,2,1,1,0,0,0,0,1,0,0,6,0,0,6,0,0,13,0,0,3,0,2],[68,94,0.7234,0.53121,0.23208,0.39286,0.49979,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,8,0,0,7,0,0,2,0,0,6,0,1],[72,94,0.766,0.5625,0.21706,0.42857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,13,0,0,5,0,0,5,0,0,4,0,2],[76,94,0.8085,0.6205,0.2615,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,5,0,0,4,0,0,9,0,0,2,0,0,6,0,5],[80,94,0.8511,0.49552,0.21122,0.28593,0.42859,0.60714,0.14286,0.85714,0,0,0,0,0,3,0,0,6,0,0,8,0,0,7,0,0,4,0,0,4,0,0],[84,94,0.8936,0.51337,0.21682,0.42857,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,11,0,0,6,0,0,4,0,0,2,0,2],[88,94,0.9362,0.55357,0.21943,0.42857,0.5,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,10,0,0,5,0,0,4,0,0,6,0,1],[92,94,0.9787,0.55352,0.17767,0.42857,0.4286,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,8,0,0,1,0,0,5,0,1],[94,94,1.0,0.45088,0.17168,0.39286,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,17,0,0,3,0,0,2,0,0,1,0,1]]}]},{"i":"a97a41dd45cd8a05","q":"For any positive integer  $n{}$  define  $a_n=\\{n/s(n)\\}$  where  $s(\\cdot)$  denotes the sum of the digits and  $\\{\\cdot\\}$  denotes the fractional part.[list=a]\n[*]Prove that there exist infinitely many positive integers  $n$  such that  $a_n=1/2.$ [*]Determine the smallest positive integer  $n$  such that  $a_n=1/6.$ [/list]\n*Marius Burtea*","t":[{"b":6,"e":0.57143,"k":"falling","v":0.49999,"x":0.85268,"p":[[0,171,0.0,0.80356,0.15871,0.857,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,26,0,0],[4,171,0.0234,0.81249,0.15746,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0],[8,171,0.0468,0.76783,0.22799,0.85711,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,26,0,0],[12,171,0.0702,0.79017,0.2082,0.85714,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0],[16,171,0.0936,0.79911,0.17076,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,27,0,0],[20,171,0.117,0.75446,0.2506,0.85711,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,0],[24,171,0.1404,0.81696,0.1525,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0],[28,171,0.1637,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[32,171,0.1871,0.79463,0.16344,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,25,0,0],[36,171,0.2105,0.83929,0.04725,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0],[40,171,0.2339,0.75446,0.25313,0.85714,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,26,0,0],[44,171,0.2573,0.7991,0.20781,0.85714,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0],[48,171,0.2807,0.7589,0.21561,0.71429,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,23,0,0],[52,171,0.3041,0.76337,0.21612,0.82143,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,24,0,0],[56,171,0.3275,0.78124,0.17124,0.82143,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,24,0,0],[60,171,0.3509,0.70982,0.28901,0.78561,0.85714,0.85714,0.0,0.85714,4,0,4,4,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,24,0,0],[64,171,0.3743,0.73212,0.28291,0.857,0.85714,0.85714,0.0,0.85714,4,0,4,4,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0],[68,171,0.3977,0.78123,0.17853,0.85711,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,25,0,0],[72,171,0.4211,0.77677,0.16729,0.71429,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,22,0,0],[76,171,0.4444,0.75446,0.21498,0.71429,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,22,0,0],[80,171,0.4678,0.80803,0.16213,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,28,0,0],[84,171,0.4912,0.7857,0.2113,0.85714,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0],[88,171,0.5146,0.70982,0.25873,0.57143,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,21,0,0],[92,171,0.538,0.75446,0.21793,0.71429,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,22,0,0],[96,171,0.5614,0.80356,0.17036,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,28,0,0],[100,171,0.5848,0.73213,0.25443,0.71429,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,22,0,0],[104,171,0.6082,0.70982,0.23551,0.57143,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,19,0,0],[108,171,0.6316,0.78571,0.17128,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,25,0,0],[112,171,0.655,0.74106,0.25365,0.82143,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,24,0,0],[116,171,0.6784,0.66071,0.28064,0.57143,0.85707,0.85714,0.0,0.85714,4,0,4,4,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,17,0,0],[120,171,0.7018,0.74998,0.22588,0.82132,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,24,0,0],[124,171,0.7251,0.6875,0.31224,0.67857,0.85714,0.85714,0.0,0.85714,5,0,5,5,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,23,0,0],[128,171,0.7485,0.5982,0.35792,0.42825,0.85714,0.85714,0.0,0.85714,8,0,8,8,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,18,0,0],[132,171,0.7719,0.73661,0.2372,0.71429,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,23,0,0],[136,171,0.7953,0.73659,0.2345,0.71429,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,23,0,0],[140,171,0.8187,0.76785,0.12241,0.67857,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,20,0,0],[144,171,0.8421,0.71875,0.19718,0.57143,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,0,19,0,0],[148,171,0.8655,0.70978,0.15356,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,6,0,0,14,0,0],[152,171,0.8889,0.56694,0.20666,0.5354,0.57143,0.60714,0.0,0.85714,2,0,1,2,0,0,0,0,1,0,0,5,0,0,16,0,0,2,0,0,6,0,0],[156,171,0.9123,0.63838,0.20198,0.57132,0.57143,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,6,0,0,12,0,0,1,0,0,12,0,0],[160,171,0.9357,0.55357,0.25938,0.42857,0.57143,0.85704,0.0,0.85714,4,0,4,4,0,0,0,0,0,0,0,6,0,0,13,0,0,0,0,0,9,0,0],[164,171,0.9591,0.49999,0.27199,0.42857,0.57143,0.57143,0.0,0.85714,6,0,6,6,0,0,0,0,0,0,0,4,0,0,16,0,0,0,0,0,6,0,0],[168,171,0.9825,0.53124,0.24284,0.5354,0.57143,0.57143,0.0,0.85714,4,0,4,4,0,0,0,0,1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0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,27,0,0],[168,189,0.8889,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[172,189,0.9101,0.82142,0.08748,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,27,0,0],[176,189,0.9312,0.83481,0.0724,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0],[180,189,0.9524,0.8482,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[184,189,0.9735,0.77679,0.12846,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,21,0,0],[188,189,0.9947,0.76783,0.13245,0.67857,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,21,0,0],[189,189,1.0,0.81249,0.12078,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,28,0,0]]}]},{"i":"41039aacac3cd070","q":"For nine different positive integers $d_{1}, d_{2}, \\ldots, d_{9}$, consider the polynomial $P(n)=\\left(n+d_{1}\\right)\\left(n+d_{2}\\right) \\cdot \\ldots \\cdot\\left(n+d_{9}\\right)$.\nShow that there exists an integer $N$ with the following property:\nFor all integers $n \\geq N$, the number $P(n)$ is divisible by a prime number greater than 20.","t":[{"b":1,"e":0.0,"k":"falling","v":0.26335,"x":0.65618,"p":[[0,31,0.0,0.65618,0.27168,0.571,0.57143,0.85714,0.0,1.0,2,7,1,2,0,1,0,0,1,0,0,0,0,0,14,0,0,3,0,0,4,0,7],[4,31,0.129,0.52232,0.24643,0.57143,0.57143,0.57143,0.0,1.0,4,3,0,4,0,0,0,0,2,0,0,0,0,0,23,0,0,0,0,0,0,0,3],[8,31,0.2581,0.5223,0.19102,0.571,0.57143,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,1,0,0,3,0,0,23,0,0,0,0,0,1,0,1],[12,31,0.3871,0.47316,0.20956,0.28571,0.57143,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,4,0,0,0,0,0,22,0,0,0,0,0,0,0,1],[16,31,0.5161,0.44642,0.23622,0.28571,0.57143,0.57143,0.0,0.85714,5,0,0,5,0,2,0,0,2,0,0,1,0,0,20,0,0,1,0,0,1,0,0],[20,31,0.6452,0.42856,0.21428,0.2857,0.57143,0.57143,0.0,0.57143,4,0,0,4,0,3,0,0,3,0,0,1,0,0,21,0,0,0,0,0,0,0,0],[24,31,0.7742,0.53123,0.15249,0.57143,0.57143,0.57143,0.0,0.857,2,0,0,2,0,0,0,0,0,0,0,3,0,0,26,0,0,0,0,0,1,0,0],[28,31,0.9032,0.49106,0.2257,0.53539,0.57143,0.57143,0.0,1.0,4,1,0,4,0,0,0,0,3,0,0,1,0,0,22,0,0,0,0,0,1,0,1],[31,31,1.0,0.26335,0.25023,0.0,0.2857,0.57111,0.0,0.57143,13,0,0,13,0,2,0,0,5,0,0,1,0,0,11,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"flat","v":0.54015,"x":0.74997,"p":[[0,39,0.0,0.7366,0.23988,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,3,0,0,4,0,11],[4,39,0.1026,0.54015,0.29824,0.42857,0.57143,0.60714,0.0,1.0,5,5,0,5,0,0,0,0,2,0,0,2,0,0,15,0,0,2,0,0,1,0,5],[8,39,0.2051,0.55802,0.24836,0.57143,0.57143,0.57143,0.0,1.0,3,4,0,3,0,0,0,0,2,0,0,1,0,0,21,0,0,0,0,0,1,0,4],[12,39,0.3077,0.63392,0.22286,0.57143,0.57143,0.57143,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,1,0,0,22,0,0,0,0,0,0,0,7],[16,39,0.4103,0.62054,0.24643,0.57143,0.57143,0.71429,0.0,1.0,2,6,0,2,0,0,0,0,1,0,0,2,0,0,17,0,0,3,0,0,1,0,6],[20,39,0.5128,0.71426,0.19887,0.57143,0.57143,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,1,0,10],[24,39,0.6154,0.74552,0.22514,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,13,0,0,3,0,0,2,0,12],[28,39,0.7179,0.66963,0.26351,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,1,0,0,15,0,0,1,0,0,1,0,10],[32,39,0.8205,0.58034,0.2922,0.57132,0.57143,0.60714,0.0,1.0,4,7,0,4,0,0,0,0,1,0,0,2,0,0,17,0,0,1,0,0,0,0,7],[36,39,0.9231,0.74997,0.27434,0.57143,0.78571,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,0,0,0,13,0,0,0,0,0,0,0,16],[39,39,1.0,0.74105,0.25365,0.57143,0.78564,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,2,0,0,3,0,13]]}]},{"i":"aed28eaa954ae4c9","q":"Given  $n\\ge 2$  a natural number,  $(K,+,\\cdot )$  a body with commutative property that  $\\underbrace{1+...+}_{m}1\\ne 0,m=2,...,n,f\\in K[X]$  a polynomial of degree  $n$  and  $G$  a subgroup of the additive group  $(K,+,\\cdot )$ ,  $G\\ne K.$ Show that there is  $a\\in K$  so $f(a)\\notin G$ .","t":[{"b":4,"e":0.71429,"k":"falling","v":0.46428,"x":0.90625,"p":[[0,32,0.0,0.90625,0.23038,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,27],[4,32,0.125,0.83929,0.28959,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,22],[8,32,0.25,0.79909,0.28764,0.53539,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,20],[12,32,0.375,0.68302,0.28062,0.42857,0.64286,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,5,0,0,7,0,0,3,0,0,2,0,11],[16,32,0.5,0.67856,0.32143,0.42857,0.78571,1.0,0.0,1.0,1,13,0,1,0,1,0,0,5,0,0,5,0,0,3,0,0,1,0,0,3,0,13],[20,32,0.625,0.58482,0.30169,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,5,0,0,6,0,0,4,0,0,2,0,0,4,0,7],[24,32,0.75,0.54464,0.29329,0.28571,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,11,0,0,7,0,0,4,0,0,0,0,0,1,0,8],[28,32,0.875,0.46428,0.24744,0.28571,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,14,0,0,8,0,0,3,0,0,1,0,0,1,0,4],[32,32,1.0,0.50445,0.19556,0.39286,0.49979,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,8,0,0,10,0,0,4,0,0,0,0,2]]},{"b":6,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,23,0.0,0.83482,0.2969,0.89286,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,24],[4,23,0.1739,0.86161,0.24868,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,2,0,0,1,0,23],[8,23,0.3478,0.875,0.26184,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,25],[12,23,0.5217,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,23,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1ee1ccedf32fbdc5","q":"Find all real numbers  $x_1, \\dots, x_{2016}$  that satisfy the following equation for each  $1 \\le i \\le 2016$ . (Here  $x_{2017} = x_1$ .)\n\\[ x_i^2 + x_i - 1 = x_{i+1} \\]","t":[{"b":3,"e":0.28571,"k":"flat","v":0.26339,"x":0.72321,"p":[[0,50,0.0,0.26339,0.10779,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,3,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.625,0.34022,0.28571,0.42857,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,14],[8,50,0.16,0.72321,0.3387,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,19],[12,50,0.24,0.51339,0.32115,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,20,0,0,2,0,0,0,0,0,0,0,0,1,0,9],[16,50,0.32,0.59821,0.32427,0.28571,0.42857,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,13,0,0,5,0,0,1,0,0,1,0,0,0,0,12],[20,50,0.4,0.42857,0.2369,0.28571,0.28571,0.46429,0.2857,1.0,0,4,0,0,0,0,0,0,20,0,0,4,0,0,4,0,0,0,0,0,0,0,4],[24,50,0.48,0.48214,0.28065,0.28571,0.28571,0.60714,0.2857,1.0,0,6,0,0,0,0,0,0,18,0,0,5,0,0,1,0,0,1,0,0,1,0,6],[28,50,0.56,0.38839,0.22371,0.28571,0.28571,0.42857,0.0,1.0,1,3,0,1,0,0,0,0,21,0,0,4,0,0,3,0,0,0,0,0,0,0,3],[32,50,0.64,0.49107,0.28333,0.28571,0.28571,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,16,0,0,5,0,0,1,0,0,1,0,0,3,0,5],[36,50,0.72,0.42412,0.2355,0.28571,0.28571,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,20,0,0,5,0,0,3,0,0,0,0,0,0,0,4],[40,50,0.8,0.36607,0.14258,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,5,0,0,3,0,0,1,0,0,1,0,0],[44,50,0.88,0.32143,0.08748,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,26,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[48,50,0.96,0.28571,0.03571,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.28125,0.02485,0.28571,0.28571,0.28571,0.1429,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"rising","v":0.25893,"x":0.5625,"p":[[0,21,0.0,0.25893,0.10972,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,4,0,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.50893,0.29653,0.28571,0.28571,0.67857,0.2857,1.0,0,8,0,0,0,0,0,0,17,0,0,4,0,0,3,0,0,0,0,0,0,0,8],[8,21,0.381,0.51786,0.30462,0.28571,0.28571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,18,0,0,2,0,0,3,0,0,0,0,0,1,0,8],[12,21,0.5714,0.5625,0.31122,0.28571,0.42859,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,10],[16,21,0.7619,0.49553,0.3174,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,9],[20,21,0.9524,0.4375,0.27418,0.28571,0.28571,0.42857,0.2857,1.0,0,6,0,0,0,0,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,6],[21,21,1.0,0.45089,0.29257,0.28571,0.28571,0.42858,0.2857,1.0,0,7,0,0,0,0,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,7]]}]},{"i":"d7f23f568486f7c5","q":"Given 1997 points inside a circle of radius 1, one of them the center of the circle. For each point take the distance to the closest (distinct) point. Show that the sum of the squares of the resulting distances is at most 9 .","t":[{"b":1,"e":0.0,"k":"falling","v":0.21429,"x":0.88393,"p":[[0,93,0.0,0.88393,0.2299,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,24],[4,93,0.043,0.86161,0.2435,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,2,0,22],[8,93,0.086,0.79911,0.31106,0.57143,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,20],[12,93,0.129,0.70982,0.32239,0.57143,0.71429,1.0,0.0,1.0,3,14,0,3,0,1,0,0,0,0,0,2,0,0,7,0,0,4,0,0,1,0,14],[16,93,0.172,0.60713,0.35355,0.42859,0.57143,1.0,0.0,1.0,5,10,0,5,0,1,0,0,1,0,0,4,0,0,6,0,0,3,0,0,2,0,10],[20,93,0.2151,0.60268,0.38255,0.35714,0.64286,1.0,0.0,1.0,7,10,0,7,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,0,4,0,10],[24,93,0.2581,0.52231,0.33996,0.28571,0.57143,0.71429,0.0,1.0,6,7,0,6,0,1,0,0,2,0,0,5,0,0,7,0,0,4,0,0,0,0,7],[28,93,0.3011,0.60714,0.40564,0.14286,0.64286,1.0,0.0,1.0,7,14,0,7,0,2,0,0,0,0,0,2,0,0,5,0,0,2,0,0,0,0,14],[32,93,0.3441,0.54018,0.41455,0.0,0.57143,1.0,0.0,1.0,10,11,0,10,0,0,0,0,0,0,0,4,0,0,5,0,0,0,0,0,2,0,11],[36,93,0.3871,0.63839,0.39767,0.39286,0.78571,1.0,0.0,1.0,7,14,0,7,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,2,0,14],[40,93,0.4301,0.43304,0.39039,0.0,0.42857,0.85714,0.0,1.0,9,7,0,9,0,5,0,0,1,0,0,4,0,0,3,0,0,1,0,0,2,0,7],[44,93,0.4731,0.58479,0.32997,0.42857,0.57143,0.89286,0.0,1.0,4,8,0,4,0,2,0,0,1,0,0,3,0,0,10,0,0,2,0,0,2,0,8],[48,93,0.5161,0.58036,0.32525,0.42857,0.57143,0.85714,0.0,1.0,5,7,0,5,0,0,0,0,1,0,0,5,0,0,9,0,0,2,0,0,3,0,7],[52,93,0.5591,0.37501,0.26184,0.10714,0.4286,0.57143,0.0,0.85714,8,0,0,8,0,2,0,0,1,0,0,10,0,0,6,0,0,4,0,0,1,0,0],[56,93,0.6022,0.31701,0.25441,0.0,0.42857,0.57143,0.0,0.57143,12,0,0,12,0,0,0,0,1,0,0,7,0,0,12,0,0,0,0,0,0,0,0],[60,93,0.6452,0.31249,0.26349,0.0,0.42857,0.57143,0.0,0.85714,11,0,0,11,0,3,0,0,0,0,0,7,0,0,10,0,0,0,0,0,1,0,0],[64,93,0.6882,0.23214,0.27141,0.0,0.0,0.4286,0.0,0.85714,17,0,0,17,0,1,0,0,1,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[68,93,0.7312,0.27231,0.24576,0.0,0.28571,0.4642,0.0,0.71429,13,0,0,13,0,0,0,0,5,0,0,6,0,0,7,0,0,1,0,0,0,0,0],[72,93,0.7742,0.21429,0.27664,0.0,0.0,0.46429,0.0,0.71429,19,0,0,19,0,1,0,0,0,0,0,4,0,0,5,0,0,3,0,0,0,0,0],[76,93,0.8172,0.31696,0.26662,0.0,0.42857,0.57143,0.0,0.85714,11,0,0,11,0,1,0,0,3,0,0,8,0,0,7,0,0,0,0,0,2,0,0],[80,93,0.8602,0.32588,0.28623,0.0,0.42857,0.57143,0.0,1.0,11,1,0,11,0,2,0,0,2,0,0,7,0,0,7,0,0,1,0,0,1,0,1],[84,93,0.9032,0.2767,0.32919,0.0,0.07,0.57143,0.0,1.0,16,2,0,16,0,2,0,0,1,0,0,4,0,0,4,0,0,2,0,0,1,0,2],[88,93,0.9462,0.30357,0.29827,0.0,0.35714,0.57143,0.0,1.0,14,1,0,14,0,1,0,0,1,0,0,4,0,0,9,0,0,2,0,0,0,0,1],[92,93,0.9892,0.25893,0.22711,0.0,0.42857,0.42857,0.0,0.57143,13,0,0,13,0,1,0,0,1,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[93,93,1.0,0.30804,0.27458,0.0,0.42857,0.57143,0.0,0.71429,13,0,0,13,0,1,0,0,0,0,0,7,0,0,8,0,0,3,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"falling","v":0.51784,"x":0.90625,"p":[[0,43,0.0,0.90625,0.23585,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,27],[4,43,0.093,0.78572,0.30304,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,1,0,0,6,0,0,3,0,0,0,0,19],[8,43,0.186,0.78124,0.31337,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,1,0,0,1,0,0,5,0,0,2,0,0,1,0,19],[12,43,0.2791,0.70536,0.36585,0.53571,1.0,1.0,0.0,1.0,4,17,0,4,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,0,0,0,17],[16,43,0.3721,0.72768,0.31412,0.57143,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,1,0,0,1,0,0,8,0,0,2,0,0,3,0,14],[20,43,0.4651,0.79016,0.29231,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,0,1,0,19],[24,43,0.5581,0.64283,0.26245,0.57132,0.57143,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,2,0,0,2,0,0,12,0,0,4,0,0,4,0,6],[28,43,0.6512,0.6205,0.17718,0.57142,0.57143,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,3,0,0,20,0,0,2,0,0,3,0,3],[32,43,0.7442,0.57143,0.22016,0.42859,0.57143,0.60714,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,7,0,0,13,0,0,2,0,0,3,0,3],[36,43,0.8372,0.5893,0.21353,0.4286,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,8,0,0,12,0,0,3,0,0,2,0,4],[40,43,0.9302,0.51784,0.2165,0.42857,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,5,0,0,5,0,0,16,0,0,1,0,0,0,0,3],[43,43,1.0,0.51786,0.13716,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,7,0,0,16,0,0,3,0,0,1,0,0]]}]},{"i":"a22c0a5a74a1027d","q":"Given a  $2n \\times 2m$  table  $(m,n \\in \\mathbb{N})$  with one of two signs \u201d+\u201d or \u201d-\u201d in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain \u201d\u2212\u201d and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn\u2019t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time.\nFind the number of all attainable tables.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.01339,"x":0.18751,"p":[[0,53,0.0,0.18751,0.09739,0.14286,0.1429,0.28571,0.0,0.42857,3,0,2,3,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,53,0.1509,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,53,0.2264,0.08036,0.13803,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,53,0.3019,0.09375,0.11633,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,53,0.3774,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,53,0.4528,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,53,0.5283,0.06232,0.07067,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,53,0.6038,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,53,0.6792,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,53,0.7547,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,53,0.8302,0.06241,0.09399,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,53,0.9057,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,53,0.9811,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.19634,"x":0.31249,"p":[[0,19,0.0,0.19634,0.09284,0.14286,0.14286,0.28571,0.0,0.42857,1,0,1,1,0,20,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.2767,0.11269,0.25,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,18,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.29911,0.16506,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,7,0,0,14,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[12,19,0.6316,0.28125,0.12619,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,6,0,0,21,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[16,19,0.8421,0.31249,0.15333,0.24999,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,14,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[19,19,1.0,0.27232,0.10926,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,9,0,0,19,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"785f2a4392cd70c3","q":"For each positive integer  $k,$  let  $A(k)$  be the number of odd divisors of  $k$  in the interval  $\\left[1,\\sqrt{2k}\\right).$  Evaluate: \\[\\sum_{k=1}^{\\infty}(-1)^{k-1}\\frac{A(k)}k.\\]","t":[{"b":0,"e":0.1429,"k":"falling","v":0.14732,"x":0.79018,"p":[[0,162,0.0,0.69196,0.37134,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,18],[4,162,0.0247,0.75446,0.26301,0.53571,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,6,0,0,2,0,0,7,0,0,0,0,15],[8,162,0.0494,0.67856,0.31944,0.39286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,4,0,0,2,0,0,2,0,0,7,0,0,0,0,13],[12,162,0.0741,0.76786,0.28291,0.53571,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,4,0,0,1,0,17],[16,162,0.0988,0.77679,0.27879,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,8,0,0,3,0,15],[20,162,0.1235,0.75893,0.29329,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,5,0,0,0,0,17],[24,162,0.1481,0.79018,0.25501,0.71429,0.85714,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,10,0,0,0,0,16],[28,162,0.1728,0.625,0.33072,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,5,0,0,3,0,0,2,0,0,3,0,0,7,0,0,1,0,10],[32,162,0.1975,0.58036,0.32525,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,8,0,0,7,0,0,2,0,0,1,0,0,0,0,11],[36,162,0.2222,0.52232,0.36702,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,12,0,0,3,0,0,2,0,0,2,0,0,3,0,0,0,0,10],[40,162,0.2469,0.56696,0.35801,0.24999,0.5,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,6,0,0,2,0,0,1,0,0,4,0,0,0,0,11],[44,162,0.2716,0.58036,0.31731,0.28571,0.64286,0.78571,0.0,1.0,1,8,0,1,0,5,0,0,3,0,0,5,0,0,2,0,0,8,0,0,0,0,8],[48,162,0.2963,0.43304,0.31029,0.14286,0.28571,0.71429,0.14286,1.0,0,5,0,0,0,10,0,0,10,0,0,1,0,0,1,0,0,5,0,0,0,0,5],[52,162,0.321,0.64286,0.31744,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,2,0,0,6,0,0,0,0,0,8,0,0,0,0,11],[56,162,0.3457,0.60714,0.3312,0.28571,0.64286,1.0,0.0,1.0,1,10,0,1,0,3,0,0,6,0,0,5,0,0,1,0,0,4,0,0,2,0,10],[60,162,0.3704,0.62052,0.32066,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,2,0,0,7,0,0,3,0,0,2,0,0,6,0,0,1,0,10],[64,162,0.3951,0.62054,0.32264,0.39286,0.64286,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,3,0,0,6,0,0,2,0,0,5,0,0,0,0,11],[68,162,0.4198,0.55804,0.33761,0.28571,0.5,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,6,0,0,3,0,0,2,0,0,4,0,0,1,0,9],[72,162,0.4444,0.60714,0.30929,0.35714,0.71429,0.75,0.14286,1.0,0,7,0,0,0,8,0,0,0,0,0,2,0,0,3,0,0,11,0,0,1,0,7],[76,162,0.4691,0.5,0.32537,0.14286,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,9,0,0,5,0,0,5,0,0,1,0,0,5,0,0,0,0,7],[80,162,0.4938,0.45982,0.26663,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,6,0,0,8,0,0,2,0,0,5,0,0,1,0,3],[84,162,0.5185,0.44643,0.27836,0.2857,0.28571,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,12,0,0,3,0,0,2,0,0,5,0,0,0,0,4],[88,162,0.5432,0.40179,0.28669,0.14286,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,11,0,0,8,0,0,4,0,0,2,0,0,3,0,0,0,0,4],[92,162,0.5679,0.34375,0.28763,0.14286,0.21428,0.42857,0.14286,1.0,0,4,0,0,0,16,0,0,7,0,0,2,0,0,2,0,0,1,0,0,0,0,4],[96,162,0.5926,0.48661,0.31715,0.14286,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,11,0,0,3,0,0,3,0,0,3,0,0,6,0,0,1,0,5],[100,162,0.6173,0.41518,0.29093,0.14286,0.28571,0.71429,0.14286,1.0,0,3,0,0,0,10,0,0,10,0,0,2,0,0,1,0,0,4,0,0,2,0,3],[104,162,0.642,0.37052,0.27165,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,13,0,0,5,0,0,3,0,0,3,0,0,5,0,0,0,0,2],[108,162,0.6667,0.44641,0.32291,0.14286,0.28571,0.60714,0.0,1.0,1,6,0,1,0,10,0,0,6,0,0,2,0,0,5,0,0,2,0,0,0,0,6],[112,162,0.6914,0.38393,0.2911,0.14286,0.28571,0.46431,0.14286,1.0,0,4,0,0,0,13,0,0,7,0,0,4,0,0,1,0,0,3,0,0,0,0,4],[116,162,0.716,0.44643,0.31084,0.14286,0.35714,0.60714,0.14286,1.0,0,6,0,0,0,10,0,0,6,0,0,6,0,0,2,0,0,2,0,0,0,0,6],[120,162,0.7407,0.30804,0.21461,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,15,0,0,8,0,0,3,0,0,3,0,0,2,0,0,0,0,1],[124,162,0.7654,0.31696,0.25935,0.14286,0.14286,0.35714,0.14286,1.0,0,2,0,0,0,18,0,0,6,0,0,0,0,0,3,0,0,3,0,0,0,0,2],[128,162,0.7901,0.3125,0.2635,0.14286,0.1429,0.42857,0.0,1.0,1,3,0,1,0,16,0,0,5,0,0,6,0,0,0,0,0,1,0,0,0,0,3],[132,162,0.8148,0.28572,0.26244,0.14286,0.14286,0.32143,0.0,1.0,1,2,0,1,0,21,0,0,2,0,0,2,0,0,1,0,0,3,0,0,0,0,2],[136,162,0.8395,0.33929,0.28291,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,16,0,0,4,0,0,4,0,0,1,0,0,3,0,0,0,0,3],[140,162,0.8642,0.29464,0.25738,0.14286,0.21428,0.28571,0.0,1.0,1,3,0,1,0,15,0,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[144,162,0.8889,0.30357,0.26666,0.14286,0.14286,0.32143,0.0,1.0,1,2,0,1,0,18,0,0,5,0,0,2,0,0,1,0,0,2,0,0,1,0,2],[148,162,0.9136,0.25001,0.20825,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,16,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[152,162,0.9383,0.32143,0.31135,0.14286,0.14286,0.32143,0.0,1.0,1,5,0,1,0,19,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,5],[156,162,0.963,0.26339,0.21461,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,21,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,1],[160,162,0.9877,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[162,162,1.0,0.16071,0.07784,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.67857,"x":0.93304,"p":[[0,67,0.0,0.79018,0.33879,0.57143,1.0,1.0,0.0,1.0,2,22,0,2,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,22],[4,67,0.0597,0.8125,0.27994,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,5,0,0,2,0,19],[8,67,0.1194,0.76786,0.26426,0.71429,0.71429,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,10,0,0,0,0,15],[12,67,0.1791,0.76786,0.27837,0.71429,0.85714,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,3,0,0,0,0,0,9,0,0,0,0,16],[16,67,0.2388,0.67857,0.30723,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,4,0,0,4,0,0,1,0,0,7,0,0,1,0,12],[20,67,0.2985,0.83036,0.24074,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,0,1,0,19],[24,67,0.3582,0.74554,0.22513,0.67857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,13,0,0,0,0,11],[28,67,0.4179,0.72772,0.26083,0.42964,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,7,0,0,2,0,0,8,0,0,0,0,13],[32,67,0.4776,0.71875,0.27545,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,5,0,0,3,0,0,6,0,0,1,0,13],[36,67,0.5373,0.81696,0.1996,0.71429,0.78571,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,12,0,0,1,0,15],[40,67,0.597,0.79464,0.24206,0.71429,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,10,0,0,0,0,16],[44,67,0.6567,0.82143,0.18558,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,0,1,0,15],[48,67,0.7164,0.84375,0.19678,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,1,0,18],[52,67,0.7761,0.85714,0.17857,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,4,0,17],[56,67,0.8358,0.875,0.18123,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,1,0,20],[60,67,0.8955,0.81696,0.18293,0.71429,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,0,0,0,15],[64,67,0.9552,0.85268,0.18723,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,18],[67,67,1.0,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24]]}]},{"i":"a85621f33e206898","q":"Given a polynomial  $P$ , assume that  $L = \\{z \\in \\mathbb{C}: |P(z)| = 1\\}$  is a Jordan curve. Show that the zeros of  $P'$  are in the interior of  $L$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.20089,"x":0.32132,"p":[[0,26,0.0,0.22319,0.30914,0.0,0.07143,0.32143,0.0,1.0,16,2,2,16,0,6,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,2],[4,26,0.1538,0.32132,0.2173,0.14286,0.28571,0.32143,0.0,1.0,2,1,0,2,0,7,0,0,15,0,0,3,0,0,2,0,0,1,0,0,1,0,1],[8,26,0.3077,0.30804,0.22335,0.25,0.28571,0.32143,0.0,1.0,4,2,0,4,0,4,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,2],[12,26,0.4615,0.25893,0.21558,0.14286,0.2857,0.28571,0.0,1.0,5,1,0,5,0,8,0,0,15,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[16,26,0.6154,0.28571,0.26964,0.14286,0.2857,0.32143,0.0,1.0,7,3,0,7,0,6,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[20,26,0.7692,0.29911,0.20316,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,10,0,0,7,0,0,3,0,0,2,0,0,0,0,0],[24,26,0.9231,0.25446,0.18808,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,5,0,0,17,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[26,26,1.0,0.20089,0.11214,0.14286,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,7,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.20973,"x":0.49107,"p":[[0,35,0.0,0.20973,0.26485,0.0,0.14286,0.32143,0.0,1.0,13,1,1,13,0,9,0,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[4,35,0.1143,0.49107,0.3008,0.28571,0.28571,0.75,0.0,1.0,1,6,0,1,0,1,0,0,15,0,0,4,0,0,2,0,0,1,0,0,2,0,6],[8,35,0.2286,0.41964,0.3173,0.2857,0.28571,0.50002,0.0,1.0,3,5,0,3,0,4,0,0,13,0,0,4,0,0,0,0,0,1,0,0,2,0,5],[12,35,0.3429,0.42409,0.2382,0.28571,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,2,0,0,15,0,0,9,0,0,2,0,0,0,0,0,0,0,4],[16,35,0.4571,0.32143,0.20203,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,10,0,0,10,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[20,35,0.5714,0.40622,0.23447,0.2857,0.28571,0.4286,0.14286,1.0,0,3,0,0,0,5,0,0,12,0,0,8,0,0,3,0,0,1,0,0,0,0,3],[24,35,0.6857,0.38391,0.30396,0.24999,0.28571,0.46418,0.0,1.0,4,4,0,4,0,4,0,0,14,0,0,2,0,0,1,0,0,2,0,0,1,0,4],[28,35,0.8,0.34375,0.2693,0.14286,0.28571,0.42857,0.0,1.0,3,3,0,3,0,7,0,0,13,0,0,3,0,0,1,0,0,2,0,0,0,0,3],[32,35,0.9143,0.41964,0.2765,0.2857,0.28571,0.4286,0.14286,1.0,0,5,0,0,0,7,0,0,10,0,0,8,0,0,2,0,0,0,0,0,0,0,5],[35,35,1.0,0.25,0.14726,0.14286,0.28571,0.32143,0.0,0.4286,6,0,0,6,0,4,0,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ff361cef869ab895","q":"For any  $ a\\in\\mathbb{Z}_{\\ge 0} $  make the notation  $ a\\mathbb{Z}_{\\ge 0} =\\{ an| n\\in\\mathbb{Z}_{\\ge 0} \\} . $  Prove that the following relations are equivalent: $ \\text{(1)} a\\mathbb{Z}_{\\ge 0} \\setminus b\\mathbb{Z}_{\\ge 0}\\subset c\\mathbb{Z}_{\\ge 0} \\setminus d\\mathbb{Z}_{\\ge 0} $  $ \\text{(2)} b|a\\text{ or } (c|a\\text{ and } \\text{lcm} (a,b) |\\text{lcm} (a,d)) $ *Marin Tolosi* and *Cosmin Nitu*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.91069,"x":1.0,"p":[[0,14,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,14,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,14,0.5714,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,14,0.8571,0.93749,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[14,14,1.0,0.91069,0.12246,0.85711,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19]]},{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,28,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,28,0.1429,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"36516d5980693736","q":"Given a sequence  $ (c_n) $  of natural numbers defined recursively:  $ c_1 = 2 $ ,  $ c_{n+1} = \\left[ \\frac{3}{2}c_n\\right] $ . Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.98661,"p":[[0,28,0.0,0.88839,0.24675,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,26],[4,28,0.1429,0.98214,0.05924,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,28,0.2857,0.92634,0.17811,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,3,0,0,1,0,26],[12,28,0.4286,0.90628,0.17344,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,0,0,24],[16,28,0.5714,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,28,0.7143,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,28,0.8571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,28,1.0,0.88839,0.20743,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,2,0,23]]},{"b":6,"e":0.85714,"k":"flat","v":0.94196,"x":0.99107,"p":[[0,24,0.0,0.96429,0.11845,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[4,24,0.1667,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,28],[8,24,0.3333,0.95536,0.16535,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[12,24,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,24,0.6667,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,24,0.8333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,24,1.0,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25]]}]},{"i":"ccaf51a15bc3fbd6","q":"Given a positive integer  $k > 1$ , find all positive integers  $n$  such that the polynomial   $$ P(z) = z^n + \\sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \\cdots +z^{2^k-2} + z^n $$   has a complex root  $w$  such that  $|w| = 1$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.89286,"x":0.98214,"p":[[0,99,0.0,0.92857,0.17128,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,26],[4,99,0.0404,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,99,0.0808,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[12,99,0.1212,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,99,0.1616,0.93304,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[20,99,0.202,0.94196,0.14665,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[24,99,0.2424,0.96428,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[28,99,0.2828,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,99,0.3232,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[36,99,0.3636,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[40,99,0.404,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[44,99,0.4444,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[48,99,0.4848,0.89286,0.20825,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,23],[52,99,0.5253,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[56,99,0.5657,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,99,0.6061,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[64,99,0.6465,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[68,99,0.6869,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[72,99,0.7273,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[76,99,0.7677,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[80,99,0.8081,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[84,99,0.8485,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[88,99,0.8889,0.95522,0.10405,1.0,1.0,1.0,0.71,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[92,99,0.9293,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[96,99,0.9697,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[99,99,1.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23]]},{"b":7,"e":0.71429,"k":"flat","v":0.875,"x":0.98661,"p":[[0,90,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,90,0.0444,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,90,0.0889,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,90,0.1333,0.94183,0.09387,0.85714,1.0,1.0,0.71,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[16,90,0.1778,0.93302,0.13359,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[20,90,0.2222,0.93304,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[24,90,0.2667,0.92857,0.14285,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,24],[28,90,0.3111,0.90624,0.13179,0.82143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[32,90,0.3556,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[36,90,0.4,0.875,0.23623,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,21],[40,90,0.4444,0.92411,0.15561,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[44,90,0.4889,0.91517,0.12807,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,20],[48,90,0.5333,0.88393,0.1729,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,3,0,20],[52,90,0.5778,0.91518,0.12299,0.82143,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[56,90,0.6222,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[60,90,0.6667,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[64,90,0.7111,0.93304,0.14719,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[68,90,0.7556,0.94651,0.06906,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[72,90,0.8,0.92411,0.15146,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[76,90,0.8444,0.93303,0.13825,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,22],[80,90,0.8889,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[84,90,0.9333,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[88,90,0.9778,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[90,90,1.0,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21]]}]},{"i":"0481392a2e57a8d5","q":"Given a quadratic trinomial  $f\\left(x\\right)=x^2+ax+b$ . Assume that the equation  $f\\left(f\\left(x\\right)\\right)=0$  has four different real solutions, and that the sum of two of these solutions is  $-1$ . Prove that  $b\\leq -\\frac14$ 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a triangle  $ABC$ ,  $A',B',C'$  are the midpoints of  $\\overline{BC},\\overline{AC},\\overline{AB}$ , respectively.  $B^*,C^*$  lie in  $\\overline{AC},\\overline{AB}$ , respectively, such that  $\\overline{BB^*},\\overline{CC^*}$  are the altitudes of the triangle  $ABC$ . Let  $B^{\\#},C^{\\#}$  be the midpoints of  $\\overline{BB^*},\\overline{CC^*}$ , respectively.  $\\overline{B'B^{\\#}}$  and  $\\overline{C'C^{\\#}}$  meet at  $K$ , and  $\\overline{AK}$  and  $\\overline{BC}$  meet at  $L$ . Prove that  $\\angle{BAL}=\\angle{CAA'}$","t":[{"b":5,"e":1.0,"k":"flat","v":0.83034,"x":0.99554,"p":[[0,123,0.0,0.83034,0.29111,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,20],[4,123,0.0325,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[8,123,0.065,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,123,0.0976,0.92857,0.20203,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[16,123,0.1301,0.91963,0.20186,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,27],[20,123,0.1626,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[24,123,0.1951,0.91518,0.23107,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[28,123,0.2276,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[32,123,0.2602,0.88839,0.25187,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,25],[36,123,0.2927,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[40,123,0.3252,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[44,123,0.3577,0.91071,0.1948,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,25],[48,123,0.3902,0.93749,0.17838,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,27],[52,123,0.4228,0.92411,0.17852,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,26],[56,123,0.4553,0.95982,0.12992,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[60,123,0.4878,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,123,0.5203,0.90625,0.21011,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,25],[68,123,0.5528,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[72,123,0.5854,0.95089,0.1048,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[76,123,0.6179,0.91071,0.1915,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,1,0,25],[80,123,0.6504,0.84822,0.28333,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,24],[84,123,0.6829,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[88,123,0.7154,0.8482,0.22287,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,0,0,21],[92,123,0.748,0.91964,0.18877,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,27],[96,123,0.7805,0.89286,0.21129,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,22],[100,123,0.813,0.88838,0.22514,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,24],[104,123,0.8455,0.91964,0.17474,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,25],[108,123,0.878,0.93301,0.14283,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[112,123,0.9106,0.95089,0.16213,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[116,123,0.9431,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[120,123,0.9756,0.87053,0.23244,0.82132,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,23],[123,123,1.0,0.90624,0.21314,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.80804,"x":1.0,"p":[[0,114,0.0,0.80804,0.24382,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,6,0,0,2,0,17],[4,114,0.0351,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,114,0.0702,0.94196,0.19186,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[12,114,0.1053,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,114,0.1404,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[20,114,0.1754,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,114,0.2105,0.95089,0.16213,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[28,114,0.2456,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,114,0.2807,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,114,0.3158,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,114,0.3509,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,114,0.386,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[48,114,0.4211,0.94196,0.14223,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[52,114,0.4561,0.91518,0.21387,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[56,114,0.4912,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,114,0.5263,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[64,114,0.5614,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[68,114,0.5965,0.95536,0.14032,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[72,114,0.6316,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[76,114,0.6667,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[80,114,0.7018,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[84,114,0.7368,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[88,114,0.7719,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[92,114,0.807,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[96,114,0.8421,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[100,114,0.8772,0.94197,0.12807,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[104,114,0.9123,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[108,114,0.9474,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[112,114,0.9825,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[114,114,1.0,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24]]}]},{"i":"69b4cf0a793861b1","q":"Given an acute triangle $A B C$. Point $D$ is the foot of the perpendicular from $A$ to $B C$. Point $E$ lies on the segment $A D$ and satisfies the equation\n\n$$\n\\frac{|A E|}{|E D|}=\\frac{|C D|}{|D B|}\n$$\n\nPoint $F$ is the foot of the perpendicular from $D$ to $B E$. Prove that $\\angle A F C=90^{\\circ}$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.04464,"x":0.12937,"p":[[0,46,0.0,0.12937,0.20934,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[4,46,0.087,0.07134,0.09442,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.06679,0.0711,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.06696,0.09439,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.06696,0.08737,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.05795,0.07863,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04018,"x":0.12054,"p":[[0,48,0.0,0.12054,0.13882,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,4,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.06687,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.07134,0.10095,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"21d7fb7d390c2e22","q":"Given an acute-angled triangle  $ABC$ , let points  $A' , B' , C'$  be located as follows:   $A'$  is the point where altitude from  $A$  on  $BC$  meets the outwards-facing semicircle on  $BC$  as diameter. Points  $B', C'$  are located similarly.\r\nProve that  $A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$  where  $A[ABC]$  is the area of triangle  $ABC$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.70076,"x":0.87051,"p":[[0,58,0.0,0.82142,0.21129,0.857,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,15,0,10],[4,58,0.069,0.85712,0.12375,0.857,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,17,0,9],[8,58,0.1379,0.8705,0.15308,0.85711,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,12,0,14],[12,58,0.2069,0.82142,0.21428,0.857,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,17,0,9],[16,58,0.2759,0.84372,0.15719,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,17,0,9],[20,58,0.3448,0.83033,0.1615,0.85714,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,17,0,8],[24,58,0.4138,0.83926,0.19151,0.857,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,10],[28,58,0.4828,0.75891,0.25615,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,0,0,0,5,0,0,4,0,0,9,0,10],[32,58,0.5517,0.78124,0.25501,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,0,0,0,6,0,0,1,0,0,8,0,13],[36,58,0.6207,0.87051,0.13058,0.85714,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,14,0,12],[40,58,0.6897,0.82589,0.19799,0.85713,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,16,0,9],[44,58,0.7586,0.8482,0.17474,0.82132,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,11,0,13],[48,58,0.8276,0.70076,0.246,0.57132,0.71429,0.85714,0.14,1.0,0,7,0,0,0,2,0,0,1,0,0,2,0,0,10,0,0,2,0,0,8,0,7],[52,58,0.8966,0.78571,0.23146,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,0,14,0,8],[56,58,0.9655,0.70979,0.28005,0.57143,0.85714,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,0,0,0,9,0,0,1,0,0,11,0,7],[58,58,1.0,0.70087,0.25345,0.57143,0.78564,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,4,0,0,1,0,0,5,0,0,5,0,0,11,0,5]]},{"b":7,"e":0.85714,"k":"flat","v":0.74561,"x":0.94194,"p":[[0,76,0.0,0.88392,0.14916,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,11,0,16],[4,76,0.0526,0.74561,0.30671,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,2,0,0,1,0,0,4,0,0,1,0,0,9,0,12],[8,76,0.1053,0.90624,0.1499,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,11,0,18],[12,76,0.1579,0.80356,0.27374,0.857,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,13,0,12],[16,76,0.2105,0.88839,0.20743,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,17],[20,76,0.2632,0.89284,0.13364,0.85714,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,17,0,13],[24,76,0.3158,0.92411,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,17],[28,76,0.3684,0.9241,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,17],[32,76,0.4211,0.90624,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[36,76,0.4737,0.89283,0.09455,0.85714,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,19,0,11],[40,76,0.5263,0.91071,0.13716,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,19],[44,76,0.5789,0.89283,0.15571,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,13,0,16],[48,76,0.6316,0.91516,0.08646,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[52,76,0.6842,0.91963,0.07937,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[56,76,0.7368,0.91517,0.08646,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[60,76,0.7895,0.91517,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[64,76,0.8421,0.93749,0.07937,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[68,76,0.8947,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[72,76,0.9474,0.89732,0.11971,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,16],[76,76,1.0,0.94194,0.0936,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21]]}]},{"i":"42499f9b6255277a","q":"Given are $n$ real numbers $\\mathrm{x}_{1}, \\mathrm{x}_{2}, \\ldots, \\mathrm{x}_{\\mathrm{n}}$ and $\\mathrm{y}_{1}, \\mathrm{y}_{2}, \\ldots, \\mathrm{y}_{\\mathrm{n}}$. The elements of an $\\mathrm{n} \\times \\mathrm{n}$ matrix A are defined as follows: ( $1 \\leq \\mathrm{i}, \\mathrm{j} \\leq \\mathrm{n}$ )\n\n$$\na_{i j}= \\begin{cases}1 & \\text { if } x_{i}+y_{j} \\geq 0 \\\\ 0 & \\text { if } x_{i}+y_{j}<0\\end{cases}\n$$\n\nFurthermore, let B be an $\\mathrm{n} \\times \\mathrm{n}$ matrix with elements 0 or 1, such that the sum of the elements in each row and each column of B is equal to the sum of the elements in the corresponding row or column of A.\nProve that then $\\mathrm{A}=\\mathrm{B}$ holds.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.21875,"x":0.43302,"p":[[0,32,0.0,0.21875,0.34065,0.0,0.0,0.32143,0.0,1.0,19,2,0,19,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,2],[4,32,0.125,0.38391,0.36321,0.0,0.42859,0.71429,0.0,1.0,13,1,0,13,0,2,0,0,0,0,0,2,0,0,4,0,0,5,0,0,5,0,1],[8,32,0.25,0.28571,0.32732,0.0,0.14286,0.42857,0.0,1.0,13,1,0,13,0,5,0,0,3,0,0,4,0,0,0,0,0,2,0,0,4,0,1],[12,32,0.375,0.43302,0.39201,0.0,0.35714,0.85714,0.0,1.0,12,1,0,12,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,0,12,0,1],[16,32,0.5,0.23204,0.30463,0.0,0.07,0.42857,0.0,0.85714,16,0,0,16,0,5,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,0],[20,32,0.625,0.25893,0.31428,0.0,0.14286,0.42858,0.0,1.0,12,1,0,12,0,10,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,1],[24,32,0.75,0.3125,0.34707,0.0,0.14286,0.71429,0.0,0.85714,12,0,0,12,0,7,0,0,2,0,0,1,0,0,1,0,0,2,0,0,7,0,0],[28,32,0.875,0.27224,0.31618,0.0,0.14286,0.42858,0.0,1.0,12,1,0,12,0,7,0,0,3,0,0,4,0,0,0,0,0,1,0,0,4,0,1],[32,32,1.0,0.24106,0.34705,0.0,0.0,0.42858,0.0,0.85714,19,0,0,19,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,7,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.11597,"x":0.31687,"p":[[0,26,0.0,0.20527,0.28782,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,4,0,0,4,0,0,2,0,0,0,0,0,2,0,0,3,0,0],[4,26,0.1538,0.31687,0.35131,0.0,0.14286,0.71429,0.0,1.0,11,2,0,11,0,8,0,0,2,0,0,2,0,0,0,0,0,3,0,0,4,0,2],[8,26,0.3077,0.19195,0.27572,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,6,0,0,2,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[12,26,0.4615,0.27229,0.32409,0.0,0.14286,0.35704,0.0,1.0,13,2,0,13,0,5,0,0,6,0,0,0,0,0,2,0,0,2,0,0,2,0,2],[16,26,0.6154,0.11597,0.18359,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[20,26,0.7692,0.21875,0.2923,0.0,0.07143,0.32143,0.0,1.0,16,1,0,16,0,5,0,0,3,0,0,2,0,0,0,0,0,5,0,0,0,0,1],[24,26,0.9231,0.1473,0.23816,0.0,0.0,0.1786,0.0,0.857,19,0,0,19,0,5,0,0,4,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[26,26,1.0,0.16063,0.23077,0.0,0.07,0.1786,0.0,0.85714,16,0,0,16,0,8,0,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,0]]}]},{"i":"53b653184cb57e78","q":"Given a triangle   $ABC$  with  $BC=a$ ,   $CA=b$ ,   $AB=c$ ,  $\\angle BAC = \\alpha$ ,  $\\angle CBA = \\beta$ ,  $\\angle ACB = \\gamma$ . Prove that   $$  a \\sin(\\beta-\\gamma) + b \\sin(\\gamma-\\alpha) +c\\sin(\\alpha-\\beta) = 0. $$","t":[{"b":1,"e":0.28571,"k":"flat","v":0.28571,"x":0.40625,"p":[[0,32,0.0,0.40625,0.05187,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.34375,0.13296,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[8,32,0.25,0.28571,0.0,0.28571,0.28571,0.28571,0.28571,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.30804,0.05187,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.29465,0.03458,0.28571,0.28571,0.28571,0.28571,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.30803,0.05187,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.29911,0.04164,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.29464,"x":0.44642,"p":[[0,29,0.0,0.44642,0.09941,0.42857,0.42857,0.42857,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,27,0,0,1,0,0,1,0,0,1,0,0],[4,29,0.1379,0.40625,0.05187,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.32589,0.08917,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[12,29,0.4138,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.2991,0.04164,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.31251,0.05576,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"212dc974fdf9229b","q":"Given any four distinct positive real numbers, show that one can choose three numbers  $A,B,C$  from among them, such that all three quadratic equations \\begin{eqnarray*} Bx^2 + x + C &=& 0 Cx^2 + x + A &=& 0  Ax^2 + x +B &=& 0 \\end{eqnarray*} have only real roots, or all three equations have only imaginary roots.","t":[{"b":2,"e":1.0,"k":"rising","v":0.62054,"x":1.0,"p":[[0,18,0.0,0.62054,0.32263,0.28571,0.4286,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,8,0,0,9,0,0,1,0,0,0,0,0,0,0,13],[4,18,0.2222,0.83929,0.2714,0.75001,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,1,0,23],[8,18,0.4444,0.94643,0.16269,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":0.28571,"k":"falling","v":0.3616,"x":0.89732,"p":[[0,36,0.0,0.68749,0.36499,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,18],[4,36,0.1111,0.68304,0.32876,0.39286,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,7,0,0,7,0,0,0,0,0,1,0,0,0,0,16],[8,36,0.2222,0.82143,0.26,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,1,0,0,4,0,19],[12,36,0.3333,0.89732,0.21498,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,3,0,24],[16,36,0.4444,0.77678,0.3008,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,6,0,0,4,0,0,0,0,0,1,0,0,2,0,19],[20,36,0.5556,0.77232,0.30693,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,7,0,0,3,0,0,0,0,0,1,0,0,2,0,19],[24,36,0.6667,0.70982,0.34715,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,18],[28,36,0.7778,0.54018,0.33643,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,16,0,0,2,0,0,0,0,0,1,0,0,1,0,10],[32,36,0.8889,0.41964,0.25738,0.28571,0.28571,0.42857,0.28571,1.0,0,5,0,0,0,0,0,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,5],[36,36,1.0,0.3616,0.2082,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"f891e556eecdbee9","q":"Given an equilateral triangle  $ABC$  and a point  $M$  in the plane ( $ABC$ ). Let  $A', B', C'$  be respectively the symmetric through  $M$  of  $A, B, C$ .\r\n\r**I.** Prove that there exists a unique point  $P$  equidistant from  $A$  and  $B'$ , from  $B$  and  $C'$  and from  $C$  and  $A'$ .\r**II.** Let  $D$  be the midpoint of the side  $AB$ . When  $M$  varies ( $M$  does not coincide with  $D$ ), prove that the circumcircle of triangle  $MNP$  ( $N$  is the intersection of the line  $DM$  and  $AP$ ) pass through a fixed point.","t":[{"b":2,"e":0.4286,"k":"rising","v":0.14732,"x":0.58929,"p":[[0,131,0.0,0.14732,0.15764,0.0,0.14286,0.2857,0.0,0.57143,15,0,2,15,0,4,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,131,0.0305,0.52232,0.17717,0.42857,0.42859,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,13,0,0,6,0,0,4,0,0,4,0,0],[8,131,0.0611,0.58929,0.15871,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,14,0,0,4,0,0,10,0,0,4,0,0],[12,131,0.0916,0.53572,0.17128,0.42857,0.50001,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,12,0,0,8,0,0,4,0,0,4,0,0],[16,131,0.1221,0.58927,0.18813,0.42857,0.57143,0.71429,0.1429,0.85714,0,0,0,0,0,1,0,0,3,0,0,6,0,0,8,0,0,9,0,0,5,0,0],[20,131,0.1527,0.52677,0.20652,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,6,0,0,11,0,0,6,0,0,3,0,0],[24,131,0.1832,0.54908,0.17536,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,10,0,0,7,0,0,11,0,0,1,0,0],[28,131,0.2137,0.50448,0.16357,0.42857,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,4,0,0,9,0,0,13,0,0,4,0,0,1,0,0],[32,131,0.2443,0.43302,0.19392,0.28571,0.42857,0.57111,0.0,0.85714,2,0,0,2,0,1,0,0,6,0,0,14,0,0,5,0,0,2,0,0,2,0,0],[36,131,0.2748,0.49551,0.21423,0.42857,0.42859,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,3,0,0,15,0,0,5,0,0,2,0,0,5,0,0],[40,131,0.3053,0.57142,0.19233,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,12,0,0,4,0,0,10,0,0,4,0,0],[44,131,0.3359,0.42862,0.18898,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,6,0,0,13,0,0,4,0,0,4,0,0,1,0,0],[48,131,0.3664,0.49999,0.17856,0.42857,0.42857,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,13,0,0,4,0,0,5,0,0,3,0,0],[52,131,0.3969,0.49109,0.18533,0.28571,0.5005,0.60714,0.14286,0.857,0,0,0,0,0,2,0,0,7,0,0,7,0,0,8,0,0,7,0,0,1,0,0],[56,131,0.4275,0.51344,0.17805,0.42857,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,15,0,0,7,0,0,4,0,0,3,0,0],[60,131,0.458,0.53115,0.15231,0.42857,0.4286,0.71107,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,16,0,0,5,0,0,7,0,0,2,0,0],[64,131,0.4885,0.45089,0.19597,0.39286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,15,0,0,4,0,0,2,0,0,3,0,0],[68,131,0.5191,0.4777,0.14984,0.42857,0.4286,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,13,0,0,7,0,0,4,0,0,1,0,0],[72,131,0.5496,0.45536,0.1729,0.39286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,14,0,0,6,0,0,2,0,0,2,0,0],[76,131,0.5802,0.48215,0.12753,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,19,0,0,6,0,0,3,0,0,1,0,0],[80,131,0.6107,0.47763,0.1737,0.42857,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,1,0,0,6,0,0,14,0,0,6,0,0,2,0,0,3,0,0],[84,131,0.6412,0.42844,0.16342,0.42857,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,20,0,0,2,0,0,1,0,0,2,0,0],[88,131,0.6718,0.41523,0.08269,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,24,0,0,3,0,0,0,0,0,0,0,0],[92,131,0.7023,0.43749,0.13332,0.39286,0.42857,0.4286,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,18,0,0,3,0,0,2,0,0,1,0,0],[96,131,0.7328,0.40161,0.13079,0.28571,0.42857,0.4286,0.14,0.71,0,0,0,0,0,3,0,0,7,0,0,16,0,0,5,0,0,1,0,0,0,0,0],[100,131,0.7634,0.42855,0.1336,0.39286,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,18,0,0,4,0,0,1,0,0,1,0,0],[104,131,0.7939,0.41967,0.07085,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,24,0,0,3,0,0,0,0,0,0,0,0],[108,131,0.8244,0.42862,0.12372,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,22,0,0,2,0,0,1,0,0,1,0,0],[112,131,0.855,0.40179,0.0974,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,19,0,0,2,0,0,1,0,0,0,0,0],[116,131,0.8855,0.44201,0.13533,0.42857,0.42857,0.4286,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,19,0,0,4,0,0,0,0,0,2,0,0],[120,131,0.916,0.44643,0.1171,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,22,0,0,3,0,0,3,0,0,0,0,0],[124,131,0.9466,0.38837,0.08913,0.2857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[128,131,0.9771,0.36161,0.09438,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[131,131,1.0,0.38393,0.06622,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.14723,"x":0.55355,"p":[[0,63,0.0,0.14723,0.16935,0.0,0.14286,0.2857,0.0,0.71429,13,0,3,13,0,10,0,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[4,63,0.0635,0.43302,0.15764,0.28571,0.42857,0.57141,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,10,0,0,7,0,0,2,0,0,1,0,0],[8,63,0.127,0.46428,0.26963,0.28571,0.42857,0.71429,0.0,1.0,2,1,0,2,0,3,0,0,9,0,0,5,0,0,4,0,0,3,0,0,5,0,1],[12,63,0.1905,0.45536,0.14032,0.42857,0.42857,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,17,0,0,4,0,0,3,0,0,1,0,0],[16,63,0.254,0.45089,0.14773,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,12,0,0,7,0,0,4,0,0,0,0,0],[20,63,0.3175,0.45981,0.18808,0.42857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,2,0,0,17,0,0,4,0,0,5,0,0,1,0,0],[24,63,0.381,0.43759,0.19212,0.42857,0.42857,0.42895,0.0,0.85714,2,0,0,2,0,1,0,0,3,0,0,20,0,0,2,0,0,1,0,0,3,0,0],[28,63,0.4444,0.4107,0.1115,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,18,0,0,4,0,0,1,0,0,0,0,0],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polynomial  $P(x)=1-\\frac{1}{3}x+\\frac{1}{6}x^2$ , define \\[ Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \\sum\\limits_{i=0}^{50}a_ix^i. \\] Then  $\\sum\\limits_{i=0}^{50}|a_i|=\\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,55,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,55,0.0727,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,55,0.1455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,55,0.2182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,55,0.2909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,55,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,55,0.4364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,55,0.5818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,55,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,55,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,55,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,55,0.9455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,5,0.0,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[4,5,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[5,5,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"af1e6972961bf760","q":"Given is a trapezium $A B C D$ with $B C \\| A D$. Assume that the bisectors of the angles $B A D$ and $C D A$ intersect on the perpendicular bisector of segment $B C$. Prove that $|A B|=|C D|$ or $|A B|+|C D|=|A 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0,0,3,0,7,0,0,5,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[152,173,0.8786,0.18973,0.17822,0.0,0.14286,0.42857,0.0,0.4286,12,0,0,12,1,5,0,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[156,173,0.9017,0.2009,0.18162,0.0,0.14288,0.42857,0.0,0.4286,12,0,0,12,0,5,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[160,173,0.9249,0.17402,0.16265,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,10,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[164,173,0.948,0.1875,0.16917,0.0,0.14286,0.32143,0.0,0.4286,11,0,0,11,0,8,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[168,173,0.9711,0.19415,0.17521,0.0,0.14286,0.42857,0.0,0.42857,11,0,0,11,1,6,0,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[172,173,0.9942,0.25223,0.17586,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,8,0,0,2,0,1,14,0,0,0,0,0,0,0,0,0,0,0],[173,173,1.0,0.19197,0.17353,0.0,0.14286,0.42857,0.0,0.4286,10,0,0,10,0,11,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d4a2bd1b358dab5e","q":"Given a triangle  $ABC$  in which  $BC = 2AB$ . The point  $D$  is the midpoint of the side  $BC$ , the point  $K$  is the midpoint of the segment  $BD$ . Prove that  $AC = 2AK$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.03571,"x":0.29018,"p":[[0,38,0.0,0.18303,0.16065,0.0,0.2857,0.28571,0.0,0.4286,13,0,0,13,0,1,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.16964,0.15335,0.0,0.21428,0.28571,0.0,0.42857,13,0,0,13,0,3,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.09821,0.14032,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,1,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.29018,0.0977,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,1,0,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.22767,0.12807,0.24999,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,1,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.1875,0.13092,0.0,0.2857,0.28571,0.0,0.42857,9,0,0,9,0,5,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.18294,0.14829,0.0,0.2857,0.28571,0.0,0.42857,12,0,0,12,0,1,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.1384,0.14501,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,2,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.14286,0.15152,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,2,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.21428,0.15152,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,3,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.13393,"x":0.51339,"p":[[0,23,0.0,0.19196,0.14555,0.0,0.28571,0.28571,0.0,0.42857,11,0,0,11,0,1,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.13393,0.18536,0.0,0.0,0.28571,0.0,0.71429,19,0,0,19,0,1,0,0,10,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,23,0.3478,0.24107,0.16536,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,0,0,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[12,23,0.5217,0.41964,0.3008,0.2857,0.28571,0.42857,0.0,1.0,1,6,0,1,0,4,0,0,17,0,0,3,0,0,0,0,0,1,0,0,0,0,6],[16,23,0.6957,0.51339,0.36919,0.28571,0.28571,1.0,0.0,1.0,4,11,0,4,0,0,0,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,11],[20,23,0.8696,0.37053,0.24964,0.28571,0.28571,0.28571,0.0,1.0,1,4,0,1,0,2,0,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[23,23,1.0,0.37053,0.25218,0.2857,0.28571,0.28571,0.0,1.0,2,4,0,2,0,0,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,4]]}]},{"i":"e9d3b5ff1064190f","q":"Given a positive integer  $n,$  what is the largest  $k$  such that the numbers  $1,2,\\dots,n$  can be put into  $k$  boxes so that the sum of the numbers in each box is the same?\n\n[When  $n=8,$  the example  $\\{1,2,3,6\\},\\{4,8\\},\\{5,7\\}$  shows that the largest  $k$  is *at least* 3.]","t":[{"b":0,"e":0.85714,"k":"flat","v":0.86607,"x":0.95089,"p":[[0,101,0.0,0.90178,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[4,101,0.0396,0.90178,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[8,101,0.0792,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[12,101,0.1188,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[16,101,0.1584,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[20,101,0.198,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[24,101,0.2376,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[28,101,0.2772,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[32,101,0.3168,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[36,101,0.3564,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[40,101,0.396,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[44,101,0.4356,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[48,101,0.4752,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[52,101,0.5149,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[56,101,0.5545,0.88393,0.06622,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,7],[60,101,0.5941,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[64,101,0.6337,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[68,101,0.6733,0.86607,0.07087,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,4],[72,101,0.7129,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[76,101,0.7525,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[80,101,0.7921,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[84,101,0.8317,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[88,101,0.8713,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[92,101,0.9109,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[96,101,0.9505,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[100,101,0.9901,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[101,101,1.0,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9]]},{"b":3,"e":0.85714,"k":"flat","v":0.89732,"x":0.9375,"p":[[0,41,0.0,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[4,41,0.0976,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[8,41,0.1951,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[12,41,0.2927,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,41,0.3902,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[20,41,0.4878,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[24,41,0.5854,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[28,41,0.6829,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[32,41,0.7805,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[36,41,0.878,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[40,41,0.9756,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[41,41,1.0,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9]]}]},{"i":"9eeb323565064ba2","q":"For a permutation  $ \\sigma\\in S_n$  with  $ (1,2,\\dots,n)\\mapsto(i_1,i_2,\\dots,i_n)$ , define\r\n\\[ D(\\sigma) \\equal{} \\sum_{k \\equal{} 1}^n |i_k \\minus{} k|\r\n\\]\r\nLet\r\n\\[ Q(n,d) \\equal{} \\left|\\left\\{\\sigma\\in S_n : D(\\sigma) \\equal{} d\\right\\}\\right|\r\n\\]\r\nShow that when  $ d \\geq 2n$ ,  $ Q(n,d)$  is an even number.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.27678,"p":[[0,32,0.0,0.08928,0.13243,0.0,0.0,0.2857,0.0,0.28571,22,0,1,22,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,32,0.5,0.2232,0.32129,0.0,0.0,0.32143,0.0,1.0,18,3,0,18,0,2,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,3],[20,32,0.625,0.14732,0.20355,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,3,0,0,7,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[24,32,0.75,0.27678,0.34244,0.0,0.07143,0.57141,0.0,1.0,16,3,0,16,0,2,0,0,3,0,0,2,0,0,4,0,0,1,0,0,1,0,3],[28,32,0.875,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.01339,"x":0.11607,"p":[[0,41,0.0,0.11607,0.15746,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,41,0.0976,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,41,0.1951,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.04018,0.15663,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,41,0.3902,0.06696,0.21124,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,41,0.4878,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.07589,0.1636,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[32,41,0.7805,0.05804,0.13296,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.05348,0.10557,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ddc82104a506951d","q":"Given a prime number  $p$ , how many  $4$ -tuples  $(a, b, c, d)$  of positive integers with  $0 \\le a, b, c, d \\le p-1$  satisfy  $ad = bc$  mod  $p$ ?","t":[{"b":0,"e":1.0,"k":"rising","v":0.76339,"x":1.0,"p":[[0,24,0.0,0.76339,0.36876,0.39286,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,22],[4,24,0.1667,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[8,24,0.3333,0.88839,0.28735,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[12,24,0.5,0.92411,0.20198,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,28],[16,24,0.6667,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30]]},{"b":6,"e":1.0,"k":"rising","v":0.56696,"x":0.94643,"p":[[0,33,0.0,0.71429,0.34256,0.39286,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,18],[4,33,0.1212,0.63839,0.3813,0.28571,0.85714,1.0,0.0,1.0,3,16,0,3,0,0,0,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,16],[8,33,0.2424,0.56696,0.42929,0.10714,0.42857,1.0,0.0,1.0,8,15,0,8,0,1,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,15],[12,33,0.3636,0.88393,0.22142,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,24],[16,33,0.4848,0.8125,0.23266,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,18],[20,33,0.6061,0.84822,0.20183,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,9,0,0,0,0,19],[24,33,0.7273,0.64286,0.26964,0.42857,0.71429,0.78571,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,10,0,0,0,0,0,11,0,0,0,0,8],[28,33,0.8485,0.7813,0.23947,0.71429,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,10,0,0,0,0,15],[32,33,0.9697,0.92857,0.15972,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,26],[33,33,1.0,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29]]}]},{"i":"91b566b91183dc4d","q":"For a positive integer  $n$ , let  $a_1, a_2, \\ldots a_n$  be nonnegative real numbers such that for all real numbers  $x_1>x_2>\\ldots>x_n>0$  with  $x_1+x_2+\\ldots+x_n<1$ , the inequality  $\\sum_{k=1}^na_kx_k^3<1$  holds. Show that \\[na_1+(n-1)a_2+\\ldots+(n-j+1)a_j+\\ldots+a_n\\leqslant\\frac{n^2(n+1)^2}{4}.\\]","t":[{"b":4,"e":0.85714,"k":"flat","v":0.84375,"x":0.93304,"p":[[0,38,0.0,0.93304,0.17852,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,23],[4,38,0.1053,0.88839,0.17762,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,17],[8,38,0.2105,0.89286,0.17497,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,8,0,19],[12,38,0.3158,0.86607,0.15947,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,14,0,13],[16,38,0.4211,0.88839,0.21049,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,11,0,18],[20,38,0.5263,0.84375,0.22406,0.85714,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,14,0,13],[24,38,0.6316,0.85268,0.14934,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,17,0,10],[28,38,0.7368,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[32,38,0.8421,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[36,38,0.9474,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[38,38,1.0,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13]]},{"b":6,"e":0.85714,"k":"flat","v":0.88393,"x":0.94643,"p":[[0,87,0.0,0.92857,0.14286,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[4,87,0.046,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[8,87,0.092,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[12,87,0.1379,0.91964,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[16,87,0.1839,0.91518,0.12807,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,19],[20,87,0.2299,0.92411,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[24,87,0.2759,0.90625,0.12682,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,17],[28,87,0.3218,0.89732,0.09606,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,18,0,12],[32,87,0.3678,0.91518,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,17],[36,87,0.4138,0.88393,0.1448,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,14,0,14],[40,87,0.4598,0.89286,0.13363,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,15],[44,87,0.5057,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[48,87,0.5517,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[52,87,0.5977,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[56,87,0.6437,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[60,87,0.6897,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[64,87,0.7356,0.92411,0.11285,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,18],[68,87,0.7816,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[72,87,0.8276,0.88839,0.12234,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,15,0,13],[76,87,0.8736,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[80,87,0.9195,0.91518,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16],[84,87,0.9655,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[87,87,1.0,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13]]}]},{"i":"32d02f81dc6ddce6","q":"Find minimal natural  $n$  for which there exist integers  $a_1, a_2,\\ldots, a_n$  such that quadratic trinom  $$ x^2-2(a_1+a_2+\\cdots+a_n)^2x+(a_1^4+a_2^4+\\cdots+a_n^4+1) $$  has at least one integral root.","t":[{"b":1,"e":0.28571,"k":"falling","v":0.12054,"x":0.74107,"p":[[0,106,0.0,0.64286,0.22588,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,3,0,0,0,0,0,7,0,0,13,0,0,5,0,2],[4,106,0.0377,0.74107,0.29974,0.71429,0.85714,1.0,0.0,1.0,3,11,3,3,0,0,0,0,1,0,0,2,0,0,0,0,0,9,0,0,6,0,11],[8,106,0.0755,0.72767,0.27048,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,12,0,0,5,0,9],[12,106,0.1132,0.66964,0.32427,0.39286,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,1,0,0,2,0,0,0,0,0,4,0,0,11,0,7],[16,106,0.1509,0.71427,0.25001,0.57143,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,0,7,0,7],[20,106,0.1887,0.63393,0.333,0.24999,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,2,0,0,0,0,0,2,0,0,5,0,0,8,0,7],[24,106,0.2264,0.58482,0.30169,0.28571,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,7,0,0,2,0,0,4,0,0,1,0,0,8,0,0,6,0,4],[28,106,0.2642,0.59375,0.31966,0.28571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,4,0,0,3,0,0,0,0,0,10,0,0,2,0,7],[32,106,0.3019,0.58482,0.31003,0.28571,0.71429,0.74996,0.0,1.0,1,4,0,1,0,6,0,0,4,0,0,0,0,0,0,0,0,13,0,0,4,0,4],[36,106,0.3396,0.58027,0.35355,0.14286,0.71429,0.85714,0.0,1.0,2,7,1,2,0,8,0,0,1,0,0,0,0,0,4,0,0,5,0,0,5,0,7],[40,106,0.3774,0.66964,0.34337,0.25,0.85714,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,1,0,0,0,0,0,3,0,0,2,0,0,8,0,10],[44,106,0.4151,0.57588,0.3416,0.14286,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,9,0,0,3,0,0,1,0,0,3,0,0,5,0,0,3,0,8],[48,106,0.4528,0.50432,0.34799,0.14286,0.57121,0.85714,0.0,1.0,2,5,0,2,0,9,0,0,4,0,0,0,0,0,4,0,0,3,0,0,5,0,5],[52,106,0.4906,0.58469,0.33756,0.24999,0.71429,0.85714,0.0,1.0,1,7,0,1,0,7,0,0,3,0,0,2,0,0,1,0,0,7,0,0,4,0,7],[56,106,0.5283,0.62946,0.30275,0.28571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,6,0,0,3,0,0,1,0,0,1,0,0,9,0,0,7,0,5],[60,106,0.566,0.57128,0.32137,0.2857,0.64071,0.85714,0.0,1.0,1,5,0,1,0,6,0,0,4,0,0,2,0,0,3,0,0,5,0,0,6,0,5],[64,106,0.6038,0.52231,0.28032,0.25,0.71429,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,3,0,0,2,0,0,2,0,0,13,0,0,3,0,1],[68,106,0.6415,0.57141,0.29234,0.28571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,5,0,0,7,0,0,1,0,0,1,0,0,9,0,0,6,0,3],[72,106,0.6792,0.4865,0.3658,0.14286,0.42836,0.75,0.0,1.0,4,6,0,4,0,8,0,0,4,0,0,0,0,0,1,0,0,7,0,0,2,0,6],[76,106,0.717,0.58482,0.34692,0.14286,0.71429,0.85714,0.0,1.0,1,6,0,1,0,8,0,0,3,0,0,1,0,0,0,0,0,6,0,0,7,0,6],[80,106,0.7547,0.30356,0.30877,0.14286,0.14286,0.2857,0.0,1.0,2,4,0,2,0,20,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[84,106,0.7925,0.19187,0.16216,0.14286,0.14286,0.17857,0.0,1.0,2,1,0,2,0,22,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[88,106,0.8302,0.1384,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,106,0.8679,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,106,0.9057,0.14286,0.07986,0.14286,0.14286,0.14286,0.0,0.28571,5,0,1,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,106,0.9434,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,106,0.9811,0.13839,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[106,106,1.0,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.62052,"x":0.84375,"p":[[0,200,0.0,0.67857,0.2369,0.71429,0.71429,0.85714,0.0,1.0,1,3,1,1,0,2,0,0,1,0,0,1,0,0,2,0,0,16,0,0,6,0,3],[4,200,0.02,0.80357,0.24936,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,9,0,0,5,0,14],[8,200,0.04,0.83929,0.21943,0.71429,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,6,0,15],[12,200,0.06,0.69641,0.26427,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,11,0,0,5,0,7],[16,200,0.08,0.70534,0.26712,0.67857,0.71429,0.85714,0.0,1.0,2,5,2,2,0,1,0,0,1,0,0,1,0,0,3,0,0,9,0,0,10,0,5],[20,200,0.1,0.68302,0.31081,0.57132,0.71429,1.0,0.0,1.0,2,9,0,2,0,3,0,0,1,0,0,1,0,0,2,0,0,10,0,0,4,0,9],[24,200,0.12,0.67402,0.31806,0.60714,0.71429,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,2,0,0,0,0,0,0,0,0,10,0,0,6,0,8],[28,200,0.14,0.68301,0.29176,0.57132,0.71429,0.89286,0.0,1.0,1,8,1,1,0,4,0,0,0,0,0,1,0,0,5,0,0,8,0,0,5,0,8],[32,200,0.16,0.81241,0.18396,0.71429,0.85714,1.0,0.14,1.0,0,10,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,8,0,10],[36,200,0.18,0.83482,0.12428,0.71429,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,7,0,10],[40,200,0.2,0.80803,0.18073,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,9,0,9],[44,200,0.22,0.76339,0.16982,0.71429,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,10,0,5],[48,200,0.24,0.83036,0.14032,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,14,0,8],[52,200,0.26,0.83482,0.11904,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,12,0,8],[56,200,0.28,0.79463,0.15544,0.71429,0.78571,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,13,0,0,9,0,7],[60,200,0.3,0.79463,0.15128,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,15,0,0,8,0,7],[64,200,0.32,0.79004,0.14726,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,15,0,0,9,0,6],[68,200,0.34,0.82143,0.17128,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,0,0,7,0,11],[72,200,0.36,0.77678,0.19212,0.71429,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,11,0,0,8,0,8],[76,200,0.38,0.84375,0.13997,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,10,0,11],[80,200,0.4,0.77677,0.17475,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,14,0,0,9,0,6],[84,200,0.42,0.81691,0.17221,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,10,0,10],[88,200,0.44,0.78571,0.14725,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0,0,8,0,6],[92,200,0.46,0.83035,0.15746,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,6,0,12],[96,200,0.48,0.77679,0.18536,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,16,0,0,7,0,7],[100,200,0.5,0.7857,0.26965,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,14],[104,200,0.52,0.84375,0.17627,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,13],[108,200,0.54,0.83036,0.13092,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,7,0,10],[112,200,0.56,0.77232,0.20158,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,13,0,0,5,0,9],[116,200,0.58,0.72768,0.24578,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,12,0,0,7,0,7],[120,200,0.6,0.8125,0.14032,0.71429,0.78564,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,7,0,9],[124,200,0.62,0.74553,0.198,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,13,0,0,5,0,7],[128,200,0.64,0.75893,0.23538,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,7,0,0,10,0,8],[132,200,0.66,0.74107,0.25614,0.71429,0.78571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,3,0,0,2,0,0,1,0,0,9,0,0,7,0,9],[136,200,0.68,0.71874,0.22725,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,11,0,0,10,0,4],[140,200,0.7,0.79018,0.18205,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,5,0,10],[144,200,0.72,0.78124,0.19229,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,8,0,0,10,0,8],[148,200,0.74,0.82588,0.16652,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,7,0,12],[152,200,0.76,0.82589,0.18466,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,9,0,12],[156,200,0.78,0.79016,0.18554,0.71429,0.78571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,0,6,0,10],[160,200,0.8,0.75446,0.23483,0.57143,0.85707,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,6,0,0,8,0,9],[164,200,0.82,0.65179,0.24984,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,5,0,0,0,0,0,1,0,0,4,0,0,12,0,0,8,0,2],[168,200,0.84,0.69185,0.20883,0.57143,0.71429,0.85714,0.14,1.0,0,6,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,10,0,0,3,0,6],[172,200,0.86,0.68301,0.17401,0.57132,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,0,7,0,2],[176,200,0.88,0.76339,0.18073,0.67857,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,9,0,0,8,0,7],[180,200,0.9,0.7366,0.22899,0.57143,0.78564,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,2,0,0,0,0,0,6,0,0,7,0,0,10,0,6],[184,200,0.92,0.62052,0.17718,0.57132,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,4,0,0,10,0,0,11,0,0,2,0,2],[188,200,0.94,0.74552,0.15042,0.57143,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,10,0,0,7,0,5],[192,200,0.96,0.67409,0.23211,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,11,0,0,8,0,3],[196,200,0.98,0.62944,0.21387,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,0,0,0,0,0,0,10,0,0,12,0,0,5,0,1],[200,200,1.0,0.73214,0.17035,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,12,0,0,6,0,5]]}]},{"i":"52005d33109d3ee1","q":"For a positive integer  $K$ , de\ffine a sequence,  $\\{a_n\\}$ , as following:  $a_1 = K$  and $a_{n+1} =a_n -1$  if  $a_n$  is even $a_{n+1} =\\frac{a_n - 1}{2}$  if  $a_n$  is odd ,  for all  $n \\ge 1$ . \nFind the smallest value of  $K$ , which makes  $a_{2005}$  the \ffirst term equal to  $0$ .","t":[{"b":1,"e":0.571,"k":"flat","v":0.68749,"x":0.91964,"p":[[0,39,0.0,0.88839,0.16263,0.85711,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,6,0,19],[4,39,0.1026,0.70089,0.37348,0.39285,0.85714,1.0,0.0,1.0,3,15,1,3,0,4,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,15],[8,39,0.2051,0.76337,0.31866,0.53539,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,0,0,0,4,0,0,2,0,0,1,0,0,4,0,17],[12,39,0.3077,0.68749,0.3223,0.42857,0.78571,1.0,0.0,1.0,1,13,0,1,0,3,0,0,2,0,0,3,0,0,6,0,0,1,0,0,3,0,13],[16,39,0.4103,0.90625,0.17717,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,24],[20,39,0.5128,0.74107,0.32818,0.42859,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,1,0,0,4,0,0,2,0,0,2,0,0,2,0,17],[24,39,0.6154,0.83929,0.27374,0.78571,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,22],[28,39,0.7179,0.79464,0.26471,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,2,0,0,1,0,18],[32,39,0.8205,0.91964,0.20183,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,25],[36,39,0.9231,0.87946,0.2372,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,23],[39,39,1.0,0.76786,0.22232,0.57143,0.64286,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,1,0,0,0,0,15]]},{"b":7,"e":1.0,"k":"flat","v":0.55802,"x":0.97321,"p":[[0,90,0.0,0.85268,0.2461,0.85714,0.92857,1.0,0.0,1.0,2,16,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,16],[4,90,0.0444,0.55802,0.32803,0.25,0.4998,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,2,0,0,6,0,0,4,0,0,0,0,0,5,0,7],[8,90,0.0889,0.70536,0.35703,0.42857,0.85714,1.0,0.0,1.0,1,14,1,1,0,6,0,0,0,0,0,4,0,0,0,0,0,0,0,0,7,0,14],[12,90,0.1333,0.77678,0.31731,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,19],[16,90,0.1778,0.71427,0.3134,0.53539,0.85714,1.0,0.0,1.0,1,14,0,1,0,3,0,0,0,0,0,4,0,0,6,0,0,1,0,0,3,0,14],[20,90,0.2222,0.88393,0.20959,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,22],[24,90,0.2667,0.76785,0.29396,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,2,0,0,1,0,0,4,0,0,1,0,0,6,0,15],[28,90,0.3111,0.69196,0.36265,0.39286,0.85714,1.0,0.0,1.0,2,14,0,2,0,5,0,0,1,0,0,1,0,0,3,0,0,1,0,0,5,0,14],[32,90,0.3556,0.87054,0.22689,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,0,5,0,21],[36,90,0.4,0.72321,0.32525,0.42859,0.85714,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,1,0,0,3,0,0,3,0,0,0,0,0,6,0,14],[40,90,0.4444,0.87946,0.18935,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,19],[44,90,0.4889,0.70982,0.34899,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,3,0,0,4,0,0,3,0,0,0,0,0,0,0,18],[48,90,0.5333,0.82589,0.28512,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,22],[52,90,0.5778,0.87491,0.25717,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,24],[56,90,0.6222,0.94196,0.15093,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,27],[60,90,0.6667,0.83929,0.27837,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,0,0,0,1,0,23],[64,90,0.7111,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[68,90,0.7556,0.84821,0.25238,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,0,1,0,22],[72,90,0.8,0.85713,0.25001,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,22],[76,90,0.8444,0.86607,0.23128,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,22],[80,90,0.8889,0.86161,0.25874,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,23],[84,90,0.9333,0.93304,0.1988,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,28],[88,90,0.9778,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[90,90,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"dec6b6e1a4d42169","q":"For real numbers  $(x, y, z)$  satisfying the following equations, \ufb01nd all possible values of  $x+y+z$  $x^2y+y^2z+z^2x=-1$  $xy^2+yz^2+zx^2=5$  $xyz=-2$","t":[{"b":2,"e":0.42857,"k":"flat","v":0.40625,"x":0.44643,"p":[[0,94,0.0,0.41518,0.12556,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,25,0,0,0,0,0,0,0,0,0,0,1],[4,94,0.0426,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[8,94,0.0851,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[12,94,0.1277,0.42857,0.06186,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,1,0,0,0,0,0],[16,94,0.1702,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[20,94,0.2128,0.44643,0.09942,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,1],[24,94,0.2553,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[28,94,0.2979,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,94,0.3404,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[36,94,0.383,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[40,94,0.4255,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[44,94,0.4681,0.43304,0.10999,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,1],[48,94,0.5106,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[52,94,0.5532,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[56,94,0.5957,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[60,94,0.6383,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[64,94,0.6809,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[68,94,0.7234,0.44197,0.10326,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[72,94,0.766,0.42857,0.03571,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[76,94,0.8085,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[80,94,0.8511,0.43304,0.10999,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,1],[84,94,0.8936,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[88,94,0.9362,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[92,94,0.9787,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[94,94,1.0,0.40625,0.06298,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.41071,"x":0.45982,"p":[[0,78,0.0,0.41071,0.04725,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[4,78,0.0513,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[8,78,0.1026,0.45982,0.14167,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,2],[12,78,0.1538,0.4375,0.07936,0.42857,0.42857,0.42857,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,1,0,0],[16,78,0.2051,0.45982,0.14167,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,2],[20,78,0.2564,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[24,78,0.3077,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[28,78,0.359,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[32,78,0.4103,0.4375,0.10677,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,1],[36,78,0.4615,0.41965,0.03458,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[40,78,0.5128,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[44,78,0.5641,0.44197,0.10326,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[48,78,0.6154,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[52,78,0.6667,0.41964,0.0497,0.42857,0.42857,0.42857,0.1429,0.42857,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[56,78,0.7179,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[60,78,0.7692,0.41518,0.04164,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[64,78,0.8205,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[68,78,0.8718,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[72,78,0.9231,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[76,78,0.9744,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[78,78,1.0,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"47ee2d87f2452cee","q":"If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of  $1/a$ ,  $a>0$ , to use the iteration   $$ x_{k+1}=x_k(2-ax_k), \\quad \\quad k=0,1,2,\\dots , $$   where  $x_0$  is a selected \u201cstarting\u201d value. Find the limitations, if any, on the starting values  $x_0$ , in order that the above iteration converges to the desired value  $1/a$ .","t":[{"b":6,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,24,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,24,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,24,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,28,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,28,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"fbc2dad23c71e5bb","q":"If  $a$ ,  $b$ ,  $c>0$  and  $abc=1$ ,  $\\alpha = max\\{a,b,c\\}$ ;  $f,g : (0, +\\infty )\\to \\mathbb{R}$ , where  $f(x)=\\frac{2(x+1)^2}{x}$  and  $g(x)= (x+1)\\left (\\frac{1}{\\sqrt{x}}+1\\right )^2$ , then   $$ (a+1)(b+1)(c+1)\\geq min\\{ \\{f(x),g(x) \\}\\ |\\ x\\in\\{a,b,c\\} \\backslash \\{ \\alpha \\}\\}  $$","t":[{"b":2,"e":0.0,"k":"falling","v":0.23214,"x":0.91518,"p":[[0,19,0.0,0.91518,0.21086,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,26],[4,19,0.2105,0.91518,0.18851,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[8,19,0.4211,0.875,0.21943,0.82143,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,22],[12,19,0.6316,0.81696,0.33548,0.85714,1.0,1.0,0.0,1.0,3,21,0,3,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,21],[16,19,0.8421,0.65625,0.40067,0.28571,0.85714,1.0,0.0,1.0,6,15,0,6,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,4,0,15],[19,19,1.0,0.23214,0.36025,0.0,0.0,0.5,0.0,1.0,22,2,0,22,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,2]]},{"b":6,"e":0.85714,"k":"flat","v":0.83034,"x":0.96429,"p":[[0,76,0.0,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[4,76,0.0526,0.95981,0.1143,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[8,76,0.1053,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[12,76,0.1579,0.94643,0.15465,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,28],[16,76,0.2105,0.91518,0.24448,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[20,76,0.2632,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[24,76,0.3158,0.91518,0.21387,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,26],[28,76,0.3684,0.93527,0.14656,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,1,25],[32,76,0.4211,0.87946,0.1992,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,4,0,21],[36,76,0.4737,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[40,76,0.5263,0.89286,0.16366,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,9,0,18],[44,76,0.5789,0.95535,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[48,76,0.6316,0.875,0.19149,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,7,0,19],[52,76,0.6842,0.90179,0.1448,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,10,0,18],[56,76,0.7368,0.91963,0.14701,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,22],[60,76,0.7895,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[64,76,0.8421,0.90178,0.11538,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0,14],[68,76,0.8947,0.87052,0.09007,0.857,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,19,0,8],[72,76,0.9474,0.85714,0.10102,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,24,0,5],[76,76,1.0,0.83034,0.10374,0.85714,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,29,0,0]]}]},{"i":"b61afc967d7aab6b","q":"Find all triples  $(x,n,p)$  of positive integers  $x$  and  $n$  and primes  $p$  for which the following holds  $x^3 + 3x + 14 = 2 p^n$","t":[{"b":5,"e":0.8571,"k":"rising","v":0.5445,"x":0.94196,"p":[[0,289,0.0,0.5445,0.26583,0.28571,0.5,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,1,0,0,3,0,0,2,0,0,10,0,1],[4,289,0.0138,0.8616,0.13592,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,19,0,9],[8,289,0.0277,0.87054,0.12556,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,15,0,11],[12,289,0.0415,0.82142,0.11293,0.82132,0.85714,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,21,0,3],[16,289,0.0554,0.86607,0.11259,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,12,0,11],[20,289,0.0692,0.86607,0.12846,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,14,0,11],[24,289,0.083,0.82141,0.16368,0.82132,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,17,0,7],[28,289,0.0969,0.89285,0.14726,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,14,0,15],[32,289,0.1107,0.87499,0.13243,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,15,0,12],[36,289,0.1246,0.87053,0.13534,0.82132,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,11,0,13],[40,289,0.1384,0.86607,0.14258,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,14,0,12],[44,289,0.1522,0.87946,0.11355,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,17,0,11],[48,289,0.1661,0.85714,0.13832,0.82132,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,13,0,11],[52,289,0.1799,0.87946,0.13415,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,14,0,13],[56,289,0.1938,0.89731,0.12992,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,14,0,15],[60,289,0.2076,0.86161,0.12619,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,15,0,10],[64,289,0.2215,0.86607,0.17104,0.82143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,8,0,16],[68,289,0.2353,0.85268,0.17307,0.82143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,11,0,13],[72,289,0.2491,0.85714,0.15152,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,11,0,12],[76,289,0.263,0.87944,0.11907,0.85714,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,12],[80,289,0.2768,0.89284,0.11845,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,14],[84,289,0.2907,0.85713,0.14286,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,19,0,9],[88,289,0.3045,0.86607,0.15126,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,12,0,13],[92,289,0.3183,0.83915,0.12256,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,16,0,7],[96,289,0.3322,0.83928,0.14174,0.85711,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,17,0,8],[100,289,0.346,0.83929,0.13243,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,15,0,8],[104,289,0.3599,0.88838,0.09269,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,11],[108,289,0.3737,0.85267,0.13115,0.857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,16,0,9],[112,289,0.3875,0.87054,0.13534,0.85714,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,18,0,10],[116,289,0.4014,0.80357,0.15872,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,10,0,8],[120,289,0.4152,0.86607,0.15126,0.85711,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,12,0,13],[124,289,0.4291,0.8616,0.16554,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,13,0,13],[128,289,0.4429,0.87053,0.1488,0.857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,12,0,13],[132,289,0.4567,0.86607,0.1234,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,16,0,10],[136,289,0.4706,0.86161,0.13115,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,16,0,10],[140,289,0.4844,0.90624,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[144,289,0.4983,0.89286,0.11294,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,14],[148,289,0.5121,0.89286,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[152,289,0.526,0.90624,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[156,289,0.5398,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[160,289,0.5536,0.93303,0.08738,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[164,289,0.5675,0.88392,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,11],[168,289,0.5813,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[172,289,0.5952,0.91518,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,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 $\\triangle{ABC}$  with  $\\angle{BAC}=60^\\circ,$  points  $D, E,$  and  $F$  lie on  $BC, AC,$  and  $AB$  respectively, such that  $D$  is the midpoint of  $BC$  and  $\\triangle{DEF}$  is equilateral. If  $BF=1$  and  $EC=13,$  then the area of  $\\triangle{DEF}$  can be written as  $\\tfrac{a\\sqrt{b}}{c},$  where  $a$  and  $c$  are relatively prime positive integers and  $b$  is not divisible by a square of a prime. Compute  $a+b+c.$","t":[{"b":5,"e":1.0,"k":"flat","v":0.87052,"x":1.0,"p":[[0,162,0.0,0.87052,0.2609,0.82143,1.0,1.0,0.0,1.0,2,23,2,2,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,23],[4,162,0.0247,0.89286,0.24223,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,25],[8,162,0.0494,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,162,0.0741,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,162,0.0988,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,162,0.1235,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,162,0.1481,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,162,0.1728,0.94196,0.20782,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[32,162,0.1975,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,162,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,162,0.2469,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,162,0.2716,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,162,0.2963,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[52,162,0.321,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,162,0.3457,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[60,162,0.3704,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,162,0.3951,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,162,0.4198,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,162,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,162,0.4691,0.92857,0.21129,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[80,162,0.4938,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,162,0.5185,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,162,0.5432,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,162,0.5679,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,162,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,162,0.6173,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,162,0.642,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,162,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,162,0.6914,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[116,162,0.716,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,162,0.7407,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,162,0.7654,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,162,0.7901,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,162,0.8148,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[136,162,0.8395,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[140,162,0.8642,0.96875,0.15039,1.0,1.0,1.0,0.1429,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[144,162,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[148,162,0.9136,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[152,162,0.9383,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[156,162,0.963,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[160,162,0.9877,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[162,162,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,113,0.0,0.83482,0.3409,1.0,1.0,1.0,0.0,1.0,3,25,2,3,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[4,113,0.0354,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,113,0.0708,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[12,113,0.1062,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[16,113,0.1416,0.95536,0.15746,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[20,113,0.177,0.9375,0.16728,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[24,113,0.2124,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,113,0.2478,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[32,113,0.2832,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,113,0.3186,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,113,0.354,0.95536,0.15746,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[44,113,0.3894,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,113,0.4248,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,113,0.4602,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,113,0.4956,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,113,0.531,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,113,0.5664,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,113,0.6018,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,113,0.6372,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[76,113,0.6726,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,113,0.708,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,113,0.7434,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[88,113,0.7788,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[92,113,0.8142,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,113,0.8496,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[100,113,0.885,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,113,0.9204,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[108,113,0.9558,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,113,0.9912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[113,113,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"a81bac9ed3bc2f98","q":"In a chess festival that is held in a school with  $2017$  students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most  $22$  matches. What is the maximum possible number of matches in this event?","t":[{"b":1,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,38,0.0,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[4,38,0.1053,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[8,38,0.2105,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[12,38,0.3158,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[16,38,0.4211,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,38,0.5263,0.99553,0.02486,1.0,1.0,1.0,0.8571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,38,0.6316,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,38,0.7368,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[38,38,1.0,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27]]},{"b":7,"e":1.0,"k":"flat","v":0.89732,"x":1.0,"p":[[0,58,0.0,0.89732,0.14827,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,21],[4,58,0.069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,58,0.1379,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,58,0.2069,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,58,0.2759,0.97098,0.09081,1.0,1.0,1.0,0.64286,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,29],[20,58,0.3448,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,58,0.4138,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,58,0.4828,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,58,0.5517,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,58,0.6207,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,58,0.6897,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[44,58,0.7586,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,58,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,58,0.8966,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,58,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[58,58,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"34bfdfe2684a0303","q":"In a circle  $C$  with centre  $O$  and radius  $r$ , let  $C_1$ ,  $C_2$  be two circles with centres  $O_1$ ,  $O_2$  and radii  $r_1$ ,  $r_2$  respectively, so that each circle  $C_i$  is internally tangent to  $C$  at  $A_i$  and so that  $C_1$ ,  $C_2$  are externally tangent to each other at  $A$ .\r\n\r\nProve that the three lines  $OA$ ,  $O_1 A_2$ , and  $O_2 A_1$  are concurrent.","t":[{"b":6,"e":1.0,"k":"rising","v":0.23661,"x":1.0,"p":[[0,47,0.0,0.23661,0.3773,0.0,0.0,0.14286,0.0,1.0,17,6,0,17,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[4,47,0.0851,0.62054,0.44408,0.14286,1.0,1.0,0.0,1.0,7,18,0,7,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,18],[8,47,0.1702,0.41964,0.44022,0.0,0.14286,1.0,0.0,1.0,11,11,0,11,0,7,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,11],[12,47,0.2553,0.36161,0.45595,0.0,0.07143,1.0,0.0,1.0,16,10,0,16,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[16,47,0.3404,0.46429,0.45034,0.0,0.14286,1.0,0.0,1.0,10,12,0,10,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,12],[20,47,0.4255,0.44197,0.47294,0.0,0.14286,1.0,0.0,1.0,15,13,0,15,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,13],[24,47,0.5106,0.84821,0.35344,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[28,47,0.5957,0.87054,0.28873,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,25],[32,47,0.6809,0.66518,0.45261,0.0,1.0,1.0,0.0,1.0,9,20,0,9,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,20],[36,47,0.766,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[40,47,0.8511,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[44,47,0.9362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[47,47,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.48214,"p":[[0,73,0.0,0.26786,0.3989,0.0,0.0,0.32142,0.0,1.0,17,7,0,17,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[4,73,0.0548,0.41964,0.46144,0.0,0.14286,1.0,0.0,1.0,14,12,0,14,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,12],[8,73,0.1096,0.17857,0.31744,0.0,0.07143,0.14286,0.0,1.0,16,4,0,16,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,73,0.1644,0.28562,0.41191,0.0,0.0,0.53571,0.0,1.0,17,7,0,17,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[16,73,0.2192,0.375,0.4598,0.0,0.07143,1.0,0.0,1.0,16,11,0,16,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11],[20,73,0.274,0.43304,0.45525,0.0,0.14286,1.0,0.0,1.0,12,12,0,12,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,12],[24,73,0.3288,0.41965,0.46144,0.0,0.14286,1.0,0.0,1.0,14,12,0,14,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,12],[28,73,0.3836,0.38839,0.44354,0.0,0.14286,1.0,0.0,1.0,14,10,0,14,0,5,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,10],[32,73,0.4384,0.48214,0.48806,0.0,0.14286,1.0,0.0,1.0,14,15,0,14,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[36,73,0.4932,0.16964,0.32031,0.0,0.0,0.14286,0.0,1.0,21,3,0,21,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,3],[40,73,0.5479,0.09822,0.2911,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[44,73,0.6027,0.11161,0.29609,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[48,73,0.6575,0.12946,0.31209,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[52,73,0.7123,0.26339,0.41359,0.0,0.0,0.46429,0.0,1.0,22,7,0,22,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,7],[56,73,0.7671,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,73,0.8219,0.05357,0.19805,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[64,73,0.8767,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,73,0.9315,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"07457ca60a0ce9c8","q":"If  $p=2^{16}+1$  is a prime, find the maximum possible number of elements in a set  $S$  of positive integers less than  $p$  so no two distinct  $a,b$  in  $S$  satisfy   $$ a^2\\equiv b\\pmod{p}. $$","t":[{"b":3,"e":0.14286,"k":"falling","v":0.1875,"x":0.92857,"p":[[0,38,0.0,0.88839,0.21349,0.85714,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[4,38,0.1053,0.92857,0.20825,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[8,38,0.2105,0.90625,0.22759,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[12,38,0.3158,0.90179,0.22428,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,24],[16,38,0.4211,0.76786,0.31288,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,5,0,0,2,0,0,0,0,0,1,0,0,5,0,17],[20,38,0.5263,0.79018,0.33309,0.42857,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,22],[24,38,0.6316,0.67411,0.38172,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,6,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,18],[28,38,0.7368,0.52232,0.38731,0.14286,0.28571,1.0,0.0,1.0,1,11,0,1,0,10,0,0,7,0,0,0,0,0,0,0,0,2,0,0,1,0,11],[32,38,0.8421,0.35268,0.3174,0.14286,0.28571,0.32143,0.0,1.0,3,5,0,3,0,10,0,0,11,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[36,38,0.9474,0.31679,0.319,0.14286,0.14286,0.28571,0.0,1.0,3,4,0,3,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0,2,0,4],[38,38,1.0,0.1875,0.08328,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.77679,"x":1.0,"p":[[0,92,0.0,0.77679,0.3443,0.67857,1.0,1.0,0.0,1.0,4,18,4,4,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,18],[4,92,0.0435,0.94196,0.17445,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[8,92,0.087,0.90625,0.23585,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[12,92,0.1304,0.875,0.28291,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[16,92,0.1739,0.85268,0.28231,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,25],[20,92,0.2174,0.94643,0.17768,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[24,92,0.2609,0.87946,0.25532,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,25],[28,92,0.3043,0.78571,0.30929,0.39288,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,8,0,0,1,0,0,0,0,0,1,0,0,2,0,20],[32,92,0.3478,0.83482,0.29257,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,2,0,23],[36,92,0.3913,0.89286,0.20825,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,3,0,23],[40,92,0.4348,0.95089,0.16602,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[44,92,0.4783,0.90625,0.22759,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[48,92,0.5217,0.94643,0.15465,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,25],[52,92,0.5652,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[56,92,0.6087,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[60,92,0.6522,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[64,92,0.6957,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[68,92,0.7391,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[72,92,0.7826,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[76,92,0.8261,0.94643,0.15871,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[80,92,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,92,0.913,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[88,92,0.9565,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[92,92,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b3a91cc050518ea0","q":"If the fraction  $\\frac{an + b}{cn + d}$  may be simplified using  $2$  (as a common divisor ), show that the number  $ad - bc$  is even.  ( $a, b, c, d, n$  are natural numbers and the  $cn + d$  different from zero).","t":[{"b":0,"e":1.0,"k":"flat","v":0.18304,"x":0.48214,"p":[[0,47,0.0,0.48214,0.48936,0.0,0.21428,1.0,0.0,1.0,15,15,0,15,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[4,47,0.0851,0.41516,0.45084,0.0,0.14285,1.0,0.0,1.0,16,10,0,16,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,10],[8,47,0.1702,0.21425,0.33308,0.0,0.0,0.35704,0.0,1.0,20,3,0,20,0,2,0,0,2,0,0,0,0,0,4,0,0,1,0,0,0,0,3],[12,47,0.2553,0.29911,0.41705,0.0,0.0,0.57143,0.0,1.0,17,8,0,17,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[16,47,0.3404,0.29463,0.42096,0.0,0.0,0.71429,0.0,1.0,20,7,0,20,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,7],[20,47,0.4255,0.24107,0.36323,0.0,0.0,0.28571,0.0,1.0,18,5,0,18,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[24,47,0.5106,0.25889,0.33582,0.0,0.07143,0.571,0.0,1.0,16,3,0,16,0,4,0,0,2,0,0,1,0,0,4,0,0,2,0,0,0,0,3],[28,47,0.5957,0.20982,0.35172,0.0,0.0,0.17857,0.0,1.0,20,4,0,20,0,4,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,4],[32,47,0.6809,0.25,0.34069,0.0,0.0,0.32143,0.0,1.0,17,4,0,17,0,1,0,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,4],[36,47,0.766,0.33481,0.38895,0.0,0.2857,0.60682,0.0,1.0,15,6,0,15,0,0,0,0,6,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[40,47,0.8511,0.18304,0.33737,0.0,0.0,0.1786,0.0,1.0,22,4,0,22,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,4],[44,47,0.9362,0.23661,0.37561,0.0,0.0,0.28571,0.0,1.0,20,5,0,20,0,2,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[47,47,1.0,0.42854,0.42856,0.0,0.42857,1.0,0.0,1.0,14,9,0,14,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,0,0,0,9]]},{"b":3,"e":1.0,"k":"volatile","v":0.16518,"x":0.51775,"p":[[0,35,0.0,0.33036,0.46213,0.0,0.0,1.0,0.0,1.0,21,10,0,21,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10],[4,35,0.1143,0.51775,0.44292,0.0,0.4998,1.0,0.0,1.0,10,13,0,10,0,3,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,13],[8,35,0.2286,0.17856,0.34991,0.0,0.0,0.03572,0.0,1.0,24,4,0,24,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[12,35,0.3429,0.38839,0.4768,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[16,35,0.4571,0.22767,0.37432,0.0,0.0,0.32143,0.0,1.0,21,5,0,21,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,5],[20,35,0.5714,0.16518,0.32754,0.0,0.0,0.14287,0.0,1.0,22,4,0,22,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,35,0.6857,0.36607,0.44598,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,9],[28,35,0.8,0.20982,0.37113,0.0,0.0,0.1786,0.0,1.0,22,5,0,22,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[32,35,0.9143,0.29463,0.39598,0.0,0.0,0.60682,0.0,1.0,18,6,0,18,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,6],[35,35,1.0,0.39284,0.46565,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,11]]}]},{"i":"0b1fab26b775a4fc","q":"In a cyclic hexagon $A B C D E F$, $A B \\perp B D$ and $|B C|=|E F|$. Let $P$ be the intersection of $B C$ and $A D$ and let $Q$ be the intersection of $E F$ and $A D$. Assume that $P$ and $Q$ both lie on the side of $D$ where $A$ does not lie. Let $S$ be the midpoint of $A D$. Let $K$ and $L$ be the centers of the inscribed circles of $\\triangle B P S$ and $\\triangle E Q S$, respectively. Prove that $\\angle K D L=90^{\\circ}$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.12491,"x":0.20971,"p":[[0,67,0.0,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,4,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.16071,0.07784,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,67,0.1194,0.16062,0.05925,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.20971,0.1712,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,27,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[16,67,0.2388,0.17393,0.1056,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,67,0.2985,0.17856,0.11289,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,26,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,67,0.3582,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.1607,0.07777,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,67,0.4776,0.18741,0.10975,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,26,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,67,0.5373,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.17848,0.11848,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,27,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[52,67,0.7761,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,67,0.8358,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,67,0.9552,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.16509,0.0883,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.14723,"x":0.19197,"p":[[0,64,0.0,0.16946,0.14036,0.14286,0.14286,0.14286,0.0,0.85714,2,0,2,2,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,64,0.0625,0.16518,0.09523,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,25,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,64,0.125,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,64,0.1875,0.16518,0.08073,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,64,0.25,0.16955,0.09064,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,26,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,64,0.3125,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.1517,0.04973,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.17411,0.11701,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[36,64,0.5625,0.15626,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.18741,0.10975,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,26,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,64,0.6875,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.16964,0.08328,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.16963,0.09056,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,64,0.875,0.19197,0.11071,0.14286,0.14286,0.14292,0.14286,0.57143,0,0,0,0,0,25,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[60,64,0.9375,0.16054,0.04731,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.15179,0.04972,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b0f4475173019d1a","q":"Given is a quadratic polynomial $P(x)$ with two distinct real roots. For all real numbers $a$ and $b$ with $|a|,|b| \\geq 2017$ it holds that $P\\left(a^{2}+b^{2}\\right) \\geq P(2 a b)$. Prove that at least one of the roots of $P$ is negative.","t":[{"b":0,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,38,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,38,0.1053,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,38,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,38,0.3158,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,35,0.0,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[4,35,0.1143,0.91964,0.15947,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,25],[8,35,0.2286,0.90179,0.14914,0.82143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,21],[12,35,0.3429,0.875,0.18123,0.78561,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,4,0,20],[16,35,0.4571,0.89732,0.15663,0.82143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,3,0,21],[20,35,0.5714,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[24,35,0.6857,0.90179,0.15746,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,21],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"880345aadc28f275","q":"In a school class with $3 n$ children, any two children make a common present to exactly one other child. Prove that for all odd $n$ it is possible that the following holds:\n\nFor any three children $A, B$ and $C$ in the class, if $A$ and $B$ make a present to $C$ then $A$ and $C$ make a present to $B$.","t":[{"b":3,"e":0.14286,"k":"volatile","v":0.13839,"x":0.74552,"p":[[0,21,0.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.74552,0.28288,0.53539,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,4,0,0,7,0,12],[8,21,0.381,0.39283,0.31338,0.14286,0.14286,0.71429,0.14286,1.0,0,2,0,0,0,18,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,2],[12,21,0.5714,0.56696,0.37199,0.14286,0.64286,1.0,0.14286,1.0,0,10,0,0,0,11,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,10],[16,21,0.7619,0.50892,0.33868,0.14286,0.42859,0.74996,0.14286,1.0,0,6,0,0,0,11,0,0,4,0,0,2,0,0,0,0,0,7,0,0,2,0,6],[20,21,0.9524,0.74103,0.25365,0.57132,0.78564,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,7,0,0,6,0,10],[21,21,1.0,0.63838,0.29663,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,4,0,0,6,0,0,1,0,0,7,0,0,2,0,9]]},{"b":7,"e":0.85714,"k":"rising","v":0.14268,"x":0.83927,"p":[[0,27,0.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.63839,0.32924,0.2857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,4,0,0,1,0,0,1,0,0,8,0,0,2,0,10],[8,27,0.2963,0.62489,0.34035,0.28571,0.71429,1.0,0.14,1.0,0,11,0,0,0,5,0,0,6,0,0,3,0,0,1,0,0,3,0,0,3,0,11],[12,27,0.4444,0.61597,0.33599,0.25,0.71429,0.89286,0.14,1.0,0,8,0,0,0,8,0,0,2,0,0,2,0,0,1,0,0,6,0,0,5,0,8],[16,27,0.5926,0.75883,0.28689,0.67846,0.85714,1.0,0.14,1.0,0,12,0,0,0,3,0,0,2,0,0,2,0,0,1,0,0,3,0,0,9,0,12],[20,27,0.7407,0.79016,0.25251,0.857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,1,0,0,16,0,9],[24,27,0.8889,0.83927,0.17767,0.85714,0.85714,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,22,0,7],[27,27,1.0,0.7991,0.25965,0.82132,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,2,0,0,12,0,12]]}]},{"i":"de5155065423b1f3","q":"In a right triangle  $A B C$ , an altitude  $C H$  is drawn to the hypotenuse  $A B$ . In triangles  $A C H$  and  $B C H$ , the bisectors  $C K$  and  $C L$  are drawn, respectively. It turns out that the point  $L$  is the midpoint of the hypotenuse  $A B$ . Find the ratio  $\\frac{2CH-AK}{KL}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.93749,"x":1.0,"p":[[0,129,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,129,0.031,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,129,0.062,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,129,0.093,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,129,0.124,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,129,0.155,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,129,0.186,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,129,0.2171,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,129,0.2481,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,129,0.2791,0.93749,0.17108,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[40,129,0.3101,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,129,0.3411,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[48,129,0.3721,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[52,129,0.4031,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[56,129,0.4341,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,129,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,129,0.4961,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,129,0.5271,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,129,0.5581,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,129,0.5891,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[80,129,0.6202,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[84,129,0.6512,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,129,0.6822,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[92,129,0.7132,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[96,129,0.7442,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[100,129,0.7752,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,129,0.8062,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,129,0.8372,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[112,129,0.8682,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[116,129,0.8992,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[120,129,0.9302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,129,0.9612,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,129,0.9922,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[129,129,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":0.85714,"k":"flat","v":0.87946,"x":1.0,"p":[[0,172,0.0,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[4,172,0.0233,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,172,0.0465,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,172,0.0698,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,172,0.093,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,172,0.1163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,172,0.1395,0.94643,0.17768,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[28,172,0.1628,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,172,0.186,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[36,172,0.2093,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,172,0.2326,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,172,0.2558,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[48,172,0.2791,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[52,172,0.3023,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,172,0.3256,0.94196,0.1551,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[60,172,0.3488,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[64,172,0.3721,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,172,0.3953,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,172,0.4186,0.88393,0.15746,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,20],[76,172,0.4419,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[80,172,0.4651,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[84,172,0.4884,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[88,172,0.5116,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[92,172,0.5349,0.91964,0.13333,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,23],[96,172,0.5581,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[100,172,0.5814,0.91964,0.15126,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,1,0,24],[104,172,0.6047,0.9107,0.16659,0.85711,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,23],[108,172,0.6279,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[112,172,0.6512,0.95089,0.14555,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[116,172,0.6744,0.92857,0.17497,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[120,172,0.6977,0.96874,0.12238,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[124,172,0.7209,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[128,172,0.7442,0.96426,0.11303,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[132,172,0.7674,0.9375,0.1234,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25],[136,172,0.7907,0.91964,0.15947,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,1,0,24],[140,172,0.814,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[144,172,0.8372,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[148,172,0.8605,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[152,172,0.8837,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[156,172,0.907,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[160,172,0.9302,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[164,172,0.9535,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[168,172,0.9767,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[172,172,1.0,0.87946,0.13882,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,1,0,18]]}]},{"i":"f625faa86481c081","q":"In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.","t":[{"b":2,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,28,0.0,0.85714,0.30514,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,26],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,31,0.0,0.82143,0.33882,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[4,31,0.129,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,31,0.2581,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"9587528d0501f68f","q":"In a  $\\triangle {ABC}$ , consider a point  $E$  in  $BC$  such that  $AE \\perp BC$ . Prove that  $AE=\\frac{bc}{2r}$ , where  $r$  is the radio of the circle circumscripte,  $b=AC$  and  $c=AB$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.24107,"x":0.68304,"p":[[0,34,0.0,0.24107,0.32623,0.0,0.14286,0.17857,0.0,1.0,11,4,0,11,0,13,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[4,34,0.1176,0.65616,0.41177,0.14286,1.0,1.0,0.0,1.0,5,17,0,5,0,4,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,17],[8,34,0.2353,0.55804,0.39506,0.14286,0.64286,1.0,0.14286,1.0,0,12,0,0,0,14,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,12],[12,34,0.3529,0.68304,0.33832,0.46429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,13],[16,34,0.4706,0.59821,0.39518,0.14286,0.71429,1.0,0.0,1.0,2,13,0,2,0,10,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,13],[20,34,0.5882,0.51777,0.38099,0.14286,0.50001,1.0,0.0,1.0,1,10,0,1,0,13,0,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,10],[24,34,0.7059,0.53114,0.34309,0.14286,0.64286,0.74996,0.14,1.0,0,6,0,0,0,13,0,0,0,0,0,0,0,0,3,0,0,8,0,0,2,0,6],[28,34,0.8235,0.32142,0.32142,0.14286,0.14286,0.32143,0.0,1.0,1,5,0,1,0,21,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,5],[32,34,0.9412,0.28571,0.29014,0.14286,0.14286,0.2857,0.0,1.0,2,3,0,2,0,21,0,0,2,0,0,1,0,0,0,0,0,3,0,0,0,0,3],[34,34,1.0,0.29018,0.26119,0.14286,0.14286,0.46429,0.0,1.0,4,1,0,4,0,16,0,0,2,0,0,2,0,0,3,0,0,4,0,0,0,0,1]]},{"b":5,"e":0.28571,"k":"flat","v":0.21875,"x":0.38393,"p":[[0,23,0.0,0.25446,0.34019,0.0,0.14286,0.17857,0.0,1.0,10,5,0,10,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[4,23,0.1739,0.21875,0.34253,0.0,0.07143,0.14286,0.0,1.0,16,4,0,16,0,9,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,4],[8,23,0.3478,0.31696,0.20119,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,6,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[12,23,0.5217,0.38393,0.22428,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,24,0,0,1,0,0,2,0,0,1,0,0,0,0,3],[16,23,0.6957,0.28573,0.03571,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.36607,0.3071,0.14286,0.2857,0.28571,0.0,1.0,1,5,0,1,0,11,0,0,13,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[23,23,1.0,0.29464,0.15126,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,3,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,1]]}]},{"i":"0ebf887663a35b7b","q":"In a triangle  $ABC$  let  $K$  be a point on the median  $BM$   such that  $CK=CM$ . It appears that  $\\angle CBM = 2 \\angle ABM$ . Prove that  $BC=MK$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.47318,"x":0.80355,"p":[[0,123,0.0,0.80355,0.26186,0.571,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,6,0,0,3,0,0,2,0,0,1,0,19],[4,123,0.0325,0.59814,0.29329,0.42857,0.571,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,2,0,0,7,0,0,6,0,0,3,0,0,3,0,7],[8,123,0.065,0.66512,0.28709,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,4,0,0,5,0,0,5,0,0,5,0,8],[12,123,0.0976,0.47318,0.31223,0.25001,0.42857,0.71429,0.0,1.0,2,5,0,2,0,6,0,0,5,0,0,7,0,0,3,0,0,2,0,0,2,0,5],[16,123,0.1301,0.61159,0.30143,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,7,0,0,3,0,0,4,0,0,3,0,8],[20,123,0.1626,0.6473,0.29877,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,4,0,0,5,0,0,3,0,0,4,0,0,4,0,9],[24,123,0.1951,0.66516,0.28033,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,3,0,0,3,0,0,5,0,0,4,0,0,7,0,7],[28,123,0.2276,0.61608,0.29056,0.28571,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,7,0,0,4,0,1,2,0,0,4,1,0,4,0,7],[32,123,0.2602,0.56463,0.29483,0.28571,0.57143,0.85704,0.14,1.0,0,5,0,0,0,5,0,0,6,0,0,2,0,0,5,1,0,4,0,0,4,0,5],[36,123,0.2927,0.65621,0.32116,0.42857,0.71429,1.0,0.0,1.0,3,11,0,3,0,0,0,0,2,0,0,6,0,0,4,0,0,4,0,0,2,0,11],[40,123,0.3252,0.54463,0.26107,0.39286,0.4286,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,5,0,0,9,0,0,4,0,0,4,0,0,3,0,4],[44,123,0.3577,0.55356,0.34764,0.2857,0.4286,1.0,0.0,1.0,2,9,0,2,0,5,0,0,4,0,0,6,0,0,2,0,0,2,0,0,2,0,9],[48,123,0.3902,0.60713,0.32143,0.39286,0.64286,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,3,0,0,4,0,0,4,0,0,5,0,0,3,0,8],[52,123,0.4228,0.66961,0.30187,0.39286,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,6,0,0,3,0,0,2,0,0,6,0,0,2,0,11],[56,123,0.4553,0.61605,0.31225,0.42857,0.64264,1.0,0.0,1.0,1,9,0,1,0,3,0,0,3,0,0,7,0,0,2,0,0,5,0,0,2,0,9],[60,123,0.4878,0.57588,0.3184,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,3,0,0,4,0,0,3,0,0,5,0,0,3,0,7],[64,123,0.5203,0.54462,0.30396,0.2857,0.42859,0.85704,0.0,1.0,1,6,0,1,0,4,0,0,4,0,0,9,0,0,2,0,0,3,0,0,3,0,6],[68,123,0.5528,0.61383,0.30032,0.28571,0.57141,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,7,0,0,5,0,1,2,0,0,4,0,0,2,0,9],[72,123,0.5854,0.49994,0.34068,0.25,0.42859,0.74996,0.0,1.0,3,7,0,3,0,5,0,0,6,0,0,3,0,0,4,0,0,3,0,0,1,0,7],[76,123,0.6179,0.54463,0.2889,0.28571,0.4998,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,7,0,0,6,0,0,2,0,0,8,0,0,1,0,5],[80,123,0.6504,0.57143,0.32094,0.28571,0.42857,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,6,0,1,7,0,0,1,1,0,2,0,0,0,0,10],[84,123,0.6829,0.68749,0.31981,0.42857,0.82136,1.0,0.14286,1.0,0,13,0,0,0,3,0,1,2,0,0,7,0,0,0,0,0,2,1,0,3,0,13],[88,123,0.7154,0.70089,0.29528,0.42857,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,6,0,0,2,0,0,3,0,0,4,0,12],[92,123,0.748,0.71874,0.29121,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,5,0,0,3,0,0,3,0,0,2,0,14],[96,123,0.7805,0.69642,0.33455,0.28571,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,7,0,0,3,0,0,0,0,0,2,0,0,4,0,14],[100,123,0.813,0.79909,0.23653,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,0,6,0,14],[104,123,0.8455,0.79464,0.25489,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,6,0,0,4,0,15],[108,123,0.878,0.72765,0.27977,0.42857,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,5,0,0,4,0,0,2,0,0,4,0,13],[112,123,0.9106,0.74997,0.27895,0.42859,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,2,0,0,3,0,15],[116,123,0.9431,0.78345,0.24969,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,1,4,0,0,4,0,0,5,0,14],[120,123,0.9756,0.79017,0.24482,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,5,0,0,2,0,16],[123,123,1.0,0.68969,0.27815,0.42859,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,5,0,1,4,0,0,5,0,0,2,0,11]]},{"b":5,"e":0.1429,"k":"falling","v":0.16963,"x":0.69193,"p":[[0,80,0.0,0.69193,0.32706,0.42857,0.82135,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,2,0,0,5,0,1,3,0,0,0,1,0,1,0,15],[4,80,0.05,0.52229,0.34229,0.14286,0.4643,0.85714,0.0,1.0,1,7,0,1,1,7,0,0,4,0,0,3,0,1,3,0,0,2,0,0,3,0,7],[8,80,0.1,0.40402,0.28828,0.14286,0.28571,0.57143,0.0,1.0,1,4,0,1,1,7,0,0,10,0,0,4,0,0,2,0,0,3,0,0,0,0,4],[12,80,0.15,0.49774,0.29852,0.28571,0.42857,0.74996,0.0,1.0,2,4,0,2,0,4,0,0,5,0,0,7,0,1,4,0,0,1,0,0,4,0,4],[16,80,0.2,0.46853,0.32206,0.14286,0.4998,0.71429,0.0,1.0,4,4,0,4,0,6,0,0,3,0,0,3,0,0,5,0,0,6,0,0,1,0,4],[20,80,0.25,0.40625,0.32948,0.14286,0.28571,0.71429,0.0,1.0,5,3,0,5,0,7,0,0,6,0,0,3,0,0,1,0,0,4,0,0,3,0,3],[24,80,0.3,0.5446,0.27534,0.28571,0.571,0.75,0.14286,1.0,0,4,0,0,0,4,0,0,6,0,0,5,0,0,6,0,0,3,0,0,4,0,4],[28,80,0.35,0.57142,0.30722,0.42857,0.42859,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,1,0,0,12,0,0,1,0,0,4,0,0,1,0,8],[32,80,0.4,0.50901,0.27348,0.28571,0.5355,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,5,0,0,4,0,2,6,0,0,5,0,0,2,0,3],[36,80,0.45,0.45965,0.27609,0.24999,0.42857,0.60682,0.0,1.0,1,3,0,1,0,7,0,0,5,0,0,5,0,0,6,0,0,4,0,0,1,0,3],[40,80,0.5,0.51781,0.30251,0.2857,0.4998,0.71429,0.0,1.0,1,5,0,1,0,6,0,0,4,0,0,5,0,0,5,0,0,4,0,0,2,0,5],[44,80,0.55,0.48214,0.29396,0.2857,0.28571,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,12,0,0,1,0,0,2,0,0,7,0,0,1,0,4],[48,80,0.6,0.36161,0.31941,0.0,0.35714,0.60714,0.0,1.0,9,2,0,9,0,5,0,0,2,0,0,5,0,0,3,0,0,5,0,0,1,0,2],[52,80,0.65,0.33259,0.31011,0.0,0.28571,0.57143,0.0,1.0,9,2,0,9,0,5,0,0,5,0,1,2,0,0,5,0,0,1,0,0,2,0,2],[56,80,0.7,0.39286,0.36245,0.10714,0.28571,0.71429,0.0,1.0,8,5,0,8,0,5,0,0,6,0,0,3,0,0,0,0,0,3,0,0,2,0,5],[60,80,0.75,0.38389,0.33962,0.14286,0.28571,0.57111,0.0,1.0,7,5,0,7,0,6,0,0,5,0,0,3,0,0,4,0,0,2,0,0,0,0,5],[64,80,0.8,0.28345,0.26148,0.14286,0.21428,0.42857,0.0,1.0,7,1,0,7,0,9,0,0,6,0,1,3,0,0,2,0,0,2,0,0,1,0,1],[68,80,0.85,0.3348,0.31663,0.14286,0.2143,0.571,0.0,1.0,7,3,0,7,0,9,0,0,4,0,0,3,0,0,3,0,0,2,0,0,1,0,3],[72,80,0.9,0.36148,0.3312,0.105,0.2857,0.60714,0.0,1.0,8,2,0,8,0,7,0,0,4,0,0,1,0,0,4,0,0,3,0,0,3,0,2],[76,80,0.95,0.17847,0.17854,0.0,0.14286,0.28571,0.0,0.571,12,0,0,12,0,8,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[80,80,1.0,0.16963,0.24334,0.0,0.07143,0.1786,0.0,0.857,16,0,0,16,0,8,0,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,0]]}]},{"i":"df14f05d95432970","q":"For which prime numbers  $p$  can we find three positive integers  $n$ ,  $x$  and  $y$  such that  $p^n = x^3 + y^3$ ?","t":[{"b":0,"e":1.0,"k":"flat","v":0.91963,"x":0.99107,"p":[[0,25,0.0,0.91963,0.19867,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,27],[4,25,0.16,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[8,25,0.32,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[12,25,0.48,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[16,25,0.64,0.9464,0.14624,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,28],[20,25,0.8,0.97321,0.12596,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,25,0.96,0.96428,0.12372,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[25,25,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,49,0.0,0.91964,0.2111,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[4,49,0.0816,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,49,0.1633,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,49,0.2449,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,49,0.3265,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,49,0.4082,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,49,0.4898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,49,0.5714,0.94643,0.17768,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[32,49,0.6531,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,49,0.7347,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,49,0.8163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,49,0.898,0.95535,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[48,49,0.9796,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[49,49,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9826b9a658d9c324","q":"If  $a$ ,  $b$  and  $c$  are the roots of the equation  $x^3 - x^2 - x - 1 = 0$ ,\n(i) show that  $a$ ,  $b$  and  $c$  are distinct:\n(ii) show that\n\\[\\frac{a^{1982} - b^{1982}}{a - b} + \\frac{b^{1982} - c^{1982}}{b - c} + \\frac{c^{1982} - a^{1982}}{c - a}\\]\nis an integer.","t":[{"b":3,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,22,0.0,0.93304,0.2082,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[4,22,0.1818,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[8,22,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,22,0.5455,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.28571,"k":"falling","v":0.33482,"x":0.95536,"p":[[0,32,0.0,0.80357,0.30671,0.60714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,22],[4,32,0.125,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,32,0.25,0.54911,0.2372,0.42857,0.42857,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,6,0,0,13,0,0,4,0,0,3,0,0,1,0,5],[12,32,0.375,0.625,0.26426,0.42857,0.57143,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,4,0,0,11,0,0,5,0,0,2,0,0,1,0,9],[16,32,0.5,0.45982,0.21939,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,14,0,0,9,0,0,3,0,0,3,0,0,0,0,3],[20,32,0.625,0.45982,0.20743,0.28571,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,9,0,0,1,0,0,5,0,0,2,0,1],[24,32,0.75,0.38838,0.17939,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,21,0,0,5,0,0,3,0,0,1,0,0,1,0,1],[28,32,0.875,0.34375,0.11214,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[32,32,1.0,0.33482,0.13175,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,3,0,0,0,0,0]]}]},{"i":"1e68aa8c60422d8a","q":"If  $n$  is an integer greater than  $7$ , prove that  ${n \\choose 7} - \\left[ \\frac{n}{7} \\right]$  is divisible by  $7$ .","t":[{"b":0,"e":0.71429,"k":"rising","v":0.69196,"x":0.92411,"p":[[0,21,0.0,0.72768,0.34136,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,19],[4,21,0.1905,0.69196,0.31564,0.42857,0.71429,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,16],[8,21,0.381,0.91071,0.15047,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,23],[12,21,0.5714,0.9241,0.14279,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,1,0,24],[16,21,0.7619,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[20,21,0.9524,0.90179,0.20652,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,25],[21,21,1.0,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22]]},{"b":4,"e":1.0,"k":"rising","v":0.64732,"x":1.0,"p":[[0,24,0.0,0.75446,0.33165,0.42857,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,20],[4,24,0.1667,0.69197,0.2969,0.42857,0.57143,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,13,0,0,2,0,0,0,0,0,0,0,15],[8,24,0.3333,0.64732,0.34623,0.42857,0.42857,1.0,0.0,1.0,1,15,0,1,0,3,0,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,15],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"52715e249011136e","q":"Given  $ x,y,z\\in (0,1)$  satisfying that\r $ \\sqrt{\\frac{1 \\minus{} x}{yz}} \\plus{} \\sqrt{\\frac{1 \\minus{} y}{xz}} \\plus{} \\sqrt{\\frac{1 \\minus{} z}{xy}} \\equal{} 2$ .\r\nFind the maximum value of  $ xyz$ .","t":[{"b":0,"e":0.4286,"k":"falling","v":0.39732,"x":1.0,"p":[[0,95,0.0,0.88393,0.21558,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,6,0,21],[4,95,0.0421,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,95,0.0842,0.88393,0.25614,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,25],[12,95,0.1263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,95,0.1684,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[20,95,0.2105,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[24,95,0.2526,0.94643,0.17405,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[28,95,0.2947,0.86161,0.24868,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,23],[32,95,0.3368,0.83036,0.28221,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,22],[36,95,0.3789,0.85714,0.26964,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,24],[40,95,0.4211,0.71875,0.34899,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,8,0,0,0,0,0,0,0,0,1,0,0,3,0,17],[44,95,0.4632,0.60713,0.36246,0.28571,0.57121,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,9,0,0,1,0,0,3,0,0,0,0,0,0,0,14],[48,95,0.5053,0.63393,0.36585,0.28571,0.78571,1.0,0.14286,1.0,0,14,0,0,0,6,0,0,6,0,0,3,0,0,0,0,0,1,0,0,2,0,14],[52,95,0.5474,0.64287,0.37796,0.24999,0.85714,1.0,0.14286,1.0,0,14,0,0,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,0,4,0,14],[56,95,0.5895,0.46875,0.32972,0.2857,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,13,0,0,1,0,0,2,0,0,0,0,0,2,0,7],[60,95,0.6316,0.66071,0.37244,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,17],[64,95,0.6737,0.58482,0.33381,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,14,0,0,3,0,0,1,0,0,0,0,0,2,0,11],[68,95,0.7158,0.47321,0.34151,0.2857,0.28571,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,9],[72,95,0.7579,0.59375,0.32559,0.28571,0.42857,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,14,0,0,4,0,0,1,0,0,0,0,0,2,0,11],[76,95,0.8,0.55357,0.35848,0.28571,0.28571,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,12],[80,95,0.8421,0.62045,0.35116,0.28571,0.57143,1.0,0.14,1.0,0,13,0,0,0,2,0,0,13,0,0,1,0,0,0,0,0,1,0,0,2,0,13],[84,95,0.8842,0.48661,0.35149,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,10],[88,95,0.9263,0.41516,0.22967,0.28571,0.28571,0.46418,0.14286,1.0,0,3,0,0,0,3,0,0,15,0,0,6,0,0,4,0,0,1,0,0,0,0,3],[92,95,0.9684,0.39732,0.09933,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,18,0,0,2,0,0,1,0,0,0,0,0],[95,95,1.0,0.41518,0.10926,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,19,0,0,4,0,0,1,0,0,0,0,0]]},{"b":1,"e":1.0,"k":"falling","v":0.33036,"x":0.95982,"p":[[0,95,0.0,0.84375,0.24836,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,3,0,20],[4,95,0.0421,0.90625,0.20079,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,25],[8,95,0.0842,0.91964,0.2141,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[12,95,0.1263,0.89286,0.22016,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,3,0,24],[16,95,0.1684,0.91964,0.18877,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,25],[20,95,0.2105,0.79464,0.28333,0.42859,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,4,0,0,5,0,0,1,0,0,0,0,0,3,0,19],[24,95,0.2526,0.82589,0.2618,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,3,0,0,0,0,21],[28,95,0.2947,0.89732,0.20896,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,25],[32,95,0.3368,0.87946,0.26513,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[36,95,0.3789,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[40,95,0.4211,0.87946,0.23987,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,2,0,24],[44,95,0.4632,0.82589,0.26422,0.67857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,21],[48,95,0.5053,0.87054,0.27049,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,25],[52,95,0.5474,0.83036,0.30606,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,24],[56,95,0.5895,0.83487,0.26746,0.67857,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,1,0,22],[60,95,0.6316,0.83034,0.26352,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,5,0,19],[64,95,0.6737,0.79018,0.30301,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,4,0,0,1,0,0,2,0,0,1,0,0,3,0,19],[68,95,0.7158,0.80357,0.2714,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,1,0,0,2,0,19],[72,95,0.7579,0.82143,0.26964,0.57143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,1,0,21],[76,95,0.8,0.73214,0.32093,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,8,0,0,1,0,0,2,0,0,0,0,0,4,0,16],[80,95,0.8421,0.65625,0.3423,0.28571,0.78571,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,6,0,0,0,0,0,3,0,0,2,0,0,4,0,12],[84,95,0.8842,0.76338,0.32066,0.57132,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,2,0,0,1,0,0,4,0,0,0,0,0,3,0,18],[88,95,0.9263,0.74552,0.30875,0.53539,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0,4,0,0,2,0,16],[92,95,0.9684,0.62945,0.32706,0.28571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,8,0,0,3,0,0,1,0,0,4,0,0,2,0,11],[95,95,1.0,0.33036,0.18708,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,9,0,0,13,0,0,4,0,0,5,0,0,0,0,0,0,0,1]]}]},{"i":"fed3beb6cd52fc3d","q":"Given is a triangle  $ABC$  with the property that  $|AB| + |AC| = 3|BC|$ . Let  $T$  be the point on segment  $AC$  such that  $|AC| = 4|AT|$ . Let  $K$  and  $L$  be points on the interior of line segments  $AB$  and  $AC$  respectively such that  $KL \\parallel BC$  and  $KL$  is tangent to the inscribed circle of  $\\vartriangle ABC$ . Let  $S$  be the intersection of  $BT$  and  $KL$ . Determine the ratio  $\\frac{|SL|}{|KL|}$","t":[{"b":3,"e":0.42857,"k":"falling","v":0.29902,"x":0.95982,"p":[[0,49,0.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,49,0.0816,0.88393,0.15746,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,13],[8,49,0.1633,0.86161,0.20355,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,10,0,15],[12,49,0.2449,0.73214,0.31693,0.64286,0.85714,1.0,0.0,1.0,1,10,0,1,0,4,0,0,1,0,0,2,0,0,0,0,0,2,0,0,12,0,10],[16,49,0.3265,0.73661,0.30117,0.85714,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,19,0,6],[20,49,0.4082,0.43304,0.30823,0.14286,0.35714,0.85714,0.0,1.0,1,2,0,1,0,10,0,0,5,0,0,7,0,0,0,0,0,0,0,0,7,0,2],[24,49,0.4898,0.43304,0.33213,0.14286,0.28571,0.85714,0.14286,1.0,0,3,0,0,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,0,7,0,3],[28,49,0.5714,0.41518,0.3524,0.14286,0.14286,0.85714,0.0,1.0,1,4,0,1,0,17,0,0,0,0,0,4,0,0,0,0,0,0,0,0,6,0,4],[32,49,0.6531,0.50446,0.37284,0.14286,0.42857,0.85714,0.0,1.0,2,6,0,2,0,11,0,0,2,0,0,3,0,0,0,0,0,1,0,0,7,0,6],[36,49,0.7347,0.50893,0.35881,0.14286,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,13,0,0,1,0,0,5,0,0,0,0,0,0,0,0,7,0,6],[40,49,0.8163,0.42411,0.35443,0.14286,0.21429,0.85714,0.0,1.0,1,5,0,1,0,15,0,0,3,0,0,3,0,0,0,0,0,0,0,0,5,0,5],[44,49,0.898,0.34375,0.25218,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,15,0,0,4,0,0,8,0,0,0,0,0,1,0,0,3,0,1],[48,49,0.9796,0.36607,0.26949,0.14286,0.35714,0.42857,0.14286,1.0,0,2,0,0,0,15,0,0,1,0,0,11,0,0,0,0,0,0,0,0,3,0,2],[49,49,1.0,0.29902,0.22412,0.14286,0.14286,0.42857,0.14,1.0,0,2,0,0,0,18,0,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,2]]},{"b":4,"e":0.85714,"k":"flat","v":0.85268,"x":0.96875,"p":[[0,54,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,54,0.0741,0.88839,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[8,54,0.1481,0.875,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,6],[12,54,0.2222,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[16,54,0.2963,0.89732,0.08917,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12],[20,54,0.3704,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[24,54,0.4444,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[28,54,0.5185,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[32,54,0.5926,0.90179,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,12],[36,54,0.6667,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[40,54,0.7407,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[44,54,0.8148,0.88393,0.0974,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,19,0,10],[48,54,0.8889,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[52,54,0.963,0.89286,0.08748,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11],[54,54,1.0,0.85268,0.09771,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,20,0,6]]}]},{"i":"ec10f27ffaef4d24","q":"In a sports tournament of  $n$  players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order  $P_1,P_2, .... , P_n$  such that  $P_i$  defeats  $P_{i+1}$  for all  $1 \\le i \\le n-1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,41,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,41,0.0976,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,41,0.1951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,41,0.2927,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,23,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,23,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,23,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c11256770f5649c7","q":"Four faces of tetrahedron  $ABCD$  are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are  $a < b < c$ , determine the radius of the sphere circumscribed about the tetrahedron as a function on  $a, b$ , and  $c$ . What is the ratio  $c/a$  if  $R = a \\ ?$","t":[{"b":0,"e":0.28571,"k":"falling","v":0.28125,"x":1.0,"p":[[0,221,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,221,0.0181,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,221,0.0362,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,221,0.0543,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,221,0.0724,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,221,0.0905,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,221,0.1086,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,221,0.1267,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,221,0.1448,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,221,0.1629,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,221,0.181,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[44,221,0.1991,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,221,0.2172,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,221,0.2353,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,221,0.2534,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,221,0.2715,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[64,221,0.2896,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,221,0.3077,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[72,221,0.3258,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,221,0.3439,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,221,0.362,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,221,0.3801,0.93304,0.17491,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,5,0,25],[88,221,0.3982,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[92,221,0.4163,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[96,221,0.4344,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[100,221,0.4525,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[104,221,0.4706,0.94196,0.17445,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[108,221,0.4887,0.87946,0.25781,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[112,221,0.5068,0.83482,0.29257,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,2,0,23],[116,221,0.5249,0.86161,0.25626,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,4,0,22],[120,221,0.543,0.91071,0.22517,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[124,221,0.5611,0.87946,0.24772,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,3,0,24],[128,221,0.5792,0.86161,0.25626,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,4,0,22],[132,221,0.5973,0.82143,0.25505,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,9,0,16],[136,221,0.6154,0.79911,0.2809,0.82143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,8,0,16],[140,221,0.6335,0.78125,0.2923,0.53571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,1,0,0,5,0,17],[144,221,0.6516,0.76786,0.30878,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,5,0,17],[148,221,0.6697,0.58929,0.30462,0.28571,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,15,0,0,0,0,0,2,0,0,3,0,0,5,0,7],[152,221,0.6878,0.74554,0.30667,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,8,0,14],[156,221,0.7059,0.68303,0.32876,0.28571,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,10,0,0,1,0,0,2,0,0,1,0,0,3,0,14],[160,221,0.724,0.66518,0.33996,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,1,0,0,3,0,14],[164,221,0.7421,0.70982,0.32436,0.28571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,10,0,0,0,0,0,0,0,0,1,0,0,7,0,13],[168,221,0.7602,0.67857,0.30929,0.28571,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,2,0,0,6,0,11],[172,221,0.7783,0.67857,0.32927,0.28571,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,7,0,12],[176,221,0.7964,0.78125,0.29663,0.60714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,2,0,0,5,0,17],[180,221,0.8145,0.61607,0.33396,0.28571,0.5,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,15,0,0,1,0,0,1,0,0,1,0,0,2,0,12],[184,221,0.8326,0.64732,0.34253,0.28571,0.85714,1.0,0.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real numbers  $ p$ ,  $ q$ ,  $ r$ ,  $ s$  satisfy  $ p+q+r+s = 9$  and  $ p^{2}+q^{2}+r^{2}+s^{2}= 21$ . Prove that there exists a permutation  $ \\left(a,b,c,d\\right)$  of  $ \\left(p,q,r,s\\right)$  such that  $ ab-cd \\geq 2$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.212,"x":0.4196,"p":[[0,47,0.0,0.212,0.23515,0.0,0.14286,0.28571,0.0,0.71429,12,0,1,12,1,6,0,0,7,0,0,0,0,0,3,0,0,3,0,0,0,0,0],[4,47,0.0851,0.2991,0.18679,0.28571,0.28571,0.28579,0.0,0.71429,5,0,0,5,0,1,0,0,19,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[8,47,0.1702,0.4196,0.20803,0.28571,0.28571,0.60714,0.0,0.71429,1,0,0,1,0,2,0,0,15,0,0,2,0,0,4,0,0,8,0,0,0,0,0],[12,47,0.2553,0.29018,0.21274,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,10,0,0,8,0,0,0,0,0,2,0,0,1,0,0],[16,47,0.3404,0.35714,0.25754,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,5,0,0,11,0,0,5,0,0,0,0,0,5,0,0,1,0,1],[20,47,0.4255,0.33036,0.19377,0.2857,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,3,0,0,11,0,0,10,0,0,2,0,0,1,0,0,1,0,0],[24,47,0.5106,0.36606,0.20804,0.2857,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,5,0,0,11,0,0,7,0,0,1,0,0,6,0,0,0,0,0],[28,47,0.5957,0.32143,0.21129,0.28571,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,1,0,0,13,0,0,7,0,0,1,0,0,4,0,0,0,0,0],[32,47,0.6809,0.28571,0.16366,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,5,0,0,13,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[36,47,0.766,0.28572,0.19233,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,7,0,0,13,0,0,0,0,0,0,0,0,1,0,0],[40,47,0.8511,0.30356,0.19148,0.14286,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,5,0,0,8,0,0,11,0,0,1,0,0,2,0,0,0,0,0],[44,47,0.9362,0.35268,0.12364,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,9,0,0,17,0,0,0,0,0,1,0,0,0,0,0],[47,47,1.0,0.29018,0.15355,0.14286,0.28571,0.42857,0.0,0.42857,4,0,0,4,0,6,0,0,7,0,0,15,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.16518,"x":0.3259,"p":[[0,186,0.0,0.25893,0.25862,0.0,0.14288,0.42857,0.0,0.71429,10,0,0,10,0,8,0,0,4,0,0,4,0,0,0,0,0,6,0,0,0,0,0],[4,186,0.0215,0.30804,0.18935,0.25,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,4,0,0,13,0,0,8,0,0,1,0,0,1,0,0,1,0,0],[8,186,0.043,0.31696,0.18466,0.2857,0.28571,0.42857,0.0,0.85714,4,0,1,4,0,2,0,0,15,0,0,8,0,0,1,0,0,1,0,0,1,0,0],[12,186,0.0645,0.32143,0.13832,0.28571,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,2,0,0,12,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[16,186,0.086,0.3259,0.1931,0.25,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,4,0,0,9,0,0,12,0,0,1,0,0,1,0,0,1,0,0],[20,186,0.1075,0.26784,0.17032,0.21427,0.28571,0.42857,0.0,0.571,8,0,0,8,0,0,0,0,13,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[24,186,0.129,0.23214,0.18123,0.10714,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,4,0,0,16,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[28,186,0.1505,0.25,0.15567,0.14286,0.28571,0.42857,0.0,0.42857,6,0,0,6,0,6,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[32,186,0.172,0.26339,0.17169,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,5,0,0,6,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[36,186,0.1935,0.26339,0.20858,0.10714,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,4,0,0,11,0,0,5,0,0,3,0,0,0,0,0,1,0,0],[40,186,0.2151,0.24553,0.15663,0.10714,0.28571,0.32143,0.0,0.4286,8,0,0,8,0,1,0,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[44,186,0.2366,0.27679,0.15947,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,5,0,0,10,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[48,186,0.2581,0.27678,0.15126,0.2857,0.28571,0.42857,0.0,0.42857,6,0,0,6,0,1,0,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[52,186,0.2796,0.20094,0.151,0.0,0.2857,0.28571,0.0,0.43,9,0,0,9,0,6,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[56,186,0.3011,0.25,0.15152,0.14286,0.28571,0.32143,0.0,0.4286,7,0,0,7,0,2,0,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[60,186,0.3226,0.30795,0.16802,0.2857,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,2,0,0,14,0,0,11,0,0,0,0,0,0,0,0,1,0,0],[64,186,0.3441,0.26339,0.14335,0.24999,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,3,0,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[68,186,0.3656,0.28125,0.2004,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,2,0,0,13,0,0,8,0,0,0,0,0,1,0,0,1,0,0],[72,186,0.3871,0.29018,0.15355,0.28571,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,1,0,0,16,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[76,186,0.4086,0.27233,0.13997,0.2857,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,2,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[80,186,0.4301,0.25,0.15972,0.14286,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,7,0,0,13,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[84,186,0.4516,0.29009,0.16172,0.25,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,4,0,0,14,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[88,186,0.4731,0.19196,0.17717,0.0,0.28571,0.28571,0.0,0.71429,12,0,0,12,0,3,0,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[92,186,0.4946,0.28125,0.16164,0.24999,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,3,0,0,14,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[96,186,0.5161,0.26786,0.20124,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,0,0,0,10,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[100,186,0.5376,0.28125,0.17672,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,9,0,0,12,0,0,0,0,0,1,0,0,0,0,0],[104,186,0.5591,0.25,0.15567,0.14286,0.28571,0.42857,0.0,0.42857,7,0,0,7,0,3,0,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[108,186,0.5806,0.28571,0.15152,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,5,0,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[112,186,0.6022,0.25004,0.17133,0.0,0.28571,0.42857,0.0,0.43,9,0,0,9,0,1,0,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[116,186,0.6237,0.26786,0.17405,0.14289,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,2,0,0,13,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[120,186,0.6452,0.23661,0.16982,0.0,0.28571,0.42857,0.0,0.42857,9,0,0,9,0,3,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[124,186,0.6667,0.32143,0.15152,0.28571,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,2,0,0,18,0,0,8,0,0,1,0,0,0,0,0,1,0,0],[128,186,0.6882,0.23661,0.15815,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[132,186,0.7097,0.23205,0.14179,0.14286,0.28571,0.28571,0.0,0.4286,6,0,0,6,0,6,0,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[136,186,0.7312,0.26786,0.17405,0.24999,0.28571,0.32143,0.0,0.71429,7,0,0,7,0,1,0,0,16,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[140,186,0.7527,0.19643,0.17768,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,3,0,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[144,186,0.7742,0.24554,0.17215,0.10714,0.28571,0.32143,0.0,0.57143,8,0,0,8,0,3,0,0,13,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[148,186,0.7957,0.25446,0.1665,0.14286,0.28571,0.32143,0.0,0.71429,6,0,0,6,0,5,0,0,13,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[152,186,0.8172,0.27679,0.11811,0.24999,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,19,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[156,186,0.8387,0.16518,0.15612,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,5,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[160,186,0.8602,0.20089,0.15916,0.0,0.28571,0.28571,0.0,0.4286,10,0,0,10,0,5,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[164,186,0.8817,0.20981,0.16743,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,0,0,0,18,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[168,186,0.9032,0.20982,0.14279,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,5,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[172,186,0.9247,0.18304,0.14827,0.0,0.14288,0.28571,0.0,0.42857,10,0,0,10,0,7,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[176,186,0.9462,0.21875,0.17122,0.0,0.28571,0.42857,0.0,0.4286,10,0,0,10,0,4,0,0,9,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[180,186,0.9677,0.1875,0.16146,0.0,0.2857,0.28571,0.0,0.42857,12,0,0,12,0,3,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[184,186,0.9892,0.16965,0.15335,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,9,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[186,186,1.0,0.17402,0.1546,0.0,0.14288,0.28571,0.0,0.4286,12,0,0,12,0,5,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"40ed2f58ff29b5e8","q":"Initially Alex, Betty, and Charlie had a total of  $444$  peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had form a geometric progression. Alex eats 5 of his peanuts, Betty eats 9 of her peanuts, and Charlie eats 25 of his peanuts. Now the three numbers of peanuts that each person has form an arithmetic progression. Find the number of peanuts Alex had initially.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.66518,"x":0.875,"p":[[0,64,0.0,0.875,0.16269,0.82132,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,8,0,16],[4,64,0.0625,0.78571,0.11845,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,2,0,7],[8,64,0.125,0.74107,0.12078,0.71429,0.71429,0.75,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,0,5,0,3],[12,64,0.1875,0.7366,0.19269,0.71429,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,18,0,0,5,0,5],[16,64,0.25,0.73214,0.15872,0.71429,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,3,0,3],[20,64,0.3125,0.72767,0.13055,0.71429,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,25,0,0,0,0,4],[24,64,0.375,0.70089,0.15303,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,22,0,0,0,0,4],[28,64,0.4375,0.71875,0.08364,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,2,0,1],[32,64,0.5,0.67411,0.21793,0.71429,0.71429,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,23,0,0,1,0,3],[36,64,0.5625,0.66518,0.17717,0.4286,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,17,0,0,4,0,2],[40,64,0.625,0.66964,0.1448,0.71429,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,24,0,0,1,0,1],[44,64,0.6875,0.66964,0.16536,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,22,0,0,1,0,2],[48,64,0.75,0.73661,0.13882,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,22,0,0,3,0,4],[52,64,0.8125,0.70089,0.1488,0.71429,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,23,0,0,1,0,3],[56,64,0.875,0.71875,0.12619,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,25,0,0,1,0,3],[60,64,0.9375,0.66964,0.10971,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,25,0,0,1,0,0],[64,64,1.0,0.66518,0.13175,0.67857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,22,0,0,1,0,1]]},{"b":6,"e":0.71429,"k":"falling","v":0.69643,"x":0.88839,"p":[[0,24,0.0,0.88839,0.12234,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,16],[4,24,0.1667,0.77232,0.12299,0.71429,0.71429,0.71429,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,0,0,7],[8,24,0.3333,0.72768,0.15714,0.71429,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,2,0,3],[12,24,0.5,0.73214,0.13716,0.71429,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,26,0,0,2,0,3],[16,24,0.6667,0.69643,0.24419,0.71429,0.71429,0.71429,0.0,1.0,3,4,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,3,0,4],[20,24,0.8333,0.71875,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0],[24,24,1.0,0.72321,0.14698,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,5,0,1]]}]},{"i":"5bf93a526695c326","q":"In a country, there are n cities. Any two cities are always connected either by a highway or by a train line. Show that one of the two means of transportation allows any city to be connected to any other.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,54,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.02223,0.07225,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.12946,"p":[[0,30,0.0,0.08929,0.19805,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,30,0.1333,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.06688,0.11279,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.10268,0.14826,0.0,0.0,0.2857,0.0,0.57143,20,0,0,20,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,30,0.6667,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.08036,0.12339,0.0,0.0,0.1786,0.0,0.28571,22,0,0,22,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.12946,0.12555,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.10714,0.13363,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"082cf8c726233e77","q":"In the quadrilateral  $ABCD$ , the angles  $B$  and  $D$  are right . The diagonal  $AC$  forms with the side  $AB$  the angle of  $40^o$ , as well with side  $AD$  an angle of  $30^o$ . Find the acute angle between the  diagonals  $AC$  and  $BD$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.68302,"x":0.95757,"p":[[0,207,0.0,0.95089,0.14555,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[4,207,0.0193,0.83033,0.19705,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,9,0,14],[8,207,0.0386,0.79464,0.23402,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,0,0,0,5,0,15],[12,207,0.058,0.79463,0.21112,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,2,0,0,4,0,14],[16,207,0.0773,0.8571,0.21134,0.57143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,1,0,21],[20,207,0.0966,0.83033,0.20028,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,7,0,15],[24,207,0.1159,0.80803,0.249,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,1,0,0,8,0,15],[28,207,0.1353,0.83479,0.2176,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,2,0,0,5,0,17],[32,207,0.1546,0.76786,0.20748,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,4,0,0,3,0,12],[36,207,0.1739,0.77231,0.24449,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,0,3,0,14],[40,207,0.1932,0.76784,0.29828,0.57143,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,6,0,14],[44,207,0.2126,0.79909,0.22265,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,4,0,15],[48,207,0.2319,0.80803,0.21011,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,2,0,0,6,0,14],[52,207,0.2512,0.80355,0.21055,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,6,0,14],[56,207,0.2705,0.75443,0.2606,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,0,0,0,7,0,0,2,0,0,5,0,13],[60,207,0.2899,0.77677,0.2447,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,0,0,0,10,0,0,1,0,0,3,0,15],[64,207,0.3092,0.74996,0.24746,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,1,0,0,11,0,0,0,0,0,4,0,13],[68,207,0.3285,0.83033,0.22712,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,0,0,0,4,0,18],[72,207,0.3478,0.79016,0.21125,0.57143,0.85707,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,3,0,0,5,0,13],[76,207,0.3671,0.69192,0.26029,0.5713,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,2,0,0,13,0,0,1,0,0,3,0,10],[80,207,0.3865,0.70981,0.28457,0.57143,0.57143,1.0,0.0,1.0,1,13,0,1,0,0,0,0,4,0,0,0,0,0,12,0,0,0,0,0,2,0,13],[84,207,0.4058,0.68302,0.30037,0.57143,0.57143,1.0,0.0,1.0,2,12,0,2,0,0,0,0,3,0,0,1,0,0,12,0,0,0,0,0,2,0,12],[88,207,0.4251,0.72762,0.27286,0.57143,0.857,1.0,0.0,1.0,2,10,0,2,0,0,0,0,1,0,0,0,0,0,11,0,0,1,0,0,7,0,10],[92,207,0.4444,0.6919,0.28599,0.571,0.57143,1.0,0.0,1.0,1,11,0,1,0,0,0,0,5,0,0,0,0,0,11,0,0,0,0,0,4,0,11],[96,207,0.4638,0.73659,0.26753,0.57143,0.78564,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,1,0,0,11,0,0,1,0,0,3,0,13],[100,207,0.4831,0.71425,0.25255,0.5714,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,0,0,0,10,0,0,2,0,0,5,0,10],[104,207,0.5024,0.78125,0.22864,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,1,0,0,6,0,13],[108,207,0.5217,0.76337,0.25905,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,2,0,0,6,0,13],[112,207,0.5411,0.8169,0.22661,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,0,0,0,2,0,18],[116,207,0.5604,0.73657,0.26992,0.571,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,4,0,0,2,0,0,10,0,0,0,0,0,1,0,15],[120,207,0.5797,0.88839,0.21939,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,3,0,23],[124,207,0.599,0.88838,0.2223,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,24],[128,207,0.6184,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[132,207,0.6377,0.79461,0.30711,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,5,0,18],[136,207,0.657,0.89285,0.23146,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,24],[140,207,0.6763,0.91069,0.21655,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,25],[144,207,0.6957,0.79016,0.29231,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,4,0,0,0,0,0,5,0,0,2,0,0,1,0,19],[148,207,0.715,0.90179,0.20025,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,4,0,23],[152,207,0.7343,0.91515,0.16703,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,25],[156,207,0.7536,0.95757,0.09599,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,1,25],[160,207,0.7729,0.88838,0.23074,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,2,0,24],[164,207,0.7923,0.87945,0.19922,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,2,0,22],[168,207,0.8116,0.83482,0.27919,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,0,1,0,22],[172,207,0.8309,0.83036,0.22711,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,3,0,0,1,0,19],[176,207,0.8502,0.83034,0.28447,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,22],[180,207,0.8696,0.90174,0.19052,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0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the triangle  $ABC$  the relationship  $AB+AC = 2BC$  holds. Let  $I$  and  $M$  be the incenter and intersection point of the medians of triangle  $ABC$  respectively,  $AL$  its angle bisector, and point  $P$  the orthocenter of triangle  $BIC$ . Prove that the points  $L, M, P$  lie on a straight line.\n\n(Matvii Kurskyi)","t":[{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.2767,"p":[[0,73,0.0,0.2767,0.14266,0.14289,0.28571,0.32143,0.0,0.57143,3,0,0,3,0,6,0,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[4,73,0.0548,0.08928,0.12752,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,73,0.1096,0.12928,0.13052,0.0,0.14143,0.2857,0.0,0.42857,14,0,0,14,0,8,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,73,0.1644,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.11152,0.13234,0.0,0.07,0.14287,0.0,0.4286,16,0,0,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.06696,0.10705,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,73,0.3288,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,73,0.3836,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,73,0.4384,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,73,0.5479,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,73,0.6027,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,73,0.6575,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,73,0.7123,0.08482,0.16698,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,5,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[56,73,0.7671,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.08482,0.11214,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,73,0.9315,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.19178,"p":[[0,180,0.0,0.19178,0.13182,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,15,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,180,0.0222,0.16062,0.13244,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,7,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,180,0.0444,0.125,0.14617,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,4,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,180,0.0667,0.15179,0.15126,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,5,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,180,0.0889,0.15625,0.14445,0.0,0.14286,0.2857,0.0,0.57143,10,0,0,10,0,13,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,180,0.1111,0.12054,0.11355,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,14,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,180,0.1333,0.1517,0.15947,0.0,0.14143,0.28571,0.0,0.42857,15,0,0,15,0,4,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,180,0.1556,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,180,0.1778,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,180,0.2,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,180,0.2222,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,180,0.2444,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,180,0.2667,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,180,0.2889,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,180,0.3111,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,180,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,180,0.3556,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,180,0.3778,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,180,0.4,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,180,0.4222,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,180,0.4444,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,180,0.4667,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,180,0.4889,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,180,0.5111,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,180,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,180,0.5556,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,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a triangle  $ABC$ ,  $\\angle A = 2 \\cdot \\angle B$ . Prove that  $a^2 = b (b+c)$ .","t":[{"b":4,"e":0.28571,"k":"rising","v":0.15625,"x":0.40624,"p":[[0,21,0.0,0.16962,0.25858,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,1,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,1],[4,21,0.1905,0.15625,0.18336,0.0,0.07143,0.2857,0.0,0.57143,16,0,0,16,0,3,0,0,10,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,21,0.381,0.31694,0.23616,0.14286,0.28571,0.571,0.0,0.85714,6,0,0,6,0,4,0,0,12,0,0,1,0,0,7,0,0,0,0,0,2,0,0],[12,21,0.5714,0.40624,0.24251,0.2857,0.28571,0.57143,0.0,1.0,1,3,0,1,0,4,0,0,13,0,0,4,0,0,7,0,0,0,0,0,0,0,3],[16,21,0.7619,0.37053,0.17807,0.2857,0.28571,0.46429,0.14286,1.0,0,1,0,0,0,2,0,0,21,0,0,1,0,0,6,0,0,1,0,0,0,0,1],[20,21,0.9524,0.33035,0.20957,0.24999,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,7,0,0,17,0,0,1,0,0,3,0,0,0,0,0,3,0,0],[21,21,1.0,0.37499,0.15463,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,20,0,0,1,0,0,9,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.125,"x":0.21872,"p":[[0,26,0.0,0.21872,0.24477,0.0,0.07143,0.571,0.0,0.57143,16,0,0,16,0,1,0,0,6,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[4,26,0.1538,0.14284,0.21426,0.0,0.0,0.28571,0.0,0.71429,20,0,0,20,0,2,0,0,5,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[8,26,0.3077,0.18304,0.24284,0.0,0.0,0.35714,0.0,0.57143,19,0,0,19,0,1,0,0,4,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[12,26,0.4615,0.12946,0.19678,0.0,0.0,0.2857,0.0,0.57143,20,0,0,20,0,3,0,0,5,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[16,26,0.6154,0.15176,0.19535,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,4,0,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[20,26,0.7692,0.15177,0.19862,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,2,0,0,8,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[24,26,0.9231,0.125,0.18123,0.0,0.0,0.2857,0.0,0.57143,19,0,0,19,0,4,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[26,26,1.0,0.12944,0.22114,0.0,0.0,0.17857,0.0,0.85714,21,0,0,21,0,3,0,0,4,0,0,0,0,0,3,0,0,0,0,0,1,0,0]]}]},{"i":"0be62bf782a31a42","q":"In a country consisting of  $2015$  cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.","t":[{"b":1,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,31,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,31,0.129,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,31,0.2581,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,85,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,85,0.0471,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,85,0.0941,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,85,0.1412,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,85,0.1882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,85,0.2353,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,85,0.2824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,85,0.3294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,85,0.3765,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,85,0.4235,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,85,0.4706,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,85,0.5176,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,85,0.5647,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,85,0.6118,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[56,85,0.6588,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,85,0.7059,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[64,85,0.7529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,85,0.8,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[72,85,0.8471,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[76,85,0.8941,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[80,85,0.9412,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[84,85,0.9882,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[85,85,1.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]}]},{"i":"866c61593496c38c","q":"In triangle  $ABC$  we have  $|AB| \\ne |AC|$ . The bisectors of  $\\angle ABC$  and  $\\angle  ACB$  meet  $AC$  and  $AB$  at  $E$  and  $F$ , respectively, and intersect at I. If  $|EI| = |FI|$  find the measure of  $\\angle  BAC$ 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a triangle  $ABC$  ,  $D$  is a point on  $BC$  such that  $AD$  is the internal bisector of  $\\angle A$  . Now Suppose  $\\angle B$ = $2\\angle C$  and  $CD=AB$  . Prove that  $\\angle A=72^0$ .","t":[{"b":4,"e":0.57143,"k":"flat","v":0.34816,"x":0.51786,"p":[[0,25,0.0,0.40174,0.26105,0.10714,0.57143,0.57143,0.0,0.85714,8,0,0,8,0,2,0,0,0,0,0,2,0,0,19,0,0,0,0,0,1,0,0],[4,25,0.16,0.34816,0.25485,0.0,0.42857,0.57143,0.0,0.71429,10,0,0,10,0,1,0,0,1,0,0,6,0,0,13,0,0,1,0,0,0,0,0],[8,25,0.32,0.39284,0.22302,0.25,0.49979,0.57143,0.0,0.57143,6,0,0,6,0,2,0,0,2,0,0,6,0,0,16,0,0,0,0,0,0,0,0],[12,25,0.48,0.43304,0.21866,0.39286,0.57143,0.57143,0.0,0.57143,5,0,0,5,0,2,0,0,1,0,0,3,0,0,21,0,0,0,0,0,0,0,0],[16,25,0.64,0.51786,0.1171,0.53571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,6,0,0,24,0,0,0,0,0,0,0,0],[20,25,0.8,0.49999,0.15971,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,0,0,0,6,0,0,22,0,0,1,0,0,0,0,0],[24,25,0.96,0.37944,0.2438,0.10714,0.5712,0.57143,0.0,0.57143,8,0,0,8,0,1,0,0,3,0,0,2,0,0,18,0,0,0,0,0,0,0,0],[25,25,1.0,0.4508,0.17912,0.42857,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,3,0,0,2,0,0,6,0,0,19,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.38391,"x":0.55801,"p":[[0,38,0.0,0.48658,0.19515,0.53539,0.57143,0.57143,0.0,0.85714,3,0,0,3,0,1,0,0,2,0,0,2,0,0,23,0,0,0,0,0,1,0,0],[4,38,0.1053,0.38391,0.27533,0.0,0.57143,0.57143,0.0,0.85714,10,0,0,10,0,1,0,0,0,0,0,1,0,0,19,0,0,0,0,0,1,0,0],[8,38,0.2105,0.44641,0.20747,0.42857,0.57143,0.57143,0.0,0.57143,5,0,0,5,0,0,0,0,2,0,0,4,0,0,21,0,0,0,0,0,0,0,0],[12,38,0.3158,0.38393,0.25364,0.0,0.57143,0.57143,0.0,0.57143,9,0,0,9,0,1,0,0,0,0,0,3,0,0,19,0,0,0,0,0,0,0,0],[16,38,0.4211,0.55801,0.11495,0.57143,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,1,0,0],[20,38,0.5263,0.50888,0.1554,0.571,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,1,0,0,0,0,0,3,0,0,26,0,0,0,0,0,0,0,0],[24,38,0.6316,0.46874,0.21199,0.57132,0.57143,0.57143,0.0,0.71429,4,0,0,4,0,2,0,0,1,0,0,0,0,0,24,0,0,1,0,0,0,0,0],[28,38,0.7368,0.42408,0.24085,0.35714,0.57143,0.57143,0.0,0.71429,7,0,0,7,0,1,0,0,0,0,0,3,0,0,20,0,0,1,0,0,0,0,0],[32,38,0.8421,0.49551,0.21123,0.57132,0.57143,0.57143,0.0,0.85714,4,0,0,4,0,1,0,0,0,0,0,1,0,0,24,0,0,1,0,0,1,0,0],[36,38,0.9474,0.51338,0.14222,0.57132,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,0,0,0,5,0,0,25,0,0,0,0,0,0,0,0],[38,38,1.0,0.38842,0.25812,0.0,0.57143,0.57143,0.0,0.71429,9,0,0,9,0,1,0,0,0,0,0,3,0,0,18,0,0,1,0,0,0,0,0]]}]},{"i":"dae9643f054e1d90","q":"It is said that two permutations $a_{1}, \\ldots, a_{4035}$ and $b_{1}, \\ldots, b_{4035}$ of the integers $1, \\ldots, 4035$ intersect if there exists an integer $k \\leqslant 4035$ such that $a_{k}=b_{k}$. An ensemble $E$ of permutations is said to be unavoidable if every permutation of the integers $1, \\ldots, 4035$ intersects a permutation belonging to $E$.\na) Show that there exists an unavoidable ensemble containing 2018 permutations.\nb) Does there exist an unavoidable ensemble containing 2017 permutations?","t":[{"b":1,"e":0.71429,"k":"rising","v":0.79896,"x":0.99554,"p":[[0,49,0.0,0.79896,0.30698,0.71321,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,17],[4,49,0.0816,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,49,0.1633,0.90625,0.17353,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,1,0,24],[12,49,0.2449,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,49,0.3265,0.87947,0.1992,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,4,0,21],[20,49,0.4082,0.92856,0.13833,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,23],[24,49,0.4898,0.90177,0.15748,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,21],[28,49,0.5714,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[32,49,0.6531,0.91504,0.14242,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[36,49,0.7347,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[40,49,0.8163,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[44,49,0.898,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,49,0.9796,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[49,49,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":4,"e":1.0,"k":"flat","v":0.77679,"x":0.99107,"p":[[0,62,0.0,0.77679,0.26711,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,0,0,0,4,0,0,2,0,0,4,0,0,7,0,13],[4,62,0.0645,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,62,0.129,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,62,0.1935,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,62,0.2581,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,62,0.3226,0.95982,0.08918,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[24,62,0.3871,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[28,62,0.4516,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[32,62,0.5161,0.94643,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[36,62,0.5806,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,62,0.6452,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,62,0.7097,0.94642,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[48,62,0.7742,0.875,0.18123,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,5,0,19],[52,62,0.8387,0.91516,0.11773,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,18],[56,62,0.9032,0.84374,0.22121,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,0,7,0,17],[60,62,0.9677,0.83927,0.21945,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,0,5,0,17],[62,62,1.0,0.78568,0.24225,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,0,8,0,12]]}]},{"i":"14160401e326ec19","q":"Let  $ ABC $  be a triangle and let  $ AD, BE, CF $  be cevians of the triangle which are concurrent at  $ G $ . Prove that if  $ CF \\cdot BE \\ge AF \\cdot EC + AE \\cdot BF + BC \\cdot FE $  then  $ AG \\le GD $ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.28125,"x":0.47321,"p":[[0,43,0.0,0.47321,0.39679,0.14289,0.28571,1.0,0.0,1.0,6,10,0,6,0,3,0,0,11,0,0,0,0,0,0,0,0,1,0,0,1,0,10],[4,43,0.093,0.38839,0.15663,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,1],[8,43,0.186,0.41965,0.24984,0.25001,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,8,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,4],[12,43,0.2791,0.36157,0.18217,0.25002,0.42857,0.42857,0.0,1.0,2,1,0,2,0,6,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,1],[16,43,0.3721,0.37947,0.16213,0.39286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,1],[20,43,0.4651,0.28125,0.14934,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,15,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.34822,0.18536,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,2,0,0,18,0,0,1,0,0,0,0,0,0,0,1],[28,43,0.6512,0.34375,0.18161,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,1],[32,43,0.7442,0.3125,0.14914,0.14286,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,9,0,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.39277,0.19896,0.35714,0.42857,0.42857,0.14,1.0,0,2,0,0,0,8,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,2],[40,43,0.9302,0.29911,0.16115,0.14286,0.42857,0.42857,0.0,0.4286,4,0,0,4,0,7,0,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.38839,0.11971,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,2,0,0,4,0,0,24,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.13839,"x":0.5,"p":[[0,47,0.0,0.47768,0.34554,0.2857,0.28571,0.78571,0.0,1.0,4,8,0,4,0,2,0,0,11,0,0,2,0,0,4,0,0,1,0,0,0,0,8],[4,47,0.0851,0.5,0.26,0.42857,0.42857,0.42858,0.0,1.0,1,6,0,1,0,2,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,6],[8,47,0.1702,0.38837,0.27017,0.14286,0.42857,0.42857,0.0,1.0,4,3,0,4,0,5,0,0,3,0,0,15,0,0,1,0,0,0,0,0,1,0,3],[12,47,0.2553,0.30357,0.18814,0.14286,0.2857,0.42857,0.0,1.0,1,1,0,1,0,13,0,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,1],[16,47,0.3404,0.4196,0.24994,0.24999,0.42857,0.42857,0.14,1.0,0,4,0,0,0,8,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,4],[20,47,0.4255,0.36161,0.11837,0.39286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,7,0,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.33483,0.13175,0.14289,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,10,0,0,1,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.33027,0.22436,0.14286,0.42857,0.42857,0.0,1.0,1,2,0,1,0,13,0,0,1,0,0,15,0,0,0,0,0,0,0,0,0,0,2],[32,47,0.6809,0.33036,0.23266,0.14286,0.42857,0.42857,0.0,1.0,3,2,0,3,0,10,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,2],[36,47,0.766,0.3125,0.19377,0.14286,0.35714,0.42857,0.0,1.0,1,1,0,1,0,13,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,1],[40,47,0.8511,0.31696,0.1461,0.14286,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,5,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[44,47,0.9362,0.30358,0.15465,0.14286,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,11,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.13839,0.12619,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,19,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab0c721253835968","q":"Given positive real numbers $a_{1}, a_{2}, \\ldots, a_{n}$ with $n \\geq 2$ for which $a_{1} a_{2} \\cdots a_{n}=1$. Prove that\n\n$$\n\\left(\\frac{a_{1}}{a_{2}}\\right)^{n-1}+\\left(\\frac{a_{2}}{a_{3}}\\right)^{n-1}+\\ldots+\\left(\\frac{a_{n-1}}{a_{n}}\\right)^{n-1}+\\left(\\frac{a_{n}}{a_{1}}\\right)^{n-1} \\geq a_{1}^{2}+a_{2}^{2}+\\ldots+a_{n}^{2}\n$$\n\nand determine when equality holds.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.22768,"x":0.29911,"p":[[0,56,0.0,0.22768,0.1551,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,56,0.0714,0.29911,0.19678,0.28571,0.28571,0.28571,0.0,1.0,2,2,0,2,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,56,0.1429,0.29911,0.1488,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,2,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,56,0.2143,0.24991,0.07998,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.26339,0.05187,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.2991,0.13054,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,56,0.5714,0.25893,0.05576,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.26339,0.05187,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.28571,0.13832,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,56,0.7857,0.27232,0.06546,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.29018,0.06667,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.29464,0.04972,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.25,"x":0.30803,"p":[[0,76,0.0,0.25893,0.20341,0.14286,0.21428,0.28571,0.14286,1.0,0,2,0,0,0,16,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,76,0.0526,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,76,0.1053,0.28125,0.06667,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,76,0.1579,0.30357,0.13243,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,76,0.2105,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,76,0.2632,0.25891,0.09057,0.2857,0.28571,0.28571,0.0,0.571,1,0,0,1,0,6,0,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,76,0.3158,0.25,0.07143,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,76,0.3684,0.28571,0.20825,0.2857,0.28571,0.28571,0.0,1.0,4,2,0,4,0,2,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,76,0.4211,0.30803,0.15612,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[36,76,0.4737,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,76,0.5263,0.29911,0.13054,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,76,0.5789,0.27678,0.04971,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.25893,0.05575,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.26339,0.05187,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.26339,0.06298,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,6,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.25893,0.05576,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.27232,0.04164,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.27232,0.04164,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e651efdb44f0eb75","q":"Let  $ ABC$  be an equilateral triangle and  $ \\Gamma$  the semicircle drawn exteriorly to the triangle, having  $ BC$  as diameter. Show that if a line passing through  $ A$  trisects  $ BC,$  it also trisects the arc  $ \\Gamma.$","t":[{"b":3,"e":1.0,"k":"flat","v":0.86161,"x":0.99554,"p":[[0,48,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,48,0.0833,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[8,48,0.1667,0.90625,0.21609,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,23],[12,48,0.25,0.9108,0.20122,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,25],[16,48,0.3333,0.9107,0.18124,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,6,0,22],[20,48,0.4167,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,48,0.5,0.91964,0.19541,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,24],[28,48,0.5833,0.90178,0.20959,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,5,0,23],[32,48,0.6667,0.93749,0.15129,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[36,48,0.75,0.86161,0.30823,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,24],[40,48,0.8333,0.92856,0.16751,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[44,48,0.9167,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[48,48,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.87946,"x":0.97768,"p":[[0,25,0.0,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,25,0.16,0.92856,0.20827,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[8,25,0.32,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,25,0.48,0.9241,0.13825,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,22],[16,25,0.64,0.89062,0.21203,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,1,0,0,0,2,0,0,3,0,23],[20,25,0.8,0.87946,0.21756,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,11,0,18],[24,25,0.96,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[25,25,1.0,0.95535,0.13571,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27]]}]},{"i":"61e0722c8bca2555","q":"In triangle  $ABC$ ,  $\\angle B=90^\\circ$ ,  $AB>BC$ , and  $P$  is the point such that  $BP=BC$  and  $\\angle APB=90^\\circ$ , where  $P$  and  $C$  lie on the same side of  $AB$ . Let  $Q$  be the point on  $AB$  such that  $AP=AQ$ , and let  $M$  be the midpoint of  $QC$ . Prove that the line through  $M$  parallel to  $AP$  passes through the midpoint of  $AB$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.00446,"x":0.10714,"p":[[0,41,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.05348,0.10557,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.10714,0.25,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,41,0.5854,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.02223,0.0806,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.07143,0.18898,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,41,0.9756,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[41,41,1.0,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.03125,"p":[[0,9,0.0,0.03116,0.07758,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"84334dfff9b4a802","q":"It is known that the merchant\u2019s   $n$  clients live in locations laid along the ring road. Of these,   $k$  customers have debts to the merchant for  $a_1,a_2,...,a_k$  rubles, and the merchant owes the remaining   $n-k$  clients, whose debts are  $b_1,b_2,...,b_{n-k}$  rubles, moreover,  $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$ . Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.14286,"x":0.1875,"p":[[0,59,0.0,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.16072,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,59,0.4068,0.1875,0.16146,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[28,59,0.4746,0.16062,0.06919,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.16063,0.06918,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14277,"x":0.16956,"p":[[0,34,0.0,0.16956,0.08331,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.1517,0.04974,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"11f3af6a993f97a9","q":"From a point $A$ on a circle with center $O$, a tangent to this circle is drawn, and two points $B$ and $C$ are taken on this tangent, with $C$ between $A$ and $B$. From $B$ and $C$, $(B D)$ and $(C E)$ are drawn tangent to the circle. Prove that $\\widehat{B O C}=\\widehat{D A E}$.","t":[{"b":4,"e":0.42857,"k":"flat","v":0.33482,"x":0.40177,"p":[[0,23,0.0,0.40177,0.25861,0.2857,0.28571,0.28571,0.0,1.0,1,4,0,1,0,0,0,0,24,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[4,23,0.1739,0.35713,0.25504,0.2857,0.28571,0.28571,0.0,1.0,3,3,0,3,0,1,0,0,22,0,0,0,0,0,1,0,0,2,0,0,0,0,3],[8,23,0.3478,0.33482,0.22477,0.2857,0.28571,0.28571,0.0,1.0,2,3,0,2,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,23,0.5217,0.38839,0.26782,0.2857,0.28571,0.28571,0.0,1.0,1,5,0,1,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,23,0.6957,0.36159,0.24217,0.2857,0.28571,0.28571,0.0,1.0,1,3,0,1,0,2,0,0,24,0,0,0,0,0,1,0,0,0,0,0,1,0,3],[20,23,0.8696,0.35267,0.22011,0.2857,0.28571,0.28571,0.0,1.0,1,3,0,1,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[23,23,1.0,0.33928,0.22232,0.2857,0.28571,0.28571,0.0,1.0,3,2,0,3,0,0,0,0,23,0,0,2,0,0,0,0,0,2,0,0,0,0,2]]},{"b":7,"e":0.0,"k":"flat","v":0.33049,"x":0.51787,"p":[[0,25,0.0,0.33049,0.22708,0.2857,0.28571,0.28571,0.0,1.0,2,3,0,2,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,25,0.16,0.36159,0.23685,0.2857,0.28571,0.28571,0.0,1.0,2,3,0,2,0,0,0,0,24,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[8,25,0.32,0.40178,0.27302,0.2857,0.28571,0.28571,0.0,1.0,1,5,0,1,0,0,0,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[12,25,0.48,0.37946,0.3126,0.2857,0.28571,0.28571,0.0,1.0,4,6,0,4,0,1,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,25,0.64,0.51787,0.38257,0.2857,0.28571,1.0,0.0,1.0,3,12,0,3,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[20,25,0.8,0.45536,0.35434,0.28571,0.28571,0.78571,0.0,1.0,5,8,0,5,0,0,0,0,16,0,0,0,0,0,1,0,0,2,0,0,0,0,8],[24,25,0.96,0.44643,0.35848,0.2857,0.28571,1.0,0.0,1.0,4,9,0,4,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[25,25,1.0,0.39286,0.35175,0.2857,0.28571,0.39286,0.0,1.0,7,7,0,7,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,7]]}]},{"i":"c883c6eba9a890ab","q":"In triangle  $ABC$ , the altitude  $AH$  passes through midpoint  of the median  $BM$ . Prove that in the triangle  $BMC$  also one of the altitudes passes through the midpoint  of one of the medians.","t":[{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.05357,"p":[[0,34,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.143,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.03125,"x":0.08929,"p":[[0,54,0.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.08929,0.10564,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.07812,0.12797,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,1,10,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,54,0.5926,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.08036,0.09407,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6df49a24a21198f1","q":"Given two positive integers  $n$  and  $m$  and a function  $f : \\mathbb{Z} \\times \\mathbb{Z} \\to \\left\\{0,1\\right\\}$  with the property that\n\\begin{align*}\nf\\left(i, j\\right) = f\\left(i+n, j\\right) = f\\left(i, j+m\\right) \\qquad \\text{for all } \\left(i, j\\right) \\in \\mathbb{Z} \\times \\mathbb{Z} .\n\\end{align*}\nLet  $\\left[k\\right] = \\left\\{1,2,\\ldots,k\\right\\}$  for each positive integer  $k$ .\nLet  $a$  be the number of all  $\\left(i, j\\right) \\in \\left[n\\right] \\times \\left[m\\right]$  satisfying\n\\begin{align*}\nf\\left(i, j\\right) = f\\left(i+1, j\\right) = f\\left(i, j+1\\right) .\n\\end{align*}\nLet  $b$  be the number of all  $\\left(i, j\\right) \\in \\left[n\\right] \\times \\left[m\\right]$  satisfying\n\\begin{align*}\nf\\left(i, j\\right) = f\\left(i-1, j\\right) = f\\left(i, j-1\\right) .\n\\end{align*}\nProve that  $a = b$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.09375,"x":0.1875,"p":[[0,70,0.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,11,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,70,0.0571,0.16518,0.12428,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,70,0.1143,0.16518,0.12428,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,70,0.1714,0.16946,0.12599,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,70,0.2286,0.17402,0.13238,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[20,70,0.2857,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.1875,0.17655,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[36,70,0.5143,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.17857,0.15567,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,70,0.6286,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,70,0.7429,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.12054,"x":0.16964,"p":[[0,36,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,5,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,36,0.7778,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"637c8c585c715105","q":"In triangle  $ABC$  medians from  $B$  and  $C$  are perpendicular. Prove that  $\\frac{\\sin(B+C)}{\\sin B \\cdot \\sin C} \\geq \\frac{2}{3}.$","t":[{"b":3,"e":0.4286,"k":"flat","v":0.50446,"x":0.62945,"p":[[0,36,0.0,0.53569,0.15971,0.42857,0.57143,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,3,0,0,7,0,0,15,0,0,5,0,0,0,0,1],[4,36,0.1111,0.55353,0.15464,0.5354,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,6,0,0,21,0,0,1,0,0,0,0,2],[8,36,0.2222,0.58927,0.19805,0.42857,0.57143,0.60714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,7,0,0,14,0,0,3,0,0,1,0,4],[12,36,0.3333,0.62945,0.1984,0.57143,0.57143,0.60714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,19,0,0,2,0,0,0,0,6],[16,36,0.4444,0.53123,0.14389,0.42857,0.57121,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,13,0,0,16,0,0,0,0,0,0,0,2],[20,36,0.5556,0.58035,0.17105,0.4286,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,7,0,0,16,0,0,4,0,0,0,0,3],[24,36,0.6667,0.58477,0.13054,0.57132,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,24,0,0,2,0,0,0,0,2],[28,36,0.7778,0.50446,0.10092,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,10,0,0,18,0,0,1,0,0,0,0,0],[32,36,0.8889,0.53117,0.08914,0.5354,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,6,0,0,23,0,0,1,0,0,0,0,0],[36,36,1.0,0.54014,0.08551,0.571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,4,0,0,25,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.50442,"x":0.62054,"p":[[0,37,0.0,0.50442,0.28567,0.28571,0.4998,0.57143,0.14286,1.0,0,6,0,0,0,7,0,0,3,0,0,6,0,0,10,0,0,0,0,0,0,0,6],[4,37,0.1081,0.55357,0.18814,0.42857,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,9,0,0,14,0,0,3,0,0,0,0,3],[8,37,0.2162,0.62054,0.19759,0.42859,0.57143,0.60714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,9,0,0,15,0,0,2,0,0,0,0,6],[12,37,0.3243,0.54908,0.14334,0.42857,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,7,0,0,21,0,0,0,0,0,0,0,2],[16,37,0.4324,0.54909,0.17536,0.42857,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,6,0,0,19,0,0,0,0,0,0,0,3],[20,37,0.5405,0.53124,0.11425,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,0,0,0,0,0,1],[24,37,0.6486,0.50446,0.08737,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,11,0,0,19,0,0,0,0,0,0,0,0],[28,37,0.7568,0.52234,0.09176,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,21,0,0,1,0,0,0,0,0],[32,37,0.8649,0.50443,0.08733,0.42857,0.57121,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,11,0,0,19,0,0,0,0,0,0,0,0],[36,37,0.973,0.52676,0.0906,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,19,0,0,2,0,0,0,0,0],[37,37,1.0,0.52227,0.09179,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,21,0,0,1,0,0,0,0,0]]}]},{"i":"4221377d225d6aa0","q":"Given the polynomial  $p(x) = x^2 + x - 70$ , do there exist integers  $0<m<n$ , so that  $p(m)$  is divisible by  $n$  and  $p(m+1)$  is divisible by  $n+1$ ?\n\n*Proposed by Nairy Sedrakyan*","t":[{"b":4,"e":0.0,"k":"flat","v":0.44643,"x":0.91964,"p":[[0,238,0.0,0.49554,0.33117,0.14286,0.42857,0.71429,0.0,1.0,3,7,0,3,0,6,0,0,1,0,0,9,0,0,3,0,0,3,0,0,0,0,7],[4,238,0.0168,0.74107,0.36323,0.39286,1.0,1.0,0.0,1.0,3,20,1,3,0,0,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,20],[8,238,0.0336,0.82142,0.30306,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,22],[12,238,0.0504,0.89286,0.19885,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,0,0,24],[16,238,0.0672,0.8348,0.27459,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,20],[20,238,0.084,0.75893,0.36846,0.39286,1.0,1.0,0.0,1.0,3,21,0,3,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,21],[24,238,0.1008,0.77679,0.34058,0.67857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,1,0,20],[28,238,0.1176,0.91518,0.20782,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,26],[32,238,0.1345,0.80357,0.34395,0.71429,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,23],[36,238,0.1513,0.68749,0.40633,0.42857,1.0,1.0,0.0,1.0,7,18,0,7,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,18],[40,238,0.1681,0.79017,0.30091,0.57142,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,1,0,19],[44,238,0.1849,0.73661,0.33714,0.57143,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,4,0,0,0,0,0,2,0,0,5,0,0,2,0,16],[48,238,0.2017,0.83482,0.28818,0.67857,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,1,0,22],[52,238,0.2185,0.76339,0.3246,0.57143,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,1,0,18],[56,238,0.2353,0.79016,0.32925,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,2,0,0,1,0,0,1,0,0,4,0,0,1,0,20],[60,238,0.2521,0.87499,0.27608,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,25],[64,238,0.2689,0.75893,0.32031,0.53571,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,5,0,0,1,0,0,2,0,0,1,0,0,4,0,17],[68,238,0.2857,0.86159,0.25377,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,22],[72,238,0.3025,0.78125,0.3214,0.57143,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,1,0,19],[76,238,0.3193,0.80802,0.32657,0.78571,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,21],[80,238,0.3361,0.83482,0.28372,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,21],[84,238,0.3529,0.91518,0.20159,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,25],[88,238,0.3697,0.82589,0.25935,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,0,0,0,4,0,0,4,0,0,2,0,19],[92,238,0.3866,0.86607,0.27185,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,1,0,23],[96,238,0.4034,0.81247,0.28448,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,3,0,0,1,0,0,4,0,0,1,0,0,2,0,20],[100,238,0.4202,0.77232,0.30901,0.64286,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,2,0,0,4,0,0,0,0,0,4,0,0,3,0,17],[104,238,0.437,0.82141,0.32342,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,23],[108,238,0.4538,0.83927,0.24681,0.67857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,2,0,0,1,0,21],[112,238,0.4706,0.88391,0.20961,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,1,0,23],[116,238,0.4874,0.91518,0.2392,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[120,238,0.5042,0.91964,0.17835,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,25],[124,238,0.521,0.79018,0.31132,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,4,0,17],[128,238,0.5378,0.74554,0.35667,0.57143,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,0,0,19],[132,238,0.5546,0.70087,0.35601,0.5354,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,3,0,0,1,0,0,5,0,0,1,0,0,3,0,15],[136,238,0.5714,0.62498,0.37585,0.28571,0.71429,1.0,0.0,1.0,4,14,0,4,0,0,0,0,8,0,0,1,0,0,2,0,0,3,0,0,0,0,14],[140,238,0.5882,0.71427,0.33121,0.39286,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,6,0,0,1,0,0,5,0,0,1,0,0,0,0,17],[144,238,0.605,0.62054,0.42948,0.24999,0.92857,1.0,0.0,1.0,7,16,0,7,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0,2,0,16],[148,238,0.6218,0.56695,0.37026,0.28571,0.57143,1.0,0.0,1.0,4,10,0,4,0,2,0,0,8,0,0,0,0,0,3,0,0,3,0,0,2,0,10],[152,238,0.6387,0.56696,0.35262,0.28571,0.57143,1.0,0.0,1.0,2,9,0,2,0,3,0,0,10,0,0,0,0,0,2,0,0,3,0,0,3,0,9],[156,238,0.6555,0.57141,0.40406,0.28571,0.64264,1.0,0.0,1.0,7,12,0,7,0,0,0,0,6,0,0,2,0,0,1,0,0,2,0,0,2,0,12],[160,238,0.6723,0.64732,0.35712,0.28571,0.78571,1.0,0.0,1.0,3,12,0,3,0,1,0,0,6,0,0,2,0,0,2,0,0,2,0,0,4,0,12],[164,238,0.6891,0.74998,0.33693,0.67846,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,4,0,0,0,0,0,1,0,0,4,0,0,4,0,16],[168,238,0.7059,0.88393,0.20959,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,1,0,23],[172,238,0.7227,0.71427,0.35715,0.28571,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,6,0,0,0,0,0,2,0,0,3,0,0,1,0,17],[176,238,0.7395,0.76786,0.29397,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,16],[180,238,0.7563,0.79464,0.2922,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,3,0,18],[184,238,0.7731,0.625,0.41611,0.25,0.85714,1.0,0.0,1.0,7,14,0,7,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,0,4,0,14],[188,238,0.7899,0.61607,0.38037,0.28571,0.71429,1.0,0.0,1.0,4,13,0,4,0,1,0,0,7,0,0,2,0,0,1,0,0,2,0,0,2,0,13],[192,238,0.8067,0.5878,0.3855,0.28571,0.61905,1.0,0.0,1.0,5,12,0,5,0,0,0,0,9,0,0,0,0,0,2,0,1,1,0,0,2,0,12],[196,238,0.8235,0.67857,0.34256,0.39286,0.78571,1.0,0.0,1.0,2,14,0,2,0,1,0,0,5,0,0,4,0,0,1,0,0,3,0,0,2,0,14],[200,238,0.8403,0.74554,0.30457,0.39293,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,8,0,0,1,0,0,3,0,0,0,0,0,4,0,16],[204,238,0.8571,0.70088,0.31413,0.39286,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,7,0,0,2,0,0,3,0,0,2,0,0,4,0,13],[208,238,0.8739,0.51784,0.41764,0.0,0.49979,1.0,0.0,1.0,9,12,0,9,0,0,0,0,6,0,0,1,0,0,3,0,0,1,0,0,0,0,12],[212,238,0.8908,0.56696,0.37709,0.28571,0.57143,1.0,0.0,1.0,6,9,0,6,0,1,0,0,4,0,0,3,0,0,3,0,0,2,0,0,4,0,9],[216,238,0.9076,0.44643,0.33835,0.2857,0.28571,0.75,0.0,1.0,5,5,0,5,0,2,0,0,11,0,0,3,0,0,1,0,0,2,0,0,3,0,5],[220,238,0.9244,0.5848,0.38193,0.25,0.57143,1.0,0.0,1.0,4,11,0,4,0,4,0,0,4,0,0,1,0,0,4,0,0,1,0,0,3,0,11],[224,238,0.9412,0.55356,0.38258,0.2857,0.64286,1.0,0.0,1.0,6,9,0,6,0,1,0,0,7,0,0,0,0,0,2,0,0,4,0,0,3,0,9],[228,238,0.958,0.49107,0.34614,0.2857,0.28571,0.85714,0.0,1.0,4,6,0,4,0,1,0,0,12,0,0,3,0,0,1,0,0,0,0,0,5,0,6],[232,238,0.9748,0.45982,0.36199,0.25,0.35714,0.75,0.0,1.0,7,7,0,7,0,1,0,0,8,0,0,3,0,0,3,0,0,2,0,0,1,0,7],[236,238,0.9916,0.56247,0.33491,0.28571,0.57143,0.85714,0.0,1.0,5,3,0,5,0,1,0,0,4,0,0,2,0,0,5,0,0,2,0,0,10,0,3],[238,238,1.0,0.50437,0.33509,0.2857,0.35714,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,11,0,0,2,0,0,3,0,0,1,0,0,4,0,6]]},{"b":5,"e":0.28571,"k":"falling","v":0.03571,"x":0.90625,"p":[[0,106,0.0,0.60266,0.28735,0.42857,0.57143,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,4,0,0,5,0,0,6,0,0,5,0,0,2,0,7],[4,106,0.0377,0.90625,0.22477,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,25],[8,106,0.0755,0.62499,0.41304,0.28571,0.85714,1.0,0.0,1.0,7,16,1,7,0,0,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,16],[12,106,0.1132,0.80357,0.35129,0.82143,1.0,1.0,0.0,1.0,3,23,0,3,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[16,106,0.1509,0.89286,0.24485,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,25],[20,106,0.1887,0.88839,0.24415,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,25],[24,106,0.2264,0.8125,0.32031,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,1,0,22],[28,106,0.2642,0.75594,0.32771,0.53539,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,1,1,0,3,0,0,3,0,0,1,0,0,2,0,18],[32,106,0.3019,0.79911,0.27632,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,1,0,0,5,0,0,2,0,0,4,0,17],[36,106,0.3396,0.77456,0.31089,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,1,0,0,2,0,0,3,0,0,2,1,0,3,0,17],[40,106,0.3774,0.75446,0.36811,0.67857,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,18],[44,106,0.4151,0.71429,0.35714,0.57143,0.92857,1.0,0.0,1.0,4,16,0,4,0,1,0,0,1,0,0,1,0,0,5,0,0,2,0,0,2,0,16],[48,106,0.4528,0.70982,0.34531,0.39286,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,5,0,0,2,0,0,2,0,0,2,0,0,2,0,16],[52,106,0.4906,0.83929,0.26426,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,3,0,20],[56,106,0.5283,0.65623,0.33476,0.28571,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,7,0,0,0,0,0,3,0,0,6,0,0,1,0,12],[60,106,0.566,0.87946,0.21162,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,0,0,0,23],[64,106,0.6038,0.78571,0.30514,0.67857,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,17],[68,106,0.6415,0.84821,0.25238,0.82143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,20],[72,106,0.6792,0.85712,0.23959,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,0,21],[76,106,0.717,0.44643,0.41304,0.0,0.28571,1.0,0.0,1.0,11,9,0,11,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,0,1,0,9],[80,106,0.7547,0.09366,0.18421,0.0,0.0,0.14071,0.0,0.85714,23,0,0,23,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[84,106,0.7925,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,106,0.8302,0.07134,0.12367,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,106,0.8679,0.07589,0.18205,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[96,106,0.9057,0.08929,0.19804,0.0,0.0,0.03572,0.0,1.0,24,1,0,24,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[100,106,0.9434,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,106,0.9811,0.06696,0.11837,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[106,106,1.0,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"98301934befc6950","q":"If  $ A,B,C,D$  are four points in space, such that\r\n\\[ \\angle ABC\\equal{}\\angle BCD\\equal{}\\angle CDA\\equal{}\\angle DAB\\equal{}\\pi/2,\r\n\\]\r\nprove that  $ A,B,C,D$  lie in a plane.","t":[{"b":2,"e":1.0,"k":"flat","v":0.88393,"x":0.99107,"p":[[0,38,0.0,0.88393,0.19045,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,4,0,21],[4,38,0.1053,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,38,0.2105,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,38,0.3158,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,38,0.4211,0.90179,0.18013,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,1,0,23],[20,38,0.5263,0.94643,0.11152,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[24,38,0.6316,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[28,38,0.7368,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,38,0.8421,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[36,38,0.9474,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[38,38,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.90622,"x":1.0,"p":[[0,49,0.0,0.90622,0.14993,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,22],[4,49,0.0816,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[8,49,0.1633,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,49,0.2449,0.96875,0.08553,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,49,0.3265,0.96428,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[20,49,0.4082,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,49,0.4898,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,49,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,49,0.6531,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,49,0.7347,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,49,0.8163,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[44,49,0.898,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,49,0.9796,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[49,49,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"53bdae7e322fc911","q":"Is there a power of  $2$  such that it is possible to rearrange the digits, giving another power of  $2$ ?","t":[{"b":0,"e":0.71429,"k":"flat","v":0.76339,"x":0.90625,"p":[[0,46,0.0,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[4,46,0.087,0.875,0.15047,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,4,0,17],[8,46,0.1739,0.90179,0.11538,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,17],[12,46,0.2609,0.8125,0.11538,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,8,0,7],[16,46,0.3478,0.77679,0.11259,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,5,0,5],[20,46,0.4348,0.80357,0.1171,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,6,0,7],[24,46,0.5217,0.83035,0.13093,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,7,0,10],[28,46,0.6087,0.81237,0.10984,0.71429,0.78571,0.85714,0.71,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,10,0,6],[32,46,0.6957,0.79464,0.10677,0.71429,0.71429,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,8,0,5],[36,46,0.7826,0.86161,0.12619,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,7,0,13],[40,46,0.8696,0.80357,0.1171,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,6,0,7],[44,46,0.9565,0.80357,0.09942,0.71429,0.78564,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,12,0,4],[46,46,1.0,0.76339,0.09182,0.71429,0.71429,0.75,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,5,0,3]]},{"b":4,"e":0.71429,"k":"flat","v":0.87499,"x":0.91071,"p":[[0,6,0.0,0.87499,0.1171,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,13],[4,6,0.6667,0.87946,0.20238,0.82132,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,19],[6,6,1.0,0.91071,0.1171,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,19]]}]},{"i":"89314f37a7ce2de8","q":"Let  $ ABP, BCQ, CAR$  be three non-overlapping triangles erected outside of acute triangle  $ ABC$ . Let  $ M$  be the midpoint of segment  $ AP$ . Given that  $ \\angle PAB \\equal{} \\angle CQB \\equal{} 45^\\circ$ ,  $ \\angle ABP \\equal{} \\angle QBC \\equal{} 75^\\circ$ ,  $ \\angle RAC \\equal{} 105^\\circ$ , and  $ RQ^2 \\equal{} 6CM^2$ , compute  $ AC^2/AR^2$ .\r\n\r\n*Zuming Feng.*","t":[{"b":1,"e":0.42857,"k":"rising","v":0.09822,"x":0.71429,"p":[[0,293,0.0,0.09822,0.18708,0.0,0.0,0.14286,0.0,0.71429,23,0,4,23,0,3,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,293,0.0137,0.70088,0.25345,0.71429,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,1,0,0,1,0,0,1,0,0,17,0,0,2,0,7],[8,293,0.0273,0.71429,0.21724,0.71429,0.71429,0.75,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,19,0,0,2,0,6],[12,293,0.041,0.60268,0.26179,0.57143,0.71429,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,2,0,0,0,0,0,2,0,0,20,0,0,1,0,2],[16,293,0.0546,0.70536,0.21998,0.71429,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,19,0,0,0,0,7],[20,293,0.0683,0.6741,0.2321,0.42857,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,7,0,0,5,0,0,7,0,0,5,0,6],[24,293,0.0819,0.70982,0.19061,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,17,0,0,5,0,4],[28,293,0.0956,0.5625,0.26471,0.28571,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,4,0,0,1,0,0,14,0,0,1,0,3],[32,293,0.1092,0.58928,0.30462,0.28571,0.71429,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,4,0,0,0,0,0,1,0,0,14,0,0,3,0,4],[36,293,0.1229,0.50445,0.2624,0.2857,0.42859,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,5,0,0,6,0,0,2,0,0,9,0,0,2,0,2],[40,293,0.1365,0.61158,0.24804,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,3,0,0,2,0,0,14,0,0,3,0,3],[44,293,0.1502,0.69196,0.18935,0.57143,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,17,0,0,1,0,5],[48,293,0.1638,0.58032,0.23128,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,0,0,0,8,0,0,4,0,0,13,0,0,0,0,3],[52,293,0.1775,0.58927,0.21651,0.42859,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,7,0,0,4,0,0,14,0,0,1,0,2],[56,293,0.1911,0.60712,0.23419,0.57132,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,1,0,0,3,0,0,19,0,0,2,0,1],[60,293,0.2048,0.52677,0.27993,0.25001,0.71429,0.71429,0.0,1.0,1,2,0,1,0,7,0,0,2,0,0,3,0,0,2,0,0,14,0,0,1,0,2],[64,293,0.2184,0.59822,0.21852,0.42859,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,1,0,0,4,0,0,2,0,0,19,0,0,1,0,1],[68,293,0.2321,0.60267,0.25935,0.42857,0.71429,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,0,0,0,5,0,0,2,0,0,15,0,0,2,0,3],[72,293,0.2457,0.58928,0.25443,0.39286,0.71429,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,1,0,0,3,0,0,16,0,0,2,0,2],[76,293,0.2594,0.50893,0.2878,0.28571,0.57143,0.71429,0.0,1.0,3,2,0,3,0,4,0,0,3,0,0,4,0,0,4,0,0,10,0,0,2,0,2],[80,293,0.273,0.53125,0.25313,0.39286,0.64286,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,3,0,0,7,0,0,1,0,0,14,0,0,0,0,2],[84,293,0.2867,0.49107,0.23128,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,5,0,0,9,0,0,3,0,0,9,0,0,1,0,1],[88,293,0.3003,0.54015,0.28061,0.28571,0.57121,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,3,0,0,5,0,0,3,0,0,11,0,0,0,0,4],[92,293,0.314,0.50447,0.20198,0.42857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,13,0,0,2,0,0,10,0,0,0,0,1],[96,293,0.3276,0.61158,0.21498,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,5,0,0,3,0,0,16,0,0,3,0,1],[100,293,0.3413,0.59375,0.2769,0.39286,0.71429,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,5,0,0,5,0,0,0,0,0,13,0,0,1,0,5],[104,293,0.3549,0.4509,0.19597,0.28571,0.42857,0.50002,0.14286,0.85714,0,0,0,0,0,3,0,0,7,0,0,14,0,0,0,0,0,6,0,0,2,0,0],[108,293,0.3686,0.4643,0.20203,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,3,0,0,16,0,0,2,0,0,5,0,0,1,0,1],[112,293,0.3823,0.46874,0.20896,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,5,0,0,12,0,0,2,0,0,8,0,0,0,0,1],[116,293,0.3959,0.52232,0.21902,0.39286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,4,0,0,6,0,0,5,0,0,12,0,0,0,0,1],[120,293,0.4096,0.53575,0.17854,0.42857,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,15,0,0,2,0,0,11,0,0,0,0,1],[124,293,0.4232,0.55357,0.21943,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,1,0,0,10,0,0,4,0,0,11,0,0,2,0,1],[128,293,0.4369,0.57141,0.22303,0.42857,0.4998,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,12,0,0,1,0,0,10,0,0,3,0,2],[132,293,0.4505,0.46875,0.24284,0.28571,0.42859,0.71429,0.0,0.71429,3,0,0,3,0,3,0,0,3,0,0,9,0,0,1,0,0,13,0,0,0,0,0],[136,293,0.4642,0.58479,0.24053,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,4,0,0,3,0,0,15,0,0,3,0,1],[140,293,0.4778,0.58036,0.21706,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,9,0,0,2,0,0,13,0,0,3,0,1],[144,293,0.4915,0.6116,0.18293,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,11,0,0,1,0,0,17,0,0,0,0,2],[148,293,0.5051,0.52678,0.24337,0.42857,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,11,0,0,1,0,0,10,0,0,0,0,3],[152,293,0.5188,0.65176,0.17475,0.57132,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,19,0,0,1,0,2],[156,293,0.5324,0.52669,0.2328,0.42857,0.4286,0.71429,0.14,1.0,0,2,0,0,0,4,0,0,3,0,0,10,0,0,1,0,0,12,0,0,0,0,2],[160,293,0.5461,0.54908,0.20858,0.42857,0.571,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,9,0,0,2,0,0,13,0,0,1,0,1],[164,293,0.5597,0.51338,0.20158,0.42857,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,11,0,0,2,0,0,11,0,0,0,0,1],[168,293,0.5734,0.61606,0.22429,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,5,0,0,2,0,0,16,0,0,1,0,3],[172,293,0.587,0.51786,0.22232,0.42857,0.42857,0.71429,0.14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an acute triangle  $ABC$  the bisector of  $\\angle BAC$  crosses  $BC$  at  $D$ . Points  $P$  and  $Q$  are orthogonal projections of  $D$  on lines  $AB$  and  $AC$ . Prove that  $[APQ]=[BCQP]$  if and only if the circumcenter of  $ABC$  lies on  $PQ$ 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an island there are  $n$  castles, and each castle is in country  $A$  or  $B$ . There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them.\nProve that the following two statements are equivalent:\n(1) If some commanders from country  $B$  move to attack an adjacent castle in country  $A$ , some commanders from country  $A$  could appropriately move in defense to adjacent castles in country  $A$  so that in every castle of country  $A$ , the number of country  $A$ 's commanders defending that castle is not less than the number of country  $B$ 's commanders attacking that castle. (Each commander can defend or attack only one castle at a time.)\n(2) For any arbitrary set  $X$  of castles in country  $A$ , the number of country  $A$ 's castles that are in  $X$  or adjacent to at least one of the castle in  $X$  is not less than the number of country  $B$ 's castles that are adjacent to at least one of the castles in  $X$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.95535,"x":0.99554,"p":[[0,22,0.0,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[4,22,0.1818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,22,0.3636,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.98212,0.07792,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,22,0.9091,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[22,22,1.0,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23]]},{"b":4,"e":1.0,"k":"flat","v":0.91963,"x":0.9866,"p":[[0,14,0.0,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[4,14,0.2857,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,14,0.5714,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,14,0.8571,0.91963,0.1595,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,23],[14,14,1.0,0.94642,0.0928,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23]]}]},{"i":"13349c837eb6f0e2","q":"In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a  $\\tfrac{2}{3}$  probability of voting for Alice and a  $\\tfrac{1}{3}$  probability of voting for Celia.\n\nWhat is the probability that Alice wins the election (by having more votes than Celia)?","t":[{"b":4,"e":0.85714,"k":"flat","v":0.93304,"x":0.99554,"p":[[0,61,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,61,0.0656,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,61,0.1311,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,61,0.1967,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[16,61,0.2623,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,61,0.3279,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,61,0.3934,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,61,0.459,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,61,0.5246,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,61,0.5902,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,61,0.6557,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,61,0.7213,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,61,0.7869,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[52,61,0.8525,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[56,61,0.918,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[60,61,0.9836,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[61,61,1.0,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18]]},{"b":7,"e":1.0,"k":"flat","v":0.93304,"x":0.99107,"p":[[0,61,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,61,0.0656,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,61,0.1311,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,61,0.1967,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,61,0.2623,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,61,0.3279,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,61,0.3934,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,61,0.459,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,61,0.5246,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,61,0.5902,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,61,0.6557,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,61,0.7213,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,61,0.7869,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,61,0.8525,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,61,0.918,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[60,61,0.9836,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[61,61,1.0,0.93304,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22]]}]},{"i":"961216ff4e0e3ef0","q":"In an exotic country, the National Bank issues coins that can take any value in the interval  $[0, 1]$ . Find the smallest constant  $c > 0$  such that the following holds, no matter the situation in that country:\n\n*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than  $1000$ , can split those coins into  $100$  boxes, such that the total value inside each box is at most  $c$ .*","t":[{"b":0,"e":0.28571,"k":"falling","v":0.14286,"x":0.48659,"p":[[0,46,0.0,0.39286,0.11294,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,5,0,0,22,0,0,1,0,0,1,0,0,0,0,0],[4,46,0.087,0.48659,0.23381,0.42857,0.57121,0.57143,0.0,1.0,2,2,0,2,0,3,0,0,2,0,0,8,0,0,11,0,0,4,0,0,0,0,2],[8,46,0.1739,0.33929,0.27374,0.10714,0.42857,0.57143,0.0,0.85714,8,0,0,8,0,6,0,0,1,0,0,7,0,0,4,0,0,5,0,0,1,0,0],[12,46,0.2609,0.35713,0.30092,0.10714,0.35714,0.57143,0.0,1.0,8,2,0,8,0,5,0,0,3,0,0,5,0,0,6,0,0,2,0,0,1,0,2],[16,46,0.3478,0.24107,0.22711,0.0,0.14286,0.42857,0.0,0.71429,10,0,0,10,0,7,0,0,6,0,0,3,0,0,4,0,0,2,0,0,0,0,0],[20,46,0.4348,0.30347,0.21059,0.105,0.42857,0.42858,0.0,0.57143,8,0,0,8,0,3,0,0,4,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[24,46,0.5217,0.31696,0.23619,0.0,0.42857,0.57143,0.0,0.71429,9,0,0,9,0,3,0,0,2,0,0,9,0,0,8,0,0,1,0,0,0,0,0],[28,46,0.6087,0.19196,0.20079,0.0,0.14286,0.32143,0.0,0.57143,12,0,0,12,0,9,0,0,3,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[32,46,0.6957,0.34374,0.31915,0.0,0.35714,0.57143,0.0,1.0,11,1,0,11,0,4,0,0,1,0,0,3,0,0,8,0,0,1,0,0,3,0,1],[36,46,0.7826,0.45981,0.21349,0.39286,0.42857,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,4,0,0,10,0,0,10,0,0,2,0,0,1,0,1],[40,46,0.8696,0.23661,0.21902,0.10714,0.14286,0.32143,0.0,0.71429,8,0,0,8,0,11,0,0,5,0,0,2,0,0,4,0,0,2,0,0,0,0,0],[44,46,0.9565,0.20981,0.19877,0.0,0.14286,0.32143,0.0,0.57143,9,0,0,9,0,12,0,0,3,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[46,46,1.0,0.14286,0.16366,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,7,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.16072,"x":0.4732,"p":[[0,33,0.0,0.29464,0.15126,0.28571,0.28571,0.42857,0.0,0.57143,5,0,2,5,0,1,0,0,14,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.4732,0.2511,0.39286,0.4286,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,1,0,0,9,0,0,8,0,0,3,0,0,3,0,1],[8,33,0.2424,0.33482,0.26392,0.14286,0.28571,0.57143,0.0,1.0,5,1,0,5,0,9,0,0,4,0,0,5,0,0,5,0,0,2,0,0,1,0,1],[12,33,0.3636,0.33035,0.23264,0.10714,0.42857,0.57141,0.0,0.71429,8,0,0,8,0,3,0,0,3,0,0,8,0,0,9,0,0,1,0,0,0,0,0],[16,33,0.4848,0.37052,0.27861,0.14286,0.42857,0.57143,0.0,1.0,7,2,0,7,0,5,0,0,0,0,0,9,0,0,8,0,0,1,0,0,0,0,2],[20,33,0.6061,0.45979,0.18116,0.39286,0.571,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,4,0,0,7,0,0,15,0,0,1,0,0,1,0,0],[24,33,0.7273,0.35715,0.20516,0.24999,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,3,0,0,5,0,0,10,0,0,8,0,0,1,0,0,0,0,0],[28,33,0.8485,0.33925,0.23072,0.14286,0.42857,0.57143,0.0,0.71429,6,0,0,6,0,6,0,0,3,0,0,5,0,0,11,0,0,1,0,0,0,0,0],[32,33,0.9697,0.19643,0.12242,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[33,33,1.0,0.16072,0.11152,0.14286,0.14286,0.1786,0.0,0.42857,6,0,0,6,0,18,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1fe3d0750bf18521","q":"In the convex quadrilateral  $ABCD$  angle  $\\angle{BAD}=90$ , $\\angle{BAC}=2\\cdot\\angle{BDC}$  and  $\\angle{DBA}+\\angle{DCB}=180$ . Then find the angle  $\\angle{DBA}$","t":[{"b":0,"e":0.0,"k":"flat","v":0.04464,"x":0.1517,"p":[[0,185,0.0,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,185,0.0216,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,185,0.0432,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,185,0.0649,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,185,0.0865,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,185,0.1081,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,185,0.1297,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,185,0.1514,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,185,0.173,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,185,0.1946,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,185,0.2162,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,185,0.2378,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,185,0.2595,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,185,0.2811,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,185,0.3027,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,185,0.3243,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,185,0.3459,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,185,0.3676,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,185,0.3892,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,185,0.4108,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,185,0.4324,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,185,0.4541,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,185,0.4757,0.12946,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,185,0.4973,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,185,0.5189,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,185,0.5405,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,185,0.5622,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,185,0.5838,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,185,0.6054,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,185,0.627,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,185,0.6486,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,185,0.6703,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,185,0.6919,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,1,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,185,0.7135,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,185,0.7351,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.143,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,185,0.7568,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,185,0.7784,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,185,0.8,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,185,0.8216,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,185,0.8432,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[160,185,0.8649,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[164,185,0.8865,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,185,0.9081,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[172,185,0.9297,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[176,185,0.9514,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[180,185,0.973,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[184,185,0.9946,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[185,185,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.13384,"x":0.1875,"p":[[0,37,0.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.17857,0.09449,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.17857,0.11845,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,37,0.7568,0.16072,0.06916,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.1875,0.14032,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[36,37,0.973,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5f81f1dbf1ebf228","q":"How many distinct sets are there such that each set contains only non-negative powers of  $2$  or  $3$  and sum of its elements is  $2014$ ? $ \n\\textbf{(A)}\\ 64\n\\qquad\\textbf{(B)}\\ 60\n\\qquad\\textbf{(C)}\\ 54\n\\qquad\\textbf{(D)}\\ 48\n\\qquad\\textbf{(E)}\\ \\text{None of the preceding}\n$","t":[{"b":2,"e":0.28571,"k":"falling","v":0.18304,"x":0.98661,"p":[[0,323,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,323,0.0124,0.87946,0.1992,0.71429,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,19],[8,323,0.0248,0.87499,0.16659,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,6,0,17],[12,323,0.0372,0.82589,0.29175,0.82132,1.0,1.0,0.0,1.0,2,18,1,2,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,18],[16,323,0.0495,0.77676,0.28781,0.71429,0.85714,1.0,0.0,1.0,3,12,2,3,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,12],[20,323,0.0619,0.72768,0.33381,0.71429,0.85714,1.0,0.0,1.0,4,12,1,4,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,0,6,0,12],[24,323,0.0743,0.81695,0.22655,0.71429,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,5,0,14],[28,323,0.0867,0.75,0.33882,0.71429,0.85714,1.0,0.0,1.0,4,15,1,4,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,15],[32,323,0.0991,0.65624,0.37773,0.42859,0.78571,1.0,0.0,1.0,6,12,2,6,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,0,4,0,12],[36,323,0.1115,0.69643,0.37923,0.39286,0.92857,1.0,0.0,1.0,5,16,3,5,0,0,0,0,3,0,0,2,0,0,0,0,0,4,0,0,2,0,16],[40,323,0.1238,0.73652,0.34845,0.53572,0.85714,1.0,0.0,1.0,3,15,2,3,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,6,0,15],[44,323,0.1362,0.67411,0.42142,0.21427,0.92857,1.0,0.0,1.0,8,16,3,8,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,16],[48,323,0.1486,0.81696,0.26543,0.71429,0.85714,1.0,0.0,1.0,2,15,1,2,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,7,0,15],[52,323,0.161,0.72766,0.356,0.67846,0.85714,1.0,0.0,1.0,4,14,1,4,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,14],[56,323,0.1734,0.66518,0.40186,0.25,0.85714,1.0,0.0,1.0,6,15,0,6,0,2,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,15],[60,323,0.1858,0.47767,0.41282,0.0,0.57143,0.85714,0.0,1.0,12,5,2,12,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,0,7,0,5],[64,323,0.1981,0.71428,0.37115,0.60714,0.92857,1.0,0.0,1.0,4,16,0,4,0,2,0,0,2,0,0,0,0,0,0,0,0,6,0,0,2,0,16],[68,323,0.2105,0.55804,0.42762,0.0,0.71429,1.0,0.0,1.0,10,11,2,10,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,0,3,0,11],[72,323,0.2229,0.69195,0.37135,0.42857,0.85714,1.0,0.0,1.0,5,14,3,5,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,0,5,0,14],[76,323,0.2353,0.75,0.34626,0.71429,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,4,0,16],[80,323,0.2477,0.58927,0.3989,0.10714,0.71429,1.0,0.0,1.0,8,9,4,8,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,6,0,9],[84,323,0.2601,0.52223,0.37908,0.14214,0.50001,0.85714,0.0,1.0,7,7,2,7,0,2,0,0,3,0,0,4,0,0,1,0,0,4,0,0,4,0,7],[88,323,0.2724,0.54911,0.41666,0.10714,0.64286,1.0,0.0,1.0,8,11,2,8,0,3,0,0,2,0,0,0,0,0,3,0,0,3,0,0,2,0,11],[92,323,0.2848,0.61606,0.43218,0.10714,0.85714,1.0,0.0,1.0,8,14,4,8,0,3,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,14],[96,323,0.2972,0.55804,0.39018,0.10714,0.71429,0.85714,0.0,1.0,8,7,4,8,0,2,0,0,1,0,0,1,0,0,0,0,0,9,0,0,4,0,7],[100,323,0.3096,0.72768,0.32996,0.71429,0.78571,1.0,0.0,1.0,3,13,0,3,0,2,0,0,1,0,0,0,0,0,0,0,0,10,0,0,3,0,13],[104,323,0.322,0.74554,0.32681,0.71429,0.85714,1.0,0.0,1.0,4,11,1,4,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,11],[108,323,0.3344,0.6607,0.39408,0.46396,0.85714,1.0,0.0,1.0,7,12,2,7,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,12],[112,323,0.3467,0.66964,0.41101,0.24999,1.0,1.0,0.0,1.0,6,17,0,6,0,2,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,17],[116,323,0.3591,0.57589,0.40952,0.10714,0.71429,1.0,0.0,1.0,8,9,2,8,0,3,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,9],[120,323,0.3715,0.68304,0.36723,0.28593,0.85714,1.0,0.0,1.0,4,13,1,4,0,1,0,0,4,0,0,1,0,0,0,0,0,4,0,0,5,0,13],[124,323,0.3839,0.51786,0.43704,0.0,0.71429,1.0,0.0,1.0,11,10,5,11,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,10],[128,323,0.3963,0.58036,0.39113,0.1429,0.71429,1.0,0.0,1.0,4,11,0,4,0,5,0,0,4,0,0,1,0,0,1,0,0,3,0,0,3,0,11],[132,323,0.4087,0.62053,0.41743,0.14286,0.85714,1.0,0.0,1.0,7,13,0,7,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,5,0,13],[136,323,0.4211,0.49991,0.39131,0.14214,0.57143,0.89286,0.0,1.0,7,8,1,7,0,4,0,0,4,0,0,0,0,0,3,0,0,4,0,0,2,0,8],[140,323,0.4334,0.41071,0.38754,0.0,0.28571,0.71429,0.0,1.0,12,4,5,12,0,2,0,0,3,0,0,0,0,0,2,0,0,6,0,0,3,0,4],[144,323,0.4458,0.66071,0.3973,0.2857,0.85714,1.0,0.0,1.0,6,14,0,6,0,1,0,0,3,0,0,0,0,0,0,0,0,5,0,0,3,0,14],[148,323,0.4582,0.53125,0.40442,0.0,0.57121,0.89286,0.0,1.0,9,8,0,9,0,0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,6,0,8],[152,323,0.4706,0.55348,0.38928,0.14214,0.71429,0.89286,0.0,1.0,7,8,1,7,0,3,0,0,2,0,0,0,0,0,2,0,0,7,0,0,3,0,8],[156,323,0.483,0.59362,0.41968,0.10714,0.71429,1.0,0.0,1.0,8,12,1,8,0,3,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,12],[160,323,0.4954,0.60714,0.42708,0.14286,0.85714,1.0,0.0,1.0,7,14,0,7,0,3,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,14],[164,323,0.5077,0.39723,0.37926,0.0,0.28571,0.85704,0.0,1.0,9,6,2,9,0,5,0,0,3,0,0,6,0,0,0,0,0,0,0,0,3,0,6],[168,323,0.5201,0.33482,0.37048,0.0,0.14288,0.71429,0.0,1.0,11,5,1,11,0,6,0,0,5,0,0,1,0,0,0,0,0,3,0,0,1,0,5],[172,323,0.5325,0.35713,0.36069,0.0,0.2857,0.57111,0.0,1.0,11,6,1,11,0,2,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,6],[176,323,0.5449,0.48213,0.4237,0.0,0.35714,1.0,0.0,1.0,10,10,0,10,0,2,0,0,4,0,0,1,0,0,2,0,0,1,0,0,2,0,10],[180,323,0.5573,0.42409,0.38379,0.0,0.28571,0.85714,0.0,1.0,9,5,0,9,0,5,0,0,3,0,0,1,0,0,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.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[44,66,0.6667,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[48,66,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,66,0.7879,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,66,0.8485,0.90178,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[60,66,0.9091,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[64,66,0.9697,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[66,66,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"2d82c243a40feaea","q":"Let  $ P(x)$  be the real polynomial function,  $ P(x) \\equal{} ax^3 \\plus{} bx^2 \\plus{} cx \\plus{} d.$  Prove that if  $ |P(x)| \\leq 1$  for all  $ x$  such that  $ |x| \\leq 1,$  then\r\n\r\n\\[ |a| \\plus{} |b| \\plus{} |c| \\plus{} |d| \\leq 7.\\]","t":[{"b":3,"e":0.14286,"k":"falling","v":0.13393,"x":0.41071,"p":[[0,33,0.0,0.40624,0.27457,0.10714,0.49979,0.60714,0.0,0.71429,8,0,0,8,0,1,0,0,3,0,0,4,0,0,8,0,0,8,0,0,0,0,0],[4,33,0.1212,0.41071,0.27374,0.14286,0.42857,0.57143,0.0,1.0,3,2,0,3,0,6,0,0,6,0,0,5,0,0,7,0,0,1,0,0,2,0,2],[8,33,0.2424,0.38392,0.35072,0.0,0.28571,0.60714,0.0,1.0,10,4,0,10,0,4,0,0,3,0,0,0,0,0,7,0,0,4,0,0,0,0,4],[12,33,0.3636,0.35268,0.29877,0.10714,0.28571,0.57143,0.0,1.0,8,2,0,8,0,4,0,0,6,0,0,3,0,0,6,0,0,2,0,0,1,0,2],[16,33,0.4848,0.33481,0.30848,0.0,0.28571,0.57143,0.0,1.0,9,3,0,9,0,5,0,0,4,0,0,4,0,0,6,0,0,1,0,0,0,0,3],[20,33,0.6061,0.32366,0.32043,0.0,0.14286,0.57143,0.0,1.0,9,2,0,9,0,9,0,0,1,0,0,1,0,1,6,0,0,1,0,0,2,0,2],[24,33,0.7273,0.375,0.23891,0.14289,0.35714,0.57143,0.0,0.85714,4,0,0,4,0,5,0,0,7,0,0,4,0,0,9,0,0,1,0,0,2,0,0],[28,33,0.8485,0.27232,0.25843,0.0,0.21428,0.57141,0.0,1.0,10,1,0,10,0,6,0,0,5,0,0,2,0,0,8,0,0,0,0,0,0,0,1],[32,33,0.9697,0.16965,0.22711,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,16,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[33,33,1.0,0.13393,0.15542,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,13,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"falling","v":0.16071,"x":0.51338,"p":[[0,86,0.0,0.51338,0.25218,0.42857,0.57143,0.60714,0.0,1.0,4,2,0,4,0,1,0,0,1,0,0,4,0,0,14,0,0,6,0,0,0,0,2],[4,86,0.0465,0.22768,0.23921,0.0,0.14286,0.4286,0.0,0.71429,13,0,0,13,0,6,0,0,2,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[8,86,0.093,0.44197,0.30169,0.14286,0.50001,0.60714,0.0,1.0,6,2,0,6,0,3,0,0,3,0,0,4,0,0,8,0,0,4,0,0,2,0,2],[12,86,0.1395,0.40624,0.32362,0.0,0.57143,0.71429,0.0,1.0,10,1,0,10,0,2,0,0,2,0,0,0,0,0,8,0,0,8,0,0,1,0,1],[16,86,0.186,0.36604,0.26709,0.14286,0.42857,0.57143,0.0,1.0,7,1,0,7,0,3,0,0,5,0,0,6,0,0,6,0,0,4,0,0,0,0,1],[20,86,0.2326,0.25,0.28571,0.0,0.14286,0.32143,0.0,1.0,12,2,0,12,0,6,0,0,6,0,0,1,0,0,4,0,0,1,0,0,0,0,2],[24,86,0.2791,0.36604,0.30499,0.0,0.4998,0.57143,0.0,1.0,10,2,0,10,0,3,0,0,1,0,0,2,0,0,13,0,0,1,0,0,0,0,2],[28,86,0.3256,0.40624,0.35374,0.10714,0.35714,0.60714,0.0,1.0,8,5,0,8,0,5,0,0,3,0,0,3,0,0,5,0,0,2,0,0,1,0,5],[32,86,0.3721,0.43304,0.32632,0.14286,0.42859,0.57143,0.0,1.0,4,5,0,4,0,8,0,0,3,0,0,2,0,0,8,0,0,2,0,0,0,0,5],[36,86,0.4186,0.30357,0.28959,0.10714,0.21435,0.46429,0.0,1.0,8,1,0,8,0,8,0,0,6,0,0,2,0,0,1,0,0,5,0,0,1,0,1],[40,86,0.4651,0.3214,0.26484,0.14286,0.28571,0.57143,0.0,0.85714,7,0,0,7,0,8,0,0,4,0,0,1,0,0,8,0,0,3,0,0,1,0,0],[44,86,0.5116,0.31919,0.26904,0.10714,0.28571,0.57143,0.0,1.0,8,1,0,8,0,6,0,0,3,0,1,5,0,0,5,0,0,3,0,0,0,0,1],[48,86,0.5581,0.44196,0.30797,0.14286,0.5,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,6,0,0,1,0,0,7,0,0,4,0,0,3,0,2],[52,86,0.6047,0.34375,0.309,0.10714,0.21429,0.57143,0.0,1.0,8,2,0,8,0,8,0,0,1,0,0,3,0,0,7,0,0,2,0,0,1,0,2],[56,86,0.6512,0.21427,0.23144,0.0,0.14286,0.32143,0.0,0.85714,13,0,0,13,0,5,0,0,6,0,0,3,0,0,4,0,0,0,0,0,1,0,0],[60,86,0.6977,0.20978,0.21713,0.0,0.14286,0.32143,0.0,0.57143,11,0,0,11,0,10,0,0,3,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[64,86,0.7442,0.25893,0.24856,0.0,0.14286,0.46431,0.0,0.85714,11,0,0,11,0,6,0,0,3,0,0,4,0,0,7,0,0,0,0,0,1,0,0],[68,86,0.7907,0.30357,0.27374,0.10714,0.2857,0.57143,0.0,1.0,8,1,0,8,0,7,0,0,6,0,0,1,0,0,7,0,0,1,0,0,1,0,1],[72,86,0.8372,0.29686,0.26969,0.14286,0.1429,0.57111,0.0,1.0,7,1,0,7,0,10,0,0,3,0,1,2,0,0,6,0,0,1,0,0,1,0,1],[76,86,0.8837,0.20089,0.23107,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,7,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[80,86,0.9302,0.16518,0.22047,0.0,0.14286,0.17857,0.0,0.71429,15,0,0,15,0,9,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[84,86,0.9767,0.18741,0.2156,0.0,0.14143,0.32143,0.0,0.57143,15,0,0,15,0,5,0,0,4,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[86,86,1.0,0.16071,0.18471,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,6,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"fa3af6245ed36ce4","q":"Let  $ F$  be the midpoint of the side  $ BC$  of a triangle  $ ABC$ . Isosceles right-angled triangles  $ ABD$  and  $ ACE$  are constructed externally on  $ AB$  and  $ AC$  with the right angles at  $ D$  and  $ E$ . Prove that the triangle  $ DEF$  is right-angled and isosceles.","t":[{"b":4,"e":0.42857,"k":"flat","v":0.41517,"x":0.53122,"p":[[0,60,0.0,0.4821,0.13241,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0,0,0,0,0,0,0],[4,60,0.0667,0.48658,0.11212,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,7,0,0,19,0,0,0,0,0,0,0,0],[8,60,0.1333,0.45085,0.12424,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,7,0,0,15,0,0,0,0,0,0,0,0],[12,60,0.2,0.49997,0.11843,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,2,0,0,23,0,0,0,0,0,0,0,0],[16,60,0.2667,0.53122,0.12992,0.5354,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,4,0,0,23,0,0,0,0,0,0,0,1],[20,60,0.3333,0.45979,0.12232,0.28571,0.49979,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[24,60,0.4,0.5089,0.11257,0.5354,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,2,0,0,24,0,0,0,0,0,0,0,0],[28,60,0.4667,0.49103,0.15124,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,1,0,0,21,0,0,0,0,0,1,0,0],[32,60,0.5333,0.50441,0.11282,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,3,0,0,23,0,0,0,0,0,0,0,0],[36,60,0.6,0.47766,0.12681,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,3,0,0,20,0,0,0,0,0,0,0,0],[40,60,0.6667,0.48209,0.12749,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,2,0,0,21,0,0,0,0,0,0,0,0],[44,60,0.7333,0.50442,0.10702,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,5,0,0,22,0,0,0,0,0,0,0,0],[48,60,0.8,0.5089,0.10675,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,4,0,0,23,0,0,0,0,0,0,0,0],[52,60,0.8667,0.50444,0.11284,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,3,0,0,23,0,0,0,0,0,0,0,0],[56,60,0.9333,0.48212,0.14172,0.42857,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,6,0,0,4,0,0,21,0,0,0,0,0,0,0,0],[60,60,1.0,0.41517,0.17626,0.2857,0.42859,0.57143,0.0,0.57143,2,0,0,2,0,3,0,0,5,0,0,8,0,0,14,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"flat","v":0.37053,"x":0.51783,"p":[[0,99,0.0,0.51783,0.11151,0.57132,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,0,0,0,26,0,0,0,0,0,0,0,0],[4,99,0.0404,0.46426,0.1675,0.28571,0.571,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,10,0,0,4,0,0,16,0,0,0,0,0,0,0,1],[8,99,0.0808,0.41967,0.13332,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,4,0,0,13,0,0,0,0,0,0,0,0],[12,99,0.1212,0.41963,0.12845,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,6,0,0,12,0,0,0,0,0,0,0,0],[16,99,0.1616,0.44196,0.12037,0.28571,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,10,0,0,9,0,0,13,0,0,0,0,0,0,0,0],[20,99,0.202,0.37946,0.10479,0.28571,0.35714,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[24,99,0.2424,0.41071,0.12242,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,8,0,0,10,0,0,0,0,0,0,0,0],[28,99,0.2828,0.40625,0.12428,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,7,0,0,10,0,0,0,0,0,0,0,0],[32,99,0.3232,0.41515,0.16885,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,2,0,0,11,0,0,0,0,0,0,0,1],[36,99,0.3636,0.42411,0.11564,0.28571,0.42857,0.57141,0.2857,0.57143,0,0,0,0,0,0,0,0,11,0,0,11,0,0,10,0,0,0,0,0,0,0,0],[40,99,0.404,0.38838,0.10851,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[44,99,0.4444,0.37053,0.1063,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[48,99,0.4848,0.38389,0.10367,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[52,99,0.5253,0.42411,0.12103,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,9,0,0,11,0,0,0,0,0,0,0,0],[56,99,0.5657,0.39731,0.12232,0.28571,0.35714,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,7,0,0,9,0,0,0,0,0,0,0,0],[60,99,0.6061,0.4107,0.11708,0.28571,0.42857,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,10,0,0,9,0,0,0,0,0,0,0,0],[64,99,0.6465,0.37053,0.12807,0.28571,0.28571,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,22,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[68,99,0.6869,0.47326,0.14912,0.39286,0.42929,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,9,0,0,14,0,0,0,0,0,0,0,1],[72,99,0.7273,0.44643,0.13243,0.28571,0.49999,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,4,0,0,16,0,0,0,0,0,0,0,0],[76,99,0.7677,0.46427,0.10713,0.42857,0.42859,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,12,0,0,14,0,0,0,0,0,0,0,0],[80,99,0.8081,0.49551,0.14717,0.42857,0.57141,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,6,0,0,18,0,0,0,0,0,0,0,1],[84,99,0.8485,0.51338,0.10012,0.42859,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,5,0,0,23,0,0,0,0,0,0,0,0],[88,99,0.8889,0.47313,0.12071,0.42857,0.5705,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,9,0,0,17,0,0,0,0,0,0,0,0],[92,99,0.9293,0.50445,0.10704,0.42857,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,5,0,0,5,0,0,22,0,0,0,0,0,0,0,0],[96,99,0.9697,0.49107,0.1234,0.39286,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,2,0,0,22,0,0,0,0,0,0,0,0],[99,99,1.0,0.50446,0.11285,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,3,0,0,23,0,0,0,0,0,0,0,0]]}]},{"i":"c29f8a98d59d0666","q":"Let  $ \\{x_n\\}_{n\\geq 1}$  be a sequences, given by  $ x_1 \\equal{} 1$ ,  $ x_2 \\equal{} 2$  and\r\n\\[ x_{n \\plus{} 2} \\equal{} \\frac { x_{n \\plus{} 1}^2 \\plus{} 3 }{x_n} .\r\n\\]\r\nProve that  $ x_{2008}$  is the sum of two perfect squares.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.56696,"x":0.88392,"p":[[0,78,0.0,0.56696,0.30195,0.42857,0.57143,0.74996,0.0,1.0,2,6,0,2,0,3,0,0,2,0,0,8,0,0,3,0,0,6,0,0,2,0,6],[4,78,0.0513,0.73214,0.21651,0.57143,0.78571,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,7,0,0,11,0,5],[8,78,0.1026,0.74554,0.27603,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,4,0,0,8,0,11],[12,78,0.1538,0.82141,0.22017,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,10,0,13],[16,78,0.2051,0.84821,0.20806,0.82143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,8,0,16],[20,78,0.2564,0.7589,0.24858,0.57132,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,5,0,0,7,0,11],[24,78,0.3077,0.80802,0.20396,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,10,0,11],[28,78,0.359,0.86161,0.15355,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,9,0,14],[32,78,0.4103,0.85714,0.18558,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,5,0,17],[36,78,0.4615,0.84822,0.19541,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,4,0,17],[40,78,0.5128,0.83927,0.21055,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,6,0,16],[44,78,0.5641,0.84375,0.20316,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,7,0,16],[48,78,0.6154,0.77232,0.23381,0.67857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,4,0,0,10,0,10],[52,78,0.6667,0.79909,0.17077,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,16,0,6],[56,78,0.7179,0.8125,0.16146,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,11,0,9],[60,78,0.7692,0.83928,0.1171,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,21,0,5],[64,78,0.8205,0.7991,0.17077,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,17,0,6],[68,78,0.8718,0.88392,0.13573,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,15],[72,78,0.9231,0.75446,0.25312,0.57143,0.857,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,6,0,0,6,0,11],[76,78,0.9744,0.64286,0.22303,0.4286,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,5,0,0,8,0,0,5,0,0,6,0,4],[78,78,1.0,0.66963,0.20652,0.57132,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,6,0,0,12,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.47313,"x":0.79018,"p":[[0,61,0.0,0.52679,0.34151,0.28571,0.42857,0.85714,0.0,1.0,3,7,0,3,0,3,0,0,6,0,0,7,0,0,1,0,0,1,0,0,4,0,7],[4,61,0.0656,0.74552,0.22228,0.67857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,10,0,0,5,0,9],[8,61,0.1311,0.79018,0.17491,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,12,0,7],[12,61,0.1967,0.77231,0.18852,0.57143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,14,0,6],[16,61,0.2623,0.70534,0.24985,0.57142,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,2,0,0,6,0,0,3,0,0,12,0,5],[20,61,0.3279,0.72766,0.1968,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,11,0,0,11,0,3],[24,61,0.3934,0.67411,0.22654,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,0,0,0,1,0,0,6,0,0,12,0,0,8,0,2],[28,61,0.459,0.67409,0.19961,0.57143,0.71429,0.75,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,13,0,0,5,0,3],[32,61,0.5246,0.66963,0.22711,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,10,0,0,9,0,2],[36,61,0.5902,0.64286,0.18558,0.57143,0.71429,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,4,0,0,8,0,0,9,0,0,7,0,1],[40,61,0.6557,0.64284,0.18898,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,0,0,0,9,0,0,15,0,0,5,0,0],[44,61,0.7213,0.65624,0.20782,0.57132,0.71429,0.75,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,0,6,0,2],[48,61,0.7869,0.63393,0.17835,0.53572,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,14,0,0,4,0,1],[52,61,0.8525,0.57587,0.22155,0.53539,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,0,0,0,4,0,0,12,0,0,6,0,0,6,0,0],[56,61,0.918,0.48205,0.23363,0.42857,0.4286,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,1,0,0,10,0,0,5,0,0,8,0,0,2,0,0],[60,61,0.9836,0.47313,0.20666,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,2,0,0,8,0,0,10,0,0,7,0,0,0,0,0],[61,61,1.0,0.52679,0.20341,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,3,0,0,2,0,0,5,0,0,9,0,0,12,0,0,0,0,0]]}]},{"i":"f569d2a94585f6d2","q":"Given the natural numbers  $a$  and  $b$ , with  $1 \\le a <b$ , prove that there exist natural numbers  $n_1<n_2< ...<n_k$ , with  $k \\le a$  such that   $$ \\frac{a}{b}=\\frac{1}{n_1}+\\frac{1}{n_2}+...+\\frac{1}{n_k} $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.89731,"x":0.96428,"p":[[0,19,0.0,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[4,19,0.2105,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[8,19,0.4211,0.93748,0.07089,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,19,0.6316,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[16,19,0.8421,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[19,19,1.0,0.89731,0.08918,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12]]},{"b":1,"e":0.857,"k":"flat","v":0.82589,"x":0.92397,"p":[[0,33,0.0,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[4,33,0.1212,0.87497,0.12246,0.85714,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,17,0,11],[8,33,0.2424,0.92397,0.08009,0.85714,0.92857,1.0,0.71,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[12,33,0.3636,0.84821,0.11811,0.82132,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,16,0,8],[16,33,0.4848,0.83468,0.07262,0.85711,0.85714,0.85714,0.71,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,23,0,2],[20,33,0.6061,0.8616,0.08364,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,24,0,5],[24,33,0.7273,0.83479,0.09522,0.71429,0.85707,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,17,0,5],[28,33,0.8485,0.84375,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,21,0,4],[32,33,0.9697,0.82589,0.07771,0.71429,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,21,0,2],[33,33,1.0,0.83928,0.09279,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,18,0,5]]}]},{"i":"90c60484ebc6554e","q":"Given triangle  $ABC$  and a point  $P$  inside it,  $\\angle BAP=18^\\circ$ ,  $\\angle CAP=30^\\circ$ ,  $\\angle ACP=48^\\circ$ , and  $AP=BC$ .  If  $\\angle BCP=x^\\circ$ , find  $x$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,45,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,10,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cd13a3de518f2b5f","q":"Let  $(A,+,\\cdot)$  a 9 elements ring. Prove that the following assertions are equivalent:\n\n(a) For any  $x\\in A\\backslash\\{0\\}$  there are two numbers  $a\\in \\{-1,0,1\\}$  and  $b\\in \\{-1,1\\}$  such that  $x^2+ax+b=0$ .\n\n(b)  $(A,+,\\cdot)$  is a field.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.74777,"x":0.99107,"p":[[0,18,0.0,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[4,18,0.2222,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,18,0.4444,0.94642,0.12243,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[12,18,0.6667,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,18,0.8889,0.85714,0.20516,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,6,0,18],[18,18,1.0,0.74777,0.20592,0.57143,0.78564,0.875,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,0,8,1,7]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,16,0.0,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,16,0.25,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,16,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,16,0.75,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,16,1.0,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"cfb630452f470305","q":"Given the digits  $1$  through  $7$ , one can form  $7!=5040$  numbers by forming different permutations of the  $7$  digits (for example,  $1234567$  and  $6321475$  are two such permutations).  If the  $5040$  numbers are then placed in ascending order, what is the  $2013^{\\text{th}}$  number?","t":[{"b":2,"e":0.71429,"k":"flat","v":0.69643,"x":0.72321,"p":[[0,93,0.0,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[4,93,0.043,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[8,93,0.086,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[12,93,0.129,0.70089,0.07457,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[16,93,0.172,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[20,93,0.2151,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[24,93,0.2581,0.69643,0.05923,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0],[28,93,0.3011,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[32,93,0.3441,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[36,93,0.3871,0.70089,0.07457,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[40,93,0.4301,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[44,93,0.4731,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[48,93,0.5161,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[52,93,0.5591,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[56,93,0.6022,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[60,93,0.6452,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[64,93,0.6882,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[68,93,0.7312,0.70089,0.07457,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[72,93,0.7742,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[76,93,0.8172,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[80,93,0.8602,0.71429,0.07143,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,1],[84,93,0.9032,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[88,93,0.9462,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[92,93,0.9892,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[93,93,1.0,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1]]},{"b":7,"e":0.71429,"k":"flat","v":0.70981,"x":0.74107,"p":[[0,54,0.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[4,54,0.0741,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,1],[8,54,0.1481,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[12,54,0.2222,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[16,54,0.2963,0.72321,0.04971,0.71429,0.71429,0.71429,0.714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[20,54,0.3704,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[24,54,0.4444,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[28,54,0.5185,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[32,54,0.5926,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,54,0.6667,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[40,54,0.7407,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[44,54,0.8148,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[48,54,0.8889,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[52,54,0.963,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[54,54,1.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0]]}]},{"i":"5befab21160f00a3","q":"Let  $ f(x)$ and  $ g(y)$  be two monic polynomials of degree= $ n$  having complex coefficients.\r\nWe know that there exist complex numbers  $ a_i,b_i,c_i \\forall 1\\le i \\le n$ , such that\r $ f(x)\\minus{}g(y)\\equal{}\\prod_{i\\equal{}1}^n{(a_ix\\plus{}b_iy\\plus{}c_i)}$ .\r\nProve that there exists  $ a,b,c\\in\\mathbb{C}$  such that \r $ f(x)\\equal{}(x\\plus{}a)^n\\plus{}c\\text{ and }g(y)\\equal{}(y\\plus{}b)^n\\plus{}c$ .","t":[{"b":5,"e":1.0,"k":"rising","v":0.43303,"x":0.90625,"p":[[0,63,0.0,0.43303,0.2977,0.14286,0.42857,0.71429,0.0,1.0,5,1,2,5,0,4,0,0,5,0,0,5,0,0,2,0,0,7,0,0,3,0,1],[4,63,0.0635,0.82589,0.20119,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,0,12,0,11],[8,63,0.127,0.83482,0.19597,0.85711,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,14,0,11],[12,63,0.1905,0.80357,0.23076,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,5,0,0,6,0,14],[16,63,0.254,0.78571,0.22016,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,10,0,0,6,0,11],[20,63,0.3175,0.83034,0.22144,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,6,0,16],[24,63,0.381,0.84821,0.17105,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,13,0,12],[28,63,0.4444,0.875,0.17035,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,5,0,18],[32,63,0.5079,0.85268,0.17672,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,9,0,15],[36,63,0.5714,0.90625,0.14987,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,8,0,19],[40,63,0.6349,0.88393,0.15746,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,9,0,17],[44,63,0.6984,0.85268,0.17307,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,16],[48,63,0.7619,0.84375,0.19019,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,6,0,16],[52,63,0.8254,0.86607,0.15126,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,5,0,16],[56,63,0.8889,0.88393,0.14914,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,17],[60,63,0.9524,0.89286,0.14286,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,17],[63,63,1.0,0.87946,0.11904,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,12,0,13]]},{"b":7,"e":0.85714,"k":"rising","v":0.4107,"x":0.95982,"p":[[0,45,0.0,0.4107,0.31288,0.14286,0.35714,0.57143,0.0,1.0,5,3,2,5,0,5,0,0,6,0,0,5,0,0,4,0,0,1,0,0,3,0,3],[4,45,0.0889,0.90625,0.17354,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,20],[8,45,0.1778,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[12,45,0.2667,0.91071,0.15465,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,6,0,21],[16,45,0.3556,0.82589,0.18466,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,8,0,13],[20,45,0.4444,0.80804,0.20395,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,7,0,11],[24,45,0.5333,0.89286,0.15567,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,9,0,18],[28,45,0.6222,0.89286,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[32,45,0.7111,0.87946,0.16016,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,3,0,19],[36,45,0.8,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[40,45,0.8889,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[44,45,0.9778,0.94197,0.11769,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[45,45,1.0,0.76785,0.23891,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,0,8,0,11]]}]},{"i":"4c41b30bc64de045","q":"Let  $1\\le k\\le n$  be integers. At most how many  $k$ -element subsets can we select from  $\\{1,2,\\dots,n\\}$  such that for any two selected subsets, one of the subsets consists of the  $k$  smallest elements of their union?","t":[{"b":2,"e":1.0,"k":"flat","v":0.84375,"x":1.0,"p":[[0,29,0.0,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,29,0.1379,0.92857,0.20516,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[8,29,0.2759,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,29,0.4138,0.875,0.28959,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,26],[16,29,0.5517,0.91964,0.20497,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,27],[20,29,0.6897,0.84375,0.27747,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,23],[24,29,0.8276,0.91964,0.18877,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,27],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.89284,"x":1.0,"p":[[0,66,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,66,0.0606,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,66,0.1212,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[12,66,0.1818,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[16,66,0.2424,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,66,0.303,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,66,0.3636,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,66,0.4242,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,66,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,66,0.5455,0.94643,0.17035,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[40,66,0.6061,0.95089,0.16213,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[44,66,0.6667,0.89284,0.23147,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,25],[48,66,0.7273,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[52,66,0.7879,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,66,0.8485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,66,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,66,0.9697,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[66,66,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"49a92b0e7f6043f2","q":"Let  $(x_n)_{n\\ge 1}$  and  $(y_n)_{n\\ge 1}$  a sequence of positive real numbers, such that:\n\n\\[x_{n+1}\\ge \\frac{x_n+y_n}{2},\\ y_{n+1}\\ge \\sqrt{\\frac{x_n^2+y_n^2}{2}},\\ (\\forall)n\\in \\mathbb{N}^*\\]\n\na) Prove that the sequences  $(x_n+y_n)_{n\\ge 1}$  and  $(x_ny_n)_{n\\ge 1}$  have limit.\n\nb) Prove that the sequences  $(x_n)_{n\\ge 1}$  and  $(y_n)_{n\\ge 1}$  have limit and that their limits are equal.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.18303,"x":0.48212,"p":[[0,57,0.0,0.46424,0.39928,0.0,0.4993,0.85714,0.0,1.0,11,6,0,11,0,1,0,0,2,0,0,2,0,0,3,0,0,3,0,0,4,0,6],[4,57,0.0702,0.41495,0.3632,0.105,0.28571,0.60714,0.0,1.0,8,6,0,8,0,4,0,0,5,0,0,2,0,0,5,0,0,1,0,0,1,0,6],[8,57,0.1404,0.48212,0.36899,0.14289,0.35716,0.85714,0.0,1.0,6,7,0,6,0,3,0,0,7,0,0,1,0,0,3,0,0,3,0,0,2,0,7],[12,57,0.2105,0.35714,0.31542,0.0,0.28571,0.57143,0.0,1.0,11,1,0,11,0,0,0,0,7,0,0,1,0,0,7,0,0,2,0,0,3,0,1],[16,57,0.2807,0.2991,0.32607,0.0,0.2143,0.60682,0.0,1.0,14,1,0,14,0,2,0,0,4,0,0,3,0,0,1,0,0,5,0,0,2,0,1],[20,57,0.3509,0.18303,0.2402,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,2,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[24,57,0.4211,0.41067,0.37071,0.0,0.28571,0.71429,0.0,1.0,11,4,0,11,0,1,0,0,5,0,0,0,0,0,5,0,0,3,0,0,3,0,4],[28,57,0.4912,0.22764,0.28758,0.0,0.14286,0.32143,0.0,1.0,15,2,0,15,0,4,0,0,5,0,0,1,0,0,5,0,0,0,0,0,0,0,2],[32,57,0.5614,0.23661,0.29366,0.0,0.07143,0.42858,0.0,1.0,16,1,0,16,0,2,0,0,4,0,0,4,0,0,3,0,0,0,0,0,2,0,1],[36,57,0.6316,0.33929,0.32093,0.0,0.28571,0.46431,0.0,1.0,11,2,0,11,0,2,0,0,4,0,0,7,0,0,1,0,0,3,0,0,2,0,2],[40,57,0.7018,0.26339,0.2699,0.0,0.2857,0.42857,0.0,1.0,12,1,0,12,0,2,0,0,9,0,0,3,0,0,3,0,0,1,0,0,1,0,1],[44,57,0.7719,0.29018,0.32436,0.0,0.21428,0.46431,0.0,1.0,14,2,0,14,0,2,0,0,4,0,0,4,0,0,3,0,0,1,0,0,2,0,2],[48,57,0.8421,0.40177,0.35253,0.0,0.35716,0.71429,0.0,1.0,9,3,0,9,0,5,0,0,2,0,0,1,0,0,6,0,0,3,0,0,3,0,3],[52,57,0.9123,0.25893,0.25862,0.0,0.2857,0.35714,0.0,1.0,11,1,0,11,0,4,0,0,9,0,0,0,0,0,6,0,0,1,0,0,0,0,1],[56,57,0.9825,0.33032,0.24069,0.14286,0.2857,0.57111,0.0,0.857,4,0,0,4,0,8,0,0,9,0,0,1,0,0,7,0,0,1,0,0,2,0,0],[57,57,1.0,0.24103,0.24068,0.0,0.2143,0.42857,0.0,0.85714,12,0,0,12,0,4,0,0,7,0,0,2,0,0,6,0,0,0,0,0,1,0,0]]},{"b":3,"e":0.4286,"k":"rising","v":0.24105,"x":0.71424,"p":[[0,66,0.0,0.40625,0.34462,0.10714,0.28571,0.71429,0.0,1.0,8,3,0,8,0,3,0,0,6,0,0,4,0,0,1,0,0,3,0,0,4,0,3],[4,66,0.0606,0.52231,0.37048,0.14289,0.57143,0.85714,0.0,1.0,7,7,0,7,0,2,0,0,3,0,0,1,0,0,5,0,0,5,0,0,2,0,7],[8,66,0.1212,0.55355,0.27606,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,9,0,0,2,0,0,8,0,0,4,0,0,2,0,5],[12,66,0.1818,0.38382,0.33781,0.0,0.35714,0.57143,0.0,1.0,11,2,0,11,0,1,0,0,4,0,0,1,0,0,8,0,0,2,0,0,3,0,2],[16,66,0.2424,0.32139,0.29447,0.0,0.28571,0.57143,0.0,0.85714,11,0,0,11,0,4,0,0,3,0,0,1,0,0,7,0,0,5,0,0,1,0,0],[20,66,0.303,0.24105,0.27762,0.0,0.14286,0.4286,0.0,1.0,15,1,0,15,0,2,0,0,5,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[24,66,0.3636,0.34374,0.2786,0.0,0.35714,0.57143,0.0,0.85714,9,0,0,9,0,4,0,0,3,0,0,3,0,0,8,0,0,4,0,0,1,0,0],[28,66,0.4242,0.38829,0.36116,0.0,0.28571,0.71429,0.0,1.0,10,3,0,10,0,5,0,0,2,0,0,1,0,0,4,0,0,4,0,0,3,0,3],[32,66,0.4848,0.28122,0.24865,0.0,0.28571,0.32143,0.0,0.85714,10,0,0,10,0,1,0,0,13,0,0,1,0,0,3,0,0,3,0,0,1,0,0],[36,66,0.5455,0.29017,0.31839,0.0,0.2143,0.57111,0.0,1.0,14,2,0,14,0,2,0,0,4,0,0,3,0,0,3,0,0,4,0,0,0,0,2],[40,66,0.6061,0.37048,0.31307,0.0,0.35714,0.57143,0.0,1.0,10,2,0,10,0,1,0,0,5,0,0,3,0,0,7,0,0,3,0,0,1,0,2],[44,66,0.6667,0.35712,0.30092,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,0,0,0,11,0,0,2,0,0,3,0,0,3,0,0,3,0,1],[48,66,0.7273,0.35698,0.32323,0.0,0.28571,0.57143,0.0,1.0,9,2,0,9,0,4,0,0,6,0,0,1,0,0,6,0,0,1,0,0,3,0,2],[52,66,0.7879,0.36159,0.33499,0.0,0.35714,0.57143,0.0,1.0,11,3,0,11,0,2,0,0,3,0,0,5,0,0,4,0,0,3,0,0,1,0,3],[56,66,0.8485,0.37053,0.32312,0.0,0.28571,0.71429,0.0,1.0,9,1,0,9,0,5,0,0,3,0,0,2,0,0,3,0,0,7,0,0,2,0,1],[60,66,0.9091,0.558,0.28651,0.28571,0.57121,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,7,0,0,3,0,0,7,0,0,2,0,0,7,0,3],[64,66,0.9697,0.58926,0.26426,0.4286,0.64286,0.74999,0.0,1.0,3,1,0,3,0,1,0,0,1,0,0,4,0,0,7,0,0,8,0,0,7,0,1],[66,66,1.0,0.71424,0.25507,0.57132,0.85707,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,3,0,0,1,0,0,8,0,0,2,0,0,10,0,7]]}]},{"i":"3847963ec81e6e70","q":"Let  $ x_1,  x_2, \\cdots, x_n$  be  positive real  numbers . Prove that:  $$ \\sum_ {i = 1}^n x_i ^2\\geq \\frac {1} {n + 1} \\left (\\sum_ {i = 1}^n x_i \\right)^2+\\frac{12(\\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}.  $$","t":[{"b":2,"e":0.42857,"k":"falling","v":0.16069,"x":0.57143,"p":[[0,157,0.0,0.41963,0.33299,0.10714,0.42857,0.71429,0.0,1.0,8,2,0,8,0,2,0,0,5,0,0,4,0,0,4,0,0,2,0,0,5,0,2],[4,157,0.0255,0.47322,0.3787,0.14286,0.42857,0.85714,0.0,1.0,7,6,0,7,0,4,0,0,4,0,0,3,0,0,1,0,0,3,0,0,4,0,6],[8,157,0.051,0.5,0.38299,0.10714,0.50001,0.85714,0.0,1.0,8,7,0,8,0,1,0,0,5,0,0,2,0,0,2,0,0,4,0,0,3,0,7],[12,157,0.0764,0.57143,0.36943,0.2857,0.57143,1.0,0.0,1.0,4,10,0,4,0,3,0,0,5,0,0,2,0,0,3,0,0,3,0,0,2,0,10],[16,157,0.1019,0.48659,0.35689,0.24999,0.49979,0.85714,0.0,1.0,7,5,0,7,0,1,0,0,6,0,0,2,0,0,4,0,0,3,0,0,4,0,5],[20,157,0.1274,0.41518,0.32608,0.14286,0.42857,0.60714,0.0,1.0,6,4,0,6,0,5,0,0,4,0,0,5,0,0,4,0,0,3,0,0,1,0,4],[24,157,0.1529,0.51339,0.34416,0.25,0.57143,0.75,0.0,1.0,6,5,0,6,0,2,0,0,3,0,0,3,0,0,5,0,0,5,0,0,3,0,5],[28,157,0.1783,0.45982,0.30249,0.14286,0.57141,0.71429,0.0,1.0,4,2,0,4,0,6,0,0,2,0,0,3,0,0,8,0,0,4,0,0,3,0,2],[32,157,0.2038,0.4375,0.37276,0.10714,0.28571,0.85714,0.0,1.0,8,6,0,8,0,3,0,0,6,0,0,1,0,0,5,0,0,0,0,0,3,0,6],[36,157,0.2293,0.42853,0.31133,0.25001,0.42857,0.60714,0.0,1.0,6,3,0,6,0,2,0,0,7,0,0,4,0,0,5,0,0,3,0,0,2,0,3],[40,157,0.2548,0.44639,0.3025,0.14289,0.28571,0.71429,0.0,1.0,2,2,0,2,0,7,0,0,8,0,0,1,0,0,5,0,0,2,0,0,5,0,2],[44,157,0.2803,0.38823,0.30547,0.14286,0.28571,0.57143,0.0,1.0,6,3,0,6,0,5,0,0,6,0,0,3,0,0,5,0,0,4,0,0,0,0,3],[48,157,0.3057,0.52231,0.35103,0.28571,0.57121,0.78571,0.0,1.0,4,8,0,4,0,3,0,0,7,0,0,1,0,0,4,0,0,5,0,0,0,0,8],[52,157,0.3312,0.48661,0.32115,0.28571,0.42857,0.71429,0.0,1.0,4,5,0,4,0,3,0,0,6,0,0,4,0,0,4,0,0,5,0,0,1,0,5],[56,157,0.3567,0.37052,0.23919,0.24999,0.42857,0.57111,0.0,0.85714,5,0,0,5,0,3,0,0,7,0,0,8,0,0,5,0,0,2,0,0,2,0,0],[60,157,0.3822,0.25447,0.30459,0.0,0.14286,0.4286,0.0,1.0,14,2,0,14,0,4,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,2],[64,157,0.4076,0.31696,0.28061,0.0,0.28571,0.46429,0.0,1.0,9,1,0,9,0,2,0,0,11,0,0,2,0,0,2,0,0,4,0,0,1,0,1],[68,157,0.4331,0.375,0.36553,0.0,0.28571,0.71429,0.0,1.0,12,4,0,12,0,1,0,0,5,0,0,2,0,0,3,0,0,3,0,0,2,0,4],[72,157,0.4586,0.37052,0.30484,0.0,0.42857,0.57111,0.0,1.0,10,2,0,10,0,0,0,0,3,0,0,10,0,0,4,0,0,1,0,0,2,0,2],[76,157,0.4841,0.34374,0.28088,0.14286,0.28571,0.57111,0.0,1.0,7,1,0,7,0,4,0,0,9,0,0,3,0,0,2,0,0,5,0,0,1,0,1],[80,157,0.5096,0.35714,0.29451,0.14286,0.28571,0.57143,0.0,1.0,7,1,0,7,0,6,0,0,4,0,0,5,0,0,3,0,0,4,0,0,2,0,1],[84,157,0.535,0.26344,0.29693,0.0,0.14286,0.32143,0.0,1.0,10,3,0,10,0,8,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[88,157,0.5605,0.38393,0.31224,0.10714,0.42857,0.57143,0.0,1.0,8,3,0,8,0,2,0,0,5,0,0,7,0,0,5,0,0,0,0,0,2,0,3],[92,157,0.586,0.30808,0.25284,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,3,0,0,12,0,0,5,0,0,2,0,0,0,0,0,2,0,1],[96,157,0.6115,0.31695,0.21937,0.14286,0.28571,0.4286,0.0,1.0,4,1,0,4,0,7,0,0,8,0,0,8,0,0,3,0,0,1,0,0,0,0,1],[100,157,0.6369,0.26786,0.26666,0.0,0.2857,0.32143,0.0,0.85714,11,0,0,11,0,3,0,0,10,0,0,2,0,0,1,0,0,3,0,0,2,0,0],[104,157,0.6624,0.38381,0.29116,0.14214,0.35714,0.60714,0.0,0.85714,7,0,0,7,0,4,0,0,5,0,0,3,0,0,5,0,0,5,0,0,3,0,0],[108,157,0.6879,0.26338,0.24249,0.0,0.2143,0.42857,0.0,0.857,9,0,0,9,0,7,0,0,6,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[112,157,0.7134,0.36604,0.28777,0.14286,0.28571,0.4642,0.0,1.0,5,2,0,5,0,5,0,0,9,0,0,5,0,0,3,0,0,0,0,0,3,0,2],[116,157,0.7389,0.33482,0.29366,0.0,0.28571,0.4286,0.0,1.0,9,2,0,9,0,3,0,0,5,0,0,8,0,0,2,0,0,2,0,0,1,0,2],[120,157,0.7643,0.25,0.31135,0.0,0.14288,0.28571,0.0,1.0,15,2,0,15,0,2,0,0,8,0,0,0,0,0,2,0,0,2,0,0,1,0,2],[124,157,0.7898,0.29911,0.26332,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,7,0,0,7,0,0,5,0,0,3,0,0,0,0,0,2,0,1],[128,157,0.8153,0.28124,0.30195,0.0,0.28571,0.42857,0.0,1.0,13,2,0,13,0,2,0,0,6,0,0,4,0,0,2,0,0,3,0,0,0,0,2],[132,157,0.8408,0.22321,0.28107,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,5,0,0,6,0,0,3,0,0,0,0,0,1,0,0,2,0,1],[136,157,0.8662,0.16069,0.17399,0.0,0.14286,0.28571,0.0,0.571,13,0,0,13,0,9,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[140,157,0.8917,0.2634,0.33714,0.0,0.0,0.42858,0.0,1.0,17,3,0,17,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,3],[144,157,0.9172,0.32579,0.31187,0.0,0.28571,0.42858,0.0,1.0,9,2,0,9,0,4,0,0,8,0,0,4,0,0,1,0,0,1,0,0,3,0,2],[148,157,0.9427,0.22768,0.31514,0.0,0.0,0.42857,0.0,1.0,18,2,0,18,0,2,0,0,2,0,0,4,0,0,2,0,0,1,0,0,1,0,2],[152,157,0.9682,0.27678,0.27879,0.0,0.2857,0.42857,0.0,1.0,9,3,0,9,0,5,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[156,157,0.9936,0.25893,0.24856,0.10714,0.14288,0.42857,0.0,1.0,8,1,0,8,0,9,0,0,6,0,0,4,0,0,3,0,0,0,0,0,1,0,1],[157,157,1.0,0.1875,0.22711,0.0,0.07143,0.42857,0.0,0.71429,16,0,0,16,0,4,0,0,3,0,0,6,0,0,1,0,0,2,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"flat","v":0.4866,"x":0.72767,"p":[[0,164,0.0,0.4866,0.36044,0.14286,0.42857,0.85714,0.0,1.0,6,5,0,6,0,4,0,0,3,0,0,4,0,0,3,0,0,2,0,0,5,0,5],[4,164,0.0244,0.54016,0.33831,0.28571,0.49979,0.85714,0.0,1.0,3,7,0,3,0,2,0,0,8,0,0,3,0,0,4,0,0,1,0,0,4,0,7],[8,164,0.0488,0.56696,0.2461,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,5,0,0,6,0,0,5,0,0,9,0,0,2,0,3],[12,164,0.0732,0.60714,0.22587,0.53571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,4,0,0,12,0,0,6,0,0,2,0,4],[16,164,0.0976,0.70534,0.19866,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,9,0,3],[20,164,0.122,0.64283,0.28572,0.53539,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,4,0,0,1,0,0,7,0,0,6,0,0,4,0,7],[24,164,0.1463,0.70533,0.22287,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,2,0,0,7,0,0,10,0,0,5,0,6],[28,164,0.1707,0.63839,0.22299,0.53572,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,4,0,0,7,0,0,11,0,0,1,0,5],[32,164,0.1951,0.54462,0.28669,0.28571,0.57143,0.71429,0.0,1.0,3,3,0,3,0,2,0,0,4,0,0,1,0,0,11,0,0,4,0,0,4,0,3],[36,164,0.2195,0.67411,0.25812,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,1,0,0,6,0,0,9,0,0,5,0,6],[40,164,0.2439,0.64286,0.25754,0.42859,0.71429,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,5,0,0,4,0,0,8,0,0,8,0,3],[44,164,0.2683,0.64732,0.28568,0.57143,0.71429,0.85714,0.0,1.0,2,5,0,2,0,2,0,0,2,0,0,1,0,0,5,0,0,9,0,0,6,0,5],[48,164,0.2927,0.6607,0.23891,0.57142,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,1,0,0,3,0,0,9,0,0,6,0,0,7,0,4],[52,164,0.3171,0.61161,0.212,0.57142,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,4,0,0,13,0,0,6,0,0,3,0,3],[56,164,0.3415,0.58706,0.2878,0.42857,0.57143,0.85714,0.0,1.0,3,3,0,3,0,0,0,0,4,0,0,5,0,0,5,0,0,4,1,0,7,0,3],[60,164,0.3659,0.66069,0.23077,0.5354,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,5,0,0,7,0,0,8,0,0,3,0,6],[64,164,0.3902,0.70536,0.18877,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,10,0,0,11,0,2],[68,164,0.4146,0.6607,0.23891,0.42859,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,7,0,0,6,0,5],[72,164,0.439,0.66071,0.20438,0.57143,0.71429,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,12,0,0,5,0,3],[76,164,0.4634,0.62946,0.22548,0.42859,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,4,0,0,7,0,0,6,0,0,8,0,2],[80,164,0.4878,0.69197,0.27917,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,2,0,0,5,0,0,8,0,0,2,0,10],[84,164,0.5122,0.6763,0.23008,0.57143,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,1,0,0,1,0,0,9,0,0,8,1,0,6,0,4],[88,164,0.5366,0.58034,0.3071,0.39286,0.57143,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,3,0,0,3,0,0,7,0,0,3,0,0,7,0,4],[92,164,0.561,0.63393,0.29653,0.42857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,4,0,0,2,0,0,6,0,0,6,0,0,4,0,7],[96,164,0.5854,0.64061,0.26696,0.42857,0.57143,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,5,0,0,7,1,0,3,0,0,4,0,7],[100,164,0.6098,0.65177,0.2878,0.53539,0.71429,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,2,0,0,2,0,0,5,0,0,8,0,0,4,0,7],[104,164,0.6341,0.63837,0.27661,0.5354,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,4,0,0,2,0,0,7,0,0,7,0,0,4,0,6],[108,164,0.6585,0.62497,0.25191,0.57132,0.57143,0.85702,0.0,1.0,2,3,0,2,0,1,0,0,1,0,0,2,0,0,11,0,0,6,0,0,6,0,3],[112,164,0.6829,0.64731,0.3009,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,4,0,0,3,0,0,5,0,0,5,0,0,3,0,9],[116,164,0.7073,0.59805,0.25358,0.5354,0.57143,0.75,0.0,1.0,2,3,0,2,0,0,0,0,4,0,0,2,0,0,11,0,0,5,0,0,5,0,3],[120,164,0.7317,0.6918,0.26752,0.5354,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,4,0,0,3,0,0,7,0,0,7,0,7],[124,164,0.7561,0.54911,0.28146,0.28571,0.64286,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,5,0,0,4,0,0,3,0,0,10,0,0,4,0,2],[128,164,0.7805,0.60713,0.29451,0.42859,0.64286,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,2,0,0,4,0,0,6,0,0,5,0,0,7,0,4],[132,164,0.8049,0.64284,0.2342,0.5354,0.71429,0.75,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,4,0,0,6,0,0,10,0,0,4,0,4],[136,164,0.8293,0.65179,0.23128,0.42859,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,4,0,0,3,0,0,10,0,0,7,0,3],[140,164,0.8537,0.64732,0.25996,0.57143,0.64286,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,4,0,0,0,0,0,10,0,0,5,0,0,6,0,5],[144,164,0.878,0.65625,0.21975,0.53572,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,6,0,0,5,0,0,10,0,0,6,0,3],[148,164,0.9024,0.58482,0.34876,0.28571,0.64286,0.85714,0.0,1.0,5,5,0,5,0,2,0,0,2,0,0,2,0,0,5,0,0,2,0,0,9,0,5],[152,164,0.9268,0.72767,0.2269,0.57143,0.78571,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,5,0,0,10,0,6],[156,164,0.9512,0.67857,0.26,0.57143,0.71429,0.85714,0.0,1.0,3,3,0,3,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,9,0,3],[160,164,0.9756,0.54462,0.25862,0.39286,0.57121,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,6,0,0,5,0,0,6,0,0,6,0,1],[164,164,1.0,0.58477,0.23517,0.42859,0.57143,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,3,0,0,4,0,0,9,0,0,9,0,0,3,0,2]]}]},{"i":"94dafa682b7bf149","q":"Let  $A$ ,  $B$  and  $C$  be three points on a line (in this order).\nFor each circle  $k$  through the points  $B$  and  $C$ , let  $D$  be one point of intersection of the perpendicular bisector of  $BC$  with the circle  $k$ . Further, let  $E$  be the second point of intersection of the line  $AD$  with  $k$ .\nShow that for each circle  $k$ , the ratio of lengths  $\\overline{BE}:\\overline{CE}$  is the same.","t":[{"b":3,"e":0.71429,"k":"falling","v":0.29908,"x":0.79464,"p":[[0,117,0.0,0.79464,0.3387,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,22],[4,117,0.0342,0.54017,0.39566,0.14286,0.57121,1.0,0.14286,1.0,0,12,0,0,0,15,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,12],[8,117,0.0684,0.61159,0.37155,0.14286,0.78564,1.0,0.14286,1.0,0,10,0,0,0,11,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,10],[12,117,0.1026,0.61604,0.38537,0.14289,0.71421,1.0,0.0,1.0,1,13,0,1,0,10,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,13],[16,117,0.1368,0.65175,0.3676,0.14286,0.85714,1.0,0.0,1.0,1,13,0,1,0,8,0,0,0,0,0,1,0,0,5,0,0,0,0,0,4,0,13],[20,117,0.1709,0.48211,0.37414,0.14286,0.35714,1.0,0.0,1.0,1,9,0,1,0,14,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,9],[24,117,0.2051,0.5981,0.39529,0.14286,0.71429,1.0,0.0,1.0,2,12,0,2,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,12],[28,117,0.2393,0.66963,0.36147,0.46396,0.85714,1.0,0.0,1.0,2,13,0,2,0,6,0,0,0,0,0,0,0,0,6,0,0,1,0,0,4,0,13],[32,117,0.2735,0.61596,0.3969,0.14286,0.71429,1.0,0.0,1.0,1,15,0,1,0,10,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,15],[36,117,0.3077,0.50445,0.37624,0.14286,0.42857,0.85714,0.0,1.0,1,7,0,1,0,14,0,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,7],[40,117,0.3419,0.75892,0.37019,0.46397,1.0,1.0,0.14286,1.0,0,22,0,0,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,22],[44,117,0.3761,0.5356,0.40256,0.14286,0.4998,1.0,0.0,1.0,1,12,0,1,0,14,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,12],[48,117,0.4103,0.72311,0.35896,0.35716,0.92857,1.0,0.14,1.0,0,16,0,0,0,8,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,16],[52,117,0.4444,0.67411,0.37836,0.14289,0.85714,1.0,0.14286,1.0,0,15,0,0,0,10,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,15],[56,117,0.4786,0.76339,0.36001,0.5,1.0,1.0,0.14286,1.0,0,21,0,0,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,21],[60,117,0.5128,0.73658,0.36442,0.46397,1.0,1.0,0.14286,1.0,0,19,0,0,0,8,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,19],[64,117,0.547,0.71874,0.35263,0.39285,1.0,1.0,0.14286,1.0,0,17,0,0,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0,3,0,17],[68,117,0.5812,0.70088,0.37689,0.14286,0.92857,1.0,0.0,1.0,1,16,0,1,0,8,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,16],[72,117,0.6154,0.6606,0.37428,0.14289,0.85714,1.0,0.14,1.0,0,14,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,14],[76,117,0.6496,0.61603,0.37191,0.14286,0.71429,1.0,0.0,1.0,1,11,0,1,0,9,0,0,1,0,0,0,0,0,5,0,0,0,0,0,5,0,11],[80,117,0.6838,0.79464,0.32915,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,20],[84,117,0.7179,0.76338,0.35285,0.42859,1.0,1.0,0.14286,1.0,0,21,0,0,0,6,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,21],[88,117,0.7521,0.71419,0.38478,0.14289,1.0,1.0,0.0,1.0,1,18,0,1,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,18],[92,117,0.7863,0.54018,0.39566,0.14286,0.57143,1.0,0.0,1.0,4,9,0,4,0,9,0,0,0,0,0,1,0,0,4,0,0,0,0,0,5,0,9],[96,117,0.8205,0.59373,0.39141,0.14286,0.71429,1.0,0.0,1.0,4,11,0,4,0,6,0,0,1,0,0,2,0,0,2,0,0,2,0,0,4,0,11],[100,117,0.8547,0.62042,0.3654,0.14286,0.85714,1.0,0.0,1.0,2,9,0,2,0,7,0,0,1,0,0,1,0,0,4,0,0,0,0,0,8,0,9],[104,117,0.8889,0.52228,0.37561,0.14286,0.57121,1.0,0.0,1.0,5,9,0,5,0,6,0,0,1,0,0,1,0,0,8,0,0,1,0,0,1,0,9],[108,117,0.9231,0.50446,0.38129,0.14286,0.4998,1.0,0.0,1.0,3,10,0,3,0,10,0,0,1,0,0,2,0,0,5,0,0,1,0,0,0,0,10],[112,117,0.9573,0.47769,0.38566,0.14286,0.35714,0.85714,0.0,1.0,4,6,0,4,0,11,0,0,1,0,0,1,0,0,2,0,0,1,0,0,6,0,6],[116,117,0.9915,0.40609,0.36268,0.14214,0.1429,0.71429,0.0,1.0,7,3,0,7,0,10,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,3],[117,117,1.0,0.29908,0.34873,0.0,0.14286,0.4642,0.0,1.0,12,2,0,12,0,7,0,0,3,0,0,2,0,0,1,0,0,0,0,0,5,0,2]]},{"b":5,"e":0.4286,"k":"falling","v":0.31239,"x":0.75446,"p":[[0,61,0.0,0.75446,0.35217,0.57132,1.0,1.0,0.0,1.0,1,20,0,1,0,5,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,20],[4,61,0.0656,0.5625,0.39437,0.14286,0.57143,1.0,0.14286,1.0,0,12,0,0,0,14,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,12],[8,61,0.1311,0.62498,0.37923,0.14289,0.71429,1.0,0.14286,1.0,0,14,0,0,0,10,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,14],[12,61,0.1967,0.4955,0.39606,0.14286,0.42835,1.0,0.0,1.0,3,10,0,3,0,12,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,10],[16,61,0.2623,0.45086,0.36614,0.14286,0.35714,0.67857,0.0,1.0,3,8,0,3,0,12,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,8],[20,61,0.3279,0.42844,0.3572,0.14286,0.14286,0.64286,0.0,1.0,1,7,0,1,0,16,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,7],[24,61,0.3934,0.57132,0.37296,0.14286,0.57143,1.0,0.0,1.0,1,10,0,1,0,11,0,0,0,0,0,0,0,0,6,0,0,1,0,0,3,0,10],[28,61,0.459,0.39732,0.33068,0.14286,0.14288,0.71429,0.0,1.0,1,3,0,1,0,17,0,0,1,0,0,1,0,0,3,0,0,2,0,0,4,0,3],[32,61,0.5246,0.41955,0.35888,0.14286,0.14286,0.64286,0.0,1.0,1,7,0,1,0,17,0,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,7],[36,61,0.5902,0.42847,0.37463,0.14286,0.14286,0.89275,0.14,1.0,0,8,0,0,0,19,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,8],[40,61,0.6557,0.31691,0.32086,0.14286,0.14286,0.571,0.0,1.0,3,4,0,3,0,19,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,4],[44,61,0.7213,0.33024,0.27538,0.14286,0.14286,0.57143,0.0,1.0,4,2,0,4,0,13,0,0,2,0,0,2,0,0,8,0,0,1,0,0,0,0,2],[48,61,0.7869,0.31694,0.34759,0.14286,0.14286,0.571,0.0,1.0,7,4,0,7,0,14,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,4],[52,61,0.8525,0.31249,0.33203,0.14286,0.14286,0.57111,0.0,1.0,7,4,0,7,0,14,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,4],[56,61,0.918,0.31239,0.30399,0.14214,0.14286,0.57111,0.0,1.0,7,3,0,7,0,12,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,3],[60,61,0.9836,0.47758,0.38075,0.14286,0.42857,0.89286,0.0,1.0,6,8,0,6,0,7,0,0,0,0,0,4,0,0,4,0,0,2,0,0,1,0,8],[61,61,1.0,0.42409,0.2912,0.14286,0.42859,0.57143,0.0,1.0,6,2,0,6,0,4,0,0,1,0,0,6,0,0,9,0,0,3,0,0,1,0,2]]}]},{"i":"5322d38fe04777e3","q":"In acute-angled triangle  $ABC, I$  is the center of the inscribed circle and holds  $| AC | + | AI | = | BC |$ . Prove that  $\\angle BAC = 2 \\angle ABC$ .","t":[{"b":4,"e":0.14,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,8,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.05804,"p":[[0,29,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.05804,0.17807,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,29,0.2759,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"695b525e3ea58408","q":"Let  $ABC$  be a triangle and  $A'$  be the reflection of  $A$  about  $BC$ . Let  $P$  and  $Q$  be points on  $AB$  and  $AC$ , respectively, such that  $PA'=PC$  and  $QA'=QB$ . Prove that the perpendicular from  $A'$  to  $PQ$  passes through the circumcenter of  $\\triangle ABC$ .\n\n*Fedir Yudin*","t":[{"b":4,"e":0.0,"k":"flat","v":0.08482,"x":0.1875,"p":[[0,109,0.0,0.16956,0.15337,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,11,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,109,0.0367,0.17857,0.17496,0.0,0.14286,0.42857,0.0,0.42857,12,0,0,12,0,9,0,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,109,0.0734,0.13393,0.17105,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,10,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[12,109,0.1101,0.18304,0.18638,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,11,0,0,2,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[16,109,0.1468,0.16071,0.18814,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,13,0,0,2,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[20,109,0.1835,0.1383,0.15355,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,13,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,109,0.2202,0.15179,0.17474,0.0,0.14286,0.1786,0.0,0.57143,14,0,0,14,0,10,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[28,109,0.2569,0.11607,0.15336,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,9,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,109,0.2936,0.11161,0.13709,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,15,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,109,0.3303,0.15625,0.17985,0.0,0.14286,0.21429,0.0,0.57143,14,0,0,14,0,10,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[40,109,0.367,0.16071,0.17405,0.0,0.14286,0.32143,0.0,0.42857,14,0,0,14,0,8,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[44,109,0.4037,0.1875,0.20025,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,8,0,0,1,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[48,109,0.4404,0.16518,0.17169,0.0,0.14286,0.32143,0.0,0.42857,13,0,0,13,0,9,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[52,109,0.4771,0.12054,0.17169,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,6,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,109,0.5138,0.11607,0.18013,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,7,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[60,109,0.5505,0.16955,0.22429,0.0,0.14143,0.32143,0.0,1.0,15,1,0,15,0,8,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[64,109,0.5872,0.10268,0.1439,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,109,0.6239,0.16518,0.16793,0.0,0.14286,0.32143,0.0,0.42857,12,0,0,12,0,11,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[72,109,0.6606,0.10715,0.14286,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,10,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,109,0.6972,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,7,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[80,109,0.7339,0.14737,0.16172,0.0,0.14286,0.14286,0.0,0.43,13,0,0,13,0,12,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[84,109,0.7706,0.13393,0.16728,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,11,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[88,109,0.8073,0.12054,0.14773,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[92,109,0.844,0.11161,0.16263,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[96,109,0.8807,0.15625,0.16888,0.0,0.14286,0.21432,0.0,0.42857,13,0,0,13,0,11,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[100,109,0.9174,0.1384,0.16554,0.0,0.14286,0.14287,0.0,0.4286,15,0,0,15,0,10,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[104,109,0.9541,0.11161,0.1504,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[108,109,0.9908,0.12054,0.16016,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,9,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[109,109,1.0,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,10,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.571,"k":"rising","v":0.10268,"x":0.33928,"p":[[0,70,0.0,0.14732,0.14934,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,11,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,70,0.0571,0.16964,0.20025,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,7,0,0,1,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[8,70,0.1143,0.10268,0.16065,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,10,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,70,0.1714,0.12054,0.15612,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,10,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,70,0.2286,0.21429,0.19561,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,9,0,0,3,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[20,70,0.2857,0.15625,0.17261,0.0,0.14286,0.17857,0.0,0.57143,13,0,0,13,0,11,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[24,70,0.3429,0.21875,0.22299,0.0,0.14286,0.42857,0.0,0.57143,13,0,0,13,0,5,0,0,4,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[28,70,0.4,0.2232,0.21996,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,7,0,0,1,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[32,70,0.4571,0.29454,0.23945,0.0,0.2857,0.46418,0.0,0.71429,9,0,0,9,0,5,0,0,3,0,0,7,0,0,6,0,0,2,0,0,0,0,0],[36,70,0.5143,0.27676,0.24465,0.0,0.21428,0.42857,0.0,0.85714,10,0,0,10,0,6,0,0,1,0,0,8,0,0,6,0,0,0,0,0,1,0,0],[40,70,0.5714,0.25893,0.23266,0.0,0.21431,0.42857,0.0,0.71429,11,0,0,11,0,5,0,0,2,0,0,8,0,0,5,0,0,1,0,0,0,0,0],[44,70,0.6286,0.33928,0.19805,0.14286,0.42857,0.42857,0.0,0.71429,5,0,0,5,0,4,0,0,4,0,0,13,0,0,5,0,0,1,0,0,0,0,0],[48,70,0.6857,0.26784,0.21649,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,4,0,0,3,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[52,70,0.7429,0.27223,0.20006,0.14286,0.21428,0.42857,0.0,0.57143,6,0,0,6,0,10,0,0,2,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[56,70,0.8,0.33034,0.21259,0.14286,0.42857,0.46418,0.0,0.57143,6,0,0,6,0,6,0,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0],[60,70,0.8571,0.24103,0.21254,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,7,0,0,3,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[64,70,0.9143,0.27679,0.2257,0.10714,0.21431,0.57143,0.0,0.57143,8,0,0,8,0,8,0,0,3,0,0,4,0,0,9,0,0,0,0,0,0,0,0],[68,70,0.9714,0.24098,0.20963,0.0,0.14286,0.42857,0.0,0.57143,9,0,0,9,0,9,0,0,2,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[70,70,1.0,0.31695,0.21348,0.10714,0.42857,0.4286,0.0,0.57143,8,0,0,8,0,3,0,0,1,0,0,14,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"19f22c4b3814afed","q":"Let  $AB$  be a diameter of the circunference  $\\omega$ , let  $C$  and  $D$  be point in this circunference, such that  $CD$  is perpedicular to  $AB$ . Let  $E$  be the point of intersection of the segment  $CD$  and the segment  $AB$ , and a point  $P$  that is in the segment  $CD, P$  is different of  $E$ . The lines  $AP$  and  $BP$  intersects  $\\omega$ , in  $F$  and  $G$  respectively. If  $O$  is the circumcenter of triangle  $EFG$ , show that the area of triangle  $OCD$  is invariant, independent of the position of the point  $P$ 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,2,0,23],[244,248,0.9839,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[248,248,1.0,0.94196,0.15916,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28]]}]},{"i":"8114f3d5ae75a9a8","q":"Let  $ABC$  be a triangle with  $AB > AC$  with incenter  $I{}$ . The internal bisector of the angle  $BAC$  intersects the  $BC$  at the point  $D{}$ . Let  $M{}$  the midpoint of the segment  $AD{}$ , and let  $F{}$  be the second intersection point of  $MB$  with the circumcircle of the triangle  $BIC$ . Prove that  $AF$  is perpendicular to  $FC$ 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 $ x$ ,  $ y$ , and  $ z$  be integers satisfying the equation \\[ \\dfrac{2008}{41y^2}\\equal{}\\dfrac{2z}{2009}\\plus{}\\dfrac{2007}{2x^2}.\\] Determine the greatest value that  $ z$  can take.\r\n*Budi Surodjo, Jogjakarta*","t":[{"b":4,"e":1.0,"k":"rising","v":0.66518,"x":0.95982,"p":[[0,205,0.0,0.70089,0.33381,0.57143,0.71429,1.0,0.0,1.0,4,14,1,4,0,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0,1,0,14],[4,205,0.0195,0.84822,0.27879,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,22],[8,205,0.039,0.88392,0.24856,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,21],[12,205,0.0585,0.88392,0.24856,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,21],[16,205,0.078,0.85268,0.29555,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,23],[20,205,0.0976,0.90178,0.20652,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,22],[24,205,0.1171,0.88393,0.24074,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,21],[28,205,0.1366,0.86159,0.29557,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,23],[32,205,0.1561,0.83482,0.32559,0.85714,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,21],[36,205,0.1756,0.79908,0.31108,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,18],[40,205,0.1951,0.79464,0.32525,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,18],[44,205,0.2146,0.87945,0.2746,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,25],[48,205,0.2341,0.80357,0.27607,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,8,0,15],[52,205,0.2537,0.83036,0.31428,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,21],[56,205,0.2732,0.80356,0.30463,0.78571,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,0,6,0,18],[60,205,0.2927,0.66518,0.35644,0.42857,0.78571,1.0,0.0,1.0,4,13,0,4,0,1,0,0,1,0,0,5,0,0,3,0,0,2,0,0,3,0,13],[64,205,0.3122,0.83926,0.22802,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,0,4,0,18],[68,205,0.3317,0.80345,0.31309,0.82132,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,18],[72,205,0.3512,0.7991,0.3131,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,3,0,19],[76,205,0.3707,0.87946,0.25533,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,22],[80,205,0.3902,0.81696,0.31387,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,21],[84,205,0.4098,0.76339,0.33619,0.71429,1.0,1.0,0.0,1.0,4,17,0,4,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,17],[88,205,0.4293,0.91517,0.18851,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,21],[92,205,0.4488,0.76339,0.37048,0.71429,1.0,1.0,0.0,1.0,5,18,0,5,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,18],[96,205,0.4683,0.79018,0.32534,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,18],[100,205,0.4878,0.79464,0.31731,0.67857,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,19],[104,205,0.5073,0.7545,0.33353,0.57132,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,0,0,0,3,0,0,3,0,0,2,0,0,3,0,17],[108,205,0.5268,0.70535,0.3976,0.53571,0.92857,1.0,0.0,1.0,7,16,0,7,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,16],[112,205,0.5463,0.79463,0.32329,0.67857,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,19],[116,205,0.5659,0.78125,0.34807,0.67857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,20],[120,205,0.5854,0.875,0.28738,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,24],[124,205,0.6049,0.74999,0.36247,0.57132,1.0,1.0,0.0,1.0,4,19,0,4,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,19],[128,205,0.6244,0.78571,0.38299,0.82143,1.0,1.0,0.0,1.0,5,23,0,5,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[132,205,0.6439,0.86161,0.25626,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,21],[136,205,0.6634,0.85714,0.29014,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,24],[140,205,0.6829,0.82589,0.28288,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,19],[144,205,0.7024,0.90179,0.17655,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,1,0,23],[148,205,0.722,0.92857,0.16366,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,26],[152,205,0.7415,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[156,205,0.761,0.91518,0.13767,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,23],[160,205,0.7805,0.92857,0.13832,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,25],[164,205,0.8,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[168,205,0.8195,0.92856,0.13836,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,25],[172,205,0.839,0.87945,0.18598,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,1,0,21],[176,205,0.8585,0.92411,0.13825,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,24],[180,205,0.878,0.92857,0.14286,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,25],[184,205,0.8976,0.91964,0.15542,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,25],[188,205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 $ABC$  be a triangle with  $AB=5$ ,  $BC=6$ ,  $CA=7$ . Let  $D$  be a point on ray  $AB$  beyond  $B$  such that  $BD=7$ ,  $E$  be a point on ray  $BC$  beyond  $C$  such that  $CE=5$ , and  $F$  be a point on ray  $CA$  beyond  $A$  such that  $AF=6$ . Compute the area of the circumcircle of  $DEF$ .\n\n*Proposed by James Lin.*","t":[{"b":4,"e":0.14286,"k":"flat","v":0.1383,"x":0.22759,"p":[[0,88,0.0,0.22759,0.15515,0.14286,0.14286,0.32143,0.0,0.57143,3,0,0,3,0,17,0,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[4,88,0.0455,0.16062,0.05925,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,88,0.0909,0.18304,0.1684,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,88,0.1364,0.1875,0.14032,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,25,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[16,88,0.1818,0.17857,0.11294,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,28,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,88,0.2273,0.16071,0.06916,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,88,0.2727,0.20089,0.19186,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,25,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[28,88,0.3182,0.16955,0.09742,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,27,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,88,0.3636,0.17411,0.08552,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,88,0.4091,0.16964,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,88,0.4545,0.20536,0.19541,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[44,88,0.5,0.20089,0.16311,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[48,88,0.5455,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,88,0.5909,0.18295,0.14393,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,27,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[56,88,0.6364,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,88,0.6818,0.18295,0.0961,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[64,88,0.7273,0.15179,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,88,0.7727,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,88,0.8182,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,88,0.8636,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,88,0.9091,0.15616,0.06548,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,88,0.9545,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,88,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.01786,"x":0.24545,"p":[[0,112,0.0,0.21875,0.14279,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,17,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,112,0.0357,0.24545,0.22938,0.14286,0.14286,0.17857,0.14,1.0,0,2,0,0,0,24,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[8,112,0.0714,0.16964,0.08328,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,112,0.1071,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,112,0.1429,0.18294,0.11424,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,26,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,112,0.1786,0.17848,0.1288,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,24,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,112,0.2143,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,112,0.25,0.16518,0.15612,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,23,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[32,112,0.2857,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,112,0.3214,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,112,0.3571,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,112,0.3929,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,112,0.4286,0.15179,0.14698,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[52,112,0.4643,0.20089,0.19186,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,21,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[56,112,0.5,0.13839,0.07563,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,112,0.5357,0.15179,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[64,112,0.5714,0.17411,0.13236,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,23,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[68,112,0.6071,0.12937,0.07455,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,112,0.6429,0.16071,0.12242,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,22,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[76,112,0.6786,0.15179,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[80,112,0.7143,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,112,0.75,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,112,0.7857,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,112,0.8214,0.1383,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[96,112,0.8571,0.16072,0.05923,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,112,0.8929,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,112,0.9286,0.13839,0.07563,0.14286,0.14286,0.14286,0.0,0.4286,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,112,0.9643,0.10268,0.08917,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[112,112,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b610d90b7d234c00","q":"Let  $ABC$  be a triangle with  $H$  its orthocenter. The circle with diameter  $[AC]$  cuts the circumcircle of triangle  $ABH$  at  $K$ . Prove that the point of intersection of the lines  $CK$  and  $BH$  is the midpoint of the segment  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 $ABC$  be a triangle, and let  $D$  be the foot of the  $A-$ altitude. Points  $P, Q$  are chosen on  $BC$  such that  $DP = DQ = DA$ . Suppose  $AP$  and  $AQ$  intersect the circumcircle of  $ABC$  again at  $X$  and  $Y$ . Prove that the perpendicular bisectors of the lines  $PX$ ,  $QY$ , and  $BC$  are concurrent.\n\n*Proposed by Pranjal Srivastava*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.12947,"x":0.90625,"p":[[0,104,0.0,0.90625,0.15815,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,23],[4,104,0.0385,0.82143,0.23146,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,4,0,15],[8,104,0.0769,0.71428,0.27432,0.71429,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,1,0,0,1,0,0,1,0,0,15,0,0,2,0,9],[12,104,0.1154,0.84821,0.15947,0.71429,0.85712,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,3,0,15],[16,104,0.1538,0.75,0.23419,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,13,0,0,4,0,9],[20,104,0.1923,0.69197,0.2372,0.57143,0.71429,0.75,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,4,0,0,3,0,0,15,0,0,1,0,7],[24,104,0.2308,0.77679,0.2111,0.71429,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,1,0,11],[28,104,0.2692,0.64729,0.31539,0.57132,0.71429,0.89286,0.0,1.0,3,8,0,3,0,3,0,0,0,0,0,0,0,0,6,0,0,10,0,0,2,0,8],[32,104,0.3077,0.74106,0.29111,0.67857,0.71429,1.0,0.0,1.0,1,13,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,9,0,0,2,0,13],[36,104,0.3462,0.78125,0.3214,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,2,0,0,0,0,0,1,0,0,5,0,0,1,0,19],[40,104,0.3846,0.70089,0.29529,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,3,0,0,1,0,0,2,0,0,3,0,0,8,0,0,4,0,10],[44,104,0.4231,0.6875,0.35072,0.53571,0.71429,1.0,0.0,1.0,3,14,0,3,0,3,0,0,1,0,0,1,0,0,2,0,0,8,0,0,0,0,14],[48,104,0.4615,0.59375,0.33141,0.35714,0.71429,0.85714,0.0,1.0,3,6,0,3,0,5,0,0,0,0,0,3,0,0,1,0,0,11,0,0,3,0,6],[52,104,0.5,0.57134,0.38143,0.24999,0.64286,1.0,0.0,1.0,5,11,0,5,0,3,0,0,4,0,0,1,0,0,3,0,0,5,0,0,0,0,11],[56,104,0.5385,0.69633,0.28083,0.67857,0.71429,1.0,0.14,1.0,0,10,0,0,0,4,0,0,2,0,0,0,0,0,2,0,0,14,0,0,0,0,10],[60,104,0.5769,0.72768,0.32996,0.57143,0.85714,1.0,0.14286,1.0,0,16,0,0,0,6,0,0,1,0,0,0,0,0,2,0,0,7,0,0,0,0,16],[64,104,0.6154,0.65179,0.31122,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,3,0,0,5,0,0,2,0,0,6,0,0,1,0,11],[68,104,0.6538,0.53572,0.35174,0.14286,0.57143,0.89286,0.0,1.0,2,8,0,2,0,8,0,0,2,0,0,3,0,0,3,0,0,5,0,0,1,0,8],[72,104,0.6923,0.43749,0.37616,0.14286,0.2143,0.71429,0.0,1.0,5,7,0,5,0,11,0,0,1,0,0,0,0,0,4,0,0,4,0,0,0,0,7],[76,104,0.7308,0.46427,0.36943,0.14286,0.35714,0.75,0.0,1.0,3,7,0,3,0,12,0,0,1,0,0,1,0,0,3,0,0,4,0,0,1,0,7],[80,104,0.7692,0.49551,0.33116,0.14286,0.4998,0.71429,0.0,1.0,2,5,0,2,0,9,0,0,2,0,0,3,0,0,2,0,0,8,0,0,1,0,5],[84,104,0.8077,0.47768,0.39707,0.14286,0.42857,1.0,0.0,1.0,6,9,0,6,0,8,0,0,1,0,0,3,0,0,0,0,0,5,0,0,0,0,9],[88,104,0.8462,0.34821,0.38785,0.0,0.14286,0.71429,0.0,1.0,11,6,0,11,0,8,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,6],[92,104,0.8846,0.51339,0.34967,0.14289,0.64286,0.71429,0.0,1.0,4,6,0,4,0,5,0,0,5,0,0,1,0,0,1,0,0,9,0,0,1,0,6],[96,104,0.9231,0.1875,0.24338,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,18,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,1],[100,104,0.9615,0.24554,0.27487,0.0,0.14286,0.32144,0.0,1.0,9,1,0,9,0,13,0,0,2,0,0,1,0,0,3,0,0,2,0,0,1,0,1],[104,104,1.0,0.12947,0.16506,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,19,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0]]},{"b":1,"e":0.71429,"k":"flat","v":0.68302,"x":0.87946,"p":[[0,56,0.0,0.87946,0.18935,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,20],[4,56,0.0714,0.8125,0.17655,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,0,2,0,13],[8,56,0.1429,0.84375,0.13997,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,4,0,13],[12,56,0.2143,0.73659,0.25282,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,1,0,0,1,0,0,15,0,0,3,0,9],[16,56,0.2857,0.78571,0.26,0.71429,0.85707,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,8,0,0,2,0,15],[20,56,0.3571,0.79465,0.27417,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,8,0,0,1,0,17],[24,56,0.4286,0.77231,0.24965,0.67857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,9,0,0,1,0,14],[28,56,0.5,0.80803,0.17353,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,15,0,0,2,0,12],[32,56,0.5714,0.77232,0.23381,0.67857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,9,0,0,2,0,13],[36,56,0.6429,0.76337,0.16217,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,19,0,0,1,0,8],[40,56,0.7143,0.69196,0.2055,0.71429,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,19,0,0,4,0,3],[44,56,0.7857,0.68302,0.24676,0.57143,0.71429,0.71429,0.0,1.0,2,6,0,2,0,1,0,0,0,0,0,0,0,0,6,0,0,16,0,0,1,0,6],[48,56,0.8571,0.70089,0.25843,0.57143,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,0,0,0,3,0,0,6,0,0,11,0,0,1,0,9],[52,56,0.9286,0.70536,0.16728,0.71429,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,23,0,0,0,0,4],[56,56,1.0,0.74554,0.09932,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,4]]}]},{"i":"c16c000f7325d09b","q":"Let  $ABC$  be an acute triangle and  $D$  an interior point of segment  $BC$ . Points  $E$  and  $F$  lie in the half-plane determined by the line  $BC$  containing  $A$  such that  $DE$  is perpendicular to  $BE$  and  $DE$  is tangent to the circumcircle of  $ACD$ , while  $DF$  is perpendicular to  $CF$  and  $DF$  is tangent to the circumcircle of  $ABD$ . Prove that the points  $A, D, E$  and  $F$  are concyclic.","t":[{"b":3,"e":0.571,"k":"rising","v":0.19643,"x":0.39732,"p":[[0,97,0.0,0.20536,0.15947,0.0,0.28571,0.28571,0.0,0.57143,11,0,0,11,0,0,0,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,97,0.0412,0.27678,0.20497,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,0,0,0,14,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[8,97,0.0825,0.21875,0.17491,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,2,0,0,16,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,97,0.1237,0.20982,0.20198,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,1,0,0,13,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[16,97,0.1649,0.19643,0.14617,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,97,0.2062,0.3125,0.16917,0.28571,0.28571,0.32143,0.0,0.57143,4,0,0,4,0,1,0,0,19,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[24,97,0.2474,0.23214,0.15872,0.10714,0.2857,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,18,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[28,97,0.2887,0.24553,0.17582,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,17,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[32,97,0.3299,0.23214,0.14617,0.14289,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,2,0,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,97,0.3711,0.25,0.20516,0.0,0.28571,0.32143,0.0,0.57143,10,0,0,10,0,2,0,0,12,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[40,97,0.4124,0.25,0.19561,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,0,0,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[44,97,0.4536,0.28571,0.17128,0.2857,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,0,0,0,19,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[48,97,0.4948,0.25,0.19885,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,0,0,0,16,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[52,97,0.5361,0.25001,0.11845,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,3,0,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,97,0.5773,0.26784,0.21051,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,1,0,0,15,0,0,0,0,0,6,0,0,1,0,0,0,0,0],[60,97,0.6186,0.31696,0.17399,0.2857,0.28571,0.35714,0.0,0.57143,4,0,0,4,0,1,0,0,19,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[64,97,0.6598,0.27677,0.17102,0.2857,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,1,0,0,19,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[68,97,0.701,0.29911,0.22689,0.0,0.28571,0.57143,0.0,0.71429,9,0,0,9,0,1,0,0,10,0,0,3,0,0,8,0,0,1,0,0,0,0,0],[72,97,0.7423,0.26339,0.21162,0.0,0.2857,0.35714,0.0,0.57143,9,0,0,9,0,3,0,0,12,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[76,97,0.7835,0.34821,0.12846,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,21,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[80,97,0.8247,0.29464,0.20806,0.10714,0.28571,0.46431,0.0,0.57143,8,0,0,8,0,1,0,0,12,0,0,3,0,0,8,0,0,0,0,0,0,0,0],[84,97,0.866,0.25889,0.18701,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[88,97,0.9072,0.39732,0.15866,0.28571,0.42857,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,12,0,0,7,0,0,11,0,0,0,0,0,0,0,0],[92,97,0.9485,0.24999,0.20201,0.0,0.28571,0.32144,0.0,0.57143,9,0,0,9,0,4,0,0,11,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[96,97,0.9897,0.33035,0.17655,0.2857,0.28571,0.4643,0.0,0.57143,4,0,0,4,0,1,0,0,16,0,0,3,0,0,8,0,0,0,0,0,0,0,0],[97,97,1.0,0.36157,0.16355,0.2857,0.28571,0.571,0.0,0.57143,2,0,0,2,0,1,0,0,17,0,0,2,0,0,10,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.12491,"x":0.33482,"p":[[0,116,0.0,0.25,0.12372,0.2857,0.28571,0.28571,0.0,0.57143,5,0,2,5,0,1,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,116,0.0345,0.31251,0.19377,0.2857,0.28571,0.57143,0.0,0.57143,6,0,0,6,0,0,0,0,17,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[8,116,0.069,0.25893,0.18707,0.0,0.28571,0.32143,0.0,0.57143,9,0,0,9,0,0,0,0,15,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[12,116,0.1034,0.32588,0.13937,0.28571,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,0,0,0,23,0,0,1,0,0,6,0,0,0,0,0,0,0,0],[16,116,0.1379,0.24553,0.17215,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,16,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[20,116,0.1724,0.28125,0.2004,0.21427,0.28571,0.32143,0.0,0.71429,8,0,0,8,0,0,0,0,16,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[24,116,0.2069,0.31249,0.2065,0.2857,0.28571,0.57143,0.0,0.57143,7,0,0,7,0,0,0,0,15,0,0,0,0,0,10,0,0,0,0,0,0,0,0],[28,116,0.2414,0.26784,0.17403,0.2857,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,0,0,0,20,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[32,116,0.2759,0.33482,0.18766,0.2857,0.28571,0.57143,0.0,0.57143,5,0,0,5,0,0,0,0,15,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[36,116,0.3103,0.23214,0.19149,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,2,0,0,15,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[40,116,0.3448,0.20982,0.19556,0.0,0.2857,0.28571,0.0,0.57143,13,0,1,13,0,0,0,0,14,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[44,116,0.3793,0.27678,0.18189,0.2857,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,0,0,0,19,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[48,116,0.4138,0.24553,0.18638,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,1,0,0,17,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[52,116,0.4483,0.24552,0.16455,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,3,0,0,17,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[56,116,0.4828,0.22321,0.18189,0.0,0.28571,0.28571,0.0,0.57143,11,0,0,11,0,0,0,0,16,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[60,116,0.5172,0.23214,0.18472,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,1,0,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[64,116,0.5517,0.21875,0.15965,0.0,0.2857,0.28571,0.0,0.57143,9,0,0,9,0,2,0,0,18,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[68,116,0.5862,0.14732,0.16935,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,1,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[72,116,0.6207,0.20089,0.20782,0.0,0.21428,0.28571,0.0,0.57143,14,0,0,14,0,2,0,0,10,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[76,116,0.6552,0.22321,0.17473,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,1,0,0,17,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[80,116,0.6897,0.18303,0.12992,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,3,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,116,0.7241,0.19643,0.14616,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,116,0.7586,0.17848,0.12879,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,4,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,116,0.7931,0.14732,0.14053,0.0,0.2143,0.28571,0.0,0.28571,15,0,0,15,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,116,0.8276,0.20089,0.17445,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,3,0,0,15,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[100,116,0.8621,0.19197,0.18423,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,2,0,0,13,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[104,116,0.8966,0.1875,0.16917,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,0,0,0,17,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[108,116,0.931,0.12491,0.13715,0.0,0.0,0.28571,0.0,0.28571,17,0,0,17,0,2,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,116,0.9655,0.24998,0.15969,0.14286,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,4,0,0,17,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[116,116,1.0,0.14286,0.13832,0.0,0.14288,0.28571,0.0,0.28571,15,0,0,15,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e87f281f5f41f833","q":"Let  $ABC$  be an acute triangle. Let  $D$ ,  $E$  and  $F$  be the feet of the altitudes from  $A$ ,  $B$  and  $C$   respectively and let  $H$  be the orthocenter of  $\\triangle ABC$ . Let  $X$  be an arbitrary point on the circumcircle of  $\\triangle DEF$  and let the circumcircles of  $\\triangle EHX$  and  $\\triangle FHX$  intersect the second time the lines  $CF$  and  $BE$   second  at  $Y$  and  $Z$ , respectively. Prove that the line  $YZ$  passes through the midpoint of  $BC$ .","t":[{"b":1,"e":0.42857,"k":"rising","v":0.11598,"x":0.36155,"p":[[0,77,0.0,0.11598,0.11536,0.0,0.14286,0.1786,0.0,0.28571,14,0,2,14,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.33928,0.19148,0.25,0.42857,0.4286,0.0,0.57143,5,0,0,5,0,3,0,0,6,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[8,77,0.1039,0.34374,0.18508,0.24999,0.42857,0.4286,0.0,0.57143,4,0,0,4,0,4,0,0,6,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[12,77,0.1558,0.33033,0.18704,0.24999,0.28571,0.46418,0.0,0.57143,4,0,0,4,0,4,0,0,10,0,0,6,0,0,8,0,0,0,0,0,0,0,0],[16,77,0.2078,0.33472,0.19441,0.14289,0.42857,0.4286,0.0,0.57143,5,0,0,5,0,4,0,0,5,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[20,77,0.2597,0.29015,0.19057,0.14286,0.2857,0.42858,0.0,0.57143,4,0,0,4,0,10,0,0,5,0,0,7,0,0,6,0,0,0,0,0,0,0,0],[24,77,0.3117,0.29909,0.14878,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,7,0,0,12,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[28,77,0.3636,0.33025,0.15344,0.24999,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,7,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[32,77,0.4156,0.29453,0.1854,0.14286,0.28571,0.4286,0.0,0.57143,5,0,0,5,0,6,0,0,8,0,0,8,0,0,5,0,0,0,0,0,0,0,0],[36,77,0.4675,0.29464,0.19865,0.14286,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,6,0,0,6,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[40,77,0.5195,0.27677,0.21408,0.0,0.2857,0.42857,0.0,0.57143,9,0,0,9,0,4,0,0,5,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[44,77,0.5714,0.21428,0.16363,0.14286,0.1429,0.32143,0.0,0.571,7,0,0,7,0,11,0,0,6,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[48,77,0.6234,0.25445,0.17759,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,10,0,0,7,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[52,77,0.6753,0.33929,0.18123,0.14289,0.35714,0.4643,0.0,0.57143,2,0,0,2,0,8,0,0,6,0,0,8,0,0,8,0,0,0,0,0,0,0,0],[56,77,0.7273,0.32588,0.18636,0.14286,0.42857,0.4286,0.0,0.57143,4,0,0,4,0,6,0,0,5,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[60,77,0.7792,0.3125,0.19045,0.14286,0.28571,0.42858,0.0,0.57143,5,0,0,5,0,5,0,0,7,0,0,9,0,0,6,0,0,0,0,0,0,0,0],[64,77,0.8312,0.36155,0.19566,0.25,0.35714,0.57143,0.0,0.57143,3,0,0,3,0,5,0,0,8,0,0,4,0,0,12,0,0,0,0,0,0,0,0],[68,77,0.8831,0.30801,0.18932,0.14286,0.28571,0.4286,0.0,0.57143,5,0,0,5,0,5,0,0,8,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[72,77,0.9351,0.32585,0.17209,0.14286,0.28571,0.4286,0.0,0.57143,2,0,0,2,0,7,0,0,10,0,0,6,0,0,7,0,0,0,0,0,0,0,0],[76,77,0.987,0.28113,0.14937,0.14286,0.28571,0.32143,0.0,0.57143,1,0,0,1,0,11,0,0,12,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[77,77,1.0,0.33024,0.1615,0.2857,0.28571,0.4642,0.0,0.57143,1,0,0,1,0,6,0,0,15,0,0,2,0,0,8,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"flat","v":0.16498,"x":0.3838,"p":[[0,55,0.0,0.16498,0.13881,0.0,0.14286,0.2857,0.0,0.571,9,0,0,9,0,12,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,55,0.0727,0.30349,0.18479,0.14286,0.35714,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,4,0,0,13,0,0,2,0,0,1,0,0,0,0,0],[8,55,0.1455,0.3838,0.16544,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,3,0,0,6,0,0,14,0,0,6,0,0,1,0,0,0,0,0],[12,55,0.2182,0.32578,0.20594,0.14286,0.42857,0.4642,0.0,0.57143,6,0,0,6,0,4,0,0,5,0,0,9,0,0,8,0,0,0,0,0,0,0,0],[16,55,0.2909,0.34362,0.17812,0.28571,0.42857,0.42857,0.0,0.71429,4,0,0,4,0,3,0,0,6,0,0,15,0,0,3,0,0,1,0,0,0,0,0],[20,55,0.3636,0.3348,0.20076,0.14286,0.42857,0.42858,0.0,0.71429,5,0,0,5,0,5,0,0,3,0,0,13,0,0,5,0,0,1,0,0,0,0,0],[24,55,0.4364,0.24996,0.20196,0.10714,0.2143,0.42857,0.0,0.57143,8,0,0,8,0,8,0,0,5,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[28,55,0.5091,0.24106,0.17287,0.14286,0.2857,0.42857,0.0,0.571,7,0,0,7,0,8,0,0,6,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[32,55,0.5818,0.22768,0.17445,0.10714,0.21428,0.42857,0.0,0.57143,8,0,0,8,0,8,0,0,6,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[36,55,0.6545,0.29908,0.17623,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,9,0,0,6,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[40,55,0.7273,0.22768,0.17807,0.0,0.2857,0.42857,0.0,0.4286,10,0,0,10,0,4,0,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.2142,0.15976,0.105,0.21428,0.32143,0.0,0.4286,8,0,0,8,0,8,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.24106,0.16915,0.14286,0.21428,0.42857,0.0,0.571,6,0,0,6,0,10,0,0,5,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[52,55,0.9455,0.24989,0.17497,0.14286,0.2857,0.42857,0.0,0.571,6,0,0,6,0,9,0,0,6,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[55,55,1.0,0.27678,0.17471,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,5,0,0,12,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"ee97bf46cb8b0a6c","q":"Let  $ABCD$  be a convex quadrilateral with  $AB$  parallel to  $CD$ . Let  $P$  and  $Q$  be the midpoints of  $AC$  and  $BD$ , respectively. Prove that if  $\\angle ABP=\\angle CBD$ , then  $\\angle BCQ=\\angle ACD$ 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 $ABCD$  be a quadrilateral such that  $\\angle A=\\angle C=90^{\\circ}$ . If  $A, D$  and the midpoints of  $BA, BC$  are concyclic, show that the midpoints of  $AD, DC$  and  $B, C$  are concyclic.","t":[{"b":1,"e":0.4286,"k":"flat","v":0.71428,"x":0.86607,"p":[[0,109,0.0,0.74768,0.30583,0.57143,0.85714,1.0,0.0,1.0,1,14,1,1,0,3,0,0,0,0,0,3,0,0,2,0,0,5,0,0,3,1,14],[4,109,0.0367,0.83927,0.15465,0.857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,16,0,9],[8,109,0.0734,0.80803,0.22191,0.82132,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,14,0,10],[12,109,0.1101,0.85713,0.15155,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,15,0,11],[16,109,0.1468,0.82142,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,13,0,9],[20,109,0.1835,0.79911,0.24964,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,13,0,10],[24,109,0.2202,0.76339,0.29148,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,0,11,0,11],[28,109,0.2569,0.80804,0.15815,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,11,0,8],[32,109,0.2936,0.79463,0.16343,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,15,0,6],[36,109,0.3303,0.71428,0.24484,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,4,0,0,4,0,0,7,0,0,9,0,6],[40,109,0.367,0.86606,0.13336,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,15,0,11],[44,109,0.4037,0.80802,0.20079,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,14,0,8],[48,109,0.4404,0.74776,0.29933,0.625,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,2,0,0,1,0,0,2,1,0,1,0,0,12,0,10],[52,109,0.4771,0.82361,0.1547,0.83918,0.85714,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,1,0,19,0,5],[56,109,0.5138,0.82587,0.19801,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,13,0,10],[60,109,0.5505,0.76337,0.21903,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,15,0,5],[64,109,0.5872,0.77231,0.22264,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,1,0,0,12,0,9],[68,109,0.6239,0.7857,0.21429,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,14,0,7],[72,109,0.6606,0.82586,0.1847,0.857,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,18,0,7],[76,109,0.6972,0.79017,0.18893,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,14,0,7],[80,109,0.7339,0.83709,0.1831,0.82143,0.85714,1.0,0.357,1.0,0,12,0,0,0,0,0,0,0,0,1,2,0,0,2,0,0,3,0,0,12,0,12],[84,109,0.7706,0.81249,0.25614,0.857,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,15,0,11],[88,109,0.8073,0.75891,0.27535,0.71429,0.85714,1.0,0.0,1.0,2,9,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,12,0,9],[92,109,0.844,0.80355,0.19806,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,0,15,0,8],[96,109,0.8807,0.83033,0.22991,0.857,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,12,0,13],[100,109,0.9174,0.77675,0.16731,0.67857,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,17,0,4],[104,109,0.9541,0.84375,0.20628,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,0,12,0,13],[108,109,0.9908,0.86607,0.1234,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,19,0,9],[109,109,1.0,0.80801,0.17355,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,13,0,8]]},{"b":2,"e":0.57143,"k":"flat","v":0.65169,"x":0.88838,"p":[[0,188,0.0,0.78571,0.28122,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,3,0,0,5,0,16],[4,188,0.0213,0.84372,0.14883,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,14,0,10],[8,188,0.0426,0.84373,0.17987,0.857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,14,0,11],[12,188,0.0638,0.88391,0.15748,0.82132,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,18],[16,188,0.0851,0.82364,0.14397,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,7,0,0,13,0,8],[20,188,0.1064,0.87496,0.12759,0.85714,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,15,0,12],[24,188,0.1277,0.88838,0.14167,0.85711,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,16],[28,188,0.1489,0.83704,0.18665,0.82143,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,1,9],[32,188,0.1702,0.84821,0.18189,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,7,0,15],[36,188,0.1915,0.86158,0.14939,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,14,0,12],[40,188,0.2128,0.73209,0.18476,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,7,0,0,7,0,6],[44,188,0.234,0.77453,0.17778,0.71429,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,1,0,16,0,4],[48,188,0.2553,0.75445,0.23211,0.57143,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,2,0,0,6,0,0,1,0,0,15,0,6],[52,188,0.2766,0.87051,0.15305,0.85711,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,12,0,14],[56,188,0.2979,0.83034,0.18365,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,10,0,12],[60,188,0.3191,0.76337,0.1772,0.57143,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,11,0,6],[64,188,0.3404,0.78792,0.15922,0.71429,0.82135,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,1,0,9,0,7],[68,188,0.3617,0.79908,0.15097,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,11,0,7],[72,188,0.383,0.77227,0.1919,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,4,0,0,12,0,7],[76,188,0.4043,0.82589,0.13236,0.71429,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,15,0,7],[80,188,0.4255,0.81692,0.13479,0.71429,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,16,0,6],[84,188,0.4468,0.82586,0.15463,0.85714,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,20,0,6],[88,188,0.4681,0.72763,0.21239,0.57143,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,1,0,0,6,0,0,8,0,0,11,0,4],[92,188,0.4894,0.7857,0.15567,0.71429,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,18,0,3],[96,188,0.5106,0.808,0.14991,0.71429,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,11,0,8],[100,188,0.5319,0.7857,0.16753,0.71429,0.85707,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,10,0,7],[104,188,0.5532,0.72094,0.1807,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,6,1,0,11,0,3],[108,188,0.5745,0.79017,0.11839,0.71429,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,9,0,5],[112,188,0.5957,0.83481,0.13415,0.82132,0.85714,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,17,0,7],[116,188,0.617,0.77677,0.17106,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,14,0,5],[120,188,0.6383,0.79463,0.14699,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,14,0,5],[124,188,0.6596,0.79463,0.21111,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,11,0,9],[128,188,0.6809,0.74549,0.20119,0.71421,0.857,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,7,0,0,15,0,3],[132,188,0.7021,0.79015,0.16361,0.71429,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,18,0,4],[136,188,0.7234,0.78121,0.16363,0.71429,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,16,0,4],[140,188,0.7447,0.81249,0.15745,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,17,0,5],[144,188,0.766,0.76339,0.18423,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,0,8,0,7],[148,188,0.7872,0.76784,0.23078,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,16,0,6],[152,188,0.8085,0.73213,0.21355,0.67857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,0,7,0,6],[156,188,0.8298,0.74552,0.1703,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,0,11,0,4],[160,188,0.8511,0.75444,0.2059,0.71429,0.85707,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,0,14,0,4],[164,188,0.8723,0.81682,0.13003,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,17,0,5],[168,188,0.8936,0.76338,0.16983,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,14,0,4],[172,188,0.9149,0.78125,0.15145,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,0,9,0,6],[176,188,0.9362,0.72763,0.15718,0.57143,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,0,10,0,3],[180,188,0.9574,0.71424,0.19886,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,0,13,0,2],[184,188,0.9787,0.78571,0.15048,0.71429,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,4,2,0,8,0,0,11,0,6],[188,188,1.0,0.65169,0.16347,0.571,0.64286,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,7,0,0,9,0,0]]}]},{"i":"c20121e34e128cf1","q":"Let  $ABCDEF$  be a convex hexagon in which diagonals  $AD, BE, CF$  are concurrent at   $O$ . Suppose  $[OAF]$  is geometric mean of  $[OAB]$  and  $[OEF]$  and  $[OBC]$  is geometric mean of  $[OAB]$  and  $[OCD]$ . Prove that  $[OED]$  is the geometric mean of  $[OCD]$  and  $[OEF]$ .\n(Here  $[XYZ]$  denotes are of  $\\triangle XYZ$ )","t":[{"b":3,"e":0.14286,"k":"falling","v":0.23205,"x":0.72765,"p":[[0,83,0.0,0.72765,0.28204,0.42857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,8,0,0,4,0,0,0,0,0,2,0,15],[4,83,0.0482,0.42856,0.2945,0.14286,0.35714,0.57143,0.0,1.0,1,5,0,1,0,8,0,0,7,0,0,6,0,0,4,0,0,1,0,0,0,0,5],[8,83,0.0964,0.28123,0.1535,0.14286,0.2857,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,9,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[12,83,0.1446,0.26784,0.13713,0.14286,0.2857,0.42857,0.0,0.571,1,0,0,1,0,13,0,0,8,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[16,83,0.1928,0.33927,0.20437,0.14286,0.28571,0.4286,0.0,0.85714,2,0,0,2,0,8,0,0,8,0,0,8,0,0,3,0,0,2,0,0,1,0,0],[20,83,0.241,0.36597,0.17115,0.2857,0.42857,0.42857,0.14,0.85714,0,0,0,0,0,7,0,0,8,0,0,12,0,0,3,0,0,1,0,0,1,0,0],[24,83,0.2892,0.3214,0.15148,0.24999,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,9,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[28,83,0.3373,0.33035,0.22988,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,6,0,0,8,0,0,3,0,0,0,0,0,0,0,2],[32,83,0.3855,0.35712,0.19881,0.14289,0.42857,0.42858,0.0,0.857,1,0,0,1,0,9,0,0,5,0,0,11,0,0,3,0,0,2,0,0,1,0,0],[36,83,0.4337,0.31695,0.18464,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,12,0,0,4,0,0,11,0,0,3,0,0,0,0,0,1,0,0],[40,83,0.4819,0.23205,0.12759,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,15,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[44,83,0.5301,0.317,0.14611,0.14286,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,6,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[48,83,0.5783,0.2634,0.13415,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,13,0,0,9,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[52,83,0.6265,0.33034,0.1491,0.2857,0.35714,0.4286,0.0,0.57143,2,0,0,2,0,5,0,0,9,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[56,83,0.6747,0.30804,0.20238,0.14286,0.28571,0.42858,0.0,0.85714,3,0,0,3,0,9,0,0,8,0,0,7,0,0,3,0,0,1,0,0,1,0,0],[60,83,0.7229,0.24999,0.12874,0.14286,0.2857,0.28571,0.0,0.571,1,0,0,1,0,14,0,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[64,83,0.7711,0.29016,0.1576,0.14286,0.28571,0.42858,0.0,0.57143,1,0,0,1,0,12,0,0,8,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[68,83,0.8193,0.28572,0.15972,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,11,0,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[72,83,0.8675,0.31249,0.17651,0.14286,0.28571,0.42857,0.14286,0.857,0,0,0,0,0,13,0,0,6,0,0,9,0,0,3,0,0,0,0,0,1,0,0],[76,83,0.9157,0.29463,0.1554,0.14286,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,6,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[80,83,0.9639,0.27232,0.13054,0.14286,0.2857,0.42857,0.0,0.4286,1,0,0,1,0,12,0,0,8,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[83,83,1.0,0.33929,0.1171,0.28571,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,12,0,0,15,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.24543,"x":0.5625,"p":[[0,42,0.0,0.5625,0.25985,0.42857,0.42857,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,17,0,0,2,0,0,1,0,0,1,0,7],[4,42,0.0952,0.36607,0.27649,0.14286,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,13,0,0,7,0,0,6,0,0,1,0,0,0,0,0,2,0,3],[8,42,0.1905,0.33033,0.25859,0.14286,0.28571,0.42857,0.0,1.0,5,2,0,5,0,7,0,0,7,0,0,6,0,0,4,0,0,1,0,0,0,0,2],[12,42,0.2857,0.3973,0.22792,0.2857,0.35714,0.571,0.0,1.0,1,1,0,1,0,6,0,0,9,0,0,7,0,0,4,0,0,3,0,0,1,0,1],[16,42,0.381,0.31249,0.16914,0.14286,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,11,0,0,6,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[20,42,0.4762,0.34375,0.25719,0.14286,0.28571,0.42857,0.0,1.0,2,3,0,2,0,9,0,0,10,0,0,5,0,0,3,0,0,0,0,0,0,0,3],[24,42,0.5714,0.24554,0.13475,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,13,0,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[28,42,0.6667,0.28124,0.16162,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,9,0,0,9,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[32,42,0.7619,0.28569,0.15147,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,12,0,0,8,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[36,42,0.8571,0.28116,0.15363,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,9,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[40,42,0.9524,0.24543,0.14393,0.14286,0.14286,0.32143,0.0,0.57143,1,0,0,1,0,17,0,0,6,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[42,42,1.0,0.29015,0.23,0.14286,0.2143,0.42857,0.0,1.0,3,1,0,3,0,13,0,0,6,0,0,5,0,0,3,0,0,0,0,0,1,0,1]]}]},{"i":"7ef539ec530a8db2","q":"Let  $ABCD$  be a cyclic quadrilateral. The bisectors of angles  $BAD$  and  $BCD$  intersect in point  $K$  such that  $K \\in BD$ . Let  $M$  be the midpoint of  $BD$ . A line passing through point  $C$  and parallel to  $AD$  intersects  $AM$  in point  $P$ . Prove that triangle  $\\triangle DPC$  is isosceles.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.13821,"x":0.19643,"p":[[0,70,0.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,70,0.0571,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,70,0.1143,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,70,0.1714,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,70,0.2286,0.14714,0.05631,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.15152,0.03466,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.15616,0.05489,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.16956,0.08331,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.19643,0.1171,0.14286,0.14286,0.1786,0.0,0.57143,1,0,0,1,0,23,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[64,70,0.9143,0.17393,0.07778,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.17848,0.08752,0.14286,0.14286,0.1429,0.0,0.42857,1,0,0,1,0,24,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13366,"x":0.16045,"p":[[0,156,0.0,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,156,0.0256,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,156,0.0513,0.1517,0.04973,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,156,0.0769,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,156,0.1026,0.16045,0.05932,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,156,0.1282,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,156,0.1538,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,156,0.1795,0.15617,0.05488,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,156,0.2051,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,156,0.2308,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,156,0.2564,0.14715,0.04354,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,156,0.2821,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,156,0.3077,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,156,0.3333,0.1383,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,156,0.359,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,156,0.3846,0.14277,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,156,0.4103,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,156,0.4359,0.14268,0.08748,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,156,0.4615,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,156,0.4872,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,156,0.5128,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,156,0.5385,0.13366,0.03452,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,156,0.5641,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,156,0.5897,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2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 $A_1$ ,  $A_2$ ,  $\\cdots$ ,  $A_m$   be  $m$  subsets of a set of size  $n$ . Prove that   $$  \\sum_{i=1}^{m} \\sum_{j=1}^{m}|A_i|\\cdot |A_i \\cap A_j|\\geq \\frac{1}{mn}\\left(\\sum_{i=1}^{m}|A_i|\\right)^3. $$","t":[{"b":3,"e":0.28571,"k":"flat","v":0.33482,"x":0.42854,"p":[[0,72,0.0,0.33482,0.10479,0.28571,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,0,0,0,20,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[4,72,0.0556,0.39286,0.23958,0.28571,0.28571,0.71429,0.0,0.71429,4,0,1,4,0,1,0,0,14,0,0,2,0,0,2,0,0,9,0,0,0,0,0],[8,72,0.1111,0.42854,0.22014,0.28571,0.28571,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,14,0,0,2,0,0,4,0,0,9,0,0,0,0,0],[12,72,0.1667,0.42411,0.17672,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,4,0,0,3,0,0,7,0,0,0,0,0],[16,72,0.2222,0.35714,0.13363,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,23,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[20,72,0.2778,0.42634,0.18939,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,4,0,0,4,1,0,4,0,0,0,0,1],[24,72,0.3333,0.42411,0.20973,0.28571,0.28571,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,19,0,0,1,0,0,1,0,0,10,0,0,0,0,0],[28,72,0.3889,0.41071,0.1915,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,15,0,0,6,0,0,2,0,0,7,0,0,0,0,0],[32,72,0.4444,0.36607,0.14258,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,23,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[36,72,0.5,0.36606,0.12844,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,19,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[40,72,0.5556,0.41518,0.18336,0.28571,0.28571,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,20,0,0,3,0,0,1,0,0,8,0,0,0,0,0],[44,72,0.6111,0.39285,0.15568,0.28571,0.28571,0.46429,0.2857,0.71429,0,0,0,0,0,0,0,0,20,0,0,4,0,0,4,0,0,4,0,0,0,0,0],[48,72,0.6667,0.35268,0.14719,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,19,0,0,6,0,0,3,0,0,2,0,0,0,0,0],[52,72,0.7222,0.34821,0.14698,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,20,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[56,72,0.7778,0.40177,0.18706,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,19,0,0,2,0,0,4,0,0,6,0,0,0,0,0],[60,72,0.8333,0.37499,0.16267,0.28571,0.28571,0.46418,0.0,0.71429,1,0,0,1,0,0,0,0,20,0,0,3,0,0,5,0,0,3,0,0,0,0,0],[64,72,0.8889,0.39732,0.17762,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,19,0,0,1,0,0,7,0,0,4,0,0,0,0,0],[68,72,0.9444,0.36161,0.13356,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,23,0,0,3,0,0,4,0,0,2,0,0,0,0,0],[72,72,1.0,0.33482,0.11633,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,26,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.30804,"x":0.48661,"p":[[0,90,0.0,0.33481,0.08455,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[4,90,0.0444,0.42857,0.19885,0.28571,0.28571,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,16,0,0,5,0,0,1,0,0,9,0,0,0,0,0],[8,90,0.0889,0.46875,0.22654,0.28571,0.42857,0.71429,0.0,0.71429,2,0,1,2,0,0,0,0,12,0,0,4,0,0,1,0,0,13,0,0,0,0,0],[12,90,0.1333,0.46875,0.20897,0.28571,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,13,0,0,5,0,0,1,0,0,12,0,0,0,0,0],[16,90,0.1778,0.41964,0.17835,0.28571,0.35714,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,15,0,0,5,0,0,6,0,0,5,0,0,0,0,0],[20,90,0.2222,0.48661,0.1984,0.28571,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,10,0,0,6,0,0,4,0,0,11,0,0,0,0,0],[24,90,0.2667,0.41518,0.16888,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,5,0,0,3,0,0,6,0,0,0,0,0],[28,90,0.3111,0.39286,0.16752,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,21,0,0,4,0,0,1,0,0,6,0,0,0,0,0],[32,90,0.3556,0.37945,0.15405,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,17,0,0,8,0,0,3,0,0,3,0,0,0,0,0],[36,90,0.4,0.39731,0.17027,0.28571,0.28571,0.57111,0.0,0.71429,1,0,0,1,0,0,0,0,17,0,0,5,0,0,5,0,0,4,0,0,0,0,0],[40,90,0.4444,0.35266,0.12867,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[44,90,0.4889,0.33482,0.17353,0.28571,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,0,0,0,20,0,0,4,0,0,2,0,0,3,0,0,0,0,0],[48,90,0.5333,0.41518,0.17261,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,3,0,0,4,0,0,6,0,0,0,0,0],[52,90,0.5778,0.36381,0.15291,0.28571,0.28571,0.44643,0.0,0.71429,1,0,0,1,0,0,0,0,21,0,0,2,0,1,5,0,0,2,0,0,0,0,0],[56,90,0.6222,0.45534,0.16145,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,8,0,0,6,0,0,6,0,0,0,0,0],[60,90,0.6667,0.41071,0.17035,0.28571,0.28571,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,19,0,0,4,0,0,3,0,0,6,0,0,0,0,0],[64,90,0.7111,0.37945,0.14985,0.28571,0.28571,0.46418,0.2857,0.71429,0,0,0,0,0,0,0,0,22,0,0,2,0,0,5,0,0,3,0,0,0,0,0],[68,90,0.7556,0.38616,0.16058,0.28571,0.28571,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,21,0,1,3,0,0,2,0,0,5,0,0,0,0,0],[72,90,0.8,0.40625,0.16409,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,4,0,0,4,0,0,5,0,0,0,0,0],[76,90,0.8444,0.42411,0.19719,0.28571,0.28571,0.60714,0.0,0.71429,1,0,0,1,0,0,0,0,17,0,0,3,0,0,3,0,0,8,0,0,0,0,0],[80,90,0.8889,0.41517,0.1726,0.28571,0.28571,0.46418,0.28571,0.71429,0,0,0,0,0,0,0,0,18,0,0,6,0,0,1,0,0,7,0,0,0,0,0],[84,90,0.9333,0.33482,0.09852,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[88,90,0.9778,0.35714,0.13832,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[90,90,1.0,0.30804,0.08073,0.28571,0.285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 $ABC$  be an equilateral triangle with side length  $10$  and  $P$  be a point inside the triangle such that  $|PA|^2+ |PB|^2 + |PC|^2 = 128$ . What is the area of a triangle with side lengths  $|PA|,|PB|,|PC|$ ? $ \n\\textbf{(A)}\\ 6\\sqrt 3\n\\qquad\\textbf{(B)}\\ 7 \\sqrt 3\n\\qquad\\textbf{(C)}\\ 8 \\sqrt 3\n\\qquad\\textbf{(D)}\\ 9 \\sqrt 3\n\\qquad\\textbf{(E)}\\ 10 \\sqrt 3\n$","t":[{"b":0,"e":0.28571,"k":"flat","v":0.5982,"x":0.75,"p":[[0,214,0.0,0.64286,0.26,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,10,0,0,0,0,0,8,0,0,3,0,7],[4,214,0.0187,0.63839,0.21424,0.42857,0.71429,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,10,0,0,0,0,0,17,0,0,0,0,4],[8,214,0.0374,0.66964,0.21852,0.42857,0.71429,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,11,0,0,1,0,7],[12,214,0.0561,0.67411,0.20896,0.42857,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,12,0,0,0,0,7],[16,214,0.0748,0.66964,0.19704,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,13,0,0,0,0,6],[20,214,0.0935,0.61159,0.18638,0.42857,0.64286,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,12,0,0,1,0,3],[24,214,0.1121,0.71429,0.19885,0.57143,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,15,0,0,0,0,8],[28,214,0.1308,0.60268,0.17029,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,12,0,0,1,0,2],[32,214,0.1495,0.66517,0.20395,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,14,0,0,0,0,6],[36,214,0.1682,0.70313,0.15983,0.625,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,4,1,0,18,0,0,0,0,5],[40,214,0.1869,0.72768,0.18681,0.67857,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,16,0,0,0,0,8],[44,214,0.2056,0.72322,0.21997,0.4286,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,13,0,0,0,0,10],[48,214,0.2243,0.63393,0.19865,0.42857,0.64286,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,11,0,0,0,0,5],[52,214,0.243,0.65179,0.2257,0.42857,0.71429,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,9,0,0,1,0,7],[56,214,0.2617,0.68304,0.1665,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,17,0,0,3,0,3],[60,214,0.2804,0.70536,0.2111,0.4286,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,13,0,0,1,0,8],[64,214,0.2991,0.67854,0.22305,0.53539,0.71429,0.71429,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,7,0,0,2,0,0,15,0,0,1,0,6],[68,214,0.3178,0.70088,0.1902,0.57132,0.71429,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,14,0,0,2,0,6],[72,214,0.3364,0.69196,0.18935,0.53571,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,17,0,0,0,0,6],[76,214,0.3551,0.65175,0.18538,0.5354,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,13,0,0,1,0,4],[80,214,0.3738,0.66517,0.20705,0.42857,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,14,0,0,0,0,6],[84,214,0.3925,0.67411,0.1996,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,15,0,0,0,0,6],[88,214,0.4112,0.75,0.18211,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,17,0,0,0,0,9],[92,214,0.4299,0.66962,0.20026,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,14,0,0,0,0,6],[96,214,0.4486,0.70536,0.20497,0.53572,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,14,0,0,0,0,8],[100,214,0.4673,0.63826,0.20193,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,13,0,0,0,0,5],[104,214,0.486,0.66072,0.21943,0.42857,0.71429,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,9,0,0,1,0,7],[108,214,0.5047,0.67857,0.17857,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,16,0,0,0,0,5],[112,214,0.5234,0.63393,0.16342,0.4286,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,16,0,0,1,0,2],[116,214,0.5421,0.72768,0.24053,0.67857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,6,0,0,1,0,0,13,0,0,1,0,10],[120,214,0.5607,0.64286,0.18557,0.42859,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,15,0,0,0,0,4],[124,214,0.5794,0.60713,0.2342,0.42857,0.71429,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,1,0,0,9,0,0,2,0,0,14,0,0,0,0,4],[128,214,0.5981,0.68303,0.19475,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,16,0,0,0,0,6],[132,214,0.6168,0.70087,0.19021,0.67857,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,18,0,0,0,0,6],[136,214,0.6355,0.69642,0.1627,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,20,0,0,1,0,4],[140,214,0.6542,0.70536,0.17834,0.71429,0.71429,0.71429,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,21,0,0,0,0,5],[144,214,0.6729,0.6875,0.22142,0.4286,0.71429,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,14,0,0,1,0,7],[148,214,0.6916,0.66964,0.27067,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,7,0,0,3,0,0,9,0,0,1,0,9],[152,214,0.7103,0.67409,0.24546,0.5354,0.71429,0.75,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,6,0,0,3,0,0,13,0,0,1,0,7],[156,214,0.729,0.65625,0.1984,0.42857,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,8,0,0,2,0,0,16,0,0,1,0,4],[160,214,0.7477,0.71429,0.17128,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,19,0,0,2,0,5],[164,214,0.7664,0.67857,0.18211,0.53571,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,17,0,0,0,0,5],[168,214,0.785,0.74554,0.21349,0.67857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,13,0,0,0,0,11],[172,214,0.8037,0.70089,0.19018,0.64286,0.71429,0.714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 $ABCD$  be a square and let  $P$  be a point on side  $AB$ . The point  $Q$  lies outside the square such that  $\\angle ABQ = \\angle ADP$  and  $\\angle AQB = 90^{\\circ}$ . The point  $R$  lies on the side  $BC$  such that  $\\angle BAR = \\angle ADQ$ . Prove that the lines  $AR, CQ$  and  $DP$  pass through a common point.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.63393,"x":0.71429,"p":[[0,100,0.0,0.66505,0.14106,0.71429,0.71429,0.71429,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,2,0,0,2,0,0,27,0,0,0,0,0],[4,100,0.04,0.69194,0.06302,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[8,100,0.08,0.6607,0.12243,0.67857,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,23,0,0,1,0,0],[12,100,0.12,0.71429,0.03571,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0],[16,100,0.16,0.69647,0.05902,0.71429,0.71429,0.71429,0.43,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0],[20,100,0.2,0.71426,0.05058,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,2,0,0],[24,100,0.24,0.68287,0.05904,0.71321,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0],[28,100,0.28,0.69642,0.05925,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0],[32,100,0.32,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[36,100,0.36,0.69196,0.08073,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,29,0,0,0,0,0],[40,100,0.4,0.70089,0.06546,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,1,0,0],[44,100,0.44,0.63393,0.17473,0.67857,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,24,0,0,0,0,0],[48,100,0.48,0.69195,0.08829,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,27,0,0,1,0,0],[52,100,0.52,0.67411,0.08918,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,25,0,0,0,0,0],[56,100,0.56,0.6875,0.07523,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,25,0,0,1,0,0],[60,100,0.6,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[64,100,0.64,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[68,100,0.68,0.70982,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0],[72,100,0.72,0.67857,0.08748,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,27,0,0,0,0,0],[76,100,0.76,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[80,100,0.8,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[84,100,0.84,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[88,100,0.88,0.6875,0.07523,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,28,0,0,0,0,0],[92,100,0.92,0.70088,0.05489,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[96,100,0.96,0.69197,0.0724,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0],[100,100,1.0,0.70534,0.03463,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.68304,"x":0.70982,"p":[[0,86,0.0,0.70089,0.06546,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,1,0,0],[4,86,0.0465,0.70534,0.03464,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[8,86,0.093,0.70088,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[12,86,0.1395,0.6875,0.08328,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,28,0,0,0,0,0],[16,86,0.186,0.69194,0.06303,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[20,86,0.2326,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[24,86,0.2791,0.70088,0.04168,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[28,86,0.3256,0.70085,0.04177,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[32,86,0.3721,0.69196,0.0724,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0],[36,86,0.4186,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[40,86,0.4651,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[44,86,0.5116,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[48,86,0.5581,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[52,86,0.6047,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[56,86,0.6512,0.70088,0.04168,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[60,86,0.6977,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[64,86,0.7442,0.70536,0.04971,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[68,86,0.7907,0.69642,0.04728,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[72,86,0.8372,0.69642,0.04728,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[76,86,0.8837,0.69197,0.06297,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[80,86,0.9302,0.69196,0.06297,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[84,86,0.9767,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[86,86,1.0,0.68304,0.07771,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,27,0,0,0,0,0]]}]},{"i":"d70604ce4782367e","q":"Let  $F$  be the set of all polynomials  $\\Gamma$  such that all the coefficients of  $\\Gamma (x)$  are integers and  $\\Gamma (x) = 1$  has integer roots. Given a positive intger  $k$ , find the smallest integer  $m(k) > 1$  such that there exist  $\\Gamma \\in F$  for which  $\\Gamma (x) = m(k)$  has exactly  $k$  distinct integer roots.","t":[{"b":1,"e":0.28571,"k":"falling","v":0.16518,"x":0.45982,"p":[[0,51,0.0,0.45982,0.44996,0.0,0.35714,1.0,0.0,1.0,14,11,12,14,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,11],[4,51,0.0784,0.17857,0.27894,0.0,0.0,0.14287,0.0,1.0,17,1,0,17,0,8,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[8,51,0.1569,0.23661,0.3001,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,8,0,0,4,0,0,0,0,0,2,0,0,3,0,0,0,0,2],[12,51,0.2353,0.24553,0.20896,0.14286,0.14286,0.28571,0.0,0.85714,6,0,0,6,0,11,0,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[16,51,0.3137,0.20536,0.23941,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,7,0,0,8,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[20,51,0.3922,0.26786,0.24419,0.10714,0.2857,0.42857,0.0,1.0,8,1,0,8,0,7,0,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,1],[24,51,0.4706,0.16518,0.23987,0.0,0.07143,0.2857,0.0,1.0,16,1,0,16,0,6,0,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[28,51,0.549,0.21428,0.19233,0.0,0.21428,0.28571,0.0,0.71429,10,0,0,10,0,6,0,0,10,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[32,51,0.6275,0.16964,0.1448,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,12,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,51,0.7059,0.17848,0.1557,0.0,0.14286,0.2857,0.0,0.57143,9,0,0,9,0,11,0,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,51,0.7843,0.19196,0.13651,0.14286,0.14288,0.28571,0.0,0.57143,7,0,0,7,0,10,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,51,0.8627,0.26784,0.16654,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,6,0,0,12,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[48,51,0.9412,0.22768,0.15093,0.14286,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,8,0,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[51,51,1.0,0.25446,0.21048,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,8,0,0,3,0,0,5,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.1429,"k":"falling","v":0.10259,"x":0.34821,"p":[[0,86,0.0,0.34821,0.42399,0.0,0.0,0.75,0.0,1.0,18,7,17,18,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,7],[4,86,0.0465,0.25447,0.25187,0.10714,0.2143,0.28571,0.0,1.0,8,1,0,8,0,8,0,0,10,0,0,2,0,0,0,0,0,2,0,0,1,0,1],[8,86,0.093,0.23214,0.26666,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,8,0,0,7,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[12,86,0.1395,0.19196,0.21312,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,9,0,0,5,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[16,86,0.186,0.25893,0.2976,0.0,0.14286,0.42857,0.0,1.0,12,2,0,12,0,7,0,0,3,0,0,3,0,0,4,0,0,0,0,0,1,0,2],[20,86,0.2326,0.26339,0.24772,0.0,0.28571,0.32143,0.0,0.85714,9,0,0,9,0,6,0,0,9,0,0,3,0,0,0,0,0,4,0,0,1,0,0],[24,86,0.2791,0.16071,0.23623,0.0,0.14286,0.1786,0.0,1.0,15,1,0,15,0,9,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[28,86,0.3256,0.22766,0.22827,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,8,0,0,7,0,0,2,0,0,3,0,0,1,0,0,1,0,0],[32,86,0.3721,0.34375,0.36572,0.0,0.14286,0.64286,0.0,1.0,9,4,0,9,0,9,0,0,3,0,0,2,0,0,1,0,0,0,0,0,4,0,4],[36,86,0.4186,0.25446,0.2618,0.0,0.2857,0.28571,0.0,1.0,10,1,0,10,0,5,0,0,11,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[40,86,0.4651,0.24554,0.24284,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,8,0,0,5,0,0,2,0,0,5,0,0,1,0,0,1,0,0],[44,86,0.5116,0.20527,0.20808,0.105,0.14286,0.2857,0.0,1.0,8,1,0,8,0,11,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[48,86,0.5581,0.33036,0.33204,0.0,0.14286,0.60714,0.0,1.0,10,1,0,10,0,7,0,0,3,0,0,1,0,0,3,0,0,3,0,0,4,0,1],[52,86,0.6047,0.24553,0.23483,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,10,0,0,7,0,0,1,0,0,3,0,0,2,0,0,1,0,0],[56,86,0.6512,0.29911,0.28652,0.0,0.28571,0.57143,0.0,1.0,10,2,0,10,0,4,0,0,7,0,0,2,0,0,6,0,0,1,0,0,0,0,2],[60,86,0.6977,0.26786,0.28516,0.0,0.21428,0.32143,0.0,1.0,10,2,0,10,0,6,0,0,8,0,0,3,0,0,1,0,0,1,0,0,1,0,2],[64,86,0.7442,0.21429,0.25754,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,7,0,0,5,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[68,86,0.7907,0.28571,0.25,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,4,0,0,11,0,0,4,0,0,2,0,0,1,0,0,1,0,1],[72,86,0.8372,0.25892,0.21556,0.14286,0.2857,0.28571,0.0,0.85714,6,0,0,6,0,9,0,0,10,0,0,3,0,0,1,0,0,2,0,0,1,0,0],[76,86,0.8837,0.27232,0.26812,0.0,0.21428,0.42857,0.0,0.85714,11,0,0,11,0,5,0,0,5,0,0,4,0,0,2,0,0,4,0,0,1,0,0],[80,86,0.9302,0.19196,0.14555,0.0,0.2143,0.28571,0.0,0.42857,9,0,0,9,0,7,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[84,86,0.9767,0.10259,0.12489,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[86,86,1.0,0.12946,0.13997,0.0,0.14286,0.2857,0.0,0.42857,15,0,0,15,0,7,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0ce281c234b67e90","q":"Let  $D$  and  $E$  be points on side  $[AB]$  of a right triangle with  $m(\\widehat{C})=90^\\circ$  such that  $|AD|=|AC|$  and  $|BE|=|BC|$ . Let  $F$  be the second intersection point of the circumcircles of triangles  $AEC$  and  $BDC$ . If  $|CF|=2$ , what is  $|ED|$ ? $ \n\\textbf{(A)}\\ \\sqrt 2\n\\qquad\\textbf{(B)}\\ 1+\\sqrt 2\n\\qquad\\textbf{(C)}\\ 2\n\\qquad\\textbf{(D)}\\ 2\\sqrt 2\n\\qquad\\textbf{(E)}\\ \\text{None of above}\n$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13839,"x":0.1875,"p":[[0,91,0.0,0.1875,0.12595,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,91,0.044,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,91,0.0879,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,91,0.1319,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,91,0.1758,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,91,0.2198,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,91,0.2637,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,91,0.3077,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,91,0.3516,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,91,0.3956,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,91,0.4396,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,91,0.4835,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,91,0.5275,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,91,0.5714,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,91,0.6154,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,91,0.6593,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,91,0.7033,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,91,0.7473,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,91,0.7912,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,91,0.8352,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,91,0.8791,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,91,0.9231,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,91,0.967,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[91,91,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.13393,"x":0.17411,"p":[[0,267,0.0,0.17411,0.09932,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,267,0.015,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,267,0.03,0.15616,0.07459,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,267,0.0449,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,267,0.0599,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,267,0.0749,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,267,0.0899,0.15616,0.07459,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,267,0.1049,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,267,0.1199,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,267,0.1348,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,267,0.1498,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,267,0.1648,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,267,0.1798,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,267,0.1948,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,267,0.2097,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,267,0.2247,0.16509,0.0883,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,267,0.2397,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,267,0.2547,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,267,0.2697,0.13839,0.02486,0.14286,0.14286,0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 $H$  be the orthocenter of triangle  $ABC$ . Let  $D$  and  $E$  be points on  $AB$  and  $AC$  such that  $DE$  is parallel to  $CH$ . If the circumcircle of triangle  $BDH$  passes through  $M$ , the midpoint of  $DE$ , then prove that  $\\angle ABM=\\angle ACM$","t":[{"b":1,"e":0.42857,"k":"flat","v":0.08482,"x":0.55357,"p":[[0,132,0.0,0.31697,0.26423,0.14286,0.2857,0.4286,0.0,1.0,5,2,1,5,0,10,0,0,3,0,0,9,0,0,1,0,0,2,0,0,0,0,2],[4,132,0.0303,0.55357,0.37754,0.25,0.57143,1.0,0.0,1.0,6,10,0,6,0,2,0,0,2,0,0,6,0,0,0,0,0,6,0,0,0,0,10],[8,132,0.0606,0.36607,0.31529,0.0,0.28571,0.46431,0.0,1.0,9,3,0,9,0,1,0,0,7,0,0,7,0,0,1,0,0,3,0,0,1,0,3],[12,132,0.0909,0.22322,0.28107,0.0,0.14286,0.28571,0.0,1.0,14,2,0,14,0,4,0,0,7,0,0,4,0,0,0,0,0,0,0,0,1,0,2],[16,132,0.1212,0.30804,0.33524,0.0,0.14288,0.42857,0.0,1.0,10,4,0,10,0,7,0,0,5,0,0,3,0,0,0,0,0,3,0,0,0,0,4],[20,132,0.1515,0.14732,0.18029,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,8,0,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[24,132,0.1818,0.18304,0.22371,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,4,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[28,132,0.2121,0.09821,0.1448,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,132,0.2424,0.11161,0.21049,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[36,132,0.2727,0.14284,0.19882,0.0,0.0,0.28571,0.0,0.71429,19,0,0,19,0,2,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[40,132,0.303,0.1875,0.21852,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,4,0,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[44,132,0.3333,0.08929,0.20748,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[48,132,0.3636,0.09821,0.1357,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,132,0.3939,0.11607,0.16146,0.0,0.0,0.1429,0.0,0.71429,17,0,0,17,0,8,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,132,0.4242,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,132,0.4545,0.20982,0.24996,0.0,0.14286,0.32143,0.0,1.0,13,1,0,13,0,7,0,0,4,0,0,5,0,0,0,0,0,2,0,0,0,0,1],[64,132,0.4848,0.13393,0.2111,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,2,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[68,132,0.5152,0.15178,0.22851,0.0,0.0,0.2857,0.0,1.0,17,1,0,17,0,6,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[72,132,0.5455,0.15625,0.26812,0.0,0.0,0.2857,0.0,1.0,20,2,0,20,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[76,132,0.5758,0.20981,0.23139,0.0,0.14286,0.42857,0.0,0.85714,15,0,0,15,0,2,0,0,5,0,0,7,0,0,2,0,0,0,0,0,1,0,0],[80,132,0.6061,0.18304,0.15663,0.14286,0.14286,0.17857,0.0,0.57143,7,0,0,7,0,17,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[84,132,0.6364,0.20973,0.16364,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,16,0,0,4,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[88,132,0.6667,0.23661,0.18074,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,18,0,0,2,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[92,132,0.697,0.25447,0.20434,0.14286,0.14286,0.42857,0.0,0.71429,5,0,0,5,0,13,0,0,4,0,0,7,0,0,0,0,0,3,0,0,0,0,0],[96,132,0.7273,0.24991,0.20521,0.14214,0.1429,0.42857,0.0,0.71429,7,0,0,7,0,10,0,0,4,0,0,8,0,0,1,0,0,2,0,0,0,0,0],[100,132,0.7576,0.31251,0.25862,0.14286,0.14295,0.42857,0.0,1.0,1,3,0,1,0,16,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,3],[104,132,0.7879,0.24107,0.22428,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,15,0,0,5,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[108,132,0.8182,0.22768,0.21086,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,14,0,0,2,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[112,132,0.8485,0.24554,0.1794,0.14286,0.14288,0.42857,0.0,0.71429,5,0,0,5,0,13,0,0,2,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[116,132,0.8788,0.21875,0.19556,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,14,0,0,2,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[120,132,0.9091,0.21875,0.15561,0.14286,0.14286,0.42857,0.0,0.57143,4,0,0,4,0,17,0,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[124,132,0.9394,0.23661,0.19103,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,20,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[128,132,0.9697,0.18304,0.12993,0.14286,0.14286,0.2857,0.0,0.4286,5,0,0,5,0,18,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[132,132,1.0,0.20527,0.11263,0.14286,0.14286,0.2857,0.0,0.4286,1,0,0,1,0,21,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.21866,"x":0.45981,"p":[[0,108,0.0,0.21866,0.20202,0.14286,0.14286,0.17857,0.0,1.0,3,1,0,3,0,21,0,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[4,108,0.037,0.45981,0.36375,0.14286,0.42859,0.71429,0.0,1.0,7,6,0,7,0,4,0,0,3,0,0,4,0,0,2,0,0,5,0,0,1,0,6],[8,108,0.0741,0.37054,0.29636,0.14286,0.42857,0.4286,0.0,1.0,6,3,0,6,0,5,0,0,4,0,0,11,0,0,0,0,0,2,0,0,1,0,3],[12,108,0.1111,0.30804,0.32164,0.0,0.2143,0.42857,0.0,1.0,10,4,0,10,0,6,0,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,4],[16,108,0.1481,0.24553,0.26782,0.0,0.21428,0.28571,0.0,1.0,11,2,0,11,0,5,0,0,9,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[20,108,0.1852,0.25446,0.25935,0.0,0.21428,0.42857,0.0,1.0,11,1,0,11,0,5,0,0,6,0,0,6,0,0,0,0,0,3,0,0,0,0,1],[24,108,0.2222,0.25,0.27664,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,3,0,0,3,0,0,7,0,0,1,0,0,3,0,0,0,0,1],[28,108,0.2593,0.29018,0.29339,0.0,0.2857,0.42857,0.0,1.0,10,2,0,10,0,5,0,0,7,0,0,4,0,0,0,0,0,4,0,0,0,0,2],[32,108,0.2963,0.23214,0.28065,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,5,0,0,7,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[36,108,0.3333,0.21875,0.2789,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,5,0,0,3,0,0,5,0,0,0,0,0,2,0,0,1,0,1],[40,108,0.3704,0.26339,0.22899,0.0,0.2857,0.42857,0.0,0.71429,9,0,0,9,0,5,0,0,8,0,0,6,0,0,0,0,0,4,0,0,0,0,0],[44,108,0.4074,0.3125,0.20959,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,11,0,0,3,0,0,11,0,0,0,0,0,4,0,0,0,0,0],[48,108,0.4444,0.24107,0.21852,0.0,0.21428,0.42857,0.0,0.71429,11,0,0,11,0,5,0,0,3,0,0,11,0,0,0,0,0,2,0,0,0,0,0],[52,108,0.4815,0.25005,0.20829,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,12,0,0,4,0,0,7,0,0,0,0,0,3,0,0,0,0,0],[56,108,0.5185,0.22768,0.16698,0.14286,0.14286,0.42857,0.0,0.57143,6,0,0,6,0,12,0,0,4,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[60,108,0.5556,0.29465,0.2141,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,11,0,0,4,0,0,9,0,0,0,0,0,4,0,0,0,0,0],[64,108,0.5926,0.33036,0.24598,0.14286,0.35714,0.42857,0.0,1.0,4,1,0,4,0,10,0,0,2,0,0,11,0,0,0,0,0,4,0,0,0,0,1],[68,108,0.6296,0.29464,0.25985,0.14286,0.21429,0.42857,0.0,1.0,7,1,0,7,0,9,0,0,2,0,0,10,0,0,0,0,0,2,0,0,1,0,1],[72,108,0.6667,0.31697,0.23887,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,11,0,0,6,0,0,7,0,0,0,0,0,4,0,0,0,0,1],[76,108,0.7037,0.29911,0.22689,0.14286,0.35714,0.42857,0.0,1.0,5,1,0,5,0,10,0,0,1,0,0,13,0,0,1,0,0,1,0,0,0,0,1],[80,108,0.7407,0.33929,0.23891,0.14286,0.42857,0.42857,0.0,0.71429,5,0,0,5,0,8,0,0,1,0,0,12,0,0,0,0,0,6,0,0,0,0,0],[84,108,0.7778,0.23661,0.23854,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,9,0,0,1,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[88,108,0.8148,0.33927,0.24935,0.14286,0.42857,0.42857,0.0,1.0,5,1,0,5,0,7,0,0,3,0,0,12,0,0,1,0,0,2,0,0,1,0,1],[92,108,0.8519,0.3304,0.21857,0.14286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,3,0,0,14,0,0,0,0,0,2,0,0,0,0,1],[96,108,0.8889,0.35259,0.22872,0.14286,0.35714,0.42857,0.0,1.0,2,1,0,2,0,9,0,0,5,0,0,11,0,0,0,0,0,4,0,0,0,0,1],[100,108,0.9259,0.34822,0.23941,0.14286,0.35714,0.42858,0.0,1.0,3,1,0,3,0,9,0,0,4,0,0,10,0,0,1,0,0,4,0,0,0,0,1],[104,108,0.963,0.30803,0.15612,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,9,0,0,10,0,0,0,0,0,2,0,0,0,0,0],[108,108,1.0,0.32143,0.22304,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,11,0,0,6,0,0,9,0,0,0,0,0,3,0,0,0,0,1]]}]},{"i":"c3e9c408c77dd51a","q":"Let  $\\lambda$  be a positive real number satisfying  $\\lambda=\\lambda^{2/3}+1$ . Show that there exists a positive integer  $M$  such that  $|M-\\lambda^{300}|<4^{-100}$ .\n\n*Proposed by Evan Chen*","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,26,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,26,0.1538,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,26],[8,26,0.3077,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,26,0.4615,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,26,0.6154,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[20,26,0.7692,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25]]},{"b":3,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,9,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,9,0.4444,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[8,9,0.8889,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[9,9,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"56d8fb2added09cc","q":"Let  $\\ell$  be a line, and let   $\\gamma$  and  $\\gamma'$  be two circles. The line   $\\ell$  meets  $\\gamma$  at points  $A$  and  $B$ , and   $\\gamma'$  at points  $A'$  and  $B'$ . The tangents to  $\\gamma$  at  $A$  and   $B$  meet at point   $C$ , and the tangents to  $\\gamma'$  at   $A'$  and   $B'$  meet at point   $C'$ . The lines   $\\ell$  and   $CC'$  meet at point  $P$ . Let   $\\lambda$  be a variable line through   $P$  and let   $X$  be one of the points where   $\\lambda$  meets  $\\gamma$ , and  $X'$  be one of the points where   $\\lambda$  meets  $\\gamma'$ . Prove that the point of intersection of the lines   $CX$  and  $C'X'$  lies on a fixed circle. \n\n*Gazeta Matematica*","t":[{"b":0,"e":0.0,"k":"flat","v":0.1383,"x":0.48214,"p":[[0,66,0.0,0.21855,0.24483,0.0,0.14143,0.42857,0.0,0.857,14,0,5,14,0,4,0,0,5,0,0,4,0,0,3,0,0,1,0,0,1,0,0],[4,66,0.0606,0.21875,0.31539,0.0,0.14286,0.28571,0.0,1.0,15,3,0,15,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[8,66,0.1212,0.21874,0.30088,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,4,0,0,5,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[12,66,0.1818,0.16964,0.30185,0.0,0.0,0.17857,0.0,1.0,21,2,0,21,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,2],[16,66,0.2424,0.24107,0.36672,0.0,0.0,0.32143,0.0,1.0,19,5,0,19,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,5],[20,66,0.303,0.48214,0.41764,0.0,0.28571,1.0,0.0,1.0,9,10,0,9,0,2,0,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,10],[24,66,0.3636,0.39732,0.36198,0.0,0.28571,0.71429,0.0,1.0,9,5,0,9,0,3,0,0,6,0,0,3,0,0,2,0,0,2,0,0,2,0,5],[28,66,0.4242,0.30356,0.34021,0.0,0.2857,0.42858,0.0,1.0,12,4,0,12,0,3,0,0,8,0,0,2,0,0,1,0,0,1,0,0,1,0,4],[32,66,0.4848,0.30803,0.40265,0.0,0.0,0.74996,0.0,1.0,18,5,0,18,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,3,0,5],[36,66,0.5455,0.1875,0.32623,0.0,0.0,0.17868,0.0,1.0,19,4,0,19,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[40,66,0.6061,0.38829,0.39003,0.0,0.28571,0.71429,0.0,1.0,13,6,0,13,0,1,0,0,3,0,0,2,0,0,4,0,0,2,0,0,1,0,6],[44,66,0.6667,0.29911,0.36309,0.0,0.14286,0.50002,0.0,1.0,14,4,0,14,0,5,0,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,4],[48,66,0.7273,0.28125,0.38875,0.0,0.0,0.39286,0.0,1.0,17,5,0,17,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,5],[52,66,0.7879,0.41963,0.40396,0.0,0.28571,0.85714,0.0,1.0,11,7,0,11,0,3,0,0,4,0,0,0,0,0,3,0,0,2,0,0,2,0,7],[56,66,0.8485,0.32589,0.38834,0.0,0.14286,0.71429,0.0,1.0,15,5,0,15,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,0,2,0,5],[60,66,0.9091,0.40623,0.38977,0.0,0.28571,0.85704,0.0,1.0,11,6,0,11,0,2,0,0,4,0,0,4,0,0,1,0,0,1,0,0,3,0,6],[64,66,0.9697,0.29015,0.3526,0.0,0.0,0.571,0.0,1.0,17,2,0,17,0,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0,3,0,2],[66,66,1.0,0.1383,0.29555,0.0,0.0,0.14071,0.0,1.0,23,2,0,23,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,2]]},{"b":2,"e":0.0,"k":"flat","v":0.07142,"x":0.27232,"p":[[0,30,0.0,0.20981,0.25748,0.0,0.14286,0.32143,0.0,0.85714,15,0,0,15,0,5,0,0,4,0,0,2,0,0,3,0,0,2,0,0,1,0,0],[4,30,0.1333,0.16071,0.29397,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,4,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[8,30,0.2667,0.27232,0.37518,0.0,0.0,0.32144,0.0,1.0,17,5,0,17,0,2,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,5],[12,30,0.4,0.15625,0.28651,0.0,0.0,0.17857,0.0,1.0,21,2,0,21,0,3,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,2],[16,30,0.5333,0.16956,0.32032,0.0,0.0,0.14287,0.0,1.0,21,3,0,21,0,5,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[20,30,0.6667,0.07143,0.19233,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,30,0.8,0.07142,0.17494,0.0,0.0,0.0,0.0,0.857,25,0,0,25,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[28,30,0.9333,0.10695,0.22863,0.0,0.0,0.14071,0.0,1.0,22,1,0,22,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[30,30,1.0,0.10268,0.18293,0.0,0.0,0.14287,0.0,0.71429,22,0,0,22,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"042fabec7bf198ae","q":"Let  $\\star$  be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers  $x$  and  $y$ ,\n(i)  $(x + 1)\\star 0 = (0\\star x) + 1$ (ii)  $0\\star (y + 1) = (y\\star 0) + 1$ (iii)  $(x + 1)\\star (y + 1) = (x\\star y) + 1$ .\nIf  $123\\star 456 = 789$ , find  $246\\star 135$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,29,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,36,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,36,0.1111,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,36,0.2222,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,36,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,36,0.7778,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,36,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"93369e514f612d3d","q":"Let  $\\mathbb{R}^{+}$  be the set of positive real numbers. Find all real numbers  $a$  for which there exists a function  $f :\\mathbb{R}^{+} \\to \\mathbb{R}^{+}$  such that  $3(f(x))^{2}=2f(f(x))+ax^{4}$ , for all  $x \\in \\mathbb{R}^{+}$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.82576,"x":0.96429,"p":[[0,91,0.0,0.91068,0.14178,0.85711,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[4,91,0.044,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[8,91,0.0879,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[12,91,0.1319,0.92409,0.15564,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[16,91,0.1758,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[20,91,0.2198,0.90625,0.16982,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,24],[24,91,0.2637,0.91517,0.16312,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[28,91,0.3077,0.88838,0.17402,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,1,0,22],[32,91,0.3516,0.82576,0.18126,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,4,0,14],[36,91,0.3956,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[40,91,0.4396,0.84375,0.17261,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,5,0,15],[44,91,0.4835,0.89285,0.15152,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,20],[48,91,0.5275,0.89732,0.16457,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,22],[52,91,0.5714,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[56,91,0.6154,0.91517,0.16314,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[60,91,0.6593,0.93302,0.16363,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,26],[64,91,0.7033,0.86606,0.18191,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,3,0,19],[68,91,0.7473,0.84372,0.19023,0.71429,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,1,0,18],[72,91,0.7912,0.94642,0.12243,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[76,91,0.8352,0.85268,0.20355,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,3,0,19],[80,91,0.8791,0.91964,0.15542,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[84,91,0.9231,0.89731,0.16843,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,22],[88,91,0.967,0.84375,0.15714,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,5,0,14],[91,91,1.0,0.83481,0.19271,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,7,0,15]]},{"b":6,"e":1.0,"k":"flat","v":0.8125,"x":0.97321,"p":[[0,92,0.0,0.91071,0.15465,0.85711,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,23],[4,92,0.0435,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,92,0.087,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,92,0.1304,0.93304,0.13825,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[16,92,0.1739,0.9241,0.15146,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,25],[20,92,0.2174,0.93304,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[24,92,0.2609,0.95088,0.12684,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[28,92,0.3043,0.91071,0.15047,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,23],[32,92,0.3478,0.91071,0.16656,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,24],[36,92,0.3913,0.94643,0.12242,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[40,92,0.4348,0.85714,0.17857,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,2,0,18],[44,92,0.4783,0.91071,0.13716,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,22],[48,92,0.5217,0.92856,0.13367,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,24],[52,92,0.5652,0.88838,0.17031,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,22],[56,92,0.6087,0.8125,0.18707,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,3,0,14],[60,92,0.6522,0.90178,0.17655,0.857,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,23],[64,92,0.6957,0.83479,0.16413,0.71429,0.78571,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,1,0,15],[68,92,0.7391,0.83482,0.18249,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,2,0,16],[72,92,0.7826,0.83034,0.20654,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,3,0,17],[76,92,0.8261,0.91071,0.15872,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,22],[80,92,0.8696,0.85266,0.19721,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,2,0,19],[84,92,0.913,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,22],[88,92,0.9565,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[92,92,1.0,0.96428,0.10715,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27]]}]},{"i":"0c0486459c41d337","q":"Let  $\\dots, a_{-1}, a_0, a_1, a_2, \\dots$  be a sequence of positive integers satisfying the folloring relations:  $a_n = 0$  for  $n < 0$ ,  $a_0 = 1$ , and for  $n \\ge 1$ ,\n\\[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\\]\nCompute\n\\[\\sum_{n \\ge 0} \\frac{10^n a_n}{n!}.\\]","t":[{"b":4,"e":0.42857,"k":"falling","v":0.62943,"x":1.0,"p":[[0,159,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,159,0.0252,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,159,0.0503,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,159,0.0755,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[16,159,0.1006,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,159,0.1258,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[24,159,0.1509,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,159,0.1761,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[32,159,0.2013,0.91518,0.21387,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[36,159,0.2264,0.90179,0.24856,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[40,159,0.2516,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,159,0.2767,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,159,0.3019,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,159,0.327,0.95982,0.1439,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[56,159,0.3522,0.96429,0.15152,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[60,159,0.3774,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,159,0.4025,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,159,0.4277,0.92411,0.2448,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[72,159,0.4528,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[76,159,0.478,0.875,0.28738,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[80,159,0.5031,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[84,159,0.5283,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[88,159,0.5535,0.78571,0.37627,0.85714,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,21],[92,159,0.5786,0.76786,0.37754,0.71429,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,21],[96,159,0.6038,0.875,0.30252,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,26],[100,159,0.6289,0.91518,0.21975,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[104,159,0.6541,0.87054,0.31816,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,26],[108,159,0.6792,0.83929,0.34209,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,24],[112,159,0.7044,0.80804,0.35104,0.92857,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[116,159,0.7296,0.83929,0.34022,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[120,159,0.7547,0.83036,0.36147,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,25],[124,159,0.7799,0.84821,0.32525,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,25],[128,159,0.805,0.72768,0.43793,0.21429,1.0,1.0,0.0,1.0,8,23,0,8,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[132,159,0.8302,0.78571,0.37627,0.71429,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,23],[136,159,0.8553,0.90179,0.24856,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,27],[140,159,0.8805,0.79464,0.29653,0.53572,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,2,0,0,1,0,20],[144,159,0.9057,0.84375,0.30589,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,24],[148,159,0.9308,0.73661,0.34462,0.42857,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,18],[152,159,0.956,0.79018,0.25501,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,8,0,0,1,0,16],[156,159,0.9811,0.88838,0.20121,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,22],[159,159,1.0,0.62943,0.21087,0.42857,0.64286,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,10,0,0,5,0,0,11,0,0,0,0,5]]},{"b":7,"e":0.0,"k":"falling","v":0.04911,"x":1.0,"p":[[0,95,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,95,0.0421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,95,0.0842,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,95,0.1263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,95,0.1684,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,95,0.2105,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[24,95,0.2526,0.94196,0.19186,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[28,95,0.2947,0.89286,0.26726,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,25],[32,95,0.3368,0.80804,0.37561,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,25],[36,95,0.3789,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,95,0.4211,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[44,95,0.4632,0.84821,0.33108,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,25],[48,95,0.5053,0.91964,0.24727,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[52,95,0.5474,0.88393,0.2683,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,24],[56,95,0.5895,0.86607,0.29001,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,25],[60,95,0.6316,0.80804,0.38896,1.0,1.0,1.0,0.0,1.0,6,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[64,95,0.6737,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[68,95,0.7158,0.83482,0.36089,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[72,95,0.7579,0.94196,0.21086,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[76,95,0.8,0.83482,0.34462,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,25],[80,95,0.8421,0.71429,0.43301,0.21429,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,21],[84,95,0.8842,0.69196,0.40581,0.28571,1.0,1.0,0.0,1.0,6,18,0,6,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,1,0,18],[88,95,0.9263,0.80804,0.35644,0.92857,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,24],[92,95,0.9684,0.54018,0.44426,0.0,0.28571,1.0,0.0,1.0,9,15,0,9,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[95,95,1.0,0.04911,0.18423,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"47e1bf0227f44d7d","q":"Let  $\\mathcal{A}$  be the set of finite sequences of positive integers  $a_1,a_2,\\dots,a_k$  such that  $|a_n-a_{n-1}|=a_{n-2}$  for all  $3\\leqslant n\\leqslant k$ . If  $a_1=a_2=1$ , and  $k=18$ , determine the number of elements of  $\\mathcal{A}$ .","t":[{"b":0,"e":0.14286,"k":"falling","v":0.15608,"x":0.96875,"p":[[0,203,0.0,0.9375,0.19541,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[4,203,0.0197,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,203,0.0394,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,203,0.0591,0.90179,0.22428,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,25],[16,203,0.0788,0.92857,0.17497,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,27],[20,203,0.0985,0.92411,0.18893,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[24,203,0.1182,0.92411,0.15966,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,25],[28,203,0.1379,0.65179,0.38621,0.14286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,10,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,16],[32,203,0.1576,0.64286,0.37287,0.25,0.71429,1.0,0.14286,1.0,0,15,0,0,0,8,0,0,4,0,0,0,0,0,2,0,0,3,0,0,0,0,15],[36,203,0.1773,0.70982,0.40007,0.14286,1.0,1.0,0.0,1.0,1,20,0,1,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,20],[40,203,0.197,0.64732,0.37284,0.14286,0.71429,1.0,0.0,1.0,1,15,0,1,0,8,0,0,0,0,0,2,0,0,4,0,0,2,0,0,0,0,15],[44,203,0.2167,0.66518,0.38234,0.24999,0.92857,1.0,0.0,1.0,1,16,0,1,0,7,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,16],[48,203,0.2365,0.70536,0.34244,0.5,0.78571,1.0,0.14286,1.0,0,15,0,0,0,7,0,0,1,0,0,0,0,0,2,0,0,6,0,0,1,0,15],[52,203,0.2562,0.62946,0.37604,0.14286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,10,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,13],[56,203,0.2759,0.64714,0.3995,0.14286,1.0,1.0,0.14,1.0,0,17,0,0,0,11,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,17],[60,203,0.2956,0.61606,0.37362,0.14286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,10,0,0,2,0,0,0,0,0,3,0,0,3,0,0,1,0,13],[64,203,0.3153,0.66518,0.34554,0.28571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,7,0,0,2,0,0,1,0,0,3,0,0,4,0,0,2,0,13],[68,203,0.335,0.5892,0.39579,0.14286,0.71429,1.0,0.0,1.0,2,13,0,2,0,9,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,13],[72,203,0.3547,0.69634,0.38767,0.25,1.0,1.0,0.0,1.0,2,17,1,2,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,17],[76,203,0.3744,0.6517,0.36071,0.25001,0.71429,1.0,0.14,1.0,0,14,0,0,0,8,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,14],[80,203,0.3941,0.58481,0.39827,0.14286,0.71414,1.0,0.14286,1.0,0,14,0,0,0,12,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,14],[84,203,0.4138,0.53561,0.36606,0.14286,0.42857,1.0,0.14,1.0,0,9,0,0,0,11,0,0,4,0,0,2,0,0,1,0,0,2,0,0,3,0,9],[88,203,0.4335,0.74554,0.33069,0.53571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,2,0,0,1,0,0,1,0,0,4,0,0,2,0,17],[92,203,0.4532,0.54455,0.38048,0.14286,0.71429,1.0,0.0,1.0,1,9,0,1,0,13,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,9],[96,203,0.4729,0.41071,0.35848,0.14286,0.14286,0.75,0.0,1.0,1,7,0,1,0,16,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,7],[100,203,0.4926,0.48214,0.3549,0.14286,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,12,0,0,6,0,0,1,0,0,1,0,0,2,0,0,3,0,7],[104,203,0.5123,0.39723,0.31292,0.14286,0.14286,0.71429,0.14,1.0,0,3,0,0,0,17,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,3],[108,203,0.532,0.44634,0.35498,0.14286,0.28571,0.85714,0.14,1.0,0,7,0,0,0,14,0,0,6,0,0,0,0,0,2,0,0,1,0,0,2,0,7],[112,203,0.5517,0.37054,0.32313,0.14286,0.14288,0.57143,0.14286,1.0,0,5,0,0,0,17,0,0,6,0,0,0,0,0,2,0,0,1,0,0,1,0,5],[116,203,0.5714,0.35259,0.30096,0.14286,0.14286,0.46429,0.14,1.0,0,4,0,0,0,17,0,0,6,0,0,1,0,0,1,0,0,3,0,0,0,0,4],[120,203,0.5911,0.3883,0.38344,0.14286,0.14286,1.0,0.14,1.0,0,9,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[124,203,0.6108,0.36607,0.34244,0.14286,0.14286,0.39286,0.14286,1.0,0,6,0,0,0,19,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,6],[128,203,0.6305,0.37929,0.33058,0.14286,0.21428,0.50002,0.14,1.0,0,6,0,0,0,16,0,0,7,0,0,1,0,0,0,0,0,2,0,0,0,0,6],[132,203,0.6502,0.4642,0.34635,0.14286,0.28571,0.78571,0.14,1.0,0,8,0,0,0,12,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,8],[136,203,0.67,0.36152,0.33123,0.14286,0.14286,0.35714,0.14,1.0,0,6,0,0,0,18,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[140,203,0.6897,0.43277,0.3722,0.14286,0.14286,0.89286,0.0,1.0,1,8,0,1,0,16,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,8],[144,203,0.7094,0.39277,0.33888,0.14286,0.14286,0.60714,0.14,1.0,0,6,0,0,0,17,0,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[148,203,0.7291,0.38384,0.34157,0.14286,0.14286,0.60714,0.14,1.0,0,6,0,0,0,18,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,6],[152,203,0.7488,0.4508,0.36273,0.14286,0.14286,0.75,0.14,1.0,0,7,0,0,0,17,0,0,1,0,0,1,0,0,1,0,0,4,0,0,1,0,7],[156,203,0.7685,0.46429,0.37287,0.14286,0.28571,0.85714,0.0,1.0,1,7,0,1,0,14,0,0,4,0,0,0,0,0,0,0,0,3,0,0,3,0,7],[160,203,0.7882,0.46857,0.38519,0.14286,0.14286,1.0,0.14,1.0,0,9,0,0,0,17,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,9],[164,203,0.8079,0.41071,0.36202,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,17,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,8],[168,203,0.8276,0.29911,0.30797,0.14286,0.14286,0.17857,0.0,1.0,1,4,0,1,0,23,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[172,203,0.8473,0.27223,0.22973,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,23,0,0,2,0,0,0,0,0,2,0,0,4,0,0,1,0,0],[176,203,0.867,0.21429,0.17857,0.14286,0.14286,0.17857,0.14286,1.0,0,1,0,0,0,24,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[180,203,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 $\\triangle ABC$  have  $AB=9$  and  $AC=10$ . A semicircle is inscribed in  $\\triangle ABC$  with its center on segment  $BC$  such that it is tangent  $AB$  at point  $D$  and  $AC$  at point  $E$ . If  $AD=2DB$  and  $r$  is the radius of the semicircle,  $r^2$  can be expressed as  $\\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .\n\n*Proposed by Andy Xu*","t":[{"b":1,"e":0.42857,"k":"falling","v":0.46875,"x":0.76339,"p":[[0,75,0.0,0.76339,0.249,0.53571,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,3,0,0,0,0,16],[4,75,0.0533,0.58482,0.15303,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,4,0,0,15,0,0,8,0,0,1,0,1],[8,75,0.1067,0.58929,0.17035,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,8,0,0,2,0,1],[12,75,0.16,0.5357,0.17857,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,4,0,0,13,0,0,8,0,0,1,0,0],[16,75,0.2133,0.52678,0.12595,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,4,0,0,19,0,0,4,0,0,0,0,0],[20,75,0.2667,0.54017,0.11142,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,10,0,0,17,0,0,3,0,0,1,0,0],[24,75,0.32,0.54018,0.12234,0.42857,0.57143,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,10,0,0,8,0,0,0,0,0],[28,75,0.3733,0.49554,0.11837,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,18,0,0,7,0,0,5,0,0,0,0,0],[32,75,0.4267,0.54018,0.14167,0.42857,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,10,0,0,11,0,0,7,0,0,1,0,0],[36,75,0.48,0.50446,0.12869,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,12,0,0,13,0,0,4,0,0,0,0,0],[40,75,0.5333,0.54464,0.12078,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,9,0,0,14,0,0,7,0,0,0,0,0],[44,75,0.5867,0.54018,0.1504,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,10,0,0,12,0,0,6,0,0,0,0,1],[48,75,0.64,0.50893,0.15947,0.42857,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,18,0,0,8,0,0,2,0,0,0,0,2],[52,75,0.6933,0.46875,0.08917,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,20,0,0,9,0,0,1,0,0,0,0,0],[56,75,0.7467,0.51786,0.12242,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,16,0,0,10,0,0,4,0,0,1,0,0],[60,75,0.8,0.48661,0.11769,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,17,0,0,8,0,0,4,0,0,0,0,0],[64,75,0.8533,0.49554,0.12364,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,16,0,0,8,0,0,5,0,0,0,0,0],[68,75,0.9067,0.50446,0.11285,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,13,0,0,0,0,0,0,0,1],[72,75,0.96,0.51339,0.09354,0.42857,0.5,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,13,0,0,3,0,0,0,0,0],[75,75,1.0,0.47322,0.0974,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,20,0,0,8,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.51339,"x":0.6875,"p":[[0,65,0.0,0.6875,0.25862,0.42857,0.57143,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,13,0,0,5,0,0,1,0,0,1,0,12],[4,65,0.0615,0.51339,0.20473,0.28571,0.57143,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,8,0,0,4,0,0,11,0,0,6,0,0,1,0,1],[8,65,0.1231,0.53571,0.21429,0.28571,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,4,0,0,8,0,0,5,0,0,4,0,1],[12,65,0.1846,0.59821,0.14914,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,4,0,0,1,0,0,14,0,0,11,0,0,2,0,0],[16,65,0.2462,0.57589,0.16935,0.57143,0.57143,0.60714,0.28571,1.0,0,2,0,0,0,0,0,0,5,0,0,1,0,0,18,0,0,6,0,0,0,0,2],[20,65,0.3077,0.62054,0.12682,0.57143,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,22,0,0,7,0,0,0,0,2],[24,65,0.3692,0.60268,0.11701,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,5,0,0,1,0,1],[28,65,0.4308,0.60264,0.10556,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,4,0,0,1,0,1],[32,65,0.4923,0.62052,0.07669,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,9,0,0,1,0,0],[36,65,0.5538,0.61607,0.11538,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,22,0,0,7,0,0,1,0,1],[40,65,0.6154,0.625,0.09279,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,9,0,0,0,0,1],[44,65,0.6769,0.61607,0.08328,0.57143,0.57143,0.60714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0],[48,65,0.7385,0.62946,0.09354,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,10,0,0,0,0,1],[52,65,0.8,0.60268,0.11143,0.57143,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,7,0,0,0,0,1],[56,65,0.8615,0.62054,0.10479,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,9,0,0,2,0,0],[60,65,0.9231,0.63393,0.09407,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,0,0,0,1],[64,65,0.9846,0.62053,0.06785,0.57143,0.57143,0.71429,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0],[65,65,1.0,0.59375,0.1017,0.57143,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,0,0,0,21,0,0,9,0,0,0,0,0]]}]},{"i":"c23d4d9433dbf2ed","q":"Let  $a$  be a positive real number.  Find the value of  $a$  such that the definite integral \\[\\int_a^{a^2} \\dfrac{dx}{x+\\sqrt{x}}\\] achieves its smallest possible value.","t":[{"b":5,"e":1.0,"k":"flat","v":0.85714,"x":0.99554,"p":[[0,40,0.0,0.85714,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,5,0,14],[4,40,0.1,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[8,40,0.2,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[12,40,0.3,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,40,0.4,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[20,40,0.5,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[24,40,0.6,0.91071,0.14174,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[28,40,0.7,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[32,40,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,40,0.9,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,40,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":6,"e":1.0,"k":"flat","v":0.87946,"x":0.99554,"p":[[0,38,0.0,0.87946,0.11355,0.82143,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,13],[4,38,0.1053,0.91964,0.12846,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[8,38,0.2105,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[12,38,0.3158,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[16,38,0.4211,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,38,0.5263,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,38,0.6316,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,38,0.7368,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[32,38,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,38,0.9474,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[38,38,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"af2371953119cc44","q":"Let  $a$  be a permutation on  $\\{0,1,\\ldots ,2015\\}$  and  $b,c$  are also permutations on  $\\{1,2,\\ldots ,2015\\}$ . For all  $x\\in \\{1,2,\\ldots ,2015\\}$ , the following conditions are satisfied:\n\n(i)  $a(x)-a(x-1)\\neq 1$ ,\n(ii) if  $b(x)\\neq x$ , then  $c(x)=x$ ,\n\nProve that the number of  $a$ 's is equal to the number of ordered pairs of  $(b,c)$ .","t":[{"b":0,"e":0.28571,"k":"rising","v":0.48661,"x":0.68304,"p":[[0,38,0.0,0.5,0.27894,0.28571,0.35714,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,16,0,0,6,0,0,1,0,0,2,0,0,1,0,6],[4,38,0.1053,0.57141,0.29451,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,10,0,0,10,0,0,1,0,0,1,0,0,1,0,9],[8,38,0.2105,0.56696,0.31029,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,13,0,0,7,0,0,0,0,0,2,0,0,0,0,10],[12,38,0.3158,0.62946,0.33093,0.28571,0.42857,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,14],[16,38,0.4211,0.5625,0.31327,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,14,0,0,6,0,0,0,0,0,2,0,0,0,0,10],[20,38,0.5263,0.53125,0.29931,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,9],[24,38,0.6316,0.59375,0.31157,0.28571,0.42857,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,10,0,0,0,0,0,0,0,0,1,0,11],[28,38,0.7368,0.55357,0.30671,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,10,0,0,0,0,0,0,0,0,0,0,10],[32,38,0.8421,0.48661,0.28316,0.28571,0.42857,0.4286,0.14286,1.0,0,7,0,0,0,2,0,0,11,0,0,12,0,0,0,0,0,0,0,0,0,0,7],[36,38,0.9474,0.68304,0.34207,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,17],[38,38,1.0,0.65625,0.32705,0.28571,0.42857,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,9,0,0,8,0,0,0,0,0,0,0,0,0,0,15]]},{"b":4,"e":1.0,"k":"flat","v":0.45535,"x":0.6875,"p":[[0,40,0.0,0.45535,0.25112,0.2857,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,15,0,0,4,0,0,5,0,0,2,0,0,0,0,4],[4,40,0.1,0.55356,0.28065,0.28571,0.42857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,9,0,0,12,0,0,2,0,0,0,0,0,1,0,8],[8,40,0.2,0.63393,0.32525,0.28571,0.42857,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,11,0,0,6,0,0,0,0,0,1,0,0,1,0,13],[12,40,0.3,0.60268,0.30875,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,8,0,0,9,0,0,1,0,0,2,0,0,0,0,11],[16,40,0.4,0.53124,0.32189,0.28571,0.42857,1.0,0.0,1.0,1,9,0,1,0,1,0,0,12,0,0,7,0,0,1,0,0,0,0,0,1,0,9],[20,40,0.5,0.61607,0.29329,0.42857,0.42857,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,7,0,0,11,0,0,1,0,0,1,0,0,2,0,10],[24,40,0.6,0.55357,0.29397,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,9],[28,40,0.7,0.58482,0.31615,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,13,0,0,6,0,0,0,0,0,1,0,0,2,0,10],[32,40,0.8,0.67411,0.30978,0.42857,0.4286,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,15],[36,40,0.9,0.6875,0.3223,0.39286,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,8,0,0,7,0,0,0,0,0,1,0,0,0,0,16],[40,40,1.0,0.59822,0.32229,0.28571,0.42857,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,12,0,0,7,0,0,0,0,0,1,0,0,0,0,12]]}]},{"i":"0fbc3a4c3380086b","q":"Let  $\\{a_n\\}_{n=0}^{\\infty}$  be the sequence defined by the recurrence relation  $a_{n+3}=2a_{n+2} - 23a_{n+1}+3a_n$  for all  $n \\ge 0,$  with initial conditions  $a_0=20, a_1=0,$  and  $a_2=23.$  Let  $b_n=a_n^3$  for all  $n \\ge 0.$  There exists a unique positive integer  $k$  and constants  $c_0, \\ldots, c_{k-1}$  with  $c_0 \\neq 0$  and  $c_{k-1} \\neq 0$  such that for all sufficiently large  $n,$  we have the recurrence relation  $b_{n+k} = \\sum_{t=0}^{k-1}  c_t b_{n+t}.$  Find  $k+\\sqrt{|c_{k-1}|}+\\sqrt{|c_0|}.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,56,0.0,0.98438,0.04966,1.0,1.0,1.0,0.78571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,29],[4,56,0.0714,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,56,0.1429,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,56,0.2143,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,56,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,56,0.3571,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,56,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,56,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,56,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,56,0.6429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,56,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,56,0.7857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,56,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.96424,"x":1.0,"p":[[0,69,0.0,0.96424,0.16293,1.0,1.0,1.0,0.07,1.0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,29],[4,69,0.058,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,69,0.1159,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,69,0.1739,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,69,0.2319,0.98438,0.04276,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,28],[20,69,0.2899,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,69,0.3478,0.99777,0.01243,1.0,1.0,1.0,0.92857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[28,69,0.4058,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,69,0.4638,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,69,0.5217,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,69,0.5797,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,69,0.6377,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,69,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,69,0.7536,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,69,0.8116,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,69,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,69,0.9275,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,69,0.9855,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[69,69,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"88cd77265ac4220a","q":"Let  $a$  be a fixed positive integer and  $(e_n)$  the sequence, which is defined by  $e_0=1$  and  $$  e_n=a + \\prod_{k=0}^{n-1} e_k $$  for  $n \\geq 1$ .\n\nProve that\n(a) There exist infinitely many prime numbers that divide one element of the sequence.\n(b) There exists one prime number that does not divide an element of the sequence.\n\n(Theresia Eisenk\u00f6lbl)","t":[{"b":0,"e":0.857,"k":"falling","v":0.48652,"x":0.98661,"p":[[0,68,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,68,0.0588,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[8,68,0.1176,0.89731,0.24022,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,26],[12,68,0.1765,0.81696,0.26301,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,5,0,0,1,0,0,2,0,0,3,0,19],[16,68,0.2353,0.82143,0.30514,0.71429,1.0,1.0,0.0,1.0,1,22,1,1,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,22],[20,68,0.2941,0.75444,0.28624,0.57132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,4,0,0,1,0,16],[24,68,0.3529,0.78121,0.24483,0.57143,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,5,0,0,2,0,15],[28,68,0.4118,0.63837,0.2766,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,5,0,0,6,0,0,4,0,0,3,0,8],[32,68,0.4706,0.58925,0.29396,0.42857,0.4998,0.85714,0.14286,1.0,0,7,0,0,0,5,0,0,0,0,0,11,0,0,4,0,0,1,0,0,4,0,7],[36,68,0.5294,0.71868,0.2934,0.5354,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,2,0,0,3,0,0,3,0,0,6,0,0,2,0,13],[40,68,0.5882,0.66961,0.26592,0.5354,0.64286,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,0,0,0,5,0,0,8,0,0,5,0,0,2,0,9],[44,68,0.6471,0.62945,0.29203,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,0,0,0,10,0,0,3,0,0,4,0,0,2,0,9],[48,68,0.7059,0.62944,0.24965,0.42857,0.57121,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,12,0,0,4,0,0,6,0,0,0,0,8],[52,68,0.7647,0.51789,0.22229,0.42857,0.4286,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,12,0,0,8,0,0,1,0,0,2,0,3],[56,68,0.8235,0.54907,0.30745,0.39286,0.571,0.71429,0.0,1.0,1,7,0,1,0,6,0,0,1,0,0,7,0,0,5,0,0,5,0,0,0,0,7],[60,68,0.8824,0.51339,0.26692,0.42857,0.42859,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,1,0,0,13,0,0,1,0,0,6,0,0,1,0,4],[64,68,0.9412,0.5491,0.29257,0.42857,0.4286,0.71429,0.0,1.0,1,7,0,1,0,4,0,0,1,0,0,12,0,0,3,0,0,4,0,0,0,0,7],[68,68,1.0,0.48652,0.26463,0.39285,0.4286,0.71429,0.14,1.0,0,4,0,0,0,7,0,0,1,0,0,13,0,0,2,0,0,5,0,0,0,0,4]]},{"b":1,"e":0.57143,"k":"flat","v":0.88393,"x":0.99554,"p":[[0,27,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,27,0.1481,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,27,0.2963,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[12,27,0.4444,0.96427,0.11299,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[16,27,0.5926,0.94197,0.13767,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[20,27,0.7407,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[24,27,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[27,27,1.0,0.88393,0.17655,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,7,0,19]]}]},{"i":"87a5f468ed77fda7","q":"Let  $a,b,c$  be coprime nonzero integers. Prove that for any coprime integers  $u,v,w$  with  $au+bv+cw = 0$  there exist integers  $m,n, p$  such that   $$ \\begin{cases} a = nw- pv  b = pu-mw  c = mv-nu \\end{cases} $$","t":[{"b":0,"e":1.0,"k":"rising","v":0.40177,"x":0.85267,"p":[[0,54,0.0,0.40177,0.36146,0.14286,0.28571,0.60682,0.0,1.0,7,7,0,7,0,5,0,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,7],[4,54,0.0741,0.69643,0.3809,0.28571,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,19],[8,54,0.1481,0.84821,0.30291,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,24],[12,54,0.2222,0.77232,0.329,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,21],[16,54,0.2963,0.71429,0.35892,0.39286,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,18],[20,54,0.3704,0.73661,0.35912,0.42857,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,19],[24,54,0.4444,0.68749,0.39357,0.28571,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,2,0,17],[28,54,0.5185,0.82588,0.31286,0.89275,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,24],[32,54,0.5926,0.72319,0.37955,0.39286,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,20],[36,54,0.6667,0.70982,0.37369,0.39286,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,3,0,0,3,0,0,0,0,0,2,0,0,1,0,18],[40,54,0.7407,0.5982,0.41717,0.14289,0.7855,1.0,0.0,1.0,4,16,0,4,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,16],[44,54,0.8148,0.85267,0.29339,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,23],[48,54,0.8889,0.66964,0.37701,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,3,0,0,6,0,0,2,0,0,0,0,0,1,0,0,2,0,16],[52,54,0.963,0.6741,0.37836,0.28571,1.0,1.0,0.0,1.0,2,17,0,2,0,4,0,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,17],[54,54,1.0,0.73214,0.38258,0.28571,1.0,1.0,0.0,1.0,2,21,0,2,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,21]]},{"b":1,"e":1.0,"k":"rising","v":0.36161,"x":0.88393,"p":[[0,41,0.0,0.36161,0.41952,0.0,0.14286,0.89286,0.0,1.0,14,8,0,14,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,8],[4,41,0.0976,0.74553,0.38088,0.39288,1.0,1.0,0.0,1.0,2,21,0,2,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,21],[8,41,0.1951,0.74554,0.36199,0.42857,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,20],[12,41,0.2927,0.88393,0.25111,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,25],[16,41,0.3902,0.6875,0.36323,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,8,0,0,1,0,0,0,0,0,2,0,0,2,0,16],[20,41,0.4878,0.77232,0.29636,0.53572,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,3,0,0,4,0,0,3,0,0,0,0,0,4,0,17],[24,41,0.5854,0.75891,0.33965,0.42857,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,20],[28,41,0.6829,0.54241,0.42101,0.14286,0.5,1.0,0.0,1.0,7,13,0,7,0,3,0,0,4,0,1,1,0,0,2,0,0,0,0,0,1,0,13],[32,41,0.7805,0.83036,0.29974,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,23],[36,41,0.878,0.82143,0.32927,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,24],[40,41,0.9756,0.8125,0.33964,0.82132,1.0,1.0,0.0,1.0,3,23,0,3,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,23],[41,41,1.0,0.76338,0.36703,0.49968,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,22]]}]},{"i":"0420a942acb60647","q":"Let  $a, b, c$  be positive real numbers such that  $$ a^2 + b^2 + c^2 = \\frac{1}{4}. $$  Prove that  $$ \\frac{1}{\\sqrt{b^2 + c^2}} + \\frac{1}{\\sqrt{c^2 + a^2}} + \\frac{1}{\\sqrt{a^2 + b^2}} \\le \\frac{\\sqrt{2}}{(a + b)(b + c)(c + a)}. $$  *Proposed by Petar Filipovski, Macedonia*","t":[{"b":4,"e":1.0,"k":"flat","v":0.20089,"x":0.99554,"p":[[0,77,0.0,0.875,0.31894,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[4,77,0.0519,0.54018,0.457,0.0,0.57143,1.0,0.0,1.0,10,15,0,10,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,15],[8,77,0.1039,0.51786,0.45422,0.0,0.64286,1.0,0.0,1.0,10,13,0,10,0,5,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,13],[12,77,0.1558,0.58482,0.45085,0.10714,0.85714,1.0,0.0,1.0,8,16,0,8,0,5,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,16],[16,77,0.2078,0.66518,0.43244,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,17],[20,77,0.2597,0.48661,0.46546,0.0,0.5,1.0,0.0,1.0,14,12,0,14,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,12],[24,77,0.3117,0.51339,0.44587,0.0,0.5,1.0,0.0,1.0,11,13,0,11,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,13],[28,77,0.3636,0.45536,0.42773,0.0,0.42857,1.0,0.0,1.0,12,10,0,12,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,0,0,0,10],[32,77,0.4156,0.48661,0.42985,0.10714,0.35714,1.0,0.0,1.0,8,11,0,8,0,7,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,11],[36,77,0.4675,0.50893,0.45448,0.0,0.49999,1.0,0.0,1.0,10,12,0,10,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,12],[40,77,0.5195,0.42857,0.46566,0.0,0.14286,1.0,0.0,1.0,15,12,0,15,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,12],[44,77,0.5714,0.49107,0.42848,0.0,0.57143,1.0,0.0,1.0,10,10,0,10,0,4,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,10],[48,77,0.6234,0.53125,0.46323,0.0,0.71429,1.0,0.0,1.0,11,14,0,11,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,14],[52,77,0.6753,0.32143,0.44032,0.0,0.0,0.85714,0.0,1.0,20,7,0,20,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,7],[56,77,0.7273,0.20089,0.34043,0.0,0.0,0.14287,0.0,1.0,20,3,0,20,0,5,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,3],[60,77,0.7792,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[64,77,0.8312,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[68,77,0.8831,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,77,0.9351,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[76,77,0.987,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[77,77,1.0,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21]]},{"b":7,"e":0.0,"k":"falling","v":0.01339,"x":0.96429,"p":[[0,42,0.0,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,42,0.0952,0.55358,0.4267,0.10714,0.64286,1.0,0.0,1.0,8,12,0,8,0,4,0,0,1,0,0,0,0,0,3,0,0,2,0,0,2,0,12],[8,42,0.1905,0.54911,0.46581,0.0,0.71429,1.0,0.0,1.0,12,15,0,12,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,15],[12,42,0.2857,0.44643,0.45562,0.0,0.21428,1.0,0.0,1.0,14,11,0,14,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,11],[16,42,0.381,0.41071,0.43412,0.0,0.21429,1.0,0.0,1.0,14,9,0,14,0,2,0,0,2,0,0,0,0,0,3,0,0,1,0,0,1,0,9],[20,42,0.4762,0.53125,0.48344,0.0,0.85714,1.0,0.0,1.0,13,15,0,13,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,15],[24,42,0.5714,0.60268,0.42966,0.10714,0.78571,1.0,0.0,1.0,8,14,0,8,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,14],[28,42,0.6667,0.37946,0.46099,0.0,0.0,1.0,0.0,1.0,18,10,0,18,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,10],[32,42,0.7619,0.18304,0.34669,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,3],[36,42,0.8571,0.15179,0.32525,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,3],[40,42,0.9524,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[42,42,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"57f9c272bf674625","q":"Let  $a, b$ , and  $c$  be real numbers such that\n\\[\\frac{1}{bc-a^2} + \\frac{1}{ca-b^2}+\\frac{1}{ab-c^2} = 0.\\]\nProve that\n\\[\\frac{a}{(bc-a^2)^2} + \\frac{b}{(ca-b^2)^2}+\\frac{c}{(ab-c^2)^2} = 0.\\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.29911,"x":0.63391,"p":[[0,85,0.0,0.37945,0.31865,0.14286,0.28571,0.57111,0.0,1.0,3,4,0,3,0,10,0,0,8,0,0,2,0,0,2,0,0,1,0,0,2,0,4],[4,85,0.0471,0.39286,0.31135,0.14286,0.28571,0.46429,0.0,1.0,3,5,0,3,0,6,0,0,12,0,0,3,0,0,2,0,0,0,0,0,1,0,5],[8,85,0.0941,0.38394,0.24598,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,6,0,0,12,0,0,7,0,0,0,0,0,3,0,0,1,0,2],[12,85,0.1412,0.50893,0.34244,0.2857,0.28571,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,11,0,0,0,0,0,4,0,0,2,0,0,1,0,8],[16,85,0.1882,0.47321,0.30186,0.28571,0.42857,0.60714,0.0,1.0,1,6,0,1,0,4,0,0,10,0,0,7,0,0,2,0,0,1,0,0,1,0,6],[20,85,0.2353,0.61161,0.32972,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,9,0,0,3,0,0,1,0,0,4,0,0,1,0,11],[24,85,0.2824,0.51786,0.32878,0.28571,0.42857,1.0,0.0,1.0,1,9,0,1,0,3,0,0,11,0,0,5,0,0,2,0,0,1,0,0,0,0,9],[28,85,0.3294,0.47768,0.36876,0.14286,0.28571,0.85714,0.0,1.0,4,7,0,4,0,5,0,0,9,0,0,2,0,0,0,0,0,1,0,0,4,0,7],[32,85,0.3765,0.53115,0.35766,0.25,0.42857,1.0,0.0,1.0,2,10,0,2,0,6,0,0,5,0,0,5,0,0,2,0,0,2,0,0,0,0,10],[36,85,0.4235,0.52232,0.34921,0.28571,0.42857,1.0,0.0,1.0,2,9,0,2,0,5,0,0,5,0,0,9,0,0,0,0,0,0,0,0,2,0,9],[40,85,0.4706,0.51339,0.35149,0.28571,0.42857,0.85714,0.0,1.0,5,6,0,5,0,1,0,0,8,0,0,3,0,0,3,0,0,1,0,0,5,0,6],[44,85,0.5176,0.29911,0.21535,0.14286,0.28571,0.28571,0.0,1.0,1,2,0,1,0,11,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,2],[48,85,0.5647,0.625,0.33264,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,9,0,0,2,0,0,2,0,0,2,0,0,3,0,11],[52,85,0.6118,0.47321,0.33396,0.28571,0.28571,0.85714,0.0,1.0,2,6,0,2,0,5,0,0,11,0,0,2,0,0,2,0,0,1,0,0,3,0,6],[56,85,0.6588,0.48659,0.34966,0.14286,0.42857,0.85704,0.0,1.0,4,6,0,4,0,6,0,0,3,0,0,5,0,0,3,0,0,2,0,0,3,0,6],[60,85,0.7059,0.42411,0.34531,0.14286,0.28571,0.75,0.0,1.0,3,6,0,3,0,8,0,0,9,0,0,2,0,0,1,0,0,1,0,0,2,0,6],[64,85,0.7529,0.46429,0.34626,0.14286,0.42857,0.75,0.0,1.0,3,6,0,3,0,8,0,0,4,0,0,5,0,0,1,0,0,3,0,0,2,0,6],[68,85,0.8,0.63391,0.2878,0.28571,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,9,0,0,4,0,0,4,0,0,3,0,0,3,0,9],[72,85,0.8471,0.45982,0.23887,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,10,0,0,10,0,0,2,0,0,2,0,0,3,0,2],[76,85,0.8941,0.49107,0.26711,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,12,0,0,7,0,0,2,0,0,4,0,0,0,0,5],[80,85,0.9412,0.48214,0.25191,0.28571,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,12,0,0,10,0,0,2,0,0,1,0,0,2,0,4],[84,85,0.9882,0.36161,0.20198,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,5,0,0,16,0,0,6,0,0,3,0,0,0,0,0,0,0,2],[85,85,1.0,0.33929,0.17035,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,17,0,0,7,0,0,1,0,0,1,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"falling","v":0.23214,"x":0.52232,"p":[[0,196,0.0,0.52232,0.34369,0.25,0.42857,0.85714,0.0,1.0,2,7,1,2,0,6,0,0,5,0,0,5,0,0,1,0,0,3,0,0,3,0,7],[4,196,0.0204,0.28571,0.23146,0.14286,0.28571,0.28571,0.0,1.0,3,2,0,3,0,10,0,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[8,196,0.0408,0.30804,0.22899,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,7,0,0,12,0,0,4,0,0,1,0,0,3,0,0,0,0,1],[12,196,0.0612,0.38393,0.3102,0.14286,0.28571,0.42857,0.0,1.0,1,6,0,1,0,9,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[16,196,0.0816,0.25893,0.24338,0.14286,0.14286,0.28571,0.0,1.0,5,2,0,5,0,12,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[20,196,0.102,0.31695,0.2737,0.14286,0.28571,0.32143,0.0,1.0,4,3,0,4,0,9,0,0,11,0,0,2,0,0,2,0,0,1,0,0,0,0,3],[24,196,0.1224,0.34821,0.21998,0.28571,0.28571,0.42857,0.0,1.0,1,1,0,1,0,6,0,0,16,0,0,3,0,0,3,0,0,0,0,0,2,0,1],[28,196,0.1429,0.41516,0.32015,0.14286,0.28571,0.60714,0.0,1.0,3,4,0,3,0,7,0,0,9,0,0,3,0,0,2,0,0,1,0,0,3,0,4],[32,196,0.1633,0.28125,0.26119,0.14286,0.28571,0.42857,0.0,1.0,7,2,0,7,0,7,0,0,9,0,0,6,0,0,0,0,0,0,0,0,1,0,2],[36,196,0.1837,0.34375,0.28763,0.14286,0.28571,0.42857,0.0,1.0,4,4,0,4,0,6,0,0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,4],[40,196,0.2041,0.44643,0.32093,0.25,0.28571,0.60714,0.0,1.0,2,6,0,2,0,6,0,0,9,0,0,5,0,0,2,0,0,1,0,0,1,0,6],[44,196,0.2245,0.40177,0.28445,0.14286,0.28571,0.57111,0.0,1.0,1,3,0,1,0,8,0,0,11,0,0,3,0,0,2,0,0,2,0,0,2,0,3],[48,196,0.2449,0.39286,0.32537,0.14286,0.28571,0.60714,0.0,1.0,3,4,0,3,0,11,0,0,5,0,0,3,0,0,2,0,0,2,0,0,2,0,4],[52,196,0.2653,0.40179,0.34707,0.14286,0.28571,0.60714,0.0,1.0,4,6,0,4,0,10,0,0,4,0,0,5,0,0,1,0,0,1,0,0,1,0,6],[56,196,0.2857,0.30804,0.2969,0.14286,0.28571,0.28571,0.0,1.0,5,4,0,5,0,10,0,0,10,0,0,1,0,0,2,0,0,0,0,0,0,0,4],[60,196,0.3061,0.29018,0.2382,0.14286,0.28571,0.28571,0.0,1.0,4,2,0,4,0,8,0,0,14,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[64,196,0.3265,0.35268,0.27196,0.14286,0.28571,0.42857,0.0,1.0,1,4,0,1,0,10,0,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,4],[68,196,0.3469,0.28571,0.2369,0.14286,0.28571,0.28571,0.0,1.0,5,1,0,5,0,7,0,0,14,0,0,2,0,0,0,0,0,2,0,0,1,0,1],[72,196,0.3673,0.33705,0.2733,0.14286,0.28571,0.28571,0.0,1.0,1,3,0,1,1,10,0,0,13,0,0,1,0,0,1,0,0,1,0,0,1,0,3],[76,196,0.3878,0.35265,0.30509,0.14286,0.28571,0.46418,0.0,1.0,3,4,0,3,0,11,0,0,8,0,0,2,0,0,3,0,0,0,0,0,1,0,4],[80,196,0.4082,0.41964,0.32328,0.24999,0.28571,0.46429,0.0,1.0,1,7,0,1,0,7,0,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,7],[84,196,0.4286,0.33482,0.27804,0.14286,0.28571,0.42857,0.0,1.0,2,3,0,2,0,12,0,0,9,0,0,3,0,0,0,0,0,3,0,0,0,0,3],[88,196,0.449,0.37946,0.30433,0.14286,0.28571,0.42857,0.0,1.0,3,4,0,3,0,7,0,0,11,0,0,4,0,0,1,0,0,0,0,0,2,0,4],[92,196,0.4694,0.375,0.30462,0.24999,0.28571,0.46429,0.0,1.0,4,5,0,4,0,4,0,0,15,0,0,1,0,0,3,0,0,0,0,0,0,0,5],[96,196,0.4898,0.23214,0.1915,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,12,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[100,196,0.5102,0.40179,0.28669,0.14286,0.28571,0.60714,0.14286,1.0,0,1,0,0,0,12,0,0,7,0,0,4,0,0,1,0,0,1,0,0,6,0,1],[104,196,0.5306,0.26339,0.20858,0.14286,0.28571,0.28571,0.0,1.0,3,1,0,3,0,12,0,0,12,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[108,196,0.551,0.32143,0.22304,0.24999,0.28571,0.32143,0.0,1.0,3,1,0,3,0,5,0,0,16,0,0,4,0,0,0,0,0,2,0,0,1,0,1],[112,196,0.5714,0.34821,0.27184,0.14286,0.28571,0.42857,0.0,1.0,4,2,0,4,0,6,0,0,11,0,0,5,0,0,1,0,0,1,0,0,2,0,2],[116,196,0.5918,0.42411,0.33213,0.14286,0.28571,0.64286,0.0,1.0,4,5,0,4,0,5,0,0,9,0,0,5,0,0,1,0,0,0,0,0,3,0,5],[120,196,0.6122,0.33929,0.31491,0.14286,0.28571,0.46429,0.0,1.0,6,3,0,6,0,9,0,0,6,0,0,3,0,0,2,0,0,1,0,0,2,0,3],[124,196,0.6327,0.32142,0.31742,0.14286,0.21429,0.42857,0.0,1.0,6,4,0,6,0,10,0,0,7,0,0,2,0,0,2,0,0,0,0,0,1,0,4],[128,196,0.6531,0.35714,0.30929,0.14286,0.28571,0.28571,0.0,1.0,1,5,0,1,0,12,0,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,5],[132,196,0.6735,0.41964,0.34615,0.14286,0.28571,0.75,0.0,1.0,4,4,0,4,0,9,0,0,6,0,0,1,0,0,2,0,0,2,0,0,4,0,4],[136,196,0.6939,0.37053,0.30693,0.14286,0.28571,0.42857,0.0,1.0,5,4,0,5,0,5,0,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,4],[140,196,0.7143,0.30357,0.26666,0.14286,0.28571,0.28571,0.0,1.0,5,3,0,5,0,7,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,3],[144,196,0.7347,0.25,0.13832,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,7,0,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[148,196,0.7551,0.4375,0.33107,0.14286,0.28571,0.71429,0.0,1.0,2,6,0,2,0,8,0,0,8,0,0,4,0,0,1,0,0,2,0,0,1,0,6],[152,196,0.7755,0.35268,0.29447,0.14286,0.28571,0.42857,0.0,1.0,4,4,0,4,0,7,0,0,11,0,0,3,0,0,2,0,0,1,0,0,0,0,4],[156,196,0.7959,0.375,0.28959,0.14286,0.28571,0.46429,0.0,1.0,2,4,0,2,0,8,0,0,12,0,0,2,0,0,2,0,0,2,0,0,0,0,4],[160,196,0.8163,0.33482,0.26392,0.14286,0.28571,0.46429,0.0,1.0,5,2,0,5,0,5,0,0,12,0,0,2,0,0,5,0,0,0,0,0,1,0,2],[164,196,0.8367,0.31249,0.23537,0.14286,0.28571,0.32143,0.0,1.0,3,1,0,3,0,8,0,0,13,0,0,3,0,0,2,0,0,0,0,0,2,0,1],[168,196,0.8571,0.50893,0.34244,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,10,0,0,5,0,0,2,0,0,4,0,0,2,0,0,1,0,8],[172,196,0.8776,0.33482,0.27804,0.14286,0.28571,0.32143,0.0,1.0,3,3,0,3,0,9,0,0,12,0,0,1,0,0,3,0,0,0,0,0,1,0,3],[176,196,0.898,0.37054,0.28983,0.14286,0.28571,0.46429,0.0,1.0,3,3,0,3,0,9,0,0,7,0,0,5,0,0,2,0,0,2,0,0,1,0,3],[180,196,0.9184,0.35266,0.2201,0.24999,0.28571,0.42857,0.0,1.0,2,1,0,2,0,6,0,0,11,0,0,7,0,0,3,0,0,1,0,0,1,0,1],[184,196,0.9388,0.33482,0.18423,0.25,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,13,0,0,7,0,0,2,0,0,1,0,0,0,0,1],[188,196,0.9592,0.32143,0.16752,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,16,0,0,6,0,0,2,0,0,0,0,0,0,0,1],[192,196,0.9796,0.29018,0.10999,0.14289,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,9,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[196,196,1.0,0.32141,0.11291,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,16,0,0,9,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"b4c6f2580abfebfd","q":"Let  $M_2(\\mathbb{Z})$  be the set of  $2 \\times 2$  matrices with integer entries. Let  $A \\in M_2(\\mathbb{Z})$  such that   $$ A^2+5I=0, $$   where  $I \\in M_2(\\mathbb{Z})$  and  $0 \\in M_2(\\mathbb{Z})$  denote the identity and null matrices, respectively. Prove that there exists an invertible matrix  $C \\in M_2(\\mathbb{Z})$  with  $C^{-1} \\in M_2(\\mathbb{Z})$  such that   $$ CAC^{-1} = \\begin{pmatrix} 1 & 2 -3 & -1 \\end{pmatrix} \\text{ ou } CAC^{-1} = \\begin{pmatrix} 0 & 1 -5 & 0 \\end{pmatrix}. $$","t":[{"b":1,"e":1.0,"k":"flat","v":0.85268,"x":0.99107,"p":[[0,7,0.0,0.85268,0.19719,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,15],[4,7,0.5714,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[7,7,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,19,0.0,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21],[4,19,0.2105,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,19,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,19,0.6316,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,19,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[19,19,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"7cce7600b31b7bdf","q":"Let  $K$  be a field having  $q=p^n$  elements, where  $p$  is a prime and  $n\\ge 2$  is an arbitrary integer number. For any  $a\\in K$ , one defines the polynomial  $f_a=X^q-X+a$ . Show that: $a)$   $f=(X^q-X)^q-(X^q-X)$  is divisible by  $f_1$ ; $b)$   $f_a$  has at least  $p^{n-1}$  essentially different irreducible factors  $K[X]$ .","t":[{"b":3,"e":0.28571,"k":"falling","v":0.50446,"x":0.95982,"p":[[0,31,0.0,0.91071,0.21943,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,25],[4,31,0.129,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[8,31,0.2581,0.85267,0.16937,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,4,0,16],[12,31,0.3871,0.87947,0.14334,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,10,0,15],[16,31,0.5161,0.82589,0.20119,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,15],[20,31,0.6452,0.74999,0.22869,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,6,0,0,2,0,12],[24,31,0.7742,0.75446,0.20277,0.67857,0.78564,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,8,0,0,9,0,7],[28,31,0.9032,0.71426,0.22305,0.571,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,7,0,7],[31,31,1.0,0.50446,0.18893,0.39286,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,7,0,0,8,0,0,8,0,0,1,0,0]]},{"b":5,"e":0.42857,"k":"falling","v":0.56694,"x":0.95536,"p":[[0,34,0.0,0.95536,0.10971,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[4,34,0.1176,0.91517,0.15916,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,23],[8,34,0.2353,0.91964,0.12846,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,20],[12,34,0.3529,0.80355,0.19151,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,8,0,11],[16,34,0.4706,0.74997,0.1856,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,9,0,0,7,0,7],[20,34,0.5882,0.70982,0.22155,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,4,0,0,7,0,7],[24,34,0.7059,0.59372,0.19269,0.42857,0.57143,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,11,0,0,8,0,0,3,0,0,7,0,1],[28,34,0.8235,0.56694,0.22724,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,2,0,0,9,0,0,5,0,0,8,0,0,6,0,0],[32,34,0.9412,0.6205,0.21609,0.53539,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,6,0,0,9,0,0,8,0,0,5,0,2],[34,34,1.0,0.65179,0.20183,0.57143,0.71429,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,11,0,0,6,0,2]]}]},{"i":"5017f06c9511daa0","q":"Let  $P$  be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of  $P$  in such a way that the sum of these numbers is  $1$ . Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most  $\\frac{3}{8}$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.44643,"x":0.58482,"p":[[0,24,0.0,0.44643,0.26426,0.28571,0.42857,0.57143,0.0,1.0,2,4,0,2,0,2,0,0,11,0,0,4,0,0,9,0,0,0,0,0,0,0,4],[4,24,0.1667,0.54911,0.37476,0.24999,0.57143,1.0,0.0,1.0,6,9,0,6,0,2,0,0,4,0,0,1,0,0,4,0,0,5,0,0,1,0,9],[8,24,0.3333,0.58482,0.39506,0.28571,0.57143,1.0,0.0,1.0,4,14,0,4,0,2,0,0,9,0,0,0,0,0,2,0,0,1,0,0,0,0,14],[12,24,0.5,0.46874,0.30353,0.28571,0.35714,0.71429,0.0,1.0,3,5,0,3,0,2,0,0,11,0,0,2,0,0,5,0,0,4,0,0,0,0,5],[16,24,0.6667,0.45089,0.34276,0.24999,0.28571,0.71429,0.0,1.0,6,6,0,6,0,2,0,0,9,0,0,0,0,0,6,0,0,3,0,0,0,0,6],[20,24,0.8333,0.53571,0.3093,0.28571,0.5,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,11,0,0,2,0,0,3,0,0,4,0,0,4,0,5],[24,24,1.0,0.48659,0.23106,0.28571,0.49979,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,12,0,0,2,0,0,5,0,0,8,0,0,3,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.55802,"x":0.97768,"p":[[0,35,0.0,0.65178,0.31529,0.42857,0.57143,1.0,0.0,1.0,2,12,0,2,0,1,0,0,2,0,0,5,0,0,9,0,0,0,0,0,1,0,12],[4,35,0.1143,0.67411,0.40126,0.28571,1.0,1.0,0.0,1.0,5,17,0,5,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,17],[8,35,0.2286,0.55802,0.41398,0.14286,0.57143,1.0,0.0,1.0,6,13,0,6,0,4,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,13],[12,35,0.3429,0.66071,0.34395,0.28571,0.71429,1.0,0.0,1.0,2,14,0,2,0,0,0,0,8,0,0,2,0,0,3,0,0,2,0,0,1,0,14],[16,35,0.4571,0.58027,0.38959,0.14286,0.64286,1.0,0.0,1.0,4,12,0,4,0,5,0,0,3,0,0,2,0,0,2,0,0,3,0,0,1,0,12],[20,35,0.5714,0.59375,0.3409,0.28571,0.57143,1.0,0.0,1.0,2,10,0,2,0,3,0,0,6,0,0,2,0,0,4,0,0,4,0,0,1,0,10],[24,35,0.6857,0.74999,0.29015,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,2,0,0,5,0,0,4,0,0,2,0,15],[28,35,0.8,0.78116,0.3427,0.67857,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,4,0,19],[32,35,0.9143,0.78124,0.30091,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,5,0,0,1,0,0,2,0,0,1,0,0,5,0,17],[35,35,1.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]}]},{"i":"c00603836acddeec","q":"Let  $OFT$  and  $NOT$  be two similar triangles (with the same orientation) and let  $FANO$  be a parallelogram. Show that\n\\[\\vert OF\\vert \\cdot \\vert ON\\vert=\\vert OA\\vert \\cdot \\vert OT\\vert.\\]","t":[{"b":0,"e":0.14286,"k":"flat","v":0.05357,"x":0.17857,"p":[[0,85,0.0,0.14732,0.10403,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,85,0.0471,0.13839,0.09771,0.10714,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,85,0.0941,0.14732,0.0977,0.14286,0.14286,0.1786,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,85,0.1412,0.11607,0.09061,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,85,0.1882,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,85,0.2353,0.14277,0.10101,0.105,0.14286,0.17857,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,85,0.2824,0.09366,0.10475,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,85,0.3294,0.1384,0.1838,0.0,0.14286,0.14287,0.0,1.0,12,1,0,12,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,85,0.3765,0.11608,0.10972,0.0,0.14286,0.1429,0.0,0.42857,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,85,0.4235,0.17857,0.17857,0.14286,0.14286,0.2857,0.0,1.0,7,1,0,7,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,85,0.4706,0.12054,0.1078,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,16,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,85,0.5176,0.12491,0.09941,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,85,0.5647,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,85,0.6118,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,85,0.6588,0.10714,0.10101,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,85,0.7059,0.09375,0.09182,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,85,0.7529,0.11161,0.17762,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,85,0.8,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,85,0.8471,0.09375,0.10479,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.09375,0.09852,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.08474,0.0935,0.0,0.07,0.14286,0.0,0.286,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.06697,"x":0.18303,"p":[[0,69,0.0,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.125,0.08564,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.1384,0.10403,0.0,0.14286,0.1786,0.0,0.28571,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.11607,0.12595,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,14,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,69,0.2319,0.14277,0.09449,0.14214,0.14286,0.1429,0.0,0.28571,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.11607,0.09062,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.12938,0.09005,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.16072,0.18123,0.0,0.14286,0.1786,0.0,1.0,9,1,0,9,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,69,0.4638,0.1383,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.16518,0.16409,0.0,0.14286,0.2857,0.0,0.85714,9,0,0,9,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,69,0.5797,0.16964,0.10374,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.16509,0.1825,0.0,0.14286,0.2857,0.0,1.0,9,1,0,9,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,69,0.6957,0.1517,0.09408,0.14286,0.14286,0.1786,0.0,0.28571,6,0,0,6,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.17857,0.09449,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.15625,0.09689,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.16071,0.18471,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,69,0.9275,0.18303,0.1794,0.14286,0.14286,0.2857,0.0,1.0,7,1,0,7,0,14,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,69,0.9855,0.13821,0.09094,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.1383,0.10402,0.0,0.14286,0.1786,0.0,0.28571,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"138cb825a8b4f0d2","q":"Let  $O$  be the set of odd numbers between 0 and 100.  Let  $T$  be the set of subsets of  $O$  of size  $25$ .  For any finite subset of integers  $S$ , let  $P(S)$  be the product of the elements of  $S$ .  Define  $n=\\textstyle{\\sum_{S \\in T}} P(S)$ .  If you divide  $n$  by 17, what is the remainder?","t":[{"b":1,"e":0.71429,"k":"flat","v":0.69196,"x":0.875,"p":[[0,88,0.0,0.85714,0.15152,0.85711,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,13,0,12],[4,88,0.0455,0.875,0.09279,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,8],[8,88,0.0909,0.83928,0.15871,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,14,0,10],[12,88,0.1364,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[16,88,0.1818,0.76786,0.16656,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,11,0,0,7,0,7],[20,88,0.2273,0.75446,0.20896,0.67857,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,10,0,7],[24,88,0.2727,0.79017,0.16746,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,13,0,6],[28,88,0.3182,0.85268,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,11,0,12],[32,88,0.3636,0.86161,0.145,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,13,0,12],[36,88,0.4091,0.76335,0.16607,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,8,0,0,13,0,4],[40,88,0.4545,0.77679,0.15126,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,15,0,3],[44,88,0.5,0.82589,0.13709,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,12,0,8],[48,88,0.5455,0.74554,0.20743,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,7,0,0,5,0,9],[52,88,0.5909,0.74107,0.22428,0.57143,0.78571,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,7,0,0,8,0,8],[56,88,0.6364,0.73214,0.1915,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,10,0,0,8,0,5],[60,88,0.6818,0.69196,0.25281,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,7,0,0,4,0,0,6,0,0,5,0,8],[64,88,0.7273,0.83482,0.19597,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,9,0,14],[68,88,0.7727,0.78125,0.20511,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,0,12,0,8],[72,88,0.8182,0.76786,0.19804,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,5,0,0,8,0,9],[76,88,0.8636,0.76339,0.15407,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,8,0,6],[80,88,0.9091,0.74107,0.24337,0.57143,0.78571,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,6,0,0,7,0,9],[84,88,0.9545,0.74554,0.19144,0.57143,0.78571,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,6,0,0,10,0,6],[88,88,1.0,0.73214,0.19805,0.57143,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,4,0,0,13,0,4]]},{"b":7,"e":0.71429,"k":"flat","v":0.79018,"x":0.875,"p":[[0,89,0.0,0.86607,0.19212,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,13,0,14],[4,89,0.0449,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[8,89,0.0899,0.83929,0.07784,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,25,0,2],[12,89,0.1348,0.83929,0.14617,0.85714,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,5],[16,89,0.1798,0.83036,0.10374,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,14,0,6],[20,89,0.2247,0.84375,0.10926,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,13,0,8],[24,89,0.2697,0.79018,0.11837,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,14,0,3],[28,89,0.3146,0.86607,0.08702,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,23,0,6],[32,89,0.3596,0.83928,0.09942,0.85714,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,24,0,3],[36,89,0.4045,0.81697,0.12492,0.85714,0.85714,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,22,0,3],[40,89,0.4494,0.82143,0.13363,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,18,0,5],[44,89,0.4944,0.85268,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,2],[48,89,0.5393,0.82143,0.14725,0.85714,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,23,0,3],[52,89,0.5843,0.8125,0.10374,0.8214,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,23,0,1],[56,89,0.6292,0.84821,0.07936,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,3],[60,89,0.6742,0.85714,0.08748,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,23,0,5],[64,89,0.7191,0.83036,0.07523,0.82143,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,22,0,2],[68,89,0.764,0.84375,0.08268,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,24,0,3],[72,89,0.809,0.83482,0.08828,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,22,0,3],[76,89,0.8539,0.8482,0.08707,0.85714,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,23,0,4],[80,89,0.8989,0.84375,0.07457,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,23,0,3],[84,89,0.9438,0.81696,0.12993,0.85714,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,22,0,3],[88,89,0.9888,0.82589,0.08552,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,25,0,1],[89,89,1.0,0.79018,0.13825,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,17,0,3]]}]},{"i":"9b96a850b1e0ba1c","q":"Let  $S\\subset \\mathbb{R}$  be a closed set and  $f:\\mathbb{R}^{2n}\\to \\mathbb{R}$  be a continuous function. Define a graph  $G$  as follows: Let  $x$  be a vertex of  $G$  iff  $x\\in \\mathbb{R}^{n}$  and  $f(x,x)\\not\\in S$ , then connect the vertices  $x$  and  $y$  by an edge in  $G$  iff  $f(x,y)\\in S$  or  $f(y,x)\\in S$ . Show that the chromatic number of  $G$  is countable.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.62277,"x":0.92857,"p":[[0,26,0.0,0.62277,0.40651,0.24999,0.85714,1.0,0.0,1.0,4,15,0,4,1,3,0,0,4,0,0,1,0,0,2,0,0,0,0,0,2,0,15],[4,26,0.1538,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[8,26,0.3077,0.85268,0.19061,0.82143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,10,0,14],[12,26,0.4615,0.88839,0.15865,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,14],[16,26,0.6154,0.85267,0.11564,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,19,0,7],[20,26,0.7692,0.82589,0.12745,0.82143,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,19,0,5],[24,26,0.9231,0.83929,0.09279,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,21,0,4],[26,26,1.0,0.83481,0.09526,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,23,0,3]]},{"b":3,"e":1.0,"k":"rising","v":0.76786,"x":0.99554,"p":[[0,21,0.0,0.76786,0.37072,0.64286,1.0,1.0,0.0,1.0,2,21,0,2,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,21],[4,21,0.1905,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,21,0.381,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,21,0.5714,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,21,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,21,0.9524,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[21,21,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"2ff05244467d8c85","q":"Let  $\\mathcal{F}$  be the set of all functions  $f : (0,\\infty)\\to (0,\\infty)$  such that  $f(3x) \\geq f( f(2x) )+x$  for all  $x$ . Find the largest  $A$  such that  $f(x) \\geq A x$  for all  $f\\in\\mathcal{F}$  and all  $x$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.63837,"x":0.79911,"p":[[0,30,0.0,0.79911,0.16311,0.82143,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,20,0,4],[4,30,0.1333,0.68749,0.25863,0.57132,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,7,0,0,0,0,0,8,0,0,0,0,0,11,0,6],[8,30,0.2667,0.79463,0.21112,0.85714,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,19,0,6],[12,30,0.4,0.69643,0.14174,0.57143,0.64286,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,3,0,0,13,0,0],[16,30,0.5333,0.66071,0.12753,0.57143,0.57143,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,5,0,0,8,0,0],[20,30,0.6667,0.66964,0.14914,0.57143,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,2,0,0,11,0,0],[24,30,0.8,0.64732,0.15146,0.57143,0.57143,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,1,0,0,4,0,3],[28,30,0.9333,0.63837,0.13357,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,4,0,0,5,0,1],[30,30,1.0,0.66964,0.1357,0.57143,0.57143,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,3,0,0,10,0,0]]},{"b":4,"e":0.85714,"k":"flat","v":0.75,"x":0.86607,"p":[[0,32,0.0,0.75,0.25505,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,0,0,0,14,0,8],[4,32,0.125,0.76339,0.25155,0.67857,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,1,0,0,15,0,8],[8,32,0.25,0.81695,0.1964,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,16,0,9],[12,32,0.375,0.83036,0.12595,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,21,0,5],[16,32,0.5,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[20,32,0.625,0.8616,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,2],[24,32,0.75,0.80804,0.11633,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,25,0,1],[28,32,0.875,0.82143,0.09449,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,28,0,0],[32,32,1.0,0.83929,0.09279,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,27,0,2]]}]},{"i":"317ec5e62106e999","q":"Let  $S$  be a set of size  $11$ . A random  $12$ -tuple  $(s_1, s_2, . . . , s_{12})$  of elements of  $S$  is chosen uniformly at random. Moreover, let  $\\pi : S \\to S$  be a permutation of  $S$  chosen uniformly at random. The probability that  $s_{i+1}\\ne \\pi (s_i)$  for all  $1 \\le i \\le 12$  (where  $s_{13} = s_1$ ) can be written as  $\\frac{a}{b}$  where  $a$  and  $b$  are relatively prime positive integers. Compute  $a$ .","t":[{"b":2,"e":0.42857,"k":"falling","v":0.41071,"x":0.95089,"p":[[0,59,0.0,0.82143,0.22588,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,8,0,14],[4,59,0.0678,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[8,59,0.1356,0.82143,0.28121,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,3,0,0,4,0,0,1,0,0,1,0,21],[12,59,0.2034,0.84375,0.26573,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,21],[16,59,0.2712,0.79464,0.3213,0.57143,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,1,0,0,3,0,0,2,0,0,0,0,0,3,0,20],[20,59,0.339,0.80357,0.26666,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,1,0,0,3,0,18],[24,59,0.4068,0.83036,0.27067,0.82143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,0,5,0,19],[28,59,0.4746,0.79464,0.32915,0.67857,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,21],[32,59,0.5424,0.79911,0.34043,0.67857,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,22],[36,59,0.6102,0.58482,0.32411,0.42857,0.42857,1.0,0.0,1.0,2,10,0,2,0,3,0,0,0,0,0,13,0,0,1,0,0,3,0,0,0,0,10],[40,59,0.678,0.5,0.21724,0.42857,0.42857,0.42857,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,1,0,4],[44,59,0.7458,0.45982,0.23619,0.42857,0.42857,0.42857,0.0,1.0,3,3,0,3,0,0,0,0,1,0,0,23,0,0,0,0,0,1,0,0,1,0,3],[48,59,0.8136,0.45536,0.21558,0.42857,0.42857,0.42857,0.0,1.0,3,2,0,3,0,0,0,0,0,0,0,23,0,0,2,0,0,1,0,0,1,0,2],[52,59,0.8814,0.47768,0.18766,0.42857,0.42857,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,2,0,2],[56,59,0.9492,0.4375,0.12846,0.42857,0.42857,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,1],[59,59,1.0,0.41071,0.07784,0.42857,0.42857,0.42857,0.0,0.42857,1,0,0,1,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.79018,"x":1.0,"p":[[0,61,0.0,0.79018,0.23954,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,1,0,0,3,0,16],[4,61,0.0656,0.89732,0.17215,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,22],[8,61,0.1311,0.92857,0.14725,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[12,61,0.1967,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,61,0.2623,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,61,0.3279,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,61,0.3934,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,61,0.459,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[32,61,0.5246,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[36,61,0.5902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,61,0.6557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,61,0.7213,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,61,0.7869,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,61,0.8525,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,61,0.918,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,61,0.9836,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[61,61,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"d5fff9e4ea0a85da","q":"Let  $a$  be positive real number such that  $a^{3}=6(a+1)$ . Prove that the equation  $x^{2}+ax+a^{2}-6=0$  has no real solution.","t":[{"b":4,"e":0.2857,"k":"flat","v":0.36607,"x":0.43304,"p":[[0,51,0.0,0.38393,0.14914,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,17,0,0,9,0,0,1,0,0,4,0,0,0,0,0],[4,51,0.0784,0.36607,0.07087,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,14,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[8,51,0.1569,0.43304,0.14934,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,15,0,0,0,0,0,6,0,0,0,0,0],[12,51,0.2353,0.37054,0.11214,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,13,0,0,0,0,0,2,0,0,0,0,0],[16,51,0.3137,0.40179,0.1357,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,14,0,0,0,0,0,4,0,0,0,0,0],[20,51,0.3922,0.42411,0.15765,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,11,0,0,1,0,0,6,0,0,0,0,0],[24,51,0.4706,0.375,0.11152,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,14,0,0,0,0,0,2,0,0,0,0,0],[28,51,0.549,0.42411,0.16554,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,10,0,0,0,0,0,7,0,0,0,0,0],[32,51,0.6275,0.38393,0.12595,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,3,0,0,0,0,0],[36,51,0.7059,0.41071,0.14617,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,13,0,0,0,0,0,5,0,0,0,0,0],[40,51,0.7843,0.37946,0.10479,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,14,0,0,2,0,0,1,0,0,0,0,0],[44,51,0.8627,0.39285,0.12877,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,13,0,0,1,0,0,3,0,0,0,0,0],[48,51,0.9412,0.36607,0.09407,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[51,51,1.0,0.41071,0.13243,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,16,0,0,0,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.34826,"x":0.46875,"p":[[0,47,0.0,0.38393,0.12595,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,3,0,0,0,0,0],[4,47,0.0851,0.38393,0.13092,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,14,0,0,14,0,0,0,0,0,3,0,0,0,0,0],[8,47,0.1702,0.38393,0.12596,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,3,0,0,0,0,0],[12,47,0.2553,0.34826,0.09411,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,20,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[16,47,0.3404,0.40179,0.12078,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,17,0,0,0,0,0,3,0,0,0,0,0],[20,47,0.4255,0.38839,0.12492,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,14,0,0,0,0,0,3,0,0,0,0,0],[24,47,0.5106,0.41518,0.11495,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,20,0,0,0,0,0,3,0,0,0,0,0],[28,47,0.5957,0.45536,0.16146,0.39286,0.42857,0.4286,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,17,0,0,2,0,0,4,0,0,0,0,1],[32,47,0.6809,0.41071,0.1171,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,19,0,0,0,0,0,3,0,0,0,0,0],[36,47,0.766,0.46875,0.15251,0.42857,0.42857,0.50002,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,17,0,0,0,0,0,8,0,0,0,0,0],[40,47,0.8511,0.45534,0.16535,0.28571,0.42857,0.60682,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,12,0,0,1,0,0,8,0,0,0,0,0],[44,47,0.9362,0.41964,0.14698,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,13,0,0,1,0,0,5,0,0,0,0,0],[47,47,1.0,0.42411,0.15355,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,13,0,0,0,0,0,6,0,0,0,0,0]]}]},{"i":"50f708dbc64a5419","q":"Let  $a$ ,  $b$ ,  $c$  be integers with  $a^3 + b^3 + c^3$  divisible by  $18$ .  Prove that  $abc$  is divisible by  $6$ .\n\n(Karl Czakler)","t":[{"b":0,"e":1.0,"k":"flat","v":0.95535,"x":0.99554,"p":[[0,23,0.0,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[4,23,0.1739,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[8,23,0.3478,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,23,0.5217,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,23,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,23,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[23,23,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.9241,"x":0.97768,"p":[[0,26,0.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,26,0.1538,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,26,0.3077,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,26,0.4615,0.95536,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[16,26,0.6154,0.9375,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[20,26,0.7692,0.9241,0.13355,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,21],[24,26,0.9231,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[26,26,1.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"720b117ef9f48830","q":"Let  $a$  and  $b$  be real numbers such that  $$ \\left(8^a+2^{b+7}\\right)\\left(2^{a+3}+8^{b-2}\\right)=4^{a+b+2}. $$  The value of the product  $ab$  can be written as  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .\n\n*Proposed by **stayhomedomath***","t":[{"b":5,"e":0.71429,"k":"falling","v":0.66518,"x":0.85714,"p":[[0,17,0.0,0.85268,0.17307,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,0,0,18],[4,17,0.2353,0.79464,0.19212,0.57143,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,0,0,14],[8,17,0.4706,0.85714,0.18557,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,19],[12,17,0.7059,0.73213,0.18124,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,10,0,0,0,0,9],[16,17,0.9412,0.66518,0.07668,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,22,0,0,0,0,0],[17,17,1.0,0.68304,0.06902,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.67856,"x":0.88393,"p":[[0,44,0.0,0.84821,0.17835,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,0,0,18],[4,44,0.0909,0.83929,0.1915,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,0,0,18],[8,44,0.1818,0.88393,0.15746,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,20],[12,44,0.2727,0.72767,0.17262,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,9,0,0,3,0,7],[16,44,0.3636,0.8125,0.18013,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,4,0,13],[20,44,0.4545,0.71428,0.16366,0.57143,0.71429,0.85711,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,8,0,0,3,0,6],[24,44,0.5455,0.78572,0.19885,0.57143,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,7,0,0,2,0,13],[28,44,0.6364,0.73661,0.19597,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,7,0,0,1,0,10],[32,44,0.7273,0.70534,0.18537,0.57143,0.64286,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,5,0,0,5,0,6],[36,44,0.8182,0.74115,0.20347,0.57143,0.78571,0.895,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,2,0,0,8,0,8],[40,44,0.9091,0.67856,0.19562,0.57143,0.57143,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,3,0,0,0,0,8],[44,44,1.0,0.77679,0.18536,0.57143,0.78571,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,5,0,0,6,0,10]]}]},{"i":"0bd28c2584ca28a8","q":"Let  $X$  be a set which consists from  $8$  consecutive positive integers. Set  $X$  is divided on two disjoint subsets  $A$  and  $B$  with equal number of elements. If sum of squares of elements from set  $A$  is equal to sum of squares of elements from set  $B$ , prove that sum of elements of set  $A$  is equal to sum of elements of set  $B$ .","t":[{"b":4,"e":0.42857,"k":"rising","v":0.60711,"x":0.87053,"p":[[0,53,0.0,0.60711,0.2369,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,4,0,0,6,0,0,8,0,0,6,0,0,2,0,5],[4,53,0.0755,0.64732,0.26241,0.42857,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,8,0,0,4,0,0,2,0,0,6,0,7],[8,53,0.1509,0.67857,0.27433,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,5,0,0,4,0,0,3,0,0,6,0,0,4,0,9],[12,53,0.2264,0.77232,0.23107,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,5,0,0,6,0,12],[16,53,0.3019,0.80355,0.21056,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,4,0,0,5,0,14],[20,53,0.3774,0.82589,0.21349,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,5,0,16],[24,53,0.4528,0.85714,0.17128,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,16],[28,53,0.5283,0.8125,0.23538,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,0,6,0,15],[32,53,0.6038,0.82588,0.18118,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,11,0,0,5,0,13],[36,53,0.6792,0.87053,0.16115,0.82132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,16],[40,53,0.7547,0.73213,0.20749,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,8,0,0,7,0,7],[44,53,0.8302,0.75,0.20825,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,7,0,0,4,0,10],[48,53,0.9057,0.79464,0.18536,0.57143,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,7,0,0,4,0,12],[52,53,0.9811,0.74998,0.18212,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,11,0,0,4,0,8],[53,53,1.0,0.77232,0.16698,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,8,0,7]]},{"b":7,"e":0.57143,"k":"flat","v":0.56248,"x":0.82143,"p":[[0,25,0.0,0.70533,0.23943,0.5354,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,9,0,0,6,0,7],[4,25,0.16,0.73659,0.23176,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,5,0,0,5,0,10],[8,25,0.32,0.82143,0.20516,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,5,0,15],[12,25,0.48,0.78124,0.23687,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,3,0,0,1,0,16],[16,25,0.64,0.77675,0.26714,0.57132,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,2,0,0,7,0,0,2,0,0,1,0,17],[20,25,0.8,0.66964,0.21558,0.53571,0.64286,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,5,0,0,6,0,5],[24,25,0.96,0.59821,0.21261,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,5,0,0,10,0,0,7,0,0,1,0,4],[25,25,1.0,0.56248,0.21705,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,3,0,0,9,0,0,8,0,0,2,0,2]]}]},{"i":"82513aa14758aa39","q":"Let  $\\Omega$  be the  $A$ -excircle of triangle  $ABC$ , and suppose that  $\\Omega$  is tangent to lines  $BC$ ,  $CA$ , and  $AB$  at points  $D$ ,  $E$ , and  $F$ , respectively. Let  $M$  be the midpoint of segment  $EF$ . Two more points  $P$  and  $Q$  are on  $\\Omega$  such that  $EP$  and  $FQ$  are both parallel to  $DM$ . Let  $BP$  meet  $CQ$  at point  $X$ . Prove that the line  $AM$  is the angle bisector of  $\\angle XAD$ . \n\n*Proposed by Shuang-Yen Lee*","t":[{"b":1,"e":0.2857,"k":"flat","v":0.30121,"x":0.78121,"p":[[0,38,0.0,0.30121,0.20656,0.14286,0.2857,0.4286,0.0,0.71429,3,0,0,3,1,9,0,0,8,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[4,38,0.1053,0.78121,0.21428,0.57143,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,6,0,0,3,0,13],[8,38,0.2105,0.71861,0.29358,0.571,0.78564,1.0,0.0,1.0,1,11,0,1,0,3,0,0,0,0,0,2,0,0,5,0,0,5,0,0,5,0,11],[12,38,0.3158,0.66291,0.25015,0.4286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,5,0,1,5,0,0,6,0,0,4,0,7],[16,38,0.4211,0.64726,0.25997,0.5354,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,4,0,0,2,0,0,8,0,0,5,0,0,5,0,6],[20,38,0.5263,0.58927,0.26902,0.39286,0.57143,0.74996,0.0,1.0,1,5,0,1,0,1,0,0,6,0,0,3,0,0,8,0,0,5,0,0,3,0,5],[24,38,0.6316,0.52229,0.21008,0.42857,0.4998,0.71429,0.1429,1.0,0,1,0,0,0,2,0,0,5,0,0,9,0,0,7,0,0,5,0,0,3,0,1],[28,38,0.7368,0.59149,0.268,0.39288,0.64286,0.74999,0.0,1.0,2,3,0,2,0,0,0,0,6,0,0,2,0,1,5,0,0,8,0,0,5,0,3],[32,38,0.8421,0.56693,0.23818,0.39286,0.57143,0.74996,0.1429,1.0,0,1,0,0,0,2,0,0,6,0,0,5,0,0,6,0,0,5,0,0,7,0,1],[36,38,0.9474,0.47987,0.25393,0.28571,0.42859,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,8,0,0,5,0,1,7,0,0,3,0,0,1,0,3],[38,38,1.0,0.35265,0.15142,0.2857,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,4,0,0,17,0,0,5,0,0,4,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.26762,"x":0.68738,"p":[[0,50,0.0,0.26762,0.16276,0.14286,0.17859,0.357,0.071,0.71429,0,0,0,0,1,15,0,1,6,0,2,3,0,0,3,0,0,1,0,0,0,0,0],[4,50,0.08,0.68738,0.26849,0.5354,0.71429,1.0,0.14,1.0,0,10,0,0,0,1,0,0,4,0,0,3,0,0,7,0,0,4,0,0,3,0,10],[8,50,0.16,0.65175,0.27649,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,6,0,0,3,0,0,4,0,0,5,0,0,7,0,6],[12,50,0.24,0.55563,0.27077,0.28571,0.571,0.857,0.14,1.0,0,2,0,0,0,5,0,0,4,0,0,3,0,1,8,0,0,1,0,0,8,0,2],[16,50,0.32,0.58015,0.26501,0.39286,0.57143,0.71429,0.14,1.0,0,4,0,0,0,4,0,0,4,0,0,3,0,0,7,0,0,7,0,0,3,0,4],[20,50,0.4,0.4821,0.32682,0.1429,0.4286,0.71429,0.0,1.0,2,6,0,2,0,8,0,0,3,0,0,4,0,0,6,0,0,2,0,0,1,0,6],[24,50,0.48,0.4886,0.30641,0.2857,0.5355,0.74996,0.0,1.0,3,2,0,3,1,3,0,0,6,0,0,2,0,1,5,1,0,2,0,0,6,0,2],[28,50,0.56,0.5288,0.28081,0.28571,0.57121,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,6,0,0,2,0,0,8,1,0,4,0,0,2,0,4],[32,50,0.64,0.51098,0.29627,0.25,0.571,0.60714,0.0,1.0,1,5,0,1,0,7,0,0,3,0,0,1,0,1,11,0,0,2,0,0,1,0,5],[36,50,0.72,0.62708,0.28292,0.42859,0.71414,0.85714,0.0714,1.0,0,6,0,0,1,3,0,0,3,0,0,2,0,0,6,0,0,7,0,0,4,0,6],[40,50,0.8,0.56449,0.24668,0.42859,0.5712,0.71429,0.14,1.0,0,3,0,0,0,4,0,0,3,0,0,3,0,1,10,0,0,5,0,0,3,0,3],[44,50,0.88,0.57808,0.28925,0.28571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,7,0,1,1,0,0,6,0,0,5,0,0,4,0,5],[48,50,0.96,0.33459,0.24128,0.14286,0.28571,0.571,0.0,0.85714,3,0,0,3,0,12,0,0,3,0,0,4,0,0,6,0,0,3,0,0,1,0,0],[50,50,1.0,0.33481,0.24052,0.1429,0.2857,0.42857,0.0,1.0,2,2,0,2,0,8,0,2,9,0,0,6,0,0,1,0,0,2,0,0,0,0,2]]}]},{"i":"532d547843418329","q":"Let  $a, b$  and  $c$  be the sidelengths of a triangle and  $S$  be the area of that triangle. Prove that  $ab+bc+ac \\ge 4\\sqrt3 S$ .","t":[{"b":2,"e":0.28571,"k":"falling","v":0.12054,"x":0.70982,"p":[[0,23,0.0,0.70982,0.43225,0.32143,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,21],[4,23,0.1739,0.48214,0.49196,0.0,0.21429,1.0,0.0,1.0,16,15,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15],[8,23,0.3478,0.3482,0.45588,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,10],[12,23,0.5217,0.25893,0.41101,0.0,0.0,0.42857,0.0,1.0,22,7,0,22,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,7],[16,23,0.6957,0.15179,0.30501,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[20,23,0.8696,0.13839,0.30196,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[23,23,1.0,0.12054,0.18249,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,0,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.52679,"x":0.99107,"p":[[0,33,0.0,0.62498,0.41611,0.25,0.85707,1.0,0.0,1.0,7,15,0,7,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,15],[4,33,0.1212,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,33,0.2424,0.78125,0.4134,1.0,1.0,1.0,0.0,1.0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[12,33,0.3636,0.52679,0.48107,0.0,0.85714,1.0,0.0,1.0,14,14,0,14,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,14],[16,33,0.4848,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,33,0.6061,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,33,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,33,0.8485,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,33,0.9697,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[33,33,1.0,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25]]}]},{"i":"118883f9c467f239","q":"Let  $a, b, c$  be positive integers such that  $$ \\gcd(a, b) + \\text{lcm}(a, b) = \\gcd(a, c) + \\text{lcm}(a, c). $$  Does it follow from this that  $b = c$ ?","t":[{"b":5,"e":0.71429,"k":"flat","v":0.87946,"x":0.94196,"p":[[0,7,0.0,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,25],[4,7,0.5714,0.92857,0.15972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,25],[7,7,1.0,0.87946,0.17896,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,7,0,18]]},{"b":6,"e":1.0,"k":"flat","v":0.90179,"x":0.98214,"p":[[0,57,0.0,0.90625,0.15815,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,4,0,21],[4,57,0.0702,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[8,57,0.1404,0.90179,0.23808,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[12,57,0.2105,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[16,57,0.2807,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,20],[20,57,0.3509,0.94643,0.13716,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[24,57,0.4211,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[28,57,0.4912,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,57,0.5614,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,57,0.6316,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,57,0.7018,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[44,57,0.7719,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[48,57,0.8421,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[52,57,0.9123,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[56,57,0.9825,0.96875,0.08553,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[57,57,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"12672425585b07df","q":"Let  $a<b<c<d$  be positive integers which satisfy  $ad=bc.$  Prove that  $2a+\\sqrt{a}+\\sqrt{d}<b+c+1.$ *Marius M\u00eenea*","t":[{"b":0,"e":0.4286,"k":"flat","v":0.33482,"x":0.54017,"p":[[0,67,0.0,0.33482,0.17717,0.2857,0.42857,0.42857,0.0,0.71429,4,0,0,4,0,3,0,0,8,0,0,13,0,0,3,0,0,1,0,0,0,0,0],[4,67,0.0597,0.53568,0.20824,0.42857,0.57143,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,2,0,0,9,0,0,9,0,0,8,0,0,1,0,1],[8,67,0.1194,0.50444,0.19556,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,14,0,0,6,0,0,5,0,0,0,0,2],[12,67,0.1791,0.50444,0.22299,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,2,0,0,7,0,0,9,0,0,6,0,0,4,0,0,2,0,2],[16,67,0.2388,0.51342,0.18848,0.42857,0.57121,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,8,0,0,7,0,0,9,0,0,1,0,0],[20,67,0.2985,0.42857,0.22868,0.2857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,7,0,0,8,0,0,5,0,0,4,0,0,1,0,1],[24,67,0.3582,0.41962,0.18187,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,8,0,0,9,0,0,7,0,0,4,0,0,0,0,0],[28,67,0.4179,0.54017,0.26662,0.28571,0.4998,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,6,0,0,2,0,0,8,0,0,3,0,3],[32,67,0.4776,0.44639,0.20121,0.28571,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,3,0,0,9,0,0,8,0,0,6,0,0,0,0,0],[36,67,0.5373,0.5134,0.21975,0.39286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,10,0,0,1,0,0,11,0,0,1,0,1],[40,67,0.597,0.46429,0.27433,0.2857,0.42857,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,7,0,0,5,0,0,3,0,0,7,0,0,3,0,1],[44,67,0.6567,0.4732,0.22989,0.28571,0.4286,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,7,0,0,9,0,0,4,0,0,7,0,0,1,0,1],[48,67,0.7164,0.47768,0.26632,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,7,0,0,5,0,0,7,0,0,4,0,0,2,0,0,6,0,1],[52,67,0.7761,0.45087,0.22333,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,5,0,0,11,0,0,4,0,0,5,0,0,1,0,1],[56,67,0.8358,0.49997,0.25253,0.28571,0.4286,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,4,0,0,8,0,0,5,0,0,6,0,0,2,0,2],[60,67,0.8955,0.4732,0.24598,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,8,0,0,5,0,0,5,0,0,6,0,0,1,0,2],[64,67,0.9552,0.46427,0.18557,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,10,0,0,6,0,0,4,0,0,2,0,0],[67,67,1.0,0.4732,0.24598,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,12,0,0,5,0,0,2,0,0,5,0,0,4,0,1]]},{"b":6,"e":0.71429,"k":"flat","v":0.35714,"x":0.62054,"p":[[0,66,0.0,0.35714,0.15152,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,3,0,0,8,0,0,16,0,0,2,0,0,1,0,0,0,0,0],[4,66,0.0606,0.55353,0.2165,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,3,0,0,2,0,0,10,0,0,11,0,0,1,0,1],[8,66,0.1212,0.60712,0.20517,0.42859,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,6,0,0,7,0,0,13,0,0,0,0,3],[12,66,0.1818,0.62052,0.18074,0.5354,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,0,13,0,0,5,0,0],[16,66,0.2424,0.58925,0.21053,0.42857,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,2,0,0,7,0,0,13,0,0,1,0,2],[20,66,0.303,0.58926,0.1915,0.42857,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,4,0,0,5,0,0,16,0,0,2,0,0],[24,66,0.3636,0.61606,0.24338,0.42857,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,4,0,0,1,0,0,5,0,0,3,0,0,10,0,0,8,0,1],[28,66,0.4242,0.5,0.18211,0.39286,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,10,0,0,4,0,0,9,0,0,1,0,0],[32,66,0.4848,0.54911,0.22619,0.39286,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,7,0,0,2,0,0,10,0,0,5,0,0],[36,66,0.5455,0.4732,0.23807,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,6,0,0,7,0,0,4,0,0,8,0,0,1,0,1],[40,66,0.6061,0.54016,0.23887,0.39286,0.49979,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,6,0,0,8,0,0,5,0,0,6,0,0,2,0,3],[44,66,0.6667,0.59821,0.23538,0.42857,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,2,0,0,4,0,0,3,0,0,15,0,0,5,0,0],[48,66,0.7273,0.54018,0.26662,0.2857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,4,0,0,4,0,0,2,0,0,10,0,0,5,0,1],[52,66,0.7879,0.62054,0.17717,0.42859,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,14,0,0,3,0,1],[56,66,0.8485,0.57586,0.23002,0.42857,0.64286,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,4,0,0,5,0,0,10,0,0,6,0,0],[60,66,0.9091,0.58482,0.22405,0.42857,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,10,0,0,2,0,0,11,0,0,6,0,0],[64,66,0.9697,0.61158,0.2321,0.42857,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,7,0,0,6,0,0,6,0,0,10,0,0],[66,66,1.0,0.47317,0.23263,0.42857,0.42857,0.60682,0.0,1.0,3,1,0,3,0,0,0,0,4,0,0,14,0,0,3,0,0,5,0,0,2,0,1]]}]},{"i":"8579be04a48ad2c3","q":"Let  $a,b,c$  be positive real numbers such that  $a+b+c=10$  and  $ab+bc+ca=25$ .  Let  $m=\\min\\{ab,bc,ca\\}$ .  Find the largest possible value of  $m$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.79911,"x":0.98661,"p":[[0,170,0.0,0.8482,0.28335,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,23],[4,170,0.0235,0.92411,0.15146,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,24],[8,170,0.0471,0.93749,0.11816,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[12,170,0.0706,0.91071,0.1915,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,1,0,25],[16,170,0.0941,0.92857,0.16752,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[20,170,0.1176,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[24,170,0.1412,0.87946,0.20858,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,1,0,23],[28,170,0.1647,0.88393,0.19704,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,0,0,23],[32,170,0.1882,0.87054,0.21237,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,22],[36,170,0.2118,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[40,170,0.2353,0.91964,0.18189,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,26],[44,170,0.2588,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[48,170,0.2824,0.89286,0.19885,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,1,0,24],[52,170,0.3059,0.87054,0.21237,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,6,0,0,0,0,22],[56,170,0.3294,0.82143,0.24223,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,2,0,0,2,0,19],[60,170,0.3529,0.84375,0.21829,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,3,0,0,5,0,18],[64,170,0.3765,0.91964,0.16728,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,24],[68,170,0.4,0.89285,0.19233,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,4,0,22],[72,170,0.4235,0.85714,0.20825,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,1,0,20],[76,170,0.4471,0.79911,0.25719,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,1,0,0,3,0,18],[80,170,0.4706,0.89286,0.13363,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,18],[84,170,0.4941,0.88393,0.15335,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,18],[88,170,0.5176,0.91518,0.13296,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,21],[92,170,0.5412,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[96,170,0.5647,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[100,170,0.5882,0.87054,0.17985,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,5,0,18],[104,170,0.6118,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[108,170,0.6353,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[112,170,0.6588,0.89732,0.20589,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,23],[116,170,0.6824,0.90179,0.14914,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,6,0,20],[120,170,0.7059,0.84375,0.17985,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,4,0,16],[124,170,0.7294,0.92409,0.15149,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[128,170,0.7529,0.89732,0.17215,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,7,0,20],[132,170,0.7765,0.89284,0.15568,0.857,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,7,0,19],[136,170,0.8,0.91518,0.11769,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,20],[140,170,0.8235,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[144,170,0.8471,0.86607,0.18877,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,4,0,19],[148,170,0.8706,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[152,170,0.8941,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[156,170,0.9176,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[160,170,0.9412,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[164,170,0.9647,0.95535,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[168,170,0.9882,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[170,170,1.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]},{"b":7,"e":0.42857,"k":"falling","v":0.46429,"x":0.94643,"p":[[0,91,0.0,0.92411,0.18893,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[4,91,0.044,0.91071,0.15465,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,1,0,23],[8,91,0.0879,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,25],[12,91,0.1319,0.87052,0.2063,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,2,0,21],[16,91,0.1758,0.91518,0.19186,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,25],[20,91,0.2198,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[24,91,0.2637,0.88393,0.21261,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,2,0,23],[28,91,0.3077,0.89286,0.22016,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,22],[32,91,0.3516,0.87946,0.1992,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,2,0,22],[36,91,0.3956,0.875,0.23623,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,22],[40,91,0.4396,0.89286,0.19885,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,3,0,23],[44,91,0.4835,0.86607,0.20183,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,3,0,20],[48,91,0.5275,0.88839,0.22227,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,2,0,0,0,0,25],[52,91,0.5714,0.85713,0.19563,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,2,0,19],[56,91,0.6154,0.82589,0.25935,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,1,0,20],[60,91,0.6593,0.90625,0.20705,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,26],[64,91,0.7033,0.83036,0.23538,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,4,0,18],[68,91,0.7473,0.81695,0.24805,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,0,0,20],[72,91,0.7912,0.83929,0.26426,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,0,0,22],[76,91,0.8352,0.83929,0.22232,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,1,0,20],[80,91,0.8791,0.82589,0.25439,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,4,0,0,1,0,20],[84,91,0.9231,0.89284,0.1786,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,2,0,22],[88,91,0.967,0.79911,0.27861,0.53572,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,3,0,0,0,0,20],[91,91,1.0,0.46429,0.11845,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,29,0,0,0,0,0,2,0,0,0,0,1]]}]},{"i":"2e3f840d78b7295f","q":"Let  $\\Phi$  and  $\\Psi$  denote the Euler totient and Dedekind\u2018s totient respectively. Determine all  $n$  such that  $\\Phi(n)$  divides  $n +\\Psi (n)$ .","t":[{"b":2,"e":0.857,"k":"flat","v":0.84807,"x":1.0,"p":[[0,90,0.0,0.84807,0.1335,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,7,0,12],[4,90,0.0444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,90,0.0889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,90,0.1333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,90,0.1778,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,90,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,90,0.2667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,90,0.3111,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,90,0.3556,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,90,0.4,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[40,90,0.4444,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[44,90,0.4889,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,90,0.5333,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[52,90,0.5778,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,90,0.6222,0.94642,0.1171,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[60,90,0.6667,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[64,90,0.7111,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,90,0.7556,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[72,90,0.8,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[76,90,0.8444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,90,0.8889,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[84,90,0.9333,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[88,90,0.9778,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[90,90,1.0,0.9375,0.17835,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,26]]},{"b":7,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,53,0.0,0.86161,0.18029,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,16],[4,53,0.0755,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,53,0.1509,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,53,0.2264,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,53,0.3019,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,53,0.3774,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,53,0.4528,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,53,0.5283,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,53,0.6038,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,53,0.6792,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,53,0.7547,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[44,53,0.8302,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[48,53,0.9057,0.9375,0.14258,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[52,53,0.9811,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[53,53,1.0,0.90624,0.13651,0.85711,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,19]]}]},{"i":"b71af5e913c6d416","q":"Let  $a,b,c$  be positive integers such that  $a|b^5, b|c^5$  and  $c|a^5$ . Prove that  $abc|(a+b+c)^{31}$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.04464,"x":0.1875,"p":[[0,36,0.0,0.14723,0.33213,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[4,36,0.1111,0.15624,0.33759,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[8,36,0.2222,0.14286,0.32927,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,36,0.3333,0.15179,0.333,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[16,36,0.4444,0.1875,0.37362,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[20,36,0.5556,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[24,36,0.6667,0.16071,0.36202,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[28,36,0.7778,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[32,36,0.8889,0.09375,0.249,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[36,36,1.0,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"flat","v":0.11161,"x":0.29911,"p":[[0,20,0.0,0.16964,0.3597,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,20,0.2,0.19643,0.38755,0.0,0.0,0.03571,0.0,1.0,24,6,0,24,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[8,20,0.4,0.20536,0.37617,0.0,0.0,0.14286,0.0,1.0,22,5,0,22,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[12,20,0.6,0.29911,0.44228,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[16,20,0.8,0.15625,0.33189,0.0,0.0,0.03571,0.0,1.0,24,4,0,24,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[20,20,1.0,0.11161,0.29175,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"1aaae5cd7f3a2892","q":"Let  $a_1, a_2, ..., a_m$  be positive integers, none of which is equal to  $10$ , such that  $a_1 + a_2 + ...+ a_m = 10m$ . Prove that  $(a_1a_2a_3 \\cdot ...\\cdot  a_m)^{1/m} \\le 3\\sqrt{11}$ .","t":[{"b":0,"e":0.71429,"k":"rising","v":0.65178,"x":0.92411,"p":[[0,10,0.0,0.65178,0.24983,0.42857,0.57143,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,2,0,0,7,0,0,8,0,0,4,0,0,2,0,8],[4,10,0.4,0.875,0.15047,0.85714,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,14,0,13],[8,10,0.8,0.90179,0.16536,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,9,0,19],[10,10,1.0,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19]]},{"b":1,"e":0.71429,"k":"flat","v":0.7567,"x":0.86161,"p":[[0,19,0.0,0.7567,0.25308,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,2,0,1,4,0,0,5,0,0,8,0,10],[4,19,0.2105,0.86161,0.20355,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,1,0,0,5,0,19],[8,19,0.4211,0.86161,0.18029,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,7,0,16],[12,19,0.6316,0.79463,0.21999,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,5,0,0,6,0,13],[16,19,0.8421,0.78125,0.23415,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,6,0,0,5,0,13],[19,19,1.0,0.76779,0.19811,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,5,0,0,8,0,9]]}]},{"i":"67c9850750751298","q":"Let  $a_1,a_2,\\cdots,a_6,a_7$  be seven positive integers. Let  $S$  be the set of all numbers of the form  $a_i^2+a_j^2$  where  $1\\leq i<j\\leq 7$ . \nProve that there exist two elements of  $S$  which have the same remainder on dividing by  $36$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.06697,"x":0.38391,"p":[[0,50,0.0,0.24107,0.29329,0.10714,0.14286,0.17857,0.0,1.0,8,1,0,8,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,1],[4,50,0.08,0.37054,0.38608,0.10714,0.14286,0.75,0.0,1.0,8,7,0,8,0,10,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,7],[8,50,0.16,0.38391,0.33774,0.14286,0.28571,0.5711,0.0,1.0,6,4,0,6,0,8,0,0,3,0,0,6,0,0,2,0,0,0,0,0,3,0,4],[12,50,0.24,0.35713,0.34992,0.14286,0.14286,0.57111,0.0,1.0,4,6,0,4,0,15,0,0,2,0,0,2,0,0,2,0,0,1,0,0,0,0,6],[16,50,0.32,0.19196,0.28259,0.0,0.14286,0.14286,0.0,1.0,12,3,0,12,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[20,50,0.4,0.3125,0.3597,0.0,0.14286,0.46431,0.0,1.0,11,5,1,11,0,9,0,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,5],[24,50,0.48,0.28571,0.35174,0.0,0.14286,0.42857,0.0,1.0,9,5,1,9,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,5],[28,50,0.56,0.20973,0.29449,0.0,0.14286,0.17857,0.0,1.0,12,3,1,12,0,12,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[32,50,0.64,0.1875,0.23266,0.14286,0.14286,0.14286,0.0,1.0,7,2,0,7,0,20,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[36,50,0.72,0.20089,0.25218,0.14286,0.14286,0.14286,0.0,1.0,6,2,0,6,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[40,50,0.8,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.10259,"x":0.46429,"p":[[0,47,0.0,0.27232,0.32803,0.0,0.14286,0.32143,0.0,1.0,9,3,0,9,0,14,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,3],[4,47,0.0851,0.46429,0.36943,0.14286,0.42857,0.89286,0.0,1.0,3,8,0,3,0,11,0,0,0,0,0,7,0,0,0,0,0,2,0,0,1,0,8],[8,47,0.1702,0.41517,0.35778,0.14286,0.21428,0.85704,0.0,1.0,3,5,0,3,0,13,0,0,2,0,0,4,0,0,0,0,0,1,0,0,4,0,5],[12,47,0.2553,0.45535,0.38038,0.14286,0.28571,0.89286,0.0,1.0,3,8,1,3,0,12,0,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,8],[16,47,0.3404,0.30339,0.31297,0.14286,0.14286,0.32143,0.0,1.0,5,4,1,5,0,14,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,4],[20,47,0.4255,0.23652,0.2566,0.14286,0.14286,0.2857,0.0,1.0,5,2,0,5,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[24,47,0.5106,0.19643,0.19805,0.14286,0.14286,0.17857,0.0,1.0,5,1,1,5,0,19,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[28,47,0.5957,0.25669,0.26836,0.14286,0.14286,0.28571,0.0,1.0,5,1,1,5,0,17,0,1,2,0,0,2,0,0,0,0,0,2,0,0,2,0,1],[32,47,0.6809,0.17857,0.21129,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[36,47,0.766,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.10259,0.07344,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"13267d56cc6d01f0","q":"Let  $d(n)$  be the number of all positive divisors of a natural number  $n \\ge 2$ .\nDetermine all natural numbers  $n \\ge 3$  such that   $d(n -1) + d(n) + d(n + 1) \\le  8$ .\n\n*Proposed by Richard Henner*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.58034,"x":0.76338,"p":[[0,59,0.0,0.74105,0.22429,0.57143,0.857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,4,0,0,9,0,8],[4,59,0.0678,0.76338,0.19433,0.71429,0.78564,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,10,0,0,9,0,7],[8,59,0.1356,0.69643,0.26184,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,8,0,0,1,0,0,6,0,0,5,0,9],[12,59,0.2034,0.70535,0.24206,0.53571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,5,0,0,8,0,7],[16,59,0.2712,0.6741,0.22933,0.57142,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,3,0,0,3,0,0,6,0,0,7,0,0,8,0,4],[20,59,0.339,0.67408,0.23754,0.57132,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,2,0,0,8,0,0,5,0,0,9,0,4],[24,59,0.4068,0.69185,0.26266,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,2,0,0,6,0,0,9,0,0,2,0,9],[28,59,0.4746,0.69641,0.22517,0.571,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,11,0,0,5,0,6],[32,59,0.5424,0.70536,0.22286,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,13,0,0,4,0,6],[36,59,0.6102,0.64732,0.2575,0.42859,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,0,4,0,0,5,0,0,6,0,0,8,0,4],[40,59,0.678,0.66517,0.15816,0.57142,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,10,0,0,5,0,2],[44,59,0.7458,0.58034,0.26229,0.42857,0.57143,0.75,0.14286,1.0,0,3,0,0,0,5,0,0,2,0,0,3,0,0,9,0,0,5,0,0,5,0,3],[48,59,0.8136,0.70088,0.26573,0.57132,0.78571,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,1,0,0,3,0,0,5,0,0,4,0,0,9,0,7],[52,59,0.8814,0.72318,0.19867,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,7,0,0,9,0,5],[56,59,0.9492,0.65183,0.16354,0.571,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,8,0,0,5,0,2],[59,59,1.0,0.60265,0.24675,0.5354,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,3,0,0,11,0,0,6,0,0,3,0,4]]},{"b":6,"e":0.85714,"k":"flat","v":0.65625,"x":0.79017,"p":[[0,62,0.0,0.65625,0.2392,0.4286,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,5,0,0,6,0,0,6,0,0,6,0,5],[4,62,0.0645,0.71874,0.20039,0.67857,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,9,0,0,12,0,3],[8,62,0.129,0.78569,0.15154,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,13,0,5],[12,62,0.1935,0.71873,0.1838,0.71429,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,13,0,0,9,0,3],[16,62,0.2581,0.70536,0.18536,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,9,0,0,11,0,2],[20,62,0.3226,0.75,0.14286,0.71429,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,18,0,0,8,0,3],[24,62,0.3871,0.74999,0.16368,0.71429,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,13,0,0,11,0,3],[28,62,0.4516,0.79017,0.11837,0.71429,0.85714,0.85714,0.5714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,15,0,3],[32,62,0.5161,0.72321,0.19212,0.71429,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,11,0,0,11,0,3],[36,62,0.5806,0.73214,0.1915,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,9,0,0,9,0,5],[40,62,0.6452,0.75446,0.12492,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,0,14,0,1],[44,62,0.7097,0.72321,0.15126,0.71429,0.71429,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,19,0,0,5,0,3],[48,62,0.7742,0.79016,0.12366,0.71429,0.85707,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,15,0,3],[52,62,0.8387,0.75004,0.15962,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,0,10,0,4],[56,62,0.9032,0.76339,0.13651,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,17,0,0,8,0,4],[60,62,0.9677,0.73218,0.18806,0.71429,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,13,0,0,9,0,4],[62,62,1.0,0.76785,0.12752,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,11,0,3]]}]},{"i":"457f769895fe3c04","q":"Let  $a_n = n(2n+1)$ .  Evaluate\n\\[\n  \\biggl | \\sum_{1 \\le j < k \\le 36} \\sin\\bigl( \\frac{\\pi}{6}(a_k-a_j) \\bigr) \\biggr |.\n\\]","t":[{"b":3,"e":0.42857,"k":"falling","v":0.0625,"x":0.70536,"p":[[0,42,0.0,0.52677,0.33204,0.28571,0.4998,0.85714,0.0,1.0,5,6,2,5,0,1,0,0,3,0,0,7,0,0,5,0,0,2,0,0,3,0,6],[4,42,0.0952,0.70536,0.2788,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,4,0,0,1,0,0,4,0,0,8,0,0,2,0,11],[8,42,0.1905,0.60714,0.38465,0.24999,0.71429,1.0,0.0,1.0,5,12,0,5,0,3,0,0,1,0,0,4,0,0,2,0,0,3,0,0,2,0,12],[12,42,0.2857,0.49098,0.36244,0.24999,0.35714,1.0,0.0,1.0,4,9,0,4,0,4,0,0,8,0,0,2,0,0,4,0,0,1,0,0,0,0,9],[16,42,0.381,0.67411,0.38834,0.39285,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,16],[20,42,0.4762,0.49545,0.37293,0.14214,0.50001,0.85714,0.0,1.0,7,7,0,7,0,2,0,0,5,0,0,2,0,0,3,0,0,4,0,0,2,0,7],[24,42,0.5714,0.45088,0.36789,0.14286,0.28571,0.75,0.0,1.0,6,7,0,6,0,5,0,0,6,0,0,1,0,0,4,0,0,2,0,0,1,0,7],[28,42,0.6667,0.47768,0.35644,0.14286,0.5,0.71429,0.0,1.0,7,6,0,7,0,2,0,0,5,0,0,2,0,0,3,0,0,7,0,0,0,0,6],[32,42,0.7619,0.18741,0.27068,0.0,0.07,0.28571,0.0,1.0,16,2,0,16,0,5,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[36,42,0.8571,0.13393,0.2141,0.0,0.0,0.1786,0.0,1.0,18,1,0,18,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[40,42,0.9524,0.08929,0.18472,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[42,42,1.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.29464,"x":0.6875,"p":[[0,212,0.0,0.58036,0.31931,0.42857,0.57143,0.85714,0.0,1.0,4,6,2,4,0,1,0,0,2,0,0,5,0,0,6,0,0,4,0,0,4,0,6],[4,212,0.0189,0.67408,0.215,0.5354,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,11,0,0,4,0,5],[8,212,0.0377,0.6875,0.31019,0.42857,0.71429,1.0,0.0,1.0,1,12,0,1,0,2,0,0,3,0,0,4,0,0,2,0,0,6,0,0,2,0,12],[12,212,0.0566,0.55793,0.32619,0.28571,0.71429,0.71429,0.0,1.0,5,5,0,5,0,2,0,0,2,0,0,1,0,0,5,0,0,11,0,0,1,0,5],[16,212,0.0755,0.62945,0.33286,0.39285,0.71429,1.0,0.0,1.0,2,10,0,2,0,3,0,0,3,0,0,4,0,0,2,0,0,6,0,0,2,0,10],[20,212,0.0943,0.58929,0.29827,0.39286,0.71429,0.71429,0.0,1.0,1,5,0,1,0,6,0,0,1,0,0,2,0,0,4,0,0,11,0,0,2,0,5],[24,212,0.1132,0.5357,0.29451,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,5,0,0,5,0,0,2,0,0,7,0,0,6,0,0,1,0,5],[28,212,0.1321,0.60713,0.32341,0.42857,0.64286,1.0,0.0,1.0,3,9,0,3,0,2,0,0,2,0,0,4,0,0,5,0,0,7,0,0,0,0,9],[32,212,0.1509,0.47768,0.33619,0.14286,0.42857,0.71429,0.0,1.0,5,4,0,5,0,5,0,0,2,0,0,5,0,0,3,0,0,5,0,0,3,0,4],[36,212,0.1698,0.29464,0.27879,0.0,0.28571,0.46431,0.0,1.0,9,1,0,9,0,5,0,0,9,0,0,1,0,0,3,0,0,3,0,0,1,0,1],[40,212,0.1887,0.51785,0.29612,0.28571,0.4998,0.71429,0.0,1.0,2,5,0,2,0,3,0,0,6,0,0,5,0,0,5,0,0,5,0,0,1,0,5],[44,212,0.2075,0.53571,0.29451,0.28571,0.57143,0.71429,0.0,1.0,3,3,0,3,0,3,0,0,3,0,0,4,0,0,5,0,0,8,0,0,3,0,3],[48,212,0.2264,0.42858,0.30304,0.14286,0.42859,0.60714,0.0,1.0,2,3,0,2,0,11,0,0,2,0,0,2,0,0,7,0,0,4,0,0,1,0,3],[52,212,0.2453,0.54911,0.31765,0.28571,0.57143,0.75,0.0,1.0,3,5,0,3,0,3,0,0,4,0,0,4,0,0,3,0,0,7,0,0,3,0,5],[56,212,0.2642,0.4106,0.32099,0.14286,0.42857,0.71429,0.0,1.0,6,3,0,6,0,6,0,0,3,0,0,4,0,0,4,0,0,5,0,0,1,0,3],[60,212,0.283,0.52232,0.33808,0.2857,0.49999,0.78571,0.0,1.0,2,8,0,2,0,5,0,0,7,0,0,2,0,0,4,0,0,4,0,0,0,0,8],[64,212,0.3019,0.54016,0.30875,0.28571,0.57143,0.75,0.0,1.0,2,4,0,2,0,4,0,0,6,0,0,1,0,0,5,0,0,6,0,0,4,0,4],[68,212,0.3208,0.41963,0.30709,0.14289,0.28571,0.71429,0.0,1.0,4,3,0,4,0,5,0,0,8,0,0,4,0,0,2,0,0,4,0,0,2,0,3],[72,212,0.3396,0.4732,0.32229,0.14286,0.42857,0.71429,0.0,1.0,4,3,0,4,0,5,0,0,5,0,0,3,0,0,2,0,0,7,0,0,3,0,3],[76,212,0.3585,0.33929,0.25692,0.14286,0.28571,0.46431,0.0,1.0,5,1,0,5,0,7,0,0,6,0,0,6,0,0,4,0,0,2,0,0,1,0,1],[80,212,0.3774,0.49553,0.30926,0.24999,0.42857,0.71429,0.0,1.0,1,4,0,1,0,7,0,0,6,0,0,3,0,0,3,0,0,5,0,0,3,0,4],[84,212,0.3962,0.46866,0.2994,0.14289,0.42857,0.71429,0.0,1.0,1,3,0,1,0,8,0,0,5,0,0,5,0,0,2,0,0,5,0,0,3,0,3],[88,212,0.4151,0.41516,0.24835,0.28571,0.28571,0.71429,0.0,0.85714,1,0,0,1,0,6,0,0,11,0,0,3,0,0,2,0,0,6,0,0,3,0,0],[92,212,0.434,0.42856,0.26963,0.24999,0.42857,0.71429,0.0,1.0,4,1,0,4,0,4,0,0,5,0,0,6,0,0,4,0,0,7,0,0,1,0,1],[96,212,0.4528,0.47765,0.27341,0.2857,0.4998,0.71429,0.0,1.0,1,3,0,1,0,5,0,0,8,0,0,2,0,0,7,0,0,5,0,0,1,0,3],[100,212,0.4717,0.38838,0.27018,0.14286,0.28571,0.57111,0.0,1.0,1,3,0,1,0,8,0,0,11,0,0,3,0,0,4,0,0,1,0,0,1,0,3],[104,212,0.4906,0.48661,0.30693,0.25,0.57143,0.71429,0.0,1.0,3,3,0,3,0,5,0,0,6,0,0,0,0,0,6,0,0,7,0,0,2,0,3],[108,212,0.5094,0.49107,0.30291,0.2857,0.42859,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,6,0,0,4,0,0,0,0,0,11,0,0,1,0,3],[112,212,0.5283,0.49106,0.34981,0.14286,0.57143,0.75,0.0,1.0,6,4,0,6,0,4,0,0,3,0,0,1,0,0,5,0,0,5,0,0,4,0,4],[116,212,0.5472,0.51339,0.30064,0.2857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,5,0,0,5,0,0,2,0,0,5,0,0,8,0,0,1,0,4],[120,212,0.566,0.45536,0.30397,0.14286,0.42857,0.71429,0.0,1.0,2,4,1,2,0,7,0,0,5,0,0,6,0,0,2,0,0,5,0,0,1,0,4],[124,212,0.5849,0.48661,0.25719,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,9,0,0,4,0,0,1,0,0,9,0,0,3,0,1],[128,212,0.6038,0.50445,0.31132,0.28571,0.42857,0.71429,0.0,1.0,2,5,0,2,0,4,0,0,8,0,0,3,0,0,2,0,0,7,0,0,1,0,5],[132,212,0.6226,0.42855,0.31134,0.14286,0.35714,0.71429,0.0,1.0,4,3,0,4,0,6,0,0,6,0,0,3,0,0,4,0,0,4,0,0,2,0,3],[136,212,0.6415,0.54017,0.30458,0.28571,0.57143,0.71429,0.0,1.0,3,5,0,3,0,3,0,0,4,0,0,1,0,0,9,0,0,6,0,0,1,0,5],[140,212,0.6604,0.41964,0.27879,0.14286,0.35714,0.71429,0.0,1.0,4,1,0,4,0,5,0,0,7,0,0,1,0,0,6,0,0,7,0,0,1,0,1],[144,212,0.6792,0.56696,0.28456,0.28571,0.64286,0.71429,0.0,1.0,1,5,0,1,0,4,0,0,4,0,0,3,0,0,4,0,0,11,0,0,0,0,5],[148,212,0.6981,0.46429,0.32537,0.28571,0.28571,0.75,0.0,1.0,3,5,0,3,0,3,0,0,12,0,0,2,0,0,2,0,0,2,0,0,3,0,5],[152,212,0.717,0.41964,0.30501,0.14289,0.28571,0.71429,0.0,1.0,5,1,0,5,0,5,0,0,7,0,0,2,0,0,0,0,0,10,0,0,2,0,1],[156,212,0.7358,0.50446,0.31539,0.25,0.5,0.71429,0.0,1.0,2,5,0,2,0,6,0,0,5,0,0,3,0,0,2,0,0,9,0,0,0,0,5],[160,212,0.7547,0.54911,0.30328,0.28571,0.57143,0.75,0.0,1.0,2,4,0,2,0,3,0,0,6,0,0,3,0,0,3,0,0,7,0,0,4,0,4],[164,212,0.7736,0.51783,0.27836,0.2857,0.49979,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,9,0,0,3,0,0,2,0,0,9,0,0,2,0,3],[168,212,0.7925,0.50893,0.29437,0.2857,0.5,0.71429,0.14286,1.0,0,4,0,0,0,7,0,0,7,0,0,2,0,0,2,0,0,9,0,0,1,0,4],[172,212,0.8113,0.50446,0.27195,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,11,0,0,4,0,0,4,0,0,4,0,0,2,0,4],[176,212,0.8302,0.46874,0.32189,0.24999,0.42857,0.71429,0.0,1.0,4,3,0,4,0,4,0,0,7,0,0,3,0,0,2,0,0,5,0,0,4,0,3],[180,212,0.8491,0.52232,0.25407,0.28571,0.64286,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,7,0,0,4,0,0,1,0,0,13,0,0,2,0,1],[184,212,0.8679,0.43749,0.28107,0.14286,0.28571,0.71429,0.0,1.0,1,2,0,1,0,8,0,0,8,0,0,2,0,0,2,0,0,8,0,0,1,0,2],[188,212,0.8868,0.44195,0.26572,0.28571,0.42857,0.60714,0.0,1.0,3,1,0,3,0,3,0,0,8,0,0,5,0,0,5,0,0,4,0,0,3,0,1],[192,212,0.9057,0.45088,0.28818,0.2857,0.28571,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,11,0,0,2,0,0,4,0,0,4,0,0,2,0,3],[196,212,0.9245,0.44643,0.25442,0.25,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,8,0,0,6,0,0,5,0,0,4,0,0,7,0,0,0,0,2],[200,212,0.9434,0.47321,0.3102,0.2857,0.35714,0.71429,0.0,1.0,3,4,0,3,0,3,0,0,10,0,0,3,0,0,0,0,0,8,0,0,1,0,4],[204,212,0.9623,0.49106,0.29653,0.28571,0.49979,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,8,0,0,2,0,0,5,0,0,6,0,0,1,0,4],[208,212,0.9811,0.49554,0.30511,0.25001,0.42857,0.71429,0.0,1.0,1,5,0,1,0,7,0,0,4,0,0,6,0,0,3,0,0,5,0,0,1,0,5],[212,212,1.0,0.32143,0.17496,0.2857,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,4,0,0,22,0,0,0,0,0,2,0,0,2,0,0,1,0,0]]}]},{"i":"18556cf345ac52e7","q":"Let  $f(x)=\\frac{1+\\cos(2 \\pi x)}{2}$ , for  $x \\in \\mathbb{R}$ , and  $f^n=\\underbrace{ f \\circ \\cdots \\circ f}_{n}$ . Is it true that for Lebesgue almost every  $x$ ,  $\\lim_{n \\to \\infty} f^n(x)=1$ ?","t":[{"b":2,"e":0.57143,"k":"rising","v":0.09375,"x":0.70981,"p":[[0,23,0.0,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.50447,0.29877,0.28571,0.4286,0.85714,0.0,1.0,4,1,0,4,0,2,0,0,3,0,0,9,0,0,2,0,0,3,0,0,8,0,1],[8,23,0.3478,0.54017,0.30458,0.28571,0.64286,0.71429,0.0,1.0,3,3,0,3,0,2,0,0,7,0,0,1,0,0,3,0,0,9,0,0,4,0,3],[12,23,0.5217,0.59375,0.28818,0.42859,0.57143,0.85714,0.0,1.0,3,3,0,3,0,1,0,0,3,0,0,2,0,0,8,0,0,5,0,0,7,0,3],[16,23,0.6957,0.66963,0.30397,0.42857,0.78571,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,3,0,0,4,0,0,2,0,0,4,0,0,9,0,7],[20,23,0.8696,0.69193,0.23988,0.571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,5,0,0,4,0,0,7,0,0,10,0,4],[23,23,1.0,0.70981,0.18724,0.71429,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,13,0,0,11,0,1]]},{"b":7,"e":1.0,"k":"volatile","v":0.06687,"x":0.69196,"p":[[0,23,0.0,0.06687,0.09432,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.5312,0.32583,0.24999,0.571,0.85714,0.0,1.0,2,5,0,2,0,6,0,0,4,0,0,2,0,0,5,0,0,4,0,0,4,0,5],[8,23,0.3478,0.44197,0.26088,0.2857,0.4286,0.57143,0.0,0.85714,4,0,0,4,0,2,0,0,6,0,0,6,0,0,8,0,0,1,0,0,5,0,0],[12,23,0.5217,0.37945,0.23312,0.28571,0.42857,0.4642,0.0,1.0,4,1,0,4,0,3,0,0,7,0,0,10,0,0,5,0,0,1,0,0,1,0,1],[16,23,0.6957,0.32126,0.27444,0.14,0.28571,0.46431,0.0,1.0,7,1,0,7,0,7,0,0,5,0,0,5,0,0,3,0,0,3,0,0,1,0,1],[20,23,0.8696,0.6607,0.26904,0.57132,0.71429,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,1,0,0,3,0,0,6,0,0,6,0,0,9,0,4],[23,23,1.0,0.69196,0.20858,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,8,0,0,8,0,4]]}]},{"i":"eff55033fe23beef","q":"Let  $f,g:[a,b]\\to [0,\\infty)$  be two continuous and non-decreasing functions such that each  $x\\in [a,b]$  we have \r\n\\[ \\int^x_a \\sqrt { f(t) }\\ dt \\leq \\int^x_a \\sqrt { g(t) }\\ dt  \\ \\  \\textrm{and}\\ \\int^b_a \\sqrt {f(t)}\\ dt = \\int^b_a \\sqrt { g(t)}\\ dt. \\]\r\n\r\nProve that \r\n\\[ \\int^b_a \\sqrt { 1+ f(t) }\\ dt \\geq \\int^b_a \\sqrt { 1 + g(t) }\\ dt. \\]","t":[{"b":1,"e":0.4286,"k":"flat","v":0.38397,"x":0.56018,"p":[[0,29,0.0,0.39732,0.23887,0.2857,0.35714,0.42857,0.14286,1.0,0,3,0,0,0,6,0,0,10,0,0,12,0,0,0,0,0,0,0,0,1,0,3],[4,29,0.1379,0.45759,0.22578,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,7,0,1,9,0,0,5,0,0,3,0,0,1,0,2],[8,29,0.2759,0.56018,0.25276,0.42857,0.5,0.75,0.14,1.0,0,3,0,0,0,2,0,1,4,0,0,9,0,0,4,0,0,4,0,0,5,0,3],[12,29,0.4138,0.47758,0.28043,0.28571,0.42857,0.57143,0.0,1.0,1,5,0,1,0,4,0,0,6,0,0,11,0,0,3,0,0,1,0,0,1,0,5],[16,29,0.5517,0.38397,0.15022,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,1,15,0,1,8,0,0,4,0,0,1,0,0,1,0,0],[20,29,0.6897,0.43527,0.20852,0.28571,0.28571,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,17,0,1,6,0,0,2,0,0,1,0,0,5,0,0],[24,29,0.8276,0.48213,0.19479,0.28571,0.42857,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,9,0,0,9,0,0,6,0,0,5,0,0,1,0,1],[28,29,0.9655,0.45982,0.20434,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,12,0,0,8,0,0,5,0,0,3,0,0,2,0,1],[29,29,1.0,0.45981,0.17398,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,13,0,0,8,0,0,0,0,0,3,0,0]]},{"b":2,"e":0.85714,"k":"rising","v":0.37053,"x":0.65176,"p":[[0,39,0.0,0.37053,0.23381,0.24999,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,8,0,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,3],[4,39,0.1026,0.5,0.26,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,11,0,0,3,0,0,3,0,0,3,0,3],[8,39,0.2051,0.62054,0.26149,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,7,0,0,8,0,0,3,0,0,1,0,8],[12,39,0.3077,0.65176,0.25739,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,7,0,0,5,0,0,6,0,0,2,0,8],[16,39,0.4103,0.64731,0.27661,0.42857,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,5,0,0,6,0,0,4,0,0,5,0,0,2,0,9],[20,39,0.5128,0.56473,0.23714,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,3,0,1,13,0,0,3,0,0,4,0,0,3,0,4],[24,39,0.6154,0.58035,0.24984,0.42857,0.50001,0.74996,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,12,0,0,5,0,0,3,0,0,3,0,5],[28,39,0.7179,0.46428,0.17495,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,19,0,0,3,0,0,1,0,0,2,0,1],[32,39,0.8205,0.44197,0.13533,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,24,0,0,2,0,0,1,0,0,0,0,1],[36,39,0.9231,0.46875,0.20589,0.42857,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,20,0,0,1,0,0,2,0,0,0,0,3],[39,39,1.0,0.57141,0.19884,0.42857,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,16,0,0,6,0,0,3,0,0,3,0,3]]}]},{"i":"40a6b58e69723411","q":"Let  $k\\ge 0$  be an integer and suppose the integers  $a_1,a_2,\\dots,a_n$  give at least  $2k$  different residues upon division by  $(n+k)$ . Show that there are some  $a_i$  whose sum is divisible by  $n+k$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.12946,"x":0.33036,"p":[[0,53,0.0,0.12946,0.09688,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.18741,0.20343,0.0,0.14286,0.2857,0.0,1.0,9,1,0,9,0,13,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[8,53,0.1509,0.22765,0.18156,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,12,0,0,7,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[12,53,0.2264,0.23661,0.21312,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,11,0,0,7,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[16,53,0.3019,0.33036,0.24856,0.14286,0.28571,0.42858,0.0,1.0,4,1,0,4,0,9,0,0,5,0,0,8,0,0,2,0,0,2,0,0,1,0,1],[20,53,0.3774,0.25444,0.20115,0.14286,0.2143,0.28571,0.0,0.71429,5,0,0,5,0,11,0,0,10,0,0,0,0,0,4,0,0,2,0,0,0,0,0],[24,53,0.4528,0.22098,0.178,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,14,0,0,3,0,0,7,0,1,0,0,0,1,0,0,0,0,0],[28,53,0.5283,0.19187,0.16605,0.105,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,13,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[32,53,0.6038,0.19643,0.17405,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,15,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[36,53,0.6792,0.17402,0.16264,0.0,0.14286,0.28571,0.0,0.57143,11,0,0,11,0,9,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[40,53,0.7547,0.29018,0.23551,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,7,0,0,6,0,0,6,0,0,3,0,0,2,0,0,1,0,0],[44,53,0.8302,0.2142,0.21132,0.14214,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,15,0,0,4,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[48,53,0.9057,0.32588,0.25312,0.14286,0.28571,0.4642,0.0,0.85714,6,0,0,6,0,7,0,0,5,0,0,6,0,0,4,0,0,2,0,0,2,0,0],[52,53,0.9811,0.22317,0.18529,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,9,0,0,8,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[53,53,1.0,0.17847,0.14284,0.14214,0.14286,0.28571,0.0,0.571,7,0,0,7,0,15,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.0892,"x":0.22768,"p":[[0,37,0.0,0.0892,0.09274,0.0,0.14143,0.14286,0.0,0.28571,15,0,1,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.14063,0.10486,0.14286,0.14286,0.14287,0.0,0.42857,7,0,0,7,0,20,0,1,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.22768,0.21086,0.14286,0.14286,0.28571,0.0,0.85714,6,0,0,6,0,13,0,0,8,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[12,37,0.3243,0.20535,0.15123,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,18,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[16,37,0.4324,0.19188,0.19436,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,13,0,0,5,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[20,37,0.5405,0.17848,0.19235,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,12,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[24,37,0.6486,0.16498,0.15197,0.0,0.14286,0.2857,0.0,0.571,10,0,0,10,0,12,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,37,0.7568,0.1875,0.1357,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,15,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.15625,0.12037,0.0,0.14286,0.2857,0.0,0.4286,9,0,0,9,0,12,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.16738,0.14566,0.05357,0.14286,0.2857,0.0,0.571,8,0,0,8,1,13,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[37,37,1.0,0.19632,0.16656,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,9,0,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"5884a798375bbf50","q":"Let  $f: \\mathbb{R}\\rightarrow ]0,+\\infty[$  be an increasing differentiable function with  $\\lim_{x\\rightarrow+\\infty}f(x)=+\\infty$  and  $f'$  is bounded, and let  $F(x)=\\int^x_0 f(t) dt$ .\r\nDefine the sequence  $(a_n)$  recursively by  $a_0=1,a_{n+1}=a_n+\\frac1{f(a_n)}$ \r\nDefine the sequence  $(b_n)$  by  $b_n=F^{-1}(n)$ .\r\nProve that  $\\lim_{x\\rightarrow+\\infty}(a_n-b_n)=0$ .","t":[{"b":0,"e":0.71429,"k":"falling","v":0.37947,"x":0.65621,"p":[[0,59,0.0,0.58927,0.24156,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,7,0,0,6,0,0,7,0,0,4,0,3],[4,59,0.0678,0.54909,0.26027,0.28571,0.57121,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,6,0,0,6,0,0,5,0,0,6,0,0,2,0,4],[8,59,0.1356,0.63392,0.29653,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,4,0,0,4,0,0,6,0,0,3,0,8],[12,59,0.2034,0.63836,0.29448,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,3,0,0,4,0,0,8,0,0,1,0,0,5,0,8],[16,59,0.2712,0.56693,0.23278,0.42857,0.49979,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,5,0,0,10,0,0,3,0,0,7,0,0,3,0,3],[20,59,0.339,0.52676,0.28445,0.28571,0.57121,0.74999,0.0,1.0,1,3,0,1,0,5,0,0,4,0,0,4,0,0,8,0,0,2,0,0,5,0,3],[24,59,0.4068,0.65621,0.26931,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,5,0,0,4,0,0,6,0,0,4,0,0,4,0,8],[28,59,0.4746,0.62499,0.31894,0.39286,0.57143,1.0,0.0,1.0,1,11,0,1,0,2,0,0,5,0,0,5,0,0,5,0,0,2,0,0,1,0,11],[32,59,0.5424,0.53113,0.28857,0.28571,0.42859,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,9,0,0,5,0,0,3,0,0,5,0,0,2,0,5],[36,59,0.6102,0.48661,0.24964,0.28571,0.4286,0.60714,0.0,1.0,1,2,0,1,0,4,0,0,5,0,0,7,0,0,7,0,0,4,0,0,2,0,2],[40,59,0.678,0.51337,0.26692,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,8,0,0,8,0,0,4,0,0,2,0,0,3,0,4],[44,59,0.7458,0.46425,0.25252,0.28571,0.42857,0.57143,0.0,1.0,3,3,0,3,0,1,0,0,5,0,0,10,0,0,9,0,0,0,0,0,1,0,3],[48,59,0.8136,0.45525,0.29769,0.14289,0.42857,0.57143,0.0,1.0,3,4,0,3,0,6,0,0,2,0,0,8,0,0,6,0,0,2,0,0,1,0,4],[52,59,0.8814,0.56247,0.23673,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,3,0,0,9,0,0,5,0,0,3,0,3],[56,59,0.9492,0.57585,0.2244,0.4286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,2,0,0,5,0,0,11,0,0,5,0,0,4,0,2],[59,59,1.0,0.37947,0.17353,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,4,0,0,7,0,0,16,0,0,3,0,0,0,0,0,0,0,1]]},{"b":1,"e":0.4286,"k":"flat","v":0.48212,"x":0.67409,"p":[[0,60,0.0,0.63391,0.25739,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,4,0,0,4,0,0,5,0,0,7,0,0,5,0,5],[4,60,0.0667,0.61604,0.27765,0.42857,0.57121,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,7,0,0,4,0,0,4,0,0,2,0,8],[8,60,0.1333,0.57137,0.2812,0.39286,0.57121,0.857,0.0,1.0,1,4,0,1,0,3,0,0,4,0,0,5,0,0,6,0,0,4,0,0,5,0,4],[12,60,0.2,0.67409,0.2402,0.4286,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,4,0,0,7,0,6],[16,60,0.2667,0.61826,0.19694,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,1,7,0,0,8,0,0,8,0,0,3,0,3],[20,60,0.3333,0.62052,0.27574,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,5,0,0,8,0,0,1,0,0,3,0,8],[24,60,0.4,0.63392,0.2549,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,6,0,0,6,0,0,6,0,0,2,0,7],[28,60,0.4667,0.57137,0.25505,0.42857,0.571,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,5,0,0,6,0,0,6,0,0,7,0,0,2,0,4],[32,60,0.5333,0.58479,0.27048,0.42859,0.57143,0.857,0.0,1.0,1,5,0,1,0,3,0,0,1,0,0,6,0,0,11,0,0,1,0,0,4,0,5],[36,60,0.6,0.58921,0.22518,0.42859,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,4,0,0,11,0,0,4,0,0,4,0,3],[40,60,0.6667,0.52225,0.23312,0.42857,0.49979,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,9,0,0,9,0,0,2,0,0,2,0,3],[44,60,0.7333,0.58929,0.26904,0.42857,0.57143,0.71429,0.0,1.0,1,6,0,1,0,1,0,0,4,0,0,8,0,0,4,0,0,7,0,0,1,0,6],[48,60,0.8,0.48212,0.23076,0.39286,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,12,0,0,7,0,0,1,0,0,1,0,3],[52,60,0.8667,0.50444,0.2201,0.42857,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,12,0,0,7,0,0,2,0,0,1,0,3],[56,60,0.9333,0.49103,0.2394,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,9,0,0,7,0,0,1,0,0,2,0,3],[60,60,1.0,0.6026,0.19801,0.53539,0.57143,0.60714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,5,0,0,16,0,0,2,0,0,2,0,4]]}]},{"i":"7ddf966afc115cc7","q":"Let  $n > 3$  be a natural number. Prove that  $4^n + 1$  has a prime divisor  $> 20$ .","t":[{"b":4,"e":0.0,"k":"falling","v":0.27677,"x":0.75445,"p":[[0,21,0.0,0.73659,0.2769,0.5354,0.85707,1.0,0.0,1.0,1,11,0,1,0,1,0,0,1,0,0,5,0,0,1,0,0,6,0,0,6,0,11],[4,21,0.1905,0.70984,0.28228,0.57132,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,2,0,0,6,0,0,6,0,0,3,0,11],[8,21,0.381,0.75445,0.27948,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,3,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,12],[12,21,0.5714,0.67853,0.21726,0.67846,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,18,0,0,1,0,5],[16,21,0.7619,0.74094,0.24339,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,0,0,0,3,0,0,15,0,0,1,0,10],[20,21,0.9524,0.29018,0.3164,0.0,0.2143,0.50002,0.0,1.0,12,2,0,12,0,4,0,0,7,0,0,1,0,0,0,0,0,6,0,0,0,0,2],[21,21,1.0,0.27677,0.28778,0.0,0.2857,0.57111,0.0,1.0,13,1,0,13,0,1,0,0,9,0,0,0,0,0,4,0,0,4,0,0,0,0,1]]},{"b":7,"e":0.57143,"k":"falling","v":0.51327,"x":0.76338,"p":[[0,29,0.0,0.76338,0.23039,0.67857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,10,0,0,2,0,12],[4,29,0.1379,0.6071,0.31944,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,2,0,0,2,0,0,4,0,0,8,0,0,0,0,9],[8,29,0.2759,0.54463,0.30605,0.28571,0.64286,0.71429,0.0,1.0,3,4,0,3,0,3,0,0,4,0,0,3,0,0,3,0,0,10,0,0,2,0,4],[12,29,0.4138,0.57587,0.30406,0.24999,0.64286,0.71429,0.0,1.0,1,5,0,1,0,7,0,0,1,0,0,0,0,0,7,0,0,9,0,0,2,0,5],[16,29,0.5517,0.62945,0.31311,0.39287,0.71429,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,2,0,0,1,0,0,4,0,0,9,0,0,2,0,8],[20,29,0.6897,0.54908,0.36615,0.14286,0.57143,1.0,0.0,1.0,3,9,0,3,0,8,0,0,1,0,0,0,0,0,5,0,0,6,0,0,0,0,9],[24,29,0.8276,0.58033,0.28334,0.28571,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,4,0,0,7,0,0,1,0,0,2,0,0,12,0,0,1,0,5],[28,29,0.9655,0.62051,0.2412,0.57143,0.71429,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,1,0,0,1,0,0,10,0,0,13,0,0,0,0,4],[29,29,1.0,0.51327,0.20469,0.28571,0.571,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,7,0,0,1,0,0,12,0,0,10,0,0,0,0,0]]}]},{"i":"f103ab549892c0c1","q":"Let  $r, s,$  and  $t$  be the distinct roots of  $x^3- 2022x^2 + 2022x + 2022.$  Compute   $$ \\frac{1}{1-r^2} + \\frac{1}{1-s^2} + \\frac{1}{1-t^2}. $$","t":[{"b":2,"e":0.42857,"k":"flat","v":0.21429,"x":0.32143,"p":[[0,48,0.0,0.32143,0.25254,0.14286,0.14286,0.57143,0.14286,1.0,0,2,0,0,0,19,0,0,2,0,0,1,0,0,8,0,0,0,0,0,0,0,2],[4,48,0.0833,0.26339,0.16016,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,18,0,0,6,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[8,48,0.1667,0.25893,0.16146,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,18,0,0,8,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[12,48,0.25,0.22321,0.15542,0.14286,0.14286,0.17857,0.14286,0.57143,0,0,0,0,0,24,0,0,3,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[16,48,0.3333,0.24098,0.14039,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,18,0,0,10,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[20,48,0.4167,0.25893,0.16536,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,19,0,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0],[24,48,0.5,0.24554,0.16065,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,4,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[28,48,0.5833,0.26339,0.16409,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,18,0,0,7,0,0,1,0,0,6,0,0,0,0,0,0,0,0],[32,48,0.6667,0.26786,0.18123,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,19,0,0,6,0,0,0,0,0,6,0,0,1,0,0,0,0,0],[36,48,0.75,0.28125,0.16164,0.14286,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,15,0,0,9,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[40,48,0.8333,0.24107,0.15335,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,20,0,0,7,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[44,48,0.9167,0.21429,0.11845,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[48,48,1.0,0.23214,0.14174,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,20,0,0,8,0,0,0,0,0,4,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.20982,"x":0.34821,"p":[[0,33,0.0,0.34821,0.2765,0.14286,0.14286,0.57143,0.14286,1.0,0,3,0,0,0,17,0,0,4,0,0,1,0,0,6,0,0,1,0,0,0,0,3],[4,33,0.1212,0.25893,0.16536,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,19,0,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0],[8,33,0.2424,0.24554,0.17582,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,22,0,0,4,0,0,0,0,0,5,0,0,1,0,0,0,0,0],[12,33,0.3636,0.3125,0.19704,0.14286,0.14286,0.57143,0.14286,0.57143,0,0,0,0,0,17,0,0,3,0,0,1,0,0,11,0,0,0,0,0,0,0,0],[16,33,0.4848,0.24107,0.15335,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,20,0,0,7,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[20,33,0.6061,0.24107,0.16536,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,22,0,0,4,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[24,33,0.7273,0.23214,0.15465,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,22,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[28,33,0.8485,0.21875,0.12869,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,8,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[32,33,0.9697,0.20982,0.11285,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[33,33,1.0,0.21429,0.13363,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,23,0,0,5,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"f71cc3fc6e1ed22b","q":"Let  $n \\geq 2$  be a given integer $a)$  Prove that one can arrange all the subsets of the set  $\\{1,2... ,n\\}$  as a sequence of subsets  $A_{1}, A_{2},\\cdots , A_{2^{n}}$ , such that  $|A_{i+1}| = |A_{i}| + 1$  or  $|A_{i}| - 1$  where  $i = 1,2,3,\\cdots , 2^{n}$  and  $A_{2^{n} + 1} = A_{1}$  $b)$  Determine all possible values of the sum  $\\sum \\limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$  where  $S(A_{i})$  denotes the sum of all elements in  $A_{i}$  and  $S(\\emptyset) = 0$ , for any subset sequence  $A_{1},A_{2},\\cdots ,A_{2^n}$  satisfying the condition in  $a)$","t":[{"b":2,"e":0.71429,"k":"flat","v":0.96429,"x":1.0,"p":[[0,69,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,69,0.058,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,69,0.1159,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,69,0.1739,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,69,0.2319,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,69,0.2899,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,69,0.3478,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,69,0.4058,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,69,0.4638,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[36,69,0.5217,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[40,69,0.5797,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,69,0.6377,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,69,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,69,0.7536,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[56,69,0.8116,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,69,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,69,0.9275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,69,0.9855,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[69,69,1.0,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]},{"b":7,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,28,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,28,0.4286,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,28,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,28,0.7143,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,28,0.8571,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,28,1.0,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25]]}]},{"i":"649b3bf5e6a6f126","q":"Let  $n$  and  $k$  be two positive integers such that  $1\\leq n \\leq k$ . Prove that, if  $d^k+k$  is a prime number for each positive divisor  $d$  of  $n$ , then  $n+k$  is a prime number.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.1517,"x":1.0,"p":[[0,37,0.0,0.44196,0.36133,0.14286,0.14288,0.78571,0.14286,1.0,0,8,0,0,0,17,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,8],[4,37,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,37,0.2162,0.90179,0.23266,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,26],[12,37,0.3243,0.81696,0.32972,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,24],[16,37,0.4324,0.81241,0.33793,0.89286,1.0,1.0,0.14,1.0,0,24,0,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,24],[20,37,0.5405,0.74553,0.37069,0.28571,1.0,1.0,0.14286,1.0,0,21,0,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,21],[24,37,0.6486,0.76786,0.37072,0.5,1.0,1.0,0.0,1.0,1,22,0,1,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,22],[28,37,0.7568,0.61161,0.38338,0.14286,0.57143,1.0,0.14286,1.0,0,15,0,0,0,9,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,15],[32,37,0.8649,0.27677,0.29218,0.14286,0.14286,0.14287,0.14286,1.0,0,4,0,0,0,26,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4],[36,37,0.973,0.24098,0.25865,0.14286,0.14286,0.14287,0.0,1.0,1,3,0,1,0,25,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[37,37,1.0,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.25446,"x":0.98214,"p":[[0,73,0.0,0.40624,0.37134,0.14286,0.14286,0.78571,0.0,1.0,1,8,0,1,0,18,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,8],[4,73,0.0548,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[8,73,0.1096,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[12,73,0.1644,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,73,0.2192,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[20,73,0.274,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[24,73,0.3288,0.95981,0.16461,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[28,73,0.3836,0.94642,0.1777,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[32,73,0.4384,0.83929,0.28738,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,23],[36,73,0.4932,0.89732,0.22654,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,26],[40,73,0.5479,0.86607,0.26471,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,25],[44,73,0.6027,0.89284,0.25002,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,26],[48,73,0.6575,0.83036,0.3102,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,24],[52,73,0.7123,0.78572,0.31339,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,21],[56,73,0.7671,0.82134,0.31155,0.78571,1.0,1.0,0.14,1.0,0,23,0,0,0,3,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,23],[60,73,0.8219,0.79464,0.31931,0.53571,1.0,1.0,0.14286,1.0,0,22,0,0,0,4,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,22],[64,73,0.8767,0.54018,0.35126,0.14286,0.57143,1.0,0.0,1.0,1,10,0,1,0,8,0,0,4,0,0,1,0,0,8,0,0,0,0,0,0,0,10],[68,73,0.9315,0.45071,0.36456,0.14286,0.28571,1.0,0.14,1.0,0,9,0,0,0,14,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[72,73,0.9863,0.47767,0.34921,0.14286,0.28571,0.89286,0.14286,1.0,0,8,0,0,0,12,0,0,5,0,0,1,0,0,5,0,0,0,0,0,1,0,8],[73,73,1.0,0.25446,0.23072,0.14286,0.14286,0.2857,0.14286,1.0,0,2,0,0,0,23,0,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,2]]}]},{"i":"c0ca001e2c4d947d","q":"Let  $s$  and  $t$  be two parallel lines. We have marked  $k$  points on line  $s$  and  $n$  points on line  $t$   ( $k\\geq n$ ). If it is known that the total number of triangles that have their three vertices at marked points is  $220$ , find all possible values of  $k$  and  $n$ .","t":[{"b":5,"e":1.0,"k":"falling","v":0.79018,"x":0.99107,"p":[[0,60,0.0,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[4,60,0.0667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,60,0.1333,0.92856,0.15571,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,26],[12,60,0.2,0.92409,0.15969,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26],[16,60,0.2667,0.92856,0.15155,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,26],[20,60,0.3333,0.89286,0.17496,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,2,0,22],[24,60,0.4,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[28,60,0.4667,0.91518,0.15093,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,24],[32,60,0.5333,0.82588,0.22229,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,1,0,19],[36,60,0.6,0.88391,0.18015,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,1,0,22],[40,60,0.6667,0.84821,0.18877,0.67857,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,0,0,19],[44,60,0.7333,0.85265,0.19397,0.57143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,0,0,20],[48,60,0.8,0.79463,0.19866,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,0,5,0,13],[52,60,0.8667,0.87946,0.17536,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,1,0,21],[56,60,0.9333,0.83927,0.18474,0.67857,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,2,0,17],[60,60,1.0,0.79018,0.21719,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,3,0,0,2,0,15]]},{"b":7,"e":1.0,"k":"falling","v":0.66963,"x":0.99107,"p":[[0,78,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,78,0.0513,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,78,0.1026,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,78,0.1538,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,78,0.2051,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[20,78,0.2564,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[24,78,0.3077,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,28],[28,78,0.359,0.91518,0.16698,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,25],[32,78,0.4103,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[36,78,0.4615,0.84375,0.2,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,1,0,19],[40,78,0.5128,0.90625,0.17353,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,1,0,24],[44,78,0.5641,0.87054,0.20627,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,5,0,0,0,0,22],[48,78,0.6154,0.87054,0.19019,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,1,0,21],[52,78,0.6667,0.87054,0.20316,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,3,0,0,0,0,22],[56,78,0.7179,0.88393,0.18013,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,0,0,22],[60,78,0.7692,0.88392,0.18015,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,1,0,22],[64,78,0.8205,0.87053,0.18681,0.67857,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,1,0,21],[68,78,0.8718,0.76339,0.18073,0.57143,0.71429,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,10,0,0,2,0,10],[72,78,0.9231,0.78572,0.20516,0.57143,0.71429,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,6,0,0,1,0,14],[76,78,0.9744,0.71426,0.18213,0.57143,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,10,0,0,0,0,8],[78,78,1.0,0.66963,0.14033,0.57143,0.64286,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,10,0,0,4,0,2]]}]},{"i":"7fdc3148aae0746a","q":"Let  $x$  be an irrational number between 0 and 1 and  $x = 0.a_1a_2a_3\\cdots$  its decimal representation. For each  $k \\ge 1$ , let  $p(k)$  denote the number of distinct sequences  $a_{j+1} a_{j+2} \\cdots a_{j+k}$  of  $k$  consecutive digits in the decimal representation of  $x$ . Prove that  $p(k) \\ge k+1$  for every positive integer  $k$ .","t":[{"b":1,"e":0.57143,"k":"rising","v":0.23652,"x":0.54909,"p":[[0,21,0.0,0.23652,0.26152,0.105,0.14286,0.28571,0.0,1.0,8,1,0,8,0,12,0,0,7,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[4,21,0.1905,0.53122,0.23752,0.28571,0.4998,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,12,0,0,4,0,0,4,0,0,8,0,0,1,0,3],[8,21,0.381,0.54909,0.22335,0.39286,0.4286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,9,0,0,3,0,0,5,0,0,6,0,1],[12,21,0.5714,0.48659,0.22829,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,13,0,0,5,0,0,3,0,0,6,0,0,3,0,1],[16,21,0.7619,0.43074,0.17716,0.28571,0.35714,0.571,0.2857,0.85714,0,0,0,0,0,0,0,0,16,0,0,6,0,0,5,1,0,2,0,0,2,0,0],[20,21,0.9524,0.44641,0.17034,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,8,0,0,7,0,0,2,0,0,2,0,0],[21,21,1.0,0.46875,0.20277,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,12,0,0,3,0,0,3,0,0,1,0,2]]},{"b":3,"e":0.28571,"k":"flat","v":0.30804,"x":0.5625,"p":[[0,12,0.0,0.30804,0.24513,0.14286,0.2857,0.42857,0.0,0.85714,3,0,0,3,0,12,0,0,7,0,0,5,0,0,0,0,0,2,0,0,3,0,0],[4,12,0.3333,0.5625,0.23941,0.39286,0.50001,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,8,0,0,5,0,0,3,0,0,5,0,3],[8,12,0.6667,0.42855,0.1956,0.28571,0.42857,0.46418,0.14286,0.85714,0,0,0,0,0,3,0,0,10,0,0,11,0,0,3,0,0,2,0,0,3,0,0],[12,12,1.0,0.41963,0.15541,0.28571,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,11,0,0,5,0,0,2,0,0,1,0,0]]}]},{"i":"3e5bde4aaab07f2a","q":"Let  $r$  be a positive integer. Find the smallest positive integer  $m$  satisfying the condition: For all sets  $A_1, A_2, \\dots, A_r$  with  $A_i \\cap A_j = \\emptyset$ , for all  $i \\neq j$ , and  $\\bigcup_{k = 1}^{r} A_k = \\{ 1, 2, \\dots, m \\}$ , there exists  $a, b \\in A_k$  for some  $k$  such that  $1 \\leq \\frac{b}{a} \\leq 1 + \\frac{1}{2022}$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.72321,"p":[[0,33,0.0,0.41518,0.27049,0.14286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,13,0,0,1,0,0,5,0,0,8,0,0,1,0,0,2,0,2],[4,33,0.1212,0.72321,0.21706,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,6,0,0,5,0,8],[8,33,0.2424,0.66517,0.29582,0.53569,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,1,0,0,2,0,0,6,0,0,4,0,0,6,0,8],[12,33,0.3636,0.54017,0.30874,0.25,0.57143,0.74996,0.0,1.0,2,4,0,2,0,6,0,0,2,0,0,1,0,0,9,0,0,4,0,0,4,0,4],[16,33,0.4848,0.53571,0.31944,0.24999,0.57143,0.85714,0.0,1.0,2,4,0,2,0,6,0,0,2,0,0,4,0,0,7,0,0,0,0,0,7,0,4],[20,33,0.6061,0.43749,0.36932,0.10714,0.42857,0.71429,0.0,1.0,8,5,0,8,0,5,0,0,2,0,0,3,0,0,2,0,0,5,0,0,2,0,5],[24,33,0.7273,0.24999,0.29232,0.0,0.14286,0.57143,0.0,1.0,15,1,0,15,0,5,0,0,0,0,0,1,0,0,9,0,0,1,0,0,0,0,1],[28,33,0.8485,0.14286,0.27894,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,2],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.00893,"x":0.69643,"p":[[0,37,0.0,0.46418,0.29676,0.14286,0.57121,0.57143,0.14,1.0,0,5,0,0,0,12,0,0,0,0,0,3,0,0,12,0,0,0,0,0,0,0,5],[4,37,0.1081,0.65625,0.22263,0.5357,0.57143,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,7,0,0,10,0,0,6,0,0,1,0,7],[8,37,0.2162,0.69643,0.27606,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,7,0,0,4,0,0,4,0,0,3,0,11],[12,37,0.3243,0.55353,0.30878,0.39286,0.5712,0.85714,0.0,1.0,2,5,0,2,0,5,0,0,1,0,0,5,0,0,7,0,0,3,0,0,4,0,5],[16,37,0.4324,0.57131,0.30317,0.42857,0.57143,0.85704,0.0,1.0,3,4,0,3,0,3,0,0,1,0,0,4,0,0,7,0,0,5,0,0,5,0,4],[20,37,0.5405,0.4732,0.3163,0.14286,0.57143,0.71429,0.0,1.0,7,2,0,7,0,3,0,0,0,0,0,2,0,0,8,0,0,9,0,0,1,0,2],[24,37,0.6486,0.44196,0.356,0.14286,0.42857,0.75,0.0,1.0,7,4,0,7,0,5,0,0,3,0,0,2,0,0,5,0,0,2,0,0,4,0,4],[28,37,0.7568,0.25893,0.30813,0.0,0.14286,0.42857,0.0,1.0,13,2,0,13,0,7,0,0,1,0,0,4,0,0,3,0,0,1,0,0,1,0,2],[32,37,0.8649,0.21427,0.27431,0.0,0.07143,0.42858,0.0,1.0,16,1,0,16,0,5,0,0,1,0,0,3,0,0,5,0,0,1,0,0,0,0,1],[36,37,0.973,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.04464,0.17655,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0]]}]},{"i":"58eb603879f205f2","q":"Let  $x, y, z$  be positive real numbers. Prove that: $\\frac{x + 2y}{z + 2x + 3y}+\\frac{y + 2z}{x + 2y + 3z}+\\frac{z + 2x}{y + 2z + 3x} \\le \\frac{3}{2}$","t":[{"b":0,"e":1.0,"k":"rising","v":0.48214,"x":0.96429,"p":[[0,32,0.0,0.61161,0.29717,0.42857,0.42859,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,16,0,0,1,0,0,3,0,0,0,0,10],[4,32,0.125,0.69643,0.27606,0.42857,0.57144,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,2,0,0,0,0,14],[8,32,0.25,0.66964,0.27765,0.42857,0.42857,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,13],[12,32,0.375,0.57589,0.24086,0.42857,0.42857,0.71429,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,21,0,0,0,0,0,3,0,0,0,0,7],[16,32,0.5,0.48214,0.16656,0.42857,0.42857,0.42857,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,3],[20,32,0.625,0.71875,0.29121,0.42857,0.71429,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,4,0,0,0,0,15],[24,32,0.75,0.84375,0.21829,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,0,0,0,20],[28,32,0.875,0.91071,0.14617,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,22],[32,32,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]},{"b":7,"e":1.0,"k":"rising","v":0.68304,"x":0.99554,"p":[[0,38,0.0,0.68304,0.29175,0.42857,0.57143,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,13,0,0,2,0,0,1,0,0,0,0,14],[4,38,0.1053,0.74107,0.27994,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,1,0,0,0,0,17],[8,38,0.2105,0.70982,0.26362,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,4,0,0,1,0,13],[12,38,0.3158,0.79464,0.25738,0.42857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,2,0,18],[16,38,0.4211,0.77679,0.27418,0.42857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,19],[20,38,0.5263,0.75893,0.30186,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,19],[24,38,0.6316,0.875,0.22798,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,1,0,24],[28,38,0.7368,0.81696,0.26058,0.64286,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,2,0,0,3,0,19],[32,38,0.8421,0.9375,0.18877,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[36,38,0.9474,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[38,38,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"36cdde5c5dff7747","q":"Let  $p\\geq 3$  a prime number,  $a$  and  $b$  integers such that  $\\gcd(a, b)=1$ . Let  $n$  a natural number such that  $p$  divides  $a^{2^n}+b^{2^n}$ , prove that  $2^{n+1}$  divides  $p-1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,44,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,44,0.0909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,44,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,44,0.2727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,44,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,44,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,44,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,44,0.6364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,29,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"14f62e51a325b1b1","q":"Let  $a_n, b_n$  be sequences of positive reals such that,  $$ a_{n+1}= a_n + \\frac{1}{2b_n} $$     $$ b_{n+1}= b_n + \\frac{1}{2a_n} $$   for all  $n\\in\\mathbb N$ .\nProve that,  $\\text{max}\\left(a_{2018}, b_{2018}\\right) >44$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.54464,"x":0.65179,"p":[[0,8,0.0,0.65179,0.31326,0.42857,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,7,0,0,0,0,0,4,0,0,1,0,0,1,0,0,15,0,4],[4,8,0.5,0.54464,0.33012,0.14286,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,12,0,0,0,0,0,2,0,0,1,0,0,2,0,0,15,0,0],[8,8,1.0,0.61604,0.29974,0.35714,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,8,0,0,0,0,0,2,0,0,3,0,0,2,0,0,17,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.58929,"x":0.80357,"p":[[0,23,0.0,0.65625,0.30275,0.39286,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,6,0,0,2,0,0,2,0,0,1,0,0,1,0,0,18,0,2],[4,23,0.1739,0.58929,0.30252,0.24999,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,8,0,0,1,0,0,3,0,0,3,0,0,1,0,0,16,0,0],[8,23,0.3478,0.65624,0.27166,0.42857,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,2,0,0,5,0,0,0,0,0,2,0,0,19,0,0],[12,23,0.5217,0.70089,0.25595,0.53571,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0,0,1,0,0,22,0,0],[16,23,0.6957,0.62054,0.32656,0.14286,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,8,0,0,0,0,0,2,0,0,0,0,0,1,0,0,20,0,0],[20,23,0.8696,0.80357,0.14617,0.85714,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,26,0,0],[23,23,1.0,0.79015,0.16749,0.85714,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,27,0,0]]}]},{"i":"edb511a38c52c050","q":"Let  $x_1 \\le x_2 \\le \\ldots \\le x_{2n-1}$  be real numbers whose arithmetic mean equals  $A$ . Prove that  $$ 2\\sum_{i=1}^{2n-1}\\left( x_{i}-A\\right)^2 \\ge \\sum_{i=1}^{2n-1}\\left( x_{i}-x_{n}\\right)^2. $$","t":[{"b":3,"e":0.28571,"k":"falling","v":0.78125,"x":0.93303,"p":[[0,19,0.0,0.93303,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[4,19,0.2105,0.90179,0.14032,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,13,0,16],[8,19,0.4211,0.88393,0.17655,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,13,0,16],[12,19,0.6316,0.87945,0.13418,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,12,0,14],[16,19,0.8421,0.79464,0.23402,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,10,0,11],[19,19,1.0,0.78125,0.22864,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,3,0,0,13,0,9]]},{"b":5,"e":0.85714,"k":"falling","v":0.74554,"x":0.94643,"p":[[0,49,0.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[4,49,0.0816,0.83483,0.20548,0.85714,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,12,0,13],[8,49,0.1633,0.84371,0.20005,0.67857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,5,0,17],[12,49,0.2449,0.83034,0.17292,0.82143,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,14,0,10],[16,49,0.3265,0.88393,0.16536,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,9,0,17],[20,49,0.4082,0.82141,0.24745,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0,0,6,0,17],[24,49,0.4898,0.85268,0.16935,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,16,0,11],[28,49,0.5714,0.83479,0.17173,0.85714,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,15,0,10],[32,49,0.6531,0.875,0.17405,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,14,0,14],[36,49,0.7347,0.79911,0.21683,0.67857,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,11,0,11],[40,49,0.8163,0.89286,0.17128,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,20],[44,49,0.898,0.84374,0.15303,0.857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,15,0,10],[48,49,0.9796,0.79018,0.1988,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,1,0,0,14,0,8],[49,49,1.0,0.74554,0.19144,0.57143,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,3,0,0,18,0,2]]}]},{"i":"2ef212b9e5a1fa5a","q":"Let  $x,y,z$  be positive real numbers such that $x^4+y^4+z^4=1$  .\n Determine with proof the minimum value of\n  $\\frac{x^3}{1-x^8}+\\frac{y^3}{1-y^8}+\\frac{z^3}{1-z^8}$","t":[{"b":5,"e":0.42857,"k":"falling","v":0.66517,"x":0.94196,"p":[[0,51,0.0,0.86607,0.22851,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,4,0,0,1,0,22],[4,51,0.0784,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[8,51,0.1569,0.91071,0.18472,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,25],[12,51,0.2353,0.87946,0.16793,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,2,0,20],[16,51,0.3137,0.87499,0.18125,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,0,0,21],[20,51,0.3922,0.85268,0.24086,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,21],[24,51,0.4706,0.77232,0.23107,0.57143,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,6,0,0,1,0,14],[28,51,0.549,0.80804,0.22759,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,11,0,0,2,0,15],[32,51,0.6275,0.86161,0.19393,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,0,0,20],[36,51,0.7059,0.87052,0.22408,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,0,0,23],[40,51,0.7843,0.80804,0.20079,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,12,0,0,1,0,14],[44,51,0.8627,0.74107,0.22711,0.67857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,1,0,0,5,0,0,13,0,0,1,0,10],[48,51,0.9412,0.70982,0.20666,0.57143,0.71429,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,15,0,0,1,0,7],[51,51,1.0,0.66517,0.19434,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,19,0,0,1,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,140,0.0,0.85714,0.23958,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,2,0,21],[4,140,0.0286,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,27],[8,140,0.0571,0.86607,0.2765,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,25],[12,140,0.0857,0.87946,0.25282,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,25],[16,140,0.1143,0.91964,0.15947,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,25],[20,140,0.1429,0.95982,0.10853,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[24,140,0.1714,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[28,140,0.2,0.96429,0.14286,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[32,140,0.2286,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[36,140,0.2571,0.94196,0.17445,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[40,140,0.2857,0.90178,0.24074,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,26],[44,140,0.3143,0.90625,0.16213,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,23],[48,140,0.3429,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[52,140,0.3714,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[56,140,0.4,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,27],[60,140,0.4286,0.94643,0.15465,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[64,140,0.4571,0.91964,0.19212,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,26],[68,140,0.4857,0.89286,0.15972,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,2,0,21],[72,140,0.5143,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[76,140,0.5429,0.94643,0.18123,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[80,140,0.5714,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[84,140,0.6,0.86607,0.24468,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,23],[88,140,0.6286,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[92,140,0.6571,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[96,140,0.6857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[100,140,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,140,0.7429,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[108,140,0.7714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,140,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,140,0.8286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,140,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,140,0.8857,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[128,140,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,140,0.9429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[136,140,0.9714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[140,140,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"6c7d187cf9c9bf9c","q":"Let  $x$  and  $y$  be relatively prime integers. Show that  $x^2+xy+y^2$  and  $x^2+3xy+y^2$  are relatively prime.","t":[{"b":0,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,35,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,35,0.1143,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[8,35,0.2286,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.57143,"k":"flat","v":0.73659,"x":0.97321,"p":[[0,29,0.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[4,29,0.1379,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,29,0.2759,0.83035,0.18363,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,5,0,15],[12,29,0.4138,0.81249,0.18709,0.57143,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,4,0,14],[16,29,0.5517,0.79909,0.18163,0.57143,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,5,0,12],[20,29,0.6897,0.79464,0.18189,0.57143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,7,0,11],[24,29,0.8276,0.73659,0.19598,0.57143,0.64286,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,4,0,0,2,0,10],[28,29,0.9655,0.83036,0.18013,0.57143,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,7,0,14],[29,29,1.0,0.87052,0.16508,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,4,0,18]]}]},{"i":"832ab0d634d6b7dd","q":"Let  $s(n)$  be the sum of all positive divisors of  $n$ , so  $s(6) = 12$ . We say  $n$  is almost perfect if  $s(n) = 2n - 1$ . Let  $\\mod(n, k)$  denote the residue of  $n$  modulo  $k$  (in other words, the remainder of dividing  $n$  by  $k$ ). Put  $t(n) = \\mod(n, 1) + \\mod(n, 2) + \\cdots + \\mod(n, n)$ . \r\n\r\nShow that  $n$  is almost perfect if and only if  $t(n) = t(n-1)$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.25893,"x":0.97768,"p":[[0,44,0.0,0.43303,0.30406,0.28571,0.28571,0.28571,0.0,1.0,1,7,0,1,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,44,0.0909,0.47321,0.33776,0.28571,0.28571,1.0,0.0,1.0,2,9,0,2,0,0,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[8,44,0.1818,0.39286,0.27433,0.28571,0.28571,0.28571,0.0,1.0,1,5,0,1,0,2,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,5],[12,44,0.2727,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.28571,1e-05,0.2857,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.93304,0.2082,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[40,44,0.9091,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,44,1.0,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30]]},{"b":4,"e":0.2857,"k":"flat","v":0.25,"x":0.43303,"p":[[0,27,0.0,0.38392,0.28891,0.2857,0.28571,0.28571,0.0,1.0,3,5,0,3,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[4,27,0.1481,0.43303,0.31234,0.28571,0.28571,0.35714,0.0,1.0,2,7,0,2,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,7],[8,27,0.2963,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.286,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.25,0.09449,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7ad1995dfe3f6086","q":"Let  $x_i$ ,  $1\\leq i\\leq n$  be real numbers. Prove that \\[ \\sum_{1\\leq i<j\\leq n}|x_i+x_j|\\geq\\frac{n-2}{2}\\sum_{i=1}^n|x_i|. \\]\n*Discrete version by Dan Schwarz of a Putnam problem*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08037,"p":[[0,85,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,85,0.0471,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,85,0.0941,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,85,0.1412,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,85,0.1882,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,85,0.2353,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,85,0.2824,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,85,0.3294,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,85,0.3765,0.08037,0.15129,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,85,0.4235,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,85,0.4706,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,85,0.5176,0.06696,0.15966,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[48,85,0.5647,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,85,0.6118,0.08036,0.17105,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[56,85,0.6588,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,85,0.7059,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,85,0.7529,0.04018,0.12993,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[68,85,0.8,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,85,0.8471,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.06697,"p":[[0,89,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,89,0.0449,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,89,0.0899,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,89,0.1348,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,89,0.1798,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,89,0.2247,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,89,0.2697,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,89,0.3146,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,89,0.3596,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,89,0.4045,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,89,0.4494,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,89,0.4944,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,89,0.5393,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,89,0.5843,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,89,0.6292,0.0267,0.08316,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,89,0.6742,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,89,0.7191,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.06697,0.15966,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[76,89,0.8539,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4a9ea4eea580e0d9","q":"Let  $f: \\mathbb{R}\\to\\mathbb{R}$  be a function such that   $(f(x))^{n}$  is a polynomial for every integer  $n\\geq 2$ . Is  $f$  also a polynomial?","t":[{"b":2,"e":1.0,"k":"flat","v":0.81696,"x":0.94196,"p":[[0,8,0.0,0.89732,0.21498,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,24],[4,8,0.5,0.81696,0.32189,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,23],[8,8,1.0,0.94196,0.17807,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,25]]},{"b":7,"e":1.0,"k":"rising","v":0.83928,"x":1.0,"p":[[0,12,0.0,0.83928,0.28739,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,0,0,23],[4,12,0.3333,0.91071,0.1915,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"0dac3551ce634c7d","q":"Let  $x_1, x_2, \\cdots , x_n$  be positive reals with  $x_1+x_2+\\cdots+x_n=1$ . Then show that\n\\[\\sum_{i=1}^n \\frac{x_i}{2-x_i} \\ge \\frac{n}{2n-1}\\]","t":[{"b":3,"e":0.71429,"k":"flat","v":0.69196,"x":0.80357,"p":[[0,41,0.0,0.80357,0.16656,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,18,0,0,0,0,12],[4,41,0.0976,0.72321,0.15126,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,23,0,0,0,0,5],[8,41,0.1951,0.70536,0.10677,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,26,0,0,0,0,2],[12,41,0.2927,0.70536,0.13333,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,25,0,0,0,0,3],[16,41,0.3902,0.70089,0.11495,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,26,0,0,0,0,2],[20,41,0.4878,0.69196,0.14334,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,23,0,0,0,0,3],[24,41,0.5854,0.69641,0.11153,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,24,0,0,0,0,2],[28,41,0.6829,0.69643,0.09942,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,28,0,0,0,0,1],[32,41,0.7805,0.70536,0.08702,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,29,0,0,0,0,1],[36,41,0.878,0.70536,0.08702,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,29,0,0,0,0,1],[40,41,0.9756,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[41,41,1.0,0.69643,0.09279,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,27,0,0,0,0,1]]},{"b":7,"e":0.71429,"k":"flat","v":0.67856,"x":0.85714,"p":[[0,40,0.0,0.71875,0.16554,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,19,0,0,0,0,6],[4,40,0.1,0.79911,0.18509,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,15,0,0,0,0,13],[8,40,0.2,0.79464,0.17835,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,17,0,0,0,0,12],[12,40,0.3,0.85714,0.15972,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,0,0,17],[16,40,0.4,0.70089,0.09006,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,28,0,0,0,0,1],[20,40,0.5,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[24,40,0.6,0.69643,0.1171,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,25,0,0,0,0,2],[28,40,0.7,0.67856,0.10716,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,24,0,0,0,0,1],[32,40,0.8,0.71429,0.09449,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0,0,0,2],[36,40,0.9,0.72768,0.10326,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,0,0,3],[40,40,1.0,0.70982,0.07563,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,0,0,0,1]]}]},{"i":"29b37f33e506da0e","q":"Let $A B C D E$ be a convex cyclic pentagon such that $A B=B D$. Let $P$ be the intersection point of the lines (EB) and ( $A C$ ). Let $Q$ be the intersection point of the lines ( $B C$ ) and ( $D E$ ). Show that $(\\mathrm{PQ})$ and (AD) are parallel.","t":[{"b":5,"e":0.4286,"k":"flat","v":0.44192,"x":0.91516,"p":[[0,76,0.0,0.58906,0.32708,0.2857,0.64271,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,4,0,0,2,0,0,4,0,0,4,0,0,6,0,6],[4,76,0.0526,0.85266,0.26364,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,6,0,20],[8,76,0.1053,0.9107,0.18472,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,22],[12,76,0.1579,0.90177,0.19704,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,6,0,22],[16,76,0.2105,0.91516,0.17448,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,22],[20,76,0.2632,0.86605,0.25986,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,4,0,22],[24,76,0.3158,0.83927,0.28739,0.82132,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,22],[28,76,0.3684,0.75444,0.33357,0.67857,0.85714,1.0,0.0,1.0,3,15,0,3,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,6,0,15],[32,76,0.4211,0.78125,0.32924,0.82143,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,9,0,15],[36,76,0.4737,0.78113,0.2814,0.57132,1.0,1.0,0.14,1.0,0,17,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,4,0,0,2,0,17],[40,76,0.5263,0.83034,0.2933,0.82132,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,20],[44,76,0.5789,0.79016,0.28569,0.57132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,4,0,0,3,0,0,1,0,0,3,0,18],[48,76,0.6316,0.74551,0.32091,0.571,0.85714,1.0,0.0,1.0,1,15,0,1,0,3,0,0,2,0,0,1,0,0,3,0,0,2,0,0,5,0,15],[52,76,0.6842,0.81693,0.25566,0.67846,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,2,0,0,4,0,18],[56,76,0.7368,0.73214,0.3004,0.5354,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,1,0,0,4,0,0,3,0,0,3,0,0,5,0,13],[60,76,0.7895,0.79909,0.26211,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,6,0,15],[64,76,0.8421,0.68748,0.30812,0.42857,0.78564,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,4,0,0,4,0,0,1,0,0,4,0,0,5,0,11],[68,76,0.8947,0.78124,0.32534,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,5,0,18],[72,76,0.9474,0.66515,0.30642,0.39286,0.78564,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,4,0,0,2,0,0,3,0,0,3,0,0,8,0,8],[76,76,1.0,0.44192,0.32605,0.14286,0.35714,0.71429,0.0,1.0,3,4,0,3,0,8,0,0,5,0,0,3,0,0,4,0,0,2,0,0,3,0,4]]},{"b":7,"e":0.57143,"k":"flat","v":0.55579,"x":0.89731,"p":[[0,77,0.0,0.61154,0.33358,0.39285,0.57121,1.0,0.0,1.0,1,10,0,1,0,6,0,0,1,0,0,3,0,0,7,0,0,2,0,0,2,0,10],[4,77,0.0519,0.87946,0.21162,0.85711,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,4,0,21],[8,77,0.1039,0.82587,0.28736,0.857,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,16],[12,77,0.1558,0.87946,0.20858,0.85711,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,4,0,21],[16,77,0.2078,0.81693,0.25814,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,0,6,0,16],[20,77,0.2597,0.71427,0.37116,0.42857,1.0,1.0,0.0,1.0,3,17,0,3,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,0,3,0,17],[24,77,0.3117,0.89731,0.20589,0.85711,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,4,0,23],[28,77,0.3636,0.88836,0.21053,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,6,0,21],[32,77,0.4156,0.85264,0.25379,0.857,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,20],[36,77,0.4675,0.80801,0.33997,0.857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,20],[40,77,0.5195,0.80355,0.27838,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,6,0,16],[44,77,0.5714,0.82589,0.26422,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,0,6,0,18],[48,77,0.6234,0.81248,0.24599,0.71429,0.92857,1.0,0.1429,1.0,0,16,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,4,0,0,5,0,16],[52,77,0.6753,0.80802,0.29149,0.67846,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,4,0,0,1,0,0,1,0,0,4,0,19],[56,77,0.7273,0.8615,0.22186,0.85714,1.0,1.0,0.14,1.0,0,19,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,6,0,19],[60,77,0.7792,0.74554,0.3209,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,15],[64,77,0.8312,0.7232,0.32131,0.4286,0.85714,1.0,0.0,1.0,1,12,0,1,0,3,0,0,2,0,0,3,0,0,2,0,0,0,0,0,9,0,12],[68,77,0.8831,0.68295,0.34032,0.42964,0.85714,1.0,0.0,1.0,3,12,0,3,0,1,0,0,3,0,0,2,0,0,4,0,0,2,0,0,5,0,12],[72,77,0.9351,0.7098,0.30196,0.42857,0.85707,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,5,0,0,5,0,0,3,0,0,1,0,0,3,0,14],[76,77,0.987,0.65177,0.33108,0.42859,0.85707,0.85714,0.0,1.0,4,6,0,4,0,1,0,0,1,0,0,4,0,0,2,0,0,3,0,0,11,0,6],[77,77,1.0,0.55579,0.32426,0.28571,0.5712,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,2,0,0,5,0,1,4,0,0,4,0,0,2,0,7]]}]},{"i":"6e999d5faa6527f7","q":"Let $A B C$ be a triangle and let $X$ be a point inside the triangle. The lines $X A$, $X B$, and $X C$ intersect the circumcircle of $\\triangle A B C$ again at $P$, $Q$, and $R$, respectively. Let $U$ be a point on the ray $X P$ (i.e., on the line $X P$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $A B$ intersects $B Q$ at $V$. The line through $U$ parallel to $A C$ intersects $C R$ at $W$.\nProve that $Q, R, V$, and $W$ lie on one circle.","t":[{"b":4,"e":0.42857,"k":"falling","v":0.38393,"x":0.97768,"p":[[0,58,0.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[4,58,0.069,0.95089,0.17354,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[8,58,0.1379,0.875,0.28516,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,26],[12,58,0.2069,0.91964,0.23128,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[16,58,0.2759,0.86161,0.27776,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,24],[20,58,0.3448,0.66964,0.30606,0.42857,0.4286,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,12,0,0,0,0,0,0,0,0,1,0,14],[24,58,0.4138,0.66969,0.29541,0.42857,0.4293,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,14,0,0,1,0,0,0,0,0,0,0,14],[28,58,0.4828,0.75893,0.31224,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,9,0,0,0,0,0,0,0,0,1,0,19],[32,58,0.5517,0.54021,0.29174,0.28571,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,10,0,0,10,0,0,2,0,0,0,0,0,1,0,8],[36,58,0.6207,0.61607,0.28219,0.42857,0.4286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,14,0,0,0,0,0,1,0,0,3,0,9],[40,58,0.6897,0.43752,0.15946,0.42857,0.42857,0.4286,0.1429,1.0,0,2,0,0,0,1,0,0,4,0,0,25,0,0,0,0,0,0,0,0,0,0,2],[44,58,0.7586,0.40184,0.05579,0.42857,0.42857,0.4286,0.2857,0.43,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.38402,0.09067,0.39286,0.42857,0.4286,0.0,0.43,1,0,0,1,0,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.38393,0.14032,0.2857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,10,0,0,19,0,0,0,0,0,0,0,0,0,0,1],[56,58,0.9655,0.39282,0.08016,0.42857,0.42857,0.4286,0.14,0.43,0,0,0,0,0,2,0,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.39286,0.06187,0.39286,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"falling","v":0.34366,"x":0.95982,"p":[[0,97,0.0,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[4,97,0.0412,0.84821,0.33107,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[8,97,0.0825,0.66954,0.39369,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,4,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,18],[12,97,0.1237,0.8259,0.30667,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,23],[16,97,0.1649,0.8125,0.2889,0.64286,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,5,0,0,0,0,0,1,0,0,3,0,20],[20,97,0.2062,0.88839,0.26663,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,26],[24,97,0.2474,0.78569,0.31544,0.571,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,20],[28,97,0.2887,0.79017,0.32141,0.53539,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,0,3,0,20],[32,97,0.3299,0.72767,0.32803,0.42859,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,2,0,0,4,0,0,2,0,0,2,0,0,1,0,17],[36,97,0.3711,0.66965,0.35792,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,2,0,0,8,0,0,0,0,0,0,0,0,1,0,16],[40,97,0.4124,0.73659,0.35555,0.42857,1.0,1.0,0.0,1.0,2,19,0,2,0,2,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,19],[44,97,0.4536,0.65614,0.38288,0.28571,0.85707,1.0,0.0,1.0,3,15,0,3,0,4,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,15],[48,97,0.4948,0.71873,0.36155,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,5,0,0,1,0,0,4,0,0,1,0,0,0,0,0,2,0,18],[52,97,0.5361,0.61607,0.42173,0.14286,1.0,1.0,0.0,1.0,4,17,0,4,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,17],[56,97,0.5773,0.74544,0.35141,0.53569,1.0,1.0,0.0,1.0,1,18,0,1,0,5,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,18],[60,97,0.6186,0.67411,0.33165,0.42857,0.78571,1.0,0.0,1.0,1,14,0,1,0,2,0,0,3,0,0,8,0,0,1,0,0,1,0,0,2,0,14],[64,97,0.6598,0.67846,0.37471,0.2857,0.85714,1.0,0.0,1.0,2,15,0,2,0,5,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,15],[68,97,0.701,0.69187,0.36102,0.42857,0.85714,1.0,0.0,1.0,1,15,0,1,0,6,0,0,0,0,0,5,0,0,0,0,0,1,0,0,4,0,15],[72,97,0.7423,0.34366,0.28545,0.14286,0.1429,0.42857,0.14,1.0,0,4,0,0,0,17,0,0,3,0,0,7,0,0,0,0,0,1,0,0,0,0,4],[76,97,0.7835,0.43742,0.36766,0.14286,0.35714,0.89286,0.0,1.0,4,8,0,4,0,9,0,0,3,0,0,7,0,0,0,0,0,0,0,0,1,0,8],[80,97,0.8247,0.55353,0.3458,0.14286,0.571,1.0,0.0,1.0,2,9,0,2,0,7,0,0,1,0,0,4,0,0,5,0,0,4,0,0,0,0,9],[84,97,0.866,0.4775,0.39561,0.14286,0.35714,1.0,0.0,1.0,3,11,0,3,0,11,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,11],[88,97,0.9072,0.43285,0.37891,0.14286,0.14288,0.89286,0.0,1.0,3,8,0,3,0,14,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,8],[92,97,0.9485,0.37936,0.25161,0.25,0.35714,0.4286,0.0,1.0,2,3,0,2,0,6,0,0,8,0,0,11,0,0,1,0,0,1,0,0,0,0,3],[96,97,0.9897,0.39285,0.22302,0.14289,0.42857,0.4286,0.14286,1.0,0,2,0,0,0,9,0,0,2,0,0,17,0,0,1,0,0,0,0,0,1,0,2],[97,97,1.0,0.34813,0.18546,0.25,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,5,0,0,16,0,0,0,0,0,3,0,0,0,0,0]]}]},{"i":"ec6a5eb08c0fde7f","q":"Let $A B C D$ be a convex quadrilateral (i.e., all interior angles are less than $180^{\\circ}$), such that there exists a point $M$ on segment $A B$ and a point $N$ on segment $B C$ with the property that $A N$ divides the quadrilateral into two regions of equal area, and $C M$ does the same.\nProve that $M N$ bisects the diagonal $B D$.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.49106,"x":0.58027,"p":[[0,41,0.0,0.5357,0.13832,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,7,0,0,18,0,0,5,0,0,0,0,0],[4,41,0.0976,0.57142,0.12372,0.42857,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,11,0,0,11,0,0,0,0,0],[8,41,0.1951,0.58027,0.15567,0.53571,0.57143,0.71429,0.14,1.0,0,1,0,0,0,1,0,0,1,0,0,6,0,0,13,0,0,10,0,0,0,0,1],[12,41,0.2927,0.49106,0.17834,0.42857,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,2,0,0,9,0,0,10,0,0,7,0,0,0,0,0],[16,41,0.3902,0.49554,0.15966,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,17,0,0,6,0,0,7,0,0,0,0,0],[20,41,0.4878,0.51339,0.20472,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,3,0,0,3,0,0,5,0,0,9,0,0,11,0,0,0,0,0],[24,41,0.5854,0.56696,0.16554,0.42859,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,4,0,0,9,0,0,14,0,0,0,0,0],[28,41,0.6829,0.54017,0.14166,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,11,0,0,10,0,0,9,0,0,0,0,0],[32,41,0.7805,0.50893,0.19212,0.42857,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,1,0,0,10,0,0,9,0,0,9,0,0,0,0,0],[36,41,0.878,0.57588,0.13115,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,7,0,0,11,0,0,12,0,0,0,0,0],[40,41,0.9756,0.57589,0.16554,0.42857,0.64286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,0,16,0,0,0,0,0],[41,41,1.0,0.55802,0.16888,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,8,0,0,10,0,0,12,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.10706,"x":0.56249,"p":[[0,51,0.0,0.56249,0.11258,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,2,0,0,23,0,0,5,0,0,0,0,0],[4,51,0.0784,0.51784,0.16268,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,1,0,0,8,0,0,13,0,0,7,0,0,0,0,0],[8,51,0.1569,0.5312,0.16839,0.42857,0.57143,0.60714,0.14286,0.71429,0,0,0,0,0,3,0,0,2,0,0,4,0,0,15,0,0,8,0,0,0,0,0],[12,51,0.2353,0.5,0.18211,0.42857,0.57143,0.60714,0.14286,0.71429,0,0,0,0,0,4,0,0,2,0,0,8,0,0,10,0,0,8,0,0,0,0,0],[16,51,0.3137,0.41518,0.23787,0.14286,0.42857,0.60714,0.0,0.71429,1,0,0,1,0,10,0,0,2,0,0,5,0,0,6,0,0,8,0,0,0,0,0],[20,51,0.3922,0.41061,0.21364,0.14289,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,8,0,0,3,0,0,7,0,0,8,0,0,5,0,0,0,0,0],[24,51,0.4706,0.39732,0.23347,0.14286,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,5,0,0,3,0,0,7,0,0,8,0,0,5,0,0,0,0,0],[28,51,0.549,0.41963,0.23673,0.24999,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,3,0,0,1,0,0,8,0,0,10,0,0,5,0,0,0,0,0],[32,51,0.6275,0.39729,0.18115,0.28571,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,6,0,0,6,0,0,5,0,0,14,0,0,0,0,0,0,0,0],[36,51,0.7059,0.33481,0.21608,0.14286,0.42857,0.42858,0.0,0.71429,4,0,0,4,0,8,0,0,3,0,0,10,0,0,4,0,0,3,0,0,0,0,0],[40,51,0.7843,0.33927,0.24418,0.14286,0.42857,0.57143,0.0,0.71429,7,0,0,7,0,5,0,0,3,0,0,6,0,0,8,0,0,3,0,0,0,0,0],[44,51,0.8627,0.16518,0.16016,0.0,0.14286,0.17857,0.0,0.57143,10,0,0,10,0,14,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[48,51,0.9412,0.11598,0.12076,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,19,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[51,51,1.0,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d491e5d8b28057ea","q":"Let $A B C$ be a triangle with circumcircle $\\Gamma$, circumcenter $O$, and orthocenter $H$. Assume that $A B \\neq A C$ and $\\angle A \\neq 90^{\\circ}$. Let $M$ and $N$ be the midpoints of $\\overline{A B}$ and $\\overline{A C}$, respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\\triangle A B C$, respectively. Let $P$ be the intersection point of line $M N$ with the tangent line to $\\Gamma$ at $A$. Let $Q$ be the intersection point, other than $A$, of $\\Gamma$ with the circumcircle of $\\triangle A E F$. Let $R$ be the intersection point of lines $A Q$ and $E F$. Prove that $\\overline{P R} \\perp \\overline{O H}$.","t":[{"b":4,"e":0.28571,"k":"flat","v":0.08481,"x":0.21875,"p":[[0,37,0.0,0.08481,0.14218,0.0,0.0,0.14286,0.0,0.571,21,0,1,21,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,37,0.1081,0.12054,0.12931,0.0,0.07143,0.2857,0.0,0.28571,16,0,0,16,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.12491,0.13242,0.0,0.14143,0.2857,0.0,0.42857,15,0,0,15,0,7,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.20089,0.12299,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,3,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.21428,0.12877,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,6,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,37,0.5405,0.20982,0.12364,0.14286,0.28571,0.28571,0.0,0.4286,6,0,0,6,0,7,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.15179,0.14258,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,6,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.15625,0.12037,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,12,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.17411,0.14167,0.0,0.21428,0.28571,0.0,0.4286,11,0,0,11,0,5,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.13393,0.13333,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,7,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.21875,0.12364,0.14286,0.2857,0.28571,0.0,0.42857,5,0,0,5,0,8,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.22768,"p":[[0,46,0.0,0.0892,0.11706,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.22768,0.21387,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,5,0,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[8,46,0.1739,0.11607,0.13092,0.0,0.07143,0.2857,0.0,0.4286,16,0,0,16,0,7,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.12053,0.13882,0.0,0.07143,0.2857,0.0,0.4286,16,0,0,16,0,7,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.10268,0.12492,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.10268,0.12992,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.125,0.21943,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[28,46,0.6087,0.10268,0.1394,0.0,0.0,0.17857,0.0,0.4286,19,0,0,19,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"86369527425f9114","q":"Let  $x, y$  be two positive integers, with  $x> y$ , such that  $2n = x + y$ , where n is a number two-digit integer. If  $\\sqrt{xy}$  is an integer with the digits of  $n$  but in reverse order, determine the value of  $x - y$","t":[{"b":0,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,37,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,37,0.1081,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,52,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,52,0.0769,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,52,0.1538,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,52,0.2308,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,52,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,52,0.3846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,52,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,52,0.5385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,52,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,52,0.6923,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b762c1a3a3305339","q":"Let $A B C$ be an acute triangle with $D$ the foot of the altitude from $A$. The circle centered at $A$ passing through $D$ intersects the circumcircle of triangle $A B C$ at $X$ and $Y$, such that the order of the points on this circumcircle is: $A, X, B, C, Y$. Prove that $\\angle B X D=\\angle C Y D$.","t":[{"b":4,"e":0.42857,"k":"flat","v":0.12054,"x":0.3615,"p":[[0,69,0.0,0.21429,0.22868,0.0,0.2857,0.28571,0.0,1.0,13,1,4,13,0,2,0,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[4,69,0.058,0.3615,0.29669,0.0,0.35714,0.57143,0.0,1.0,9,2,0,9,0,2,0,0,5,0,0,4,0,0,7,0,0,3,0,0,0,0,2],[8,69,0.1159,0.31249,0.273,0.14286,0.28571,0.42857,0.0,1.0,7,2,0,7,0,5,0,0,10,0,0,4,0,0,1,0,0,3,0,0,0,0,2],[12,69,0.1739,0.31686,0.22518,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,7,0,0,10,0,0,5,0,0,3,0,0,1,0,0,2,0,0],[16,69,0.2319,0.30357,0.23891,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,5,0,0,7,0,0,7,0,0,4,0,0,1,0,0,0,0,1],[20,69,0.2899,0.21875,0.22011,0.0,0.21428,0.28571,0.0,0.71429,12,0,0,12,0,4,0,0,10,0,0,1,0,0,3,0,0,2,0,0,0,0,0],[24,69,0.3478,0.30356,0.21942,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,3,0,0,10,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[28,69,0.4058,0.30801,0.23717,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,7,0,0,10,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[32,69,0.4638,0.31249,0.19702,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,7,0,0,6,0,0,11,0,0,3,0,0,0,0,0,1,0,0],[36,69,0.5217,0.24107,0.21261,0.0,0.2857,0.32143,0.0,0.71429,9,0,0,9,0,6,0,0,9,0,0,5,0,0,0,0,0,3,0,0,0,0,0],[40,69,0.5797,0.27679,0.2257,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,7,0,0,8,0,0,4,0,0,4,0,0,1,0,0,1,0,0],[44,69,0.6377,0.20517,0.17108,0.14,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,13,0,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[48,69,0.6957,0.25892,0.2065,0.14286,0.2857,0.42857,0.0,0.71429,7,0,0,7,0,8,0,0,8,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[52,69,0.7536,0.28124,0.18721,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,10,0,0,6,0,0,8,0,0,3,0,0,1,0,0,0,0,0],[56,69,0.8116,0.23661,0.18766,0.0,0.2857,0.32143,0.0,0.71429,9,0,0,9,0,4,0,0,11,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[60,69,0.8696,0.27232,0.24836,0.0,0.2857,0.32143,0.0,0.85714,9,0,0,9,0,5,0,0,10,0,0,1,0,0,3,0,0,3,0,0,1,0,0],[64,69,0.9275,0.12054,0.15612,0.0,0.0,0.2857,0.0,0.42857,18,0,0,18,0,5,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.13839,0.13115,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,8,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.14732,0.17307,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,3,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.16965,"x":0.38613,"p":[[0,92,0.0,0.17634,0.20592,0.0,0.07143,0.28571,0.0,0.71429,16,0,4,16,0,2,0,0,8,0,1,2,0,0,2,0,0,1,0,0,0,0,0],[4,92,0.0435,0.25444,0.21044,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,7,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[8,92,0.087,0.29017,0.27544,0.0,0.2857,0.42858,0.0,1.0,10,1,0,10,0,4,0,0,6,0,0,6,0,0,3,0,0,0,0,0,2,0,1],[12,92,0.1304,0.25445,0.21347,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,1,0,0,14,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[16,92,0.1739,0.29462,0.27415,0.0,0.28571,0.42858,0.0,1.0,10,1,0,10,0,3,0,0,8,0,0,4,0,0,3,0,0,2,0,0,1,0,1],[20,92,0.2174,0.30357,0.22517,0.14286,0.28571,0.4286,0.0,0.85714,7,0,0,7,0,3,0,0,11,0,0,4,0,0,5,0,0,1,0,0,1,0,0],[24,92,0.2609,0.23427,0.20848,0.05325,0.2857,0.32143,0.0,1.0,8,1,0,8,1,6,0,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[28,92,0.3043,0.38613,0.28055,0.25,0.28571,0.51775,0.0,1.0,5,2,0,5,0,3,0,0,9,0,0,6,0,1,2,0,0,2,0,0,2,0,2],[32,92,0.3478,0.27231,0.20314,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,2,0,0,8,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[36,92,0.3913,0.28115,0.29773,0.0,0.21428,0.4286,0.0,1.0,12,1,0,12,0,4,0,0,5,0,0,4,0,0,2,0,0,2,0,0,2,0,1],[40,92,0.4348,0.26339,0.19597,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,4,0,0,9,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[44,92,0.4783,0.29908,0.23241,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,6,0,0,10,0,0,4,0,0,2,0,0,3,0,0,1,0,0],[48,92,0.5217,0.27008,0.16726,0.14286,0.28571,0.37489,0.0,0.71429,5,0,0,5,0,5,0,0,13,0,1,6,0,0,1,0,0,1,0,0,0,0,0],[52,92,0.5652,0.32131,0.23694,0.14214,0.28571,0.42858,0.0,0.85714,7,0,0,7,0,2,0,0,11,0,0,5,0,0,3,0,0,3,0,0,1,0,0],[56,92,0.6087,0.27232,0.23517,0.14286,0.2857,0.28571,0.0,1.0,7,1,0,7,0,5,0,0,13,0,0,4,0,0,0,0,0,1,0,0,1,0,1],[60,92,0.6522,0.35489,0.24054,0.24999,0.28571,0.44645,0.0,1.0,5,1,0,5,0,3,0,0,9,0,0,7,0,1,4,0,0,1,0,0,1,0,1],[64,92,0.6957,0.37946,0.22191,0.28571,0.42857,0.42858,0.0,1.0,3,1,0,3,0,3,0,0,9,0,0,10,0,0,4,0,0,1,0,0,1,0,1],[68,92,0.7391,0.24552,0.25058,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,6,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,1],[72,92,0.7826,0.23213,0.22797,0.0,0.2857,0.32143,0.0,0.85714,12,0,0,12,0,3,0,0,9,0,0,3,0,0,4,0,0,0,0,0,1,0,0],[76,92,0.8261,0.24097,0.16539,0.14286,0.2857,0.42857,0.0,0.571,6,0,0,6,0,9,0,0,7,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[80,92,0.8696,0.17411,0.15866,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,9,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[84,92,0.913,0.16965,0.19045,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,6,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[88,92,0.9565,0.23205,0.18128,0.14286,0.21428,0.28571,0.0,0.85714,6,0,0,6,0,10,0,0,9,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[92,92,1.0,0.26338,0.18933,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,7,0,0,10,0,0,5,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"24138354cf2b56c0","q":"Let  $x$  and  $y$  be integers satisfying both  $x^2 - 16x + 3y = 20$  and  $y^2 + 4y - x = -12$ . Find  $x + y$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,39,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,39,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,39,0.2051,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,39,0.3077,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,39,0.4103,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,39,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,39,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,39,0.7179,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,39,0.8205,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,39,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":0.99554,"p":[[0,50,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,50,0.08,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,50,0.16,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,50,0.24,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,50,0.32,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,50,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,50,0.48,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,50,0.56,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,50,0.64,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,50,0.72,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,50,0.8,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,50,0.88,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,50,0.96,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[50,50,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"28f39536f56a2c81","q":"Let  $n$  be a given even number,  $a_1,a_2,\\cdots,a_n$  be non-negative real numbers such that  $a_1+a_2+\\cdots+a_n=1.$  Find the maximum possible value of  $\\sum_{1\\le i<j\\le n}\\min\\{(i-j)^2,(n+i-j)^2\\}a_ia_j .$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13384,"x":0.15179,"p":[[0,55,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.12946,"x":0.16509,"p":[[0,47,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.16072,0.07784,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.14732,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.16509,0.07243,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.14732,0.13592,0.14286,0.14286,0.14286,0.0,0.85714,4,0,0,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,47,0.4255,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b101881ca0b41823","q":"Let  $n$  be a positive integer. Find the largest nonnegative real number  $f(n)$  (depending on  $n$ ) with the following property: whenever  $a_1,a_2,...,a_n$  are real numbers such that  $a_1+a_2+\\cdots +a_n$  is an integer, there exists some  $i$  such that   $\\left|a_i-\\frac{1}{2}\\right|\\ge f(n)$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,18,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,18,0.4444,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,18,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,18,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[18,18,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,16,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,16,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,16,0.5,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,16,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"283ed840295219f1","q":"Let  $x, y,$  and  $z$  be positive real numbers such that  $xy + yz + zx = 27$ . Prove that  $x + y + z \\ge \\sqrt{3xyz}$ .\n\nWhen does equality hold?","t":[{"b":2,"e":1.0,"k":"flat","v":0.92408,"x":1.0,"p":[[0,29,0.0,0.96428,0.11294,1.0,1.0,1.0,0.5714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[4,29,0.1379,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,0.92408,0.1336,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,22],[16,29,0.5517,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,29,0.6897,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[24,29,0.8276,0.97321,0.10374,1.0,1.0,1.0,0.5714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[28,29,0.9655,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[29,29,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.76339,"x":1.0,"p":[[0,24,0.0,0.92857,0.15152,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,26],[4,24,0.1667,0.9241,0.2172,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[8,24,0.3333,0.80803,0.13175,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,1,0,10],[12,24,0.5,0.76339,0.11633,0.71429,0.71429,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,6],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"bb10dc01f66bdf60","q":"Let  $x_1=2021$ ,  $x_n^2-2(x_n+1)x_{n+1}+2021=0$  ( $n\\geq1$ ). Prove that the sequence  ${x_n}$  converges. Find the limit  $\\lim_{n \\to \\infty} x_n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,34,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,34,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,34,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,34,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[34,34,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,13,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,13,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,13,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8ca11ee27ea4224f","q":"Let  $x,y,z$  be positive real numbers such that  $x+y+z=1$   Prove that always  $\\left( 1+\\frac1x\\right)\\times\\left(1+\\frac1y\\right)\\times\\left(1 +\\frac1z\\right)\\ge 64$ When does equality hold?","t":[{"b":4,"e":1.0,"k":"rising","v":0.77679,"x":1.0,"p":[[0,18,0.0,0.77679,0.34615,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,21],[4,18,0.2222,0.84821,0.28333,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,24],[8,18,0.4444,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.76786,"x":1.0,"p":[[0,12,0.0,0.85259,0.26627,0.92857,1.0,1.0,0.14,1.0,0,24,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,24],[4,12,0.3333,0.76786,0.32093,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,1,0,0,7,0,0,0,0,0,0,0,0,1,0,20],[8,12,0.6667,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8ac9365b07d1c4d5","q":"Let $A B C D$ be a convex quadrilateral. For any point $M$ inside or on the boundary of $A B C D$, we define\n\n$$\nf(M)=M A+M B+M C+M D .\n$$\n\nProve that, for any $M$, we have\n\n$$\nf(M) \\leqslant \\max (f(A), f(B), f(C), f(D))\n$$","t":[{"b":1,"e":0.14286,"k":"volatile","v":0.24107,"x":0.76339,"p":[[0,34,0.0,0.29018,0.24086,0.14286,0.14286,0.42857,0.0,1.0,1,2,0,1,0,19,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,2],[4,34,0.1176,0.41062,0.28967,0.14286,0.42857,0.46429,0.14,1.0,0,4,0,0,0,13,0,0,1,0,0,10,0,0,2,0,0,1,0,0,1,0,4],[8,34,0.2353,0.56696,0.33784,0.35714,0.42857,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,0,0,0,11,0,0,0,0,0,2,0,0,1,0,10],[12,34,0.3529,0.37054,0.33476,0.14286,0.14286,0.46429,0.14286,1.0,0,6,0,0,0,20,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,6],[16,34,0.4706,0.24107,0.18708,0.14286,0.14286,0.21429,0.14286,0.85714,0,0,0,0,0,24,0,0,0,0,0,5,0,0,1,0,0,1,0,0,1,0,0],[20,34,0.5882,0.76339,0.34369,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,20],[24,34,0.7059,0.49554,0.37962,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,15,0,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,10],[28,34,0.8235,0.71429,0.39286,0.14286,1.0,1.0,0.14286,1.0,0,20,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,20],[32,34,0.9412,0.25893,0.25364,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,2],[34,34,1.0,0.37053,0.32899,0.14286,0.14286,0.57143,0.14286,1.0,0,5,0,0,0,20,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,5]]},{"b":3,"e":0.14286,"k":"flat","v":0.20536,"x":0.35705,"p":[[0,44,0.0,0.34821,0.30501,0.14286,0.14286,0.46429,0.14286,1.0,0,4,0,0,0,20,0,0,0,0,0,4,0,0,3,0,0,0,0,0,1,0,4],[4,44,0.0909,0.29911,0.28203,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,23,0,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,3],[8,44,0.1818,0.35705,0.26971,0.14286,0.2857,0.42857,0.14,1.0,0,3,0,0,0,15,0,0,3,0,0,8,0,0,2,0,0,0,0,0,1,0,3],[12,44,0.2727,0.25893,0.19045,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,21,0,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[16,44,0.3636,0.30357,0.21943,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,18,0,0,1,0,0,9,0,0,2,0,0,0,0,0,1,0,1],[20,44,0.4545,0.22321,0.14698,0.14286,0.14286,0.21429,0.14286,0.71429,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[24,44,0.5455,0.23661,0.20079,0.14286,0.14286,0.14287,0.14286,1.0,0,1,0,0,0,25,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[28,44,0.6364,0.28563,0.21434,0.14286,0.14286,0.42857,0.14,1.0,0,1,0,0,0,19,0,0,2,0,0,8,0,0,1,0,0,0,0,0,1,0,1],[32,44,0.7273,0.33464,0.27817,0.14286,0.14286,0.42857,0.14,1.0,0,3,0,0,0,19,0,0,0,0,0,7,0,0,2,0,0,0,0,0,1,0,3],[36,44,0.8182,0.20536,0.13803,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[40,44,0.9091,0.27231,0.20314,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,21,0,0,0,0,0,7,0,0,3,0,0,0,0,0,0,0,1],[44,44,1.0,0.28125,0.18379,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,19,0,0,0,0,0,10,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"06162b0f6964f8cc","q":"Let  $x, y$  and  $z$  be consecutive integers such that\n\\[\\frac 1x+\\frac 1y+\\frac 1z >\\frac{1}{45}.\\]\nFind the maximum value of  $x + y + z$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,82,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,82,0.0488,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,82,0.0976,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,82,0.1463,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,82,0.1951,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,82,0.2439,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,82,0.2927,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,82,0.3415,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,82,0.3902,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,82,0.439,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,82,0.4878,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,82,0.5366,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,82,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,82,0.6341,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[56,82,0.6829,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,82,0.7317,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[64,82,0.7805,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[68,82,0.8293,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[72,82,0.878,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,82,0.9268,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[80,82,0.9756,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[82,82,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,44,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,44,0.0909,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[8,44,0.1818,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[12,44,0.2727,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,44,0.3636,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,44,0.4545,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,44,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,44,0.6364,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,44,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,44,0.8182,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,44,0.9091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,44,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"35ce0781f6cfc133","q":"Let $S$ be the set of positive integers $n$ such that the inequality\n\n$$\n\\phi(n) \\cdot \\tau(n) \\geq \\sqrt{\\frac{n^{3}}{3}}\n$$\n\nholds, where $\\phi(n)$ is the number of positive integers $k \\leq n$ that are relatively prime to $n$, and $\\tau(n)$ is the number of positive divisors of $n$. Prove that $S$ is finite.","t":[{"b":6,"e":0.71429,"k":"falling","v":0.60713,"x":0.90622,"p":[[0,21,0.0,0.90622,0.14117,0.82143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,3,0,21],[4,21,0.1905,0.79464,0.25738,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,8,0,0,2,0,16],[8,21,0.381,0.76338,0.18767,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,14,0,0,3,0,9],[12,21,0.5714,0.75892,0.24075,0.67857,0.71429,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,4,0,0,3,0,0,10,0,0,2,0,12],[16,21,0.7619,0.75,0.24743,0.67857,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,2,0,0,1,0,0,4,0,0,10,0,0,3,0,11],[20,21,0.9524,0.60713,0.24223,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,10,0,0,5,0,0,7,0,0,2,0,5],[21,21,1.0,0.63389,0.21111,0.571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,5,0,0,6,0,0,15,0,0,1,0,3]]},{"b":7,"e":0.571,"k":"falling","v":0.66958,"x":0.9375,"p":[[0,15,0.0,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[4,15,0.2667,0.7991,0.18851,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,3,0,12],[8,15,0.5333,0.80802,0.19761,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,7,0,12],[12,15,0.8,0.80801,0.22479,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,5,0,0,6,0,14],[15,15,1.0,0.66958,0.19379,0.57132,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,12,0,0,1,0,5]]}]},{"i":"5929087ed8b52c88","q":"Let $(a, b, c)$ be a Pythagorean triple, i.e., a triplet of positive integers with $a^{2}+b^{2}=c^{2}$.\n\na) Prove that $(c / a+c / b)^{2}>8$.\n\nb) Prove that there does not exist any integer $n$ for which we can find a Pythagorean triple $(a, b, c)$ satisfying $(c / a+c / b)^{2}=n$.\n\n## a) Solution 1\n\nLet $(a, b, c)$ be a Pythagorean triple. View $a, b$ as lengths of the legs of a right angled triangle with hypotenuse of length $c$; let $\\theta$ be the angle determined by the sides with lengths $a$ and $c$. Then\n\n$$\n\\begin{aligned}\n\\left(\\frac{c}{a}+\\frac{c}{b}\\right)^{2} & =\\left(\\frac{1}{\\cos \\theta}+\\frac{1}{\\sin \\theta}\\right)^{2}=\\frac{\\sin ^{2} \\theta+\\cos ^{2} \\theta+2 \\sin \\theta \\cos \\theta}{(\\sin \\theta \\cos \\theta)^{2}} \\\\\n& =4\\left(\\frac{1+\\sin 2 \\theta}{\\sin ^{2} 2 \\theta}\\right)=\\frac{4}{\\sin ^{2} 2 \\theta}+\\frac{4}{\\sin 2 \\theta}\n\\end{aligned}\n$$\n\nNote that because $0<\\theta<90^{\\circ}$, we have $0<\\sin 2 \\theta \\leq 1$, with equality only if $\\theta=45^{\\circ}$. But then $a=b$ and we obtain $\\sqrt{2}=c / a$, contradicting $a, c$ both being integers. Thus, $0<\\sin 2 \\theta<1$ which gives $(c / a+c / b)^{2}>8$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,40,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,40,0.1,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,40,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,40,0.3,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,40,0.6,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,40,0.7,0.98214,0.04725,1.0,1.0,1.0,0.8571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,40,0.8,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,38,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,38,0.1053,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,38,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,38,0.3158,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,38,0.4211,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,38,0.5263,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,38,0.6316,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,38,0.7368,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,38,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,38,0.9474,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"32ccd86b612ac2c6","q":"Let $S$ be a set of 100 positive integers having the following property:\n\n\"Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.\"\n\nProve that the set $S$ contains a number which divides each of the other 99 numbers of $S$.","t":[{"b":6,"e":0.14286,"k":"falling","v":0.29018,"x":0.65177,"p":[[0,70,0.0,0.65177,0.39276,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,11,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,17],[4,70,0.0571,0.41965,0.38949,0.14286,0.14286,1.0,0.14286,1.0,0,9,0,0,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,9],[8,70,0.1143,0.49097,0.40718,0.14286,0.14286,1.0,0.14,1.0,0,12,0,0,0,18,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,12],[12,70,0.1714,0.47767,0.40816,0.14286,0.14286,1.0,0.0,1.0,3,11,0,3,0,14,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,11],[16,70,0.2286,0.4375,0.3976,0.14286,0.14286,1.0,0.0,1.0,2,10,0,2,0,17,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,10],[20,70,0.2857,0.39732,0.36723,0.14286,0.14286,0.74996,0.0,1.0,1,7,0,1,0,19,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,7],[24,70,0.3429,0.49098,0.42104,0.14286,0.14286,1.0,0.14,1.0,0,13,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[28,70,0.4,0.49097,0.40718,0.14286,0.1429,1.0,0.14,1.0,0,12,0,0,0,18,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,12],[32,70,0.4571,0.57142,0.4072,0.14286,0.57121,1.0,0.14286,1.0,0,14,0,0,0,14,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,14],[36,70,0.5143,0.41963,0.37446,0.14286,0.14286,0.89286,0.0,1.0,1,8,0,1,0,18,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,8],[40,70,0.5714,0.47313,0.40639,0.14286,0.14286,1.0,0.14,1.0,0,11,0,0,0,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,11],[44,70,0.6286,0.49106,0.39275,0.14286,0.14286,1.0,0.14286,1.0,0,11,0,0,0,17,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,11],[48,70,0.6857,0.41518,0.36659,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,19,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,8],[52,70,0.7429,0.4374,0.39282,0.14286,0.14286,1.0,0.14,1.0,0,10,0,0,0,20,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,10],[56,70,0.8,0.31695,0.30036,0.14286,0.14286,0.57111,0.14286,1.0,0,4,0,0,0,23,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,4],[60,70,0.8571,0.37923,0.35303,0.14286,0.14286,0.57143,0.14,1.0,0,7,0,0,0,21,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,7],[64,70,0.9143,0.29018,0.31438,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[68,70,0.9714,0.37052,0.35509,0.14286,0.14286,0.57143,0.14286,1.0,0,7,0,0,0,22,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,7],[70,70,1.0,0.41071,0.3973,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10]]},{"b":7,"e":0.85714,"k":"falling","v":0.29017,"x":0.67407,"p":[[0,70,0.0,0.67407,0.38339,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,17],[4,70,0.0571,0.48659,0.4132,0.14286,0.14286,1.0,0.0,1.0,2,12,0,2,0,15,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,12],[8,70,0.1143,0.34375,0.33285,0.14286,0.14286,0.57143,0.0,1.0,5,4,0,5,0,15,0,0,0,0,0,1,0,0,4,0,0,3,0,0,0,0,4],[12,70,0.1714,0.29017,0.31436,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,23,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,4],[16,70,0.2286,0.45981,0.39404,0.14286,0.21436,1.0,0.0,1.0,3,9,0,3,0,13,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,9],[20,70,0.2857,0.3125,0.31223,0.14286,0.14286,0.21432,0.14286,1.0,0,4,0,0,0,24,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,4],[24,70,0.3429,0.37054,0.35867,0.14286,0.14286,0.60714,0.0,1.0,1,7,0,1,0,20,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,7],[28,70,0.4,0.30348,0.32688,0.14286,0.14286,0.1786,0.0,1.0,2,5,0,2,0,22,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,5],[32,70,0.4571,0.42857,0.39286,0.14286,0.14286,1.0,0.0,1.0,3,9,0,3,0,15,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,9],[36,70,0.5143,0.41963,0.37785,0.14286,0.14286,1.0,0.14286,1.0,0,9,0,0,0,19,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,9],[40,70,0.5714,0.36159,0.36418,0.14286,0.14286,0.57143,0.0,1.0,3,7,0,3,0,18,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,7],[44,70,0.6286,0.42411,0.4278,0.14286,0.14286,1.0,0.0,1.0,6,11,0,6,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11],[48,70,0.6857,0.33036,0.39031,0.14286,0.14286,0.35714,0.0,1.0,6,8,0,6,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[52,70,0.7429,0.33482,0.38234,0.14286,0.14286,0.53574,0.0,1.0,6,7,0,6,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[56,70,0.8,0.41071,0.40681,0.14286,0.14286,1.0,0.0,1.0,5,9,0,5,0,15,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,9],[60,70,0.8571,0.40179,0.38538,0.14286,0.14286,1.0,0.0,1.0,1,9,0,1,0,20,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,9],[64,70,0.9143,0.34362,0.31718,0.14286,0.14286,0.571,0.14,1.0,0,4,0,0,0,22,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,4],[68,70,0.9714,0.33033,0.33394,0.14286,0.14286,0.571,0.0,1.0,2,4,0,2,0,21,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,4],[70,70,1.0,0.45536,0.41563,0.14286,0.14286,1.0,0.0,1.0,3,11,0,3,0,16,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,11]]}]},{"i":"ee75e84a9fcc436c","q":"Let  $n$  be a positive integer. Find the smallest positive integer  $k$  with the property that for any colouring nof the squares of a  $2n$  by  $k$  chessboard with  $n$  colours, there are  $2$  columns and  $2$  rows such that the  $4$  squares in their intersections have the same colour.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.70982,"x":0.91071,"p":[[0,15,0.0,0.75446,0.343,0.57143,1.0,1.0,0.0,1.0,3,18,1,3,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,0,3,0,18],[4,15,0.2667,0.91071,0.15872,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,4,0,22],[8,15,0.5333,0.78571,0.17857,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,5,0,10],[12,15,0.8,0.72765,0.13536,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,18,0,0,7,0,2],[15,15,1.0,0.70982,0.16554,0.71429,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,17,0,0,7,0,2]]},{"b":5,"e":0.71429,"k":"falling","v":0.65177,"x":0.97768,"p":[[0,31,0.0,0.83928,0.29397,0.82143,1.0,1.0,0.0,1.0,2,22,2,2,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,22],[4,31,0.129,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,31,0.2581,0.74552,0.2961,0.57132,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,13],[12,31,0.3871,0.81248,0.16538,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,8,0,10],[16,31,0.5161,0.77676,0.1595,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,6,0,7],[20,31,0.6452,0.683,0.21351,0.571,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,9,0,0,9,0,3],[24,31,0.7742,0.70535,0.15946,0.71429,0.71429,0.71429,0.1429,1.0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,21,0,0,6,0,1],[28,31,0.9032,0.65177,0.15542,0.5354,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,0,17,0,0,5,0,0],[31,31,1.0,0.68302,0.11144,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,20,0,0,2,0,1]]}]},{"i":"abe51ad56bf1f5c4","q":"Let  $n$  be a positive integer. Using the integers from  $1$  to  $4n$  inclusive, pairs are to be formed such that the product of the numbers in each pair is a perfect square. Each number can be part of at most one pair, and the two numbers in each pair must be different. Determine, for each  $n$ , the maximum number of pairs that can be formed.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.70089,"x":0.99107,"p":[[0,64,0.0,0.70089,0.16506,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,19,0,0,2,0,4],[4,64,0.0625,0.97098,0.10693,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,29],[8,64,0.125,0.93304,0.19556,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[12,64,0.1875,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[16,64,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,64,0.3125,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,64,0.375,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,64,0.4375,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[32,64,0.5,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[36,64,0.5625,0.92857,0.12877,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,24],[40,64,0.625,0.91518,0.1551,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,0,0,24],[44,64,0.6875,0.82143,0.17496,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,14,0,0,1,0,14],[48,64,0.75,0.82588,0.23348,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,1,0,19],[52,64,0.8125,0.76339,0.22759,0.67857,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,9,0,0,3,0,12],[56,64,0.875,0.875,0.17035,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,5,0,18],[60,64,0.9375,0.8817,0.14574,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,1,0,5,0,17],[64,64,1.0,0.8125,0.14032,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,6,0,9]]},{"b":2,"e":0.42857,"k":"falling","v":0.54909,"x":0.93304,"p":[[0,65,0.0,0.75,0.18898,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,16,0,0,1,0,9],[4,65,0.0615,0.93304,0.15966,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,26],[8,65,0.1231,0.90625,0.14555,0.82143,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,3,0,21],[12,65,0.1846,0.91517,0.18512,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,26],[16,65,0.2462,0.93304,0.14719,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,26],[20,65,0.3077,0.90179,0.20958,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,23],[24,65,0.3692,0.8616,0.16936,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,3,0,17],[28,65,0.4308,0.86161,0.20972,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,3,0,20],[32,65,0.4923,0.86604,0.17839,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,4,0,18],[36,65,0.5538,0.84822,0.20805,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,4,0,18],[40,65,0.6154,0.76786,0.21354,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,12,0,0,3,0,11],[44,65,0.6769,0.66965,0.19703,0.4286,0.71429,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,12,0,0,4,0,4],[48,65,0.7385,0.75446,0.20589,0.67857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,11,0,0,3,0,10],[52,65,0.8,0.69641,0.1948,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,8,0,0,4,0,6],[56,65,0.8615,0.70532,0.20808,0.53539,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,9,0,0,4,0,7],[60,65,0.9231,0.6875,0.22428,0.42859,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,9,0,0,2,0,8],[64,65,0.9846,0.56247,0.18877,0.42857,0.42859,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,3,0,0,2,0,3],[65,65,1.0,0.54909,0.13414,0.42857,0.4286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,3,0,0,12,0,0,0,0,0]]}]},{"i":"6a2d4a9990c40763","q":"Let  $n$  be an integer such that  $n > 3$ .  Suppose that we choose three numbers from the set  $\\{1, 2, \\ldots, n\\}$ .  Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all\r\npossible combinations.\r\n(a) Show that if we choose all three numbers greater than  $\\frac{n}{2}$ , then the values of these combinations are all distinct.\r\n(b) Let  $p$  be a prime number such that  $p \\leq \\sqrt{n}$ .  Show that the number of ways of choosing three numbers so that the smallest one is  $p$  and the values of the combinations are not all distinct is precisely the number of positive divisors of  $p - 1$ .","t":[{"b":1,"e":0.85714,"k":"falling","v":0.64729,"x":0.88839,"p":[[0,105,0.0,0.84375,0.16115,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,5,0,14],[4,105,0.0381,0.85713,0.15972,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,10,0,13],[8,105,0.0762,0.81683,0.14836,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,9,0,9],[12,105,0.1143,0.88839,0.15458,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,18],[16,105,0.1524,0.86607,0.17474,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,6,0,17],[20,105,0.1905,0.83034,0.15748,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,9,0,11],[24,105,0.2286,0.83482,0.22047,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,0,8,0,15],[28,105,0.2667,0.80357,0.22232,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,8,0,13],[32,105,0.3048,0.8348,0.21463,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,15],[36,105,0.3429,0.85713,0.15153,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,10,0,13],[40,105,0.381,0.77678,0.23673,0.71429,0.78564,1.0,0.0,1.0,2,9,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,7,0,9],[44,105,0.419,0.81695,0.20898,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,6,0,0,9,0,12],[48,105,0.4571,0.78123,0.19558,0.57143,0.78564,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,0,5,0,11],[52,105,0.4952,0.79017,0.223,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,7,0,11],[56,105,0.5333,0.80801,0.16606,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,8,0,10],[60,105,0.5714,0.77229,0.14226,0.71429,0.71429,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,9,0,5],[64,105,0.6095,0.80804,0.15815,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,0,8,0,9],[68,105,0.6476,0.82586,0.15869,0.71429,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,8,0,11],[72,105,0.6857,0.82588,0.16652,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,10,0,11],[76,105,0.7238,0.81249,0.19379,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,7,0,0,9,0,11],[80,105,0.7619,0.85714,0.13832,0.71429,0.85714,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,13],[84,105,0.8,0.80357,0.18123,0.71429,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,7,0,11],[88,105,0.8381,0.78571,0.17128,0.71429,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,8,0,8],[92,105,0.8762,0.76326,0.1877,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,11,0,0,8,0,7],[96,105,0.9143,0.74107,0.18707,0.67857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,10,0,0,9,0,5],[100,105,0.9524,0.78121,0.19884,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,8,0,10],[104,105,0.9905,0.67856,0.25506,0.5354,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,0,9,0,5],[105,105,1.0,0.64729,0.15561,0.57132,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,10,0,0,7,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.59821,"x":0.84372,"p":[[0,38,0.0,0.79462,0.19866,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,8,0,0,8,0,10],[4,38,0.1053,0.83482,0.18249,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,8,0,13],[8,38,0.2105,0.84372,0.15719,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,5,0,14],[12,38,0.3158,0.81695,0.21794,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,8,0,13],[16,38,0.4211,0.77665,0.23945,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,6,0,0,5,0,0,6,0,12],[20,38,0.5263,0.71425,0.24745,0.571,0.78564,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,5,0,0,9,0,7],[24,38,0.6316,0.66518,0.25155,0.53571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,5,0,0,2,0,0,2,0,0,11,0,0,7,0,4],[28,38,0.7368,0.70533,0.23129,0.57132,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,4,0,0,6,0,0,5,0,0,10,0,5],[32,38,0.8421,0.63836,0.22864,0.5354,0.71429,0.857,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,5,0,0,6,0,0,9,0,0,7,0,2],[36,38,0.9474,0.59821,0.16145,0.4286,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,11,0,0,5,0,0,11,0,0,4,0,0],[38,38,1.0,0.65622,0.1851,0.5354,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,8,0,0,7,0,2]]}]},{"i":"befca7965437362f","q":"Let  $x_1$ ,  $x_2$ , ...,  $x_n$  be real numbers with arithmetic mean  $X$ . Prove that there is a positive integer  $K$  such that for any integer  $i$  satisfying  $0\\leq i<K$ , we have  $\\frac{1}{K-i}\\sum_{j=i+1}^{K} x_j \\leq X$ . (In other words, prove that there is a positive integer  $K$  such that the arithmetic mean of each of the lists  $\\left\\{x_1,x_2,...,x_K\\right\\}$ ,  $\\left\\{x_2,x_3,...,x_K\\right\\}$ ,  $\\left\\{x_3,...,x_K\\right\\}$ , ...,  $\\left\\{x_{K-1},x_K\\right\\}$ ,  $\\left\\{x_K\\right\\}$  is not greater than  $X$ .)","t":[{"b":2,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,25,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,25,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,25,0.32,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,25,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,25,0.64,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,25,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,25,0.96,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,32,0.0,0.91964,0.20183,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,25],[4,32,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"f951c786dc38a0b9","q":"Let $\\mathcal{S}$ be a set of size $n$, and $k$ be a positive integer. For each $1 \\leq i \\leq k n$, there is a subset $S_{i} \\subset \\mathcal{S}$ such that $\\left|S_{i}\\right|=2$. Furthermore, for each $e \\in \\mathcal{S}$, there are exactly $2 k$ values of $i$ such that $e \\in S_{i}$. Show that it is possible to choose one element from $S_{i}$ for each $1 \\leq i \\leq k n$ such that every element of $\\mathcal{S}$ is chosen exactly $k$ times.","t":[{"b":3,"e":1.0,"k":"flat","v":0.83929,"x":0.99107,"p":[[0,25,0.0,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,25,0.16,0.93304,0.22011,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[8,25,0.32,0.83929,0.30878,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,24],[12,25,0.48,0.9107,0.18474,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,25],[16,25,0.64,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,25,0.8,0.90625,0.26633,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[24,25,0.96,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[25,25,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.80356,"x":0.98661,"p":[[0,19,0.0,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,19,0.2105,0.80356,0.31895,0.57143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,22],[8,19,0.4211,0.92411,0.17491,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,26],[12,19,0.6316,0.81696,0.26782,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,0,0,0,6,0,0,3,0,0,0,0,20],[16,19,0.8421,0.89732,0.25313,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,27],[19,19,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"0787d9b15d71dca4","q":"Let $\\left(a_{n}\\right)_{n \\geqslant 0}$ be a sequence of real numbers. Suppose that $a_{n}=\\left|a_{n+1}-a_{n+2}\\right|$ for all natural numbers $n$. Furthermore, $a_{0}$ and $a_{1}$ are strictly positive and distinct. Show that the sequence $\\left(a_{n}\\right)_{n \\geqslant 0}$ is not bounded.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.15179,"x":0.33929,"p":[[0,35,0.0,0.15179,0.14698,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.26338,0.22617,0.0,0.28571,0.42857,0.0,0.71429,11,0,1,11,0,2,0,0,6,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[8,35,0.2286,0.25893,0.19377,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,5,0,0,7,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[12,35,0.3429,0.28124,0.1767,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,2,0,0,10,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[16,35,0.4571,0.21429,0.15972,0.10714,0.21428,0.32143,0.0,0.4286,8,0,0,8,0,8,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.33929,0.18123,0.28571,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,4,0,0,8,0,0,15,0,0,0,0,0,1,0,0,1,0,0],[24,35,0.6857,0.30357,0.17768,0.28571,0.28571,0.42857,0.0,1.0,4,1,0,4,0,1,0,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[28,35,0.8,0.29018,0.2004,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,6,0,0,10,0,0,8,0,0,1,0,0,1,0,0,1,0,0],[32,35,0.9143,0.28571,0.17128,0.24999,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,1,0,0,10,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[35,35,1.0,0.28125,0.12103,0.24999,0.28571,0.42857,0.0,0.42857,2,0,0,2,0,6,0,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04455,"x":0.2634,"p":[[0,81,0.0,0.17411,0.17029,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,9,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,81,0.0494,0.2634,0.19597,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,6,0,0,8,0,0,9,0,0,0,0,0,2,0,0,0,0,0],[8,81,0.0988,0.26337,0.21158,0.0,0.28571,0.42857,0.0,0.71429,10,0,0,10,0,3,0,0,5,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[12,81,0.1481,0.24553,0.18638,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,2,0,0,8,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[16,81,0.1975,0.2366,0.18763,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,5,0,0,9,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[20,81,0.2469,0.21875,0.19556,0.0,0.2857,0.28571,0.0,0.71429,11,0,0,11,0,3,0,0,12,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[24,81,0.2963,0.14732,0.14054,0.0,0.14286,0.28571,0.0,0.57143,12,0,0,12,0,9,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,81,0.3457,0.12946,0.15714,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,5,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.17857,0.20516,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,2,0,0,7,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[36,81,0.4444,0.125,0.20438,0.0,0.0,0.2857,0.0,0.85714,21,0,0,21,0,2,0,0,4,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[40,81,0.4938,0.12945,0.16111,0.0,0.0,0.28571,0.0,0.571,17,0,0,17,0,5,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,81,0.5432,0.11607,0.14914,0.0,0.0,0.28571,0.0,0.42857,19,0,0,19,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,81,0.642,0.12946,0.15303,0.0,0.0,0.28571,0.0,0.4286,18,0,0,18,0,1,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.12052,0.19266,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[60,81,0.7407,0.11161,0.16263,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,3,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,81,0.7901,0.12946,0.16115,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.09375,0.14555,0.0,0.0,0.2857,0.0,0.42857,22,0,0,22,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.04455,0.09052,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ca51c1b6885a085e","q":"Let $\\mathrm{a}, \\mathrm{b} \\geq 2$ be positive integers with $\\operatorname{gcd}(\\mathrm{a}, \\mathrm{b})=1$. Let $r$ be the smallest positive value assumed by an expression of the form $\\frac{a}{b}-\\frac{c}{d}$, where $c$ and $d$ are positive integers satisfying $\\mathbf{c} \\leq \\mathrm{a}$ and $\\mathrm{d} \\leq \\mathrm{b}$. Prove that $\\frac{1}{r}$ is an integer.","t":[{"b":5,"e":0.85714,"k":"flat","v":0.71429,"x":0.86607,"p":[[0,25,0.0,0.71429,0.25505,0.4286,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,7,0,0,2,0,11],[4,25,0.16,0.83928,0.21053,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,3,0,18],[8,25,0.32,0.76339,0.24383,0.42857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,6,0,0,4,0,13],[12,25,0.48,0.80804,0.18766,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,11,0,0,5,0,12],[16,25,0.64,0.85714,0.17496,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,6,0,16],[20,25,0.8,0.86607,0.18877,0.82132,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,0,7,0,17],[24,25,0.96,0.83482,0.10779,0.71429,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,13,0,7],[25,25,1.0,0.82589,0.12234,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,10,0,8]]},{"b":6,"e":0.42857,"k":"falling","v":0.62054,"x":0.84821,"p":[[0,17,0.0,0.84821,0.19541,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,7,0,16],[4,17,0.2353,0.74554,0.23887,0.42857,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,7,0,0,3,0,12],[8,17,0.4706,0.84375,0.17985,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,5,0,15],[12,17,0.7059,0.8125,0.19704,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,7,0,0,8,0,12],[16,17,0.9412,0.70981,0.21274,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,9,0,0,7,0,6],[17,17,1.0,0.62054,0.21011,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,13,0,0,3,0,0,8,0,0,3,0,4]]}]},{"i":"f625b640c9fd8d75","q":"Let $\\mathrm{P}, \\mathrm{Q}$ be polynomials with real coefficients such that $\\mathrm{P} \\circ \\mathrm{Q}=\\mathrm{P}^{2019}$. Suppose that all the roots of P are real. Show that they are all equal.","t":[{"b":2,"e":0.57143,"k":"falling","v":0.61606,"x":0.93304,"p":[[0,44,0.0,0.78124,0.23955,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,0,3,0,14],[4,44,0.0909,0.79464,0.26949,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,5,0,0,1,0,18],[8,44,0.1818,0.84821,0.20183,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,0,0,19],[12,44,0.2727,0.93304,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[16,44,0.3636,0.88839,0.21048,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,0,0,24],[20,44,0.4545,0.89732,0.15663,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,0,0,22],[24,44,0.5455,0.92857,0.14286,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,25],[28,44,0.6364,0.88393,0.16917,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,1,0,21],[32,44,0.7273,0.90179,0.15335,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,1,0,22],[36,44,0.8182,0.85714,0.20825,0.67857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,3,0,0,0,0,21],[40,44,0.9091,0.73661,0.18935,0.57143,0.64286,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,5,0,0,1,0,10],[44,44,1.0,0.61606,0.13092,0.57143,0.57143,0.57143,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,2,0,0,0,0,3]]},{"b":5,"e":0.57143,"k":"flat","v":0.70536,"x":0.82142,"p":[[0,54,0.0,0.74106,0.19705,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,8,0,0,1,0,10],[4,54,0.0741,0.80357,0.23623,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,5,0,0,1,0,17],[8,54,0.1481,0.82142,0.2369,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,2,0,18],[12,54,0.2222,0.79911,0.22829,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,8,0,0,2,0,15],[16,54,0.2963,0.74554,0.24415,0.57143,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,7,0,0,8,0,0,0,0,13],[20,54,0.3704,0.77679,0.20806,0.67857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,10,0,0,2,0,12],[24,54,0.4444,0.76786,0.21053,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,10,0,0,1,0,12],[28,54,0.5185,0.77232,0.23107,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,12,0,0,1,0,13],[32,54,0.5926,0.75893,0.19377,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,16,0,0,0,0,10],[36,54,0.6667,0.70982,0.23551,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,2,0,0,7,0,0,9,0,0,2,0,9],[40,54,0.7407,0.72321,0.23128,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,5,0,0,3,0,10],[44,54,0.8148,0.75446,0.22084,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,7,0,0,2,0,12],[48,54,0.8889,0.76339,0.23854,0.53572,0.78571,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,7,0,0,3,0,13],[52,54,0.963,0.70982,0.25626,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,5,0,0,1,0,12],[54,54,1.0,0.70536,0.18877,0.57143,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,15,0,0,0,0,7]]}]},{"i":"b4dad3bdfd8218db","q":"Let $0 \\leqslant a, b, c, d, e \\leqslant 1$ be real numbers. Show that\n\n$$\n(1+a+b+c+d+e)^{2} \\geqslant 4\\left(a^{2}+b^{2}+c^{2}+d^{2}+e^{2}\\right)\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.5625,"x":1.0,"p":[[0,18,0.0,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[4,18,0.2222,0.76339,0.4159,0.85714,1.0,1.0,0.0,1.0,7,24,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24],[8,18,0.4444,0.5625,0.49608,0.0,1.0,1.0,0.0,1.0,14,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.0,"k":"falling","v":0.02232,"x":0.875,"p":[[0,35,0.0,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,35,0.1143,0.79464,0.37447,0.89286,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,24],[8,35,0.2286,0.71875,0.44961,0.0,1.0,1.0,0.0,1.0,9,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[12,35,0.3429,0.40625,0.49113,0.0,0.0,1.0,0.0,1.0,19,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[16,35,0.4571,0.27232,0.43793,0.0,0.0,0.78571,0.0,1.0,23,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[20,35,0.5714,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,35,0.6857,0.13839,0.29984,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,2],[28,35,0.8,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,35,0.9143,0.11607,0.2911,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[35,35,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d25a4687bcc270ae","q":"Let $P_{k}(x)=1+x+x^{2}+\\cdots+x^{k-1}$. Show that\n\n$$\n\\sum_{k=1}^{n}\\left(\\begin{array}{l}\nn \\\\\nk\n\\end{array}\\right) P_{k}(x)=2^{n-1} P_{n}\\left(\\frac{1+x}{2}\\right)\n$$\n\nfor every real number $x$ and every positive integer $n$.","t":[{"b":1,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,15,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,15,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,31,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,31,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2d4fcc35114a2614","q":"Let $a, b$ be positive real numbers. Prove that $\\sqrt{\\frac{a^{2}+a b+b^{2}}{3}}+\\sqrt{a b} \\leq a+b$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,14,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,14,0.2857,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,14,0.5714,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,14,0.8571,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[14,14,1.0,0.96427,0.14289,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.95534,"x":1.0,"p":[[0,14,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,14,0.2857,0.95534,0.13095,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[8,14,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,14,0.8571,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[14,14,1.0,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28]]}]},{"i":"178e3e18b5b31a2d","q":"Let $\\varepsilon>0$ be given. Show that there exist integers $a, b$, and $c$, not all zero, such that the inequality $|a \\sqrt{2}+b \\sqrt{3}+c \\sqrt{5}|<\\varepsilon$ is satisfied.\nFurthermore, prove that in every triple $(a, b, c) \\neq(0,0,0)$ of integers for which this inequality holds, at least one of the absolute values $|a|,|b|,|c|$ is greater than $\\varepsilon^{-1 / 3} / \\sqrt{30}$.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.40623,"x":0.83482,"p":[[0,34,0.0,0.66071,0.35129,0.28571,0.78571,1.0,0.0,1.0,3,12,2,3,0,1,0,0,5,0,0,2,0,0,2,0,0,3,0,0,4,0,12],[4,34,0.1176,0.71874,0.35443,0.39287,0.92857,1.0,0.0,1.0,2,16,0,2,0,3,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,16],[8,34,0.2353,0.58034,0.34058,0.2857,0.57121,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,7,0,0,2,0,0,2,0,0,4,0,0,1,0,10],[12,34,0.3529,0.51786,0.31084,0.28571,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,5,0,0,10,0,0,3,0,0,2,0,0,5,0,0,0,0,7],[16,34,0.4706,0.46427,0.26486,0.2857,0.4998,0.60714,0.0,1.0,2,2,0,2,0,5,0,0,5,0,0,4,0,0,8,0,0,5,0,0,1,0,2],[20,34,0.5882,0.47768,0.34183,0.2857,0.28571,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,11,0,0,3,0,0,0,0,0,2,0,0,2,0,7],[24,34,0.7059,0.40623,0.28594,0.14286,0.28571,0.60714,0.0,1.0,1,2,0,1,0,10,0,0,7,0,0,4,0,0,2,0,0,3,0,0,3,0,2],[28,34,0.8235,0.65624,0.29203,0.53539,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,1,0,0,2,0,0,6,0,0,6,0,0,4,0,8],[32,34,0.9412,0.83482,0.23176,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,7,0,16],[34,34,1.0,0.79462,0.15948,0.71429,0.857,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,9,0,8]]},{"b":7,"e":0.85714,"k":"falling","v":0.44196,"x":0.80357,"p":[[0,33,0.0,0.65178,0.35881,0.28571,0.85707,1.0,0.0,1.0,2,12,1,2,0,3,0,0,5,0,0,3,0,0,0,0,0,2,0,0,5,0,12],[4,33,0.1212,0.67411,0.30978,0.28571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,7,0,0,2,0,0,2,0,0,4,0,0,4,0,11],[8,33,0.2424,0.80357,0.26905,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,1,0,0,7,0,16],[12,33,0.3636,0.80357,0.26426,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,3,0,0,8,0,15],[16,33,0.4848,0.68749,0.33204,0.28571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,8,0,0,3,0,0,0,0,0,0,0,0,6,0,13],[20,33,0.6061,0.74999,0.29015,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,4,0,0,0,0,0,2,0,0,5,0,0,7,0,12],[24,33,0.7273,0.65179,0.23673,0.42857,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,4,0,0,3,0,0,7,0,0,9,0,3],[28,33,0.8485,0.63835,0.25751,0.42859,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,2,0,0,3,0,0,7,0,0,12,0,1],[32,33,0.9697,0.58482,0.27747,0.28571,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,11,0,0,0,0,0,1,0,0,6,0,0,11,0,1],[33,33,1.0,0.44196,0.26812,0.28571,0.28571,0.60714,0.14286,1.0,0,3,0,0,0,3,0,0,17,0,0,3,0,0,1,0,0,2,0,0,3,0,3]]}]},{"i":"c5dc9b651505bae9","q":"Let $a$ and $b$ be positive integers such that $a!b$ ! is a multiple of $a!+b!$. Prove that $3 a \\geqslant 2 b+2$. (United Kingdom)","t":[{"b":1,"e":0.42857,"k":"rising","v":0.27679,"x":0.50447,"p":[[0,56,0.0,0.27679,0.17105,0.14286,0.14288,0.42857,0.0,0.71429,2,0,0,2,0,15,0,0,0,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[4,56,0.0714,0.37491,0.30259,0.14286,0.21428,0.4286,0.0,1.0,1,4,0,1,0,15,0,0,1,0,0,8,0,0,1,0,0,1,0,0,1,0,4],[8,56,0.1429,0.34375,0.25966,0.14286,0.1429,0.42857,0.14286,1.0,0,2,0,0,0,17,0,0,1,0,0,8,0,0,0,0,0,4,0,0,0,0,2],[12,56,0.2143,0.50447,0.34623,0.14286,0.42859,0.89286,0.14286,1.0,0,8,0,0,0,12,0,0,0,0,0,8,0,0,0,0,0,3,0,0,1,0,8],[16,56,0.2857,0.42402,0.28464,0.14286,0.42857,0.71429,0.14,1.0,0,2,0,0,0,13,0,0,1,0,0,8,0,0,0,0,0,6,0,0,2,0,2],[20,56,0.3571,0.41515,0.22118,0.24999,0.42857,0.4642,0.0,1.0,1,1,0,1,0,7,0,0,2,0,0,14,0,0,2,0,0,5,0,0,0,0,1],[24,56,0.4286,0.4954,0.2767,0.39285,0.42857,0.60682,0.14,1.0,0,4,0,0,0,7,0,0,1,0,0,13,0,0,3,0,0,1,0,0,3,0,4],[28,56,0.5,0.42411,0.25123,0.14289,0.42857,0.42858,0.0,1.0,1,2,0,1,0,8,0,0,1,0,0,15,0,0,0,0,0,4,0,0,1,0,2],[32,56,0.5714,0.33912,0.24656,0.14286,0.2857,0.42857,0.14286,1.0,0,2,0,0,0,15,0,0,5,0,0,5,0,0,3,0,0,2,0,0,0,0,2],[36,56,0.6429,0.44195,0.2681,0.14286,0.42857,0.57143,0.0,1.0,1,3,0,1,0,8,0,0,1,0,0,13,0,0,2,0,0,3,0,0,1,0,3],[40,56,0.7143,0.49999,0.30093,0.14286,0.4286,0.71429,0.0,1.0,1,4,0,1,0,8,0,0,2,0,0,6,0,0,2,0,0,8,0,0,1,0,4],[44,56,0.7857,0.38835,0.2293,0.14286,0.42857,0.4642,0.0,1.0,1,1,0,1,0,10,0,0,1,0,0,12,0,0,3,0,0,4,0,0,0,0,1],[48,56,0.8571,0.3616,0.23685,0.14286,0.42857,0.4286,0.0,1.0,1,1,0,1,0,13,0,0,0,0,0,11,0,0,2,0,0,4,0,0,0,0,1],[52,56,0.9286,0.37499,0.19148,0.14286,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,11,0,0,0,0,0,15,0,0,2,0,0,4,0,0,0,0,0],[56,56,1.0,0.47768,0.249,0.39286,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,6,0,0,2,0,0,14,0,0,2,0,0,4,0,0,1,0,3]]},{"b":3,"e":0.71429,"k":"rising","v":0.3482,"x":0.6696,"p":[[0,54,0.0,0.37947,0.18074,0.14286,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,10,0,0,0,0,0,16,0,0,3,0,0,3,0,0,0,0,0],[4,54,0.0741,0.41518,0.31209,0.14286,0.42857,0.71429,0.0,1.0,4,4,0,4,0,7,0,0,4,0,0,7,0,0,1,0,0,5,0,0,0,0,4],[8,54,0.1481,0.41518,0.25344,0.14286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,12,0,0,0,0,0,9,0,0,5,0,0,4,0,0,0,0,2],[12,54,0.2222,0.54463,0.22427,0.42857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,9,0,0,4,0,0,10,0,0,1,0,2],[16,54,0.2963,0.49098,0.2923,0.24999,0.42857,0.71429,0.14,1.0,0,5,0,0,0,8,0,0,2,0,0,11,0,0,1,0,0,4,0,0,1,0,5],[20,54,0.3704,0.4107,0.22516,0.14286,0.42857,0.46418,0.14286,1.0,0,1,0,0,0,9,0,0,3,0,0,12,0,0,1,0,0,6,0,0,0,0,1],[24,54,0.4444,0.41961,0.24204,0.14286,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,9,0,0,3,0,0,12,0,0,2,0,0,2,0,0,3,0,1],[28,54,0.5185,0.49107,0.31731,0.14286,0.42857,0.71429,0.0,1.0,2,5,0,2,0,8,0,0,0,0,0,9,0,0,1,0,0,6,0,0,1,0,5],[32,54,0.5926,0.3482,0.28331,0.14286,0.2143,0.4286,0.0,1.0,2,2,0,2,0,14,0,0,3,0,0,6,0,0,1,0,0,2,0,0,2,0,2],[36,54,0.6667,0.38841,0.24019,0.14286,0.42857,0.46525,0.14286,1.0,0,1,0,0,0,12,0,0,2,0,0,10,0,0,2,0,0,4,0,0,1,0,1],[40,54,0.7407,0.4107,0.25441,0.14286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,12,0,0,1,0,0,9,0,0,3,0,0,4,0,0,2,0,1],[44,54,0.8148,0.53125,0.29717,0.2857,0.5,0.71429,0.0,1.0,1,4,0,1,0,6,0,0,3,0,0,6,0,0,1,0,0,9,0,0,2,0,4],[48,54,0.8889,0.57589,0.24086,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,7,0,0,2,0,0,11,0,0,2,0,3],[52,54,0.963,0.6294,0.21977,0.571,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,0,0,0,3,0,0,7,0,0,14,0,0,3,0,2],[54,54,1.0,0.6696,0.2156,0.571,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,14,0,0,2,0,5]]}]},{"i":"89e6639aae22fc25","q":"Let $a, b$, and $c$ be non-zero real numbers and let $a \\geq b \\geq c$. Prove the inequality\n\n$$\n\\frac{a^{3}-c^{3}}{3} \\geq a b c\\left(\\frac{a-b}{c}+\\frac{b-c}{a}\\right)\n$$\n\nWhen does equality hold?","t":[{"b":2,"e":0.71429,"k":"flat","v":0.79911,"x":0.95536,"p":[[0,32,0.0,0.91071,0.23623,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,32,0.125,0.85713,0.25002,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,3,0,0,0,0,23],[8,32,0.25,0.91517,0.18162,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,25],[12,32,0.375,0.85268,0.23551,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,22],[16,32,0.5,0.86607,0.25238,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,24],[20,32,0.625,0.79911,0.2621,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,7,0,0,0,0,18],[24,32,0.75,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[28,32,0.875,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[32,32,1.0,0.91518,0.13767,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,23]]},{"b":3,"e":1.0,"k":"flat","v":0.83927,"x":0.98661,"p":[[0,33,0.0,0.85266,0.24612,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,5,0,0,0,0,22],[4,33,0.1212,0.86161,0.24868,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,2,0,0,1,0,23],[8,33,0.2424,0.83927,0.23892,0.71429,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,0,2,0,19],[12,33,0.3636,0.91964,0.18877,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,26],[16,33,0.4848,0.88839,0.20744,0.82132,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,1,0,23],[20,33,0.6061,0.97322,0.10971,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[24,33,0.7273,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[28,33,0.8485,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,33,0.9697,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[33,33,1.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26]]}]},{"i":"acd93bf0d5aa3075","q":"Let $S \\subset\\{1, \\ldots, n\\}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \\neq 1$ and let $d$ be its smallest divisor greater than 1 . Let $T \\subset\\{1, \\ldots, n\\}$ be a set such that $S \\subset T$ and $|T| \\geq 1+\\left[\\frac{n}{d}\\right]$. Prove that the greatest common divisor of the elements in $T$ is 1 .","t":[{"b":3,"e":0.42857,"k":"falling","v":0.47767,"x":0.73213,"p":[[0,29,0.0,0.73213,0.27837,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,1,0,0,3,0,14],[4,29,0.1379,0.70979,0.25124,0.53539,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,2,0,0,5,0,10],[8,29,0.2759,0.65621,0.26212,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,5,0,0,10,0,0,3,0,0,0,0,10],[12,29,0.4138,0.72316,0.24987,0.571,0.85714,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,1,0,0,8,0,9],[16,29,0.5517,0.5,0.28571,0.28571,0.42857,0.60714,0.14286,1.0,0,5,0,0,0,7,0,0,2,0,0,11,0,0,4,0,0,1,0,0,2,0,5],[20,29,0.6897,0.47767,0.23584,0.39286,0.42857,0.57143,0.0,1.0,1,2,0,1,0,4,0,0,3,0,0,10,0,0,9,0,0,1,0,0,2,0,2],[24,29,0.8276,0.53571,0.25254,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,12,0,0,3,0,0,6,0,0,1,0,4],[28,29,0.9655,0.5179,0.21941,0.42857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,12,0,0,4,0,0,7,0,0,1,0,2],[29,29,1.0,0.53125,0.24804,0.39286,0.5,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,5,0,0,8,0,0,7,0,0,4,0,0,1,0,4]]},{"b":7,"e":1.0,"k":"rising","v":0.68749,"x":1.0,"p":[[0,37,0.0,0.70534,0.23942,0.57132,0.64286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,3,0,0,3,0,10],[4,37,0.1081,0.68749,0.25615,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,4,0,0,4,0,9],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2689d55cb613cf14","q":"Let $P(x)$ be a given polynomial with integer coefficients. Prove that there exist two polynomials $Q(x)$ and $R(x)$, again with integer coefficients, such that (i) $P(x) Q(x)$ is a polynomial in $x^{2}$; and (ii) $P(x) R(x)$ is a polynomial in $x^{3}$.","t":[{"b":1,"e":0.42857,"k":"falling","v":0.41072,"x":0.83929,"p":[[0,31,0.0,0.83929,0.09942,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,6],[4,31,0.129,0.83929,0.15047,0.85714,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,20,0,7],[8,31,0.2581,0.71873,0.24611,0.57142,0.85707,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,11,0,6],[12,31,0.3871,0.61606,0.22429,0.42857,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,4,0,0,8,0,0,4,0,0,8,0,2],[16,31,0.5161,0.43304,0.14934,0.39286,0.42857,0.46431,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,16,0,0,6,0,0,1,0,0,1,0,0],[20,31,0.6452,0.5,0.21724,0.42857,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,15,0,0,1,0,0,3,0,0,6,0,0],[24,31,0.7742,0.45981,0.13235,0.42857,0.42857,0.57111,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,17,0,0,6,0,0,2,0,0,1,0,0],[28,31,0.9032,0.41072,0.14617,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,9,0,0,18,0,0,0,0,0,4,0,0,0,0,0],[31,31,1.0,0.41072,0.12242,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,6,0,0,22,0,0,1,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.80803,"x":0.84821,"p":[[0,24,0.0,0.84821,0.10677,0.82132,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,17,0,7],[4,24,0.1667,0.83036,0.1448,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,9,0,10],[8,24,0.3333,0.81696,0.1394,0.8214,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,20,0,4],[12,24,0.5,0.84821,0.13803,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,13,0,10],[16,24,0.6667,0.80803,0.09852,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,21,0,1],[20,24,0.8333,0.84821,0.10062,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,19,0,6],[24,24,1.0,0.81696,0.11425,0.71429,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,15,0,5]]}]},{"i":"92bb5f94338ea76e","q":"Let $a, b$ and $c$ be positive real numbers. Prove that\n\n$$\na^{3} b^{6}+b^{3} c^{6}+c^{3} a^{6}+3 a^{3} b^{3} c^{3} \\geq a b c\\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\\right)+a^{2} b^{2} c^{2}\\left(a^{3}+b^{3}+c^{3}\\right)\n$$","t":[{"b":0,"e":0.85714,"k":"flat","v":0.70534,"x":0.94643,"p":[[0,138,0.0,0.70982,0.38546,0.39286,1.0,1.0,0.0,1.0,4,19,3,4,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,19],[4,138,0.029,0.88393,0.20652,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,6,0,21],[8,138,0.058,0.94643,0.12242,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[12,138,0.087,0.85267,0.24868,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,20],[16,138,0.1159,0.91969,0.15111,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,22],[20,138,0.1449,0.90625,0.16982,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,6,0,21],[24,138,0.1739,0.89286,0.17128,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,10,0,18],[28,138,0.2029,0.86158,0.20359,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,7,0,18],[32,138,0.2319,0.87945,0.19271,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,7,0,19],[36,138,0.2609,0.83036,0.23808,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,6,0,17],[40,138,0.2899,0.82589,0.27833,0.7499,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,2,0,0,5,0,0,0,0,0,0,0,0,3,0,21],[44,138,0.3188,0.86607,0.20806,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,8,0,18],[48,138,0.3478,0.81696,0.27718,0.82143,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,0,6,0,18],[52,138,0.3768,0.85267,0.20042,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,13,0,14],[56,138,0.4058,0.87946,0.2055,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,7,0,20],[60,138,0.4348,0.75446,0.30978,0.42859,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,2,0,0,5,0,0,1,0,0,0,0,0,8,0,14],[64,138,0.4638,0.86606,0.2141,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,10,0,17],[68,138,0.4928,0.83033,0.22145,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,2,0,0,7,0,16],[72,138,0.5217,0.79018,0.2575,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,0,8,0,14],[76,138,0.5507,0.89286,0.15152,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,10,0,17],[80,138,0.5797,0.89286,0.18898,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,21],[84,138,0.6087,0.91072,0.12242,0.85714,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,17],[88,138,0.6377,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[92,138,0.6667,0.83928,0.23076,0.82132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,6,0,18],[96,138,0.6957,0.86605,0.20808,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,8,0,18],[100,138,0.7246,0.82589,0.27137,0.82143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,0,5,0,19],[104,138,0.7536,0.83927,0.23353,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,11,0,15],[108,138,0.7826,0.88839,0.14167,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,16],[112,138,0.8116,0.83036,0.21852,0.82143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,2,0,0,10,0,14],[116,138,0.8406,0.82143,0.25,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,4,0,0,5,0,17],[120,138,0.8696,0.70534,0.28558,0.42857,0.78571,1.0,0.0,1.0,1,10,0,1,0,1,0,0,2,0,0,5,0,0,3,0,0,4,0,0,6,0,10],[124,138,0.8986,0.85714,0.19233,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,13,0,14],[128,138,0.9275,0.77232,0.25218,0.4286,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,7,0,0,1,0,0,0,0,0,10,0,12],[132,138,0.9565,0.80801,0.18767,0.82132,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,16,0,8],[136,138,0.9855,0.74999,0.22017,0.67857,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,0,15,0,5],[138,138,1.0,0.7723,0.1851,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,6,0,0,13,0,6]]},{"b":7,"e":1.0,"k":"flat","v":0.6607,"x":0.92411,"p":[[0,78,0.0,0.6607,0.41149,0.28571,1.0,1.0,0.0,1.0,6,18,3,6,0,1,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,18],[4,78,0.0513,0.88392,0.2156,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,3,0,23],[8,78,0.1026,0.88839,0.23072,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,6,0,22],[12,78,0.1538,0.87946,0.18935,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,7,0,19],[16,78,0.2051,0.89732,0.14827,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,11,0,17],[20,78,0.2564,0.87052,0.20936,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,5,0,20],[24,78,0.3077,0.875,0.18472,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,9,0,17],[28,78,0.359,0.80357,0.21353,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,2,0,0,10,0,12],[32,78,0.4103,0.84821,0.24206,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,5,0,19],[36,78,0.4615,0.89285,0.18211,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,7,0,20],[40,78,0.5128,0.92411,0.1636,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,23],[44,78,0.5641,0.83927,0.25192,0.82143,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,0,5,0,19],[48,78,0.6154,0.87052,0.21831,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,6,0,20],[52,78,0.6667,0.83927,0.21945,0.82132,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,10,0,14],[56,78,0.7179,0.84822,0.23128,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,10,0,16],[60,78,0.7692,0.75,0.21724,0.67857,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,5,0,0,13,0,6],[64,78,0.8205,0.71875,0.22155,0.4286,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,7,0,0,1,0,0,4,0,0,14,0,4],[68,78,0.8718,0.71425,0.22017,0.5354,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,2,0,0,14,0,4],[72,78,0.9231,0.67411,0.22653,0.42859,0.85707,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,10,0,0,2,0,0,1,0,0,15,0,2],[76,78,0.9744,0.70536,0.24727,0.53572,0.85714,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,6,0,0,3,0,0,2,0,0,16,0,3],[78,78,1.0,0.73214,0.19479,0.57143,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,4,0,0,15,0,3]]}]},{"i":"2b093f26a6514c52","q":"Let $a, b, c, d$ be real numbers such that $0 \\leqslant a \\leqslant b \\leqslant c \\leqslant d$. Show that\n\n$$\na b^{3}+b c^{3}+c d^{3}+d a^{3} \\geqslant a^{2} b^{2}+b^{2} c^{2}+c^{2} d^{2}+d^{2} a^{2}\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.04902,"x":0.23661,"p":[[0,71,0.0,0.05804,0.17807,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,71,0.0563,0.10712,0.1556,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,11,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,71,0.1127,0.16964,0.33204,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[12,71,0.169,0.13831,0.23551,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,9,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[16,71,0.2254,0.11606,0.22708,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,8,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[20,71,0.2817,0.08929,0.1948,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[24,71,0.338,0.15622,0.24312,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,8,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0],[28,71,0.3944,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,71,0.4507,0.22991,0.31981,0.0,0.03571,0.42858,0.0,1.0,16,2,0,16,1,5,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,2],[36,71,0.507,0.11161,0.198,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,71,0.5634,0.23661,0.35285,0.0,0.07143,0.17857,0.0,1.0,16,4,0,16,0,8,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,4],[44,71,0.6197,0.12054,0.15612,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,10,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[48,71,0.6761,0.12054,0.18249,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,12,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[52,71,0.7324,0.09375,0.19102,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[56,71,0.7887,0.0625,0.10063,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,71,0.9014,0.12946,0.20935,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,9,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[68,71,0.9577,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.09375,0.17717,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,9,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.02679,"x":0.24554,"p":[[0,63,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.14732,0.21866,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,8,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[8,63,0.127,0.12054,0.21461,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[12,63,0.1905,0.16955,0.27302,0.0,0.07,0.14286,0.0,1.0,16,2,0,16,0,10,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[16,63,0.254,0.22321,0.35162,0.0,0.0,0.14286,0.0,1.0,17,4,0,17,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,4],[20,63,0.3175,0.19643,0.29827,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,12,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,2],[24,63,0.381,0.24554,0.30771,0.0,0.14286,0.28571,0.0,1.0,10,3,0,10,0,13,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,3],[28,63,0.4444,0.11161,0.17762,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,12,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[32,63,0.5079,0.14286,0.26726,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,8,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[36,63,0.5714,0.14732,0.26119,0.0,0.0,0.14286,0.0,1.0,17,2,0,17,0,10,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[40,63,0.6349,0.13393,0.2111,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[44,63,0.6984,0.11161,0.23887,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,63,0.7619,0.12054,0.25028,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[52,63,0.8254,0.07143,0.14285,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[56,63,0.8889,0.17857,0.27663,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,8,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,1],[60,63,0.9524,0.10705,0.17126,0.0,0.07,0.14286,0.0,0.71429,16,0,0,16,0,14,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[63,63,1.0,0.09375,0.11071,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"88e7dcef4e0b64cc","q":"Let $a, b, c > 0$ such that: $a + b + c = 1$. Show that:\n\n$$\n\\frac{a}{b} + \\frac{a}{c} + \\frac{b}{a} + \\frac{b}{c} + \\frac{c}{a} + \\frac{c}{b} + 6 \\geqslant 2 \\sqrt{2} \\left( \\sqrt{\\frac{1-a}{a}} + \\sqrt{\\frac{1-c}{c}} + \\sqrt{\\frac{1-b}{b}} \\right)\n$$\n\nand determine the cases of equality.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.38393,"x":0.79463,"p":[[0,89,0.0,0.72321,0.2788,0.42857,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,0,0,0,1,0,15],[4,89,0.0449,0.59375,0.16409,0.57143,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,21,0,0,0,0,0,0,0,4],[8,89,0.0899,0.73214,0.26666,0.57143,0.57143,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,4,0,0,11,0,0,0,0,0,0,0,15],[12,89,0.1348,0.71875,0.2435,0.57143,0.57143,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,0,0,0,0,0,13],[16,89,0.1798,0.66964,0.25364,0.42859,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,8,0,0,12,0,0,0,0,0,0,0,11],[20,89,0.2247,0.69196,0.26513,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,4,0,0,12,0,0,0,0,0,1,0,12],[24,89,0.2697,0.69643,0.26666,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,2,0,0,12,0,0,1,0,0,1,0,12],[28,89,0.3146,0.72768,0.26573,0.57143,0.57143,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,0,0,0,1,0,14],[32,89,0.3596,0.61607,0.24856,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,1,0,0,7,0,0,13,0,0,0,0,0,2,0,7],[36,89,0.4045,0.72321,0.2549,0.57143,0.57143,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,0,0,0,2,0,13],[40,89,0.4494,0.66963,0.29111,0.42857,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,6,0,0,8,0,0,0,0,0,0,0,13],[44,89,0.4944,0.68304,0.29175,0.42857,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,6,0,0,6,0,0,1,0,0,1,0,13],[48,89,0.5393,0.79463,0.22852,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,1,0,0,4,0,15],[52,89,0.5843,0.66516,0.29583,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,6,0,0,4,0,0,6,0,0,2,0,0,1,0,12],[56,89,0.6292,0.56249,0.24206,0.42857,0.57143,0.57143,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,2,0,0,16,0,0,2,0,0,0,0,5],[60,89,0.6742,0.47768,0.27574,0.28571,0.42857,0.57143,0.14286,1.0,0,5,0,0,0,4,0,0,11,0,0,4,0,0,7,0,0,0,0,0,1,0,5],[64,89,0.7191,0.60268,0.26422,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,5,0,0,6,0,0,11,0,0,0,0,0,1,0,8],[68,89,0.764,0.57588,0.27078,0.39286,0.57121,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,7,0,0,7,0,0,8,0,0,0,0,0,2,0,7],[72,89,0.809,0.55804,0.30589,0.28571,0.50001,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,6,0,0,6,0,0,7,0,0,0,0,0,0,0,9],[76,89,0.8539,0.55357,0.3004,0.28571,0.5,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,6,0,0,6,0,0,7,0,0,0,0,0,1,0,8],[80,89,0.8989,0.51339,0.24448,0.42857,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,3,0,0,4,0,0,12,0,0,7,0,0,0,0,0,2,0,4],[84,89,0.9438,0.48213,0.25691,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,3,0,0,6,0,0,9,0,0,8,0,0,0,0,0,1,0,4],[88,89,0.9888,0.41518,0.17985,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,9,0,0,17,0,0,2,0,0,0,0,0,0,0,2],[89,89,1.0,0.38393,0.16146,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,2,0,0,9,0,0,17,0,0,2,0,0,0,0,0,0,0,1]]},{"b":6,"e":1.0,"k":"rising","v":0.54463,"x":1.0,"p":[[0,60,0.0,0.76784,0.25693,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,0,0,0,2,0,16],[4,60,0.0667,0.54911,0.18935,0.53571,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,2,0,0,4,0,0,21,0,0,0,0,0,0,0,3],[8,60,0.1333,0.59822,0.21558,0.42859,0.57143,0.57143,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,7,0,0,17,0,0,0,0,0,0,0,6],[12,60,0.2,0.54463,0.15335,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,7,0,0,21,0,0,0,0,0,0,0,2],[16,60,0.2667,0.58929,0.21651,0.42857,0.57143,0.57143,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,8,0,0,15,0,0,0,0,0,0,0,6],[20,60,0.3333,0.56695,0.16554,0.57132,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,4,0,0,22,0,0,0,0,0,0,0,3],[24,60,0.4,0.60268,0.27371,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,13,0,0,8,0,0,0,0,0,0,0,9],[28,60,0.4667,0.62054,0.26633,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,4,0,0,14,0,0,0,0,0,0,0,9],[32,60,0.5333,0.54464,0.22142,0.42857,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,5,0,0,6,0,0,14,0,0,1,0,0,2,0,3],[36,60,0.6,0.61607,0.24074,0.42857,0.57143,0.78571,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,12,0,0,8,0,0,2,0,0,0,0,8],[40,60,0.6667,0.70088,0.28652,0.53539,0.57143,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,3,0,0,9,0,0,1,0,0,0,0,14],[44,60,0.7333,0.65177,0.27185,0.42857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,6,0,0,11,0,0,0,0,0,0,0,11],[48,60,0.8,0.88839,0.20119,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,0,0,24],[52,60,0.8667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2ff1b09bbbfc28c4","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}, k$, and $M$ be positive integers such that  $$ \\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{n}}=k \\quad \\text { and } \\quad a_{1} a_{2} \\ldots a_{n}=M $$  If $M>1$, prove that the polynomial  $$ P(x)=M(x+1)^{k}-\\left(x+a_{1}\\right)\\left(x+a_{2}\\right) \\cdots\\left(x+a_{n}\\right) $$  has no positive roots. (Trinidad and Tobago)","t":[{"b":3,"e":0.1429,"k":"volatile","v":0.22321,"x":0.91069,"p":[[0,23,0.0,0.87499,0.20126,0.57143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,23],[4,23,0.1739,0.81695,0.25565,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,20],[8,23,0.3478,0.85713,0.24745,0.57143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,23],[12,23,0.5217,0.78567,0.23694,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,1,0,0,0,0,17],[16,23,0.6957,0.91069,0.21655,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,26],[20,23,0.8696,0.59375,0.4317,0.14286,0.78571,1.0,0.0,1.0,7,16,0,7,0,3,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,16],[23,23,1.0,0.22321,0.3272,0.0,0.0,0.28571,0.0,1.0,17,4,0,17,0,2,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,4]]},{"b":7,"e":1.0,"k":"rising","v":0.65179,"x":1.0,"p":[[0,92,0.0,0.82586,0.20437,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,1,0,18],[4,92,0.0435,0.73658,0.23177,0.57143,0.57143,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,0,0,0,1,0,13],[8,92,0.087,0.71874,0.25874,0.57132,0.57143,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,5,0,0,11,0,0,0,0,0,0,0,14],[12,92,0.1304,0.70529,0.2081,0.57143,0.57143,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,0,0,0,1,0,10],[16,92,0.1739,0.81247,0.23269,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,0,0,0,0,0,19],[20,92,0.2174,0.73214,0.24419,0.57143,0.57143,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,13],[24,92,0.2609,0.77229,0.24188,0.57143,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,13,0,0,1,0,0,0,0,16],[28,92,0.3043,0.78124,0.24482,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,1,0,0,0,0,17],[32,92,0.3478,0.66963,0.24075,0.57143,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,3,0,0,17,0,0,0,0,0,0,0,10],[36,92,0.3913,0.75888,0.22145,0.57143,0.57143,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,0,0,0,1,0,14],[40,92,0.4348,0.66516,0.23313,0.57143,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,2,0,0,17,0,0,0,0,0,1,0,9],[44,92,0.4783,0.74996,0.25003,0.57132,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,4,0,0,10,0,0,1,0,0,2,0,14],[48,92,0.5217,0.75893,0.23265,0.57143,0.64286,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,1,0,0,1,0,14],[52,92,0.5652,0.69642,0.21944,0.57143,0.57143,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,18,0,0,0,0,0,1,0,10],[56,92,0.6087,0.84375,0.24578,0.57143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,0,0,0,0,0,22],[60,92,0.6522,0.66069,0.26185,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,1,0,0,18,0,0,0,0,0,0,0,10],[64,92,0.6957,0.7232,0.21111,0.57143,0.57143,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,15,0,0,3,0,0,2,0,10],[68,92,0.7391,0.73657,0.24774,0.57143,0.57143,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,14],[72,92,0.7826,0.65179,0.25985,0.53572,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,4,0,0,13,0,0,1,0,0,0,0,10],[76,92,0.8261,0.69631,0.23645,0.57143,0.57143,1.0,0.14,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,19,0,0,0,0,0,0,0,11],[80,92,0.8696,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[84,92,0.913,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[88,92,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,92,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"fcdfba095f89ba2e","q":"Let $a_{1} \\leqslant a_{2} \\leqslant \\ldots$ be a monotonically increasing sequence of positive integers. A positive integer $n$ is called reliable if there exists a positive integer index $i$ such that $n=\\frac{i}{a_{i}}$.\nProve: If 2013 is reliable, then 20 is also reliable.","t":[{"b":2,"e":0.14286,"k":"rising","v":0.16518,"x":0.66518,"p":[[0,43,0.0,0.42411,0.3416,0.14286,0.35714,0.71429,0.0,1.0,5,4,0,5,0,8,0,0,3,0,0,4,0,0,1,0,0,5,0,0,2,0,4],[4,43,0.093,0.66518,0.39546,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,9,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,17],[8,43,0.186,0.59375,0.37306,0.24999,0.64286,1.0,0.0,1.0,2,12,0,2,0,6,0,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,12],[12,43,0.2791,0.59822,0.33775,0.25,0.71429,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,2,0,0,4,0,0,0,0,0,7,0,0,2,0,9],[16,43,0.3721,0.51786,0.38589,0.14286,0.35714,1.0,0.14286,1.0,0,10,0,0,0,15,0,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,10],[20,43,0.4651,0.47768,0.40027,0.14286,0.28571,1.0,0.0,1.0,2,11,0,2,0,13,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,11],[24,43,0.5581,0.4732,0.39999,0.14286,0.42857,1.0,0.0,1.0,5,10,0,5,0,10,0,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,10],[28,43,0.6512,0.375,0.39407,0.0,0.14286,0.75,0.0,1.0,9,7,0,9,0,9,0,0,2,0,0,2,0,0,0,0,0,2,0,0,1,0,7],[32,43,0.7442,0.16518,0.16015,0.10714,0.14286,0.14287,0.0,0.71429,8,0,0,8,0,17,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[36,43,0.8372,0.16518,0.12931,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,23,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[40,43,0.9302,0.28572,0.32143,0.14286,0.14286,0.28571,0.0,1.0,5,5,0,5,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[43,43,1.0,0.61161,0.36811,0.2857,0.57143,1.0,0.14286,1.0,0,14,0,0,0,7,0,0,5,0,0,4,0,0,0,0,0,2,0,0,0,0,14]]},{"b":6,"e":0.14286,"k":"flat","v":0.14732,"x":0.50893,"p":[[0,56,0.0,0.3125,0.34707,0.10714,0.14286,0.42858,0.0,1.0,8,5,0,8,0,12,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,5],[4,56,0.0714,0.44196,0.35956,0.14286,0.21429,0.75,0.0,1.0,1,7,0,1,0,15,0,0,1,0,0,4,0,0,0,0,0,3,0,0,1,0,7],[8,56,0.1429,0.41517,0.32802,0.14286,0.21428,0.71429,0.14286,1.0,0,5,0,0,0,16,0,0,2,0,0,3,0,0,2,0,0,3,0,0,1,0,5],[12,56,0.2143,0.50893,0.39438,0.14286,0.42857,1.0,0.0,1.0,4,10,0,4,0,9,0,0,1,0,0,4,0,0,0,0,0,3,0,0,1,0,10],[16,56,0.2857,0.37054,0.3533,0.14286,0.14286,0.71429,0.0,1.0,3,6,0,3,0,16,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,6],[20,56,0.3571,0.33705,0.30735,0.14286,0.14286,0.60714,0.0,1.0,4,2,0,4,0,15,0,0,1,0,0,3,0,0,1,0,0,4,1,0,1,0,2],[24,56,0.4286,0.3259,0.29717,0.14286,0.14286,0.42857,0.0,1.0,2,3,0,2,0,18,0,0,1,0,0,4,0,0,0,0,0,4,0,0,0,0,3],[28,56,0.5,0.29911,0.34692,0.14286,0.14286,0.32143,0.0,1.0,7,5,0,7,0,15,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[32,56,0.5714,0.18295,0.18979,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,23,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[36,56,0.6429,0.31695,0.27369,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,16,0,0,4,0,0,3,0,0,2,0,0,2,0,0,1,0,2],[40,56,0.7143,0.29009,0.29125,0.14286,0.14286,0.42857,0.0,1.0,4,3,0,4,0,17,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,3],[44,56,0.7857,0.26777,0.23628,0.14286,0.14286,0.28571,0.14,1.0,0,2,0,0,0,20,0,0,7,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[48,56,0.8571,0.14732,0.0977,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,20,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.1875,0.09061,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.2009,0.12807,0.14286,0.14286,0.2857,0.0,0.71429,1,0,0,1,0,22,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"804a407d3560f323","q":"Let $c \\geqslant 1$ be an integer. Define a sequence of positive integers by $a_{1}=c$ and  $$ a_{n+1}=a_{n}^{3}-4 c \\cdot a_{n}^{2}+5 c^{2} \\cdot a_{n}+c $$  for all $n \\geqslant 1$. Prove that for each integer $n \\geqslant 2$ there exists a prime number $p$ dividing $a_{n}$ but none of the numbers $a_{1}, \\ldots, a_{n-1}$. (Austria)","t":[{"b":0,"e":0.28571,"k":"flat","v":0.12054,"x":0.21875,"p":[[0,14,0.0,0.12054,0.0724,0.14286,0.14286,0.14286,0.0,0.28571,7,0,2,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.21875,0.17122,0.14286,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,13,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[8,14,0.5714,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.16063,0.08566,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"rising","v":0.10697,"x":0.3482,"p":[[0,38,0.0,0.10697,0.06176,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.2366,0.18764,0.14286,0.14286,0.28579,0.0,0.71429,4,0,0,4,0,16,0,0,5,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[8,38,0.2105,0.1875,0.11538,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,19,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,38,0.3158,0.23658,0.15809,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,14,0,0,10,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[16,38,0.4211,0.27667,0.17107,0.14286,0.14288,0.42857,0.14,0.57143,0,0,0,0,0,18,0,0,4,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[20,38,0.5263,0.25893,0.13092,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,9,0,0,16,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,38,0.6316,0.28123,0.10399,0.2857,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,7,0,0,21,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[28,38,0.7368,0.30803,0.11356,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,20,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[32,38,0.8421,0.29909,0.11492,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,6,0,0,20,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[36,38,0.9474,0.34372,0.12802,0.2857,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,2,0,0,22,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[38,38,1.0,0.3482,0.12844,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,21,0,0,2,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"c0ec883bf488373f","q":"Let $\\mathrm{f}: \\mathbb{N}^{\\star} \\rightarrow \\mathbb{N}^{\\star}$ be a function such that for any integer $\\mathrm{n} \\geqslant 1, \\mathrm{f}(\\mathrm{f}(\\mathrm{n}))$ is equal to the number of positive divisors of $n$. Show that if $p$ is a prime number, then $f(p)$ is also a prime number.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.69194,"x":0.88839,"p":[[0,17,0.0,0.81696,0.18638,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,9,0,11],[4,17,0.2353,0.69194,0.33334,0.57142,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,0,0,0,0,0,0,8,0,0,4,0,0,3,0,12],[8,17,0.4706,0.72321,0.27879,0.57143,0.78564,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,1,0,0,4,0,0,6,0,0,6,0,10],[12,17,0.7059,0.81696,0.16457,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,11,0,0,10,0,9],[16,17,0.9412,0.88839,0.12234,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,16],[17,17,1.0,0.85265,0.14936,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,11,0,12]]},{"b":4,"e":0.4286,"k":"falling","v":0.24996,"x":0.8482,"p":[[0,23,0.0,0.8482,0.14701,0.85713,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,17,0,9],[4,23,0.1739,0.76344,0.20072,0.57143,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,4,0,0,10,0,8],[8,23,0.3478,0.48213,0.25939,0.39286,0.42857,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,2,0,0,9,0,0,6,0,0,5,0,0,3,0,1],[12,23,0.5217,0.33481,0.30847,0.0,0.28571,0.57143,0.0,0.85714,10,0,0,10,0,5,0,0,2,0,0,5,0,0,3,0,0,3,0,0,4,0,0],[16,23,0.6957,0.27231,0.25593,0.0,0.21428,0.42857,0.0,1.0,9,1,0,9,0,7,0,0,5,0,0,5,0,0,3,0,0,2,0,0,0,0,1],[20,23,0.8696,0.24996,0.25995,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,6,0,0,3,0,0,4,0,0,4,0,0,2,0,0,1,0,0],[23,23,1.0,0.38393,0.28221,0.21427,0.42857,0.57143,0.0,1.0,8,1,0,8,0,0,0,0,5,0,0,10,0,0,4,0,0,1,0,0,3,0,1]]}]},{"i":"56e148d92e4d4a2b","q":"Let $m$ and $n$ be non-zero natural numbers. Prove that\n\n$$\n\\frac{(m+n)!}{(m+n)^{m+n}}<\\frac{m!}{m^{m}} \\frac{n!}{n^{n}}\n$$","t":[{"b":2,"e":0.0,"k":"volatile","v":0.03125,"x":0.78125,"p":[[0,21,0.0,0.53571,0.47916,0.0,0.78571,1.0,0.0,1.0,13,16,0,13,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,16],[4,21,0.1905,0.78125,0.39767,0.92857,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[8,21,0.381,0.65179,0.47237,0.0,1.0,1.0,0.0,1.0,11,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,20],[12,21,0.5714,0.46875,0.49902,0.0,0.0,1.0,0.0,1.0,17,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[16,21,0.7619,0.57589,0.4864,0.0,1.0,1.0,0.0,1.0,13,18,0,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18],[20,21,0.9524,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[21,21,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.84375,"p":[[0,70,0.0,0.57589,0.47041,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,17],[4,70,0.0571,0.81696,0.38172,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[8,70,0.1143,0.67857,0.46015,0.0,1.0,1.0,0.0,1.0,10,21,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,21],[12,70,0.1714,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[16,70,0.2286,0.60714,0.47649,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[20,70,0.2857,0.65179,0.47237,0.0,1.0,1.0,0.0,1.0,11,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,20],[24,70,0.3429,0.49554,0.48443,0.0,0.35714,1.0,0.0,1.0,14,15,0,14,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15],[28,70,0.4,0.37946,0.48129,0.0,0.0,1.0,0.0,1.0,19,12,0,19,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[32,70,0.4571,0.61607,0.47034,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[36,70,0.5143,0.35714,0.45876,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[40,70,0.5714,0.47768,0.48788,0.0,0.2143,1.0,0.0,1.0,16,14,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,14],[44,70,0.6286,0.41964,0.48173,0.0,0.07143,1.0,0.0,1.0,16,13,0,16,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[48,70,0.6857,0.48214,0.48806,0.0,0.14286,1.0,0.0,1.0,14,15,0,14,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[52,70,0.7429,0.41071,0.48806,0.0,0.0,1.0,0.0,1.0,18,13,0,18,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[56,70,0.8,0.21875,0.40404,0.0,0.0,0.03571,0.0,1.0,24,6,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[60,70,0.8571,0.12054,0.29474,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[64,70,0.9143,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,70,0.9714,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[70,70,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1b5e3156b447d9c5","q":"Let $k \\geq 3$ be an integer. We define the sequence $\\left(a_{n}\\right)_{n \\geq k}$ by $a_{k}=2 k$, and\n\n$$\na_{n}= \\begin{cases}a_{n-1}+1 & \\text { if } \\operatorname{gcd}\\left(a_{n-1}, n\\right)=1 \\\\ 2 n & \\text { otherwise. }\\end{cases}\n$$\n\nShow that the sequence $\\left(a_{n+1}-a_{n}\\right)_{n \\geq k}$ has infinitely many terms that are prime numbers.","t":[{"b":0,"e":0.42857,"k":"falling","v":0.42856,"x":0.92857,"p":[[0,43,0.0,0.86161,0.31841,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[4,43,0.093,0.77678,0.35163,0.53571,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,21],[8,43,0.186,0.87054,0.26331,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,23],[12,43,0.2791,0.8058,0.29246,0.78561,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0,5,1,18],[16,43,0.3721,0.82143,0.28347,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,0,5,0,19],[20,43,0.4651,0.76339,0.33045,0.67857,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,4,0,0,0,0,0,1,0,0,3,0,0,4,0,17],[24,43,0.5581,0.75,0.3312,0.53571,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,4,0,0,1,0,0,3,0,0,1,0,0,2,0,18],[28,43,0.6512,0.72767,0.32608,0.5354,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,5,0,0,1,0,0,3,0,0,4,0,0,1,0,16],[32,43,0.7442,0.8616,0.23278,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,3,0,22],[36,43,0.8372,0.92857,0.17128,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,26],[40,43,0.9302,0.53572,0.28571,0.39286,0.42857,0.78571,0.14286,1.0,0,8,0,0,0,2,0,0,6,0,0,15,0,0,0,0,0,1,0,0,0,0,8],[43,43,1.0,0.42856,0.22867,0.28571,0.42857,0.42857,0.0,1.0,1,3,0,1,0,4,0,0,4,0,0,18,0,0,1,0,0,1,0,0,0,0,3]]},{"b":4,"e":0.14286,"k":"falling","v":0.04902,"x":0.90625,"p":[[0,33,0.0,0.90625,0.19103,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,4,0,23],[4,33,0.1212,0.82142,0.29452,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,6,0,19],[8,33,0.2424,0.72321,0.30501,0.53571,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,3,0,0,2,0,0,4,0,0,3,0,0,3,0,14],[12,33,0.3636,0.77679,0.29653,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,3,0,0,2,0,0,1,0,0,2,0,0,7,0,15],[16,33,0.4848,0.85714,0.29233,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,22],[20,33,0.6061,0.78348,0.33191,0.64286,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,3,1,19],[24,33,0.7273,0.84821,0.2765,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,3,0,22],[28,33,0.8485,0.46429,0.39447,0.14286,0.42857,0.85714,0.0,1.0,6,7,0,6,0,9,0,0,0,0,0,4,0,0,0,0,0,2,0,0,4,0,7],[32,33,0.9697,0.09813,0.16534,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[33,33,1.0,0.04902,0.11064,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"697407c84374744d","q":"Let $f$ be function defined from the set $\\{(x, y): x, y$ reals, $x y \\neq 0\\}$ in to the set of all positive real numbers such that\n\n(i) $\\quad f(x y, z)=f(x, z) f(y, z)$, for all $x, y \\neq 0$;\n\n(ii) $\\quad f(x, y z)=f(x, y) f(x, z)$, for all $x, y \\neq 0$;\n\n(iii) $\\quad f(x, 1-x)=1$, for all $x \\neq 0,1$.\n\nProve that\n\n(a) $\\quad f(x, x)=f(x,-x)=1$, for all $x \\neq 0$;\n\n(b) $\\quad f(x, y) f(y, x)=1$, for all $x, y \\neq 0$.","t":[{"b":6,"e":0.71429,"k":"rising","v":0.32134,"x":0.75446,"p":[[0,157,0.0,0.32134,0.26732,0.14286,0.14286,0.57143,0.0,1.0,2,1,0,2,0,17,0,0,2,0,0,2,0,0,2,0,0,6,0,0,0,0,1],[4,157,0.0255,0.66961,0.21559,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,2,0,0,4,0,0,16,0,0,4,0,3],[8,157,0.051,0.63393,0.23402,0.67857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,2,0,0,1,0,0,1,0,0,18,0,0,5,0,1],[12,157,0.0764,0.62052,0.27107,0.42859,0.71429,0.74996,0.0,1.0,1,4,0,1,0,4,0,0,0,0,0,4,0,0,4,0,0,11,0,0,4,0,4],[16,157,0.1019,0.75446,0.18638,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,19,0,0,3,0,7],[20,157,0.1274,0.71429,0.25505,0.67857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,10,0,0,7,0,7],[24,157,0.1529,0.70536,0.25985,0.67857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,13,0,0,2,0,9],[28,157,0.1783,0.71429,0.25,0.71429,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,0,0,0,3,0,0,0,0,0,12,0,0,9,0,5],[32,157,0.2038,0.70089,0.25344,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,12,0,0,6,0,6],[36,157,0.2293,0.64286,0.28347,0.5,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,4,0,0,0,0,0,5,0,0,8,0,0,5,0,6],[40,157,0.2548,0.67857,0.25505,0.53572,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,1,0,0,4,0,0,1,0,0,14,0,0,2,0,7],[44,157,0.2803,0.64286,0.26486,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,2,0,0,1,0,0,7,0,0,8,0,0,5,0,5],[48,157,0.3057,0.70088,0.17986,0.57143,0.71429,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,14,0,0,4,0,4],[52,157,0.3312,0.71872,0.21277,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,17,0,0,3,0,6],[56,157,0.3567,0.71428,0.23419,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,12,0,0,10,0,4],[60,157,0.3822,0.70981,0.20041,0.67857,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0,0,7,0,4],[64,157,0.4076,0.69642,0.22517,0.67857,0.71429,0.85711,0.0,1.0,1,4,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,14,0,0,6,0,4],[68,157,0.4331,0.6607,0.23892,0.5354,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,4,0,0,4,0,0,11,0,0,4,0,5],[72,157,0.4586,0.6339,0.25238,0.5354,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,3,0,0,2,0,0,15,0,0,4,0,3],[76,157,0.4841,0.62947,0.28315,0.4286,0.71429,0.75,0.0,1.0,1,5,0,1,0,4,0,0,1,0,0,4,0,0,0,0,0,14,0,0,3,0,5],[80,157,0.5096,0.58482,0.29743,0.42857,0.71429,0.75,0.0,1.0,3,2,0,3,0,4,0,0,0,0,0,2,0,0,4,0,0,11,0,0,6,0,2],[84,157,0.535,0.61159,0.25059,0.39286,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,2,0,0,2,0,0,11,0,0,8,0,1],[88,157,0.5605,0.64283,0.21429,0.571,0.71429,0.74996,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,14,0,0,8,0,0],[92,157,0.586,0.66072,0.20438,0.67857,0.71429,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,4,0,0,2,0,0,19,0,0,3,0,2],[96,157,0.6115,0.60714,0.28347,0.39286,0.71429,0.75,0.0,1.0,1,4,0,1,0,4,0,0,3,0,0,1,0,0,4,0,0,11,0,0,4,0,4],[100,157,0.6369,0.70979,0.14056,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,22,0,0,3,0,2],[104,157,0.6624,0.62053,0.22759,0.42857,0.71429,0.74999,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,6,0,0,3,0,0,11,0,0,7,0,1],[108,157,0.6879,0.62498,0.23351,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,0,0,0,5,0,0,15,0,0,5,0,1],[112,157,0.7134,0.55357,0.27837,0.2857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,2,0,0,3,0,0,2,0,0,12,0,0,5,0,1],[116,157,0.7389,0.69642,0.24157,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,14,0,0,4,0,6],[120,157,0.7643,0.5625,0.24727,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,3,0,0,6,0,0,1,0,0,14,0,0,3,0,1],[124,157,0.7898,0.66964,0.25614,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,2,0,0,2,0,0,2,0,0,11,0,0,9,0,3],[128,157,0.8153,0.6116,0.24284,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,1,0,0,5,0,0,3,0,0,12,0,0,5,0,2],[132,157,0.8408,0.53122,0.25313,0.28571,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,5,0,0,4,0,0,4,0,0,9,0,0,4,0,1],[136,157,0.8662,0.54911,0.24772,0.28571,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,6,0,0,4,0,0,0,0,0,4,0,0,15,0,0,3,0,0],[140,157,0.8917,0.55804,0.23244,0.42857,0.64286,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,2,0,0,6,0,0,4,0,0,12,0,0,4,0,0],[144,157,0.9172,0.57579,0.2779,0.28571,0.64286,0.71429,0.14,1.0,0,4,0,0,0,5,0,0,4,0,0,3,0,0,4,0,0,9,0,0,3,0,4],[148,157,0.9427,0.55802,0.22689,0.39286,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,3,0,0,4,0,0,14,0,0,3,0,0],[152,157,0.9682,0.54909,0.23175,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,1,0,0,3,0,0,6,0,0,15,0,0,0,0,1],[156,157,0.9936,0.54017,0.25688,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,4,0,0,5,0,0,4,0,0,9,0,0,3,0,2],[157,157,1.0,0.48661,0.24707,0.24999,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,8,0,0,4,0,0,2,0,0,4,0,0,13,0,0,1,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.35705,"x":0.75893,"p":[[0,114,0.0,0.35705,0.25761,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,11,0,0,7,0,0,2,0,0,3,0,0,6,0,0,0,0,1],[4,114,0.0351,0.67855,0.24485,0.53539,0.71429,0.85714,0.0,1.0,1,5,1,1,0,1,0,0,1,0,0,5,0,0,1,0,0,13,0,0,5,0,5],[8,114,0.0702,0.70535,0.27418,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,11,0,0,7,0,7],[12,114,0.1053,0.71429,0.20516,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,19,0,0,5,0,4],[16,114,0.1404,0.71875,0.17672,0.71429,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,21,0,0,3,0,4],[20,114,0.1754,0.66518,0.30222,0.71429,0.71429,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,2,0,0,0,0,0,0,0,0,15,0,0,3,0,7],[24,114,0.2105,0.62499,0.20439,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,1,0,0,3,0,0,20,0,0,3,0,0],[28,114,0.2456,0.64286,0.22588,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,2,0,0,3,0,0,18,0,0,3,0,2],[32,114,0.2807,0.72768,0.19019,0.71429,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,18,0,0,3,0,6],[36,114,0.3158,0.6964,0.21054,0.71429,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,18,0,0,2,0,5],[40,114,0.3509,0.75893,0.14032,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,19,0,0,5,0,5],[44,114,0.386,0.75446,0.20897,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,16,0,0,4,0,8],[48,114,0.4211,0.66518,0.23585,0.71429,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,21,0,0,1,0,4],[52,114,0.4561,0.65179,0.18877,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,1,0,0,0,0,0,23,0,0,2,0,1],[56,114,0.4912,0.65624,0.19516,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,17,0,0,3,0,2],[60,114,0.5263,0.69196,0.21461,0.71429,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,21,0,0,4,0,3],[64,114,0.5614,0.75446,0.11425,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,24,0,0,3,0,4],[68,114,0.5965,0.61606,0.21559,0.67857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,2,0,0,0,0,0,2,0,0,23,0,0,0,0,1],[72,114,0.6316,0.71875,0.20972,0.71429,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,20,0,0,5,0,4],[76,114,0.6667,0.6875,0.17655,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,19,0,0,6,0,1],[80,114,0.7018,0.625,0.21943,0.5357,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,0,0,0,4,0,0,1,0,0,19,0,0,3,0,1],[84,114,0.7368,0.69196,0.13415,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,26,0,0,3,0,0],[88,114,0.7719,0.68741,0.13606,0.71429,0.71429,0.71429,0.14,0.85714,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,25,0,0,3,0,0],[92,114,0.807,0.65625,0.19839,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,25,0,0,3,0,0],[96,114,0.8421,0.65179,0.18536,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,25,0,0,2,0,0],[100,114,0.8772,0.6875,0.1448,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,28,0,0,2,0,0],[104,114,0.9123,0.71874,0.09097,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0,3,0,1],[108,114,0.9474,0.70536,0.10677,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,29,0,0,2,0,0],[112,114,0.9825,0.70535,0.12845,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,24,0,0,5,0,0],[114,114,1.0,0.59374,0.22619,0.57132,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,5,0,0,2,0,0,0,0,0,3,0,0,20,0,0,2,0,0]]}]},{"i":"0e071567bcd2165e","q":"Let $a, b$, and $c$ be the side lengths of a triangle and let $R$ be its circumradius. Show that\n\n$$\n\\frac{1}{a b}+\\frac{1}{b c}+\\frac{1}{c a} \\geq \\frac{1}{R^{2}}\n$$","t":[{"b":0,"e":0.42857,"k":"falling","v":0.42857,"x":0.70982,"p":[[0,26,0.0,0.70982,0.33784,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,18],[4,26,0.1538,0.56697,0.25123,0.42857,0.42857,0.57145,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,8],[8,26,0.3077,0.45982,0.18466,0.42857,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,3],[12,26,0.4615,0.48661,0.1984,0.42857,0.42857,0.42857,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,4],[16,26,0.6154,0.4375,0.10677,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,1],[20,26,0.7692,0.44197,0.10326,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[24,26,0.9231,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"falling","v":0.42857,"x":0.73661,"p":[[0,33,0.0,0.73661,0.32165,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,4,0,0,6,0,0,0,0,0,0,0,0,3,0,17],[4,33,0.1212,0.67411,0.33737,0.42857,0.78571,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,0,0,0,10,0,0,0,0,0,1,0,0,1,0,15],[8,33,0.2424,0.60719,0.27196,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,18,0,0,0,0,0,2,0,0,1,0,9],[12,33,0.3636,0.45536,0.10972,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,27,0,0,3,0,0,0,0,0,0,0,1],[16,33,0.4848,0.49107,0.18189,0.42857,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0,1,0,3],[20,33,0.6061,0.4375,0.10677,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,1],[24,33,0.7273,0.42857,0.07986,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,28,0,0,1,0,0,1,0,0,0,0,0],[28,33,0.8485,0.44196,0.10326,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[32,33,0.9697,0.45982,0.13709,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,27,0,0,0,0,0,1,0,0,1,0,1],[33,33,1.0,0.44643,0.09942,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"5599621a5c00e3a1","q":"Let $n \\geqslant 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.97768,"p":[[0,34,0.0,0.88839,0.17399,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,1,0,21],[4,34,0.1176,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,34,0.2353,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,23],[12,34,0.3529,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[16,34,0.4706,0.91964,0.13803,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,23],[20,34,0.5882,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,34,0.7059,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[28,34,0.8235,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[32,34,0.9412,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[34,34,1.0,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22]]},{"b":7,"e":1.0,"k":"flat","v":0.82141,"x":0.99107,"p":[[0,35,0.0,0.88393,0.16536,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,2,0,20],[4,35,0.1143,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,35,0.2286,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,35,0.3429,0.95076,0.1166,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[16,35,0.4571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,35,0.5714,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,35,0.6857,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,35,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,35,0.9143,0.88839,0.21646,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,24],[35,35,1.0,0.82141,0.25002,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,3,0,0,4,0,18]]}]},{"i":"b465604ccdd22dab","q":"Let $a, b$ and $c$ be positive real numbers with $a b c=1$. Prove that\n\n$$\na+b+c \\geq \\sqrt{\\frac{(a+2)(b+2)(c+2)}{3}}\n$$","t":[{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.0625,"p":[[0,27,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,58,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b62d840ef14e630e","q":"Let $n \\geq 2$ and let $x_{1}, x_{2}, \\ldots x_{n}$ be real numbers satisfying $x_{1}+x_{2}+\\ldots+x_{n} \\geq 0$ and $x_{1}^{2}+x_{2}^{2}+\\ldots+x_{n}^{2}=1$. Let $M=\\max \\left\\{x_{1}, x_{2}, \\ldots, x_{n}\\right\\}$. Show that\n\n$$\nM \\geq \\frac{1}{\\sqrt{n(n-1)}}\n$$\n\nWhen does equality hold in (1)?","t":[{"b":2,"e":0.71429,"k":"flat","v":0.71429,"x":0.83036,"p":[[0,61,0.0,0.75892,0.22142,0.71429,0.78564,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,8,0,8],[4,61,0.0656,0.79911,0.24185,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,5,0,14],[8,61,0.1311,0.79462,0.21707,0.71421,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,7,0,0,5,0,13],[12,61,0.1967,0.77665,0.2081,0.67536,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,0,3,0,12],[16,61,0.2623,0.82143,0.17128,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,6,0,12],[20,61,0.3279,0.8125,0.21852,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,7,0,0,4,0,15],[24,61,0.3934,0.8125,0.19704,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,6,0,13],[28,61,0.459,0.83036,0.16917,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,8,0,12],[32,61,0.5246,0.75893,0.27067,0.67857,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,7,0,11],[36,61,0.5902,0.71429,0.31339,0.71429,0.85714,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,2,0,0,1,0,0,0,0,0,5,0,0,13,0,7],[40,61,0.6557,0.80804,0.18073,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,11,0,0,6,0,11],[44,61,0.7213,0.79464,0.19541,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,8,0,11],[48,61,0.7869,0.79464,0.18189,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,14,0,7],[52,61,0.8525,0.77231,0.16699,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,7,0,7],[56,61,0.918,0.76337,0.1541,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,0,9,0,5],[60,61,0.9836,0.74999,0.15567,0.71429,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,13,0,2],[61,61,1.0,0.74551,0.16265,0.67857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,10,0,0,10,0,4]]},{"b":5,"e":0.85714,"k":"flat","v":0.66071,"x":0.88392,"p":[[0,116,0.0,0.76785,0.20124,0.67857,0.78564,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,7,0,9],[4,116,0.0345,0.85714,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,9,0,14],[8,116,0.069,0.88392,0.18708,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,3,0,21],[12,116,0.1034,0.85714,0.19885,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,5,0,18],[16,116,0.1379,0.80803,0.22759,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,14],[20,116,0.1724,0.79004,0.18211,0.71321,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,4,0,11],[24,116,0.2069,0.82589,0.23072,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,4,0,16],[28,116,0.2414,0.81249,0.19047,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,7,0,12],[32,116,0.2759,0.83022,0.19712,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,4,0,15],[36,116,0.3103,0.82587,0.16652,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,7,0,12],[40,116,0.3448,0.8125,0.22428,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,3,0,16],[44,116,0.3793,0.82143,0.19562,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,5,0,14],[48,116,0.4138,0.85268,0.16935,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,9,0,14],[52,116,0.4483,0.80357,0.15046,0.71429,0.71429,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,3,0,10],[56,116,0.4828,0.78572,0.17496,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,6,0,9],[60,116,0.5172,0.79463,0.23404,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,4,0,13],[64,116,0.5517,0.82588,0.19476,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,7,0,14],[68,116,0.5862,0.72321,0.23128,0.53571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,8,0,0,6,0,8],[72,116,0.6207,0.76339,0.21609,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,12,0,0,7,0,8],[76,116,0.6552,0.79911,0.20159,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,5,0,0,11,0,10],[80,116,0.6897,0.83929,0.19805,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,3,0,17],[84,116,0.7241,0.80357,0.17767,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,8,0,10],[88,116,0.7586,0.83036,0.15335,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,8,0,11],[92,116,0.7931,0.80804,0.20395,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,8,0,12],[96,116,0.8276,0.76339,0.20079,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,6,0,9],[100,116,0.8621,0.76339,0.22759,0.67857,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,8,0,9],[104,116,0.8966,0.71429,0.26245,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,4,0,0,3,0,0,7,0,0,7,0,8],[108,116,0.931,0.7054,0.19535,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,11,0,0,9,0,3],[112,116,0.9655,0.7366,0.17896,0.67857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,11,0,0,8,0,5],[116,116,1.0,0.66071,0.21354,0.42857,0.71429,0.74996,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,9,0,0,5,0,0,9,0,0,2,0,6]]}]},{"i":"04f2bf464e391f3e","q":"One hundred and one of the squares of an  $n\u0002\\times n$  table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible  $n$ .","t":[{"b":2,"e":0.14286,"k":"rising","v":0.00893,"x":0.33482,"p":[[0,36,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,10,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.12946,0.20316,0.0,0.0,0.17857,0.0,0.85714,19,0,0,19,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,36,0.2222,0.1384,0.27545,0.0,0.0,0.03575,0.0,1.0,24,1,0,24,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,1],[12,36,0.3333,0.07589,0.15966,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,36,0.4444,0.27232,0.26332,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,5,0,0,3,0,0,2,0,0,2,0,0,1,0,1],[20,36,0.5556,0.3214,0.25751,0.14286,0.21428,0.57111,0.0,1.0,5,1,0,5,0,11,0,0,2,0,0,4,0,0,7,0,0,2,0,0,0,0,1],[24,36,0.6667,0.30804,0.17896,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,9,0,0,10,0,0,1,0,0,2,0,0,0,0,0],[28,36,0.7778,0.33482,0.19434,0.24999,0.35714,0.42857,0.0,1.0,3,1,0,3,0,5,0,0,8,0,0,13,0,0,2,0,0,0,0,0,0,0,1],[32,36,0.8889,0.29018,0.20973,0.14286,0.35714,0.42858,0.0,0.57143,7,0,0,7,0,7,0,0,2,0,0,10,0,0,6,0,0,0,0,0,0,0,0],[36,36,1.0,0.17857,0.15972,0.0,0.14286,0.2857,0.0,0.57143,9,0,0,9,0,13,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.16071,"p":[[0,42,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.05795,0.14659,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,42,0.1905,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,42,0.2857,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.09375,0.22477,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[20,42,0.4762,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,42,0.5714,0.12946,0.2,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,3,0,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[28,42,0.6667,0.10714,0.20203,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[32,42,0.7619,0.13393,0.2141,0.0,0.0,0.32143,0.0,0.71429,22,0,0,22,0,1,0,0,1,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[36,42,0.8571,0.16071,0.20748,0.0,0.07143,0.2857,0.0,0.85714,16,0,0,16,0,5,0,0,6,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[40,42,0.9524,0.08036,0.13803,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.07589,0.18205,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"32a7890bcb03fd34","q":"Let $a, b$ and $c$ be rational numbers for which $a+b c, b+a c$ and $a+b$ are all not equal to 0 and for which it holds that\n\n$$\n\\frac{1}{a+b c}+\\frac{1}{b+a c}=\\frac{1}{a+b}\n$$\n\nProve that $\\sqrt{(c-3)(c+1)}$ is rational.","t":[{"b":0,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,47,0.0,0.86161,0.2461,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,5,0,21],[4,47,0.0851,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,47,0.1702,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,47,0.2553,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,47,0.3404,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,47,0.4255,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,47,0.5106,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[28,47,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,47,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[47,47,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.28571,"k":"falling","v":0.32143,"x":0.84375,"p":[[0,33,0.0,0.75445,0.2702,0.53539,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,6,0,0,1,0,15],[4,33,0.1212,0.73661,0.33902,0.42859,1.0,1.0,0.0,1.0,2,17,2,2,0,2,0,0,2,0,0,3,0,0,0,0,0,5,0,0,1,0,17],[8,33,0.2424,0.84375,0.27747,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0,2,0,23],[12,33,0.3636,0.69195,0.30328,0.42857,0.64286,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,6,0,0,7,0,0,3,0,0,1,0,0,0,0,15],[16,33,0.4848,0.38839,0.14827,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,1],[20,33,0.6061,0.33481,0.09849,0.28571,0.28571,0.42857,0.14286,0.571,0,0,0,0,0,3,0,0,16,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.36159,0.10702,0.28571,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,14,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[28,33,0.8485,0.34375,0.07873,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,17,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.32143,0.08748,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,21,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[33,33,1.0,0.34821,0.09407,0.28571,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,3,0,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6535466af6b33723","q":"Let $a, b$ and $c$ be real numbers such that $0 \\leqslant a, b, c \\leqslant 2$. Show that\n\n$$\n(a-b)(b-c)(a-c) \\leqslant 2\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.94196,"x":0.99554,"p":[[0,25,0.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,25,0.16,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[8,25,0.32,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[12,25,0.48,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[16,25,0.64,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,25,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,25,0.96,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[25,25,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":4,"e":1.0,"k":"flat","v":0.93304,"x":0.97768,"p":[[0,5,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,5,0.8,0.95089,0.13175,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,25],[5,5,1.0,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19]]}]},{"i":"c13c98812ee98f6f","q":"Let $n$ be a positive integer and let $x_{1}, \\ldots, x_{n}$ be real numbers. Show that\n\n$$\n\\sum_{i=1}^{n} x_{i}^{2} \\geqslant \\frac{1}{n+1}\\left(\\sum_{i=1}^{n} x_{i}\\right)^{2}+\\frac{12\\left(\\sum_{i=1}^{n} i x_{i}\\right)^{2}}{n(n+1)(n+2)(3 n+1)}\n$$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.09375,"x":0.26784,"p":[[0,122,0.0,0.20982,0.22864,0.0,0.21428,0.32143,0.0,0.71429,15,0,0,15,0,1,0,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[4,122,0.0328,0.14286,0.18558,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,0,0,0,9,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,122,0.0656,0.20536,0.23673,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,2,0,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[12,122,0.0984,0.14286,0.26964,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,1],[16,122,0.1311,0.13393,0.2141,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,1,0,0,6,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[20,122,0.1639,0.16518,0.23987,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,2,0,0,5,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[24,122,0.1967,0.26784,0.3149,0.0,0.14286,0.46418,0.0,1.0,15,1,0,15,0,2,0,0,5,0,0,2,0,0,2,0,0,3,0,0,2,0,1],[28,122,0.2295,0.16964,0.2299,0.0,0.0,0.32143,0.0,0.71429,19,0,0,19,0,1,0,0,4,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[32,122,0.2623,0.16518,0.28146,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,2,0,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,1],[36,122,0.2951,0.14732,0.21866,0.0,0.0,0.2857,0.0,1.0,17,1,0,17,0,6,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[40,122,0.3279,0.13393,0.22286,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,1,0,0,3,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[44,122,0.3607,0.23214,0.2714,0.0,0.14286,0.32143,0.0,0.85714,15,0,0,15,0,2,0,0,7,0,0,2,0,0,1,0,0,4,0,0,1,0,0],[48,122,0.3934,0.16516,0.23173,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,0,0,0,8,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[52,122,0.4262,0.09821,0.16536,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[56,122,0.459,0.20982,0.22299,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,2,0,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[60,122,0.4918,0.19196,0.20079,0.0,0.14286,0.28571,0.0,0.71429,13,0,0,13,0,4,0,0,10,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[64,122,0.5246,0.16071,0.17768,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,6,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[68,122,0.5574,0.13393,0.18189,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,0,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[72,122,0.5902,0.12947,0.22406,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[76,122,0.623,0.2232,0.24466,0.0,0.21428,0.28571,0.0,1.0,13,1,0,13,0,3,0,0,9,0,0,3,0,0,2,0,0,1,0,0,0,0,1],[80,122,0.6557,0.17411,0.20743,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,2,0,0,9,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[84,122,0.6885,0.14286,0.18558,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,2,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[88,122,0.7213,0.15625,0.21829,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,3,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[92,122,0.7541,0.09375,0.15407,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[96,122,0.7869,0.19643,0.22798,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,1,0,0,8,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[100,122,0.8197,0.17857,0.21129,0.0,0.07143,0.28571,0.0,0.85714,16,0,0,16,0,1,0,0,10,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[104,122,0.8525,0.16964,0.18363,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,2,0,0,12,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[108,122,0.8852,0.15178,0.19212,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,3,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[112,122,0.918,0.11607,0.14914,0.0,0.0,0.28571,0.0,0.42857,19,0,0,19,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[116,122,0.9508,0.16518,0.22335,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,2,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[120,122,0.9836,0.17411,0.17029,0.0,0.21429,0.28571,0.0,0.71429,13,0,0,13,0,3,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[122,122,1.0,0.16964,0.16536,0.0,0.21431,0.28571,0.0,0.57143,14,0,0,14,0,2,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.02679,"x":0.29464,"p":[[0,111,0.0,0.20982,0.23686,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,6,0,0,9,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[4,111,0.036,0.29464,0.32328,0.0,0.21428,0.57143,0.0,1.0,14,2,0,14,0,2,0,0,4,0,0,2,0,0,5,0,0,2,0,0,1,0,2],[8,111,0.0721,0.08036,0.17835,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,111,0.1081,0.12054,0.25532,0.0,0.0,0.14287,0.0,1.0,23,1,0,23,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[16,111,0.1441,0.09822,0.16917,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[20,111,0.1802,0.17409,0.25185,0.0,0.0,0.32143,0.0,0.71429,20,0,0,20,0,1,0,0,3,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[24,111,0.2162,0.13393,0.18877,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,1,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[28,111,0.2523,0.08482,0.20159,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,111,0.2883,0.12054,0.22899,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[36,111,0.3243,0.1875,0.29974,0.0,0.0,0.2857,0.0,1.0,19,2,0,19,0,3,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,2],[40,111,0.3604,0.05357,0.16269,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[44,111,0.3964,0.08036,0.17105,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[48,111,0.4324,0.12944,0.24312,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,0,0,0,0],[52,111,0.4685,0.07143,0.15152,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,111,0.5045,0.09374,0.18419,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,1,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[60,111,0.5405,0.09374,0.1943,0.0,0.0,0.03572,0.0,0.85714,24,0,0,24,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[64,111,0.5766,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,111,0.6126,0.10714,0.24484,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[72,111,0.6486,0.16071,0.24419,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,3,0,0,5,0,0,1,0,0,1,0,0,2,0,0,1,0,0],[76,111,0.6847,0.14286,0.25254,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[80,111,0.7207,0.11607,0.22142,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,2,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[84,111,0.7568,0.06697,0.13356,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.10714,0.22016,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[92,111,0.8288,0.0625,0.15542,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[96,111,0.8649,0.08036,0.18536,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[100,111,0.9009,0.08929,0.15872,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[104,111,0.9369,0.05804,0.1551,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[108,111,0.973,0.05803,0.18509,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[111,111,1.0,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b76c388caac8b19","q":"Let $a, b$, and $c$ be positive real numbers. Prove:\n\n$$\n\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{a} \\leq \\frac{a^{2}}{b^{2}}+\\frac{b^{2}}{c^{2}}+\\frac{c^{2}}{a^{2}}\n$$","t":[{"b":1,"e":0.57143,"k":"falling","v":0.59373,"x":0.77232,"p":[[0,16,0.0,0.77232,0.33855,0.57143,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,3,0,19],[4,16,0.25,0.77231,0.31514,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,3,0,0,0,0,0,7,0,0,0,0,0,1,0,19],[8,16,0.5,0.65624,0.2809,0.49968,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,1,0,11],[12,16,0.75,0.66962,0.22143,0.57143,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,0,0,0,19,0,0,0,0,0,2,0,8],[16,16,1.0,0.59373,0.19596,0.57143,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,5,0,0,1,0,0,19,0,0,1,0,0,3,0,3]]},{"b":7,"e":0.57143,"k":"flat","v":0.61161,"x":0.82589,"p":[[0,41,0.0,0.79911,0.31514,0.57143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,3,0,0,0,0,0,5,0,0,0,0,0,1,0,21],[4,41,0.0976,0.82589,0.27137,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,0,0,0,7,0,0,0,0,0,1,0,21],[8,41,0.1951,0.75,0.32143,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,4,0,0,0,0,0,6,0,0,2,0,0,0,0,18],[12,41,0.2927,0.66964,0.30606,0.28571,0.57143,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,10,0,0,0,0,0,7,0,0,1,0,0,1,0,13],[16,41,0.3902,0.61161,0.27947,0.28571,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,10,0,0,0,0,0,12,0,0,0,0,0,1,0,9],[20,41,0.4878,0.66964,0.27302,0.57143,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,12,0,0,1,0,0,1,0,11],[24,41,0.5854,0.71426,0.28573,0.571,0.57143,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,0,0,0,0,0,15],[28,41,0.6829,0.68747,0.27535,0.571,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,10,0,0,1,0,0,3,0,11],[32,41,0.7805,0.79462,0.26474,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,0,0,0,8,0,0,1,0,0,0,0,19],[36,41,0.878,0.71872,0.25125,0.57143,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,13,0,0,1,0,0,2,0,12],[40,41,0.9756,0.69194,0.26991,0.57132,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,6,0,0,0,0,0,12,0,0,1,0,0,1,0,12],[41,41,1.0,0.67854,0.29882,0.28571,0.57143,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,9,0,0,0,0,0,8,0,0,1,0,0,1,0,13]]}]},{"i":"6c3486293ef0efe5","q":"Pablo was trying to solve the following problem: find the sequence $\\mathrm{x}_{0}, \\mathrm{x}_{1}, \\mathrm{x}_{2}, \\ldots, \\mathrm{x}_{2003}$ which satisfies $\\mathrm{x}_{0}=1,0 \\leq \\mathrm{x}_{\\mathrm{i}} \\leq 2 \\mathrm{x}_{\\mathrm{i}-1}$ for $1 \\leq \\mathrm{i} \\leq 2003$ and which maximises $\\mathrm{S}$. Unfortunately he could not remember the expression for $\\mathrm{S}$, but he knew that it had the form $\\mathrm{S}= \\pm \\mathrm{x}_{1} \\pm \\mathrm{x}_{2} \\pm \\ldots \\pm \\mathrm{x}_{2002}$ $+\\mathrm{x}_{2003}$. Show that he can still solve the problem.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.57143,"x":0.97321,"p":[[0,13,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,13,0.3077,0.84821,0.28333,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,23],[8,13,0.6154,0.76786,0.30462,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,8,0,0,0,0,0,1,0,0,1,0,19],[12,13,0.9231,0.57143,0.29014,0.42857,0.57143,0.75,0.14286,1.0,0,5,0,0,0,6,0,0,1,0,0,9,0,0,0,0,0,8,0,0,3,0,5],[13,13,1.0,0.57143,0.23419,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,10,0,0,3,0,0,10,0,0,0,0,4]]},{"b":5,"e":1.0,"k":"flat","v":0.71875,"x":0.97321,"p":[[0,80,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,80,0.05,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[8,80,0.1,0.93304,0.13825,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[12,80,0.15,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[16,80,0.2,0.86161,0.22156,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,19],[20,80,0.25,0.78125,0.29447,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,6,0,15],[24,80,0.3,0.74553,0.34206,0.57143,0.92857,1.0,0.0,1.0,3,16,0,3,0,2,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,16],[28,80,0.35,0.87052,0.18338,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,3,0,19],[32,80,0.4,0.80579,0.26138,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,1,3,0,0,6,0,0,5,0,15],[36,80,0.45,0.86607,0.22286,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,18],[40,80,0.5,0.90179,0.13092,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,19],[44,80,0.55,0.875,0.19805,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,5,0,20],[48,80,0.6,0.76786,0.23891,0.67857,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,9,0,0,3,0,12],[52,80,0.65,0.71875,0.24086,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,5,0,0,5,0,9],[56,80,0.7,0.88392,0.20026,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,1,0,23],[60,80,0.75,0.88393,0.21261,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,20],[64,80,0.8,0.82589,0.20434,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,0,6,0,14],[68,80,0.85,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[72,80,0.9,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[76,80,0.95,0.94642,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[80,80,1.0,0.91516,0.1781,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,24]]}]},{"i":"69236bbdb85bc5cf","q":"Positive integers  $a$ ,  $b$  and  $c$  are positive integers with greatest common divisor equal to  $1$  (i.e. they have no common divisors greater than  $1$ ), and  $$ \\frac{ab}{a-b}=c $$  Prove that  $a -b$  is a perfect square.\n\n(SL Berlov)","t":[{"b":2,"e":0.42857,"k":"falling","v":0.50444,"x":0.8125,"p":[[0,13,0.0,0.8125,0.35613,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[4,13,0.3077,0.62054,0.28928,0.42857,0.64286,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,0,0,0,12,0,0,1,0,0,4,0,0,6,0,6],[8,13,0.6154,0.52679,0.22711,0.42857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,19,0,0,0,0,0,4,0,0,4,0,2],[12,13,0.9231,0.51786,0.25692,0.42857,0.42857,0.46431,0.0,1.0,1,5,0,1,0,2,0,0,0,0,0,21,0,0,1,0,0,0,0,0,2,0,5],[13,13,1.0,0.50444,0.24218,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,0,0,0,17,0,0,5,0,0,0,0,0,3,0,3]]},{"b":4,"e":1.0,"k":"flat","v":0.625,"x":0.84821,"p":[[0,25,0.0,0.74554,0.38916,0.42857,1.0,1.0,0.0,1.0,3,22,0,3,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,22],[4,25,0.16,0.625,0.34947,0.42857,0.71429,0.85714,0.0,1.0,6,7,0,6,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,8,0,7],[8,25,0.32,0.62946,0.37091,0.35714,0.85714,1.0,0.0,1.0,4,10,0,4,0,4,0,0,0,0,0,3,0,0,4,0,0,0,0,0,7,0,10],[12,25,0.48,0.65625,0.3533,0.42857,0.85714,1.0,0.0,1.0,2,11,0,2,0,5,0,0,0,0,0,5,0,0,1,0,0,2,0,0,6,0,11],[16,25,0.64,0.74106,0.3223,0.42857,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,7,0,14],[20,25,0.8,0.84821,0.21997,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,12,0,15],[24,25,0.96,0.75893,0.23807,0.4286,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,0,0,0,9,0,11],[25,25,1.0,0.76784,0.24937,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,5,0,0,8,0,11]]}]},{"i":"4f6fd89fe692d4be","q":"Let $a_{0,1}, a_{0,2}, \\ldots, a_{0,2016}$ be positive real numbers. For $n \\geq 0$ and $1 \\leq k<2016$ set\n\n$$\na_{n+1, k}=a_{n, k}+\\frac{1}{2 a_{n, k+1}} \\quad \\text { and } \\quad a_{n+1,2016}=a_{n, 2016}+\\frac{1}{2 a_{n, 1}} .\n$$\n\nShow that $\\max _{1 \\leq k \\leq 2016} a_{2016, k}>44$.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.12937,"x":0.26339,"p":[[0,13,0.0,0.18304,0.19638,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,13,0.3077,0.16509,0.152,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,13,0.6154,0.26339,0.28146,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,2],[12,13,0.9231,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.1429,"k":"flat","v":0.1383,"x":0.19643,"p":[[0,41,0.0,0.19187,0.19105,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,41,0.0976,0.19643,0.20748,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,41,0.1951,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.17411,0.16263,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,41,0.3902,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5218d66c9531dadd","q":"Points  $A,B$  are on circle  $\\omega$ . Points  $C$  and  $D$  are moved on the arc  $AB$ , such that  $CD$  has constant length.  $I_1,I_2$  - incenters of  $ABC$  and  $ABD$ . \nProve that line  $I_1I_2$  is tangent to some fixed circle.","t":[{"b":0,"e":0.0,"k":"falling","v":0.04018,"x":0.47768,"p":[[0,65,0.0,0.47768,0.4265,0.0,0.57143,1.0,0.0,1.0,11,9,0,11,0,3,0,0,1,0,0,0,0,0,3,0,0,3,0,0,2,0,9],[4,65,0.0615,0.40847,0.39738,0.0,0.28571,0.85714,0.0,1.0,11,5,0,11,0,4,0,0,3,0,0,0,0,0,3,0,0,2,0,0,3,1,5],[8,65,0.1231,0.23205,0.34212,0.0,0.0,0.2857,0.0,1.0,17,2,0,17,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,4,0,2],[12,65,0.1846,0.25892,0.36146,0.0,0.07143,0.35714,0.0,1.0,16,3,0,16,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,3],[16,65,0.2462,0.15179,0.25238,0.0,0.0,0.2857,0.0,1.0,20,1,0,20,0,3,0,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,1],[20,65,0.3077,0.18295,0.27949,0.0,0.0,0.21432,0.0,1.0,18,1,0,18,0,6,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,1],[24,65,0.3692,0.18304,0.31589,0.0,0.0,0.17857,0.0,1.0,19,3,0,19,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[28,65,0.4308,0.165,0.24773,0.0,0.0,0.17857,0.0,0.85714,17,0,0,17,0,7,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[32,65,0.4923,0.18303,0.32386,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[36,65,0.5538,0.16506,0.25278,0.0,0.0,0.1786,0.0,0.85714,18,0,0,18,0,6,0,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,0],[40,65,0.6154,0.09812,0.16915,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,65,0.6769,0.1875,0.27067,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,4,0,0,6,0,0,0,0,0,2,0,0,1,0,0,1,0,1],[48,65,0.7385,0.12043,0.19265,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,5,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[52,65,0.8,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.07143,0.15152,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,65,0.9231,0.12946,0.24053,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[64,65,0.9846,0.04464,0.11538,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.00893,"x":0.48205,"p":[[0,47,0.0,0.39277,0.43011,0.0,0.21428,0.89286,0.0,1.0,14,8,0,14,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,3,0,8],[4,47,0.0851,0.48205,0.38926,0.105,0.42857,0.85714,0.0,1.0,8,6,0,8,0,4,0,0,2,0,0,3,0,0,0,0,0,5,0,0,4,0,6],[8,47,0.1702,0.40625,0.38649,0.0,0.35714,0.85714,0.0,1.0,11,4,0,11,0,3,0,0,2,0,0,5,0,0,0,0,0,1,0,0,6,0,4],[12,47,0.2553,0.36607,0.38621,0.0,0.14286,0.75,0.0,1.0,11,5,0,11,0,6,0,0,2,0,0,3,0,0,0,0,0,2,0,0,3,0,5],[16,47,0.3404,0.14714,0.21573,0.0,0.0,0.17857,0.0,0.85714,18,0,0,18,0,6,0,0,1,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[20,47,0.4255,0.29464,0.35344,0.0,0.14286,0.5,0.0,1.0,15,2,0,15,0,2,0,0,5,0,0,2,0,0,0,0,0,2,0,0,4,0,2],[24,47,0.5106,0.20527,0.30293,0.0,0.07,0.2857,0.0,1.0,16,1,0,16,0,7,0,0,3,0,0,1,0,0,0,0,0,1,0,0,3,0,1],[28,47,0.5957,0.25446,0.33068,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,1,0,0,3,0,0,6,0,0,0,0,0,0,0,0,3,0,2],[32,47,0.6809,0.13384,0.2394,0.0,0.0,0.1786,0.0,0.85714,22,0,0,22,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,2,0,0],[36,47,0.766,0.10268,0.19638,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[40,47,0.8511,0.12946,0.20316,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[44,47,0.9362,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a62c78ff117b805","q":"Point  $I$  is incenter of triangle  $ABC$  in which  $AB \\neq AC$ . Lines  $BI$  and  $CI$  intersect sides  $AC$  and  $AB$  in points  $D$  and  $E$ , respectively. Determine all measures of angle  $BAC$ , for which may be  $DI = EI$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.81481,"x":1.0,"p":[[0,179,0.0,0.81481,0.24655,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,1,1,0,0,2,0,0,4,0,0,7,0,15],[4,179,0.0223,0.95533,0.11548,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[8,179,0.0447,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[12,179,0.067,0.93303,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[16,179,0.0894,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[20,179,0.1117,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,179,0.1341,0.97321,0.05578,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,179,0.1564,0.96429,0.11845,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[32,179,0.1788,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,179,0.2011,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,179,0.2235,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,179,0.2458,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,179,0.2682,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[52,179,0.2905,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[56,179,0.3128,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[60,179,0.3352,0.93527,0.22114,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[64,179,0.3575,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,179,0.3799,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,179,0.4022,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,179,0.4246,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[80,179,0.4469,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,179,0.4693,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[88,179,0.4916,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[92,179,0.514,0.94642,0.10565,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[96,179,0.5363,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[100,179,0.5587,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[104,179,0.581,0.96426,0.09456,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[108,179,0.6034,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[112,179,0.6257,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[116,179,0.648,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[120,179,0.6704,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[124,179,0.6927,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[128,179,0.7151,0.97098,0.10391,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,1,28],[132,179,0.7374,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[136,179,0.7598,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[140,179,0.7821,0.9375,0.18189,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[144,179,0.8045,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[148,179,0.8268,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[152,179,0.8492,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[156,179,0.8715,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[160,179,0.8939,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[164,179,0.9162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[168,179,0.9385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[172,179,0.9609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[176,179,0.9832,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[179,179,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"rising","v":0.83928,"x":1.0,"p":[[0,145,0.0,0.83928,0.28959,0.82132,1.0,1.0,0.0,1.0,3,19,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,19],[4,145,0.0276,0.92409,0.12364,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20],[8,145,0.0552,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,145,0.0828,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,145,0.1103,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,145,0.1379,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[24,145,0.1655,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[28,145,0.1931,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[32,145,0.2207,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[36,145,0.2483,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[40,145,0.2759,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[44,145,0.3034,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[48,145,0.331,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[52,145,0.3586,0.95088,0.10484,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[56,145,0.3862,0.95981,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[60,145,0.4138,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[64,145,0.4414,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[68,145,0.469,0.96428,0.10714,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[72,145,0.4966,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[76,145,0.5241,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[80,145,0.5517,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22],[84,145,0.5793,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[88,145,0.6069,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[92,145,0.6345,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[96,145,0.6621,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[100,145,0.6897,0.98213,0.05924,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[104,145,0.7172,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[108,145,0.7448,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[112,145,0.7724,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[116,145,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[120,145,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,145,0.8552,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[128,145,0.8828,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[132,145,0.9103,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[136,145,0.9379,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[140,145,0.9655,0.96874,0.09269,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[144,145,0.9931,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[145,145,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c605eff6706429e7","q":"Prove that  $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$  is divisible by  $5$  for every  $n$ .\nProve that for no  $n$ :  $\\sum_{\\ell=1}^5 (n+\\ell)^2$  is a perfect square.\nLet  $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$ . Prove that  $S_1 \\cdot S_2$  is divisible by  $150$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,18,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[18,18,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.9866,"x":1.0,"p":[[0,35,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[35,35,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"0b1ad81ec4ab6972","q":"Prove or disprove that for all positive real numbers $a, b$ and $c$ the inequality\n\n$$\n3 \\leq \\frac{4 a+b}{a+4 b}+\\frac{4 b+c}{b+4 c}+\\frac{4 c+a}{c+4 a}<\\frac{33}{4}\n$$","t":[{"b":0,"e":0.1429,"k":"flat","v":0.6607,"x":0.94643,"p":[[0,59,0.0,0.76784,0.24937,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,0,5,0,13],[4,59,0.0678,0.91964,0.19865,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[8,59,0.1356,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[12,59,0.2034,0.90179,0.1448,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[16,59,0.2712,0.92411,0.15146,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[20,59,0.339,0.90179,0.16536,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,22],[24,59,0.4068,0.91518,0.14664,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,22],[28,59,0.4746,0.86158,0.19065,0.85711,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,8,0,17],[32,59,0.5424,0.83928,0.24679,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,6,0,18],[36,59,0.6102,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[40,59,0.678,0.81696,0.31991,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0,2,0,22],[44,59,0.7458,0.7991,0.23107,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,7,0,13],[48,59,0.8136,0.77232,0.33093,0.67857,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,3,0,0,0,0,0,2,0,0,2,0,0,5,0,17],[52,59,0.8814,0.77677,0.26712,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,9,0,11],[56,59,0.9492,0.66518,0.3246,0.53572,0.71429,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,0,0,0,3,0,0,3,0,0,7,0,0,6,0,8],[59,59,1.0,0.6607,0.34763,0.39286,0.857,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0,10,0,8]]},{"b":3,"e":0.2857,"k":"flat","v":0.75443,"x":0.95076,"p":[[0,73,0.0,0.81026,0.24975,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,3,0,0,1,0,0,0,1,0,3,0,0,9,0,14],[4,73,0.0548,0.9375,0.12846,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,24],[8,73,0.1096,0.91964,0.15947,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,23],[12,73,0.1644,0.92411,0.14719,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,25],[16,73,0.2192,0.88392,0.23266,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,22],[20,73,0.274,0.86606,0.2447,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[24,73,0.3288,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[28,73,0.3836,0.95076,0.13674,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[32,73,0.4384,0.75446,0.34853,0.57143,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,2,0,18],[36,73,0.4932,0.87052,0.2063,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,4,0,20],[40,73,0.5479,0.79463,0.29,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,0,6,0,15],[44,73,0.6027,0.82143,0.27894,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,20],[48,73,0.6575,0.83929,0.24419,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,18],[52,73,0.7123,0.86606,0.15129,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,8,0,15],[56,73,0.7671,0.83482,0.25028,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,2,0,19],[60,73,0.8219,0.84821,0.26229,0.85711,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,6,0,19],[64,73,0.8767,0.85714,0.23419,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,3,0,20],[68,73,0.9315,0.87946,0.17536,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,19],[72,73,0.9863,0.84375,0.19999,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,5,0,17],[73,73,1.0,0.75443,0.24286,0.57143,0.857,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,1,0,0,6,0,0,6,0,0,8,0,9]]}]},{"i":"cfb80d20694e2366","q":"Positive real sequences  $\\{ a_n \\}$  and  $\\{ b_n \\}$  satisfy the following conditions for all positive integers  $n$ .\n\n\n- $a_{n+1}b_{n+1}= a_n^2 + b_n^2$\n- $a_{n+1}+b_{n+1}=a_nb_n$\n- $a_n \\geq b_n$\n\nProve that there exists positive integer  $n$  such that  $\\frac{a_n}{b_n}>2023^{2023}.$","t":[{"b":0,"e":0.42857,"k":"falling","v":0.18304,"x":0.46427,"p":[[0,146,0.0,0.45534,0.2889,0.24999,0.49979,0.71429,0.0,1.0,4,1,0,4,0,4,0,0,5,0,0,3,0,0,7,0,0,4,0,0,4,0,1],[4,146,0.0274,0.39732,0.17399,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,10,0,0,7,0,0,7,0,0,3,0,0,0,0,0],[8,146,0.0548,0.4107,0.21052,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,8,0,0,7,0,0,7,0,0,2,0,0,2,0,0],[12,146,0.0822,0.41517,0.22405,0.28571,0.42857,0.57143,0.0,0.857,3,0,0,3,0,2,0,0,8,0,0,9,0,0,3,0,0,6,0,0,1,0,0],[16,146,0.1096,0.35714,0.23145,0.2857,0.28571,0.46431,0.0,0.85714,3,0,0,3,0,4,0,0,14,0,0,3,0,0,4,0,0,1,0,0,3,0,0],[20,146,0.137,0.46427,0.25505,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,9,0,0,7,0,0,5,0,0,3,0,0,1,0,3],[24,146,0.1644,0.37051,0.24705,0.14286,0.28571,0.571,0.0,1.0,2,1,0,2,0,9,0,0,6,0,0,6,0,0,4,0,0,3,0,0,1,0,1],[28,146,0.1918,0.33927,0.21649,0.14286,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,11,0,0,4,0,0,6,0,0,7,0,0,1,0,0,1,0,0],[32,146,0.2192,0.4308,0.25784,0.28571,0.42857,0.57143,0.0,1.0,2,2,0,2,0,4,0,1,8,0,0,4,0,0,7,0,0,3,0,0,1,0,2],[36,146,0.2466,0.34375,0.25218,0.14286,0.28571,0.57143,0.0,1.0,5,1,0,5,0,7,0,0,5,0,0,6,0,0,5,0,0,3,0,0,0,0,1],[40,146,0.274,0.37946,0.20395,0.2857,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,10,0,0,6,0,0,5,0,0,3,0,0,1,0,0],[44,146,0.3014,0.34374,0.23919,0.14286,0.28571,0.57111,0.0,1.0,2,1,0,2,0,11,0,0,6,0,0,4,0,0,5,0,0,3,0,0,0,0,1],[48,146,0.3288,0.40179,0.23266,0.14286,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,7,0,0,5,0,0,6,0,0,6,0,0,5,0,0,1,0,0],[52,146,0.3562,0.45536,0.2299,0.28571,0.42859,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,7,0,0,6,0,0,7,0,0,5,0,0,1,0,1],[56,146,0.3836,0.32589,0.21793,0.14286,0.28571,0.46429,0.0,0.71429,3,0,0,3,0,9,0,0,8,0,0,4,0,0,4,0,0,4,0,0,0,0,0],[60,146,0.411,0.41963,0.18535,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,8,0,0,8,0,0,6,0,0,5,0,0,0,0,0],[64,146,0.4384,0.39286,0.24484,0.14286,0.35714,0.46428,0.0,1.0,1,1,0,1,0,8,0,0,7,0,0,8,0,0,2,0,0,3,0,0,2,0,1],[68,146,0.4658,0.34374,0.26691,0.14286,0.28571,0.57111,0.0,1.0,5,1,0,5,0,8,0,0,5,0,0,5,0,0,4,0,0,3,0,0,1,0,1],[72,146,0.4932,0.33036,0.23808,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,10,0,0,4,0,0,4,0,0,7,0,0,2,0,0,1,0,0],[76,146,0.5205,0.33928,0.25191,0.14286,0.28571,0.42857,0.0,1.0,5,2,0,5,0,4,0,0,11,0,0,5,0,0,4,0,0,1,0,0,0,0,2],[80,146,0.5479,0.32589,0.25313,0.14286,0.28571,0.46429,0.0,1.0,4,1,0,4,0,9,0,0,8,0,0,3,0,0,4,0,0,2,0,0,1,0,1],[84,146,0.5753,0.35268,0.2082,0.24999,0.28571,0.46429,0.0,0.71429,3,0,0,3,0,5,0,0,10,0,0,6,0,0,4,0,0,4,0,0,0,0,0],[88,146,0.6027,0.31694,0.21936,0.14286,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,7,0,0,10,0,0,4,0,0,4,0,0,2,0,0,1,0,0],[92,146,0.6301,0.33929,0.1948,0.14286,0.28571,0.46429,0.0,0.71429,2,0,0,2,0,8,0,0,8,0,0,6,0,0,6,0,0,2,0,0,0,0,0],[96,146,0.6575,0.28572,0.19885,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,10,0,0,7,0,0,6,0,0,3,0,0,2,0,0,0,0,0],[100,146,0.6849,0.33929,0.21053,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,8,0,0,9,0,0,6,0,0,5,0,0,0,0,0,2,0,0],[104,146,0.7123,0.31249,0.2065,0.14286,0.35714,0.4642,0.0,0.57143,5,0,0,5,0,8,0,0,3,0,0,8,0,0,8,0,0,0,0,0,0,0,0],[108,146,0.7397,0.3125,0.23808,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,4,0,0,12,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[112,146,0.7671,0.28125,0.23279,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,2,0,0,6,0,0,10,0,0,2,0,0,1,0,0,1,0,0],[116,146,0.7945,0.23661,0.20705,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,11,0,0,7,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[120,146,0.8219,0.28124,0.23277,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,7,0,0,7,0,0,6,0,0,3,0,0,0,0,0,2,0,0],[124,146,0.8493,0.24107,0.18013,0.14286,0.2143,0.42857,0.0,0.57143,7,0,0,7,0,9,0,0,5,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[128,146,0.8767,0.29018,0.20973,0.14286,0.2857,0.42857,0.0,0.57143,6,0,0,6,0,9,0,0,2,0,0,8,0,0,7,0,0,0,0,0,0,0,0],[132,146,0.9041,0.23214,0.1915,0.10714,0.14286,0.42857,0.0,0.57143,8,0,0,8,0,9,0,0,6,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[136,146,0.9315,0.20089,0.17445,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,7,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[140,146,0.9589,0.2008,0.18853,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,12,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[144,146,0.9863,0.22322,0.18877,0.0,0.2143,0.42857,0.0,0.57143,10,0,0,10,0,6,0,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[146,146,1.0,0.18304,0.20896,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,6,0,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.23661,"x":0.50445,"p":[[0,157,0.0,0.49554,0.31132,0.14286,0.50001,0.71429,0.0,1.0,4,2,0,4,0,5,0,0,1,0,0,6,0,0,2,0,0,8,0,0,4,0,2],[4,157,0.0255,0.50445,0.27195,0.28571,0.49979,0.60714,0.0,1.0,1,4,0,1,0,5,0,0,3,0,0,7,0,0,8,0,0,3,0,0,1,0,4],[8,157,0.051,0.35713,0.21427,0.25,0.28571,0.57111,0.0,1.0,2,1,0,2,0,6,0,0,11,0,0,4,0,0,7,0,0,1,0,0,0,0,1],[12,157,0.0764,0.42857,0.24744,0.2857,0.42857,0.60714,0.0,0.85714,2,0,0,2,0,5,0,0,7,0,0,6,0,0,4,0,0,5,0,0,3,0,0],[16,157,0.1019,0.37497,0.23889,0.14286,0.42857,0.46418,0.0,1.0,3,1,0,3,0,6,0,0,6,0,0,9,0,0,4,0,0,2,0,0,1,0,1],[20,157,0.1274,0.4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that a triangle  $ABC$  is right-angled if and only if\n\\[\\sin A + \\sin B + \\sin C = \\cos A + \\cos B + \\cos C  + 1\\]","t":[{"b":0,"e":0.0,"k":"flat","v":0.04464,"x":0.23661,"p":[[0,116,0.0,0.08022,0.20141,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,116,0.0345,0.04464,0.12078,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,116,0.069,0.17857,0.29451,0.0,0.0,0.28571,0.0,1.0,21,2,0,21,0,0,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[12,116,0.1034,0.14732,0.29339,0.0,0.0,0.07143,0.0,1.0,24,2,0,24,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[16,116,0.1379,0.08929,0.21354,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[20,116,0.1724,0.09821,0.22428,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[24,116,0.2069,0.11161,0.29609,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[28,116,0.2414,0.13839,0.28231,0.0,0.0,0.17857,0.0,1.0,23,2,0,23,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[32,116,0.2759,0.21875,0.35532,0.0,0.0,0.28571,0.0,1.0,20,5,0,20,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[36,116,0.3103,0.09821,0.25862,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[40,116,0.3448,0.16071,0.29827,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[44,116,0.3793,0.12054,0.26513,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[48,116,0.4138,0.10714,0.2369,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[52,116,0.4483,0.09375,0.23038,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[56,116,0.4828,0.11607,0.25364,0.0,0.0,0.07143,0.0,1.0,24,2,0,24,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[60,116,0.5172,0.1875,0.37362,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[64,116,0.5517,0.22768,0.37091,0.0,0.0,0.32143,0.0,1.0,21,5,0,21,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,5],[68,116,0.5862,0.20982,0.35172,0.0,0.0,0.28571,0.0,1.0,21,4,0,21,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,4],[72,116,0.6207,0.15179,0.2922,0.0,0.0,0.17857,0.0,1.0,23,2,0,23,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[76,116,0.6552,0.09821,0.17655,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[80,116,0.6897,0.20089,0.36221,0.0,0.0,0.28571,0.0,1.0,23,4,0,23,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[84,116,0.7241,0.16518,0.30328,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[88,116,0.7586,0.23661,0.35644,0.0,0.0,0.28571,0.0,1.0,19,4,0,19,0,1,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[92,116,0.7931,0.20536,0.36059,0.0,0.0,0.32143,0.0,1.0,23,4,0,23,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,4],[96,116,0.8276,0.16518,0.31361,0.0,0.0,0.2857,0.0,1.0,23,3,0,23,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[100,116,0.8621,0.17857,0.32733,0.0,0.0,0.2857,0.0,1.0,23,3,0,23,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,3],[104,116,0.8966,0.12054,0.2699,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[108,116,0.931,0.16071,0.29179,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,1,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[112,116,0.9655,0.14732,0.30196,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[116,116,1.0,0.10714,0.25505,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,2]]},{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.25446,"p":[[0,92,0.0,0.09821,0.24856,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[4,92,0.0435,0.10714,0.28121,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[8,92,0.087,0.08036,0.20497,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,92,0.1304,0.125,0.26184,0.0,0.0,0.07143,0.0,1.0,24,2,0,24,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[16,92,0.1739,0.09821,0.25614,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[20,92,0.2174,0.25446,0.36724,0.0,0.0,0.42857,0.0,1.0,19,5,0,19,0,0,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,5],[24,92,0.2609,0.20982,0.35172,0.0,0.0,0.2857,0.0,1.0,21,4,0,21,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,4],[28,92,0.3043,0.12054,0.29474,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[32,92,0.3478,0.09821,0.17655,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[36,92,0.3913,0.07589,0.19228,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[40,92,0.4348,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,92,0.4783,0.11607,0.2976,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[48,92,0.5217,0.1875,0.35072,0.0,0.0,0.2857,0.0,1.0,23,4,0,23,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[52,92,0.5652,0.21875,0.34623,0.0,0.0,0.42857,0.0,1.0,20,4,0,20,0,2,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,4],[56,92,0.6087,0.12946,0.27747,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,1],[60,92,0.6522,0.20536,0.34244,0.0,0.0,0.2857,0.0,1.0,21,4,0,21,0,0,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[64,92,0.6957,0.14732,0.2977,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,3,0,1],[68,92,0.7391,0.08928,0.24936,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[72,92,0.7826,0.09821,0.21558,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[76,92,0.8261,0.15625,0.24317,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,1,0,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[80,92,0.8696,0.15179,0.29867,0.0,0.0,0.14286,0.0,1.0,22,3,0,22,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[84,92,0.913,0.1875,0.33776,0.0,0.0,0.28571,0.0,1.0,23,3,0,23,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,3],[88,92,0.9565,0.05357,0.18814,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[92,92,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"188daefe86dd2419","q":"Prove that for all positive real numbers  $a$ ,  $b$ , and  $c$ ,\r\n\\[ \\frac{a^3}{bc} + \\frac{b^3}{ca} + \\frac{c^3}{ab} \\geq a+b+c \\]\r\nand determine when equality occurs.","t":[{"b":1,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,31,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,31,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,31,0.2581,0.97322,0.10971,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,31,0.3871,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,31,0.5161,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,0.95087,0.15822,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[28,31,0.9032,0.93304,0.21124,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[31,31,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]},{"b":6,"e":0.71429,"k":"falling","v":0.73657,"x":1.0,"p":[[0,59,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,59,0.0678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,59,0.1356,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,59,0.2034,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,59,0.2712,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[20,59,0.339,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,59,0.4068,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[28,59,0.4746,0.95536,0.15746,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[32,59,0.5424,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,59,0.6102,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[40,59,0.678,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[44,59,0.7458,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,59,0.8136,0.9107,0.18816,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,25],[52,59,0.8814,0.94643,0.17035,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[56,59,0.9492,0.86157,0.21278,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,3,0,20],[59,59,1.0,0.73657,0.24775,0.571,0.78571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,6,0,0,6,0,10]]}]},{"i":"54ddfac4efe70a26","q":"Prove that none of the numbers  $2^{2^n}+ 1$ ,  $n = 0, 1, 2, \\dots$  is a perfect cube.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.0,"x":0.81695,"p":[[0,23,0.0,0.81695,0.34486,0.89286,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,24],[4,23,0.1739,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,23,0.3478,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.0,"k":"volatile","v":0.0,"x":0.77679,"p":[[0,16,0.0,0.77679,0.37276,0.67857,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,22],[4,16,0.25,0.22768,0.40226,0.0,0.0,0.14286,0.0,1.0,24,6,0,24,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[8,16,0.5,0.1183,0.30961,0.0,0.0,0.0,0.0,1.0,27,3,0,27,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[12,16,0.75,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b477489c5c896e7e","q":"Prove that for every prime $p>100$ and every integer $r$ there exist two integers $a$ and $b$ such that $p$ divides $a^{2}+b^{5}-r$.","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.99107,"p":[[0,31,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,31,0.129,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,31,0.2581,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[12,31,0.3871,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,31,0.5161,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,31,0.6452,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[24,31,0.7742,0.63839,0.46564,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,18],[28,31,0.9032,0.52679,0.48107,0.0,0.85714,1.0,0.0,1.0,14,14,0,14,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,14],[31,31,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,31,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,31,0.129,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,31,0.2581,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,31,0.3871,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,31,0.5161,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,31,0.6452,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,31,0.7742,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"69220c31332b63fd","q":"Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_{0}, \\ldots, k_{n}$, all strictly greater than 1 , such that $k_{0} k_{1} \\ldots k_{n}-1$ is the product of two consecutive integers.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.57133,"x":0.98661,"p":[[0,38,0.0,0.57133,0.36432,0.28571,0.42859,1.0,0.0,1.0,2,12,0,2,0,3,0,0,9,0,0,3,0,0,1,0,0,2,0,0,0,0,12],[4,38,0.1053,0.78125,0.2879,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,3,0,0,1,0,0,2,0,0,6,0,0,0,0,18],[8,38,0.2105,0.84375,0.29093,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,24],[12,38,0.3158,0.91964,0.20497,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[16,38,0.4211,0.88837,0.21352,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,7,0,21],[20,38,0.5263,0.92409,0.15149,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[24,38,0.6316,0.94642,0.15047,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[28,38,0.7368,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,26],[32,38,0.8421,0.96875,0.10555,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[36,38,0.9474,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[38,38,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":5,"e":1.0,"k":"rising","v":0.61607,"x":0.96429,"p":[[0,42,0.0,0.61607,0.35434,0.28571,0.64286,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,6,0,0,2,0,0,2,0,0,3,0,0,0,0,13],[4,42,0.0952,0.87052,0.22122,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,2,0,22],[8,42,0.1905,0.75893,0.30606,0.5,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,7,0,0,0,0,0,1,0,0,4,0,0,2,0,17],[12,42,0.2857,0.77232,0.33093,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,3,0,0,0,0,0,1,0,0,2,0,0,4,0,18],[16,42,0.381,0.93304,0.17491,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,27],[20,42,0.4762,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[24,42,0.5714,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[28,42,0.6667,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[32,42,0.7619,0.87054,0.22406,0.85714,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,8,0,19],[36,42,0.8571,0.93302,0.12873,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[40,42,0.9524,0.91963,0.16731,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,2,0,25],[42,42,1.0,0.95533,0.11547,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27]]}]},{"i":"b2d18a5a76662562","q":"Prove that for every prime number  $p$  there exist infinity many natural numbers  $n$  so that they satisfy: $2^{2^{2^{ \\dots ^{2^n}}}} \\equiv n^{2^{2^{\\dots ^{2}}}} (mod p)$ Where in both sides  $2$  appeared  $1397$  times","t":[{"b":1,"e":0.71429,"k":"flat","v":0.43526,"x":0.625,"p":[[0,38,0.0,0.44857,0.31411,0.14286,0.39279,0.71429,0.0,1.0,3,2,1,3,0,8,0,0,4,0,1,2,0,0,1,0,0,8,0,0,3,0,2],[4,38,0.1053,0.61157,0.23484,0.42859,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,0,0,0,7,0,0,5,0,0,10,0,0,5,0,2],[8,38,0.2105,0.625,0.28959,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,3,0,0,5,0,0,2,0,0,8,0,0,3,0,7],[12,38,0.3158,0.49554,0.335,0.14286,0.5,0.75,0.0,1.0,5,4,0,5,0,4,0,0,2,0,0,5,0,0,4,0,0,4,0,0,4,0,4],[16,38,0.4211,0.52007,0.20688,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,2,0,1,1,0,0,10,0,0,6,0,0,9,0,0,2,0,0],[20,38,0.5263,0.58481,0.28873,0.42857,0.57143,0.85714,0.0,1.0,2,5,0,2,0,2,0,0,2,0,0,7,0,0,5,0,0,5,0,0,4,0,5],[24,38,0.6316,0.5357,0.28121,0.28571,0.57143,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,3,0,0,3,0,0,7,0,0,8,0,0,2,0,3],[28,38,0.7368,0.49096,0.27429,0.28571,0.4998,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,5,0,0,5,0,0,6,0,0,5,0,0,3,0,2],[32,38,0.8421,0.48655,0.2876,0.25,0.4998,0.71429,0.0,1.0,2,2,0,2,0,6,0,0,3,0,0,5,0,0,6,0,0,4,0,0,4,0,2],[36,38,0.9474,0.49103,0.2671,0.2857,0.571,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,4,0,0,4,0,0,8,0,0,6,0,0,3,0,1],[38,38,1.0,0.43526,0.23031,0.26786,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,1,4,0,0,6,0,0,9,0,0,2,0,0,3,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.48201,"x":0.70981,"p":[[0,47,0.0,0.48201,0.23901,0.2857,0.4286,0.71429,0.0,0.85714,1,0,0,1,0,4,0,0,5,0,0,8,0,0,4,0,0,6,0,0,4,0,0],[4,47,0.0851,0.60712,0.26727,0.53539,0.71429,0.85714,0.0,1.0,2,2,2,2,0,2,0,0,2,0,0,2,0,0,7,0,0,8,0,0,7,0,2],[8,47,0.1702,0.65178,0.26471,0.53572,0.71429,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,3,0,0,2,0,0,3,0,0,9,0,0,9,0,3],[12,47,0.2553,0.59819,0.27765,0.42859,0.71429,0.85714,0.0,1.0,1,2,0,1,0,4,0,0,2,0,0,4,0,0,3,0,0,8,0,0,8,0,2],[16,47,0.3404,0.5357,0.31944,0.2857,0.57143,0.85714,0.0,1.0,4,1,0,4,0,3,0,0,4,0,0,2,0,0,4,0,0,4,0,0,10,0,1],[20,47,0.4255,0.61157,0.25812,0.42857,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,2,0,0,7,0,0,3,0,0,13,0,0],[24,47,0.5106,0.6071,0.26245,0.42857,0.71429,0.85704,0.0,1.0,1,2,0,1,0,3,0,0,2,0,0,4,0,0,4,0,0,9,0,0,7,0,2],[28,47,0.5957,0.66516,0.23585,0.5354,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,5,0,0,3,0,0,9,0,0,10,0,2],[32,47,0.6809,0.67188,0.16833,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,2,0,0,11,1,0,8,0,0,8,0,1],[36,47,0.766,0.70981,0.25627,0.67857,0.78571,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,8,0,0,11,0,5],[40,47,0.8511,0.57143,0.26244,0.42857,0.57143,0.85714,0.0,1.0,1,2,0,1,0,3,0,0,2,0,0,7,0,0,6,0,0,4,0,0,7,0,2],[44,47,0.9362,0.59373,0.29038,0.39286,0.64271,0.85714,0.0,1.0,2,3,0,2,0,2,0,0,4,0,0,3,0,0,5,0,0,5,0,0,8,0,3],[47,47,1.0,0.6383,0.15565,0.571,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,13,0,0,10,0,0,4,0,1]]}]},{"i":"265c87fa9489173f","q":"One point of the plane is called  $rational$  if both coordinates are rational and  $irrational$  if both coordinates are irrational. Check whether the following statements are true or false:**a)** Every point of the plane is in a line that can be defined by  $2$  rational points.**b)** Every point of the plane is in a line that can be defined by  $2$  irrational points.\n\nThis maybe is not algebra so sorry if I putted it in the wrong category!","t":[{"b":5,"e":0.85714,"k":"flat","v":0.83928,"x":0.85714,"p":[[0,11,0.0,0.83928,0.10564,0.85714,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,29,0,1],[4,11,0.3636,0.85714,2e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[8,11,0.7273,0.85714,2e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[11,11,1.0,0.85714,3e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.83928,"x":0.86161,"p":[[0,32,0.0,0.83928,0.07784,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,30,0,0],[4,32,0.125,0.84374,0.07457,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0],[8,32,0.25,0.85714,2e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[12,32,0.375,0.85713,3e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[16,32,0.5,0.85714,1e-05,0.85714,0.85714,0.85714,0.8571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[20,32,0.625,0.85267,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[24,32,0.75,0.85713,3e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[28,32,0.875,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[32,32,1.0,0.86161,0.02486,0.85714,0.85714,0.85714,0.8571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1]]}]},{"i":"3eac36128928f606","q":"Prove that for every real number  $M$  there exists an infinite arithmetical progression of positive integers such that  \n- the common difference is not divisible by  $10$ ,\n- the sum of digits of each term exceeds  $M$ .","t":[{"b":0,"e":0.57143,"k":"falling","v":0.07143,"x":0.41955,"p":[[0,57,0.0,0.30355,0.24934,0.10714,0.28571,0.57111,0.0,0.857,8,0,1,8,0,7,0,0,2,0,0,6,0,0,7,0,0,1,0,0,1,0,0],[4,57,0.0702,0.41955,0.27888,0.14286,0.42857,0.57143,0.0,1.0,3,2,0,3,0,7,0,0,3,0,0,7,0,0,6,0,0,2,0,0,2,0,2],[8,57,0.1404,0.39286,0.26486,0.14286,0.42857,0.57143,0.0,0.85714,4,0,0,4,0,7,0,0,2,0,0,9,0,0,3,0,0,4,0,0,3,0,0],[12,57,0.2105,0.35265,0.29009,0.14286,0.28571,0.57141,0.0,1.0,6,1,0,6,0,8,0,0,4,0,0,3,0,0,5,0,0,3,0,0,2,0,1],[16,57,0.2807,0.29909,0.26571,0.14286,0.14286,0.46418,0.0,0.85714,7,0,0,7,0,11,0,0,0,0,0,6,0,0,4,0,0,2,0,0,2,0,0],[20,57,0.3509,0.35268,0.27429,0.14286,0.28571,0.57143,0.0,1.0,4,1,0,4,0,10,0,0,3,0,0,6,0,0,5,0,0,0,0,0,3,0,1],[24,57,0.4211,0.39286,0.25,0.14286,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,4,0,0,4,0,0,6,0,0,9,0,0,2,0,0,2,0,0],[28,57,0.4912,0.33036,0.29545,0.14286,0.21428,0.57143,0.0,0.85714,7,0,0,7,0,9,0,0,3,0,0,3,0,0,5,0,0,0,0,0,5,0,0],[32,57,0.5614,0.27677,0.28999,0.0,0.14286,0.46418,0.0,1.0,10,1,0,10,0,8,0,0,4,0,0,2,0,0,4,0,0,1,0,0,2,0,1],[36,57,0.6316,0.21875,0.21124,0.14286,0.14286,0.32143,0.0,0.85714,7,0,0,7,0,15,0,0,2,0,0,5,0,0,1,0,0,1,0,0,1,0,0],[40,57,0.7018,0.27679,0.21706,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,9,0,0,5,0,0,8,0,0,2,0,0,1,0,0,1,0,0],[44,57,0.7719,0.12054,0.15198,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,12,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[48,57,0.8421,0.13839,0.16935,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[52,57,0.9123,0.11607,0.1448,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,10,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.11607,0.14032,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,14,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[57,57,1.0,0.07143,0.14286,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.15624,"x":0.43741,"p":[[0,49,0.0,0.27679,0.23403,0.0,0.28571,0.4286,0.0,0.71429,9,0,1,9,0,6,0,0,4,0,0,6,0,0,5,0,0,2,0,0,0,0,0],[4,49,0.0816,0.40179,0.29545,0.14286,0.28571,0.57143,0.0,1.0,3,2,0,3,0,8,0,0,6,0,0,4,0,0,5,0,0,0,0,0,4,0,2],[8,49,0.1633,0.42857,0.24223,0.24999,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,6,0,0,4,0,0,7,0,0,7,0,0,3,0,0,3,0,0],[12,49,0.2449,0.38838,0.32582,0.0,0.42857,0.57143,0.0,1.0,10,1,0,10,0,2,0,0,3,0,0,2,0,0,8,0,0,2,0,0,4,0,1],[16,49,0.3265,0.30357,0.21943,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,7,0,0,4,0,0,9,0,0,5,0,0,0,0,0,1,0,0],[20,49,0.4082,0.40625,0.26513,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,8,0,0,2,0,0,6,0,0,8,0,0,2,0,0,2,0,1],[24,49,0.4898,0.41072,0.32093,0.14286,0.42857,0.60714,0.0,1.0,4,2,0,4,0,10,0,0,1,0,0,5,0,0,4,0,0,1,0,0,5,0,2],[28,49,0.5714,0.36607,0.31326,0.14286,0.28571,0.57143,0.0,1.0,7,1,0,7,0,7,0,0,4,0,0,2,0,0,5,0,0,2,0,0,4,0,1],[32,49,0.6531,0.41964,0.24206,0.14286,0.42857,0.57143,0.0,1.0,2,1,0,2,0,7,0,0,3,0,0,6,0,0,10,0,0,2,0,0,1,0,1],[36,49,0.7347,0.43741,0.28789,0.14286,0.42857,0.57143,0.0,1.0,2,2,0,2,0,8,0,0,3,0,0,7,0,0,5,0,0,1,0,0,4,0,2],[40,49,0.8163,0.35714,0.27433,0.14286,0.28571,0.57143,0.0,1.0,3,1,0,3,0,12,0,0,2,0,0,6,0,0,3,0,0,3,0,0,2,0,1],[44,49,0.898,0.2008,0.23109,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,11,0,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[48,49,0.9796,0.15624,0.19675,0.0,0.14286,0.17857,0.0,0.71429,15,0,0,15,0,9,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[49,49,1.0,0.20088,0.26451,0.0,0.07143,0.32143,0.0,0.85714,16,0,0,16,0,5,0,0,3,0,0,3,0,0,2,0,0,1,0,0,2,0,0]]}]},{"i":"2ec6c5d89de3c6f0","q":"Let $n$ be an integer at least 5. At most how many diagonals of a regular $n$-gon can be simultaneously drawn so that no two are parallel? Prove your answer.","t":[{"b":1,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,22,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,22,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,22,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,22,0.5455,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,22,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,22,0.9091,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[22,22,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":2,"e":0.28571,"k":"falling","v":0.18302,"x":1.0,"p":[[0,30,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,30,0.1333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,30,0.2667,0.91518,0.21387,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,24],[12,30,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,30,0.5333,0.76786,0.36378,0.39286,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,22],[20,30,0.6667,0.46857,0.36999,0.14286,0.28571,1.0,0.0,1.0,5,9,0,5,0,4,0,0,8,0,0,2,0,0,4,0,0,0,0,0,0,0,9],[24,30,0.8,0.36604,0.23126,0.2857,0.28571,0.571,0.0,1.0,2,2,0,2,0,5,0,0,13,0,0,3,0,0,7,0,0,0,0,0,0,0,2],[28,30,0.9333,0.23213,0.17765,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,2,0,0,17,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[30,30,1.0,0.18302,0.16062,0.0,0.2143,0.28571,0.0,0.57143,11,0,0,11,0,5,0,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"e3245c241b6fb9d4","q":"Prove that all real numbers  $x \\ne -1$ ,  $y \\ne -1$  with  $xy = 1$  satisfy the following inequality:   $$ \\left(\\frac{2+x}{1+x}\\right)^2 + \\left(\\frac{2+y}{1+y}\\right)^2 \\ge \\frac92 $$  (Karl Czakler)","t":[{"b":1,"e":1.0,"k":"falling","v":0.66071,"x":0.875,"p":[[0,28,0.0,0.875,0.27374,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,25],[4,28,0.1429,0.79463,0.31123,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,0,2,0,20],[8,28,0.2857,0.66071,0.33645,0.28571,0.64286,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,5,0,0,3,0,0,4,0,0,1,0,0,1,0,14],[12,28,0.4286,0.70536,0.36932,0.28571,1.0,1.0,0.14286,1.0,0,19,0,0,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,19],[16,28,0.5714,0.70979,0.27777,0.57143,0.64286,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,0,0,0,11,0,0,1,0,0,3,0,12],[20,28,0.7143,0.71872,0.30197,0.57132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,3,0,0,0,0,0,9,0,0,0,0,0,3,0,14],[24,28,0.8571,0.67411,0.35398,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,7,0,0,1,0,0,3,0,0,0,0,0,1,0,16],[28,28,1.0,0.66071,0.31491,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,6,0,0,1,0,0,5,0,0,4,0,0,1,0,12]]},{"b":2,"e":1.0,"k":"rising","v":0.70089,"x":1.0,"p":[[0,46,0.0,0.74554,0.35667,0.5,1.0,1.0,0.0,1.0,2,20,0,2,0,2,0,0,4,0,0,0,0,0,3,0,0,1,0,0,0,0,20],[4,46,0.087,0.70089,0.32803,0.39286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,4,0,0,1,0,0,4,0,0,3,0,0,1,0,15],[8,46,0.1739,0.73214,0.36727,0.39286,1.0,1.0,0.0,1.0,1,20,0,1,0,5,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,20],[12,46,0.2609,0.74553,0.32485,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,19],[16,46,0.3478,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[20,46,0.4348,0.94643,0.17768,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"520e1ec1b3d8cdf1","q":"Positive integers  $m$  and  $n$  have no common divisor greater than one. What is the largest possible value of the greatest common divisor of  $m + 2000n$  and  $n + 2000m$  ?\t\n\n(S Zlobin)","t":[{"b":6,"e":0.85714,"k":"flat","v":0.91964,"x":0.97321,"p":[[0,59,0.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[4,59,0.0678,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[8,59,0.1356,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[12,59,0.2034,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,59,0.2712,0.92857,0.09449,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,18],[20,59,0.339,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[24,59,0.4068,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[28,59,0.4746,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[32,59,0.5424,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[36,59,0.6102,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[40,59,0.678,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[44,59,0.7458,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[48,59,0.8136,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[52,59,0.8814,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[56,59,0.9492,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[59,59,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":7,"e":0.85714,"k":"flat","v":0.92411,"x":1.0,"p":[[0,54,0.0,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[4,54,0.0741,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[8,54,0.1481,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,54,0.2222,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,54,0.2963,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,54,0.3704,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,54,0.4444,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,54,0.5185,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,54,0.5926,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,54,0.6667,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,54,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,54,0.8148,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,54,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,54,0.963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[54,54,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b8ed6a8fe496c5ad","q":"Let $n>1$ be a positive integer. Each cell of an $n \\times n$ table contains an integer. Suppose that the following conditions are satisfied:  (i) Each number in the table is congruent to 1 modulo $n$;  (ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$.  Let $R_{i}$ be the product of the numbers in the $i^{\\text {th }}$ row, and $C_{j}$ be the product of the numbers in the $j^{\\text {th }}$ column. Prove that the sums $R_{1}+\\cdots+R_{n}$ and $C_{1}+\\cdots+C_{n}$ are congruent modulo $n^{4}$.  (Indonesia)","t":[{"b":1,"e":0.42857,"k":"flat","v":0.49997,"x":0.558,"p":[[0,39,0.0,0.49997,0.07983,0.42857,0.571,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,14,0,0,17,0,0,0,0,0,0,0,0],[4,39,0.1026,0.54458,0.06619,0.571,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0],[8,39,0.2051,0.55356,0.04724,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[12,39,0.3077,0.55354,0.04724,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[16,39,0.4103,0.54459,0.06619,0.571,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0],[20,39,0.5128,0.54462,0.06621,0.57143,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0],[24,39,0.6154,0.53122,0.1197,0.57143,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,2,0,0,1,0,0,28,0,0,0,0,0,0,0,0],[28,39,0.7179,0.558,0.04162,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0],[32,39,0.8205,0.55353,0.04723,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[36,39,0.9231,0.54907,0.05185,0.57132,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[39,39,1.0,0.54908,0.05186,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"flat","v":0.49549,"x":0.58478,"p":[[0,52,0.0,0.49549,0.07123,0.42857,0.4286,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,17,0,0,15,0,0,0,0,0,0,0,0],[4,52,0.0769,0.54907,0.05185,0.57132,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[8,52,0.1538,0.54011,0.05902,0.571,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0],[12,52,0.2308,0.55353,0.04723,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[16,52,0.3077,0.54909,0.05186,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[20,52,0.3846,0.54908,0.06297,0.57143,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0,0,0,0],[24,52,0.4615,0.53567,0.07141,0.571,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,6,0,0,25,0,0,0,0,0,0,0,0],[28,52,0.5385,0.57139,0.08748,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,1],[32,52,0.6154,0.55798,0.04162,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0],[36,52,0.6923,0.57139,0.00012,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[40,52,0.7692,0.55356,0.09942,0.5713,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,24,0,0,0,0,0,0,0,1],[44,52,0.8462,0.58478,0.07458,0.57143,0.57143,0.57143,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,1],[48,52,0.9231,0.57137,0.00014,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[52,52,1.0,0.55354,0.05922,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0,0,0,0]]}]},{"i":"847f5cacd46034ed","q":"Prove that for any positive $x_{1}, x_{2}, \\ldots, x_{n}$ and $y_{1}, y_{2}, \\ldots, y_{n}$ the inequality\n\n$$\n\\sum_{i=1}^{n} \\frac{1}{x_{i} y_{i}} \\geq \\frac{4 n^{2}}{\\sum_{i=1}^{n}\\left(x_{i}+y_{i}\\right)^{2}}\n$$\n\nholds.","t":[{"b":0,"e":1.0,"k":"falling","v":0.71429,"x":0.99107,"p":[[0,34,0.0,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[4,34,0.1176,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,34,0.2353,0.72322,0.28107,0.53572,0.64286,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,6,0,0,8,0,0,1,0,0,0,0,15],[12,34,0.3529,0.79909,0.26212,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,0,0,0,1,0,19],[16,34,0.4706,0.79464,0.27418,0.53572,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,7,0,0,4,0,0,0,0,0,0,0,20],[20,34,0.5882,0.71429,0.31135,0.42857,0.78571,1.0,0.0,1.0,1,16,0,1,0,2,0,0,0,0,0,6,0,0,7,0,0,0,0,0,0,0,16],[24,34,0.7059,0.73661,0.26027,0.42857,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,0,0,0,1,0,15],[28,34,0.8235,0.91518,0.17807,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,26],[32,34,0.9412,0.77679,0.29437,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,7,0,0,3,0,0,0,0,0,1,0,19],[34,34,1.0,0.73214,0.26666,0.42857,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,1,0,0,1,0,15]]},{"b":2,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,31,0.0,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,31,0.129,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,31,0.2581,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[12,31,0.3871,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,31,0.5161,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,31,0.6452,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"02e856602988a298","q":"Prove that for every integer  $ n>1, 1 \\cdot 3 \\cdot 5 \\cdot ... \\cdot (2n\\minus{}1)<n^n.$","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.11607,"p":[[0,33,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.11607,0.2976,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.11607,0.2683,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[24,33,0.7273,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,33,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[33,33,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.07589,"p":[[0,19,0.0,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,19,0.2105,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,19,0.6316,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,19,0.8421,0.04018,0.12492,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bc7144c2f3290d66","q":"Let $p>7$ be a prime number and let $A$ be a subset of $\\{0,1, \\ldots, p-1\\}$ consisting of at least $\\frac{p-1}{2}$ elements. Show that for each integer $r$, there exist (not necessarily distinct) numbers $a, b, c, d \\in A$ such that\n\n$$\na b-c d \\equiv r \\quad(\\bmod p)\n$$","t":[{"b":3,"e":0.2857,"k":"falling","v":0.32589,"x":0.78124,"p":[[0,56,0.0,0.55357,0.25692,0.39286,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,4,0,0,7,0,0,6,0,0,6,0,1],[4,56,0.0714,0.70534,0.24985,0.57143,0.85714,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,2,0,0,4,0,0,4,0,0,15,0,3],[8,56,0.1429,0.70089,0.27976,0.53571,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,6,0,0,1,0,0,2,0,0,5,0,0,10,0,7],[12,56,0.2143,0.60714,0.29014,0.28571,0.78564,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,7,0,0,4,0,0,2,0,0,1,0,0,15,0,1],[16,56,0.2857,0.63615,0.28312,0.39286,0.71429,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,6,0,0,2,0,0,2,0,0,6,0,0,11,1,2],[20,56,0.3571,0.71426,0.27895,0.67846,0.85714,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,3,0,0,0,0,0,2,0,0,6,0,0,12,0,6],[24,56,0.4286,0.66955,0.24875,0.57143,0.71429,0.85714,0.14,1.0,0,3,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,6,0,0,11,0,3],[28,56,0.5,0.78124,0.23415,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,15,0,8],[32,56,0.5714,0.625,0.27375,0.39286,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,6,0,0,2,0,0,2,0,0,7,0,0,12,0,1],[36,56,0.6429,0.57589,0.30406,0.28571,0.71429,0.85714,0.0,0.85714,4,0,0,4,0,1,0,0,5,0,0,0,0,0,3,0,0,8,0,0,11,0,0],[40,56,0.7143,0.51338,0.2621,0.28571,0.49979,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,9,0,0,4,0,0,2,0,0,8,0,0,6,0,0],[44,56,0.7857,0.51337,0.26692,0.28571,0.4998,0.75,0.0,0.85714,2,0,0,2,0,1,0,0,9,0,0,4,0,0,4,0,0,4,0,0,8,0,0],[48,56,0.8571,0.32589,0.15661,0.28571,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,1,0,0,18,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[52,56,0.9286,0.39732,0.21646,0.28571,0.28571,0.46429,0.0,0.85714,1,0,0,1,0,1,0,0,19,0,0,3,0,0,3,0,0,1,0,0,4,0,0],[56,56,1.0,0.35714,0.14286,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,21,0,0,6,0,0,2,0,0,1,0,0,1,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.33036,"x":0.75891,"p":[[0,19,0.0,0.60268,0.23619,0.53571,0.57143,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,4,0,0,2,0,0,10,0,0,5,0,0,8,0,1],[4,19,0.2105,0.75891,0.22143,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,15,0,5],[8,19,0.4211,0.71875,0.25123,0.57143,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,7,0,0,0,0,0,2,0,0,4,0,0,14,0,5],[12,19,0.6316,0.62945,0.26931,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,6,0,0,4,0,0,0,0,0,11,0,4],[16,19,0.8421,0.45982,0.23887,0.28571,0.35714,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,15,0,0,6,0,0,1,0,0,3,0,0,6,0,0],[19,19,1.0,0.33036,0.09062,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,4,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"5ebac147b142b940","q":"Positive integers $1,2, \\ldots, 100,101$ are written in the cells of a $101 \\times 101$ square grid so that each number is repeated 101 times. Prove that there exists either a column or a row containing at least 11 different numbers.","t":[{"b":2,"e":1.0,"k":"rising","v":0.74552,"x":1.0,"p":[[0,41,0.0,0.74552,0.30038,0.57132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,6,0,0,0,0,0,6,0,0,0,0,0,3,0,16],[4,41,0.0976,0.88839,0.18808,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,1,0,23],[8,41,0.1951,0.88839,0.25688,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,26],[12,41,0.2927,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,41,0.3902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,41,0.4878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,41,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,46,0.0,0.83036,0.27302,0.78571,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,0,3,0,21],[4,46,0.087,0.82143,0.25254,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,6,0,0,2,0,0,2,0,19],[8,46,0.1739,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,46,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,46,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,46,0.6087,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[32,46,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"26a13c0508800340","q":"Prove that for each prime  $ P =9k+1$  ,exist natural n such that  $P|n^3-3n+1$ .","t":[{"b":1,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,12,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,12,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,30,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,30,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,30,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1196664d434a8f86","q":"Positive integers  $x,y,z \\le 100$  satisfy\n\\begin{align*}\n1099x+901y+1110z &= 59800 \n109x+991y+101z &= 44556\n\\end{align*}\nCompute  $10000x+100y+z$ .\n\n*Evan Chen*","t":[{"b":3,"e":1.0,"k":"flat","v":0.94638,"x":1.0,"p":[[0,26,0.0,0.94638,0.15476,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[4,26,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.86161,"x":1.0,"p":[[0,53,0.0,0.86161,0.2435,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,2,0,0,1,0,23],[4,53,0.0755,0.93304,0.18893,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[8,53,0.1509,0.88392,0.19379,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,1,0,22],[12,53,0.2264,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,53,0.3019,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,53,0.3774,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,53,0.4528,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,53,0.5283,0.9375,0.16727,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[32,53,0.6038,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[36,53,0.6792,0.94196,0.16698,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,28],[40,53,0.7547,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[44,53,0.8302,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[48,53,0.9057,0.91518,0.15916,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,24],[52,53,0.9811,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[53,53,1.0,0.9375,0.14698,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,25]]}]},{"i":"ab78b011d7e2e001","q":"Prove that for all positive real numbers $a, b$, and $c$,\n\n$$\n\\frac{a^{3}}{b c}+\\frac{b^{3}}{c a}+\\frac{c^{3}}{a b} \\geq a+b+c\n$$\n\nand determine when equality occurs.\n\nEach of the inequalities used in the solutions below has the property that equality holds if and only if $a=b=c$. Thus equality holds for the given inequality if and only if $a=b=c$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.74997,"x":1.0,"p":[[0,36,0.0,0.74997,0.30516,0.49968,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,8,0,0,0,0,0,4,0,0,2,0,0,0,0,18],[4,36,0.1111,0.89286,0.26,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,27],[8,36,0.2222,0.91071,0.21943,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[12,36,0.3333,0.93304,0.2082,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[16,36,0.4444,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.57143,"k":"flat","v":0.70532,"x":0.81695,"p":[[0,9,0.0,0.81695,0.28625,0.57143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,6,0,0,0,0,0,3,0,0,1,0,0,0,0,22],[4,9,0.4444,0.77679,0.31931,0.28571,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,1,0,0,0,0,21],[8,9,0.8889,0.71872,0.26843,0.57143,0.57143,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,0,0,0,12,0,0,1,0,0,0,0,14],[9,9,1.0,0.70532,0.2856,0.571,0.64286,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,7,0,0,0,0,0,9,0,0,2,0,0,0,0,14]]}]},{"i":"8e6163ed4fdecde4","q":"Let $x, y, z$ be positive real numbers such that $x+y+z=\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}$.\n\na) Prove the inequality\n\n$$\nx+y+z \\geq \\sqrt{\\frac{x y+1}{2}}+\\sqrt{\\frac{y z+1}{2}}+\\sqrt{\\frac{z x+1}{2}}\n$$\n\nb) (Added by the problem selecting committee) When does the equality hold?","t":[{"b":0,"e":1.0,"k":"flat","v":0.78123,"x":0.98661,"p":[[0,108,0.0,0.78123,0.25751,0.67836,0.85714,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,8,0,0,0,0,16],[4,108,0.037,0.9107,0.17408,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,24],[8,108,0.0741,0.88393,0.2034,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,0,22],[12,108,0.1111,0.92188,0.13758,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,1,22],[16,108,0.1481,0.88392,0.16148,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,4,0,19],[20,108,0.1852,0.90177,0.17657,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,23],[24,108,0.2222,0.90625,0.14555,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,20],[28,108,0.2593,0.87946,0.22899,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,22],[32,108,0.2963,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,108,0.3333,0.93301,0.12877,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[40,108,0.3704,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,108,0.4074,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[48,108,0.4444,0.96651,0.07782,1.0,1.0,1.0,0.64286,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5,0,26],[52,108,0.4815,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[56,108,0.5185,0.96205,0.09103,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,4,0,26],[60,108,0.5556,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[64,108,0.5926,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[68,108,0.6296,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[72,108,0.6667,0.94195,0.18853,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[76,108,0.7037,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[80,108,0.7407,0.93749,0.12344,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[84,108,0.7778,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[88,108,0.8148,0.95759,0.10543,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,1,0,27],[92,108,0.8519,0.90624,0.16984,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,22],[96,108,0.8889,0.92633,0.10334,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,7,0,20],[100,108,0.9259,0.91741,0.11725,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,1,19],[104,108,0.963,0.92856,0.13835,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,23],[108,108,1.0,0.87052,0.18683,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,10,0,16]]},{"b":4,"e":0.28571,"k":"falling","v":0.35713,"x":0.95536,"p":[[0,102,0.0,0.80354,0.18126,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,7,0,11],[4,102,0.0392,0.92409,0.14282,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,23],[8,102,0.0784,0.92411,0.14279,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,22],[12,102,0.1176,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[16,102,0.1569,0.89286,0.16366,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,2,0,21],[20,102,0.1961,0.88838,0.13712,0.82143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,17],[24,102,0.2353,0.85714,0.16751,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,4,0,16],[28,102,0.2745,0.88839,0.22794,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,23],[32,102,0.3137,0.9107,0.15875,0.85711,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,23],[36,102,0.3529,0.8683,0.18395,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,1,0,1,0,20],[40,102,0.3922,0.8482,0.20499,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,18],[44,102,0.4314,0.85713,0.1713,0.82143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,10,0,14],[48,102,0.4706,0.88839,0.15458,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,4,0,19],[52,102,0.5098,0.91518,0.16698,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,21],[56,102,0.549,0.87946,0.18249,0.82143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,18],[60,102,0.5882,0.83482,0.21756,0.82132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,8,0,16],[64,102,0.6275,0.86158,0.16169,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,3,0,17],[68,102,0.6667,0.85713,0.19235,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,18],[72,102,0.7059,0.87947,0.19597,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,1,0,22],[76,102,0.7451,0.81695,0.2059,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,2,0,16],[80,102,0.7843,0.85714,0.19233,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,18],[84,102,0.8235,0.86161,0.24087,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,4,0,0,3,0,21],[88,102,0.8627,0.78124,0.23955,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,4,0,0,6,0,13],[92,102,0.902,0.75668,0.23342,0.57142,0.85707,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,4,1,0,4,0,0,6,0,11],[96,102,0.9412,0.68972,0.29376,0.28571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,8,0,0,0,0,1,1,0,0,6,0,0,5,0,10],[100,102,0.9804,0.35714,0.16367,0.28571,0.28571,0.42858,0.14286,1.0,0,1,0,0,0,2,0,0,20,0,0,6,0,0,2,0,0,1,0,0,0,0,1],[102,102,1.0,0.35713,0.11292,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,16,0,0,10,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"80bf5b6fddf2b534","q":"Prove that  $\\sqrt 2 +\\sqrt 3 +\\sqrt{1990}$  is irrational.","t":[{"b":0,"e":1.0,"k":"rising","v":0.63839,"x":0.95536,"p":[[0,6,0.0,0.63839,0.38628,0.2857,0.85714,1.0,0.0,1.0,4,13,0,4,0,2,0,0,6,0,0,0,0,0,1,0,0,2,0,0,4,0,13],[4,6,0.6667,0.93304,0.22011,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[6,6,1.0,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30]]},{"b":7,"e":0.0,"k":"falling","v":0.04464,"x":0.57141,"p":[[0,25,0.0,0.57141,0.3896,0.2857,0.57121,1.0,0.0,1.0,5,12,0,5,0,2,0,0,6,0,0,2,0,0,2,0,0,2,0,0,1,0,12],[4,25,0.16,0.54909,0.41205,0.21427,0.57121,1.0,0.0,1.0,8,12,0,8,0,0,0,0,6,0,0,1,0,0,2,0,0,2,0,0,1,0,12],[8,25,0.32,0.16518,0.2372,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[12,25,0.48,0.07589,0.15966,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,25,0.64,0.14732,0.25874,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,1,0,0,4,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[20,25,0.8,0.12498,0.1881,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,0,0,0,8,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[24,25,0.96,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0a2aa62eeca65889","q":"Prove that for any positive real  $x$  and  $y$ , holds the inequality   $$ \\frac{1}{(x+y)^2}+\\frac{1}{x^2}+\\frac{1}{y^2} \\ge \\frac{9}{4xy} $$","t":[{"b":2,"e":0.57143,"k":"flat","v":0.73213,"x":0.91964,"p":[[0,38,0.0,0.88392,0.22713,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,25],[4,38,0.1053,0.91964,0.17105,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,0,0,26],[8,38,0.2105,0.75445,0.24021,0.57143,0.57143,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,15,0,0,0,0,0,0,0,15],[12,38,0.3158,0.73213,0.23352,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,13],[16,38,0.4211,0.76786,0.22517,0.57143,0.64286,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,1,0,0,0,0,15],[20,38,0.5263,0.84373,0.20002,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,2,0,19],[24,38,0.6316,0.86606,0.19867,0.57143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,22],[28,38,0.7368,0.875,0.20748,0.67857,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,23],[32,38,0.8421,0.86607,0.19865,0.57143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,22],[36,38,0.9474,0.83481,0.20552,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,19],[38,38,1.0,0.83481,0.20552,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,19]]},{"b":5,"e":1.0,"k":"flat","v":0.8125,"x":1.0,"p":[[0,24,0.0,0.87053,0.25344,0.96425,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,1,0,24],[4,24,0.1667,0.8125,0.31428,0.57143,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,23],[8,24,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,24,0.5,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"08716b4672f47314","q":"Let ABC be a right triangle with  $\\angle B = 90^{\\circ}$ .Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.83928,"x":0.99107,"p":[[0,147,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,147,0.0272,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,147,0.0544,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[12,147,0.0816,0.90623,0.13654,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,19],[16,147,0.1088,0.90179,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,15],[20,147,0.1361,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[24,147,0.1633,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[28,147,0.1905,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[32,147,0.2177,0.91964,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[36,147,0.2449,0.90625,0.11633,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,17],[40,147,0.2721,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[44,147,0.2993,0.89286,0.18558,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,16],[48,147,0.3265,0.87946,0.13882,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,15,0,13],[52,147,0.3537,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[56,147,0.381,0.93304,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,19],[60,147,0.4082,0.87946,0.11356,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,17,0,11],[64,147,0.4354,0.88838,0.13712,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,10,0,16],[68,147,0.4626,0.90178,0.10374,0.85714,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,14],[72,147,0.4898,0.87944,0.14778,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,9,0,16],[76,147,0.517,0.83928,0.15872,0.85711,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,7],[80,147,0.5442,0.91071,0.08565,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[84,147,0.5714,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[88,147,0.5986,0.87945,0.11904,0.85711,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,12],[92,147,0.6259,0.88835,0.13719,0.85714,0.92857,1.0,0.57,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,10,0,16],[96,147,0.6531,0.88838,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,13],[100,147,0.6803,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,13],[104,147,0.7075,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,17,0,12],[108,147,0.7347,0.88839,0.08553,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,19,0,10],[112,147,0.7619,0.89732,0.08171,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,11],[116,147,0.7891,0.83928,0.14174,0.82132,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,15,0,9],[120,147,0.8163,0.86607,0.12846,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,15,0,11],[124,147,0.8435,0.89286,0.10102,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,12],[128,147,0.8707,0.84821,0.12846,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,18,0,8],[132,147,0.898,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,17],[136,147,0.9252,0.89286,0.09449,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,19,0,11],[140,147,0.9524,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,17,0,12],[144,147,0.9796,0.84821,0.11811,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,19,0,7],[147,147,1.0,0.85267,0.11564,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,20,0,7]]},{"b":4,"e":0.571,"k":"flat","v":0.83929,"x":0.96429,"p":[[0,156,0.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[4,156,0.0256,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[8,156,0.0513,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[12,156,0.0769,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,156,0.1026,0.90625,0.14555,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,17],[20,156,0.1282,0.88839,0.18118,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,9,0,18],[24,156,0.1538,0.875,0.24419,0.85714,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,18],[28,156,0.1795,0.86607,0.21998,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,10,0,17],[32,156,0.2051,0.9241,0.13825,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,10,0,20],[36,156,0.2308,0.89286,0.15972,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,13,0,16],[40,156,0.2564,0.91963,0.13334,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,18],[44,156,0.2821,0.94642,0.09284,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[48,156,0.3077,0.85267,0.20355,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,13,0,14],[52,156,0.3333,0.90625,0.18073,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,18],[56,156,0.359,0.83929,0.27837,0.85714,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,18],[60,156,0.3846,0.9241,0.11837,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,11,0,19],[64,156,0.4103,0.89731,0.2402,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,21],[68,156,0.4359,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[72,156,0.4615,0.89732,0.20277,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,20],[76,156,0.4872,0.92855,0.09455,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,18],[80,156,0.5128,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[84,156,0.5385,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[88,156,0.5641,0.91517,0.14665,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,19],[92,156,0.5897,0.90177,0.14483,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,14,0,16],[96,156,0.6154,0.90625,0.09182,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,14],[100,156,0.641,0.88839,0.21646,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,19],[104,156,0.6667,0.86607,0.21998,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,9,0,17],[108,156,0.6923,0.88393,0.24856,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,22],[112,156,0.7179,0.88839,0.22794,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,20],[116,156,0.7436,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[120,156,0.7692,0.88392,0.20652,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,14,0,16],[124,156,0.7949,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[128,156,0.8205,0.84821,0.24468,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,12,0,15],[132,156,0.8462,0.89286,0.19562,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,17],[136,156,0.8718,0.90177,0.13574,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,18],[140,156,0.8974,0.85267,0.18382,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,11,0,14],[144,156,0.9231,0.87054,0.22689,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,7,0,19],[148,156,0.9487,0.87946,0.24772,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,20],[152,156,0.9744,0.90178,0.16917,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,10,0,19],[156,156,1.0,0.86605,0.18538,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,12,0,15]]}]},{"i":"99f3a21be5b658bc","q":"Let be a sequence of  $ 51 $  natural numbers whose sum is  $ 100. $  Show that for any natural number  $ 1\\le k<100 $  there are some consecutive numbers from this sequence whose sum is  $ k $  or  $ 100-k. $","t":[{"b":4,"e":0.85714,"k":"flat","v":0.70083,"x":0.91963,"p":[[0,58,0.0,0.70083,0.19022,0.571,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,8,0,0,6,0,5],[4,58,0.069,0.87946,0.2055,0.857,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,5,0,20],[8,58,0.1379,0.86603,0.20503,0.82132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,1,0,0,4,0,20],[12,58,0.2069,0.88393,0.25111,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,24],[16,58,0.2759,0.91963,0.15542,0.85711,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,23],[20,58,0.3448,0.86606,0.23941,0.857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,0,4,0,21],[24,58,0.4138,0.83928,0.23891,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,5,0,0,3,0,19],[28,58,0.4828,0.85266,0.22726,0.85711,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,7,0,18],[32,58,0.5517,0.78123,0.29231,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,2,0,0,2,0,18],[36,58,0.6207,0.79907,0.29638,0.67857,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,6,0,17],[40,58,0.6897,0.84149,0.25616,0.82143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,2,0,1,2,0,0,1,0,0,4,0,20],[44,58,0.7586,0.8214,0.19563,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,7,0,0,12,0,10],[48,58,0.8276,0.8973,0.16843,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,20],[52,58,0.8966,0.85265,0.18382,0.857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,15,0,12],[56,58,0.9655,0.84818,0.19216,0.82132,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,9,0,15],[58,58,1.0,0.82587,0.16652,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,11,0,10]]},{"b":6,"e":1.0,"k":"flat","v":0.69638,"x":0.98659,"p":[[0,64,0.0,0.69638,0.15468,0.57143,0.71429,0.74996,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,12,0,0,5,0,3],[4,64,0.0625,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,64,0.125,0.9464,0.1276,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[12,64,0.1875,0.92856,0.18558,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[16,64,0.25,0.92857,0.19562,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[20,64,0.3125,0.9375,0.15126,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[24,64,0.375,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[28,64,0.4375,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,64,0.5,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[36,64,0.5625,0.89732,0.21498,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,21],[40,64,0.625,0.86158,0.23006,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,8,0,18],[44,64,0.6875,0.89731,0.17942,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,20],[48,64,0.75,0.85268,0.23551,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,6,0,19],[52,64,0.8125,0.85714,0.24744,0.85711,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,19],[56,64,0.875,0.84819,0.20186,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,17],[60,64,0.9375,0.81246,0.126,0.71429,0.857,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,13,0,6],[64,64,1.0,0.74545,0.14172,0.71429,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,13,0,0,11,0,2]]}]},{"i":"21967a85faf644fb","q":"Let $x, y, z$ be real numbers (not necessarily positive) such that $x^{4}+y^{4}+z^{4}+x y z=4$. Prove that $x \\leq 2$ and  $$ \\sqrt{2-x} \\geq \\frac{y+z}{2} 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the interior point  $P$  of the convex quadrilateral  $ABCD$  be such that   $$ |\\angle PAD| = |\\angle ADP| = |\\angle CBP| = |\\angle PCB| = |\\angle CPD|. $$   Let  $O$  be the center of the circumcircle of the triangle  $CPD$ . Prove that  $|OA| = |OB|$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13384,"x":0.16955,"p":[[0,64,0.0,0.1517,0.04973,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.15179,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.15152,0.03466,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.13839,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.16955,0.06625,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.15178,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.13384,0.0612,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.125,"x":0.15625,"p":[[0,58,0.0,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,2,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.13839,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.14268,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.1383,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"efc1b9b0d711f98f","q":"Let circles $\\Gamma_{1}$ and $\\Gamma_{2}$, with centers $O_{1}$ and $O_{2}$ respectively, intersect at two distinct points $A$ and $B$. The line $O_{1} A$ intersects $\\Gamma_{2}$ again at $C$ and the line $O_{2} A$ intersects $\\Gamma_{1}$ again at $D$. The line through $B$ parallel to $A D$ intersects $\\Gamma_{1}$ again at $E$. Suppose that $O_{1} A$ is parallel to $D E$. Prove that $C D$ is perpendicular to $O_{2} C$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.08482,"x":0.30357,"p":[[0,113,0.0,0.16071,0.15465,0.0,0.14286,0.2857,0.0,0.42857,13,0,1,13,0,6,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,113,0.0354,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,113,0.0708,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,113,0.1062,0.25446,0.08552,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,113,0.1416,0.25893,0.10374,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,0,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,113,0.177,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,113,0.2124,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,113,0.2478,0.30357,0.13716,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,113,0.2832,0.25892,0.12595,0.2857,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,0,0,0,24,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,113,0.3186,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,113,0.354,0.27232,0.09006,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,1,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,113,0.3894,0.27223,0.05507,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,113,0.4248,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,113,0.4602,0.22768,0.12807,0.14289,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,3,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,113,0.4956,0.25001,0.09449,0.2857,0.28571,0.28571,0.0,0.286,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,113,0.531,0.23214,0.10564,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,113,0.5664,0.24553,0.12492,0.2857,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,2,0,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[68,113,0.6018,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,113,0.6372,0.24553,0.10853,0.2857,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,0,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,113,0.6726,0.25892,0.10972,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,113,0.708,0.20535,0.12339,0.10714,0.2857,0.2857,0.0,0.28571,8,0,0,8,0,2,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,113,0.7434,0.20536,0.13333,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,1,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,113,0.7788,0.20089,0.14664,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,1,0,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[92,113,0.8142,0.20982,0.11836,0.14286,0.2857,0.28571,0.0,0.28571,7,0,0,7,0,3,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,113,0.8496,0.23214,0.11152,0.25,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,3,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,113,0.885,0.18303,0.13474,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,113,0.9204,0.14732,0.14933,0.0,0.14285,0.28571,0.0,0.42857,16,0,0,16,0,0,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,113,0.9558,0.09822,0.12078,0.0,0.0,0.1786,0.0,0.28571,18,0,0,18,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,113,0.9912,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[113,113,1.0,0.09821,0.13092,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,2,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.28125,"p":[[0,122,0.0,0.17848,0.14727,0.0,0.21428,0.28571,0.0,0.42857,11,0,4,11,0,5,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,122,0.0328,0.24553,0.09606,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,122,0.0656,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,122,0.0984,0.25893,0.07523,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,122,0.1311,0.28125,0.13592,0.2857,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[20,122,0.1639,0.25892,0.09061,0.2857,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,1,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,122,0.1967,0.25446,0.09932,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,122,0.2295,0.25,0.09449,0.28571,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,122,0.2623,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,122,0.2951,0.25,0.09449,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,122,0.3279,0.27233,0.05486,0.2857,0.28571,0.28571,0.0,0.286,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,122,0.3607,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,122,0.3934,0.26785,0.07783,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,122,0.4262,0.22321,0.11811,0.2857,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,122,0.459,0.20089,0.12807,0.0,0.28571,0.28571,0.0,0.28571,9,0,0,9,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,122,0.4918,0.16964,0.14032,0.0,0.2857,0.28571,0.0,0.28571,13,0,0,13,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,122,0.5246,0.16964,0.14913,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,2,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,122,0.5574,0.16964,0.1448,0.0,0.2857,0.28571,0.0,0.4286,13,0,0,13,0,1,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,122,0.5902,0.16518,0.15198,0.0,0.2143,0.28571,0.0,0.57143,13,0,0,13,0,3,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,122,0.623,0.14732,0.14053,0.0,0.2143,0.28571,0.0,0.28571,15,0,0,15,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,122,0.6557,0.10714,0.13363,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,122,0.6885,0.11161,0.15458,0.0,0.0,0.28571,0.0,0.57143,20,0,0,20,0,1,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,122,0.7213,0.12053,0.16015,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,2,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[92,122,0.7541,0.08929,0.13716,0.0,0.0,0.28571,0.0,0.42857,22,0,0,22,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,122,0.7869,0.08929,0.14175,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,4,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[100,122,0.8197,0.10268,0.16065,0.0,0.0,0.2857,0.0,0.57143,22,0,0,22,0,0,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[104,122,0.8525,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,122,0.8852,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,122,0.918,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,122,0.9508,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,122,0.9836,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[122,122,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f04f14b9926d20f6","q":"Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.","t":[{"b":5,"e":1.0,"k":"volatile","v":0.83036,"x":1.0,"p":[[0,2,0.0,0.83036,0.35434,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[2,2,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.14286,"k":"falling","v":0.15625,"x":0.89732,"p":[[0,29,0.0,0.89732,0.27254,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,29,0.1379,0.86607,0.31326,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[8,29,0.2759,0.23652,0.26395,0.14286,0.14286,0.14286,0.0,1.0,3,3,0,3,0,22,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[12,29,0.4138,0.15625,0.11495,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,26,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,29,0.5517,0.20536,0.2141,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,29,0.6897,0.16964,0.08328,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,29,0.8276,0.17857,0.12877,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[28,29,0.9655,0.16955,0.12597,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[29,29,1.0,0.17411,0.16262,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"ff40b836051cf997","q":"Let points $A_{1}, A_{2}$, and $A_{3}$ lie on the circle $\\Gamma$ in counter-clockwise order, and let $P$ be a point in the same plane. For $i \\in\\{1,2,3\\}$, let $\\tau_{i}$ denote the counter-clockwise rotation of the plane centred at $A_{i}$, where the angle of the rotation is equal to the angle at vertex $A_{i}$ in $\\triangle A_{1} A_{2} A_{3}$. Further, define $P_{i}$ to be the point $\\tau_{i+2}\\left(\\tau_{i}\\left(\\tau_{i+1}(P)\\right)\\right)$, where indices are taken modulo 3 (i.e., $\\tau_{4}=\\tau_{1}$ and $\\tau_{5}=\\tau_{2}$ ).\n\nProve that the radius of the circumcircle of $\\triangle P_{1} P_{2} P_{3}$ is at most the radius of $\\Gamma$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13839,"x":0.21875,"p":[[0,49,0.0,0.16072,0.11152,0.14286,0.14286,0.14286,0.0,0.71429,2,0,1,2,0,27,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,49,0.0816,0.21875,0.2082,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,49,0.1633,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.18304,0.15251,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,49,0.5714,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.14705,0.02492,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.12938,"x":0.20982,"p":[[0,69,0.0,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,2,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.20982,0.2172,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,24,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[8,69,0.1159,0.20089,0.1551,0.14286,0.14286,0.17857,0.14286,1.0,0,1,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,69,0.1739,0.18741,0.15748,0.14286,0.14286,0.14287,0.0,1.0,1,1,0,1,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,69,0.2319,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.16063,0.05925,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.12938,0.05484,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.15598,0.04173,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.15178,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.16072,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.1383,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.16519,0.07241,0.14286,0.14286,0.14286,0.0,0.286,2,0,0,2,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e2a11e9d8bf686be","q":"Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers  $ k_1, k_2, \\ldots , k_n$ . Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with  $ k_1$  times the number on the red card plus the number on the blue card, and replaces the number on the blue card with the number originally on the red card. He repeats this process with number  $ k_2$ . (That is, he replaces the number on the red card with  $ k_2$  times the number now on the red card plus the number now on the blue card, and replaces the number on the blue card with the number that was just placed on the red card.) He then repeats this process with each of the numbers  $ k_3, \\ldots k_n$ , in this order. After has has gone through the sequence of integers, Mr. Fat then gives the cards back to Ms. Taf. How many times must Ms. Taf submit the red and blue cards in order to be able to determine the sequence of integers  $ k_1, k_2, \\ldots k_n$ ?","t":[{"b":1,"e":0.857,"k":"flat","v":0.74107,"x":0.91071,"p":[[0,20,0.0,0.83928,0.20124,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,5,0,16],[4,20,0.2,0.79017,0.17852,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,14,0,0,6,0,9],[8,20,0.4,0.85714,0.24223,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,5,0,0,2,0,21],[12,20,0.6,0.74107,0.21852,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,12,0,0,5,0,8],[16,20,0.8,0.87946,0.17896,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,2,0,20],[20,20,1.0,0.91071,0.10564,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17]]},{"b":6,"e":0.4286,"k":"falling","v":0.37054,"x":0.87054,"p":[[0,23,0.0,0.80804,0.23037,0.71429,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,5,0,14],[4,23,0.1739,0.85713,0.19235,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,18],[8,23,0.3478,0.87054,0.13997,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,4,0,16],[12,23,0.5217,0.71874,0.20667,0.67857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,14,0,0,3,0,7],[16,23,0.6957,0.69197,0.25281,0.42859,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,5,0,0,1,0,0,7,0,0,7,0,7],[20,23,0.8696,0.41517,0.13052,0.39286,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,20,0,0,1,0,0,3,0,0,0,0,0],[23,23,1.0,0.37054,0.08645,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,12,0,0,18,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"8f3b1f886dedf34c","q":"On a circle  $n \\geq 1$  real numbers are written, their sum is  $n-1$ . Prove that one can denote these numbers as  $x_1, x_2, ..., x_n$  consecutively, starting from a number and moving clockwise,  such that for any  $k$  ( $1\\leq k \\leq n$ )  $ x_1 + x_2+...+x_k \\geq k-1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,20,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,20,0.2,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,20,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.71429,"k":"flat","v":0.97767,"x":0.98661,"p":[[0,11,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,11,0.3636,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,11,0.7273,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[11,11,1.0,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]}]},{"i":"46e2a90204f972d9","q":"Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^{n}$ all of whose digits are odd.","t":[{"b":2,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,32,0.0,0.91964,0.24468,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[4,32,0.125,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,10,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,10,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,10,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[10,10,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"19cdb63caf8f6139","q":"On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most  $ k$  times. What is the minimum  $ k$  so that we can make all the numbers on the circle equal?","t":[{"b":0,"e":0.71429,"k":"rising","v":0.23205,"x":0.71875,"p":[[0,81,0.0,0.53125,0.2321,0.42859,0.57143,0.60714,0.14286,0.85714,0,0,0,0,0,6,0,0,1,0,0,3,0,0,14,0,0,2,0,0,6,0,0],[4,81,0.0494,0.58927,0.31084,0.28571,0.71429,0.85714,0.0,1.0,2,2,0,2,0,5,0,0,3,0,0,0,0,0,2,0,0,9,0,0,9,0,2],[8,81,0.0988,0.62491,0.32697,0.14286,0.71429,0.85714,0.14,1.0,0,4,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,7,0,0,11,0,4],[12,81,0.1481,0.58482,0.29093,0.28571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,6,0,0,4,0,0,1,0,0,4,0,0,7,0,0,7,0,3],[16,81,0.1975,0.71428,0.22588,0.67857,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,5,0,0,18,0,1],[20,81,0.2469,0.64286,0.27433,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,6,0,0,1,0,0,0,0,0,2,0,0,13,0,0,7,0,3],[24,81,0.2963,0.68303,0.27603,0.5,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,0,0,0,3,0,0,4,0,0,12,0,5],[28,81,0.3457,0.60268,0.32485,0.25,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,8,0,0,3,0,0,1,0,0,0,0,0,6,0,0,10,0,4],[32,81,0.3951,0.45089,0.29474,0.14286,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,11,0,0,6,0,0,1,0,0,2,0,0,6,0,0,5,0,1],[36,81,0.4444,0.40177,0.27764,0.14286,0.28571,0.60714,0.14286,1.0,0,1,0,0,0,13,0,0,6,0,0,0,0,0,5,0,0,4,0,0,3,0,1],[40,81,0.4938,0.39732,0.30875,0.14286,0.2857,0.71429,0.14286,1.0,0,2,0,0,0,15,0,0,6,0,0,0,0,0,1,0,0,4,0,0,4,0,2],[44,81,0.5432,0.35268,0.29663,0.14286,0.14286,0.60712,0.14286,1.0,0,1,0,0,0,18,0,0,5,0,0,0,0,0,1,0,0,2,0,0,5,0,1],[48,81,0.5926,0.40624,0.31965,0.14286,0.2857,0.71429,0.0,1.0,1,2,0,1,0,14,0,0,5,0,0,0,0,0,2,0,0,3,0,0,5,0,2],[52,81,0.642,0.30357,0.2519,0.14286,0.14286,0.32143,0.14286,1.0,0,1,0,0,0,19,0,0,5,0,0,2,0,0,1,0,0,2,0,0,2,0,1],[56,81,0.6914,0.23205,0.15877,0.14286,0.14288,0.28571,0.0,0.85714,2,0,0,2,0,15,0,0,12,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[60,81,0.7407,0.62054,0.21609,0.57143,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,1,0,0,13,0,0,7,0,0,5,0,2],[64,81,0.7901,0.67411,0.20589,0.57143,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,11,0,0,4,0,0,9,0,3],[68,81,0.8395,0.71426,0.13364,0.57143,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,8,0,2],[72,81,0.8889,0.71875,0.16554,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,9,0,0,11,0,2],[76,81,0.9383,0.70531,0.11815,0.57143,0.71429,0.74996,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,13,0,0,7,0,1],[80,81,0.9877,0.71875,0.12103,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,15,0,0,10,0,0],[81,81,1.0,0.68749,0.10973,0.57143,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,12,0,0,7,0,0]]},{"b":1,"e":0.1429,"k":"falling","v":0.16964,"x":0.6875,"p":[[0,46,0.0,0.52679,0.2299,0.42857,0.57143,0.57143,0.0,0.85714,1,0,1,1,0,5,0,0,0,0,0,3,0,0,16,0,0,2,0,0,5,0,0],[4,46,0.087,0.6875,0.25111,0.67857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,0,0,0,2,0,0,10,0,0,11,0,3],[8,46,0.1739,0.58482,0.27975,0.28571,0.71429,0.74996,0.14286,1.0,0,2,0,0,0,6,0,0,4,0,0,0,0,0,3,0,0,11,0,0,6,0,2],[12,46,0.2609,0.55348,0.29409,0.24999,0.64286,0.85704,0.0,1.0,1,1,0,1,0,7,0,0,2,0,0,1,0,0,5,0,0,7,0,0,8,0,1],[16,46,0.3478,0.5,0.31943,0.14286,0.5,0.85714,0.14286,1.0,0,1,0,0,0,11,0,0,4,0,0,1,0,0,2,0,0,3,0,0,10,0,1],[20,46,0.4348,0.48661,0.33476,0.14286,0.50001,0.85714,0.0,1.0,2,2,0,2,0,10,0,0,3,0,0,1,0,0,1,0,0,6,0,0,7,0,2],[24,46,0.5217,0.50893,0.35163,0.14286,0.64286,0.85714,0.0,1.0,2,3,1,2,0,11,0,0,1,0,0,1,0,0,1,0,0,5,0,0,8,0,3],[28,46,0.6087,0.46875,0.30978,0.14286,0.42859,0.71429,0.0,1.0,2,1,0,2,0,9,0,0,3,0,0,3,0,0,2,0,0,6,0,0,6,0,1],[32,46,0.6957,0.43732,0.30934,0.14286,0.28571,0.71429,0.14,1.0,0,1,0,0,0,14,0,0,4,0,0,0,0,0,1,0,0,7,0,0,5,0,1],[36,46,0.7826,0.45089,0.32164,0.14286,0.28571,0.85714,0.0,1.0,1,1,0,1,0,12,0,0,4,0,0,1,0,0,2,0,0,3,0,0,8,0,1],[40,46,0.8696,0.43304,0.32827,0.14286,0.2143,0.71429,0.0,1.0,1,2,0,1,0,15,0,0,1,0,0,1,0,0,2,0,0,5,0,0,5,0,2],[44,46,0.9565,0.16964,0.14032,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,21,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[46,46,1.0,0.19634,0.17037,0.14286,0.14286,0.14287,0.0,0.85714,2,0,0,2,0,24,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,0]]}]},{"i":"7fd072fcbb5413bc","q":"Prove that there are 100 natural number  $a_1 < a_2 < ... < a_{99} < a_{100}$  (  $ a_i < 10^6$ ) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ? $A = \\{a_1 , a_2 ,... , a_{99} ,a_{100}\\}$  $2A = \\{2a_i \\vert 1\\leq i\\leq 100\\}$  $A+A = \\{a_i + a_j \\vert 1\\leq i<j\\leq 100\\}$  $A + 2A = \\{a_i + 2a_j \\vert 1\\leq i,j\\leq 100\\}$  $2A + 2A = \\{2a_i + 2a_j \\vert 1\\leq i<j\\leq 100\\}$ (20 ponits )","t":[{"b":2,"e":0.14286,"k":"falling","v":0.10259,"x":0.4732,"p":[[0,63,0.0,0.4732,0.43218,0.0,0.42857,1.0,0.0,1.0,12,11,0,12,0,1,0,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,11],[4,63,0.0635,0.31696,0.35126,0.14286,0.14286,0.39285,0.0,1.0,5,4,1,5,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,4],[8,63,0.127,0.33027,0.33209,0.14286,0.14286,0.46429,0.0,1.0,3,3,0,3,0,19,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,3],[12,63,0.1905,0.29911,0.33571,0.14286,0.14286,0.42857,0.0,1.0,6,5,0,6,0,16,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,5],[16,63,0.254,0.36607,0.3862,0.14286,0.14286,0.75,0.0,1.0,6,7,0,6,0,15,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,7],[20,63,0.3175,0.26786,0.30462,0.14286,0.14286,0.1786,0.0,1.0,4,4,0,4,0,20,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[24,63,0.381,0.15161,0.12847,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,24,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,63,0.4444,0.20089,0.24964,0.0,0.14286,0.14286,0.0,1.0,9,2,0,9,0,16,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[32,63,0.5079,0.23659,0.25903,0.14286,0.14286,0.21429,0.0,1.0,6,2,0,6,0,18,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,2],[36,63,0.5714,0.1517,0.19542,0.105,0.14286,0.14286,0.0,1.0,8,1,0,8,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[40,63,0.6349,0.20536,0.24206,0.14286,0.14286,0.14286,0.0,1.0,5,2,0,5,0,22,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[44,63,0.6984,0.20089,0.21683,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,20,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[48,63,0.7619,0.16518,0.22335,0.14286,0.14286,0.14286,0.0,1.0,7,2,0,7,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,63,0.8254,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,63,0.8889,0.11152,0.1114,0.0,0.14286,0.14286,0.0,0.42857,13,0,1,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.10259,0.1299,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[63,63,1.0,0.12052,0.15194,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,18,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.08482,"x":0.62053,"p":[[0,84,0.0,0.62053,0.37391,0.2857,0.71429,1.0,0.0,1.0,4,10,0,4,0,3,0,0,4,0,0,1,0,0,0,0,0,5,0,0,5,0,10],[4,84,0.0476,0.31697,0.33452,0.14286,0.14286,0.42857,0.0,1.0,5,5,0,5,0,16,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,5],[8,84,0.0952,0.33473,0.3547,0.14286,0.14286,0.60714,0.0,1.0,5,5,0,5,0,17,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,5],[12,84,0.1429,0.15625,0.15714,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,84,0.1905,0.15615,0.13994,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,24,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,84,0.2381,0.22321,0.26229,0.10714,0.14286,0.17857,0.0,1.0,8,2,0,8,0,16,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[24,84,0.2857,0.1384,0.13592,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,21,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,84,0.3333,0.12946,0.19019,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,84,0.381,0.13393,0.16342,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,20,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[36,84,0.4286,0.16072,0.17035,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[40,84,0.4762,0.1517,0.18537,0.105,0.14286,0.14286,0.0,1.0,8,1,0,8,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[44,84,0.5238,0.13831,0.12619,0.105,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,21,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,84,0.5714,0.16518,0.1992,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,18,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[52,84,0.619,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,19,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,84,0.6667,0.14732,0.15351,0.0,0.14286,0.14287,0.0,0.57143,10,0,0,10,0,17,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[60,84,0.7143,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,84,0.7619,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,84,0.8095,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,84,0.8571,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,84,0.9048,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,84,0.9524,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,84,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b845d12555d4f656","q":"Prove that there exists an infinite set of points  $$ \\ldots, P_{-3}, P_{-2}, P_{-1}, P_{0}, P_{1}, P_{2}, P_{3}, \\ldots $$  in the plane with the following property: For any three distinct integers $a, b$, and $c$, points $P_{a}, P_{b}$, and $P_{c}$ are collinear if and only if $a+b+c=2014$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,33,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,33,0.1212,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,33,0.2424,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,33,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,33,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,33,0.8485,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,33,0.9697,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,37,0.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[4,37,0.1081,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]}]},{"i":"d723db624c3bf435","q":"Let $p$ be an odd prime number and $a, b, c$ be integers so that the integers\n\n$$\na^{2023}+b^{2023}, \\quad b^{2024}+c^{2024}, \\quad c^{2025}+a^{2025}\n$$\n\nare all divisible by $p$. Prove that $p$ divides each of $a, b$, and $c$.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.34821,"x":0.43303,"p":[[0,25,0.0,0.39284,0.19884,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,19,0,0,7,0,0,1,0,0,2,0,0,0,0,2],[4,25,0.16,0.40624,0.19596,0.28571,0.28571,0.42858,0.14286,1.0,0,1,0,0,0,2,0,0,15,0,0,9,0,0,1,0,0,3,0,0,1,0,1],[8,25,0.32,0.43303,0.2382,0.28571,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,1,0,0,16,0,0,9,0,0,1,0,0,1,0,0,0,0,4],[12,25,0.48,0.36604,0.17832,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,18,0,0,6,0,0,2,0,0,2,0,0,0,0,1],[16,25,0.64,0.36161,0.15966,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,18,0,0,9,0,0,1,0,0,1,0,0,0,0,1],[20,25,0.8,0.40625,0.1992,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,18,0,0,10,0,0,0,0,0,1,0,0,1,0,2],[24,25,0.96,0.39286,0.11294,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,16,0,0,1,0,0,2,0,0,0,0,0],[25,25,1.0,0.34821,0.14699,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,6,0,0,0,0,0,1,0,0,0,0,1]]},{"b":5,"e":0.42857,"k":"flat","v":0.28562,"x":0.40178,"p":[[0,36,0.0,0.40178,0.23808,0.2857,0.28571,0.42857,0.0,1.0,1,3,0,1,0,2,0,0,16,0,0,7,0,0,1,0,0,2,0,0,0,0,3],[4,36,0.1111,0.37945,0.13173,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,16,0,0,1,0,0,2,0,0,0,0,0],[8,36,0.2222,0.36607,0.13803,0.28571,0.28571,0.42857,0.1429,1.0,0,1,0,0,0,1,0,0,16,0,0,14,0,0,0,0,0,0,0,0,0,0,1],[12,36,0.3333,0.35714,0.16367,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,22,0,0,6,0,0,0,0,0,2,0,0,0,0,1],[16,36,0.4444,0.35268,0.19228,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,2,0,0,23,0,0,4,0,0,0,0,0,1,0,0,0,0,2],[20,36,0.5556,0.34822,0.15126,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,17,0,0,10,0,0,1,0,0,0,0,0,0,0,1],[24,36,0.6667,0.28562,0.08002,0.2857,0.28571,0.28571,0.14,0.4286,0,0,0,0,0,5,0,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.375,0.19805,0.2857,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,20,0,0,5,0,0,2,0,0,1,0,0,0,0,2],[32,36,0.8889,0.35714,0.13363,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,19,0,0,10,0,0,0,0,0,1,0,0,1,0,0],[36,36,1.0,0.3661,0.18536,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,19,0,0,5,0,0,1,0,0,3,0,0,0,0,1]]}]},{"i":"0e528fc24ccf8465","q":"On a circle, Alina draws  $2019$  chords, the endpoints of which are all different. A point is considered *marked* if it is either\n\n   $\\text{(i)}$  one of the  $4038$  endpoints of a chord; or\n\n   $\\text{(ii)}$  an intersection point of at least two chords.\n\n     Alina labels each marked point. Of the  $4038$  points meeting criterion  $\\text{(i)}$ , Alina labels  $2019$  points with a  $0$  and the other  $2019$  points with a  $1$ . She labels each point meeting criterion  $\\text{(ii)}$  with an arbitrary integer (not necessarily positive).\n     Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with  $k$  marked points has  $k-1$  such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. \n     Alina finds that the  $N + 1$  yellow labels take each value  $0, 1, . . . , N$  exactly once. Show that at least one blue label is a multiple of  $3$ .\n(A chord is a line segment joining two different points on a circle.)","t":[{"b":0,"e":0.4286,"k":"flat","v":0.44642,"x":0.95088,"p":[[0,75,0.0,0.58034,0.43732,0.10714,0.85714,1.0,0.0,1.0,8,14,8,8,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0,3,0,14],[4,75,0.0533,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[8,75,0.1067,0.89286,0.21129,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,23],[12,75,0.16,0.84821,0.26711,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,2,0,22],[16,75,0.2133,0.87054,0.19678,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,4,0,20],[20,75,0.2667,0.85266,0.2328,0.82132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,0,4,0,20],[24,75,0.32,0.86157,0.21578,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,0,3,0,20],[28,75,0.3733,0.78571,0.26244,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,3,0,0,5,0,15],[32,75,0.4267,0.76786,0.28065,0.53572,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,3,0,0,3,0,16],[36,75,0.48,0.80356,0.26185,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,4,0,0,2,0,18],[40,75,0.5333,0.74548,0.24155,0.571,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,5,0,0,5,0,11],[44,75,0.5867,0.72765,0.22408,0.57132,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,3,0,0,6,0,9],[48,75,0.64,0.71427,0.21725,0.57132,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,8,0,0,4,0,8],[52,75,0.6933,0.80352,0.2308,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,2,0,0,7,0,14],[56,75,0.7467,0.69643,0.26426,0.4286,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,8,0,0,4,0,0,3,0,0,3,0,11],[60,75,0.8,0.58919,0.22232,0.42857,0.571,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,7,0,0,10,0,0,3,0,0,3,0,4],[64,75,0.8533,0.6339,0.28107,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,7,0,0,5,0,0,4,0,0,3,0,8],[68,75,0.9067,0.63833,0.19557,0.53539,0.64286,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,5,0,0,8,0,0,8,0,0,6,0,2],[72,75,0.96,0.5357,0.20203,0.42857,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,12,0,0,4,0,0,5,0,0,4,0,1],[75,75,1.0,0.44642,0.12241,0.42857,0.42857,0.4642,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,17,0,0,5,0,0,3,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.66071,"x":1.0,"p":[[0,92,0.0,0.66071,0.40367,0.2857,1.0,1.0,0.0,1.0,5,18,4,5,0,0,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,18],[4,92,0.0435,0.83481,0.26029,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,0,0,1,0,21],[8,92,0.087,0.85268,0.23551,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,19],[12,92,0.1304,0.80793,0.26173,0.67857,1.0,1.0,0.14,1.0,0,18,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,4,0,0,2,0,18],[16,92,0.1739,0.82587,0.27605,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,20],[20,92,0.2174,0.88839,0.17762,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,5,0,20],[24,92,0.2609,0.83034,0.23809,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,5,0,0,3,0,18],[28,92,0.3043,0.87052,0.18338,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,4,0,19],[32,92,0.3478,0.78118,0.24744,0.571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,1,0,0,6,0,14],[36,92,0.3913,0.83478,0.19925,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,3,0,17],[40,92,0.4348,0.75,0.26,0.57143,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,6,0,0,3,0,13],[44,92,0.4783,0.79466,0.22846,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,6,0,0,4,0,14],[48,92,0.5217,0.76786,0.22232,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,5,0,0,6,0,11],[52,92,0.5652,0.67409,0.30354,0.42857,0.64286,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,3,0,0,4,0,0,6,0,0,2,0,0,2,0,12],[56,92,0.6087,0.6875,0.25111,0.53571,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,3,0,0,4,0,9],[60,92,0.6522,0.71873,0.2435,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,6,0,0,3,0,10],[64,92,0.6957,0.90179,0.12595,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,18],[68,92,0.7391,0.80802,0.19761,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,7,0,12],[72,92,0.7826,0.85267,0.19063,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,4,0,17],[76,92,0.8261,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[80,92,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,92,0.913,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[88,92,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,92,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"219cdc9f5f995547","q":"Prove that there exist infinitely many pairs of relatively prime positive integers $a, b>1$ for which $a+b$ divides $a^{b}+b^{a}$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,43,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,43,0.093,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,43,0.186,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,43,0.2791,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,43,0.3721,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,43,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,43,0.5581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,43,0.6512,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,43,0.7442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,43,0.8372,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,43,0.9302,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[43,43,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,34,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,34,0.1176,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,34,0.2353,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,34,0.3529,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[16,34,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,34,0.5882,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,34,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[34,34,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"730f54107bb05c8e","q":"Prove that there is no function  $f:\\mathbb{R}_{\\ge0}\\rightarrow\\mathbb{R}$  satisfying: $f(x+y^2)\\ge f(x)+y$  for all two nonnegative real numbers  $x,y$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.90624,"x":0.98661,"p":[[0,26,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,26,0.1538,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,26,0.3077,0.90624,0.13651,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,14,0,16],[12,26,0.4615,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,26,0.6154,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[20,26,0.7692,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,26,0.9231,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[26,26,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.91518,"x":0.98661,"p":[[0,10,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,10,0.4,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,10,0.8,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[10,10,1.0,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17]]}]},{"i":"f8e11cd832b4a212","q":"There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).\nAfter some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? \n*I. Izmest\u2019ev*","t":[{"b":1,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,30,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,30,0.1333,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,30,0.2667,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[12,30,0.4,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[16,30,0.5333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,30,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[30,30,1.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":3,"e":0.28571,"k":"falling","v":0.45089,"x":0.98214,"p":[[0,18,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,18,0.2222,0.72768,0.29744,0.42857,0.85714,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,10,0,0,1,0,0,3,0,0,0,0,16],[8,18,0.4444,0.71874,0.31438,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,9,0,0,1,0,0,0,0,0,2,0,16],[12,18,0.6667,0.67411,0.27718,0.42857,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,14,0,0,0,0,0,3,0,0,1,0,12],[16,18,0.8889,0.57142,0.26964,0.42857,0.42857,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,14,0,0,1,0,0,5,0,0,0,0,7],[18,18,1.0,0.45089,0.11356,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,26,0,0,3,0,0,0,0,0,0,0,1]]}]},{"i":"6b2d5b4d8baa0e99","q":"Prove that there exists exactly one polynomial $P(x)$ with real coefficients for which the polynomial\n\n$$\n(x+y)^{1000}-P(x)-P(y)\n$$\n\nis divisible by the polynomial $x y-x-y$.\n\n(Dusan Dukic)","t":[{"b":4,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,55,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,55,0.0727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,55,0.1455,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,55,0.2182,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,55,0.2909,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,55,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,55,0.4364,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,55,0.5818,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[36,55,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,55,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,55,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,55,0.8727,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,55,0.9455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,43,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,43,0.093,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,43,0.186,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,43,0.2791,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,43,0.3721,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[20,43,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,43,0.5581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,43,0.6512,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,43,0.7442,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,43,0.8372,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,43,0.9302,0.97766,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[43,43,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"a9df25911088ef7e","q":"Three natural numbers  $ a,b,c $  with  $ \\gcd (a,b) =1 $  define in the Diophantine plane a line  $ d: ax+by-c=0. $  Prove that:**a)** the distance between any two points from  $ d $  is at least  $ \\sqrt{a^2+b^2} . $ **b)** the restriction of  $ d $  to the first quadrant of the Diophantine plane is a finite line having at most  $ 1+\\frac{c}{ab} $  elements.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.83929,"x":0.91964,"p":[[0,22,0.0,0.83929,0.12242,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,19,0,6],[4,22,0.1818,0.90177,0.14917,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,19],[8,22,0.3636,0.91518,0.12299,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,20],[12,22,0.5455,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[16,22,0.7273,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[20,22,0.9091,0.89732,0.11425,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,15],[22,22,1.0,0.90625,0.10479,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15]]},{"b":7,"e":1.0,"k":"flat","v":0.81249,"x":0.89285,"p":[[0,28,0.0,0.83927,0.14619,0.82143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,15,0,9],[4,28,0.1429,0.83036,0.17655,0.67857,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,6,0,14],[8,28,0.2857,0.85713,0.14728,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,10,0,13],[12,28,0.4286,0.87946,0.16016,0.85713,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,8,0,17],[16,28,0.5714,0.81249,0.16148,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,11,0,9],[20,28,0.7143,0.87486,0.09965,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,9],[24,28,0.8571,0.89272,0.08776,0.85714,0.85714,1.0,0.71,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11],[28,28,1.0,0.89285,0.08748,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11]]}]},{"i":"06c5f124e369ba2b","q":"There exists a unique positive integer  $a$  for which the sum \\[U=\\sum_{n=1}^{2023}\\left\\lfloor\\dfrac{n^{2}-na}{5}\\right\\rfloor\\] is an integer strictly between  $-1000$  and  $1000$ . For that unique  $a$ , find  $a+U$ .\n\n(Note that  $\\lfloor x\\rfloor$  denotes the greatest integer that is less than or equal to  $x$ .)","t":[{"b":3,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,131,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,131,0.0305,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,131,0.0611,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,131,0.0916,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,131,0.1221,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,131,0.1527,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,131,0.1832,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,131,0.2137,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,131,0.2443,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,131,0.2748,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,131,0.3053,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,131,0.3359,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,131,0.3664,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,131,0.3969,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,131,0.4275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,131,0.458,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,131,0.4885,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,131,0.5191,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,131,0.5496,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,131,0.5802,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,131,0.6107,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,131,0.6412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,131,0.6718,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,131,0.7023,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,131,0.7328,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,131,0.7634,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,131,0.7939,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,131,0.8244,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,131,0.855,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,131,0.8855,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[120,131,0.916,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,131,0.9466,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[128,131,0.9771,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[131,131,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":4,"e":0.85714,"k":"flat","v":0.86161,"x":1.0,"p":[[0,149,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,149,0.0268,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,149,0.0537,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,149,0.0805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,149,0.1074,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,149,0.1342,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,149,0.1611,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,149,0.1879,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,149,0.2148,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,149,0.2416,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,149,0.2685,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,149,0.2953,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,149,0.3221,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,149,0.349,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,149,0.3758,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,149,0.4027,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,149,0.4295,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,149,0.4564,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[72,149,0.4832,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[76,149,0.5101,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[80,149,0.5369,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[84,149,0.5638,0.91964,0.13333,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,20],[88,149,0.5906,0.93304,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[92,149,0.6174,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[96,149,0.6443,0.91518,0.15093,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,9,0,20],[100,149,0.6711,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[104,149,0.698,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,18],[108,149,0.7248,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[112,149,0.7517,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[116,149,0.7785,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[120,149,0.8054,0.90178,0.14033,0.85714,1.0,1.0,0.5714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,19],[124,149,0.8322,0.89286,0.13832,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,17],[128,149,0.8591,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[132,149,0.8859,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[136,149,0.9128,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[140,149,0.9396,0.89732,0.12492,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,16],[144,149,0.9664,0.90625,0.11071,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[148,149,0.9933,0.90179,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,15],[149,149,1.0,0.86161,0.10999,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,16,0,9]]}]},{"i":"e7099aac129f95ec","q":"There is a complex number  $ z$  with imaginary part  $ 164$  and a positive integer  $ n$  such that \r\n\\[ \\frac {z}{z \\plus{} n} \\equal{} 4i.\r\n\\]Find  $ n$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.74107,"x":0.86607,"p":[[0,22,0.0,0.86607,0.18536,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,3,0,19],[4,22,0.1818,0.83036,0.1729,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,1,0,15],[8,22,0.3636,0.82589,0.18117,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,1,0,15],[12,22,0.5455,0.85268,0.1838,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,1,0,18],[16,22,0.7273,0.79911,0.22548,0.71429,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,10,0,0,1,0,15],[20,22,0.9091,0.75,0.17128,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,17,0,0,1,0,8],[22,22,1.0,0.74107,0.1448,0.71429,0.71429,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,19,0,0,3,0,5]]},{"b":2,"e":1.0,"k":"flat","v":0.79464,"x":0.91518,"p":[[0,40,0.0,0.91518,0.16311,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,1,0,24],[4,40,0.1,0.89285,0.17128,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,21],[8,40,0.2,0.85714,0.17496,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,3,0,17],[12,40,0.3,0.83929,0.16269,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,1,0,15],[16,40,0.4,0.88393,0.16146,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,1,0,20],[20,40,0.5,0.90179,0.17655,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,0,0,24],[24,40,0.6,0.80804,0.18766,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,1,0,14],[28,40,0.7,0.79464,0.2111,0.67857,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,9,0,0,0,0,15],[32,40,0.8,0.83036,0.16146,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,0,1,0,14],[36,40,0.9,0.85268,0.16164,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,2,0,16],[40,40,1.0,0.87054,0.16506,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,0,0,19]]}]},{"i":"eff03b0f6c1a4c3a","q":"Let be two matrices  $ A,N\\in\\mathcal{M}_2(\\mathbb{R}) $  that commute and such that  $ N $  is nilpotent. Show that:**a)**  $ \\det (A+N)=\\det (A) $ **b)** if  $ A $  is general linear, then the matrix  $ A+N $  is invertible and  $ (A+N)^{-1}=(A-N)A^{-2} . $","t":[{"b":1,"e":1.0,"k":"flat","v":0.91517,"x":0.98214,"p":[[0,19,0.0,0.92857,0.15972,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[4,19,0.2105,0.91517,0.1551,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,23],[8,19,0.4211,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[12,19,0.6316,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[16,19,0.8421,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[19,19,1.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30]]},{"b":2,"e":1.0,"k":"flat","v":0.87054,"x":0.98214,"p":[[0,31,0.0,0.87054,0.19019,0.57143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,2,0,21],[4,31,0.129,0.91518,0.15916,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24],[8,31,0.2581,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[12,31,0.3871,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[16,31,0.5161,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[20,31,0.6452,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[24,31,0.7742,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,31,0.9032,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[31,31,1.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30]]}]},{"i":"ed2003b753dbe002","q":"There is a regular  $17$ -gon  $\\mathcal{P}$  and its circumcircle  $\\mathcal{Y}$  on the plane. \nThe vertices of  $\\mathcal{P}$  are coloured in such a way that  $A,B \\in \\mathcal{P}$  are of diff\u000berent colour, if the shorter arc connecting  $A$  and  $B$  on  $\\mathcal{Y}$  has  $2^k+1$  vertices, for some  $k \\in \\mathbb{N},$  including  $A$  and  $B.$  \nWhat is the least number of colours which suffi\u000eces?","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13831,"x":0.22768,"p":[[0,44,0.0,0.22768,0.14223,0.14286,0.28571,0.28571,0.0,0.57143,5,0,1,5,0,8,0,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,44,0.0909,0.19196,0.17717,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,13,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[8,44,0.1818,0.16072,0.12242,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,23,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,44,0.2727,0.13831,0.09095,0.14214,0.14286,0.14286,0.0,0.286,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.19643,0.11152,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,22,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,44,0.4545,0.17856,0.15149,0.14286,0.14286,0.17857,0.0,0.57143,6,0,0,6,0,18,0,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[24,44,0.5455,0.22768,0.16698,0.14286,0.14288,0.28571,0.0,0.71429,3,0,1,3,0,15,0,0,11,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[28,44,0.6364,0.19647,0.16662,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,10,0,0,11,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[32,44,0.7273,0.21428,0.17128,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,16,0,0,9,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[36,44,0.8182,0.17402,0.16264,0.0,0.14286,0.2857,0.0,0.71429,9,0,1,9,0,12,0,0,9,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[40,44,0.9091,0.22759,0.16702,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,18,0,0,8,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[44,44,1.0,0.20088,0.17804,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,18,0,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.571,"k":"flat","v":0.16508,"x":0.21418,"p":[[0,22,0.0,0.2009,0.20472,0.10714,0.14286,0.28571,0.0,1.0,8,1,2,8,0,11,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[4,22,0.1818,0.16508,0.10776,0.14286,0.14286,0.14287,0.0,0.571,4,0,0,4,0,21,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,22,0.3636,0.20974,0.14723,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,20,0,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,22,0.5455,0.21418,0.17497,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,13,0,0,8,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[16,22,0.7273,0.20982,0.15966,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,12,0,0,9,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[20,22,0.9091,0.20536,0.15542,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,14,0,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[22,22,1.0,0.20982,0.14719,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,14,0,0,12,0,0,0,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"c5a2fc069c19ca37","q":"Let be the sequence  $ \\left( I_n \\right)_{n\\ge 1} $  defined as  $ I_n=\\int_0^1 \\frac{x^n}{\\sqrt{x^{2n} +1}} dx . $ **a)** Show that  $ \\left( I_n \\right)_{n\\ge 1} $  converges to  $ 0. $ **b)** Calculate  $ \\lim_{m\\to\\infty } m\\cdot I_m. $ **c)** Prove that the sequence  $ \\left( n\\left( -n\\cdot I_n +\\lim_{m\\to\\infty } m\\cdot I_m \\right) \\right)_{n\\ge 1} $  is convergent.","t":[{"b":2,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,24,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,24,0.1667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,24,0.3333,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,22,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,22,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,22,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4be7a8be9464a1eb","q":"Triangle  $ABC$  has  $BC = 7, CA = 8, AB = 9$ . Let  $D, E, F$  be the midpoints of  $BC, CA, AB$  respectively, and let  $G$  be the intersection of  $AD$  and  $BE$ .  $G'$  is the reflection of  $G$  across  $D$ . Let  $G'E$  meet  $CG$  at  $P$ , and let  $G'F$  meet  $BG$  at  $Q$ . Determine the area of  $APG'Q$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.89732,"x":1.0,"p":[[0,128,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,128,0.0312,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,128,0.0625,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,128,0.0938,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,128,0.125,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,128,0.1562,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,128,0.1875,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,128,0.2188,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,128,0.25,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,128,0.2812,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,128,0.3125,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,128,0.3438,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[48,128,0.375,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,128,0.4062,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,128,0.4375,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[60,128,0.4688,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[64,128,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,128,0.5312,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[72,128,0.5625,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[76,128,0.5938,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[80,128,0.625,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[84,128,0.6562,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,128,0.6875,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[92,128,0.7188,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[96,128,0.75,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[100,128,0.7812,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[104,128,0.8125,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[108,128,0.8438,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[112,128,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,128,0.9062,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[120,128,0.9375,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[124,128,0.9688,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[128,128,1.0,0.89732,0.11425,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16]]},{"b":7,"e":0.85714,"k":"flat","v":0.79911,"x":1.0,"p":[[0,101,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,101,0.0396,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,101,0.0792,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[12,101,0.1188,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,101,0.1584,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,101,0.198,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,101,0.2376,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,101,0.2772,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,101,0.3168,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[36,101,0.3564,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[40,101,0.396,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,101,0.4356,0.90625,0.16982,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,23],[48,101,0.4752,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,101,0.5149,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,101,0.5545,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,101,0.5941,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,101,0.6337,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[68,101,0.6733,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[72,101,0.7129,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,101,0.7525,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,101,0.7921,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[84,101,0.8317,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[88,101,0.8713,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[92,101,0.9109,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,101,0.9505,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[100,101,0.9901,0.79911,0.12807,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,4,0,8],[101,101,1.0,0.83482,0.11355,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,11,0,8]]}]},{"i":"f91c34ff1983b4a0","q":"Noah has to fit 8 species of animals into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accommodation. Prove that there is a way to assign the animals to their cages so that each species shares a cage with compatible species.","t":[{"b":5,"e":0.2857,"k":"flat","v":0.28571,"x":0.29018,"p":[[0,28,0.0,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.28572,5e-05,0.28571,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.28572,5e-05,0.28571,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.28571,"x":0.30803,"p":[[0,22,0.0,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,22,0.3636,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,22,0.5455,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.28571,1e-05,0.2857,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8cb8062152155dba","q":"Two circumferences  $\\omega_1$  and   $\\omega_2$   with centers  $O_1$  and  $O_2$  meet at  $J$  and  $M$ . The line  $BC$  is a common tangent such that  $B$  belongs to   $\\omega_1$   and C belongs to   $\\omega_2$  . The line  $BC$  meets  $O_1O_2$  at  $F$ . The line  $FM$  meets again   $\\omega_1$   and   $\\omega_2$   at  $ A$  and  $D$ , respectively. Point  $G$  is the intersection of  $AB$  and  $CD$ . Let  $O$  be the circumcenter of  $\\vartriangle AGD$ .\nProve that  $ \\angle OJM = 90^o$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.17857,"p":[[0,56,0.0,0.08027,0.10672,0.0,0.0,0.14286,0.0,0.4286,18,0,3,18,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.17857,0.15567,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,1,0,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.05348,0.10557,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.05804,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,52,0.0,0.07144,0.07987,0.0,0.0,0.14286,0.0,0.2857,17,0,1,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.10714,0.14286,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,4,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,52,0.1538,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,52,0.2308,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,52,0.3846,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ff5eca5c7d4c4452","q":"Two circles  ${{c} _ {1}}, \\, \\, {{c} _ {2}}$  pass through the center  $O$  of the circle  $c$  and touch it internally in points  $A$  and  $B$ , respectively. Prove that the line  $AB$  passes though a common point of circles  ${{c} _ {1}}, \\, \\, {{c} _ {2}} $ .","t":[{"b":0,"e":0.85714,"k":"rising","v":0.0,"x":0.87053,"p":[[0,33,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.06696,0.22011,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,33,0.2424,0.15179,0.32328,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,2],[12,33,0.3636,0.10714,0.29667,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[16,33,0.4848,0.10714,0.29667,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[20,33,0.6061,0.29018,0.42481,0.0,0.0,0.85714,0.0,1.0,21,6,0,21,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,6],[24,33,0.7273,0.3125,0.43512,0.0,0.0,0.85714,0.0,1.0,21,5,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,5],[28,33,0.8485,0.625,0.40838,0.0,0.85714,0.85714,0.0,1.0,9,6,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,6],[32,33,0.9697,0.87053,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5],[33,33,1.0,0.8616,0.12619,0.85714,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,22,0,7]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.2723,"p":[[0,48,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,48,0.0833,0.12943,0.27278,0.0,0.0,0.0,0.0,0.857,26,0,0,26,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0],[8,48,0.1667,0.18304,0.34669,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,2],[12,48,0.25,0.17857,0.34993,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,3],[16,48,0.3333,0.22768,0.37434,0.0,0.0,0.35714,0.0,1.0,22,3,0,22,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,3],[20,48,0.4167,0.21874,0.35888,0.0,0.0,0.35704,0.0,1.0,22,2,0,22,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,2],[24,48,0.5,0.17411,0.34577,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,2],[28,48,0.5833,0.2723,0.37686,0.0,0.0,0.6425,0.0,1.0,20,1,0,20,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,7,0,1],[32,48,0.6667,0.11161,0.26901,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0],[36,48,0.75,0.22321,0.36235,0.0,0.0,0.60714,0.0,1.0,23,1,0,23,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,1],[40,48,0.8333,0.21429,0.3312,0.0,0.0,0.57143,0.0,0.85714,22,0,0,22,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,3,0,0],[44,48,0.9167,0.21875,0.34439,0.0,0.0,0.60714,0.0,0.85714,22,0,0,22,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0],[48,48,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fac6c620a9c5ac20","q":"Prove that there doesn't exist any prime $p$ such that every power of $p$ is a palindrome (palindrome is a number that is read the same from the left as it is from the right; in particular, number that ends in one or more zeros cannot be a palindrome).","t":[{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.20089,"p":[[0,54,0.0,0.1116,0.13235,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,3,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.20089,0.14223,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,1,0,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.08035,0.11811,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.17857,0.13832,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,3,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.09375,0.13175,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.0267,0.06606,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.16963,"x":0.42857,"p":[[0,31,0.0,0.17411,0.15866,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,2,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,31,0.129,0.21875,0.15146,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,1,0,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,31,0.2581,0.16963,0.1801,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,5,0,0,8,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,31,0.3871,0.17411,0.18808,0.0,0.21428,0.28571,0.0,0.71429,15,0,0,15,0,1,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,31,0.5161,0.42857,0.15567,0.28571,0.42857,0.42858,0.28571,1.0,0,1,0,0,0,0,0,0,10,0,0,17,0,0,3,0,0,0,0,0,1,0,1],[20,31,0.6452,0.36161,0.09439,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[24,31,0.7742,0.36159,0.11283,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,12,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[28,31,0.9032,0.38392,0.11536,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,4,0,0,5,0,0,20,0,0,3,0,0,0,0,0,0,0,0],[31,31,1.0,0.37482,0.13275,0.28571,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,4,0,0,9,0,0,15,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"3fe3e68f9f571bf2","q":"We call a sequence of  $m$  consecutive integers a *friendly* sequence if its first term is divisible by  $1$ , the second by  $2$ , ..., the  $(m-1)^{th}$  by  $m-1$ , and in addition, the last term is divisible by  $m^2$ Does a friendly sequence exist for (a)  $m=20$  and (b)  $m=11$ ?","t":[{"b":4,"e":0.85714,"k":"flat","v":0.70967,"x":0.76786,"p":[[0,28,0.0,0.70967,0.14054,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,7,0,0,11,0,1],[4,28,0.1429,0.73214,0.15464,0.71429,0.78564,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,9,0,0,16,0,0],[8,28,0.2857,0.76339,0.1411,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,17,0,1],[12,28,0.4286,0.76786,0.13243,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,17,0,1],[16,28,0.5714,0.74554,0.1504,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,10,0,0,13,0,2],[20,28,0.7143,0.75446,0.1439,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,16,0,1],[24,28,0.8571,0.76339,0.11071,0.71429,0.78571,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,16,0,0],[28,28,1.0,0.76339,0.13175,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,18,0,0]]},{"b":5,"e":0.2857,"k":"falling","v":0.23214,"x":0.79018,"p":[[0,18,0.0,0.7098,0.14054,0.57143,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,0,12,0,1],[4,18,0.2222,0.79018,0.11837,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,23,0,0],[8,18,0.4444,0.75892,0.16147,0.57143,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,6,0,0,13,0,4],[12,18,0.6667,0.77677,0.147,0.57143,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,15,0,4],[16,18,0.8889,0.30803,0.15198,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,5,0,0,20,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[18,18,1.0,0.23214,0.09279,0.14286,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f4ea5be53a279fd1","q":"There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from the other one. Prove that there is a town such that all the remaining towns can be reached from it.","t":[{"b":3,"e":0.0,"k":"falling","v":0.0625,"x":0.28571,"p":[[0,29,0.0,0.28571,0.38465,0.0,0.0,0.46429,0.0,1.0,17,6,0,17,0,3,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,6],[4,29,0.1379,0.15179,0.26949,0.0,0.0,0.17857,0.0,1.0,21,2,0,21,0,3,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[8,29,0.2759,0.15179,0.29001,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,2,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[12,29,0.4138,0.06696,0.18893,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,29,0.5517,0.07589,0.17491,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[20,29,0.6897,0.16071,0.30252,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,2],[24,29,0.8276,0.0625,0.14698,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.12944,0.2534,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[29,29,1.0,0.08482,0.18509,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.08482,"x":0.2366,"p":[[0,36,0.0,0.2366,0.35102,0.0,0.0,0.32142,0.0,1.0,17,4,0,17,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[4,36,0.1111,0.18304,0.284,0.0,0.0,0.2857,0.0,1.0,18,2,0,18,0,5,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[8,36,0.2222,0.20088,0.34783,0.0,0.0,0.17857,0.0,1.0,21,4,0,21,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,4],[12,36,0.3333,0.08482,0.19516,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,36,0.4444,0.08927,0.1881,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[20,36,0.5556,0.16071,0.29179,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,4,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[24,36,0.6667,0.16071,0.26666,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[28,36,0.7778,0.16517,0.29473,0.0,0.0,0.21429,0.0,1.0,23,1,0,23,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[32,36,0.8889,0.21427,0.35173,0.0,0.0,0.35704,0.0,1.0,21,4,0,21,0,2,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[36,36,1.0,0.13392,0.26947,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,2,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,2]]}]},{"i":"05cc1d0346b98fb5","q":"Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.20982,"x":0.25893,"p":[[0,45,0.0,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.23214,0.11152,0.28571,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.24554,0.11971,0.2857,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,1,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,45,0.2667,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.20982,0.12364,0.10714,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.25891,0.10968,0.2857,0.28571,0.28571,0.0,0.571,4,0,0,4,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,45,0.5333,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.24107,0.10972,0.2857,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,1,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.24553,0.12992,0.2857,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,0,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,45,0.8,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.23214,0.12752,0.2857,0.28571,0.28571,0.0,0.42857,7,0,0,7,0,0,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.25,0.09449,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.1875,"x":0.25892,"p":[[0,35,0.0,0.24553,0.09606,0.28571,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.25892,0.11538,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,1,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,35,0.2286,0.23661,0.10479,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.24106,0.12592,0.28571,0.28571,0.28571,0.0,0.571,6,0,0,6,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,35,0.4571,0.20982,0.12363,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.1875,0.14032,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,1,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.21876,0.1287,0.21427,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,0,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.20983,0.13356,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.23214,0.10564,0.28571,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.19643,0.12752,0.0,0.28571,0.28571,0.0,0.28571,9,0,0,9,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b4148b1b82cdb441","q":"Two permutations  $a_1,a_2,\\dots,a_{2010}$  and  $b_1,b_2,\\dots,b_{2010}$  of the numbers  $1,2,\\dots,2010$  are said to *intersect* if  $a_k=b_k$  for some value of  $k$  in the range  $1\\le k\\le 2010$ .  Show that there exist  $1006$  permutations of the numbers  $1,2,\\dots,2010$  such that any other such permutation is guaranteed to intersect at least one of these  $1006$  permutations.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.41963,"x":0.72767,"p":[[0,45,0.0,0.64286,0.44464,0.0,0.92857,1.0,0.0,1.0,9,16,7,9,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,16],[4,45,0.0889,0.70087,0.26574,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,0,0,0,2,0,0,3,0,0,11,0,0,4,0,8],[8,45,0.1778,0.68749,0.3102,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,2,0,0,4,0,0,2,0,0,7,0,0,0,0,13],[12,45,0.2667,0.72767,0.29094,0.53539,0.71429,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,4,0,0,3,0,0,6,0,0,1,0,14],[16,45,0.3556,0.54909,0.33333,0.2857,0.57121,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,4,0,0,4,0,0,4,0,0,3,0,0,3,0,7],[20,45,0.4444,0.60716,0.33117,0.25001,0.71429,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,1,0,0,3,0,0,3,0,0,6,0,0,2,0,9],[24,45,0.5333,0.62945,0.34967,0.2857,0.71429,1.0,0.0,1.0,1,9,0,1,0,6,0,0,4,0,0,1,0,0,0,0,0,5,0,0,6,0,9],[28,45,0.6222,0.49105,0.28777,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,6,0,0,5,0,0,5,0,0,5,0,0,5,0,0,1,0,4],[32,45,0.7111,0.58926,0.28291,0.42857,0.57121,0.78571,0.14286,1.0,0,8,0,0,0,3,0,0,4,0,0,7,0,0,6,0,0,4,0,0,0,0,8],[36,45,0.8,0.56696,0.29339,0.39286,0.57143,0.71429,0.0,1.0,1,5,0,1,0,5,0,0,2,0,0,5,0,0,4,0,0,8,0,0,2,0,5],[40,45,0.8889,0.63388,0.22571,0.42857,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,5,0,0,6,0,0,11,0,0,1,0,5],[44,45,0.9778,0.46419,0.29024,0.14289,0.42859,0.71429,0.0,1.0,1,3,0,1,0,9,0,0,3,0,0,5,0,0,2,0,0,9,0,0,0,0,3],[45,45,1.0,0.41963,0.31529,0.14286,0.35714,0.71429,0.0,1.0,2,3,0,2,0,13,0,0,1,0,0,2,0,0,4,0,0,6,0,0,1,0,3]]},{"b":3,"e":1.0,"k":"rising","v":0.54912,"x":0.92411,"p":[[0,31,0.0,0.54912,0.44047,0.10714,0.71427,1.0,0.0,1.0,8,13,5,8,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,13],[4,31,0.129,0.74552,0.30249,0.67846,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,2,0,0,1,0,0,1,0,0,5,0,0,6,0,13],[8,31,0.2581,0.79016,0.30927,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,18],[12,31,0.3871,0.79016,0.27196,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,8,0,14],[16,31,0.5161,0.73213,0.32489,0.57132,0.85714,1.0,0.0,1.0,2,14,0,2,0,2,0,0,2,0,0,1,0,0,2,0,0,5,0,0,4,0,14],[20,31,0.6452,0.89732,0.15663,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,9,0,18],[24,31,0.7742,0.92411,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[28,31,0.9032,0.88839,0.17399,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,18],[31,31,1.0,0.90179,0.21261,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,23]]}]},{"i":"4490e4a7a297a988","q":"We will say that a rearrangement of the letters of a word has no *fixed letters* if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance  $HBRATA$  is a rearragnement with no fixed letter of  $BHARAT$ . How many distinguishable rearrangements with no fixed letters does  $BHARAT$  have? (The two  $A$ s are considered identical.)","t":[{"b":1,"e":1.0,"k":"falling","v":0.27232,"x":0.97321,"p":[[0,70,0.0,0.67411,0.42293,0.14286,1.0,1.0,0.0,1.0,2,20,1,2,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[4,70,0.0571,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,70,0.1143,0.89732,0.25313,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[12,70,0.1714,0.55804,0.42161,0.14286,0.35715,1.0,0.14286,1.0,0,15,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15],[16,70,0.2286,0.75438,0.3717,0.39285,1.0,1.0,0.0,1.0,1,21,0,1,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,21],[20,70,0.2857,0.74107,0.36672,0.35714,1.0,1.0,0.14286,1.0,0,20,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,20],[24,70,0.3429,0.67848,0.39782,0.14286,1.0,1.0,0.14,1.0,0,18,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,18],[28,70,0.4,0.78125,0.3694,0.67857,1.0,1.0,0.14286,1.0,0,23,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[32,70,0.4571,0.77679,0.37786,0.67857,1.0,1.0,0.0,1.0,1,23,0,1,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[36,70,0.5143,0.67411,0.40913,0.14286,1.0,1.0,0.0,1.0,1,19,0,1,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,19],[40,70,0.5714,0.66071,0.41458,0.14286,1.0,1.0,0.0,1.0,1,19,0,1,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[44,70,0.6286,0.71857,0.39228,0.14286,1.0,1.0,0.14,1.0,0,21,0,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,21],[48,70,0.6857,0.76338,0.36876,0.46396,1.0,1.0,0.14286,1.0,0,22,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,22],[52,70,0.7429,0.8125,0.3489,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,24],[56,70,0.8,0.7142,0.40261,0.14286,1.0,1.0,0.0,1.0,2,21,0,2,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,21],[60,70,0.8571,0.69197,0.41049,0.14286,1.0,1.0,0.0,1.0,1,20,0,1,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,20],[64,70,0.9143,0.60713,0.41032,0.14286,0.85714,1.0,0.14286,1.0,0,16,0,0,0,13,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,16],[68,70,0.9714,0.27232,0.28428,0.14286,0.14286,0.14287,0.14286,1.0,0,4,0,0,0,25,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[70,70,1.0,0.38393,0.36672,0.14286,0.14286,0.67857,0.14286,1.0,0,8,0,0,0,21,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,8]]},{"b":2,"e":0.28571,"k":"falling","v":0.43304,"x":1.0,"p":[[0,87,0.0,0.68295,0.38599,0.14289,1.0,1.0,0.14,1.0,0,18,0,0,0,9,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,18],[4,87,0.046,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,87,0.092,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,87,0.1379,0.8482,0.31328,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[16,87,0.1839,0.91955,0.25012,1.0,1.0,1.0,0.14,1.0,0,29,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[20,87,0.2299,0.91518,0.24964,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[24,87,0.2759,0.94634,0.20783,1.0,1.0,1.0,0.14,1.0,0,30,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,87,0.3218,0.92411,0.23686,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[32,87,0.3678,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,87,0.4138,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,87,0.4598,0.85267,0.29772,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,25],[44,87,0.5057,0.93303,0.21124,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[48,87,0.5517,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[52,87,0.5977,0.91965,0.24983,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[56,87,0.6437,0.91518,0.2392,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[60,87,0.6897,0.91518,0.24964,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[64,87,0.7356,0.91518,0.24964,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[68,87,0.7816,0.86161,0.32239,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[72,87,0.8276,0.93304,0.22864,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[76,87,0.8736,0.9375,0.20806,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[80,87,0.9195,0.89286,0.28347,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[84,87,0.9655,0.63839,0.42255,0.14286,1.0,1.0,0.0,1.0,1,18,0,1,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,18],[87,87,1.0,0.43304,0.38213,0.14286,0.14286,0.85714,0.0,1.0,2,7,0,2,0,17,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,7]]}]},{"i":"da2917a49a828854","q":"We start with any \ffinite list of distinct positive integers. We may replace any pair  $n, n + 1$  (not necessarily adjacent in the list) by the single integer  $n-2$ , now allowing negatives and repeats in the list. We may also replace any pair  $n, n + 4$  by  $n - 1$ . We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.","t":[{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.57589,"p":[[0,48,0.0,0.57589,0.35081,0.28571,0.28571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[4,48,0.0833,0.20089,0.29851,0.0,0.0,0.28571,0.0,1.0,18,3,0,18,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[8,48,0.1667,0.12946,0.20628,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,48,0.25,0.10268,0.20277,0.0,0.0,0.2857,0.0,1.0,23,1,0,23,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,48,0.3333,0.11161,0.1461,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.11161,0.20433,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,48,0.5,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.625,"p":[[0,67,0.0,0.625,0.38091,0.28571,0.64286,1.0,0.0,1.0,2,16,0,2,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,67,0.0597,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.08928,0.13243,0.0,0.0,0.2857,0.0,0.28571,22,0,0,22,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,67,0.2985,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1ee5ad0f940dcb13","q":"Triangle  $GRT$  has  $GR=5,$   $RT=12,$  and  $GT=13.$  The perpendicular bisector of  $GT$  intersects the extension of  $GR$  at  $O.$  Find  $TO.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.09375,"x":0.17857,"p":[[0,15,0.0,0.11161,0.1461,0.0,0.0,0.28571,0.0,0.4286,20,0,0,20,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.14732,0.14053,0.0,0.21428,0.28571,0.0,0.28571,15,0,0,15,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.17857,0.13832,0.0,0.28571,0.28571,0.0,0.28571,12,0,0,12,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.10268,0.1439,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,0,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.09375,"x":0.19196,"p":[[0,27,0.0,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.12946,0.1488,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,0,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.16964,0.1448,0.0,0.28571,0.28571,0.0,0.42857,13,0,0,13,0,1,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.19196,0.17353,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,0,0,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,27,0.5926,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.11607,0.14032,0.0,0.0,0.28571,0.0,0.28571,19,0,0,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.18304,0.1439,0.0,0.28571,0.28571,0.0,0.4286,12,0,0,12,0,0,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.125,0.14174,0.0,0.0,0.28571,0.0,0.28571,18,0,0,18,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"52d7d8850f4707af","q":"We define the sequence $a_{1}, a_{2}, a_{3} \\ldots$ as follows: $a_{1}=63$ and, for all integers $n \\geqslant 2, a_{n}$ is the smallest multiple of $n$ that is greater than or equal to $a_{n-1}$. Prove that the terms of our sequence are pairwise distinct.","t":[{"b":5,"e":0.42857,"k":"falling","v":0.33036,"x":0.95088,"p":[[0,36,0.0,0.95088,0.1048,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[4,36,0.1111,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[8,36,0.2222,0.92411,0.14279,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,22],[12,36,0.3333,0.9375,0.1234,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[16,36,0.4444,0.89732,0.17582,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,19],[20,36,0.5556,0.86607,0.18536,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,5,0,18],[24,36,0.6667,0.61161,0.31183,0.42859,0.57143,0.89286,0.0,1.0,3,8,0,3,0,2,0,0,0,0,0,5,0,0,8,0,0,4,0,0,2,0,8],[28,36,0.7778,0.54018,0.26422,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,7,0,0,9,0,0,4,0,0,3,0,0,2,0,5],[32,36,0.8889,0.33036,0.22142,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,9,0,0,8,0,0,9,0,0,1,0,0,1,0,0,1,0,1],[36,36,1.0,0.34375,0.14223,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,3,0,0,14,0,0,12,0,0,0,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.37498,"x":0.9375,"p":[[0,53,0.0,0.9375,0.13803,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[4,53,0.0755,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[8,53,0.1509,0.83482,0.22619,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,2,0,19],[12,53,0.2264,0.81696,0.20589,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,8,0,13],[16,53,0.3019,0.80356,0.2468,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,5,0,0,2,0,17],[20,53,0.3774,0.79018,0.23141,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,7,0,0,4,0,14],[24,53,0.4528,0.83036,0.20025,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,8,0,14],[28,53,0.5283,0.76337,0.27805,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,0,4,0,14],[32,53,0.6038,0.76338,0.2276,0.57143,0.78569,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,5,0,0,4,0,12],[36,53,0.6792,0.72764,0.23247,0.571,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,6,0,0,2,0,11],[40,53,0.7547,0.76784,0.23352,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,3,0,0,8,0,11],[44,53,0.8302,0.74107,0.20958,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,6,0,0,7,0,8],[48,53,0.9057,0.45534,0.25862,0.2857,0.42857,0.60714,0.0,1.0,1,1,0,1,0,6,0,0,7,0,0,3,0,0,7,0,0,4,0,0,3,0,1],[52,53,0.9811,0.47766,0.20078,0.42857,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,12,0,0,7,0,0,3,0,0,3,0,0],[53,53,1.0,0.37498,0.2569,0.14286,0.28571,0.57111,0.0,1.0,1,2,0,1,0,11,0,0,6,0,0,4,0,0,5,0,0,3,0,0,0,0,2]]}]},{"i":"24638af84a601419","q":"Prove that there exists a prime number  $p$  such that the minimum positive integer  $n$  such that  $p|2^n -1$  is  $3^{2013}$ .","t":[{"b":1,"e":0.85714,"k":"rising","v":0.63839,"x":0.90625,"p":[[0,13,0.0,0.67408,0.30772,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,4,0,0,2,0,0,8,0,0,0,0,0,3,0,12],[4,13,0.3077,0.63839,0.28344,0.42857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,4,0,0,2,0,0,6,0,0,4,0,0,8,0,5],[8,13,0.6154,0.81696,0.17941,0.67857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,13,0,10],[12,13,0.9231,0.82143,0.22303,0.82143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,11,0,13],[13,13,1.0,0.90625,0.14987,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,20]]},{"b":6,"e":0.57143,"k":"flat","v":0.62499,"x":0.75893,"p":[[0,26,0.0,0.69196,0.29474,0.53571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,1,0,0,3,0,0,5,0,0,5,0,0,3,0,11],[4,26,0.1538,0.64732,0.27196,0.53572,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,6,0,0,1,0,0,11,0,0,2,0,0,2,0,9],[8,26,0.3077,0.75893,0.19045,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,7,0,0,5,0,9],[12,26,0.4615,0.64732,0.22011,0.57143,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,3,0,0,14,0,0,1,0,0,7,0,4],[16,26,0.6154,0.66058,0.20745,0.57143,0.64071,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,8,0,0,3,0,5],[20,26,0.7692,0.67409,0.212,0.57143,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,5,0,0,4,0,6],[24,26,0.9231,0.63839,0.24219,0.57143,0.57143,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,2,0,0,2,0,0,12,0,0,4,0,0,6,0,4],[26,26,1.0,0.62499,0.21944,0.53539,0.57143,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,2,0,0,9,0,0,7,0,0,5,0,3]]}]},{"i":"0b34643a21701106","q":"There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.16518,"x":0.24107,"p":[[0,28,0.0,0.24089,0.2955,0.14214,0.14286,0.14286,0.0,1.0,6,3,0,6,0,20,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[4,28,0.1429,0.24107,0.2299,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[8,28,0.2857,0.21429,0.18898,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[12,28,0.4286,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.17402,0.15042,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,28,0.7143,0.19188,0.16216,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[24,28,0.8571,0.20509,0.21118,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[28,28,1.0,0.21429,0.21724,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2]]},{"b":4,"e":0.14286,"k":"flat","v":0.174,"x":0.26777,"p":[[0,20,0.0,0.26777,0.24683,0.14286,0.14286,0.21432,0.0,1.0,1,1,0,1,0,23,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,1],[4,20,0.2,0.20982,0.23141,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,26,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[8,20,0.4,0.24991,0.28351,0.14286,0.14286,0.14286,0.14,1.0,0,4,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,20,0.6,0.22741,0.25479,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[16,20,0.8,0.174,0.10553,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,27,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,20,1.0,0.25,0.25254,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"fb24464148565314","q":"We are given a triangle  $ABC$ .  Let  $M$  be the mid-point of its side  $AB$ .\n\nLet  $P$  be an interior point of the triangle.  We let  $Q$  denote the point symmetric to  $P$  with respect to  $M$ .\n\nFurthermore, let  $D$  and  $E$  be the common points of  $AP$  and  $BP$  with sides  $BC$  and  $AC$ , respectively.\n\nProve that points  $A$ ,  $B$ ,  $D$ , and  $E$  lie on a common circle if and only if  $\\angle ACP = \\angle QCB$  holds.\n\n(Karl Czakler)","t":[{"b":0,"e":0.71429,"k":"rising","v":0.28124,"x":0.61603,"p":[[0,60,0.0,0.28124,0.21862,0.10714,0.2857,0.46525,0.0,0.57143,8,0,1,8,0,6,0,0,5,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[4,60,0.0667,0.46874,0.17214,0.42857,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,2,0,0,9,0,0,14,0,0,3,0,0,0,0,0],[8,60,0.1333,0.41962,0.22568,0.24999,0.42857,0.57143,0.0,0.857,2,0,0,2,0,6,0,0,4,0,0,6,0,0,9,0,0,4,0,0,1,0,0],[12,60,0.2,0.42411,0.1838,0.39286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,3,0,0,13,0,0,8,0,0,3,0,0,0,0,0],[16,60,0.2667,0.44194,0.18334,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,3,0,0,9,0,0,10,0,0,4,0,0,0,0,0],[20,60,0.3333,0.44638,0.18119,0.28571,0.571,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,4,0,0,6,0,0,15,0,0,2,0,0,0,0,0],[24,60,0.4,0.43301,0.16933,0.28571,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,5,0,0,10,0,0,11,0,0,2,0,0,0,0,0],[28,60,0.4667,0.35694,0.21444,0.14286,0.42857,0.4642,0.0,0.71429,2,0,0,2,0,10,0,0,2,0,0,10,0,0,4,0,0,4,0,0,0,0,0],[32,60,0.5333,0.55796,0.13052,0.5354,0.57143,0.60714,0.1429,0.71429,0,0,0,0,0,1,0,0,1,0,0,6,0,0,16,0,0,8,0,0,0,0,0],[36,60,0.6,0.51783,0.19479,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,3,0,0,7,0,0,8,0,0,11,0,0,0,0,0],[40,60,0.6667,0.52228,0.13649,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,7,0,0,16,0,0,5,0,0,0,0,0],[44,60,0.7333,0.47761,0.1807,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,2,0,0,7,0,0,13,0,0,5,0,0,0,0,0],[48,60,0.8,0.55577,0.15124,0.42859,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,5,0,0,12,1,0,10,0,0,0,0,0],[52,60,0.8667,0.53123,0.1606,0.42857,0.57121,0.71429,0.1429,0.71429,0,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,0,9,0,0,0,0,0],[56,60,0.9333,0.60251,0.1545,0.42859,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,17,0,0,1,0,0],[60,60,1.0,0.61603,0.13093,0.5354,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,19,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.34374,"x":0.53567,"p":[[0,91,0.0,0.35268,0.23415,0.14286,0.42857,0.57143,0.0,0.71429,7,0,2,7,0,2,0,0,6,0,0,5,0,0,10,0,0,2,0,0,0,0,0],[4,91,0.044,0.53567,0.16365,0.571,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,2,0,0,2,0,0,18,0,0,7,0,0,0,0,0],[8,91,0.0879,0.46423,0.18207,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,2,0,0,7,0,0,15,0,0,3,0,0,0,0,0],[12,91,0.1319,0.47307,0.17994,0.42857,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,2,0,0,8,0,0,12,0,0,5,0,0,0,0,0],[16,91,0.1758,0.46871,0.1829,0.39286,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,7,0,0,12,0,0,5,0,0,0,0,0],[20,91,0.2198,0.45089,0.16791,0.42857,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,1,0,0,12,0,0,14,0,0,1,0,0,0,0,0],[24,91,0.2637,0.44195,0.16505,0.2857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,10,0,0,8,0,0,4,0,0,0,0,0],[28,91,0.3077,0.47317,0.20023,0.39286,0.571,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,4,0,0,6,0,0,12,0,0,6,0,0,0,0,0],[32,91,0.3516,0.49328,0.17799,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,3,0,0,5,0,1,13,0,0,6,0,0,0,0,0],[36,91,0.3956,0.44197,0.16888,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,11,0,0,10,0,0,3,0,0,0,0,0],[40,91,0.4396,0.46203,0.17585,0.39286,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,5,0,0,9,0,0,10,1,0,4,0,0,0,0,0],[44,91,0.4835,0.45085,0.19918,0.39285,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,3,0,0,9,0,0,10,0,0,5,0,0,0,0,0],[48,91,0.5275,0.44638,0.20121,0.39286,0.4998,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,2,0,0,8,0,0,12,0,0,4,0,0,0,0,0],[52,91,0.5714,0.48656,0.1816,0.39286,0.571,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,6,0,0,6,0,0,13,0,0,4,0,0,1,0,0],[56,91,0.6154,0.41961,0.1747,0.28571,0.42859,0.57111,0.0,0.71429,2,0,0,2,0,2,0,0,5,0,0,12,0,0,9,0,0,2,0,0,0,0,0],[60,91,0.6593,0.41963,0.20496,0.28571,0.42859,0.57143,0.0,0.71429,2,0,0,2,0,5,0,0,4,0,0,6,0,0,12,0,0,3,0,0,0,0,0],[64,91,0.7033,0.41963,0.21109,0.24999,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,2,0,0,8,0,0,10,0,0,4,0,0,0,0,0],[68,91,0.7473,0.4039,0.21483,0.14289,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,8,0,0,4,0,1,5,0,0,8,0,0,5,0,0,0,0,0],[72,91,0.7912,0.38846,0.19959,0.1429,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,3,0,0,7,0,0,12,0,0,1,0,0,0,0,0],[76,91,0.8352,0.48197,0.20421,0.39286,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,4,0,0,5,0,0,12,0,0,7,0,0,0,0,0],[80,91,0.8791,0.44639,0.17765,0.42857,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,3,0,0,10,0,0,13,0,0,2,0,0,0,0,0],[84,91,0.9231,0.34374,0.21681,0.14286,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,8,0,0,3,0,0,6,0,0,10,0,0,1,0,0,0,0,0],[88,91,0.967,0.39719,0.19807,0.2857,0.42859,0.57143,0.0,0.71429,3,0,0,3,0,3,0,0,7,0,0,5,0,0,13,0,0,1,0,0,0,0,0],[91,91,1.0,0.45522,0.18375,0.42857,0.571,0.57143,0.14,0.71429,0,0,0,0,0,7,0,0,0,0,0,8,0,0,14,0,0,3,0,0,0,0,0]]}]},{"i":"8e585e64e4113a03","q":"Which positive integers satisfy that the sum of the number\u2019s last three digits added to the number itself yields  $2029$ ?","t":[{"b":1,"e":0.85714,"k":"falling","v":0.75446,"x":0.99554,"p":[[0,48,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,48,0.0833,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,48,0.1667,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,48,0.25,0.88393,0.30186,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[16,48,0.3333,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[20,48,0.4167,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,48,0.5,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,48,0.5833,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,48,0.6667,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[36,48,0.75,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,48,0.8333,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[44,48,0.9167,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[48,48,1.0,0.75446,0.23753,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,1,0,0,7,0,0,2,0,0,10,0,9]]},{"b":7,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,22,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,22,0.1818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,22,0.3636,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,22,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,22,0.7273,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[20,22,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"f5d45b0315b356e6","q":"What is the largest possible number of subsets of the set  $\\{1, 2, \\dots , 2n+1\\}$  such that the intersection of any two subsets consists of one or several consecutive integers?","t":[{"b":0,"e":0.71429,"k":"flat","v":0.56249,"x":0.96875,"p":[[0,44,0.0,0.62044,0.20415,0.57143,0.57143,0.74996,0.14,0.85714,0,0,0,0,0,2,0,0,3,0,0,0,0,0,12,0,0,7,0,0,8,0,0],[4,44,0.0909,0.95536,0.15746,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[8,44,0.1818,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[12,44,0.2727,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[16,44,0.3636,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[20,44,0.4545,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[24,44,0.5455,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[28,44,0.6364,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[32,44,0.7273,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[36,44,0.8182,0.8482,0.1673,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,10,0,13],[40,44,0.9091,0.87946,0.16409,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,11,0,16],[44,44,1.0,0.56249,0.23941,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,4,0,0,2,0,0,8,0,0,10,0,0,2,0,2]]},{"b":1,"e":1.0,"k":"rising","v":0.63393,"x":0.98661,"p":[[0,55,0.0,0.63393,0.2549,0.57143,0.71429,0.71429,0.0,1.0,1,4,1,1,0,2,0,0,3,0,0,1,0,0,5,0,0,13,0,0,3,0,4],[4,55,0.0727,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,55,0.1455,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[12,55,0.2182,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,55,0.2909,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,55,0.3636,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[24,55,0.4364,0.92411,0.17852,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,24],[28,55,0.5091,0.95089,0.11071,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[32,55,0.5818,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,55,0.6545,0.94196,0.15093,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,27],[40,55,0.7273,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[44,55,0.8,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[48,55,0.8727,0.85265,0.17312,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,16],[52,55,0.9455,0.87945,0.16018,0.71429,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,18],[55,55,1.0,0.87053,0.13534,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,6,0,15]]}]},{"i":"847f4d1cf735637d","q":"We have a  $100\\times100$  garden and we\u2019ve plant  $10000$  trees in the  $1\\times1$  squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.","t":[{"b":1,"e":0.0,"k":"falling","v":0.00893,"x":0.98661,"p":[[0,46,0.0,0.81696,0.1996,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,6,0,14],[4,46,0.087,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,46,0.1739,0.91071,0.26666,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[12,46,0.2609,0.90625,0.25657,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,27],[16,46,0.3478,0.92857,0.20825,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[20,46,0.4348,0.89286,0.29233,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[24,46,0.5217,0.77679,0.38785,0.78571,1.0,1.0,0.0,1.0,6,22,0,6,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,22],[28,46,0.6087,0.875,0.30671,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,27],[32,46,0.6957,0.88839,0.28956,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[36,46,0.7826,0.72768,0.40305,0.39286,1.0,1.0,0.0,1.0,6,20,0,6,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,20],[40,46,0.8696,0.46429,0.46566,0.0,0.28571,1.0,0.0,1.0,14,13,0,14,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,13],[44,46,0.9565,0.04911,0.18423,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[46,46,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.79911,"x":1.0,"p":[[0,25,0.0,0.79911,0.21975,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,10,0,12],[4,25,0.16,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,25,0.32,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,25,0.48,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ff1054dfd9b2a41b","q":"Two squares on an $8 \\times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some\nsquare and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?","t":[{"b":0,"e":1.0,"k":"flat","v":0.54018,"x":1.0,"p":[[0,22,0.0,0.86161,0.29121,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[4,22,0.1818,0.58473,0.4351,0.14214,0.78571,1.0,0.0,1.0,7,15,0,7,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,15],[8,22,0.3636,0.54018,0.40835,0.0,0.57143,1.0,0.0,1.0,9,11,0,9,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,0,0,0,11],[12,22,0.5455,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[16,22,0.7273,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,22,0.9091,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":0.0,"k":"falling","v":0.04911,"x":0.82589,"p":[[0,79,0.0,0.82589,0.36023,0.96429,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,24],[4,79,0.0506,0.52231,0.4512,0.0,0.49979,1.0,0.0,1.0,10,14,0,10,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,14],[8,79,0.1013,0.81696,0.34853,0.96425,1.0,1.0,0.0,1.0,3,24,0,3,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,24],[12,79,0.1519,0.60268,0.457,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,17],[16,79,0.2025,0.49554,0.47108,0.0,0.57143,1.0,0.0,1.0,14,13,0,14,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,13],[20,79,0.2532,0.49098,0.40404,0.14286,0.42857,1.0,0.0,1.0,6,11,0,6,0,7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,11],[24,79,0.3038,0.48205,0.43271,0.0,0.42857,1.0,0.0,1.0,9,12,0,9,0,5,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,12],[28,79,0.3544,0.48214,0.40049,0.10714,0.42857,1.0,0.0,1.0,8,10,0,8,0,3,0,0,3,0,0,5,0,0,1,0,0,2,0,0,0,0,10],[32,79,0.4051,0.34375,0.40067,0.0,0.14286,0.71429,0.0,1.0,13,7,0,13,0,6,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,7],[36,79,0.4557,0.50446,0.4103,0.0,0.42859,1.0,0.0,1.0,9,11,0,9,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,11],[40,79,0.5063,0.47771,0.43829,0.0,0.42857,1.0,0.0,1.0,11,12,0,11,0,3,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,12],[44,79,0.557,0.30804,0.37306,0.0,0.14286,0.42857,0.0,1.0,14,6,0,14,0,4,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,6],[48,79,0.6076,0.54018,0.40047,0.14286,0.4286,1.0,0.0,1.0,7,12,0,7,0,2,0,0,2,0,0,6,0,0,2,0,0,1,0,0,0,0,12],[52,79,0.6582,0.52665,0.42318,0.10714,0.57143,1.0,0.0,1.0,8,12,0,8,0,4,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,12],[56,79,0.7089,0.45982,0.43848,0.0,0.42857,1.0,0.0,1.0,12,11,0,12,0,3,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,11],[60,79,0.7595,0.22767,0.32898,0.0,0.0,0.42858,0.0,1.0,18,3,0,18,0,4,0,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,3],[64,79,0.8101,0.33927,0.24418,0.10714,0.42857,0.57111,0.0,0.71429,8,0,0,8,0,4,0,0,0,0,0,11,0,0,6,0,0,3,0,0,0,0,0],[68,79,0.8608,0.38393,0.3597,0.0,0.42857,0.57143,0.0,1.0,11,6,0,11,0,1,0,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,6],[72,79,0.9114,0.19196,0.34554,0.0,0.0,0.14286,0.0,1.0,22,3,0,22,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,3],[76,79,0.962,0.12054,0.29257,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[79,79,1.0,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"11aee29113e40cb9","q":"We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_{0}, a_{1}, \\ldots, a_{n} \\in S$, all integer roots of the polynomial $a_{0}+a_{1} x+\\cdots+a_{n} x^{n}$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^{a}-2^{b}$ for positive integers $a$ and $b$. (Czech Republic)","t":[{"b":3,"e":0.0,"k":"flat","v":0.05348,"x":0.34374,"p":[[0,68,0.0,0.15625,0.10326,0.14286,0.14286,0.14286,0.0,0.4286,5,0,1,5,0,21,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.34374,0.28089,0.14286,0.28571,0.42858,0.0,1.0,5,1,0,5,0,7,0,0,8,0,0,5,0,0,2,0,0,0,0,0,4,0,1],[8,68,0.1176,0.3125,0.2299,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,10,0,0,7,0,0,8,0,0,1,0,0,1,0,0,1,0,1],[12,68,0.1765,0.32581,0.24811,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,8,0,0,7,0,0,9,0,0,0,0,0,1,0,0,2,0,1],[16,68,0.2353,0.28563,0.2173,0.14286,0.2857,0.42857,0.0,1.0,3,1,0,3,0,12,0,0,6,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[20,68,0.2941,0.26786,0.19805,0.14286,0.2857,0.42857,0.0,1.0,5,1,0,5,0,8,0,0,9,0,0,9,0,0,0,0,0,0,0,0,0,0,1],[24,68,0.3529,0.31247,0.27299,0.14286,0.2143,0.42858,0.0,1.0,6,1,0,6,0,10,0,0,3,0,0,6,0,0,3,0,0,1,0,0,2,0,1],[28,68,0.4118,0.22313,0.205,0.105,0.14288,0.28571,0.0,1.0,8,1,0,8,0,9,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[32,68,0.4706,0.20534,0.21995,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,11,0,0,5,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[36,68,0.5294,0.15624,0.17982,0.0,0.14286,0.28571,0.0,0.571,15,0,0,15,0,7,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[40,68,0.5882,0.24555,0.23753,0.0,0.14286,0.42857,0.0,0.85714,9,0,0,9,0,9,0,0,4,0,0,6,0,0,2,0,0,0,0,0,2,0,0],[44,68,0.6471,0.14955,0.17161,0.0,0.14286,0.1786,0.0,0.64286,13,0,0,13,0,11,0,0,4,0,0,2,0,0,1,1,0,0,0,0,0,0,0],[48,68,0.7059,0.23661,0.28484,0.0,0.14286,0.42857,0.0,1.0,13,1,0,13,0,7,0,0,3,0,0,4,0,0,0,0,0,3,0,0,1,0,1],[52,68,0.7647,0.1875,0.25111,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,6,0,0,5,0,0,3,0,0,1,0,0,0,0,0,1,0,1],[56,68,0.8235,0.16053,0.22799,0.0,0.14143,0.14286,0.0,0.85714,14,0,0,14,0,12,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[60,68,0.8824,0.12052,0.16405,0.0,0.0,0.17857,0.0,0.571,18,0,0,18,0,6,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[64,68,0.9412,0.06696,0.09439,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"rising","v":0.14286,"x":0.39731,"p":[[0,102,0.0,0.16071,0.13716,0.14286,0.14286,0.14286,0.0,0.57143,7,0,1,7,0,19,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,102,0.0392,0.29463,0.22849,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,8,0,0,7,0,0,9,0,0,1,0,0,0,0,0,1,0,1],[8,102,0.0784,0.32588,0.28844,0.14286,0.28571,0.42858,0.0,1.0,6,3,0,6,0,9,0,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,3],[12,102,0.1176,0.30794,0.23455,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,8,0,0,6,0,0,9,0,0,0,0,0,2,0,0,2,0,0],[16,102,0.1569,0.28115,0.23555,0.14286,0.21428,0.42857,0.0,0.85714,6,0,0,6,0,10,0,0,5,0,0,6,0,0,1,0,0,3,0,0,1,0,0],[20,102,0.1961,0.25893,0.26592,0.0,0.2857,0.32143,0.0,1.0,10,2,1,10,0,5,0,0,9,0,0,4,0,0,1,0,0,1,0,0,0,0,2],[24,102,0.2353,0.30354,0.27137,0.14286,0.14286,0.46418,0.0,1.0,5,1,0,5,0,12,0,0,5,0,0,2,0,0,4,0,0,1,0,0,2,0,1],[28,102,0.2745,0.27679,0.24206,0.14286,0.14288,0.42857,0.0,1.0,5,1,0,5,0,13,0,0,3,0,0,7,0,0,0,0,0,3,0,0,0,0,1],[32,102,0.3137,0.23215,0.14617,0.14286,0.2857,0.28571,0.0,0.57143,5,0,0,5,0,9,0,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[36,102,0.3529,0.28124,0.2636,0.0,0.2857,0.42857,0.0,1.0,9,1,0,9,0,6,0,0,4,0,0,10,0,0,0,0,0,0,0,0,2,0,1],[40,102,0.3922,0.14286,0.16366,0.0,0.14286,0.1786,0.0,0.71429,13,0,0,13,0,11,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[44,102,0.4314,0.30803,0.22619,0.14286,0.2857,0.42857,0.0,1.0,5,1,0,5,0,6,0,0,9,0,0,8,0,0,1,0,0,2,0,0,0,0,1],[48,102,0.4706,0.28124,0.22722,0.14286,0.2143,0.42857,0.0,1.0,4,1,0,4,0,12,0,0,5,0,0,8,0,0,1,0,0,0,0,0,1,0,1],[52,102,0.5098,0.26784,0.21352,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,6,0,0,10,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[56,102,0.549,0.27679,0.22851,0.14286,0.2857,0.42857,0.0,1.0,6,1,0,6,0,8,0,0,7,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[60,102,0.5882,0.35267,0.29446,0.14286,0.28571,0.42857,0.0,1.0,6,2,0,6,0,7,0,0,4,0,0,8,0,0,1,0,0,2,0,0,2,0,2],[64,102,0.6275,0.35714,0.27199,0.14286,0.2857,0.42857,0.0,1.0,2,2,0,2,0,10,0,0,7,0,0,8,0,0,0,0,0,0,0,0,3,0,2],[68,102,0.6667,0.32589,0.22933,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,10,0,0,4,0,0,11,0,0,1,0,0,0,0,0,3,0,0],[72,102,0.7059,0.35265,0.26238,0.14286,0.2857,0.571,0.0,1.0,4,1,0,4,0,8,0,0,6,0,0,5,0,0,5,0,0,1,0,0,2,0,1],[76,102,0.7451,0.29018,0.19393,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,6,0,0,14,0,0,0,0,0,0,0,0,1,0,0],[80,102,0.7843,0.31696,0.13236,0.2857,0.35714,0.42857,0.0,0.4286,2,0,0,2,0,5,0,0,9,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[84,102,0.8235,0.29464,0.24206,0.14286,0.28571,0.42857,0.0,1.0,6,2,0,6,0,7,0,0,6,0,0,11,0,0,0,0,0,0,0,0,0,0,2],[88,102,0.8627,0.28134,0.16562,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,6,0,0,7,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[92,102,0.902,0.39731,0.16649,0.28571,0.42857,0.42858,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,16,0,0,4,0,0,0,0,0,0,0,1],[96,102,0.9412,0.33482,0.16982,0.2857,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,4,0,0,7,0,0,17,0,0,0,0,0,0,0,0,1,0,0],[100,102,0.9804,0.34375,0.11769,0.28571,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,1,0,0,11,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[102,102,1.0,0.35268,0.11837,0.28571,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,1,0,0,9,0,0,20,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a8ec999351e2de0b","q":"Two people,  $A$  and  $B$ , play the following game with a deck of 32 cards. With  $A$  starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. Eventually all of the cards are chosen, and the person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?","t":[{"b":0,"e":0.85714,"k":"flat","v":0.89731,"x":0.97768,"p":[[0,52,0.0,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[4,52,0.0769,0.92411,0.07129,0.85714,0.85714,1.0,0.8571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[8,52,0.1538,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[12,52,0.2308,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[16,52,0.3077,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[20,52,0.3846,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,52,0.4615,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,52,0.5385,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,52,0.6154,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,52,0.6923,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,52,0.7692,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[44,52,0.8462,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[48,52,0.9231,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[52,52,1.0,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":1,"e":0.85714,"k":"flat","v":0.88839,"x":0.92857,"p":[[0,47,0.0,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[4,47,0.0851,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[8,47,0.1702,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,47,0.2553,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[16,47,0.3404,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[20,47,0.4255,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[24,47,0.5106,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[28,47,0.5957,0.91071,0.06917,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[32,47,0.6809,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[36,47,0.766,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[40,47,0.8511,0.90625,0.06785,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[44,47,0.9362,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[47,47,1.0,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11]]}]},{"i":"636300a8ad74b4da","q":"There is a prime number  $p$  such that  $16p+1$  is the cube of a positive integer. Find  $p$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.87946,"x":1.0,"p":[[0,68,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,68,0.0588,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,68,0.1176,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[12,68,0.1765,0.87946,0.15198,0.85714,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,11],[16,68,0.2353,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[20,68,0.2941,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,68,0.3529,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,68,0.4118,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,68,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,68,0.5294,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,68,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,68,0.6471,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,68,0.7059,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,68,0.7647,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,68,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,68,0.8824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,23,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,23,0.1739,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,23,0.3478,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c0e40950b250a914","q":"The excircles of triangle  $ABC$  touch its sides  $BC$ ,  $CA$ , and  $AB$  at points  $A_1$ ,  $B_1$ , and  $C_1$ , respectively. Let  $B_2$  and  $C_2$  be the midpoints of segments  $BB_1$  and  $CC_1$ , respectively. Line  $B_2C_2$  intersects line  $BC$  at point  $W$ . Prove that  $AW = A_1W$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.14286,"x":0.28125,"p":[[0,56,0.0,0.22321,0.17105,0.0,0.2857,0.32143,0.0,0.57143,9,0,0,9,0,5,0,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[4,56,0.0714,0.22768,0.16311,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,2,0,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[8,56,0.1429,0.24999,0.16364,0.14286,0.2857,0.28571,0.0,0.57143,7,0,0,7,0,3,0,0,15,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[12,56,0.2143,0.25,0.15567,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,2,0,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[16,56,0.2857,0.25893,0.15335,0.2857,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,0,0,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[20,56,0.3571,0.25,0.15972,0.10714,0.28571,0.42857,0.0,0.4286,8,0,0,8,0,1,0,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.20089,0.13296,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,5,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.27231,0.17624,0.21427,0.28571,0.42857,0.0,0.5714,8,0,0,8,0,0,0,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[32,56,0.5714,0.28125,0.17307,0.25,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,1,0,0,12,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[36,56,0.6429,0.21428,0.14725,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,2,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.20982,0.15561,0.0,0.28571,0.28571,0.0,0.4286,10,0,0,10,0,2,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.25446,0.13709,0.2857,0.28571,0.28571,0.0,0.42857,6,0,0,6,0,1,0,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.20089,0.1551,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,4,0,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.16072,0.16269,0.0,0.14288,0.28571,0.0,0.4286,15,0,0,15,0,2,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.14286,0.14725,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,4,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.08929,"x":0.26339,"p":[[0,47,0.0,0.17857,0.15152,0.0,0.2857,0.28571,0.0,0.4286,12,0,0,12,0,3,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.23661,0.16602,0.10714,0.28571,0.32143,0.0,0.57143,8,0,0,8,0,4,0,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[8,47,0.1702,0.22321,0.16728,0.0,0.28571,0.32143,0.0,0.42857,10,0,0,10,0,2,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.26339,0.12428,0.2857,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,3,0,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.25893,0.13092,0.2857,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,2,0,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.20527,0.1513,0.0,0.28571,0.28571,0.0,0.42857,10,0,0,10,0,2,0,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.20536,0.16342,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,4,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[28,47,0.5957,0.20089,0.15093,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,3,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.16071,0.14616,0.0,0.2857,0.28571,0.0,0.42857,14,0,0,14,0,1,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.18741,0.14035,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,4,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.1875,0.16536,0.0,0.21428,0.28571,0.0,0.4286,12,0,0,12,0,4,0,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.11161,0.14166,0.0,0.0,0.2857,0.0,0.42857,18,0,0,18,0,5,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.08929,0.12753,0.0,0.0,0.1429,0.0,0.42857,20,0,0,20,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"705a2f44f1bd46ee","q":"The points $A, B, C, D$ lie, in this order, on a circle $\\omega$, where $A D$ is a diameter of $\\omega$. Furthermore, $A B=B C=a$ and $C D=c$ for some relatively prime integers $a$ and $c$. Show that if the diameter $d$ of $\\omega$ is also an integer, then either $d$ or $2 d$ is a perfect square.","t":[{"b":0,"e":1.0,"k":"flat","v":0.75446,"x":0.84375,"p":[[0,53,0.0,0.81696,0.16457,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,7,0,11],[4,53,0.0755,0.7991,0.13767,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,12,0,6],[8,53,0.1509,0.75446,0.1996,0.71429,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,8,0,6],[12,53,0.2264,0.80804,0.1411,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,11,0,7],[16,53,0.3019,0.82143,0.20203,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,7,0,12],[20,53,0.3774,0.79462,0.17475,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,6,0,0,11,0,8],[24,53,0.4528,0.79017,0.16745,0.71429,0.857,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,9,0,8],[28,53,0.5283,0.82589,0.18117,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,14,0,9],[32,53,0.6038,0.77231,0.16699,0.67857,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,11,0,6],[36,53,0.6792,0.79909,0.20474,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,10,0,9],[40,53,0.7547,0.82589,0.13709,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,15,0,7],[44,53,0.8302,0.84375,0.1488,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,9,0,12],[48,53,0.9057,0.76338,0.14555,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,8,0,5],[52,53,0.9811,0.76336,0.1541,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,0,9,0,5],[53,53,1.0,0.77679,0.1234,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,7,0,5]]},{"b":4,"e":0.42857,"k":"flat","v":0.74107,"x":0.85267,"p":[[0,56,0.0,0.85267,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,11,0,12],[4,56,0.0714,0.76785,0.16656,0.67857,0.78564,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,10,0,6],[8,56,0.1429,0.79896,0.14674,0.71429,0.78564,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,8,0,8],[12,56,0.2143,0.79915,0.16688,0.71429,0.85714,1.0,0.43,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,9,0,9],[16,56,0.2857,0.8125,0.12078,0.71429,0.78564,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,7],[20,56,0.3571,0.80803,0.14987,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,13,0,7],[24,56,0.4286,0.82143,0.15567,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,10,0,10],[28,56,0.5,0.81696,0.1394,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,8,0,9],[32,56,0.5714,0.78569,0.14729,0.71429,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,11,0,6],[36,56,0.6429,0.75445,0.17583,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,0,9,0,6],[40,56,0.7143,0.78125,0.14279,0.71429,0.78571,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,11,0,5],[44,56,0.7857,0.80357,0.1171,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,15,0,4],[48,56,0.8571,0.75446,0.13474,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,0,8,0,4],[52,56,0.9286,0.77665,0.15952,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,5,0,8],[56,56,1.0,0.74107,0.16146,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,11,0,0,7,0,5]]}]},{"i":"68366af88343ccc2","q":"The reals  $b>0$  and  $a$  are such that the quadratic  $x^2+ax+b$  has two distinct real roots,  exactly one of which lies in the interval  $[-1;1]$ . Prove that one of the roots lies in the interval  $(-b;b)$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.84374,"x":0.97321,"p":[[0,23,0.0,0.84374,0.14445,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,8,0,12],[4,23,0.1739,0.89285,0.19562,0.85711,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,20],[8,23,0.3478,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,23,0.5217,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,23,0.6957,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[20,23,0.8696,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[23,23,1.0,0.9241,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19]]},{"b":6,"e":0.85714,"k":"flat","v":0.72767,"x":0.94643,"p":[[0,22,0.0,0.72767,0.20934,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,10,0,0,4,0,8],[4,22,0.1818,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[8,22,0.3636,0.91515,0.13774,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,22],[12,22,0.5455,0.91517,0.10631,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[16,22,0.7273,0.89732,0.11971,0.82143,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,17],[20,22,0.9091,0.91516,0.11773,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,19],[22,22,1.0,0.87053,0.12556,0.85711,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,13,0,12]]}]},{"i":"624fb2d65a2ca777","q":"We call an ordered set of integers  $(a_1,a_2,a_3,...,a_k)$  with  $a_1,a_2,\\dots,a_k\\ge2$  `` $n$ -special\" if  $a_1+a_2+a_3+...+a_k = n$ . Show that the number of distinct  $n$ -special sets is always a Fibonacci number.\n\n*Proposed by Cody Johnson*","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,36,0.0,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[4,36,0.1111,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,36,0.2222,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[12,36,0.3333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,36,0.7778,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,31,0.0,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[4,31,0.129,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,31,0.2581,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,31,0.3871,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,31,0.5161,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,31,0.9032,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"83b2dc5c9677336f","q":"The quadrilateral  $ABCD$  is inscribed in a circle with center  $O$ . The diagonals  $AC$  and  $BD$  do not pass through  $O$ . If the circumcentre of triangle  $AOC$  lies on the line  $BD$ , prove that the circumcentre of triangle  $BOD$  lies on the line  $AC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,72,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,72,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,72,0.3889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,72,0.7778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,72,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.0,"x":0.06241,"p":[[0,60,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.01786,0.04726,0.0,0.0,0.0,0.0,0.143,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b88f245a870f8af0","q":"The triangle  $ABC$  has a right angle at  $A$ . The centre of the circumcircle is called  $O$ , and the base point of the normal from  $O$  to  $AC$  is called  $D$ . The point  $E$  lies on  $AO$  with  $AE = AD$ . The angle bisector of  $\\angle CAO$  meets  $CE$  in  $Q$ . The lines  $BE$  and  $OQ$  intersect in  $F$ . Show that the lines  $CF$  and  $OE$  are parallel.","t":[{"b":1,"e":0.0,"k":"falling","v":0.30356,"x":0.81696,"p":[[0,79,0.0,0.81696,0.27254,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,0,0,20],[4,79,0.0506,0.77232,0.31514,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,16],[8,79,0.1013,0.76784,0.25192,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,0,3,0,12],[12,79,0.1519,0.65625,0.33476,0.28571,0.71429,1.0,0.0,1.0,1,12,0,1,0,4,0,0,4,0,0,1,0,0,3,0,0,6,0,0,1,0,12],[16,79,0.2025,0.62052,0.32461,0.42857,0.71429,1.0,0.0,1.0,3,10,0,3,0,1,0,0,2,0,0,7,0,0,2,0,0,7,0,0,0,0,10],[20,79,0.2532,0.50446,0.2879,0.2857,0.5,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,5,0,0,4,0,0,3,0,0,8,0,0,2,0,3],[24,79,0.3038,0.43303,0.29769,0.14286,0.42859,0.71429,0.0,1.0,1,3,0,1,0,11,0,0,3,0,0,5,0,0,2,0,0,6,0,0,1,0,3],[28,79,0.3544,0.57143,0.27664,0.42857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,3,0,0,0,0,0,8,0,0,4,0,0,9,0,0,2,0,4],[32,79,0.4051,0.5982,0.23538,0.42857,0.64286,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,4,0,0,5,0,0,11,0,0,1,0,4],[36,79,0.4557,0.54464,0.33204,0.28571,0.57143,0.89286,0.0,1.0,3,8,0,3,0,3,0,0,4,0,0,5,0,0,6,0,0,2,0,0,1,0,8],[40,79,0.5063,0.60713,0.28121,0.42857,0.57121,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,3,0,0,10,0,0,3,0,0,4,0,0,3,0,7],[44,79,0.557,0.46427,0.23689,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,10,0,0,2,0,0,7,0,0,0,0,2],[48,79,0.6076,0.45089,0.33902,0.14286,0.28571,0.71429,0.0,1.0,3,4,0,3,0,9,0,0,5,0,0,1,0,0,2,0,0,5,0,0,3,0,4],[52,79,0.6582,0.46875,0.29717,0.28571,0.42857,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,6,0,0,6,0,0,2,0,0,6,0,0,2,0,3],[56,79,0.7089,0.43303,0.30406,0.14286,0.35714,0.71429,0.0,1.0,3,3,0,3,0,7,0,0,6,0,0,3,0,0,3,0,0,6,0,0,1,0,3],[60,79,0.7595,0.36607,0.3071,0.14286,0.28571,0.42858,0.0,1.0,5,4,0,5,0,8,0,0,4,0,0,8,0,0,1,0,0,2,0,0,0,0,4],[64,79,0.8101,0.38839,0.31183,0.14286,0.28571,0.71429,0.0,1.0,4,2,0,4,0,10,0,0,4,0,0,3,0,0,1,0,0,6,0,0,2,0,2],[68,79,0.8608,0.49552,0.2879,0.28571,0.4998,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,6,0,0,3,0,0,4,0,0,7,0,0,2,0,3],[72,79,0.9114,0.42409,0.27077,0.14286,0.42857,0.60714,0.0,1.0,1,2,0,1,0,9,0,0,5,0,0,5,0,0,4,0,0,5,0,0,1,0,2],[76,79,0.962,0.40624,0.30745,0.14286,0.35714,0.57111,0.0,1.0,4,4,0,4,0,6,0,0,6,0,0,7,0,0,2,0,0,2,0,0,1,0,4],[79,79,1.0,0.30356,0.2335,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,11,0,0,4,0,0,8,0,0,2,0,0,2,0,0,0,0,1]]},{"b":2,"e":0.28571,"k":"falling","v":0.20089,"x":0.91518,"p":[[0,61,0.0,0.71874,0.28457,0.57132,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,3,0,0,4,0,0,8,0,0,0,0,13],[4,61,0.0656,0.69195,0.25029,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,1,0,0,6,0,0,10,0,0,1,0,9],[8,61,0.1311,0.91518,0.15093,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,23],[12,61,0.1967,0.625,0.30671,0.39286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,5,0,0,5,0,0,2,0,0,5,0,0,4,0,8],[16,61,0.2623,0.77676,0.19868,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,10,0,8],[20,61,0.3279,0.59374,0.26513,0.39286,0.57143,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,5,0,0,4,0,0,7,0,0,3,0,5],[24,61,0.3934,0.58926,0.28065,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,5,0,0,2,0,0,5,0,0,6,0,0,5,0,0,4,0,5],[28,61,0.459,0.5089,0.21109,0.42857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,1,0,0,12,0,0,8,0,0,3,0,0,3,0,1],[32,61,0.5246,0.43302,0.27775,0.2857,0.35714,0.60682,0.0,1.0,1,3,0,1,0,6,0,0,9,0,0,7,0,0,1,0,0,3,0,0,2,0,3],[36,61,0.5902,0.39276,0.22596,0.25,0.28571,0.5711,0.14,1.0,0,1,0,0,0,8,0,0,9,0,0,6,0,0,4,0,0,3,0,0,1,0,1],[40,61,0.6557,0.46875,0.25313,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,4,0,0,6,0,0,8,0,0,7,0,0,2,0,0,1,0,3],[44,61,0.7213,0.4866,0.20472,0.39286,0.5,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,8,0,0,10,0,0,3,0,0,3,0,0],[48,61,0.7869,0.4642,0.25765,0.24999,0.42859,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,3,0,0,7,0,0,4,0,0,7,0,0,2,0,1],[52,61,0.8525,0.20982,0.1636,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,15,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[56,61,0.918,0.25893,0.19377,0.14286,0.21428,0.28571,0.0,1.0,2,1,0,2,0,14,0,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[60,61,0.9836,0.23661,0.19759,0.14286,0.21428,0.28571,0.0,1.0,5,1,0,5,0,11,0,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[61,61,1.0,0.20089,0.1551,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,15,0,0,9,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"f98ab57258c3c5e9","q":"There are  $2006$  students and  $14$  teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least  $t.$  Find the maximum possible value of  $t.$","t":[{"b":2,"e":0.0,"k":"falling","v":0.00893,"x":0.4375,"p":[[0,26,0.0,0.20981,0.19877,0.0,0.2857,0.28571,0.0,0.57143,13,0,1,13,0,1,0,0,12,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[4,26,0.1538,0.4375,0.32915,0.21427,0.57143,0.57143,0.0,1.0,8,5,0,8,0,0,0,0,6,0,0,1,0,0,12,0,0,0,0,0,0,0,5],[8,26,0.3077,0.26339,0.28596,0.0,0.28571,0.28571,0.0,1.0,12,2,0,12,0,2,0,0,11,0,0,1,0,0,3,0,0,0,0,0,1,0,2],[12,26,0.4615,0.20536,0.26229,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,0,0,0,9,0,0,0,0,0,4,0,0,1,0,0,0,0,1],[16,26,0.6154,0.17411,0.25688,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,1,0,0,7,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[20,26,0.7692,0.0625,0.19541,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,26,0.9231,0.0625,0.15542,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[26,26,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"volatile","v":0.0,"x":0.52232,"p":[[0,18,0.0,0.13839,0.19393,0.0,0.0,0.28571,0.0,0.57143,20,0,1,20,0,0,0,0,8,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,18,0.2222,0.52232,0.35285,0.25,0.57143,0.78571,0.0,1.0,6,8,0,6,0,2,0,0,3,0,0,0,0,0,12,0,0,1,0,0,0,0,8],[8,18,0.4444,0.44196,0.32996,0.21429,0.5,0.57143,0.0,1.0,8,5,0,8,0,0,0,0,5,0,0,3,0,0,10,0,0,1,0,0,0,0,5],[12,18,0.6667,0.23661,0.27107,0.0,0.14286,0.32143,0.0,1.0,14,1,0,14,0,3,0,0,7,0,0,1,0,0,4,0,0,2,0,0,0,0,1],[16,18,0.8889,0.25893,0.35072,0.0,0.0,0.57143,0.0,1.0,18,4,0,18,0,1,0,0,3,0,0,1,0,0,5,0,0,0,0,0,0,0,4],[18,18,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"49cf4a3771ee3532","q":"The sequences of positive integers  $1,a_2,a_3,\\ldots$  and  $1,b_2,b_3,\\ldots$  are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let  $c_n=a_n+b_n$ . There is an integer  $k$  such that  $c_{k-1}=100$  and  $c_{k+1}=1000$ . Find  $c_k$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,214,0.0,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[4,214,0.0187,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,214,0.0374,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,214,0.0561,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,214,0.0748,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,214,0.0935,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,214,0.1121,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,214,0.1308,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,214,0.1495,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,214,0.1682,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,214,0.1869,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,214,0.2056,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,214,0.2243,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,214,0.243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,214,0.2617,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,214,0.2804,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,214,0.2991,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,214,0.3178,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,214,0.3364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,214,0.3551,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,214,0.3738,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,214,0.3925,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,214,0.4112,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,214,0.4299,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[96,214,0.4486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,214,0.4673,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,214,0.486,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[108,214,0.5047,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,214,0.5234,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,214,0.5421,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[120,214,0.5607,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,214,0.5794,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,214,0.5981,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[132,214,0.6168,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[136,214,0.6355,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[140,214,0.6542,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[144,214,0.6729,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[148,214,0.6916,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[152,214,0.7103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[156,214,0.729,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[160,214,0.7477,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[164,214,0.7664,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[168,214,0.785,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[172,214,0.8037,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[176,214,0.8224,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[180,214,0.8411,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[184,214,0.8598,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[188,214,0.8785,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[192,214,0.8972,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[196,214,0.9159,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[200,214,0.9346,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[204,214,0.9533,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[208,214,0.972,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[212,214,0.9907,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[214,214,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,254,0.0,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[4,254,0.0157,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,254,0.0315,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,254,0.0472,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,254,0.063,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,254,0.0787,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,254,0.0945,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,254,0.1102,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,254,0.126,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,254,0.1417,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,254,0.1575,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,254,0.1732,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,254,0.189,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,254,0.2047,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,254,0.2205,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,254,0.2362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,254,0.252,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,254,0.2677,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,254,0.2835,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,254,0.2992,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,254,0.315,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,254,0.3307,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,254,0.3465,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,254,0.3622,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,254,0.378,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,254,0.3937,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,254,0.4094,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,254,0.4252,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,254,0.4409,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[116,254,0.4567,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[120,254,0.4724,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,254,0.4882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[128,254,0.5039,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,254,0.5197,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[136,254,0.5354,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[140,254,0.5512,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[144,254,0.5669,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[148,254,0.5827,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[152,254,0.5984,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[156,254,0.6142,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[160,254,0.6299,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[164,254,0.6457,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[168,254,0.6614,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[172,254,0.6772,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[176,254,0.6929,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[180,254,0.7087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[184,254,0.7244,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[188,254,0.7402,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[192,254,0.7559,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[196,254,0.7717,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[200,254,0.7874,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[204,254,0.8031,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[208,254,0.8189,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[212,254,0.8346,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[216,254,0.8504,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[220,254,0.8661,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[224,254,0.8819,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[228,254,0.8976,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[232,254,0.9134,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[236,254,0.9291,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[240,254,0.9449,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[244,254,0.9606,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[248,254,0.9764,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[252,254,0.9921,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[254,254,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"80f8323a0d433143","q":"The coefficients  $a,b,c$  of a polynomial  $f:\\mathbb{R}\\to\\mathbb{R}, f(x)=x^3+ax^2+bx+c$  are mutually distinct integers and different from zero. Furthermore,  $f(a)=a^3$  and  $f(b)=b^3.$  Determine  $a,b$  and  $c$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.84598,"x":0.94196,"p":[[0,24,0.0,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,21],[4,24,0.1667,0.9375,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[8,24,0.3333,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[12,24,0.5,0.86607,0.10825,0.76786,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,1,0,12,1,10],[16,24,0.6667,0.90179,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[20,24,0.8333,0.85268,0.10403,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,15,0,8],[24,24,1.0,0.84598,0.10429,0.71429,0.85714,0.875,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,14,1,7]]},{"b":3,"e":1.0,"k":"flat","v":0.92857,"x":0.99554,"p":[[0,42,0.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[4,42,0.0952,0.94643,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[8,42,0.1905,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[12,42,0.2857,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[16,42,0.381,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[20,42,0.4762,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,42,0.5714,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,42,0.6667,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[32,42,0.7619,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[36,42,0.8571,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[40,42,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[42,42,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"934ed49d91649fc5","q":"The two circles $\\Gamma_{1}$ and $\\Gamma_{2}$ intersect at $P$ and $Q$. The common tangent on the side of $P$ touches the circles at $A$ and $B$ respectively. The tangent to $\\Gamma_{1}$ at $P$ intersects $\\Gamma_{2}$ again at $C$, and the tangent to $\\Gamma_{2}$ at $P$ intersects $\\Gamma_{1}$ again at $D$. The intersection of the lines $A P$ and $B C$ is called $E$, and the intersection of the lines $B P$ and $A D$ is called $F$. Let $M$ be the reflection of $P$ in the midpoint of $A B$. Prove that $A M B E Q F$ is a cyclic hexagon.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.22321,"x":0.45535,"p":[[0,41,0.0,0.26339,0.17536,0.14286,0.28571,0.28571,0.0,0.71429,4,0,1,4,0,9,0,0,12,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[4,41,0.0976,0.43749,0.20182,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,8,0,0,10,0,0,5,0,0,5,0,0,1,0,0],[8,41,0.1951,0.41964,0.2111,0.28571,0.42857,0.46431,0.0,1.0,2,1,0,2,0,1,0,0,10,0,0,11,0,0,2,0,0,5,0,0,0,0,1],[12,41,0.2927,0.45535,0.19044,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,9,0,0,3,0,0,7,0,0,1,0,0],[16,41,0.3902,0.45534,0.21558,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,13,0,0,5,0,0,5,0,0,3,0,0,4,0,0],[20,41,0.4878,0.38839,0.22084,0.2857,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,11,0,0,4,0,0,5,0,0,3,0,0,2,0,0],[24,41,0.5854,0.33927,0.25441,0.14286,0.2857,0.42858,0.0,1.0,4,1,0,4,0,7,0,0,9,0,0,5,0,0,3,0,0,1,0,0,2,0,1],[28,41,0.6829,0.36606,0.23401,0.25002,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,4,0,0,10,0,0,4,0,0,5,0,0,4,0,0,1,0,0],[32,41,0.7805,0.33482,0.21312,0.24999,0.28571,0.42858,0.0,0.85714,5,0,0,5,0,3,0,0,10,0,0,7,0,0,5,0,0,1,0,0,1,0,0],[36,41,0.878,0.40177,0.19703,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,9,0,0,6,0,0,10,0,0,1,0,0,1,0,0],[40,41,0.9756,0.29016,0.16162,0.2857,0.28571,0.32143,0.0,0.57143,5,0,0,5,0,1,0,0,18,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[41,41,1.0,0.22321,0.15542,0.10714,0.2857,0.28571,0.0,0.57143,8,0,0,8,0,4,0,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.26329,"x":0.62053,"p":[[0,65,0.0,0.26329,0.17173,0.14286,0.2857,0.42857,0.0,0.57143,4,0,3,4,0,10,0,0,9,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[4,65,0.0615,0.40178,0.23807,0.2857,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,3,0,0,9,0,0,8,0,0,3,0,0,3,0,0,3,0,0],[8,65,0.1231,0.46875,0.21792,0.28571,0.4286,0.60714,0.0,0.85714,1,0,0,1,0,3,0,0,7,0,0,6,0,0,7,0,0,6,0,0,2,0,0],[12,65,0.1846,0.39731,0.20432,0.2857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,10,0,0,7,0,0,6,0,0,3,0,0,1,0,0],[16,65,0.2462,0.3749,0.21361,0.2857,0.35714,0.4286,0.0,0.857,3,0,0,3,0,3,0,0,10,0,0,9,0,0,2,0,0,4,0,0,1,0,0],[20,65,0.3077,0.33482,0.19434,0.2857,0.28571,0.42858,0.0,0.85714,3,0,0,3,0,4,0,0,13,0,0,6,0,0,4,0,0,1,0,0,1,0,0],[24,65,0.3692,0.54463,0.19703,0.42857,0.4998,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,11,0,0,5,0,0,6,0,0,5,0,0],[28,65,0.4308,0.4554,0.21558,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,5,0,0,10,0,0,7,0,0,2,0,0,2,0,1],[32,65,0.4923,0.53572,0.25,0.42857,0.50001,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,3,0,0,10,0,0,5,0,0,4,0,0,6,0,1],[36,65,0.5538,0.53567,0.19232,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,6,0,0,11,0,0,6,0,0,3,0,0],[40,65,0.6154,0.59359,0.21154,0.42857,0.57143,0.74996,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,4,0,0,10,0,0,5,0,0,8,0,0],[44,65,0.6769,0.62053,0.16601,0.42859,0.64286,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,9,0,0,6,0,0,10,0,0,6,0,0],[48,65,0.7385,0.51337,0.18161,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,10,0,0,12,0,0,1,0,0,4,0,0],[52,65,0.8,0.53125,0.16457,0.42857,0.57143,0.60714,0.1429,0.85714,0,0,0,0,0,1,0,0,3,0,0,10,0,0,10,0,0,6,0,0,2,0,0],[56,65,0.8615,0.49552,0.19556,0.42857,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,9,0,0,12,0,0,2,0,0,3,0,0],[60,65,0.9231,0.45534,0.18362,0.28571,0.49979,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,7,0,0,6,0,0,13,0,0,2,0,0,1,0,0],[64,65,0.9846,0.4509,0.21458,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,5,0,0,9,0,0,8,0,0,2,0,0,3,0,0],[65,65,1.0,0.40179,0.15335,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,3,0,0,9,0,0,14,0,0,4,0,0,1,0,0,1,0,0]]}]},{"i":"b6ac2ff260da02ae","q":"The persons  $P_1, P_2, . . . , P_{n-1}, P_n$  sit around a table, in this order, and each one of them has a number of coins. In the start,  $P_1$  has one coin more than  $P_2, P_2$  has one coin more than  $P_3$ , etc., up to  $P_{n-1}$  who has one coin more than  $P_n$ . Now  $P_1$  gives one coin to  $P_2$ , who in turn gives two coins to  $P_3 $  etc., up to  $ Pn$  who gives n coins to  $ P_1$ . Now the process continues in the same way:  $P_1$  gives  $n+ 1$  coins to  $P_2$ ,  $P_2$  gives  $n+2$  coins to  $P_3$ ; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.","t":[{"b":1,"e":0.28571,"k":"falling","v":0.20982,"x":0.69643,"p":[[0,143,0.0,0.69643,0.36025,0.42857,0.85714,1.0,0.0,1.0,5,10,4,5,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,0,11,0,10],[4,143,0.028,0.62045,0.30236,0.42857,0.71429,0.85714,0.0,1.0,2,4,0,2,0,3,0,0,2,0,0,3,0,0,4,0,0,5,0,0,9,0,4],[8,143,0.0559,0.61607,0.2911,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,6,0,0,1,0,0,3,0,0,5,0,0,6,0,0,6,0,5],[12,143,0.0839,0.58482,0.29743,0.42857,0.71429,0.85714,0.0,1.0,3,2,0,3,0,3,0,0,1,0,0,3,0,0,5,0,0,7,0,0,8,0,2],[16,143,0.1119,0.61158,0.29284,0.39286,0.71429,0.85714,0.0,1.0,1,2,0,1,0,4,0,0,3,0,0,3,0,0,3,0,0,4,0,0,12,0,2],[20,143,0.1399,0.66516,0.26392,0.57143,0.78564,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,3,0,0,0,0,0,4,0,0,6,0,0,16,0,0],[24,143,0.1678,0.60045,0.3378,0.2857,0.71429,0.85714,0.0,1.0,2,6,0,2,0,3,0,0,6,0,1,2,0,0,0,0,0,3,0,0,9,0,6],[28,143,0.1958,0.58929,0.28065,0.39286,0.71429,0.85714,0.0,1.0,2,1,0,2,0,2,0,0,4,0,0,4,0,0,1,0,0,9,0,0,9,0,1],[32,143,0.2238,0.69642,0.27606,0.71421,0.78571,0.85714,0.0,1.0,1,4,0,1,0,3,0,0,2,0,0,0,0,0,1,0,0,9,0,0,12,0,4],[36,143,0.2517,0.55358,0.32683,0.28571,0.71429,0.85714,0.0,1.0,4,3,0,4,0,2,0,0,5,0,0,3,0,0,0,0,0,8,0,0,7,0,3],[40,143,0.2797,0.53572,0.31134,0.28571,0.50001,0.85714,0.0,1.0,3,1,0,3,0,2,0,0,7,0,0,4,0,0,1,0,0,3,0,0,11,0,1],[44,143,0.3077,0.50446,0.31539,0.14286,0.42859,0.85714,0.0,1.0,1,3,0,1,0,8,0,0,4,0,0,4,0,0,2,0,0,4,0,0,6,0,3],[48,143,0.3357,0.51778,0.33844,0.24999,0.42859,0.85714,0.0,1.0,2,3,0,2,0,6,0,0,7,0,0,2,0,0,1,0,0,1,0,0,10,0,3],[52,143,0.3636,0.58928,0.31491,0.28571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,9,0,0,1,0,0,2,0,0,5,0,0,7,0,5],[56,143,0.3916,0.54009,0.32101,0.28571,0.71429,0.75,0.0,1.0,3,3,0,3,0,4,0,0,5,0,0,2,0,0,0,0,0,10,0,0,5,0,3],[60,143,0.4196,0.51343,0.28314,0.28571,0.4293,0.85714,0.0,1.0,1,1,0,1,0,5,0,0,5,0,0,6,0,0,3,0,0,3,0,0,8,0,1],[64,143,0.4476,0.49105,0.29,0.2857,0.42857,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,11,0,0,2,0,0,5,0,0,3,0,0,4,0,3],[68,143,0.4755,0.58019,0.29214,0.28571,0.571,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,11,0,0,3,0,0,2,0,0,3,0,0,8,0,4],[72,143,0.5035,0.59374,0.34276,0.28571,0.71429,0.85714,0.0,1.0,4,5,0,4,0,0,0,0,8,0,0,1,0,0,1,0,0,3,0,0,10,0,5],[76,143,0.5315,0.57141,0.32537,0.28571,0.64264,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,9,0,0,2,0,0,1,0,0,2,0,0,10,0,4],[80,143,0.5594,0.4866,0.29851,0.2857,0.28571,0.85714,0.0,1.0,2,1,0,2,0,2,0,0,13,0,0,2,0,0,1,0,0,2,0,0,9,0,1],[84,143,0.5874,0.5311,0.3158,0.2857,0.57143,0.85704,0.0,1.0,3,3,0,3,0,1,0,0,10,0,0,1,0,0,4,0,0,2,0,0,8,0,3],[88,143,0.6154,0.49097,0.27868,0.28571,0.42929,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,8,0,0,4,0,0,3,0,0,7,0,0,3,0,2],[92,143,0.6434,0.44643,0.25939,0.28571,0.35714,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,11,0,0,5,0,0,2,0,0,4,0,0,4,0,1],[96,143,0.6713,0.55357,0.32488,0.28571,0.4286,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,10,0,0,3,0,0,1,0,0,2,0,0,6,0,6],[100,143,0.6993,0.3683,0.29403,0.14286,0.28571,0.66071,0.0,0.85714,6,0,0,6,0,4,0,0,11,0,0,1,0,0,1,1,0,3,0,0,5,0,0],[104,143,0.7273,0.36607,0.30291,0.25,0.28571,0.46429,0.0,1.0,6,2,0,6,0,2,0,0,15,0,0,1,0,0,1,0,0,1,0,0,4,0,2],[108,143,0.7552,0.29018,0.23279,0.14286,0.28571,0.28571,0.0,1.0,5,1,0,5,0,5,0,0,17,0,0,1,0,0,0,0,0,2,0,0,1,0,1],[112,143,0.7832,0.35268,0.28568,0.14286,0.28571,0.42858,0.0,1.0,5,3,0,5,0,4,0,0,13,0,0,3,0,0,2,0,0,1,0,0,1,0,3],[116,143,0.8112,0.34379,0.26454,0.14286,0.28571,0.42857,0.0,1.0,5,1,1,5,0,4,0,0,12,0,0,5,0,0,1,0,0,1,0,0,3,0,1],[120,143,0.8392,0.30357,0.24936,0.14289,0.2857,0.28571,0.0,1.0,5,1,0,5,0,4,0,0,18,0,0,1,0,0,0,0,0,0,0,0,3,0,1],[124,143,0.8671,0.36159,0.2624,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,3,0,0,19,0,0,0,0,0,1,0,0,1,0,0,4,0,1],[128,143,0.8951,0.26777,0.23628,0.14214,0.28571,0.28571,0.0,0.85714,7,0,0,7,0,6,0,0,14,0,0,0,0,0,1,0,0,2,0,0,2,0,0],[132,143,0.9231,0.35268,0.26241,0.2857,0.28571,0.4286,0.0,0.85714,6,0,0,6,0,1,0,0,14,0,0,4,0,0,1,0,0,2,0,0,4,0,0],[136,143,0.951,0.25,0.21429,0.10714,0.2857,0.28571,0.0,1.0,8,1,0,8,0,3,0,0,17,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[140,143,0.979,0.28126,0.16164,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,2,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[143,143,1.0,0.20982,0.11837,0.14286,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,3,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.82142,"p":[[0,189,0.0,0.82142,0.29666,0.85711,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,17],[4,189,0.0212,0.74106,0.2299,0.71429,0.85714,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,3,0,0,1,0,0,8,0,0,14,0,4],[8,189,0.0423,0.42393,0.32451,0.14286,0.28571,0.71429,0.0,1.0,4,1,0,4,0,9,0,0,4,0,0,1,0,0,3,0,0,4,0,0,6,0,1],[12,189,0.0635,0.24106,0.24335,0.0,0.1429,0.32143,0.0,0.85714,10,0,0,10,0,8,0,0,6,0,0,2,0,0,3,0,0,2,0,0,1,0,0],[16,189,0.0847,0.16062,0.17036,0.0,0.14286,0.2857,0.0,0.85714,10,0,0,10,0,13,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[20,189,0.1058,0.21875,0.23415,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,8,0,0,8,0,0,4,0,0,0,0,0,0,0,0,1,0,1],[24,189,0.127,0.165,0.18937,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,8,0,0,7,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[28,189,0.1481,0.1875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is the area of the figure in the complex plane enclosed by the origin and the set of all points  $\\tfrac{1}{z}$  such that  $(1-2i)z+(-2i-1)\\overline{z}=6i$ ?","t":[{"b":5,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,16,0.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[4,16,0.25,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,16,0.5,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[12,16,0.75,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,16,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.91518,"x":0.97321,"p":[[0,47,0.0,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[4,47,0.0851,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,47,0.1702,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[12,47,0.2553,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[16,47,0.3404,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[20,47,0.4255,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[24,47,0.5106,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[28,47,0.5957,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,47,0.6809,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[36,47,0.766,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[40,47,0.8511,0.91964,0.11259,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[44,47,0.9362,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[47,47,1.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]}]},{"i":"774ffe94b41ced22","q":"There are  $n$  rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number  $k$  such that any  $k$  couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.","t":[{"b":0,"e":1.0,"k":"flat","v":0.67411,"x":0.79464,"p":[[0,27,0.0,0.67857,0.34627,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,17],[4,27,0.1481,0.69195,0.30328,0.42857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,6,0,0,2,0,0,0,0,0,4,0,13],[8,27,0.2963,0.79464,0.27879,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,4,0,17],[12,27,0.4444,0.67411,0.30978,0.39286,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,6,0,0,3,0,0,1,0,0,5,0,0,5,0,10],[16,27,0.5926,0.74106,0.24599,0.4286,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,4,0,0,4,0,12],[20,27,0.7407,0.70087,0.27049,0.42857,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,5,0,0,4,0,0,2,0,0,6,0,10],[24,27,0.8889,0.79463,0.23404,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,6,0,0,3,0,15],[27,27,1.0,0.79018,0.2789,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,3,0,0,2,0,18]]},{"b":3,"e":1.0,"k":"flat","v":0.59372,"x":0.80802,"p":[[0,34,0.0,0.59372,0.3409,0.28571,0.42857,1.0,0.0,1.0,1,12,0,1,0,0,0,0,13,0,0,3,0,0,2,0,0,0,0,0,1,0,12],[4,34,0.1176,0.64284,0.28572,0.28571,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,9,0,0,3,0,0,4,0,0,4,0,0,3,0,9],[8,34,0.2353,0.7232,0.24728,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,7,0,0,3,0,0,6,0,0,3,0,11],[12,34,0.3529,0.78571,0.25254,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,5,0,0,2,0,16],[16,34,0.4706,0.69641,0.28516,0.42857,0.78564,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,0,7,0,0,3,0,0,1,0,0,4,0,12],[20,34,0.5882,0.73656,0.2205,0.571,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,0,6,0,9],[24,34,0.7059,0.72308,0.24728,0.53572,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,8,0,0,2,0,11],[28,34,0.8235,0.80802,0.23314,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,2,0,17],[32,34,0.9412,0.71427,0.28348,0.42857,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,6,0,0,2,0,0,3,0,0,3,0,13],[34,34,1.0,0.70981,0.27313,0.42857,0.57143,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,9,0,0,6,0,0,0,0,0,1,0,14]]}]},{"i":"537e978182393e1e","q":"The triangle  $ABC$  is given. Let  $A', B'$  and  $C'$  be the midpoints of the sides  $BC, CA$  and  $AB$  and  $O_a,O_b$  and  $O_c$  be the circumcenters of the triangles  $CAC', ABA'$  and  $BCB'$  respectively. Prove that the triangles  $ABC$  and  $O_aO_bO_c$  are similar.\n\n*Proposed by Don Luu (Vietnam)*","t":[{"b":4,"e":0.85714,"k":"flat","v":0.11161,"x":0.45533,"p":[[0,76,0.0,0.11161,0.19145,0.0,0.0,0.14286,0.0,0.71429,21,0,1,21,0,5,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,76,0.0526,0.38839,0.37327,0.0,0.28571,0.75,0.0,1.0,10,5,0,10,0,4,0,0,3,0,0,5,0,0,1,0,0,1,0,0,3,0,5],[8,76,0.1053,0.4464,0.35489,0.14286,0.42857,0.74996,0.0,1.0,6,5,0,6,0,6,0,0,2,0,0,5,0,0,3,0,0,2,0,0,3,0,5],[12,76,0.1579,0.28123,0.266,0.0,0.1429,0.571,0.0,0.71429,10,0,0,10,0,7,0,0,3,0,0,3,0,0,4,0,0,5,0,0,0,0,0],[16,76,0.2105,0.34818,0.34428,0.0,0.2857,0.60682,0.0,1.0,11,3,0,11,0,3,0,0,5,0,0,3,0,0,2,0,0,3,0,0,2,0,3],[20,76,0.2632,0.39731,0.33452,0.10714,0.42857,0.71429,0.0,1.0,8,3,0,8,0,5,0,0,2,0,0,5,0,0,2,0,0,6,0,0,1,0,3],[24,76,0.3158,0.32132,0.32736,0.0,0.2143,0.4642,0.0,1.0,11,2,0,11,0,5,0,0,2,0,0,6,0,0,2,0,0,1,0,0,3,0,2],[28,76,0.3684,0.41514,0.33569,0.10714,0.42857,0.71429,0.0,1.0,8,2,0,8,0,4,0,0,2,0,0,5,0,0,3,0,0,4,0,0,4,0,2],[32,76,0.4211,0.39728,0.3345,0.0,0.42857,0.60714,0.0,1.0,9,2,0,9,0,3,0,0,3,0,0,4,0,0,5,0,0,2,0,0,4,0,2],[36,76,0.4737,0.38834,0.3411,0.0,0.42857,0.571,0.0,1.0,10,5,0,10,0,1,0,0,3,0,0,8,0,0,4,0,0,1,0,0,0,0,5],[40,76,0.5263,0.37499,0.32487,0.0,0.42857,0.60714,0.0,1.0,10,3,0,10,0,2,0,0,2,0,0,8,0,0,2,0,0,5,0,0,0,0,3],[44,76,0.5789,0.28568,0.30301,0.0,0.14286,0.571,0.0,1.0,12,1,0,12,0,6,0,0,1,0,0,4,0,0,5,0,0,1,0,0,2,0,1],[48,76,0.6316,0.4419,0.35776,0.14286,0.42836,0.71429,0.0,1.0,7,5,0,7,0,5,0,0,4,0,0,0,0,0,5,0,0,5,0,0,1,0,5],[52,76,0.6842,0.36601,0.31525,0.0,0.35714,0.5711,0.0,1.0,9,1,0,9,0,4,0,0,3,0,0,4,0,0,6,0,0,1,0,0,4,0,1],[56,76,0.7368,0.26789,0.2594,0.0,0.2857,0.42858,0.0,0.85714,12,0,0,12,0,2,0,0,6,0,0,7,0,0,2,0,0,1,0,0,2,0,0],[60,76,0.7895,0.32143,0.3481,0.0,0.14288,0.57143,0.0,1.0,11,4,0,11,0,7,0,0,1,0,0,4,0,0,3,0,0,1,0,0,1,0,4],[64,76,0.8421,0.29462,0.2965,0.0,0.2143,0.46418,0.0,1.0,11,1,0,11,0,5,0,0,4,0,0,4,0,0,2,0,0,4,0,0,1,0,1],[68,76,0.8947,0.45533,0.36146,0.14286,0.42857,0.74996,0.0,1.0,6,4,0,6,0,7,0,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,4],[72,76,0.9474,0.27231,0.27047,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,7,0,0,3,0,0,6,0,0,3,0,0,1,0,0,1,0,1],[76,76,1.0,0.25,0.30723,0.0,0.07143,0.46431,0.0,0.85714,16,0,0,16,0,3,0,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,0]]},{"b":5,"e":0.14,"k":"rising","v":0.11161,"x":0.49106,"p":[[0,115,0.0,0.11161,0.15866,0.0,0.0,0.14286,0.0,0.57143,18,0,3,18,0,8,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,115,0.0348,0.3973,0.34018,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,4,0,0,1,0,0,6,0,0,3,0,0,4,0,0,2,0,3],[8,115,0.0696,0.39732,0.3402,0.10714,0.42857,0.71429,0.0,1.0,8,2,0,8,0,6,0,0,1,0,0,5,0,0,2,0,0,4,0,0,4,0,2],[12,115,0.1043,0.33482,0.33237,0.0,0.2857,0.4286,0.0,1.0,10,3,0,10,0,5,0,0,3,0,0,7,0,0,0,0,0,2,0,0,2,0,3],[16,115,0.1391,0.45085,0.3446,0.10714,0.4998,0.75,0.0,1.0,8,2,0,8,0,2,0,0,4,0,0,2,0,0,5,0,0,3,0,0,6,0,2],[20,115,0.1739,0.31694,0.33259,0.0,0.14288,0.57111,0.0,1.0,13,2,0,13,0,4,0,0,0,0,0,5,0,0,3,0,0,4,0,0,1,0,2],[24,115,0.2087,0.32142,0.33692,0.0,0.28571,0.60714,0.0,1.0,14,2,0,14,0,1,0,0,3,0,0,4,0,0,2,0,0,5,0,0,1,0,2],[28,115,0.2435,0.38391,0.33585,0.10714,0.28571,0.71429,0.0,1.0,8,3,0,8,0,5,0,0,4,0,0,4,0,0,2,0,0,4,0,0,2,0,3],[32,115,0.2783,0.36158,0.36765,0.0,0.28571,0.71429,0.0,1.0,14,2,0,14,0,2,0,0,0,0,0,3,0,0,2,0,0,6,0,0,3,0,2],[36,115,0.313,0.43304,0.35801,0.0,0.42857,0.75,0.0,1.0,9,5,0,9,0,1,0,0,3,0,0,8,0,0,2,0,0,1,0,0,3,0,5],[40,115,0.3478,0.34819,0.29651,0.10714,0.2857,0.57143,0.0,1.0,8,1,0,8,0,6,0,0,3,0,0,4,0,0,4,0,0,5,0,0,1,0,1],[44,115,0.3826,0.33482,0.34738,0.0,0.14288,0.71429,0.0,1.0,11,3,0,11,0,6,0,0,2,0,0,3,0,0,1,0,0,5,0,0,1,0,3],[48,115,0.4174,0.37499,0.30461,0.14286,0.42857,0.60682,0.0,1.0,7,2,0,7,0,6,0,0,1,0,0,9,0,0,1,0,0,5,0,0,1,0,2],[52,115,0.4522,0.39283,0.30092,0.14286,0.42857,0.60714,0.0,1.0,6,2,0,6,0,6,0,0,2,0,0,7,0,0,3,0,0,5,0,0,1,0,2],[56,115,0.487,0.38393,0.35072,0.0,0.28571,0.71429,0.0,1.0,9,4,0,9,0,5,0,0,3,0,0,3,0,0,3,0,0,4,0,0,1,0,4],[60,115,0.5217,0.40177,0.32622,0.10714,0.42857,0.60714,0.0,1.0,8,3,0,8,0,3,0,0,3,0,0,7,0,0,3,0,0,3,0,0,2,0,3],[64,115,0.5565,0.36607,0.28107,0.14286,0.35714,0.71429,0.0,0.85714,5,0,0,5,0,9,0,0,2,0,0,6,0,0,1,0,0,7,0,0,2,0,0],[68,115,0.5913,0.36159,0.30089,0.10714,0.28571,0.57111,0.0,1.0,8,2,0,8,0,3,0,0,6,0,0,6,0,0,2,0,0,4,0,0,1,0,2],[72,115,0.6261,0.43078,0.33809,0.10714,0.4998,0.66082,0.0,1.0,8,3,0,8,0,3,0,0,3,0,0,2,0,0,7,1,0,2,0,0,3,0,3],[76,115,0.6609,0.35712,0.29664,0.10714,0.28571,0.57143,0.0,1.0,8,1,0,8,0,5,0,0,4,0,0,2,0,0,8,0,0,2,0,0,2,0,1],[80,115,0.6957,0.33481,0.3285,0.0,0.14293,0.57143,0.0,1.0,9,3,0,9,0,8,0,0,2,0,0,2,0,0,4,0,0,4,0,0,0,0,3],[84,115,0.7304,0.3571,0.32534,0.0,0.35714,0.57111,0.0,1.0,10,2,0,10,0,4,0,0,2,0,0,5,0,0,4,0,0,3,0,0,2,0,2],[88,115,0.7652,0.34821,0.35344,0.0,0.2857,0.60714,0.0,1.0,11,4,0,11,0,4,0,0,4,0,0,3,0,0,2,0,0,3,0,0,1,0,4],[92,115,0.8,0.4375,0.36932,0.0,0.42857,0.71429,0.0,1.0,10,4,0,10,0,2,0,0,2,0,0,4,0,0,0,0,0,8,0,0,2,0,4],[96,115,0.8348,0.49106,0.37446,0.10714,0.5,0.75,0.0,1.0,8,6,0,8,0,2,0,0,3,0,0,3,0,0,1,0,0,7,0,0,2,0,6],[100,115,0.8696,0.39284,0.34624,0.0,0.28571,0.74996,0.0,1.0,9,1,0,9,0,4,0,0,4,0,0,3,0,0,2,0,0,2,0,0,7,0,1],[104,115,0.9043,0.48661,0.35149,0.14286,0.42857,0.75,0.0,1.0,4,6,0,4,0,6,0,0,4,0,0,4,0,0,1,0,0,5,0,0,2,0,6],[108,115,0.9391,0.44194,0.33188,0.14286,0.42857,0.71429,0.0,1.0,6,3,0,6,0,5,0,0,2,0,0,5,0,0,5,0,0,2,0,0,4,0,3],[112,115,0.9739,0.33473,0.28265,0.14286,0.2857,0.4286,0.0,1.0,5,1,0,5,0,10,0,0,4,0,0,6,0,0,0,0,0,4,0,0,2,0,1],[115,115,1.0,0.30795,0.32367,0.105,0.14286,0.42857,0.0,1.0,8,3,0,8,0,11,0,0,2,0,0,4,0,0,0,0,0,3,0,0,1,0,3]]}]},{"i":"e35d8158c174d2ee","q":"There are 2006 points marked on the surface of a sphere. Prove that the surface can be cut into 2006 congruent pieces so that each piece contains exactly one of these points inside it.","t":[{"b":3,"e":0.4286,"k":"rising","v":0.02232,"x":0.76335,"p":[[0,21,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,20,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.49554,0.25995,0.42857,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,0,0,0,2,0,0,13,0,0,4,0,0,2,0,0,7,0,0],[8,21,0.381,0.54909,0.17536,0.42857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,16,0,0,6,0,0,4,0,0,5,0,0],[12,21,0.5714,0.59819,0.16146,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,8,0,0,6,0,0,6,0,0],[16,21,0.7619,0.62051,0.26149,0.42857,0.64286,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,4,0,0,9,0,0,2,0,0,3,0,0,10,0,3],[20,21,0.9524,0.75445,0.14391,0.71429,0.85707,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,11,0,0,17,0,0],[21,21,1.0,0.76335,0.12171,0.71429,0.857,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,17,0,0]]},{"b":6,"e":0.85714,"k":"volatile","v":0.03125,"x":0.72324,"p":[[0,15,0.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,18,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.52677,0.24597,0.42857,0.49979,0.71429,0.0,1.0,3,2,0,3,0,0,0,0,1,0,0,12,0,0,7,0,0,4,0,0,3,0,2],[8,15,0.5333,0.60268,0.23347,0.42857,0.57143,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,7,0,0,6,0,0,6,0,0,8,0,1],[12,15,0.8,0.72324,0.22846,0.57132,0.85714,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,0,18,0,2],[15,15,1.0,0.66962,0.26108,0.5354,0.71429,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,0,0,0,5,0,0,2,0,0,9,0,0,10,0,3]]}]},{"i":"278fc3abfb44dee9","q":"There are relatively prime positive integers  $p$  and  $q$  such that  $\\dfrac{p}{q}=\\displaystyle\\sum_{n=3}^{\\infty} \\dfrac{1}{n^5-5n^3+4n}$ .  Find  $p+q$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.58036,"x":0.70982,"p":[[0,32,0.0,0.70536,0.18877,0.57143,0.57143,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,3,0,0,0,0,9],[4,32,0.125,0.70088,0.1902,0.57143,0.57143,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,2,0,0,0,0,9],[8,32,0.25,0.70982,0.18723,0.57143,0.57143,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,4,0,0,0,0,9],[12,32,0.375,0.59374,0.05188,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0],[16,32,0.5,0.58927,0.05923,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0],[20,32,0.625,0.58036,0.03458,0.57143,0.57143,0.57143,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[24,32,0.75,0.58929,0.05923,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0],[28,32,0.875,0.58036,0.03458,0.57143,0.57143,0.57143,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[32,32,1.0,0.58482,0.05486,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.59374,"x":0.84375,"p":[[0,34,0.0,0.69196,0.1992,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,2,0,0,0,0,9],[4,34,0.1176,0.66071,0.18814,0.57143,0.57143,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,2,0,0,0,0,7],[8,34,0.2353,0.69196,0.18595,0.57143,0.57143,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,4,0,0,0,0,8],[12,34,0.3529,0.84375,0.19678,0.57143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,1,0,19],[16,34,0.4706,0.74554,0.20119,0.57143,0.57143,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,3,0,0,0,0,12],[20,34,0.5882,0.66071,0.15465,0.57143,0.57143,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,5,0,0,0,0,5],[24,34,0.7059,0.60268,0.05906,0.57143,0.57143,0.57143,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0],[28,34,0.8235,0.59374,0.05188,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0],[32,34,0.9412,0.62054,0.07668,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,12,0,0,0,0,0],[34,34,1.0,0.5982,0.05577,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0]]}]},{"i":"653e455a98b897c3","q":"Which is greater:  $ 17091982!^2$  or  $ 17091982^{17091982}$ ?","t":[{"b":0,"e":0.14286,"k":"falling","v":0.20089,"x":0.54902,"p":[[0,16,0.0,0.53125,0.39161,0.14286,0.28571,1.0,0.14286,1.0,0,13,0,0,0,10,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[4,16,0.25,0.54902,0.37656,0.28571,0.28571,1.0,0.14,1.0,0,13,0,0,0,6,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[8,16,0.5,0.46429,0.36596,0.14286,0.28571,1.0,0.14286,1.0,0,10,0,0,0,10,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[12,16,0.75,0.34375,0.28984,0.14286,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,12,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,16,1.0,0.20089,0.07016,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.31697,"x":1.0,"p":[[0,38,0.0,0.58482,0.41705,0.14286,0.64286,1.0,0.14286,1.0,0,16,0,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,38,0.1053,0.48214,0.37923,0.14286,0.28571,1.0,0.14286,1.0,0,11,0,0,0,11,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[8,38,0.2105,0.42411,0.36506,0.14286,0.28571,1.0,0.14286,1.0,0,9,0,0,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[12,38,0.3158,0.31697,0.26663,0.14286,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[38,38,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8fe3c9a63acb38a7","q":"$2015$  points are given in a plane such that from any five points we can choose two points with distance less than  $1$  unit. Prove that  $504$  of the given points lie on a unit disc.","t":[{"b":1,"e":0.0,"k":"falling","v":0.12053,"x":0.70088,"p":[[0,26,0.0,0.70088,0.32997,0.42857,0.85714,1.0,0.0,1.0,2,16,2,2,0,1,0,0,1,0,0,7,0,0,4,0,0,1,0,0,0,0,16],[4,26,0.1538,0.61161,0.39647,0.2857,0.78571,1.0,0.0,1.0,4,14,0,4,0,3,0,0,5,0,0,3,0,0,0,0,0,1,0,0,2,0,14],[8,26,0.3077,0.34375,0.3131,0.14286,0.28571,0.32143,0.0,1.0,6,5,0,6,0,4,0,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,5],[12,26,0.4615,0.16071,0.15047,0.0,0.21428,0.28571,0.0,0.42857,14,0,0,14,0,2,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.18303,0.1684,0.0,0.2857,0.28571,0.0,0.57143,13,0,1,13,0,2,0,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,26,0.7692,0.125,0.15465,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,2,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,26,0.9231,0.16071,0.15465,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,3,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.12053,0.15198,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.26786,"x":0.73659,"p":[[0,20,0.0,0.70533,0.28108,0.42857,0.64286,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,7,0,0,6,0,0,2,0,0,0,0,14],[4,20,0.2,0.73659,0.35376,0.42857,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,19],[8,20,0.4,0.58036,0.36932,0.28571,0.42857,1.0,0.0,1.0,2,13,0,2,0,2,0,0,11,0,0,2,0,0,1,0,0,1,0,0,0,0,13],[12,20,0.6,0.45973,0.31699,0.2857,0.28571,0.60714,0.14,1.0,0,7,0,0,0,7,0,0,12,0,0,2,0,0,3,0,0,1,0,0,0,0,7],[16,20,0.8,0.41066,0.26435,0.28571,0.28571,0.46525,0.14,1.0,0,4,0,0,0,5,0,0,16,0,0,3,0,0,2,0,0,2,0,0,0,0,4],[20,20,1.0,0.26786,0.16269,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,12,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,1]]}]},{"i":"3f05a6d480b0f169","q":"Zij $k$ een positief geheel getal en geef de som van de cijfers van een positief geheel getal $n$ aan met $s(n)$. Bewijs dat er onder de positieve gehele getallen met $k$ cijfers evenveel getallen $n$ zijn die voldoen aan $s(n)<s(2 n)$ als getallen $n$ die voldoen aan $s(n)>s(2 n)$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.54464,"x":0.97768,"p":[[0,45,0.0,0.85265,0.25379,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,7,0,19],[4,45,0.0889,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[8,45,0.1778,0.91964,0.2257,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,28],[12,45,0.2667,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[16,45,0.3556,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,45,0.4444,0.93304,0.1988,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,27],[24,45,0.5333,0.95536,0.16536,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[28,45,0.6222,0.92857,0.19562,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[32,45,0.7111,0.94643,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[36,45,0.8,0.87054,0.21535,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,4,0,21],[40,45,0.8889,0.54464,0.34523,0.2857,0.42859,1.0,0.0,1.0,1,9,0,1,0,6,0,0,6,0,0,5,0,0,0,0,0,4,0,0,1,0,9],[44,45,0.9778,0.86607,0.14258,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,15,0,11],[45,45,1.0,0.90179,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,15]]},{"b":5,"e":1.0,"k":"flat","v":0.87054,"x":0.99554,"p":[[0,44,0.0,0.87054,0.24053,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,20],[4,44,0.0909,0.92411,0.20511,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,27],[8,44,0.1818,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[12,44,0.2727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,44,0.3636,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[20,44,0.4545,0.95536,0.12078,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[24,44,0.5455,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[28,44,0.6364,0.92411,0.1988,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[32,44,0.7273,0.90625,0.22192,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[36,44,0.8182,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,44,0.9091,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[44,44,1.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"d023b8ed5ee2d386","q":"given a positive integer  $n$ .\nthe set  $\\{ 1,2,..,2n \\}$  is partitioned into  $a_1<a_2<...<a_n $  and  $b_1>b_2>...>b_n$ .\nfind the value of :  $ \\sum_{i=1}^{n}|a_i - b_i| $","t":[{"b":1,"e":1.0,"k":"rising","v":0.4508,"x":0.82143,"p":[[0,47,0.0,0.4508,0.35202,0.14286,0.21428,0.71429,0.0,1.0,1,6,0,1,0,15,0,0,1,0,0,0,0,0,4,0,0,4,0,0,1,0,6],[4,47,0.0851,0.48214,0.33834,0.14286,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,15,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,5],[8,47,0.1702,0.54018,0.35845,0.14286,0.71429,0.85714,0.0,1.0,1,7,0,1,0,12,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,7],[12,47,0.2553,0.53571,0.3481,0.14286,0.57143,0.78571,0.0,1.0,1,8,0,1,0,11,0,0,0,0,0,0,0,0,7,0,0,5,0,0,0,0,8],[16,47,0.3404,0.46874,0.33737,0.14286,0.5712,0.71429,0.14286,1.0,0,6,0,0,0,15,0,0,0,0,0,0,0,0,8,0,0,2,0,0,1,0,6],[20,47,0.4255,0.45536,0.34151,0.14286,0.35714,0.71429,0.0,1.0,1,4,0,1,0,15,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,4],[24,47,0.5106,0.49107,0.34615,0.14286,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,15,0,0,0,0,0,0,0,0,2,0,0,9,0,0,0,0,6],[28,47,0.5957,0.52232,0.35823,0.14286,0.57143,0.78571,0.0,1.0,1,8,0,1,0,12,0,0,0,0,0,2,0,0,2,0,0,7,0,0,0,0,8],[32,47,0.6809,0.82143,0.20825,0.71429,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,12,0,0,0,0,16],[36,47,0.766,0.77232,0.14223,0.71429,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,0,0,8],[40,47,0.8511,0.75446,0.15663,0.71429,0.71429,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,23,0,0,0,0,7],[44,47,0.9362,0.80804,0.1411,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,0,0,11],[47,47,1.0,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5]]},{"b":6,"e":1.0,"k":"rising","v":0.38839,"x":0.86161,"p":[[0,19,0.0,0.38839,0.30978,0.14286,0.14286,0.71429,0.0,1.0,1,3,0,1,0,17,0,0,0,0,0,1,0,0,4,0,0,6,0,0,0,0,3],[4,19,0.2105,0.49554,0.34066,0.14286,0.57143,0.71429,0.0,1.0,1,6,0,1,0,12,0,0,1,0,0,1,0,0,3,0,0,8,0,0,0,0,6],[8,19,0.4211,0.79018,0.21424,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,11,0,0,0,0,14],[12,19,0.6316,0.82143,0.18898,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,0,0,1,0,14],[16,19,0.8421,0.79911,0.21975,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,13,0,0,1,0,14],[19,19,1.0,0.86161,0.17672,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,2,0,18]]}]},{"i":"b536a72fa150c32a","q":"Let  $n\\ge2$ , let  $A_1,A_2,\\ldots,A_{n+1}$  be  $n+1$  points in the\n   $n$ -dimensional Euclidean space, not lying on the same hyperplane,\n  and let  $B$  be a point strictly inside the convex hull of\n   $A_1,A_2,\\ldots,A_{n+1}$ . Prove that  $\\angle A_iBA_j&gt;90^\\circ$  holds\n  for at least  $n$  pairs  $(i,j)$  with  $\\displaystyle{1\\le i&lt;j\\le\n    n+1}$ .\n\nProposed by G\u00e9za K\u00f3s, E\u00f6tv\u00f6s University, Budapest","t":[{"b":4,"e":0.28571,"k":"falling","v":0.22768,"x":0.77679,"p":[[0,53,0.0,0.42411,0.40166,0.14286,0.14286,1.0,0.0,1.0,1,10,1,1,0,20,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,10],[4,53,0.0755,0.77679,0.36932,0.57143,1.0,1.0,0.14286,1.0,0,23,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,23],[8,53,0.1509,0.44197,0.39506,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,20,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,10],[12,53,0.2264,0.5625,0.39599,0.14286,0.4286,1.0,0.14286,1.0,0,13,0,0,0,13,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,13],[16,53,0.3019,0.36151,0.33883,0.14286,0.14286,0.57111,0.14,1.0,0,6,0,0,0,21,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,6],[20,53,0.3774,0.40179,0.34523,0.14286,0.14286,0.71429,0.14286,1.0,0,6,0,0,0,19,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,6],[24,53,0.4528,0.52232,0.39059,0.14286,0.42857,1.0,0.14286,1.0,0,12,0,0,0,14,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,12],[28,53,0.5283,0.35267,0.28789,0.14286,0.14286,0.4642,0.14286,1.0,0,3,0,0,0,18,0,0,2,0,0,4,0,0,1,0,0,4,0,0,0,0,3],[32,53,0.6038,0.44197,0.37177,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,18,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,8],[36,53,0.6792,0.37491,0.34401,0.14286,0.14286,0.50002,0.14,1.0,0,6,0,0,0,20,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,6],[40,53,0.7547,0.36607,0.3387,0.14286,0.14286,0.5,0.14286,1.0,0,6,0,0,0,20,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,6],[44,53,0.8302,0.29902,0.27521,0.14286,0.14286,0.42857,0.14,1.0,0,3,0,0,0,22,0,0,1,0,0,4,0,0,0,0,0,2,0,0,0,0,3],[48,53,0.9057,0.24107,0.19377,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,23,0,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[52,53,0.9811,0.24106,0.17288,0.14286,0.14286,0.32143,0.14286,0.71429,0,0,0,0,0,23,0,0,1,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[53,53,1.0,0.22768,0.16311,0.14286,0.14286,0.1786,0.14286,0.71429,0,0,0,0,0,24,0,0,1,0,0,5,0,0,0,0,0,2,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.24545,"x":0.70536,"p":[[0,76,0.0,0.36606,0.3498,0.14286,0.14286,0.60682,0.14286,1.0,0,6,0,0,0,22,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,6],[4,76,0.0526,0.59366,0.40591,0.14286,0.71429,1.0,0.14,1.0,0,14,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,14],[8,76,0.1053,0.70536,0.39437,0.14286,1.0,1.0,0.14286,1.0,0,20,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,20],[12,76,0.1579,0.58027,0.42257,0.14286,0.71429,1.0,0.14,1.0,0,16,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16],[16,76,0.2105,0.58036,0.40867,0.14286,0.57144,1.0,0.14286,1.0,0,15,0,0,0,13,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,15],[20,76,0.2632,0.54464,0.4141,0.14286,0.28571,1.0,0.14286,1.0,0,14,0,0,0,15,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,14],[24,76,0.3158,0.49107,0.39599,0.14286,0.14286,1.0,0.14286,1.0,0,11,0,0,0,17,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,11],[28,76,0.3684,0.52232,0.40344,0.14286,0.28574,1.0,0.14286,1.0,0,12,0,0,0,16,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,12],[32,76,0.4211,0.40178,0.3597,0.14286,0.14286,0.74996,0.0,1.0,1,6,0,1,0,18,0,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,6],[36,76,0.4737,0.53124,0.40126,0.14286,0.42857,1.0,0.14286,1.0,0,13,0,0,0,15,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,13],[40,76,0.5263,0.65179,0.42097,0.14286,1.0,1.0,0.14286,1.0,0,19,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[44,76,0.5789,0.67411,0.39324,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,10,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,18],[48,76,0.6316,0.42856,0.36245,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,18,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,8],[52,76,0.6842,0.56695,0.39687,0.14286,0.64264,1.0,0.14286,1.0,0,13,0,0,0,14,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,13],[56,76,0.7368,0.31696,0.30874,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,23,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,4],[60,76,0.7895,0.45974,0.36383,0.14286,0.21431,0.78571,0.14,1.0,0,8,0,0,0,16,0,0,1,0,0,3,0,0,0,0,0,4,0,0,0,0,8],[64,76,0.8421,0.30804,0.27458,0.14286,0.14286,0.42858,0.14286,1.0,0,2,0,0,0,22,0,0,0,0,0,4,0,0,0,0,0,3,0,0,1,0,2],[68,76,0.8947,0.25884,0.22995,0.14286,0.14286,0.2857,0.14,1.0,0,2,0,0,0,22,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[72,76,0.9474,0.31696,0.29823,0.14286,0.14286,0.42858,0.14286,1.0,0,3,0,0,0,22,0,0,1,0,0,3,0,0,0,0,0,1,0,0,2,0,3],[76,76,1.0,0.24545,0.18298,0.14286,0.14286,0.32142,0.14,0.71429,0,0,0,0,0,23,0,0,1,0,0,5,0,0,0,0,0,3,0,0,0,0,0]]}]},{"i":"a6ac8d3623fbcb2e","q":"There are exactly  $120$  Twitter subscribers from National Science High School. Statistics show that each of  $10$  given celebrities has at least  $85$  followers from National Science High School. Prove that there must be two students such that each of the  $10$  celebrities is being followed in Twitter by at least one of these students.","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.10268,"p":[[0,33,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,33,0.2424,0.07143,0.24484,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,33,0.3636,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,33,0.4848,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.04911,0.18423,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,33,0.7273,0.10268,0.29284,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[28,33,0.8485,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[33,33,1.0,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.11607,"p":[[0,28,0.0,0.07589,0.23141,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,28,0.1429,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.09375,0.249,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,28,0.4286,0.08482,0.24707,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,28,0.5714,0.04911,0.18073,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,28,0.7143,0.04911,0.18766,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,28,0.8571,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.11607,0.3102,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3]]}]},{"i":"583f249c79ae1787","q":"The sequence  $\\{a_n\\}_{n\\geq 0}$  of real numbers satisfies the relation:\r\n\\[ a_{m+n} + a_{m-n} - m + n -1 = \\frac12 (a_{2m} + a_{2n})  \\]\r\nfor all non-negative integers  $m$  and  $n$ ,  $m \\ge n$ . If  $a_1 = 3$  find  $a_{2004}$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.91518,"x":0.98661,"p":[[0,44,0.0,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[4,44,0.0909,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[8,44,0.1818,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,44,0.2727,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[16,44,0.3636,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,44,0.4545,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[24,44,0.5455,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[28,44,0.6364,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,44,0.7273,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,44,0.8182,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[40,44,0.9091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,44,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":4,"e":1.0,"k":"flat","v":0.91071,"x":0.99554,"p":[[0,40,0.0,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[4,40,0.1,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[8,40,0.2,0.94642,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[12,40,0.3,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[16,40,0.4,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[20,40,0.5,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,40,0.6,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[28,40,0.7,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,40,0.8,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,40,0.9,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,40,1.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27]]}]},{"i":"d47e86e85fb6b6e1","q":"$a,b,c,x,y,z$  are positive real numbers and  $bz+cy=a$ ,  $az+cx=b$ ,  $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\\frac{x^2}{1+x}+\\frac{y^2}{1+y}+\\frac{z^2}{1+z}$","t":[{"b":0,"e":0.28571,"k":"flat","v":0.28125,"x":0.51339,"p":[[0,160,0.0,0.39732,0.12234,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,18,0,0,0,0,0,1,0,0,1,0,0],[4,160,0.025,0.36161,0.18552,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,3,0,0,21,0,0,2,0,0,2,0,0,2,0,0,2,0,0],[8,160,0.05,0.39732,0.22794,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,2,0,0,22,0,0,1,0,0,0,0,0,2,0,0,5,0,0],[12,160,0.075,0.35268,0.16746,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,3,0,0,21,0,0,2,0,0,3,0,0,2,0,0,1,0,0],[16,160,0.1,0.36607,0.19212,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,3,0,0,21,0,0,2,0,0,1,0,0,3,0,0,2,0,0],[20,160,0.125,0.38393,0.16146,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,21,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[24,160,0.15,0.38393,0.19704,0.28571,0.28571,0.4286,0.14286,0.85714,0,0,0,0,0,2,0,0,21,0,0,2,0,0,1,0,0,4,0,0,2,0,0],[28,160,0.175,0.3125,0.11538,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,24,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[32,160,0.2,0.36161,0.18552,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,25,0,0,1,0,0,0,0,0,4,0,0,0,0,1],[36,160,0.225,0.39286,0.21429,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,18,0,0,1,0,0,2,0,0,5,0,0,2,0,0],[40,160,0.25,0.35268,0.13825,0.28571,0.28571,0.32143,0.2857,0.85714,0,0,0,0,0,0,0,0,24,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[44,160,0.275,0.30803,0.11355,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,25,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[48,160,0.3,0.3125,0.11538,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,26,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[52,160,0.325,0.31696,0.1461,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,4,0,0,23,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[56,160,0.35,0.28125,0.09771,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,5,0,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[60,160,0.375,0.33034,0.15744,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,3,0,0,23,0,0,3,0,0,1,0,0,0,0,0,2,0,0],[64,160,0.4,0.35714,0.16752,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,3,0,0,20,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[68,160,0.425,0.375,0.19805,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,26,0,0,0,0,0,2,0,0,1,0,0,2,0,1],[72,160,0.45,0.30357,0.11152,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,23,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[76,160,0.475,0.36607,0.17105,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,21,0,0,3,0,0,2,0,0,3,0,0,1,0,0],[80,160,0.5,0.29464,0.12846,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,22,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[84,160,0.525,0.37946,0.18766,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,1,0,0,23,0,0,1,0,0,2,0,0,3,0,0,2,0,0],[88,160,0.55,0.32589,0.12993,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,27,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[92,160,0.575,0.35268,0.13825,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,22,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[96,160,0.6,0.42411,0.19719,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,18,0,0,2,0,0,5,0,0,4,0,0,2,0,0],[100,160,0.625,0.30804,0.12428,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,3,0,0,25,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[104,160,0.65,0.37946,0.18423,0.28571,0.28571,0.46428,0.14286,0.85714,0,0,0,0,0,2,0,0,21,0,0,1,0,0,3,0,0,4,0,0,1,0,0],[108,160,0.675,0.30357,0.06916,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[112,160,0.7,0.33482,0.1411,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,23,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[116,160,0.725,0.33482,0.14555,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,2,0,0],[120,160,0.75,0.37053,0.16698,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,23,0,0,5,0,0,0,0,0,2,0,0,2,0,0],[124,160,0.775,0.32589,0.12993,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,1,0,0,27,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[128,160,0.8,0.36607,0.16728,0.28571,0.28571,0.32143,0.2857,0.85714,0,0,0,0,0,0,0,0,24,0,0,4,0,0,0,0,0,2,0,0,2,0,0],[132,160,0.825,0.37053,0.17445,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[136,160,0.85,0.37053,0.17445,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[140,160,0.875,0.375,0.17768,0.28571,0.28571,0.32143,0.2857,0.85714,0,0,0,0,0,0,0,0,24,0,0,3,0,0,0,0,0,3,0,0,2,0,0],[144,160,0.9,0.37053,0.17076,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,1,0,0,1,0,0,4,0,0,1,0,0],[148,160,0.925,0.36607,0.15947,0.28571,0.28571,0.32143,0.2857,0.85714,0,0,0,0,0,0,0,0,24,0,0,3,0,0,1,0,0,3,0,0,1,0,0],[152,160,0.95,0.4241,0.19719,0.28571,0.28571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,20,0,0,2,0,0,3,0,0,5,0,0,2,0,0],[156,160,0.975,0.4375,0.18536,0.28571,0.28571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,17,0,0,4,0,0,4,0,0,6,0,0,1,0,0],[160,160,1.0,0.51339,0.16311,0.39286,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,6,0,0,9,0,0,9,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.28571,"x":0.67857,"p":[[0,129,0.0,0.40624,0.12929,0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real numbers x and \u0443 satisfy the equations    $$ \\begin{cases} \\sqrt{3x}\\left(1+\\dfrac{1}{x+y}\\right)=2    \\sqrt{7y}\\left(1-\\dfrac{1}{x+y}\\right)=4\\sqrt2  \\end{cases} $$   \nFind the numerical value of the ratio  $y/x$ .","t":[{"b":5,"e":0.85714,"k":"flat","v":0.86607,"x":0.88839,"p":[[0,9,0.0,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[4,9,0.4444,0.86607,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[8,9,0.8889,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[9,9,1.0,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3]]},{"b":6,"e":0.85714,"k":"flat","v":0.85268,"x":0.88839,"p":[[0,56,0.0,0.86607,0.07087,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,4],[4,56,0.0714,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[8,56,0.1429,0.85268,0.10999,0.85714,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,28,0,3],[12,56,0.2143,0.87053,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[16,56,0.2857,0.88393,0.06622,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,7],[20,56,0.3571,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[24,56,0.4286,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[28,56,0.5,0.85714,0.0,0.85714,0.85714,0.85714,0.85714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[32,56,0.5714,0.86606,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[36,56,0.6429,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[40,56,0.7143,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[44,56,0.7857,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[48,56,0.8571,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[52,56,0.9286,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[56,56,1.0,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3]]}]},{"i":"06b5b0f7cbad23d3","q":"$f,g:\\mathbb{R}\\rightarrow\\mathbb{R}$  find all  $f,g$  satisfying  $\\forall x,y\\in \\mathbb{R}$ :\n\\[g(f(x)-y)=f(g(y))+x.\\]","t":[{"b":1,"e":0.14286,"k":"rising","v":0.49097,"x":0.97768,"p":[[0,190,0.0,0.49097,0.3273,0.14286,0.57121,0.75,0.14,1.0,0,5,0,0,0,13,0,0,0,0,0,2,0,0,7,0,0,2,0,0,3,0,5],[4,190,0.0211,0.88839,0.21049,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,23],[8,190,0.0421,0.87946,0.21461,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,23],[12,190,0.0632,0.95981,0.12997,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[16,190,0.0842,0.87054,0.21535,0.67857,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,0,0,23],[20,190,0.1053,0.85713,0.19235,0.57143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,1,0,20],[24,190,0.1263,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[28,190,0.1474,0.91964,0.16728,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[32,190,0.1684,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[36,190,0.1895,0.90179,0.16146,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,3,0,22],[40,190,0.2105,0.95089,0.1411,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[44,190,0.2316,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[48,190,0.2526,0.87938,0.22364,0.92857,1.0,1.0,0.14,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,0,0,0,24],[52,190,0.2737,0.91964,0.15126,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24],[56,190,0.2947,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[60,190,0.3158,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[64,190,0.3368,0.94195,0.12811,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[68,190,0.3579,0.92409,0.15149,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,25],[72,190,0.3789,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[76,190,0.4,0.94643,0.16269,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[80,190,0.4211,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[84,190,0.4421,0.90625,0.16214,0.85714,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,23],[88,190,0.4632,0.91518,0.19186,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,25],[92,190,0.4842,0.90179,0.18707,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,0,0,24],[96,190,0.5053,0.87496,0.2136,0.78571,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,1,0,23],[100,190,0.5263,0.88392,0.19706,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,3,0,22],[104,190,0.5474,0.90625,0.21609,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,26],[108,190,0.5684,0.91964,0.18536,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,25],[112,190,0.5895,0.94195,0.12811,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[116,190,0.6105,0.85714,0.21429,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,3,0,20],[120,190,0.6316,0.92857,0.15972,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[124,190,0.6526,0.8482,0.25239,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,2,0,21],[128,190,0.6737,0.91964,0.17105,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[132,190,0.6947,0.87947,0.23449,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,23],[136,190,0.7158,0.91071,0.18123,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,24],[140,190,0.7368,0.83034,0.27068,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,21],[144,190,0.7579,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[148,190,0.7789,0.92411,0.18205,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,27],[152,190,0.8,0.87946,0.18249,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,2,0,21],[156,190,0.8211,0.85267,0.25875,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,23],[160,190,0.8421,0.95088,0.12686,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[164,190,0.8632,0.87053,0.24053,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,20],[168,190,0.8842,0.91071,0.21354,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,26],[172,190,0.9053,0.86604,0.25491,0.857,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0,3,0,23],[176,190,0.9263,0.82142,0.26965,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,21],[180,190,0.9474,0.87499,0.24158,0.92857,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,0,0,24],[184,190,0.9684,0.90177,0.19047,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,24],[188,190,0.9895,0.86161,0.22156,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,0,2,0,21],[190,190,1.0,0.8616,0.2461,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,21]]},{"b":7,"e":0.57143,"k":"flat","v":0.48661,"x":0.91518,"p":[[0,89,0.0,0.49997,0.30929,0.14286,0.57121,0.60714,0.14286,1.0,0,6,0,0,0,11,0,0,0,0,0,3,0,0,10,0,0,2,0,0,0,0,6],[4,89,0.0449,0.83036,0.23266,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,1,0,0,0,0,20],[8,89,0.0899,0.85268,0.22442,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,4,0,0,0,0,21],[12,89,0.1348,0.87497,0.20128,0.57143,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,23],[16,89,0.1798,0.91518,0.1551,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,24],[20,89,0.2247,0.82589,0.20119,0.57143,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,2,0,17],[24,89,0.2697,0.85268,0.24086,0.67857,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,21],[28,89,0.3146,0.85268,0.21865,0.67857,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,2,0,0,2,0,20],[32,89,0.3596,0.91518,0.15916,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24],[36,89,0.4045,0.85267,0.19395,0.67857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,2,0,19],[40,89,0.4494,0.87054,0.19352,0.67857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,2,0,21],[44,89,0.4944,0.82142,0.20826,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,0,0,18],[48,89,0.5393,0.77229,0.2471,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,2,0,0,2,0,15],[52,89,0.5843,0.70982,0.21866,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,6,0,0,2,0,9],[56,89,0.6292,0.80802,0.19104,0.57143,0.78571,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,1,0,15],[60,89,0.6742,0.79463,0.20498,0.57143,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,4,0,0,1,0,15],[64,89,0.7191,0.79018,0.21424,0.57143,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,2,0,0,0,0,16],[68,89,0.764,0.71428,0.24484,0.57143,0.64286,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,11,0,0,3,0,0,2,0,11],[72,89,0.809,0.79461,0.19544,0.57143,0.857,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,4,0,13],[76,89,0.8539,0.72321,0.21998,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,11,0,0,7,0,0,1,0,10],[80,89,0.8989,0.62497,0.22232,0.57143,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,0,0,0,19,0,0,4,0,0,1,0,5],[84,89,0.9438,0.57587,0.2004,0.57132,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,2,0,0,20,0,0,2,0,0,1,0,3],[88,89,0.9888,0.49554,0.1749,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,1,0,0,9,0,0,17,0,0,1,0,0,0,0,1],[89,89,1.0,0.48661,0.18161,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,0,0,0,9,0,0,15,0,0,1,0,0,2,0,0]]}]},{"i":"d828b77bfda15552","q":"$n$  is a natural number that  $\\frac{x^{n}+1}{x+1}$  is irreducible over  $\\mathbb Z_{2}[x]$ . Consider a vector in  $\\mathbb Z_{2}^{n}$  that it has odd number of  $1$ 's (as entries) and at least one of its entries are  $0$ . Prove that these vector and its translations are a basis for  $\\mathbb Z_{2}^{n}$","t":[{"b":1,"e":1.0,"k":"flat","v":0.90625,"x":0.97768,"p":[[0,12,0.0,0.90625,0.11071,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,17],[4,12,0.3333,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,12,0.6667,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,12,1.0,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]},{"b":2,"e":1.0,"k":"flat","v":0.88392,"x":0.95982,"p":[[0,10,0.0,0.88392,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,12],[4,10,0.4,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[8,10,0.8,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[10,10,1.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23]]}]},{"i":"983d66c9caa06661","q":"$C$  is a point on the semicircle diameter  $AB$ , between  $A$  and  $B$ .  $D$  is the foot of the perpendicular from  $C$  to  $AB$ . The circle  $K_1$  is the incircle of  $ABC$ , the circle  $K_2$  touches  $CD,DA$  and the semicircle, the circle  $K_3$  touches  $CD,DB$  and the semicircle. Prove that  $K_1,K_2$  and  $K_3$  have another common tangent apart from  $AB$ 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 consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of  $n$  if the last number in the row is odd?\n\n(A Shapovalov)","t":[{"b":2,"e":0.4286,"k":"falling","v":0.11152,"x":0.7232,"p":[[0,116,0.0,0.62054,0.3126,0.28571,0.64286,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,13,0,0,2,0,0,1,0,0,2,0,0,5,0,9],[4,116,0.0345,0.66071,0.31288,0.42857,0.71429,1.0,0.0,1.0,3,9,3,3,0,0,0,0,2,0,0,5,0,0,4,0,0,4,0,0,5,0,9],[8,116,0.069,0.7232,0.33301,0.42857,1.0,1.0,0.0,1.0,2,17,2,2,0,0,0,0,4,0,0,4,0,0,3,0,0,1,0,0,1,0,17],[12,116,0.1034,0.65176,0.30081,0.42857,0.57143,1.0,0.0,1.0,1,11,1,1,0,0,0,0,5,0,0,7,0,0,4,0,0,2,0,0,2,0,11],[16,116,0.1379,0.60268,0.28735,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,6,0,0,8,0,0,4,0,0,3,0,0,2,0,8],[20,116,0.1724,0.63838,0.2696,0.42857,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,7,0,0,5,0,0,2,0,0,4,0,8],[24,116,0.2069,0.48661,0.25966,0.28571,0.42857,0.57143,0.0,1.0,2,3,0,2,0,0,0,0,9,0,0,9,0,0,5,0,0,1,0,0,3,0,3],[28,116,0.2414,0.45535,0.27301,0.28571,0.42857,0.60714,0.0,1.0,4,2,0,4,0,1,0,0,7,0,0,7,0,0,5,0,0,4,0,0,2,0,2],[32,116,0.2759,0.52679,0.29111,0.28571,0.42857,0.75,0.0,1.0,2,6,0,2,0,0,0,0,8,0,0,9,0,0,4,0,0,1,0,0,2,0,6],[36,116,0.3103,0.50888,0.2647,0.42857,0.42857,0.60714,0.0,1.0,3,3,0,3,0,0,0,0,4,0,0,11,0,0,6,0,0,2,0,0,3,0,3],[40,116,0.3448,0.52232,0.28484,0.28571,0.4286,0.74996,0.0,1.0,1,5,0,1,0,1,0,0,10,0,0,7,0,0,3,0,0,2,0,0,3,0,5],[44,116,0.3793,0.43302,0.23819,0.28571,0.42857,0.57143,0.0,1.0,3,2,0,3,0,1,0,0,7,0,0,11,0,0,6,0,0,1,0,0,1,0,2],[48,116,0.4138,0.44642,0.2829,0.28571,0.42857,0.57143,0.0,1.0,3,5,0,3,0,0,0,0,12,0,0,8,0,0,3,0,0,1,0,0,0,0,5],[52,116,0.4483,0.55808,0.3141,0.39286,0.4286,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,4,0,0,10,0,0,2,0,0,3,0,0,1,0,8],[56,116,0.4828,0.45525,0.35261,0.24999,0.42857,0.75,0.0,1.0,6,7,1,6,0,2,0,0,7,0,0,6,0,0,2,0,0,1,0,0,1,0,7],[60,116,0.5172,0.52675,0.31428,0.2857,0.4286,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,4,0,0,9,0,0,4,0,0,1,0,0,3,0,6],[64,116,0.5517,0.37053,0.36919,0.0,0.2857,0.60714,0.0,1.0,9,6,0,9,0,6,0,0,4,0,0,4,0,0,1,0,0,1,0,0,1,0,6],[68,116,0.5862,0.44643,0.37244,0.10714,0.42859,0.71429,0.0,1.0,8,6,1,8,0,4,0,0,3,0,0,3,0,0,2,0,0,5,0,0,1,0,6],[72,116,0.6207,0.4732,0.35434,0.14286,0.49979,0.71429,0.0,1.0,7,5,0,7,0,3,0,0,3,0,0,3,0,0,4,0,0,5,0,0,2,0,5],[76,116,0.6552,0.50893,0.35344,0.14286,0.5,0.75,0.0,1.0,4,7,0,4,0,6,0,0,2,0,0,4,0,0,3,0,0,5,0,0,1,0,7],[80,116,0.6897,0.47321,0.32427,0.24999,0.42857,0.75,0.0,1.0,2,6,0,2,0,6,0,0,6,0,0,7,0,0,2,0,0,1,0,0,2,0,6],[84,116,0.7241,0.4464,0.37584,0.14286,0.42857,0.71429,0.0,1.0,7,7,0,7,0,7,0,0,0,0,0,4,0,0,3,0,0,4,0,0,0,0,7],[88,116,0.7586,0.42399,0.35628,0.14286,0.28571,0.60714,0.0,1.0,6,7,0,6,0,5,0,0,6,0,0,4,0,0,3,0,0,1,0,0,0,0,7],[92,116,0.7931,0.34374,0.34229,0.0,0.28571,0.4642,0.0,1.0,10,4,0,10,0,5,0,0,3,0,0,6,0,0,1,0,0,2,0,0,1,0,4],[96,116,0.8276,0.34375,0.36746,0.0,0.2143,0.42857,0.0,1.0,10,6,0,10,0,6,0,0,4,0,0,5,0,0,0,0,0,0,0,0,1,0,6],[100,116,0.8621,0.3125,0.33964,0.0,0.14288,0.42857,0.0,1.0,12,4,0,12,0,5,0,0,1,0,0,7,0,0,1,0,0,2,0,0,0,0,4],[104,116,0.8966,0.33929,0.29179,0.14286,0.28571,0.46431,0.0,1.0,7,2,0,7,0,6,0,0,6,0,0,5,0,0,1,0,0,5,0,0,0,0,2],[108,116,0.931,0.32142,0.30513,0.0,0.35714,0.4642,0.0,1.0,11,1,0,11,0,4,0,0,1,0,0,8,0,0,2,0,0,3,0,0,2,0,1],[112,116,0.9655,0.20088,0.2365,0.0,0.14286,0.32143,0.0,1.0,11,1,0,11,0,12,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[116,116,1.0,0.11152,0.16261,0.0,0.0,0.1429,0.0,0.57143,19,0,0,19,0,6,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"falling","v":0.35714,"x":0.82589,"p":[[0,82,0.0,0.55357,0.33455,0.28571,0.42857,1.0,0.0,1.0,2,10,2,2,0,0,0,0,12,0,0,4,0,0,2,0,0,2,0,0,0,0,10],[4,82,0.0488,0.75444,0.25564,0.4286,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,1,0,0,4,0,14],[8,82,0.0976,0.82589,0.26663,0.71429,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,3,0,19],[12,82,0.1463,0.71427,0.26487,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,10,0,0,3,0,0,3,0,0,3,0,12],[16,82,0.1951,0.77232,0.29202,0.4286,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,6,0,0,1,0,0,2,0,0,3,0,17],[20,82,0.2439,0.65624,0.26211,0.42857,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,10,0,0,6,0,0,0,0,0,4,0,9],[24,82,0.2927,0.5357,0.31135,0.28571,0.42857,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,6,0,0,8,0,0,4,0,0,1,0,0,2,0,7],[28,82,0.3415,0.58927,0.27607,0.42857,0.49979,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,12,0,0,6,0,0,0,0,0,3,0,7],[32,82,0.3902,0.49545,0.26009,0.28571,0.42857,0.60714,0.14,1.0,0,4,0,0,0,3,0,0,8,0,0,10,0,0,3,0,0,2,0,0,2,0,4],[36,82,0.439,0.5848,0.29093,0.39286,0.4286,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,7,0,0,9,0,0,3,0,0,1,0,0,4,0,7],[40,82,0.4878,0.61158,0.29717,0.42857,0.4998,1.0,0.0,1.0,1,10,0,1,0,0,0,0,5,0,0,10,0,0,4,0,0,1,0,0,1,0,10],[44,82,0.5366,0.56697,0.26603,0.42857,0.42857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,6,0,0,15,0,0,1,0,0,1,0,0,2,0,7],[48,82,0.5854,0.5357,0.25505,0.42857,0.42857,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,12,0,0,4,0,0,3,0,0,1,0,5],[52,82,0.6341,0.62497,0.29179,0.42857,0.57121,1.0,0.0,1.0,1,10,1,1,0,0,0,0,5,0,0,8,0,0,4,0,0,4,0,0,0,0,10],[56,82,0.6829,0.62054,0.28259,0.42857,0.4286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,13,0,0,1,0,0,2,0,0,1,0,10]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 students take a test with  $m$  questions, where  $m,n\\ge 2$  are integers. The score given to every question is as such: for a certain question, if  $x$  students fails to answer it correctly, then those who answer it correctly scores  $x$  points, while those who answer it wrongly scores  $0$ . The score of a student is the sum of his scores for the  $m$  questions. Arrange the scores in descending order  $p_1\\ge p_2\\ge \\ldots \\ge p_n$ . Find the maximum value of  $p_1+p_n$ .","t":[{"b":5,"e":0.71429,"k":"rising","v":0.24991,"x":0.72321,"p":[[0,94,0.0,0.24991,0.14292,0.14286,0.2143,0.28571,0.14,0.71429,0,0,0,0,0,16,0,0,12,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,94,0.0426,0.66516,0.25408,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,3,0,0,5,0,0,8,0,0,2,0,8],[8,94,0.0851,0.64729,0.2743,0.53539,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,3,0,0,7,0,0,7,0,0,3,0,7],[12,94,0.1277,0.56696,0.23002,0.42857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,10,0,0,3,0,0,6,0,0,5,0,2],[16,94,0.1702,0.58032,0.24206,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,4,0,0,5,0,0,8,0,0,7,0,0,3,0,3],[20,94,0.2128,0.6607,0.27837,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,4,0,0,1,0,0,6,0,0,8,0,6],[24,94,0.2553,0.62498,0.27606,0.42857,0.57143,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,4,0,0,5,0,0,6,0,0,4,0,0,5,0,6],[28,94,0.2979,0.65177,0.24984,0.5713,0.71429,0.75,0.0,1.0,1,6,0,1,0,1,0,0,2,0,0,3,0,0,7,0,0,10,0,0,2,0,6],[32,94,0.3404,0.64729,0.26723,0.39286,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,3,0,0,4,0,0,4,0,0,7,0,6],[36,94,0.383,0.61157,0.19311,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,6,0,0,10,0,0,8,0,0,2,0,3],[40,94,0.4255,0.71426,0.26965,0.5713,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,1,0,0,6,0,0,5,0,0,5,0,10],[44,94,0.4681,0.62497,0.23623,0.42857,0.57143,0.74996,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,7,0,0,8,0,0,6,0,0,3,0,5],[48,94,0.5106,0.70088,0.24318,0.53539,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,4,0,0,3,0,0,10,0,0,2,0,9],[52,94,0.5532,0.61592,0.30182,0.39286,0.57143,1.0,0.0,1.0,1,9,0,1,0,1,0,0,6,0,0,5,0,0,5,0,0,3,0,0,2,0,9],[56,94,0.5957,0.68747,0.24339,0.571,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,5,0,0,10,0,0,3,0,0,3,0,9],[60,94,0.6383,0.62945,0.25719,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,9,0,0,5,0,0,4,0,0,3,0,7],[64,94,0.6809,0.66066,0.23893,0.571,0.57143,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,3,0,0,11,0,0,5,0,0,4,0,6],[68,94,0.7234,0.616,0.23267,0.42859,0.71429,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,6,0,0,2,0,0,6,0,0,10,0,0,4,0,3],[72,94,0.766,0.58034,0.28333,0.39286,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,4,0,0,6,0,0,5,0,0,3,0,0,5,0,5],[76,94,0.8085,0.6652,0.22187,0.42857,0.71429,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,7,0,0,2,0,0,11,0,0,4,0,5],[80,94,0.8511,0.65624,0.23105,0.53571,0.71429,0.857,0.1429,1.0,0,3,0,0,0,1,0,0,5,0,0,2,0,0,2,0,0,13,0,0,6,0,3],[84,94,0.8936,0.72321,0.23673,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,6,0,0,5,0,9],[88,94,0.9362,0.62946,0.26692,0.28571,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,10,0,0,1,0,0,3,0,0,7,0,0,6,0,5],[92,94,0.9787,0.66069,0.26667,0.4286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,4,0,0,4,0,0,8,0,0,5,0,0,0,0,10],[94,94,1.0,0.56249,0.27879,0.28571,0.57143,0.75,0.14286,1.0,0,4,0,0,0,5,0,0,4,0,0,4,0,0,6,0,0,5,0,0,4,0,4]]},{"b":7,"e":0.42857,"k":"rising","v":0.36607,"x":0.67409,"p":[[0,57,0.0,0.36607,0.30501,0.14286,0.2857,0.42857,0.14286,1.0,0,5,0,0,0,15,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,5],[4,57,0.0702,0.67409,0.30144,0.53539,0.71429,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,3,0,0,1,0,0,6,0,0,4,0,0,4,0,10],[8,57,0.1404,0.64284,0.25754,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,3,0,0,5,0,0,5,0,0,7,0,0,4,0,6],[12,57,0.2105,0.65175,0.25239,0.42859,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,7,0,0,3,0,0,10,0,0,0,0,8],[16,57,0.2807,0.64732,0.2879,0.42857,0.64286,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,5,0,0,4,0,0,5,0,0,4,0,0,3,0,9],[20,57,0.3509,0.54015,0.27603,0.28571,0.57121,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,5,0,0,5,0,0,5,0,0,7,0,0,3,0,3],[24,57,0.4211,0.51335,0.24447,0.39286,0.42859,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,5,0,0,10,0,0,4,0,0,5,0,0,3,0,2],[28,57,0.4912,0.54911,0.35912,0.2857,0.5,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,5,0,0,4,0,0,1,0,0,4,0,0,3,0,8],[32,57,0.5614,0.44195,0.29956,0.1429,0.42857,0.71429,0.0,1.0,3,2,0,3,0,6,0,0,6,0,0,4,0,0,4,0,0,3,0,0,4,0,2],[36,57,0.6316,0.66067,0.25444,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,6,0,0,3,0,0,8,0,0,2,0,8],[40,57,0.7018,0.6205,0.25657,0.42857,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,6,0,0,6,0,4],[44,57,0.7719,0.61605,0.2822,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,2,0,0,6,0,0,5,0,0,4,0,0,5,0,6],[48,57,0.8421,0.58482,0.3038,0.39286,0.57143,0.85704,0.0,1.0,1,7,0,1,0,4,0,0,3,0,0,5,0,0,5,0,0,5,0,0,2,0,7],[52,57,0.9123,0.64732,0.23954,0.42859,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,5,0,0,7,0,0,7,0,0,3,0,6],[56,57,0.9825,0.66071,0.26904,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,6,0,0,2,0,0,7,0,0,5,0,7],[57,57,1.0,0.58927,0.25692,0.42857,0.57143,0.74996,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,7,0,0,5,0,0,6,0,0,4,0,4]]}]},{"i":"a476bb931e29714a","q":"(Distributive law) Prove that $(x \\oplus y) \\odot z=x \\odot z \\oplus y \\odot z$ for all $x, y, z \\in \\mathbb{R} \\cup\\{\\infty\\}$.","t":[{"b":1,"e":1.0,"k":"volatile","v":0.34375,"x":0.98214,"p":[[0,15,0.0,0.34375,0.41011,0.0,0.0,0.71429,0.0,1.0,18,5,0,18,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5],[4,15,0.2667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,15,0.5333,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[12,15,0.8,0.96873,0.09275,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[15,15,1.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":4,"e":1.0,"k":"volatile","v":0.38392,"x":0.98661,"p":[[0,12,0.0,0.38392,0.44526,0.0,0.0,0.78571,0.0,1.0,18,8,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,8],[4,12,0.3333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,12,0.6667,0.94196,0.12807,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[12,12,1.0,0.92857,0.19885,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,27]]}]},{"i":"fbc74c3294de6f78","q":"1. (BUL 1) A pentagon $A B C D E$ inscribed in a circle for which $B C<C D$ and $A B<D E$ is the base of a pyramid with vertex $S$. If $A S$ is the longest edge starting from $S$, prove that $B S>C S$.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.42856,"x":0.54908,"p":[[0,50,0.0,0.54908,0.33902,0.28571,0.42836,1.0,0.0,1.0,2,9,1,2,0,1,0,0,13,0,0,0,0,0,3,0,0,3,0,0,1,0,9],[4,50,0.08,0.44199,0.20934,0.28571,0.4293,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,11,0,0,3,0,0,12,0,0,1,0,0,1,0,1],[8,50,0.16,0.48216,0.20435,0.28571,0.57143,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,7,0,0,2,0,0,17,0,0,1,0,0,1,0,1],[12,50,0.24,0.50442,0.23953,0.28571,0.571,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,11,0,0,2,0,0,13,0,0,0,0,0,2,0,3],[16,50,0.32,0.50441,0.21422,0.28571,0.571,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,10,0,0,2,0,0,15,0,0,1,0,0,0,0,3],[20,50,0.4,0.51783,0.14173,0.49968,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,8,0,0,0,0,0,20,0,0,4,0,0,0,0,0],[24,50,0.48,0.45533,0.18706,0.28571,0.571,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,12,0,0,2,0,0,15,0,0,1,0,0,0,0,1],[28,50,0.56,0.43302,0.15354,0.28571,0.35714,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,16,0,0,1,0,0,13,0,0,2,0,0,0,0,0],[32,50,0.64,0.48659,0.19839,0.28571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,11,0,0,1,0,0,16,0,0,1,0,0,0,0,2],[36,50,0.72,0.4553,0.1836,0.28571,0.571,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,11,0,0,0,0,0,17,0,0,1,0,0,1,0,0],[40,50,0.8,0.46875,0.15251,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,4,0,0,17,0,0,0,0,0,1,0,0],[44,50,0.88,0.44641,0.18122,0.28571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,12,0,0,0,0,0,15,0,0,3,0,0,0,0,0],[48,50,0.96,0.45533,0.16144,0.28571,0.57141,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,11,0,0,1,0,0,18,0,0,1,0,0,0,0,0],[50,50,1.0,0.42856,0.17127,0.28571,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,13,0,0,0,0,0,16,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.34374,"x":0.73661,"p":[[0,59,0.0,0.73661,0.25781,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,1,0,0,6,0,0,3,0,0,6,0,11],[4,59,0.0678,0.45979,0.17027,0.28571,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,2,0,0,14,0,0,2,0,0,1,0,0],[8,59,0.1356,0.40178,0.26829,0.25,0.35714,0.57143,0.0,1.0,5,1,0,5,0,3,0,0,8,0,0,1,0,0,11,0,0,1,0,0,2,0,1],[12,59,0.2034,0.45534,0.23538,0.28571,0.35714,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,14,0,0,2,0,0,8,0,0,3,0,0,1,0,2],[16,59,0.2712,0.41518,0.26088,0.28571,0.28571,0.57143,0.0,1.0,3,3,0,3,0,2,0,0,13,0,0,0,0,0,11,0,0,0,0,0,0,0,3],[20,59,0.339,0.41068,0.2362,0.2857,0.35714,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,10,0,0,4,0,0,7,0,0,3,0,0,1,0,1],[24,59,0.4068,0.38838,0.21497,0.28571,0.28571,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,12,0,0,2,0,0,12,0,0,0,0,0,0,0,1],[28,59,0.4746,0.41514,0.23784,0.28571,0.42857,0.57143,0.0,1.0,4,1,0,4,0,1,0,0,10,0,0,2,0,0,11,0,0,3,0,0,0,0,1],[32,59,0.5424,0.34374,0.14662,0.2857,0.28571,0.42858,0.0,0.57143,1,0,0,1,0,3,0,0,17,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[36,59,0.6102,0.51334,0.20471,0.28571,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,7,0,0,0,0,0,19,0,0,1,0,0,2,0,1],[40,59,0.678,0.4375,0.21998,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,10,0,0,5,0,0,8,0,0,3,0,0,1,0,1],[44,59,0.7458,0.40177,0.20957,0.28571,0.57143,0.57143,0.0,0.57143,4,0,0,4,0,2,0,0,8,0,0,0,0,0,18,0,0,0,0,0,0,0,0],[48,59,0.8136,0.46871,0.17938,0.28571,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,1,0,0,15,0,0,1,0,0,2,0,0],[52,59,0.8814,0.36161,0.19557,0.28571,0.28571,0.46431,0.0,1.0,2,1,0,2,0,3,0,0,14,0,0,5,0,0,7,0,0,0,0,0,0,0,1],[56,59,0.9492,0.43302,0.18028,0.28571,0.5712,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,11,0,0,1,0,0,15,0,0,2,0,0,0,0,0],[59,59,1.0,0.35266,0.13354,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,20,0,0,4,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"5d2121411de18a19","q":"$a,b>1$  - are naturals, and  $a^2+b,a+b^2$  are primes. Prove  $(ab+1,a+b)=1$","t":[{"b":2,"e":0.4286,"k":"flat","v":0.50444,"x":0.78571,"p":[[0,43,0.0,0.65625,0.31311,0.53571,0.71429,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,2,0,0,1,0,0,3,0,0,9,0,0,4,0,8],[4,43,0.093,0.78571,0.20825,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,11,0,0,3,0,12],[8,43,0.186,0.78571,0.20516,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,14,0,0,0,0,13],[12,43,0.2791,0.5714,0.15151,0.42857,0.57121,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,6,0,0,4,0,0],[16,43,0.3721,0.50448,0.11832,0.42857,0.42857,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,0,1,0,0,2,0,0],[20,43,0.4651,0.52228,0.13173,0.42857,0.42859,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,18,0,0,10,0,0,1,0,0,3,0,0],[24,43,0.5581,0.50444,0.11282,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,17,0,0,11,0,0,2,0,0,1,0,0],[28,43,0.6512,0.51338,0.13766,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,20,0,0,8,0,0,2,0,0,1,0,1],[32,43,0.7442,0.52674,0.12076,0.42857,0.571,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,1,0,0,14,0,0,12,0,0,4,0,0,1,0,0],[36,43,0.8372,0.54458,0.10969,0.42857,0.571,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,12,0,0,7,0,0,0,0,0],[40,43,0.9302,0.52229,0.1107,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,11,0,0,15,0,0,4,0,0,0,0,0],[43,43,1.0,0.52677,0.10373,0.42857,0.57121,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,15,0,0,12,0,0,5,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.46429,"x":0.74544,"p":[[0,44,0.0,0.74544,0.26921,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,2,0,0,0,0,0,2,0,0,1,0,0,13,0,0,1,0,12],[4,44,0.0909,0.58477,0.25345,0.42857,0.57143,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,1,0,0,4,0,0,8,0,0,9,0,0,4,0,2],[8,44,0.1818,0.63396,0.19537,0.53575,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,17,0,0,0,0,3],[12,44,0.2727,0.58927,0.24679,0.39286,0.64286,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,5,0,0,1,0,0,7,0,0,12,0,0,0,0,4],[16,44,0.3636,0.46429,0.18558,0.28571,0.42857,0.60714,0.0,0.71429,1,0,0,1,0,1,0,0,7,0,0,11,0,0,4,0,0,8,0,0,0,0,0],[20,44,0.4545,0.57141,0.14286,0.42857,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,10,0,0,6,0,0,14,0,0,0,0,0],[24,44,0.5455,0.59375,0.13415,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,17,0,0,0,0,0],[28,44,0.6364,0.57587,0.13115,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,8,0,0,13,0,0,0,0,0],[32,44,0.7273,0.6116,0.12492,0.57132,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,17,0,0,0,0,0],[36,44,0.8182,0.57588,0.13592,0.42857,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,13,0,0,0,0,0],[40,44,0.9091,0.58033,0.13333,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,15,0,0,0,0,0],[44,44,1.0,0.60714,0.12877,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,18,0,0,0,0,0]]}]},{"i":"fe5cacf309fe71d4","q":"$A B C D$ is a trapezium $(A D \\| B C) . P$ is the point on the line $A B$ such that $\\angle C P D$ is maximal. $Q$ is the point on the line $C D$ such that $\\angle B Q A$ is maximal. Given that $P$ lies on the segment $A B$, prove that $\\angle C P D=\\angle B Q A$.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.45086,"x":0.62499,"p":[[0,35,0.0,0.51786,0.3004,0.28571,0.42857,0.75,0.0,1.0,3,5,2,3,0,0,0,0,7,0,0,10,0,0,1,0,0,3,0,0,3,0,5],[4,35,0.1143,0.46874,0.24284,0.28571,0.28571,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,17,0,0,4,0,0,4,0,0,2,0,0,2,0,3],[8,35,0.2286,0.45086,0.2116,0.28571,0.42857,0.571,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,8,0,0,3,0,0,3,0,0,1,0,2],[12,35,0.3429,0.62498,0.23352,0.42857,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,3,0,0,11,0,0,0,0,0,9,0,3],[16,35,0.4571,0.62499,0.27139,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,3,0,0,9,0,0,0,0,0,8,0,5],[20,35,0.5714,0.48657,0.21973,0.28571,0.571,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,12,0,0,1,0,0,13,0,0,1,0,0,3,0,1],[24,35,0.6857,0.57139,0.23145,0.28571,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,4,0,0,7,0,0,3,0,0,8,0,1],[28,35,0.8,0.58927,0.24419,0.28571,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,2,0,0,10,0,0,1,0,0,7,0,3],[32,35,0.9143,0.60711,0.22304,0.42857,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,2,0,0,11,0,0,1,0,0,10,0,1],[35,35,1.0,0.59375,0.26989,0.28571,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,9,0,0,3,0,0,4,0,0,0,0,0,15,0,0]]},{"b":3,"e":0.85714,"k":"rising","v":0.41965,"x":0.82589,"p":[[0,39,0.0,0.47768,0.27107,0.28571,0.42857,0.46431,0.0,1.0,1,6,0,1,0,0,0,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,6],[4,39,0.1026,0.53571,0.19562,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,8,0,0,8,0,0,5,0,0,3,0,1],[8,39,0.2051,0.41965,0.17834,0.28571,0.28586,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,17,0,0,5,0,0,8,0,0,0,0,0,1,0,1],[12,39,0.3077,0.45087,0.23718,0.28571,0.28586,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,16,0,0,3,0,0,6,0,0,2,0,0,2,0,2],[16,39,0.4103,0.53123,0.23209,0.28571,0.4286,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,10,0,0,7,0,0,5,0,0,5,0,0,2,0,3],[20,39,0.5128,0.5982,0.2822,0.28571,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,11,0,0,3,0,0,5,0,0,1,0,0,6,0,6],[24,39,0.6154,0.65625,0.23652,0.42857,0.64286,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,6,0,0,6,0,0,4,0,0,7,0,5],[28,39,0.7179,0.60281,0.2981,0.39393,0.64286,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,5,0,0,5,0,0,3,0,0,3,0,0,9,0,4],[32,39,0.8205,0.75892,0.15337,0.57143,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,1,0,0,22,0,0],[36,39,0.9231,0.77228,0.16317,0.82143,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,24,0,0],[39,39,1.0,0.82589,0.11143,0.85714,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,29,0,0]]}]},{"i":"7fc80a8f8cb1a945","q":"$ ABC$  is a triangle with  $ \\angle BAC \\equal{} 10{}^\\circ$ ,  $ \\angle ABC \\equal{} 150{}^\\circ$ . Let  $ X$  be a point on  $ \\left[AC\\right]$  such that  $ \\left|AX\\right| \\equal{} \\left|BC\\right|$ . Find  $ \\angle BXC$ . $\\textbf{(A)}\\ 15^\\circ   \\qquad\\textbf{(B)}\\ 20^\\circ   \\qquad\\textbf{(C)}\\ 25^\\circ   \\qquad\\textbf{(D)}\\ 30^\\circ   \\qquad\\textbf{(E)}\\ 35^\\circ$","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,11,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,35,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b834969ee5739090","q":"13. 4a.(BUL 1) Let $m$ boxes be given, with some balls in each box. Let $n<m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put 1 ball in each of them. Prove: (a) If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. (b) If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.","t":[{"b":3,"e":1.0,"k":"rising","v":0.65179,"x":0.80803,"p":[[0,35,0.0,0.65179,0.17105,0.42859,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,15,0,0,5,0,1],[4,35,0.1143,0.72768,0.19351,0.67857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,12,0,0,6,0,6],[8,35,0.2286,0.79464,0.16342,0.71429,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,13,0,0,8,0,8],[12,35,0.3429,0.74107,0.18363,0.67857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,11,0,0,7,0,6],[16,35,0.4571,0.77232,0.14664,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,14,0,0,11,0,4],[20,35,0.5714,0.77684,0.14706,0.71429,0.85712,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,15,0,3],[24,35,0.6857,0.74999,0.19233,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,8,0,0,8,0,7],[28,35,0.8,0.78569,0.17499,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,6,0,0,12,0,7],[32,35,0.9143,0.76338,0.18074,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,15,0,0,6,0,7],[35,35,1.0,0.80803,0.15815,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,13,0,7]]},{"b":4,"e":1.0,"k":"rising","v":0.66072,"x":0.88393,"p":[[0,28,0.0,0.66072,0.17405,0.53572,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,15,0,0,4,0,2],[4,28,0.1429,0.66964,0.19045,0.42859,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,8,0,0,9,0,2],[8,28,0.2857,0.70979,0.21574,0.57132,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,8,0,0,5,0,7],[12,28,0.4286,0.68295,0.24172,0.42859,0.71429,0.85714,0.14,1.0,0,5,0,0,0,2,0,0,0,0,0,8,0,0,1,0,0,8,0,0,8,0,5],[16,28,0.5714,0.80801,0.16983,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,13,0,8],[20,28,0.7143,0.85714,0.16751,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,7,0,15],[24,28,0.8571,0.88393,0.18013,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,20],[28,28,1.0,0.87075,0.19008,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,7,0,18]]}]},{"i":"c2355eb5683625ed","q":"(V.Yasinsky, Ukraine) Suppose  $ X$  and  $ Y$  are the common points of two circles  $ \\omega_1$  and  $ \\omega_2$ . The third circle  $ \\omega$  is internally tangent to  $ \\omega_1$  and  $ \\omega_2$  in  $ P$  and  $ Q$  respectively. Segment  $ XY$  intersects  $ \\omega$  in points  $ M$  and  $ N$ . Rays  $ PM$  and  $ PN$  intersect  $ \\omega_1$  in points  $ A$  and  $ D$ ; rays  $ QM$  and  $ QN$  intersect  $ \\omega_2$  in points  $ B$  and  $ C$  respectively. Prove that  $ AB \\equal{} CD$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.05357,"x":0.23652,"p":[[0,57,0.0,0.14732,0.12619,0.0,0.14286,0.1786,0.0,0.57143,9,0,1,9,0,15,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,57,0.0702,0.16518,0.14334,0.0,0.14286,0.28571,0.0,0.4286,12,0,0,12,0,5,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.23652,0.14561,0.14286,0.2857,0.28571,0.0,0.4286,6,0,0,6,0,6,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.15607,0.13056,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,11,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.12045,0.12929,0.0,0.14143,0.2857,0.0,0.42857,15,0,0,15,0,8,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.13384,0.16342,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,5,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.14277,0.15972,0.0,0.07,0.28571,0.0,0.4286,16,0,0,16,0,4,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.11161,0.1461,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.05804,0.14223,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,57,0.7018,0.0759,0.1287,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.09375,0.12682,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.08928,0.12242,0.0,0.0,0.17857,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.12054,0.15198,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.0625,0.12339,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"rising","v":0.15178,"x":0.30357,"p":[[0,41,0.0,0.15178,0.15542,0.0,0.14286,0.28571,0.0,0.4286,15,0,0,15,0,3,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.23659,0.15812,0.14286,0.28571,0.32143,0.0,0.571,6,0,0,6,0,8,0,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[8,41,0.1951,0.20981,0.15142,0.10714,0.2857,0.28571,0.0,0.571,8,0,0,8,0,6,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[12,41,0.2927,0.18304,0.14827,0.0,0.14286,0.28571,0.0,0.4286,9,0,0,9,0,10,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.19197,0.14111,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,12,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.25893,0.10374,0.25,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,6,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.28571,0.10101,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,1,0,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,41,0.6829,0.29911,0.04164,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.29017,0.06667,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.30357,0.04725,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4591a2932d32fe77","q":"13. G1 (FRA) A semicircle $\\Gamma$ is drawn on one side of a straight line $l$. $C$ and $D$ are points on $\\Gamma$. The tangents to $\\Gamma$ at $C$ and $D$ meet $l$ at $B$ and $A$ respectively, with the center of the semicircle between them. Let $E$ be the point of intersection of $A C$ and $B D$, and $F$ the point on $l$ such that $E F$ is perpendicular to $l$. Prove that $E F$ bisects $\\angle C F D$.","t":[{"b":2,"e":1.0,"k":"flat","v":0.91071,"x":0.99107,"p":[[0,112,0.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,112,0.0357,0.95536,0.1448,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[8,112,0.0714,0.91071,0.24679,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[12,112,0.1071,0.99107,0.03459,1.0,1.0,1.0,0.8571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,112,0.1429,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,112,0.1786,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[24,112,0.2143,0.95089,0.18074,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[28,112,0.25,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[32,112,0.2857,0.94642,0.17038,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[36,112,0.3214,0.91962,0.18541,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,26],[40,112,0.3571,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,25],[44,112,0.3929,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[48,112,0.4286,0.94643,0.14617,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[52,112,0.4643,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[56,112,0.5,0.94196,0.18509,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[60,112,0.5357,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,112,0.5714,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[68,112,0.6071,0.91071,0.15465,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,6,0,21],[72,112,0.6429,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[76,112,0.6786,0.94643,0.12242,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[80,112,0.7143,0.94643,0.16269,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[84,112,0.75,0.95089,0.1411,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[88,112,0.7857,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[92,112,0.8214,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[96,112,0.8571,0.91963,0.17474,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,24],[100,112,0.8929,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[104,112,0.9286,0.94643,0.15872,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[108,112,0.9643,0.94642,0.12243,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[112,112,1.0,0.92856,0.13364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22]]},{"b":3,"e":0.71429,"k":"falling","v":0.80357,"x":0.98661,"p":[[0,77,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,77,0.0519,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[8,77,0.1039,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,77,0.1558,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,77,0.2078,0.94642,0.15872,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[20,77,0.2597,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,77,0.3117,0.94196,0.14665,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[28,77,0.3636,0.91963,0.17838,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,25],[32,77,0.4156,0.95544,0.12589,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[36,77,0.4675,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[40,77,0.5195,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[44,77,0.5714,0.88839,0.17399,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,4,0,20],[48,77,0.6234,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[52,77,0.6753,0.94196,0.18161,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[56,77,0.7273,0.94643,0.13243,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[60,77,0.7792,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[64,77,0.8312,0.90179,0.24338,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,26],[68,77,0.8831,0.90625,0.22192,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,25],[72,77,0.9351,0.92411,0.17491,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[76,77,0.987,0.87946,0.22899,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,23],[77,77,1.0,0.80357,0.30671,0.67857,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,20]]}]},{"i":"e9ff4ef2ee58a270","q":"14. (POL 3) Let $a, b, c$ be integers and $p$ an odd prime number. Prove that if $f(x)=a x^{2}+b x+c$ is a perfect square for $2 p-1$ consecutive integer values of $x$, then $p$ divides $b^{2}-4 a c$.","t":[{"b":3,"e":1.0,"k":"rising","v":0.5223,"x":0.74105,"p":[[0,18,0.0,0.5223,0.25154,0.42857,0.49979,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,1,0,0,9,0,0,5,0,0,6,0,0,3,0,2],[4,18,0.2222,0.72764,0.27749,0.571,0.85714,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,4,0,0,3,0,0,4,0,0,9,0,9],[8,18,0.4444,0.67856,0.25254,0.5354,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,1,0,0,5,0,0,7,0,0,4,0,0,6,0,7],[12,18,0.6667,0.74105,0.24857,0.57132,0.857,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,5,0,0,2,0,0,6,0,0,8,0,9],[16,18,0.8889,0.66071,0.24678,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,7,0,0,5,0,0,8,0,0,1,0,8],[18,18,1.0,0.70534,0.20497,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,7,0,0,6,0,6]]},{"b":7,"e":0.42857,"k":"flat","v":0.57589,"x":0.83928,"p":[[0,59,0.0,0.57589,0.26841,0.42857,0.64286,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,3,0,0,6,0,0,3,0,0,9,0,0,4,0,3],[4,59,0.0678,0.83928,0.19149,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,7,0,0,6,0,15],[8,59,0.1356,0.79017,0.23416,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,7,0,0,7,0,12],[12,59,0.2034,0.65163,0.28331,0.42857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,5,0,0,4,0,0,6,0,0,5,0,7],[16,59,0.2712,0.73213,0.25692,0.67857,0.78564,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,8,0,0,8,0,8],[20,59,0.339,0.74552,0.29176,0.57132,0.85714,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,0,0,0,3,0,0,3,0,0,2,0,0,8,0,12],[24,59,0.4068,0.75446,0.25059,0.67857,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,7,0,0,6,0,11],[28,59,0.4746,0.74105,0.2243,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,9,0,0,3,0,10],[32,59,0.5424,0.72321,0.26711,0.42859,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,6,0,0,2,0,0,3,0,0,9,0,9],[36,59,0.6102,0.74999,0.27433,0.57132,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,4,0,0,2,0,0,5,0,0,6,0,12],[40,59,0.678,0.70991,0.29343,0.42859,0.85714,0.895,0.0,1.0,1,8,0,1,0,3,0,0,0,0,0,5,0,0,0,0,0,5,0,0,10,0,8],[44,59,0.7458,0.67854,0.272,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,5,0,0,6,0,0,5,0,0,2,0,10],[48,59,0.8136,0.70973,0.2216,0.5713,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,7,0,0,7,0,6],[52,59,0.8814,0.62052,0.28484,0.39286,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,5,0,0,4,0,0,3,0,0,6,0,0,5,0,6],[56,59,0.9492,0.64282,0.28121,0.42859,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,3,0,0,2,0,0,4,0,0,8,0,0,5,0,6],[59,59,1.0,0.61159,0.24284,0.42857,0.64286,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,6,0,0,4,0,0,8,0,0,4,0,4]]}]},{"i":"4c143e3c494c15fe","q":"10. (NET 1) ${ }^{\\text {IMO6 }}$ An international society has its members in 6 different countries. The list of members contains 1978 names, numbered $1,2, \\ldots$, 1978. Prove that there is at least one member whose number is the sum of the numbers of two, not necessarily distinct, of his compatriots.","t":[{"b":3,"e":0.71429,"k":"rising","v":0.33926,"x":0.81695,"p":[[0,42,0.0,0.55801,0.3038,0.28571,0.35714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,16,0,0,1,0,0,2,0,0,4,0,0,1,0,8],[4,42,0.0952,0.33928,0.2335,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,1,0,0,20,0,0,1,0,0,1,0,0,3,0,0,1,0,1],[8,42,0.1905,0.45978,0.35665,0.21427,0.28571,0.857,0.0,1.0,8,3,0,8,0,0,0,0,9,0,0,0,0,0,2,0,0,4,0,0,6,0,3],[12,42,0.2857,0.49106,0.31529,0.28571,0.28571,0.75,0.0,1.0,2,6,0,2,0,0,0,0,17,0,0,1,0,0,1,0,0,3,0,0,2,0,6],[16,42,0.381,0.58919,0.32697,0.2857,0.64286,1.0,0.0,1.0,1,9,0,1,0,1,0,0,12,0,0,1,0,0,1,0,0,5,0,0,2,0,9],[20,42,0.4762,0.43748,0.33869,0.2857,0.28571,0.71429,0.0,1.0,6,5,0,6,0,1,0,0,12,0,0,1,0,0,1,0,0,5,0,0,1,0,5],[24,42,0.5714,0.48216,0.29611,0.28571,0.35786,0.71429,0.0,1.0,3,3,0,3,0,1,0,0,12,0,0,1,0,0,4,0,0,5,0,0,3,0,3],[28,42,0.6667,0.33926,0.26424,0.24999,0.28571,0.571,0.0,0.85714,7,0,0,7,0,1,0,0,15,0,0,0,0,0,2,0,0,5,0,0,2,0,0],[32,42,0.7619,0.62051,0.34184,0.28571,0.57121,1.0,0.0,1.0,1,13,0,1,0,0,0,0,13,0,0,0,0,0,3,0,0,2,0,0,0,0,13],[36,42,0.8571,0.79461,0.15948,0.71429,0.857,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,11,0,7],[40,42,0.9524,0.79905,0.16315,0.71429,0.857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,8,0,0,12,0,7],[42,42,1.0,0.81695,0.11973,0.71429,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,13,0,6]]},{"b":5,"e":0.2857,"k":"falling","v":0.41515,"x":0.63839,"p":[[0,36,0.0,0.63839,0.33879,0.28571,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,2,0,0,2,0,13],[4,36,0.1111,0.54016,0.33831,0.2857,0.57121,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,11,0,0,0,0,0,4,0,0,4,0,0,1,0,8],[8,36,0.2222,0.52676,0.32427,0.28571,0.49979,0.75,0.0,1.0,3,7,0,3,0,0,0,0,12,0,0,1,0,0,4,0,0,4,0,0,1,0,7],[12,36,0.3333,0.61605,0.29974,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,7,0,0,1,0,9],[16,36,0.4444,0.49549,0.31939,0.28571,0.28571,0.85704,0.0,1.0,2,6,0,2,0,1,0,0,15,0,0,1,0,0,3,0,0,1,0,0,3,0,6],[20,36,0.5556,0.44632,0.26435,0.2857,0.28571,0.60714,0.0,1.0,1,2,0,1,0,1,0,0,18,0,0,1,0,0,3,0,0,2,0,0,4,0,2],[24,36,0.6667,0.54908,0.3136,0.28571,0.42836,0.85704,0.0,1.0,1,6,0,1,0,1,0,0,14,0,0,0,0,0,2,0,0,4,0,0,4,0,6],[28,36,0.7778,0.4732,0.26591,0.28571,0.28571,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,18,0,0,1,0,0,3,0,0,4,0,0,1,0,4],[32,36,0.8889,0.44197,0.24836,0.2857,0.28571,0.60714,0.2857,1.0,0,3,0,0,0,0,0,0,22,0,0,0,0,0,2,0,0,4,0,0,1,0,3],[36,36,1.0,0.41515,0.20625,0.28571,0.28571,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,22,0,0,0,0,0,4,0,0,4,0,0,1,0,1]]}]},{"i":"9b903da9c73a5605","q":"$65$  distinct natural numbers not exceeding  $2016$  are given. Prove that among these numbers we can find four   $a,b,c,d$  such that  $a+b-c-d$  is divisible by  $2016.$","t":[{"b":4,"e":0.71429,"k":"flat","v":0.94643,"x":1.0,"p":[[0,27,0.0,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[12,27,0.4444,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,27,0.5926,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.95089,"x":0.99107,"p":[[0,36,0.0,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[4,36,0.1111,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[8,36,0.2222,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,36,0.3333,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,36,0.4444,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,36,0.5556,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,36,0.6667,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[28,36,0.7778,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[32,36,0.8889,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,36,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]}]},{"i":"eaca1433bd1ee086","q":"2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.","t":[{"b":1,"e":0.28571,"k":"rising","v":0.01339,"x":0.17857,"p":[[0,12,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0982,0.18704,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,1,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,12,0.6667,0.16071,0.19804,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,0,0,0,9,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,12,1.0,0.17857,0.22016,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,2,0,0,7,0,0,0,0,0,6,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.11161,"p":[[0,14,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.07589,0.17122,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,14,0.5714,0.06697,0.15966,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,14,0.8571,0.11161,0.16263,0.0,0.0,0.2857,0.0,0.57143,20,0,0,20,0,3,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[14,14,1.0,0.06241,0.11252,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e7071db39649723b","q":"$f$  is a function defined on integers and satisfies  $f(x)+f(x+3)=x^2$  for every integer  $x$ . If  $f(19)=94$ , then calculate  $f(94)$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.93741,"x":1.0,"p":[[0,6,0.0,0.93741,0.2084,1.0,1.0,1.0,0.14,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[4,6,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[6,6,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.94642,"x":1.0,"p":[[0,23,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,23,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,23,0.3478,0.94642,0.19152,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[12,23,0.5217,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,23,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"fb688fa6b037fd01","q":"11. (NET) Let $P(x)$ be a polynomial with real coefficients such that $P(x)>$ 0 for all $x \\geq 0$. Prove that there exists a positive integer $n$ such that $(1+x)^{n} P(x)$ is a polynomial with nonnegative coefficients.","t":[{"b":0,"e":1.0,"k":"volatile","v":0.23214,"x":0.99107,"p":[[0,28,0.0,0.73661,0.35555,0.28571,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,0,3,0,18],[4,28,0.1429,0.23214,0.31084,0.0,0.0,0.32143,0.0,1.0,17,3,0,17,0,0,0,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,3],[8,28,0.2857,0.44196,0.41088,0.0,0.42857,0.85714,0.0,1.0,12,7,0,12,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,0,2,0,7],[12,28,0.4286,0.75892,0.36498,0.71429,1.0,1.0,0.0,1.0,4,19,0,4,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0,2,0,19],[16,28,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,28,0.7143,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,28,0.8571,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[28,28,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":3,"e":0.28571,"k":"falling","v":0.04911,"x":0.85268,"p":[[0,47,0.0,0.85268,0.25123,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,3,0,21],[4,47,0.0851,0.45536,0.40317,0.0,0.42857,1.0,0.0,1.0,10,9,0,10,0,1,0,0,4,0,0,5,0,0,1,0,0,1,0,0,1,0,9],[8,47,0.1702,0.41964,0.36059,0.0,0.42857,0.71429,0.0,1.0,9,5,0,9,0,2,0,0,4,0,0,6,0,0,1,0,0,3,0,0,2,0,5],[12,47,0.2553,0.4375,0.32328,0.25,0.42857,0.71429,0.0,1.0,7,4,0,7,0,1,0,0,4,0,0,10,0,0,0,0,0,5,0,0,1,0,4],[16,47,0.3404,0.33928,0.31693,0.0,0.28571,0.42857,0.0,1.0,11,3,0,11,0,0,0,0,6,0,0,9,0,0,0,0,0,2,0,0,1,0,3],[20,47,0.4255,0.37053,0.31311,0.0,0.42857,0.57143,0.0,1.0,9,3,0,9,0,2,0,0,4,0,0,8,0,0,2,0,0,4,0,0,0,0,3],[24,47,0.5106,0.27679,0.35703,0.0,0.07143,0.42857,0.0,1.0,16,5,0,16,0,2,0,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,5],[28,47,0.5957,0.41071,0.36553,0.0,0.28571,0.71429,0.0,1.0,10,4,0,10,0,2,0,0,5,0,0,2,0,0,1,0,0,6,0,0,2,0,4],[32,47,0.6809,0.3125,0.3719,0.0,0.07143,0.60714,0.0,1.0,16,4,0,16,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,0,1,0,4],[36,47,0.766,0.28572,0.31542,0.0,0.28571,0.4286,0.0,1.0,15,2,0,15,0,0,0,0,4,0,0,6,0,0,1,0,0,4,0,0,0,0,2],[40,47,0.8511,0.15178,0.18877,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,0,0,0,10,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[44,47,0.9362,0.12053,0.19269,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,0,0,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[47,47,1.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"28537cce7c9d5291","q":"14. G2 (GRE) Three distinct points $A, B, C$ are fixed on a line in this order. Let $\\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $A C$. Denote by $P$ the intersection of the tangents to $\\Gamma$ at $A$ and $C$. Suppose $\\Gamma$ meets the segment $P B$ at $Q$. Prove that the intersection of the bisector of $\\angle A Q C$ and the line $A C$ does not depend on the choice of $\\Gamma$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.09813,"x":0.71427,"p":[[0,137,0.0,0.28571,0.16365,0.14286,0.2857,0.28571,0.0,0.857,1,0,0,1,0,9,0,0,17,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[4,137,0.0292,0.48659,0.2693,0.28571,0.4998,0.57143,0.0,1.0,1,5,0,1,0,2,0,0,11,0,0,2,0,0,11,0,0,0,0,0,0,0,5],[8,137,0.0584,0.46873,0.27718,0.28571,0.42857,0.57143,0.0,1.0,2,5,0,2,0,1,0,0,12,0,0,3,0,0,9,0,0,0,0,0,0,0,5],[12,137,0.0876,0.48658,0.27399,0.28571,0.571,0.57143,0.0,1.0,1,5,0,1,0,3,0,0,10,0,0,1,0,0,12,0,0,0,0,0,0,0,5],[16,137,0.1168,0.50892,0.29653,0.28571,0.4998,0.57143,0.14286,1.0,0,7,0,0,0,4,0,0,11,0,0,1,0,0,9,0,0,0,0,0,0,0,7],[20,137,0.146,0.47758,0.31269,0.2857,0.28571,0.57143,0.14,1.0,0,7,0,0,0,7,0,0,10,0,0,1,0,0,7,0,0,0,0,0,0,0,7],[24,137,0.1752,0.3839,0.16533,0.28571,0.28571,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,13,0,0,1,0,0,13,0,0,0,0,0,0,0,0],[28,137,0.2044,0.47321,0.27067,0.2857,0.28571,0.57143,0.0,1.0,1,5,0,1,0,1,0,0,15,0,0,0,0,0,10,0,0,0,0,0,0,0,5],[32,137,0.2336,0.45536,0.27302,0.28571,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,4,0,0,14,0,0,1,0,0,8,0,0,0,0,0,0,0,5],[36,137,0.2628,0.48213,0.28291,0.2857,0.28571,0.57143,0.14286,1.0,0,6,0,0,0,3,0,0,14,0,0,1,0,0,8,0,0,0,0,0,0,0,6],[40,137,0.292,0.47317,0.29973,0.28571,0.28571,0.57143,0.0,1.0,1,6,0,1,0,4,0,0,12,0,0,1,0,0,7,0,0,1,0,0,0,0,6],[44,137,0.3212,0.49552,0.34438,0.28571,0.28571,1.0,0.0,1.0,2,9,0,2,0,4,0,0,11,0,0,2,0,0,4,0,0,0,0,0,0,0,9],[48,137,0.3504,0.52677,0.31831,0.28571,0.35714,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,13,0,0,2,0,0,5,0,0,0,0,0,0,0,9],[52,137,0.3796,0.60266,0.30668,0.28571,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,12,0,0,2,0,0,7,0,0,0,0,0,0,0,11],[56,137,0.4088,0.60712,0.37458,0.28571,0.57143,1.0,0.0,1.0,2,14,0,2,0,4,0,0,7,0,0,0,0,0,5,0,0,0,0,0,0,0,14],[60,137,0.438,0.71427,0.31944,0.39288,0.9285,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,6,0,0,1,0,0,5,0,0,1,0,0,1,0,16],[64,137,0.4672,0.61606,0.34707,0.28571,0.57143,1.0,0.0,1.0,1,13,0,1,0,2,0,0,10,0,0,0,0,0,5,0,0,1,0,0,0,0,13],[68,137,0.4964,0.60266,0.29824,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,8,0,0,1,0,0,11,0,0,0,0,0,0,0,10],[72,137,0.5255,0.62945,0.3533,0.28571,0.57143,1.0,0.0,1.0,2,14,0,2,0,1,0,0,8,0,0,2,0,0,5,0,0,0,0,0,0,0,14],[76,137,0.5547,0.66518,0.32264,0.28571,0.57143,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,6,0,0,0,0,0,9,0,0,0,0,0,0,0,14],[80,137,0.5839,0.70534,0.35345,0.28571,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,18],[84,137,0.6131,0.55352,0.32291,0.2857,0.571,1.0,0.0,1.0,1,9,0,1,0,2,0,0,11,0,0,1,0,0,7,0,0,0,0,0,1,0,9],[88,137,0.6423,0.62499,0.3531,0.28571,0.57143,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,7,0,0,1,0,0,5,0,0,0,0,0,0,0,14],[92,137,0.6715,0.66069,0.36026,0.28571,0.78571,1.0,0.0,1.0,2,16,0,2,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,0,0,16],[96,137,0.7007,0.55803,0.35058,0.28571,0.57143,1.0,0.0,1.0,2,11,0,2,0,2,0,0,11,0,0,0,0,0,6,0,0,0,0,0,0,0,11],[100,137,0.7299,0.59375,0.37815,0.28571,0.57143,1.0,0.0,1.0,2,14,0,2,0,3,0,0,10,0,0,0,0,0,3,0,0,0,0,0,0,0,14],[104,137,0.7591,0.64732,0.37455,0.28571,0.78571,1.0,0.0,1.0,1,16,0,1,0,5,0,0,6,0,0,0,0,0,4,0,0,0,0,0,0,0,16],[108,137,0.7883,0.57143,0.36422,0.28571,0.57143,1.0,0.0,1.0,2,12,0,2,0,3,0,0,10,0,0,0,0,0,4,0,0,1,0,0,0,0,12],[112,137,0.8175,0.62501,0.34946,0.28571,0.57143,1.0,0.0,1.0,1,14,0,1,0,1,0,0,11,0,0,1,0,0,4,0,0,0,0,0,0,0,14],[116,137,0.8467,0.5357,0.35174,0.28571,0.57121,1.0,0.0,1.0,3,10,0,3,0,2,0,0,10,0,0,0,0,0,7,0,0,0,0,0,0,0,10],[120,137,0.8759,0.49106,0.31326,0.2857,0.42836,0.57143,0.0,1.0,1,7,0,1,0,5,0,0,10,0,0,0,0,0,9,0,0,0,0,0,0,0,7],[124,137,0.9051,0.29909,0.29742,0.10714,0.21428,0.5711,0.0,1.0,8,3,0,8,0,8,0,0,7,0,0,0,0,0,6,0,0,0,0,0,0,0,3],[128,137,0.9343,0.21429,0.26726,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,7,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,2],[132,137,0.9635,0.20517,0.28336,0.0,0.14286,0.2857,0.0,1.0,13,2,0,13,0,10,0,0,4,0,0,0,0,0,2,0,0,0,0,0,1,0,2],[136,137,0.9927,0.11598,0.2034,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[137,137,1.0,0.09813,0.11535,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.04911,"x":0.52677,"p":[[0,115,0.0,0.33926,0.17031,0.24999,0.28571,0.571,0.0,0.57143,1,0,0,1,0,7,0,0,12,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[4,115,0.0348,0.46872,0.27717,0.2857,0.42857,0.57143,0.0,1.0,1,5,0,1,0,4,0,0,9,0,0,4,0,0,9,0,0,0,0,0,0,0,5],[8,115,0.0696,0.45982,0.26422,0.28571,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,4,0,0,13,0,0,1,0,0,9,0,0,0,0,0,1,0,4],[12,115,0.1043,0.49997,0.2945,0.28571,0.49979,0.57143,0.14286,1.0,0,6,0,0,0,5,0,0,10,0,0,1,0,0,9,0,0,0,0,0,1,0,6],[16,115,0.1391,0.50001,0.34441,0.28571,0.28571,1.0,0.0,1.0,2,9,0,2,0,3,0,0,13,0,0,1,0,0,3,0,0,1,0,0,0,0,9],[20,115,0.1739,0.42856,0.33502,0.14286,0.28571,0.57143,0.0,1.0,5,6,0,5,0,5,0,0,7,0,0,1,0,0,8,0,0,0,0,0,0,0,6],[24,115,0.2087,0.40178,0.30186,0.24999,0.28571,0.57143,0.0,1.0,2,5,0,2,0,6,0,0,14,0,0,0,0,0,4,0,0,1,0,0,0,0,5],[28,115,0.2435,0.38393,0.25364,0.2857,0.28571,0.57143,0.0,1.0,2,3,0,2,0,4,0,0,15,0,0,1,0,0,7,0,0,0,0,0,0,0,3],[32,115,0.2783,0.48214,0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G6 (ARG) Let $A B C D$ be a convex quadrilateral with $A B$ not parallel to $C D$, let $X$ be a point inside $A B C D$ such that $\\measuredangle A D X=\\measuredangle B C X<90^{\\circ}$ and $\\measuredangle D A X=\\measuredangle C B X<90^{\\circ}$. If $Y$ is the point of intersection of the perpendicular bisectors of $A B$ and $C D$, prove that $\\measuredangle A Y B=2 \\measuredangle A D X$.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.20536,"x":0.55793,"p":[[0,70,0.0,0.24099,0.17295,0.14286,0.14286,0.28571,0.0,0.71429,1,0,1,1,0,20,0,0,4,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[4,70,0.0571,0.55353,0.32684,0.2857,0.4998,0.89286,0.14286,1.0,0,8,0,0,0,7,0,0,4,0,0,5,0,0,5,0,0,0,0,0,3,0,8],[8,70,0.1143,0.55793,0.31424,0.28571,0.4998,0.85714,0.14,1.0,0,6,0,0,0,5,0,0,8,0,0,3,0,0,1,0,0,5,0,0,4,0,6],[12,70,0.1714,0.5,0.33312,0.14297,0.28571,0.85714,0.14286,1.0,0,5,0,0,0,9,0,0,8,0,0,1,0,0,1,0,0,3,0,0,5,0,5],[16,70,0.2286,0.43739,0.29445,0.1429,0.28571,0.60682,0.14,1.0,0,4,0,0,0,9,0,0,8,0,0,5,0,0,2,0,0,2,0,0,2,0,4],[20,70,0.2857,0.35261,0.25256,0.14286,0.2857,0.42893,0.14,1.0,0,1,0,0,0,13,0,0,8,0,0,4,0,0,2,0,0,1,0,0,3,0,1],[24,70,0.3429,0.48661,0.346,0.14286,0.35714,0.85714,0.14286,1.0,0,5,0,0,0,13,0,0,3,0,0,2,0,0,1,0,0,3,0,0,5,0,5],[28,70,0.4,0.5089,0.34057,0.25,0.42857,0.85714,0.0,1.0,2,7,0,2,0,6,0,0,5,0,0,6,0,0,2,0,0,1,0,0,3,0,7],[32,70,0.4571,0.43293,0.30203,0.14286,0.28571,0.71429,0.14,1.0,0,4,0,0,0,11,0,0,7,0,0,2,0,0,3,0,0,4,0,0,1,0,4],[36,70,0.5143,0.40178,0.30185,0.14286,0.28571,0.4286,0.0,1.0,1,4,0,1,0,11,0,0,5,0,0,8,0,0,0,0,0,1,0,0,2,0,4],[40,70,0.5714,0.49553,0.31538,0.14289,0.49979,0.71429,0.14286,1.0,0,4,0,0,0,11,0,0,3,0,0,2,0,0,2,0,0,8,0,0,2,0,4],[44,70,0.6286,0.46874,0.30142,0.24999,0.35714,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,8,0,0,4,0,0,2,0,0,3,0,0,3,0,4],[48,70,0.6857,0.46426,0.26725,0.28571,0.28571,0.60714,0.14286,1.0,0,2,0,0,0,4,0,0,13,0,0,3,0,0,4,0,0,1,0,0,5,0,2],[52,70,0.7429,0.43294,0.29992,0.14286,0.28571,0.71429,0.14,1.0,0,3,0,0,0,12,0,0,5,0,0,3,0,0,2,0,0,5,0,0,2,0,3],[56,70,0.8,0.4241,0.30406,0.14286,0.28571,0.71429,0.14286,1.0,0,3,0,0,0,11,0,0,8,0,0,3,0,0,1,0,0,2,0,0,4,0,3],[60,70,0.8571,0.30804,0.23719,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,14,0,0,12,0,0,2,0,0,0,0,0,0,0,0,3,0,1],[64,70,0.9143,0.25884,0.18713,0.14286,0.14288,0.28571,0.14,0.85714,0,0,0,0,0,18,0,0,10,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[68,70,0.9714,0.26774,0.14176,0.14286,0.2857,0.28571,0.14,0.71429,0,0,0,0,0,13,0,0,14,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[70,70,1.0,0.20536,0.07936,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.1429,"k":"flat","v":0.19188,"x":0.71875,"p":[[0,70,0.0,0.21429,0.12877,0.14286,0.14286,0.2857,0.0,0.57143,1,0,1,1,0,21,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[4,70,0.0571,0.71875,0.31438,0.39286,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,5,0,0,2,0,0,0,0,0,1,0,0,10,0,11],[8,70,0.1143,0.67855,0.3134,0.42857,0.78571,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,0,0,0,6,0,0,3,0,0,2,0,0,5,0,11],[12,70,0.1714,0.4866,0.32899,0.24999,0.28571,0.85714,0.14286,1.0,0,5,0,0,0,8,0,0,9,0,0,4,0,0,0,0,0,0,0,0,6,0,5],[16,70,0.2286,0.53122,0.33164,0.24999,0.49979,0.85714,0.0,1.0,1,6,0,1,0,7,0,0,5,0,0,3,0,0,3,0,0,3,0,0,4,0,6],[20,70,0.2857,0.54463,0.34337,0.14289,0.4998,0.89286,0.14286,1.0,0,8,0,0,0,10,0,0,2,0,0,4,0,0,2,0,0,4,0,0,2,0,8],[24,70,0.3429,0.49998,0.33502,0.14289,0.49979,0.74996,0.0,1.0,2,5,0,2,0,8,0,0,4,0,0,2,0,0,3,0,0,5,0,0,3,0,5],[28,70,0.4,0.39282,0.29012,0.14286,0.28571,0.571,0.0,1.0,1,4,0,1,0,11,0,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,4],[32,70,0.4571,0.49107,0.34057,0.14286,0.28571,0.85714,0.14286,1.0,0,5,0,0,0,11,0,0,6,0,0,1,0,0,1,0,0,3,0,0,5,0,5],[36,70,0.5143,0.45969,0.30882,0.14286,0.35714,0.71429,0.14,1.0,0,4,0,0,0,11,0,0,5,0,0,2,0,0,4,0,0,4,0,0,2,0,4],[40,70,0.5714,0.36158,0.2395,0.14286,0.28571,0.571,0.14286,1.0,0,1,0,0,0,12,0,0,8,0,0,3,0,0,4,0,0,3,0,0,1,0,1],[44,70,0.6286,0.29909,0.25089,0.14286,0.14288,0.28571,0.14286,1.0,0,2,0,0,0,18,0,0,7,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[48,70,0.6857,0.22313,0.12851,0.14286,0.1429,0.28571,0.14,0.71429,0,0,0,0,0,20,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[52,70,0.7429,0.19635,0.11714,0.14286,0.14286,0.1786,0.14,0.71429,0,0,0,0,0,24,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,70,0.8,0.22322,0.10677,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,18,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[60,70,0.8571,0.19188,0.09187,0.14286,0.14286,0.1786,0.14,0.42857,0,0,0,0,0,24,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.23212,0.17762,0.14286,0.14286,0.1786,0.14286,0.71429,0,0,0,0,0,24,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[68,70,0.9714,0.23213,0.1417,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,20,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[70,70,1.0,0.20982,0.11285,0.14286,0.14286,0.28571,0.0,0.4286,1,0,0,1,0,20,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0a3a51a7298e35a7","q":"26. (VIE 2) Let $a, b, c, d$ be four nonnegative numbers satisfying $a+b+c+d=$ 1. Prove the inequality  $$ a b c+b c d+c d a+d a b \\leq \\frac{1}{27}+\\frac{176}{27} a b c d $$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.61606,"x":0.77231,"p":[[0,44,0.0,0.72322,0.22284,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,10,0,0,2,0,9],[4,44,0.0909,0.72768,0.17985,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,11,0,0,3,0,7],[8,44,0.1818,0.67856,0.21725,0.57143,0.71429,0.75,0.0,1.0,1,6,1,1,0,0,0,0,1,0,0,1,0,0,12,0,0,9,0,0,2,0,6],[12,44,0.2727,0.6875,0.25111,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,1,0,0,5,0,0,10,0,0,1,0,9],[16,44,0.3636,0.69643,0.18123,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,16,0,0,3,0,4],[20,44,0.4545,0.75,0.18211,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,16,0,0,2,0,8],[24,44,0.5455,0.77231,0.15095,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,8,0,6],[28,44,0.6364,0.75446,0.18977,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,14,0,0,2,0,9],[32,44,0.7273,0.74106,0.20026,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,7,0,0,4,0,9],[36,44,0.8182,0.69197,0.18593,0.57143,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,14,0,0,2,0,5],[40,44,0.9091,0.61606,0.19704,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,14,0,0,2,0,2],[44,44,1.0,0.65625,0.1551,0.57143,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,18,0,0,1,0,2]]},{"b":1,"e":0.57143,"k":"flat","v":0.58927,"x":0.7857,"p":[[0,86,0.0,0.63393,0.22851,0.57143,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,4,0,0,11,0,0,8,0,0,0,0,6],[4,86,0.0465,0.72768,0.20935,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,10,0,0,5,0,7],[8,86,0.093,0.7857,0.19563,0.57143,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,6,0,0,2,0,13],[12,86,0.1395,0.74552,0.2012,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,12,0,0,2,0,9],[16,86,0.186,0.71873,0.16165,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,9,0,0,5,0,5],[20,86,0.2326,0.71429,0.22304,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,0,0,0,9,0,0,9,0,0,3,0,8],[24,86,0.2791,0.70089,0.20316,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,9,0,0,4,0,6],[28,86,0.3256,0.73661,0.21162,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,7,0,0,6,0,8],[32,86,0.3721,0.74107,0.15335,0.67857,0.71429,0.85704,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,15,0,0,3,0,6],[36,86,0.4186,0.75892,0.20026,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,8,0,0,4,0,10],[40,86,0.4651,0.67411,0.20896,0.57143,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,14,0,0,0,0,6],[44,86,0.5116,0.72322,0.23941,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,1,0,0,6,0,0,9,0,0,5,0,8],[48,86,0.5581,0.71429,0.21129,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,14,0,0,4,0,6],[52,86,0.6047,0.74107,0.20958,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,12,0,0,1,0,10],[56,86,0.6512,0.70982,0.12103,0.71429,0.71429,0.71429,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,21,0,0,3,0,2],[60,86,0.6977,0.69643,0.17405,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,14,0,0,2,0,5],[64,86,0.7442,0.69196,0.19597,0.57143,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,14,0,0,1,0,6],[68,86,0.7907,0.75447,0.19637,0.67857,0.71429,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,14,0,0,0,0,10],[72,86,0.8372,0.73214,0.15465,0.57143,0.71429,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,15,0,0,2,0,6],[76,86,0.8837,0.71429,0.17496,0.57143,0.71429,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,14,0,0,2,0,6],[80,86,0.9302,0.69643,0.18472,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,10,0,0,4,0,5],[84,86,0.9767,0.58927,0.1948,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,7,0,0,11,0,0,6,0,0,3,0,2],[86,86,1.0,0.60713,0.19885,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,8,0,0,4,0,0,13,0,0,2,0,2]]}]},{"i":"0bb8b9307fa224cf","q":"23. (YUG 2) Prove that for every natural number $k(k \\geq 2)$ there exists an irrational number $r$ such that for every natural number $m$,  $$ \\left[r^{m}\\right] \\equiv-1 \\quad(\\bmod k) $$  Remark. An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.","t":[{"b":2,"e":0.0,"k":"falling","v":0.00893,"x":0.95536,"p":[[0,72,0.0,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,72,0.0556,0.79911,0.38608,0.85714,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,23],[8,72,0.1111,0.61161,0.46185,0.0,0.92857,1.0,0.0,1.0,11,16,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,16],[12,72,0.1667,0.30804,0.43757,0.0,0.0,0.85714,0.0,1.0,21,7,0,21,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,7],[16,72,0.2222,0.4375,0.46966,0.0,0.07143,1.0,0.0,1.0,16,11,0,16,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,11],[20,72,0.2778,0.32143,0.43448,0.0,0.0,0.85714,0.0,1.0,20,7,0,20,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,7],[24,72,0.3333,0.375,0.4414,0.0,0.14286,1.0,0.0,1.0,15,10,0,15,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,10],[28,72,0.3889,0.16072,0.32093,0.0,0.0,0.14286,0.0,1.0,23,3,0,23,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[32,72,0.4444,0.23661,0.36528,0.0,0.0,0.32144,0.0,1.0,19,5,0,19,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,5],[36,72,0.5,0.23214,0.38091,0.0,0.0,0.32143,0.0,1.0,21,5,0,21,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[40,72,0.5556,0.16964,0.32031,0.0,0.0,0.17857,0.0,1.0,23,3,0,23,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[44,72,0.6111,0.25,0.38132,0.0,0.0,0.46429,0.0,1.0,21,4,0,21,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,4],[48,72,0.6667,0.1607,0.31892,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[52,72,0.7222,0.125,0.29613,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[56,72,0.7778,0.20089,0.346,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,3],[60,72,0.8333,0.12946,0.25595,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[64,72,0.8889,0.125,0.26904,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[68,72,0.9444,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[72,72,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"falling","v":0.41964,"x":1.0,"p":[[0,77,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,77,0.0519,0.70982,0.42331,0.25,1.0,1.0,0.0,1.0,7,20,0,7,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,20],[8,77,0.1039,0.79464,0.37617,0.85714,1.0,1.0,0.0,1.0,5,22,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,22],[12,77,0.1558,0.74554,0.39887,0.60714,1.0,1.0,0.0,1.0,5,21,0,5,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,21],[16,77,0.2078,0.87946,0.26513,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,24],[20,77,0.2597,0.56697,0.41417,0.14286,0.57144,1.0,0.0,1.0,5,13,0,5,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,0,2,0,13],[24,77,0.3117,0.66518,0.4159,0.24999,1.0,1.0,0.0,1.0,6,17,0,6,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,17],[28,77,0.3636,0.57589,0.45244,0.0,0.85714,1.0,0.0,1.0,9,16,0,9,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,16],[32,77,0.4156,0.58473,0.44381,0.0,0.78571,1.0,0.0,1.0,9,15,0,9,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,15],[36,77,0.4675,0.70982,0.3838,0.53571,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,16],[40,77,0.5195,0.43303,0.40483,0.0,0.28571,0.89286,0.0,1.0,11,8,0,11,0,1,0,0,5,0,0,3,0,0,0,0,0,3,0,0,1,0,8],[44,77,0.5714,0.41964,0.34429,0.14286,0.28571,0.71429,0.0,1.0,5,5,0,5,0,7,0,0,5,0,0,4,0,0,2,0,0,2,0,0,2,0,5],[48,77,0.6234,0.53563,0.41196,0.14214,0.42857,1.0,0.0,1.0,7,12,0,7,0,3,0,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,12],[52,77,0.6753,0.65179,0.3976,0.25001,0.85714,1.0,0.0,1.0,4,15,0,4,0,4,0,0,3,0,0,0,0,0,1,0,0,3,0,0,2,0,15],[56,77,0.7273,0.65179,0.31731,0.42859,0.64286,1.0,0.0,1.0,2,11,0,2,0,2,0,0,2,0,0,3,0,0,7,0,0,4,0,0,1,0,11],[60,77,0.7792,0.67857,0.29451,0.53571,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,0,5,0,9],[64,77,0.8312,0.55356,0.33645,0.39286,0.4998,0.89286,0.0,1.0,4,8,0,4,0,2,0,0,2,0,0,8,0,0,3,0,0,4,0,0,1,0,8],[68,77,0.8831,0.625,0.32488,0.42857,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,2,0,0,7,0,0,2,0,0,5,0,0,3,0,9],[72,77,0.9351,0.46429,0.18558,0.39286,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,5,0,0,7,0,0,13,0,0,4,0,0,0,0,0],[76,77,0.987,0.47767,0.20704,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,2,0,0,7,0,0,11,0,0,7,0,0,0,0,0],[77,77,1.0,0.48213,0.14173,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,14,0,0,10,0,0,4,0,0,0,0,0]]}]},{"i":"54a1a4fe592430ad","q":"25. N2 (RUS) The function $\\psi$ from the set $\\mathbb{N}$ of positive integers into itself is defined by the equality  $$ \\psi(n)=\\sum_{k=1}^{n}(k, n), \\quad n \\in \\mathbb{N} $$  where $(k, n)$ denotes the greatest common divisor of $k$ and $n$. (a) Prove that $\\psi(m n)=\\psi(m) \\psi(n)$ for every two relatively prime $m, n \\in$ $\\mathbb{N}$. (b) Prove that for each $a \\in \\mathbb{N}$ the equation $\\psi(x)=a x$ has a solution. (c) Find all $a \\in \\mathbb{N}$ such that the equation $\\psi(x)=a x$ has a unique solution.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.85714,"x":0.96875,"p":[[0,44,0.0,0.85714,0.20516,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,2,0,20],[4,44,0.0909,0.91964,0.18877,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[8,44,0.1818,0.95982,0.10249,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[12,44,0.2727,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[16,44,0.3636,0.95089,0.12169,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[20,44,0.4545,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[24,44,0.5455,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[28,44,0.6364,0.94196,0.12807,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[32,44,0.7273,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,44,0.8182,0.95981,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[40,44,0.9091,0.9241,0.14279,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,23],[44,44,1.0,0.9107,0.1505,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,22]]},{"b":6,"e":1.0,"k":"rising","v":0.81696,"x":0.99107,"p":[[0,25,0.0,0.81696,0.22083,0.67857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,4,0,16],[4,25,0.16,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,25,0.32,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[12,25,0.48,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[16,25,0.64,0.94642,0.1372,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[20,25,0.8,0.95979,0.125,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[24,25,0.96,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[25,25,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"4441a75d1e2461b2","q":"A graph has  $17$  points and each point has  $4$  edges. Show that there are two points which are not joined and which are not both joined to the same point.","t":[{"b":0,"e":1.0,"k":"flat","v":0.88392,"x":0.99107,"p":[[0,48,0.0,0.88392,0.16917,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,16,0,14],[4,48,0.0833,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,48,0.1667,0.95535,0.11538,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[12,48,0.25,0.95535,0.13092,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[16,48,0.3333,0.93302,0.18208,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[20,48,0.4167,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[24,48,0.5,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,48,0.5833,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,48,0.6667,0.95981,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[36,48,0.75,0.94195,0.17448,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[40,48,0.8333,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,48,0.9167,0.91517,0.12811,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[48,48,1.0,0.90175,0.18368,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,22]]},{"b":5,"e":0.57143,"k":"falling","v":0.56249,"x":0.91071,"p":[[0,14,0.0,0.89731,0.09611,0.85714,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,18,0,12],[4,14,0.2857,0.91071,0.17768,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,23],[8,14,0.5714,0.83034,0.24339,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,17],[12,14,0.8571,0.70088,0.26573,0.4286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,3,0,0,5,0,10],[14,14,1.0,0.56249,0.23941,0.39286,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,6,0,0,7,0,0,4,0,0,4,0,3]]}]},{"i":"403d5ea792bd1d4e","q":"A flea jumps in a straight numbered line. It jumps first from point  $0$  to point  $1$ . Afterwards, if its last jump was from  $A$  to  $B$ , then the next jump is from  $B$  to one of the points  $B + (B - A) - 1$ ,  $B + (B - A)$ ,  $B + (B-A) + 1$ .\r\n\r\nProve that if the flea arrived twice at the point  $n$ ,  $n$  positive integer, then it performed at least  $\\lceil 2\\sqrt n\\rceil$  jumps.","t":[{"b":0,"e":0.42857,"k":"falling","v":0.28125,"x":0.71874,"p":[[0,70,0.0,0.65179,0.26651,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,1,1,0,0,3,0,0,1,0,0,4,0,0,10,0,0,7,1,3],[4,70,0.0571,0.4732,0.32229,0.2857,0.35714,0.71429,0.0,1.0,2,6,0,2,0,5,0,0,9,0,0,4,0,0,2,0,0,3,0,0,1,0,6],[8,70,0.1143,0.60045,0.31025,0.28571,0.67857,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,8,0,0,4,0,0,0,1,0,5,0,0,3,0,8],[12,70,0.1714,0.37055,0.23921,0.24999,0.28571,0.46461,0.0,1.0,2,1,0,2,0,6,0,0,12,0,0,4,0,0,1,0,0,6,0,0,0,0,1],[16,70,0.2286,0.48661,0.31916,0.2857,0.35716,0.71429,0.0,1.0,2,5,0,2,0,5,0,0,9,0,0,1,0,0,4,0,0,4,0,0,2,0,5],[20,70,0.2857,0.45982,0.31891,0.2857,0.35714,0.71429,0.0,1.0,4,5,0,4,0,3,0,0,9,0,0,2,0,0,4,0,0,5,0,0,0,0,5],[24,70,0.3429,0.51335,0.21682,0.42857,0.57121,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,4,0,0,6,0,0,12,0,0,5,0,0,0,0,2],[28,70,0.4,0.51786,0.20748,0.39286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,5,0,0,10,0,0,7,0,0,1,0,1],[32,70,0.4571,0.66072,0.28064,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,8,0,0,5,0,7],[36,70,0.5143,0.59819,0.28446,0.28571,0.71429,0.71429,0.0,1.0,1,6,0,1,0,3,0,0,5,0,0,0,0,0,6,0,0,11,0,0,0,0,6],[40,70,0.5714,0.66295,0.23299,0.57143,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,2,0,0,6,1,0,12,0,0,1,0,6],[44,70,0.6286,0.71874,0.24087,0.67857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,13,0,0,3,0,8],[48,70,0.6857,0.69196,0.16793,0.67857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,18,0,0,3,0,3],[52,70,0.7429,0.64729,0.26723,0.571,0.71429,0.85704,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,1,0,0,8,0,0,8,0,0,2,0,7],[56,70,0.8,0.60936,0.24871,0.5713,0.71429,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,0,0,0,2,0,1,8,0,0,12,0,0,2,0,3],[60,70,0.8571,0.55357,0.22798,0.28571,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,8,0,0,1,0,0,3,0,0,16,0,0,2,0,0],[64,70,0.9143,0.44195,0.22405,0.28571,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,19,0,0,0,0,0,3,0,0,7,0,0,1,0,1],[68,70,0.9714,0.33928,0.15872,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,25,0,0,0,0,0,2,0,0,2,0,0,1,0,0],[70,70,1.0,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.31027,"x":0.625,"p":[[0,57,0.0,0.625,0.28064,0.39286,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,5,0,0,3,0,0,1,0,0,4,0,0,7,0,0,9,0,3],[4,57,0.0702,0.56696,0.33404,0.2857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,6,0,0,3,0,0,3,0,0,3,0,0,3,0,8],[8,57,0.1404,0.55803,0.25345,0.28571,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,11,0,0,0,0,4],[12,57,0.2105,0.47768,0.26633,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,9,0,0,4,0,0,3,0,0,7,0,0,2,0,2],[16,57,0.2807,0.40402,0.28156,0.25,0.28571,0.57143,0.0,1.0,4,2,0,4,0,4,0,0,10,0,0,1,0,0,7,0,0,3,0,0,0,1,2],[20,57,0.3509,0.4308,0.26874,0.28571,0.35714,0.60714,0.0,1.0,2,1,0,2,0,5,0,0,9,0,0,4,0,0,4,0,0,4,0,0,2,1,1],[24,57,0.4211,0.39508,0.28514,0.1429,0.28571,0.57143,0.0,1.0,4,3,0,4,0,5,0,0,8,0,0,5,0,0,3,1,0,3,0,0,0,0,3],[28,57,0.4912,0.42856,0.23145,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,9,0,0,4,0,0,6,0,0,6,0,0,0,0,1],[32,57,0.5614,0.37051,0.21084,0.25,0.35714,0.4642,0.0,0.85714,2,0,0,2,0,6,0,0,8,0,0,8,0,0,4,0,0,3,0,0,1,0,0],[36,57,0.6316,0.40179,0.25862,0.2857,0.28571,0.57143,0.0,1.0,2,2,0,2,0,5,0,0,11,0,0,4,0,0,4,0,0,3,0,0,1,0,2],[40,57,0.7018,0.40625,0.26271,0.14286,0.28571,0.71429,0.0,1.0,2,1,0,2,0,7,0,0,8,0,0,4,0,0,2,0,0,7,0,0,1,0,1],[44,57,0.7719,0.39731,0.20433,0.2857,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,5,0,0,12,0,0,6,0,0,6,0,0,1,0,0,1,0,1],[48,57,0.8421,0.39512,0.26425,0.25,0.28571,0.57143,0.0,1.0,1,2,0,1,1,6,0,0,12,0,0,2,0,0,3,0,0,4,0,0,1,0,2],[52,57,0.9123,0.31919,0.19478,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,1,9,0,0,11,0,0,3,0,0,4,0,0,3,0,0,0,0,0],[56,57,0.9825,0.31027,0.24911,0.14286,0.28571,0.42857,0.0,1.0,4,2,0,4,0,8,0,1,10,0,0,3,0,0,3,0,0,1,0,0,0,0,2],[57,57,1.0,0.35268,0.18893,0.2857,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,6,0,0,13,0,0,5,0,0,3,0,0,4,0,0,0,0,0]]}]},{"i":"091720043b698a09","q":"A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations \r\n- to take 2 numbers from these, to double first one, to add the second one and to save the sum. \r\n- to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. \r\n- to repeat this procedure, until all the 2002 numbers won't be used. \r\nStudent wins the game if final product is maximum possible. \r\nFind the winning strategy and prove it.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.66069,"x":0.98661,"p":[[0,38,0.0,0.66069,0.228,0.57143,0.71429,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,3,0,0,6,0,0,15,0,0,2,0,4],[4,38,0.1053,0.9598,0.10858,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[8,38,0.2105,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[12,38,0.3158,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[16,38,0.4211,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[20,38,0.5263,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,38,0.6316,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[28,38,0.7368,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,38,0.8421,0.93303,0.09441,0.85714,1.0,1.0,0.714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[36,38,0.9474,0.9464,0.11718,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[38,38,1.0,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23]]},{"b":5,"e":1.0,"k":"rising","v":0.60713,"x":0.97768,"p":[[0,15,0.0,0.60713,0.30723,0.42859,0.64286,0.75,0.0,1.0,4,7,0,4,0,0,0,0,2,0,0,3,0,0,7,0,0,8,0,0,1,0,7],[4,15,0.2667,0.89732,0.22084,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,24],[8,15,0.5333,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[12,15,0.8,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[15,15,1.0,0.90177,0.14036,0.82143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20]]}]},{"i":"e81c29b7824f947f","q":"A subset $S$ of $\\{1,2, \\ldots, n\\}$ is called balanced if for every $a \\in S$ there exists some $b \\in S, b \\neq a$, such that $\\frac{a+b}{2} \\in S$ as well.\n\n(a) Let $k>1$ be an integer and let $n=2^{k}$. Show that every subset $S$ of $\\{1,2, \\ldots, n\\}$ with $|S|>\\frac{3 n}{4}$ is balanced.\n\n(b) Does there exist an $n=2^{k}$, with $k>1$ an integer, for which every subset $S$ of $\\{1,2, \\ldots, n\\}$ with $|S|>\\frac{2 n}{3}$ is balanced?\n\nSolution of part (a). Let $m=n-|S|$, thus $m<\\frac{n}{4}$ and (as $n$ is a multiple of 4 ) $m \\leq \\frac{n}{4}-1$. Let $a \\in S$. There are $\\frac{n}{2}-1$ elements in $\\{1,2, \\ldots, n\\}$ distinct from $a$ and with the same parity as $a$. At most $m$ of those elements are not in $S$, hence at least $\\frac{n}{2}-1-m \\geq \\frac{n}{4}$ of them are in $S$. For each such $b$, the number $\\frac{a+b}{2}$ is an integer, and all of these at least $\\frac{n}{4}$ numbers are distinct. But at most $m<\\frac{n}{4}$ of them are not in $S$, so at least one is a member of $S$. Hence $S$ is balanced.","t":[{"b":2,"e":1.0,"k":"rising","v":0.37501,"x":0.70965,"p":[[0,24,0.0,0.37501,0.21354,0.42857,0.42857,0.4286,0.0,0.71429,7,0,5,7,0,0,0,0,0,0,0,18,0,0,5,0,0,2,0,0,0,0,0],[4,24,0.1667,0.62498,0.16656,0.4286,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,9,0,0,5,0,1],[8,24,0.3333,0.62052,0.15816,0.42859,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,9,0,0,4,0,1],[12,24,0.5,0.6205,0.15408,0.4286,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,8,0,0,11,0,0,3,0,1],[16,24,0.6667,0.6116,0.18293,0.4286,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,10,0,0,5,0,0,3,0,3],[20,24,0.8333,0.65173,0.12849,0.57143,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,12,0,0,3,0,1],[24,24,1.0,0.70965,0.14937,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,15,0,0,3,0,4]]},{"b":7,"e":0.42857,"k":"rising","v":0.27231,"x":0.66057,"p":[[0,32,0.0,0.27231,0.22687,0.0,0.42857,0.42858,0.0,0.71429,12,0,8,12,0,1,0,0,1,0,0,15,0,0,2,0,0,1,0,0,0,0,0],[4,32,0.125,0.66057,0.17764,0.5354,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,6,0,0,11,0,0,4,0,3],[8,32,0.25,0.6116,0.24545,0.42857,0.71414,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,2,0,0,5,0,0,6,0,0,12,0,0,1,0,4],[12,32,0.375,0.58926,0.22232,0.42857,0.5712,0.71429,0.1429,1.0,0,4,0,0,0,1,0,0,2,0,0,11,0,0,6,0,0,6,0,0,2,0,4],[16,32,0.5,0.5982,0.27301,0.42857,0.57143,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,0,0,0,11,0,0,4,0,0,5,0,0,4,0,5],[20,32,0.625,0.55801,0.17261,0.42857,0.4998,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,15,0,0,7,0,0,6,0,0,1,0,2],[24,32,0.75,0.50891,0.20805,0.42857,0.4286,0.60714,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,16,0,0,5,0,0,5,0,0,2,0,1],[28,32,0.875,0.45536,0.1448,0.42857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,27,0,0,1,0,0,2,0,0,0,0,1],[32,32,1.0,0.42411,0.09094,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"b09b3482f9ad76d2","q":"A set  $A$  of positive integers is called *uniform* if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer  $n>1$  such that there exist a uniform set  $A$  with  $n$  elements.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.39732,"x":0.88839,"p":[[0,107,0.0,0.39732,0.29824,0.14286,0.28571,0.57143,0.0,1.0,2,2,2,2,0,14,0,0,0,0,0,1,0,0,9,0,0,2,0,0,2,0,2],[4,107,0.0374,0.84375,0.1488,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,14,0,10],[8,107,0.0748,0.74999,0.30722,0.67857,0.85714,1.0,0.0,1.0,3,10,1,3,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,12,0,10],[12,107,0.1121,0.62054,0.36001,0.35714,0.71429,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,0,0,0,2,0,0,1,0,0,7,0,0,7,0,7],[16,107,0.1495,0.65176,0.34615,0.571,0.78571,0.89286,0.0,1.0,4,8,1,4,0,3,0,0,0,0,0,0,0,0,6,0,0,3,0,0,8,0,8],[20,107,0.1869,0.69195,0.30952,0.5354,0.85707,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,0,0,0,4,0,0,3,0,0,4,0,0,9,0,8],[24,107,0.2243,0.79463,0.23402,0.71429,0.85714,0.89286,0.0,1.0,1,8,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,7,0,0,14,0,8],[28,107,0.2617,0.71428,0.32537,0.57143,0.85714,1.0,0.0,1.0,4,9,0,4,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,9],[32,107,0.2991,0.70981,0.3302,0.71429,0.85714,0.89286,0.0,1.0,3,8,1,3,0,3,0,0,0,0,0,1,0,0,0,0,0,5,0,0,12,0,8],[36,107,0.3364,0.88839,0.18466,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,16],[40,107,0.3738,0.70089,0.29957,0.57143,0.85714,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,0,10,0,7],[44,107,0.4112,0.82587,0.2362,0.857,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,16,0,10],[48,107,0.4486,0.82143,0.29667,0.82143,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,18],[52,107,0.486,0.76784,0.29397,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,0,13,0,10],[56,107,0.5234,0.77669,0.26007,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,1,0,0,2,0,0,4,0,0,13,0,9],[60,107,0.5607,0.69642,0.29827,0.57143,0.85714,0.85714,0.0,1.0,4,5,1,4,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,12,0,5],[64,107,0.5981,0.58482,0.37518,0.14286,0.78564,0.85714,0.0,1.0,5,6,1,5,0,4,0,0,2,0,0,1,0,0,2,0,0,2,0,0,10,0,6],[68,107,0.6355,0.70087,0.34876,0.57143,0.85714,1.0,0.0,1.0,5,9,0,5,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,9],[72,107,0.6729,0.70088,0.33949,0.57143,0.85714,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,0,13,0,8],[76,107,0.7103,0.79016,0.19229,0.67836,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,6,0,0,8,0,10],[80,107,0.7477,0.81694,0.18296,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,5,0,13],[84,107,0.785,0.81697,0.21498,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,0,6,0,14],[88,107,0.8224,0.71424,0.19887,0.5713,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,5,0,0,9,0,5],[92,107,0.8598,0.73659,0.22619,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,0,0,0,9,0,0,7,0,0,6,0,8],[96,107,0.8972,0.77229,0.18512,0.67857,0.78564,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,8,0,0,8,0,8],[100,107,0.9346,0.79909,0.16313,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,11,0,8],[104,107,0.972,0.77677,0.15949,0.67857,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,11,0,6],[107,107,1.0,0.76786,0.15872,0.71429,0.78571,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,11,0,0,12,0,4]]},{"b":1,"e":0.0,"k":"falling","v":0.04018,"x":0.87946,"p":[[0,224,0.0,0.36607,0.2765,0.14286,0.14286,0.57143,0.0,1.0,1,2,1,1,0,16,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,2],[4,224,0.0179,0.85713,0.19563,0.85714,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,13,0,13],[8,224,0.0357,0.77232,0.20472,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,6,0,0,12,0,7],[12,224,0.0536,0.75891,0.26352,0.67857,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,2,0,0,0,0,0,4,0,0,5,0,0,9,0,10],[16,224,0.0714,0.76786,0.23891,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,0,14,0,7],[20,224,0.0893,0.80802,0.19761,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,11,0,10],[24,224,0.1071,0.8482,0.14258,0.85711,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,18,0,8],[28,224,0.125,0.87946,0.19597,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,16],[32,224,0.1429,0.7366,0.27689,0.57143,0.85714,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,0,0,0,5,0,0,3,0,0,13,0,7],[36,224,0.1607,0.73659,0.26514,0.67857,0.85714,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,13,0,6],[40,224,0.1786,0.76338,0.25408,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,3,0,0,6,0,0,1,0,0,9,0,11],[44,224,0.1964,0.6741,0.35397,0.28571,0.85714,1.0,0.0,1.0,2,11,0,2,0,4,0,0,3,0,0,1,0,0,2,0,0,1,0,0,8,0,11],[48,224,0.2143,0.6116,0.35756,0.24999,0.78564,0.85714,0.0,1.0,4,6,0,4,0,4,0,0,1,0,0,2,0,0,2,0,0,3,0,0,10,0,6],[52,224,0.2321,0.83929,0.21651,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,10],[56,224,0.25,0.73214,0.28738,0.67857,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,0,9,0,9],[60,224,0.2679,0.66961,0.31429,0.57132,0.85714,0.85714,0.0,1.0,3,4,0,3,0,3,0,0,0,0,0,0,0,0,5,0,0,3,0,0,14,0,4],[64,224,0.2857,0.65169,0.35894,0.46429,0.85707,0.89286,0.0,1.0,4,8,0,4,0,4,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,8],[68,224,0.3036,0.64285,0.35535,0.25,0.85714,0.85714,0.0,1.0,4,4,0,4,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,0,16,0,4],[72,224,0.3214,0.58025,0.36246,0.14286,0.71429,0.85714,0.0,1.0,5,4,0,5,0,5,0,0,0,0,0,0,0,0,3,0,0,5,0,0,10,0,4],[76,224,0.3393,0.50446,0.36243,0.10714,0.57143,0.85704,0.0,1.0,8,4,0,8,0,2,0,0,1,0,0,2,0,0,5,0,0,5,0,0,5,0,4],[80,224,0.3571,0.54018,0.41762,0.14286,0.78571,0.89286,0.0,1.0,7,8,0,7,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,0,8,0,8],[84,224,0.375,0.52678,0.41255,0.0,0.71429,0.85714,0.0,1.0,9,6,0,9,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,6],[88,224,0.3929,0.58035,0.37446,0.14286,0.85707,0.85714,0.0,1.0,5,3,1,5,0,6,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,3],[92,224,0.4107,0.41962,0.39598,0.0,0.14286,0.85714,0.0,1.0,9,4,0,9,0,8,0,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,4],[96,224,0.4286,0.5714,0.31135,0.39285,0.64286,0.85714,0.0,1.0,3,3,0,3,0,3,0,0,2,0,0,5,0,0,3,0,0,5,0,0,8,0,3],[100,224,0.4464,0.63393,0.34244,0.28571,0.78571,0.85714,0.0,1.0,3,6,0,3,0,4,0,0,2,0,0,0,0,0,3,0,0,4,0,0,10,0,6],[104,224,0.4643,0.46875,0.36111,0.14286,0.42859,0.85714,0.0,1.0,6,4,0,6,0,6,0,0,2,0,0,3,0,0,2,0,0,4,0,0,5,0,4],[108,224,0.4821,0.36152,0.36773,0.0,0.14286,0.71429,0.0,1.0,12,2,0,12,0,5,0,0,1,0,0,1,0,0,2,0,0,5,0,0,4,0,2],[112,224,0.5,0.56249,0.36584,0.10714,0.71429,0.85714,0.0,1.0,8,3,0,8,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,10,0,3],[116,224,0.5179,0.39732,0.40835,0.0,0.28573,0.85714,0.0,1.0,15,4,0,15,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,4],[120,224,0.5357,0.47322,0.36323,0.14286,0.57143,0.85714,0.0,1.0,6,3,0,6,0,7,0,0,1,0,0,1,0,0,4,0,0,3,0,0,7,0,3],[124,224,0.5536,0.4107,0.35129,0.10714,0.42857,0.71429,0.0,1.0,8,2,0,8,0,7,0,0,0,0,0,3,0,0,2,0,0,6,0,0,4,0,2],[128,224,0.5714,0.44634,0.35137,0.105,0.5,0.71429,0.0,1.0,8,3,0,8,0,4,0,0,1,0,0,3,0,0,5,0,0,4,0,0,4,0,3],[132,224,0.5893,0.41071,0.38755,0.0,0.42857,0.71429,0.0,1.0,11,6,0,11,0,4,0,0,0,0,0,3,0,0,4,0,0,3,0,0,1,0,6],[136,224,0.6071,0.41071,0.34395,0.14286,0.35714,0.71429,0.0,1.0,7,3,0,7,0,6,0,0,3,0,0,3,0,0,4,0,0,2,0,0,4,0,3],[140,224,0.625,0.44196,0.38025,0.14286,0.28571,0.85714,0.0,1.0,7,5,0,7,0,6,0,0,5,0,0,0,0,0,2,0,0,2,0,0,5,0,5],[144,224,0.6429,0.51786,0.3567,0.14286,0.57143,0.85714,0.0,1.0,5,4,0,5,0,6,0,0,0,0,0,3,0,0,4,0,0,3,0,0,7,0,4],[148,224,0.6607,0.37945,0.35284,0.0,0.28571,0.71429,0.0,1.0,10,2,0,10,0,5,0,0,2,0,0,2,0,0,3,0,0,4,0,0,4,0,2],[152,224,0.6786,0.30357,0.36202,0.0,0.14286,0.71429,0.0,1.0,12,2,0,12,0,9,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,2],[156,224,0.6964,0.37499,0.33071,0.0,0.42857,0.60714,0.0,1.0,10,1,0,10,0,4,0,0,1,0,0,4,0,0,5,0,0,3,0,0,4,0,1],[160,224,0.7143,0.44642,0.32683,0.14286,0.42857,0.74996,0.0,1.0,3,3,0,3,0,8,0,0,4,0,0,6,0,0,0,0,0,3,0,0,5,0,3],[164,224,0.7321,0.27679,0.333,0.0,0.14286,0.42857,0.0,1.0,10,3,0,10,0,12,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,3],[168,224,0.75,0.35258,0.35894,0.0,0.21429,0.71429,0.0,1.0,13,2,0,13,0,3,0,0,1,0,0,3,0,0,2,0,0,5,0,0,3,0,2],[172,224,0.7679,0.24106,0.27533,0.0,0.14286,0.4286,0.0,1.0,13,1,0,13,0,7,0,0,1,0,0,4,0,0,4,0,0,2,0,0,0,0,1],[176,224,0.7857,0.29463,0.34056,0.0,0.14286,0.60682,0.0,0.85714,14,0,0,14,0,5,0,0,1,0,0,3,0,0,1,0,0,2,0,0,6,0,0],[180,224,0.8036,0.37052,0.32899,0.0,0.42857,0.57143,0.0,1.0,10,2,0,10,0,4,0,0,1,0,0,4,0,0,6,0,0,3,0,0,2,0,2],[184,224,0.8214,0.0892,0.1324,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,9,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[188,224,0.8393,0.08482,0.17076,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[192,224,0.8571,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[196,224,0.875,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[200,224,0.8929,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[204,224,0.9107,0.06697,0.07974,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[208,224,0.9286,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[212,224,0.9464,0.04911,0.13175,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[216,224,0.9643,0.05357,0.1171,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[220,224,0.9821,0.0758,0.13351,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[224,224,1.0,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"978d3beb12429ea8","q":"A stromino is a $3 \\times 1$ rectangle. Show that a $5 \\times 5$ board divided into twenty-five $1 \\times 1$ squares\ncannot be covered by 16 strominos such that each stromino covers exactly three unit squares of the board and every unit square is covered by either one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)","t":[{"b":1,"e":0.57143,"k":"rising","v":0.27232,"x":0.65625,"p":[[0,114,0.0,0.27232,0.31615,0.0,0.14286,0.57143,0.0,1.0,14,2,12,14,0,5,0,0,1,0,0,2,0,0,7,0,0,0,0,0,1,0,2],[4,114,0.0351,0.65625,0.23107,0.57143,0.64286,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,3,0,0,10,0,0,7,0,0,4,0,5],[8,114,0.0702,0.65176,0.13805,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,5,0,0,4,0,2],[12,114,0.1053,0.65173,0.17107,0.57132,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,15,0,0,6,0,0,4,0,3],[16,114,0.1404,0.59819,0.19046,0.57143,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,0,0,0,19,0,0,7,0,0,1,0,2],[20,114,0.1754,0.58925,0.20439,0.571,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,4,0,0,19,0,0,2,0,0,2,0,3],[24,114,0.2105,0.56695,0.20666,0.57143,0.57143,0.60714,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,2,0,0,18,0,0,5,0,0,1,0,2],[28,114,0.2456,0.59371,0.11356,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,22,0,0,6,0,0,0,0,1],[32,114,0.2807,0.60713,0.12877,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,21,0,0,3,0,0,3,0,1],[36,114,0.3158,0.59821,0.1448,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,2,0,0,23,0,0,2,0,0,3,0,1],[40,114,0.3509,0.62054,0.13651,0.57143,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,19,0,0,7,0,0,4,0,0],[44,114,0.386,0.62944,0.14665,0.57143,0.57143,0.60714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,4,0,0,1,0,3],[48,114,0.4211,0.5625,0.16342,0.57143,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,2,0,0,22,0,0,3,0,0,1,0,1],[52,114,0.4561,0.55802,0.16115,0.57143,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,4,0,0,23,0,0,1,0,0,0,0,2],[56,114,0.4912,0.51786,0.15872,0.57143,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,0,0,0,2,0,0,26,0,0,1,0,0,0,0,0],[60,114,0.5263,0.58034,0.13803,0.57143,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,2,0,0,26,0,0,1,0,0,0,0,2],[64,114,0.5614,0.53567,0.15567,0.57132,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,2,0,0,26,0,0,0,0,0,0,0,1],[68,114,0.5965,0.51339,0.20782,0.53571,0.57143,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,2,0,0,2,0,0,19,0,0,4,0,0,0,0,1],[72,114,0.6316,0.54016,0.09932,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,3,0,0,26,0,0,1,0,0,0,0,0],[76,114,0.6667,0.57142,0.09448,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,24,0,0,1,0,0,2,0,0],[80,114,0.7018,0.53122,0.1931,0.571,0.57143,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,0,0,0,3,0,0,21,0,0,2,0,0,1,0,1],[84,114,0.7368,0.58479,0.05486,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0],[88,114,0.7719,0.56696,0.06667,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,3,0,0,0,0,0],[92,114,0.807,0.59819,0.10972,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,0,3,0,0,3,0,0],[96,114,0.8421,0.61161,0.11971,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,5,0,0,0,0,2],[100,114,0.8772,0.55804,0.17627,0.57143,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,3,0,0,20,0,0,4,0,0,1,0,1],[104,114,0.9123,0.58927,0.12753,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,23,0,0,4,0,0,1,0,1],[108,114,0.9474,0.58481,0.11495,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,19,0,0,6,0,0,0,0,1],[112,114,0.9825,0.61604,0.11539,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0,0,14,0,0,0,0,0],[114,114,1.0,0.62051,0.1048,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,12,0,0,1,0,0]]},{"b":3,"e":0.57143,"k":"rising","v":0.24107,"x":0.70089,"p":[[0,46,0.0,0.24107,0.24337,0.0,0.14286,0.46429,0.0,0.85714,10,0,9,10,0,10,0,0,2,0,0,2,0,0,7,0,0,0,0,0,1,0,0],[4,46,0.087,0.59372,0.16793,0.57143,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,2,0,0,20,0,0,4,0,0,3,0,1],[8,46,0.1739,0.65621,0.24965,0.57143,0.64286,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,1,0,0,0,0,0,12,0,0,6,0,0,5,0,5],[12,46,0.2609,0.70089,0.21829,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,1,0,0,9,0,0,10,0,0,4,0,6],[16,46,0.3478,0.64731,0.30511,0.57132,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,1,0,0,2,0,0,8,0,0,7,0,0,1,0,9],[20,46,0.4348,0.57588,0.15355,0.57132,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,5,0,0,17,0,0,6,0,0,1,0,1],[24,46,0.5217,0.60714,0.18898,0.57143,0.57143,0.60714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,2,0,0,20,0,0,3,0,0,2,0,3],[28,46,0.6087,0.61157,0.17583,0.57143,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,1,0,0,23,0,0,3,0,0,1,0,3],[32,46,0.6957,0.5714,0.21129,0.57143,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,4,0,0,0,0,0,1,0,0,21,0,0,2,0,0,1,0,3],[36,46,0.7826,0.62052,0.13176,0.57143,0.57143,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,5,0,0,1,0,2],[40,46,0.8696,0.4732,0.20024,0.2857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,7,0,0,2,0,0,1,0,0,19,0,0,2,0,0,1,0,0],[44,46,0.9565,0.57142,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[46,46,1.0,0.58481,0.04164,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0]]}]},{"i":"bc275ec18107c7d2","q":"A non-empty subset of  $\\{1,2, ..., n\\}$  is called *arabic* if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of   $\\{1,2, ..., n\\}$   has the same parity as  $n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.69197,"x":1.0,"p":[[0,11,0.0,0.85268,0.34531,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,11,0.3636,0.69197,0.43757,0.21429,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,21],[8,11,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[11,11,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.42857,"k":"falling","v":0.29018,"x":0.77679,"p":[[0,54,0.0,0.74554,0.43114,0.64286,1.0,1.0,0.0,1.0,8,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[4,54,0.0741,0.77679,0.37105,0.42859,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,23],[8,54,0.1481,0.65625,0.44011,0.10714,1.0,1.0,0.0,1.0,8,19,0,8,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,19],[12,54,0.2222,0.73661,0.42873,0.53571,1.0,1.0,0.0,1.0,8,22,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,22],[16,54,0.2963,0.63839,0.46564,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,18],[20,54,0.3704,0.61161,0.46871,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,18],[24,54,0.4444,0.37054,0.43574,0.0,0.07143,0.89286,0.0,1.0,16,8,0,16,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,8],[28,54,0.5185,0.34822,0.44883,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,10],[32,54,0.5926,0.54911,0.42724,0.10714,0.42857,1.0,0.0,1.0,8,13,0,8,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,0,2,0,13],[36,54,0.6667,0.5,0.43301,0.0,0.42857,1.0,0.0,1.0,11,12,0,11,0,0,0,0,2,0,0,6,0,0,0,0,0,0,0,0,1,0,12],[40,54,0.7407,0.55357,0.4222,0.10714,0.42857,1.0,0.0,1.0,8,14,0,8,0,2,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,14],[44,54,0.8148,0.5625,0.38785,0.25001,0.42857,1.0,0.0,1.0,4,13,0,4,0,4,0,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,13],[48,54,0.8889,0.50447,0.41185,0.10714,0.42857,1.0,0.0,1.0,8,12,0,8,0,2,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,12],[52,54,0.963,0.29018,0.22442,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,5,0,0,6,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[54,54,1.0,0.29465,0.21997,0.10714,0.42857,0.42857,0.0,1.0,8,1,0,8,0,3,0,0,4,0,0,16,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"47338e942f237aea","q":"Acute triangle  $ABC$  is inscribed in circle  $O$ .  $P$  is the foot of altitude from  $A$  to  $BC$ , and  $D$  is the intersection of  $O$  and line  $AP$ .  $M, N$  are midpoint of  $AB, AC$  respectively.  $MP$  and  $CD$  intersects at  $Q$ , and  $NP$  and  $BD$  intersects at  $R$ . Show that  $AD, BQ, CR$  meet at one point if and only if  $AB=AC$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.04464,"x":0.25438,"p":[[0,132,0.0,0.17402,0.2105,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,15,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0],[4,132,0.0303,0.25438,0.22798,0.14214,0.14286,0.57143,0.0,0.57143,7,0,0,7,0,14,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,0],[8,132,0.0606,0.16518,0.23176,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,14,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[12,132,0.0909,0.17857,0.22588,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,15,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[16,132,0.1212,0.12947,0.16888,0.0,0.14286,0.14287,0.0,0.57143,15,0,0,15,0,11,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[20,132,0.1515,0.14732,0.21275,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,15,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[24,132,0.1818,0.11161,0.12745,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,132,0.2121,0.13393,0.15542,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,18,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[32,132,0.2424,0.12499,0.15868,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,16,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[36,132,0.2727,0.11607,0.14913,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,15,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,132,0.303,0.14723,0.1838,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,14,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[44,132,0.3333,0.14286,0.17496,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[48,132,0.3636,0.17411,0.28061,0.0,0.0,0.14287,0.0,1.0,17,2,0,17,0,9,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2],[52,132,0.3939,0.22322,0.24984,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,13,0,0,0,0,0,2,0,0,6,0,0,0,0,0,0,0,1],[56,132,0.4242,0.10705,0.1383,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[60,132,0.4545,0.11607,0.14914,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,15,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[64,132,0.4848,0.17411,0.23619,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,13,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,1],[68,132,0.5152,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,132,0.5455,0.08482,0.11214,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,132,0.5758,0.09357,0.11063,0.0,0.14143,0.14286,0.0,0.57143,14,0,0,14,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,132,0.6061,0.12947,0.15714,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,17,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[84,132,0.6364,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,132,0.6667,0.10268,0.1394,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,15,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[92,132,0.697,0.16955,0.25365,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,16,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[96,132,0.7273,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,132,0.7576,0.09375,0.1411,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,13,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[104,132,0.7879,0.09375,0.1411,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,13,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[108,132,0.8182,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,132,0.8485,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[116,132,0.8788,0.08482,0.11214,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[120,132,0.9091,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,132,0.9394,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,132,0.9697,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,132,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.13839,"x":0.625,"p":[[0,44,0.0,0.13839,0.16554,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,13,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,44,0.0909,0.20534,0.26227,0.0,0.14286,0.35702,0.0,1.0,14,1,0,14,0,9,0,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,1],[8,44,0.1818,0.43749,0.28557,0.14286,0.57143,0.57143,0.0,1.0,4,3,0,4,0,7,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,3],[12,44,0.2727,0.24999,0.26243,0.0,0.14286,0.57143,0.0,1.0,9,1,0,9,0,13,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1],[16,44,0.3636,0.37054,0.31715,0.14286,0.21431,0.57143,0.0,1.0,5,4,0,5,0,11,0,0,1,0,0,2,0,0,9,0,0,0,0,0,0,0,4],[20,44,0.4545,0.33036,0.30186,0.0,0.14286,0.57143,0.0,1.0,9,2,0,9,0,8,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,2],[24,44,0.5455,0.35265,0.30299,0.14286,0.14288,0.57143,0.0,1.0,5,3,0,5,0,12,0,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,3],[28,44,0.6364,0.61606,0.14914,0.57143,0.57143,0.57143,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,0,0,0,0,0,4],[32,44,0.7273,0.5357,0.19561,0.57143,0.57143,0.57143,0.0,1.0,2,2,0,2,0,0,0,0,3,0,0,0,0,0,25,0,0,0,0,0,0,0,2],[36,44,0.8182,0.625,0.18123,0.57143,0.57143,0.57143,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,24,0,0,1,0,0,0,0,5],[40,44,0.9091,0.62497,0.14175,0.57143,0.57143,0.57143,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,4],[44,44,1.0,0.58477,0.17262,0.57143,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,0,0,3]]}]},{"i":"fd02910f9debbe71","q":"A line parallel to the side  $BC$  of a triangle  $ABC$  meets the sides  $AB$  and  $AC$  at points  $P$  and  $Q$ , respectively. A point  $M$  is chosen inside the triangle  $APQ$ . The segments  $MB$  and  $MC$  meet the segment  $PQ$  at points  $E$  and  $F$ , respectively. Let  $N$  be the second intersection point of the circumcircles of the triangles  $PMF$  and  $QME$ . Prove that the points  $A,M,N$  are collinear.","t":[{"b":1,"e":0.0,"k":"flat","v":0.07589,"x":0.22319,"p":[[0,70,0.0,0.08928,0.15465,0.0,0.0,0.17857,0.0,0.5714,23,0,0,23,0,1,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,70,0.0571,0.18301,0.18287,0.0,0.2857,0.28571,0.0,0.57143,14,0,0,14,0,1,0,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,70,0.1143,0.15625,0.15714,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,70,0.1714,0.14732,0.16554,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,0,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,70,0.2286,0.1383,0.15764,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,1,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,70,0.2857,0.19196,0.20394,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,10,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[24,70,0.3429,0.16518,0.17169,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,70,0.4,0.19196,0.17717,0.0,0.28571,0.28571,0.0,0.57143,13,0,0,13,0,1,0,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[32,70,0.4571,0.22319,0.18531,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,3,0,0,14,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[36,70,0.5143,0.16071,0.16269,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,70,0.5714,0.13393,0.17105,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,0,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,70,0.6286,0.16516,0.17533,0.0,0.14286,0.28571,0.0,0.571,15,0,0,15,0,3,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[48,70,0.6857,0.1473,0.20034,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,1,0,0,8,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[52,70,0.7429,0.1607,0.1812,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,2,0,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[56,70,0.8,0.16072,0.1948,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,3,0,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[60,70,0.8571,0.17857,0.18558,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,3,0,0,11,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[64,70,0.9143,0.13837,0.19387,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,3,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[68,70,0.9714,0.08482,0.16698,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[70,70,1.0,0.07589,0.19556,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.24552,"p":[[0,87,0.0,0.08036,0.12846,0.0,0.0,0.14287,0.0,0.4286,22,0,0,22,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,87,0.046,0.20536,0.17835,0.0,0.28571,0.28571,0.0,0.57143,12,0,0,12,0,0,0,0,17,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,87,0.092,0.19196,0.14987,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,1,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,87,0.1379,0.17857,0.18558,0.0,0.2857,0.28571,0.0,0.57143,15,0,0,15,0,0,0,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,87,0.1839,0.2008,0.1781,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,1,0,0,16,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[20,87,0.2299,0.16518,0.13882,0.0,0.2857,0.28571,0.0,0.28571,13,0,0,13,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,87,0.2759,0.24105,0.20338,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,0,0,0,15,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[28,87,0.3218,0.2232,0.17102,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,0,0,0,19,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[32,87,0.3678,0.20535,0.17835,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,0,0,0,17,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[36,87,0.4138,0.20534,0.17471,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,2,0,0,16,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[40,87,0.4598,0.24097,0.14035,0.24999,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,2,0,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,87,0.5057,0.24552,0.19958,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,1,0,0,15,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[48,87,0.5517,0.23214,0.18471,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,1,0,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[52,87,0.5977,0.22768,0.15916,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,0,0,0,20,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[56,87,0.6437,0.19197,0.17717,0.0,0.2857,0.28571,0.0,0.71429,12,0,0,12,0,2,0,0,16,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[60,87,0.6897,0.19643,0.12752,0.0,0.2857,0.28571,0.0,0.28571,9,0,0,9,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,87,0.7356,0.20536,0.18877,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,2,0,0,14,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[68,87,0.7816,0.13384,0.16341,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,1,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[72,87,0.8276,0.15624,0.17982,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,0,0,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[76,87,0.8736,0.0758,0.12359,0.0,0.0,0.17642,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,87,0.9195,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,87,0.9655,0.08482,0.16311,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,1,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[87,87,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42f4e4126a4c3611","q":"A point  $T$  is given on the altitude through point  $C$  in the acute triangle  $ABC$  with circumcenter  $O$ , such that  $\\measuredangle TBA=\\measuredangle ACB$ . If the line  $CO$  intersects side  $AB$  at point  $K$ , prove that the perpendicular bisector of  $AB$ , the altitude through  $A$  and the segment  $KT$  are concurrent.","t":[{"b":1,"e":0.1429,"k":"falling","v":0.23206,"x":0.98214,"p":[[0,61,0.0,0.94193,0.12307,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[4,61,0.0656,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,61,0.1311,0.92856,0.15975,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[12,61,0.1967,0.95533,0.11547,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[16,61,0.2623,0.92856,0.16754,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[20,61,0.3279,0.91964,0.19212,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,25],[24,61,0.3934,0.87944,0.22622,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,23],[28,61,0.459,0.90179,0.20958,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,24],[32,61,0.5246,0.91071,0.18472,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,25],[36,61,0.5902,0.88838,0.2105,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,23],[40,61,0.6557,0.87946,0.20858,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,22],[44,61,0.7213,0.81696,0.27254,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,5,0,0,2,0,19],[48,61,0.7869,0.56697,0.32825,0.28571,0.5005,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,7,0,0,4,0,0,4,0,0,2,0,0,0,0,10],[52,61,0.8525,0.30785,0.27234,0.14286,0.2857,0.28571,0.0,1.0,2,3,0,2,0,13,0,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[56,61,0.918,0.36604,0.24204,0.24999,0.28571,0.42858,0.14286,1.0,0,3,0,0,0,8,0,0,14,0,0,3,0,0,4,0,0,0,0,0,0,0,3],[60,61,0.9836,0.27223,0.20633,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,13,0,0,11,0,0,3,0,0,0,0,0,2,0,0,0,0,1],[61,61,1.0,0.23206,0.12759,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,11,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.2499,"x":0.99554,"p":[[0,110,0.0,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,110,0.0364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,110,0.0727,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[12,110,0.1091,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[16,110,0.1455,0.91964,0.19212,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,26],[20,110,0.1818,0.93749,0.12344,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[24,110,0.2182,0.83034,0.25863,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,2,0,19],[28,110,0.2545,0.82141,0.25507,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,0,2,0,19],[32,110,0.2909,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[36,110,0.3273,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,25],[40,110,0.3636,0.87051,0.2032,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,0,3,0,20],[44,110,0.4,0.87051,0.22972,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,21],[48,110,0.4364,0.89282,0.17136,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,2,0,22],[52,110,0.4727,0.89732,0.16065,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,21],[56,110,0.5091,0.80802,0.2826,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,18],[60,110,0.5455,0.87946,0.22619,0.82132,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,0,1,0,23],[64,110,0.5818,0.87052,0.19353,0.85711,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,7,0,18],[68,110,0.6182,0.84822,0.2257,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,19],[72,110,0.6545,0.80801,0.28035,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,0,0,0,4,0,0,2,0,0,4,0,18],[76,110,0.6909,0.75442,0.29287,0.571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,4,0,0,0,0,0,5,0,0,2,0,0,4,0,15],[80,110,0.7273,0.70523,0.32346,0.42857,0.78571,1.0,0.14,1.0,0,15,0,0,0,4,0,0,3,0,0,2,0,0,4,0,0,3,0,0,1,0,15],[84,110,0.7636,0.76785,0.3067,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,5,0,0,2,0,17],[88,110,0.8,0.73658,0.29039,0.57143,0.78564,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,1,0,0,0,0,0,5,0,0,6,0,0,3,0,13],[92,110,0.8364,0.61596,0.35266,0.14289,0.71429,1.0,0.14,1.0,0,12,0,0,0,9,0,0,0,0,0,4,0,0,2,0,0,5,0,0,0,0,12],[96,110,0.8727,0.60712,0.3607,0.2857,0.71429,1.0,0.0,1.0,3,10,0,3,0,4,0,0,4,0,0,1,0,0,1,0,0,7,0,0,2,0,10],[100,110,0.9091,0.65177,0.33108,0.39286,0.71429,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,1,0,0,2,0,0,3,0,0,6,0,0,2,0,11],[104,110,0.9455,0.58034,0.34615,0.2857,0.4998,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,7,0,0,3,0,0,2,0,0,1,0,0,3,0,10],[108,110,0.9818,0.40179,0.33396,0.14286,0.2143,0.71429,0.0,1.0,1,5,0,1,0,15,0,0,4,0,0,1,0,0,2,0,0,3,0,0,1,0,5],[110,110,1.0,0.2499,0.25507,0.14286,0.14286,0.2857,0.0,1.0,3,2,0,3,0,20,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,2]]}]},{"i":"b33b34bce67805cc","q":"A positive integer is written on a blackboard. Players  $A$  and  $B$  play the following game: in each move one has to choose a proper divisor  $m$  of the number  $n$  written on the blackboard ( $1<m<n$ ) and replaces  $n$  with  $n-m$ . Player  $A$  makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player  $B$ ?","t":[{"b":4,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,71,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,71,0.0563,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,71,0.1127,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,71,0.169,0.97322,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,71,0.2254,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,71,0.2817,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,71,0.338,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,71,0.3944,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[32,71,0.4507,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,71,0.507,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,71,0.5634,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,71,0.6197,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,71,0.6761,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[52,71,0.7324,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[56,71,0.7887,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,71,0.8451,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[64,71,0.9014,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[68,71,0.9577,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[71,71,1.0,0.96428,0.15153,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.96428,"x":1.0,"p":[[0,24,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,24,0.1667,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,24,0.3333,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,24,0.5,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,24,0.6667,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,24,0.8333,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,24,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"a93bf7a7220e8c61","q":"A rectangular piece of paper has top edge  $AD$ . A line  $L$  from  $A$  to the bottom edge makes an angle  $x$  with the line  $AD$ . We want to trisect  $x$ . We take  $B$  and  $C$  on the vertical ege through  $A$  such that  $AB = BC$ . We then fold the paper so that  $C$  goes to a point  $C'$  on the line  $L$  and  $A$  goes to a point  $A'$  on the horizontal line through  $B$ . The fold takes  $B$  to  $B'$ . Show that  $AA'$  and  $AB'$  are the required trisectors.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.34143,"x":0.52683,"p":[[0,67,0.0,0.34143,0.2089,0.14289,0.28586,0.42857,0.0,0.71429,3,0,0,3,0,6,0,1,7,0,0,10,0,0,0,0,0,5,0,0,0,0,0],[4,67,0.0597,0.40176,0.13567,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,10,0,0,16,0,0,3,0,0,2,0,0,0,0,0],[8,67,0.1194,0.48214,0.17035,0.39286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,12,0,0,3,0,0,9,0,0,0,0,0],[12,67,0.1791,0.45973,0.16667,0.39286,0.42857,0.57143,0.14,0.71429,0,0,0,0,0,2,0,0,6,0,0,14,0,0,3,0,0,7,0,0,0,0,0],[16,67,0.2388,0.40848,0.13271,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,7,0,1,18,0,0,1,0,0,3,0,0,0,0,0],[20,67,0.2985,0.5,0.17128,0.39286,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,11,0,0,2,0,0,11,0,0,0,0,0],[24,67,0.3582,0.47991,0.17699,0.42857,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,16,0,1,0,0,0,8,0,0,1,0,0],[28,67,0.4179,0.48661,0.17445,0.39286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,12,0,0,2,0,0,10,0,0,0,0,0],[32,67,0.4776,0.47322,0.16146,0.42857,0.42857,0.60714,0.1429,0.71429,0,0,0,0,0,1,0,0,6,0,0,15,0,0,2,0,0,8,0,0,0,0,0],[36,67,0.5373,0.41518,0.16506,0.28571,0.42857,0.42858,0.0,0.71429,1,0,0,1,0,2,0,0,6,0,0,18,0,0,0,0,0,5,0,0,0,0,0],[40,67,0.597,0.47991,0.15177,0.42857,0.42857,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,1,15,0,0,2,0,0,8,0,0,0,0,0],[44,67,0.6567,0.42857,0.17128,0.28571,0.42857,0.46429,0.0,0.71429,1,0,0,1,0,0,0,0,11,0,0,12,0,0,2,0,0,6,0,0,0,0,0],[48,67,0.7164,0.40397,0.12971,0.28571,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,2,0,0,7,0,1,19,0,0,0,0,0,3,0,0,0,0,0],[52,67,0.7761,0.40179,0.19045,0.28571,0.42857,0.42858,0.0,0.71429,3,0,0,3,0,0,0,0,9,0,0,13,0,0,2,0,0,5,0,0,0,0,0],[56,67,0.8358,0.45533,0.13568,0.42857,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,20,0,0,4,0,0,4,0,0,0,0,0],[60,67,0.8955,0.52683,0.20649,0.42857,0.4293,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,3,0,0,12,0,0,2,0,0,12,0,0,0,0,1],[64,67,0.9552,0.45311,0.1219,0.42857,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,1,19,0,0,5,0,0,3,0,0,0,0,0],[67,67,1.0,0.35937,0.08654,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,11,0,1,18,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.33705,"x":0.48215,"p":[[0,61,0.0,0.35035,0.22123,0.26777,0.42857,0.42857,0.0,1.0,5,1,0,5,0,2,0,1,6,0,0,14,0,0,1,0,0,2,0,0,0,0,1],[4,61,0.0656,0.43303,0.14054,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,16,0,0,1,0,0,5,0,0,0,0,0],[8,61,0.1311,0.40178,0.13571,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,14,0,0,0,0,0,4,0,0,0,0,0],[12,61,0.1967,0.42409,0.12618,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,16,0,0,3,0,0,3,0,0,0,0,0],[16,61,0.2623,0.41964,0.15947,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,10,0,0,1,0,0,6,0,0,0,0,0],[20,61,0.3279,0.43304,0.13115,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,17,0,0,2,0,0,4,0,0,0,0,0],[24,61,0.3934,0.41964,0.11812,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,16,0,0,4,0,0,2,0,0,0,0,0],[28,61,0.459,0.42857,0.14726,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,16,0,0,1,0,0,5,0,0,0,0,0],[32,61,0.5246,0.33705,0.09819,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,15,0,1,14,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.42857,0.13363,0.42857,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,21,0,0,2,0,0,3,0,0,0,0,0],[40,61,0.6557,0.40178,0.12469,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,2,14,0,0,1,0,0,3,0,0,0,0,0],[44,61,0.7213,0.46429,0.15152,0.39286,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,15,0,0,2,0,0,7,0,0,0,0,0],[48,61,0.7869,0.44643,0.12753,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,21,0,0,0,0,0,5,0,0,0,0,0],[52,61,0.8525,0.47098,0.16255,0.42857,0.42857,0.50002,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,1,19,0,0,0,0,0,8,0,0,0,0,0],[56,61,0.918,0.45982,0.11701,0.42857,0.42857,0.42858,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,24,0,0,0,0,0,5,0,0,0,0,0],[60,61,0.9836,0.48215,0.15872,0.42857,0.42857,0.60714,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,20,0,0,1,0,0,8,0,0,0,0,0],[61,61,1.0,0.42411,0.14054,0.39286,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,19,0,0,1,0,0,4,0,0,0,0,0]]}]},{"i":"8586230dcbd8ab51","q":"An  $m \\times n$  chessboard where  $m \\le n$  has several black squares such that no two rows have the same pattern. Determine the largest integer  $k$  such that we can always color  $k$  columns red while still no two rows have the same pattern.","t":[{"b":1,"e":0.2857,"k":"rising","v":0.30357,"x":0.6875,"p":[[0,19,0.0,0.30357,0.18472,0.2857,0.28571,0.28571,0.0,1.0,3,1,3,3,0,0,0,0,26,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[4,19,0.2105,0.60267,0.28288,0.42857,0.42859,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,12,0,0,3,0,0,1,0,0,0,0,10],[8,19,0.4211,0.58022,0.28774,0.28571,0.42857,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,9,0,0,10,0,0,1,0,0,3,0,0,0,0,9],[12,19,0.6316,0.66964,0.30813,0.28571,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,10,0,0,2,0,0,3,0,0,2,0,0,3,0,12],[16,19,0.8421,0.6875,0.3223,0.28571,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,11,0,0,0,0,0,4,0,0,1,0,0,1,0,15],[19,19,1.0,0.61607,0.31831,0.28571,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,13,0,0,1,0,0,5,0,0,1,0,0,0,0,12]]},{"b":5,"e":0.42857,"k":"rising","v":0.33482,"x":0.5848,"p":[[0,10,0.0,0.33482,0.12169,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,22,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[4,10,0.4,0.5848,0.26332,0.42857,0.57121,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,6,0,0,7,0,0,8,0,0,2,0,0,1,0,7],[8,10,0.8,0.57139,0.29015,0.28571,0.571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,12,0,0,3,0,0,8,0,0,0,0,0,0,0,9],[10,10,1.0,0.53122,0.28174,0.28571,0.42859,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,15,0,0,2,0,0,6,0,0,2,0,0,0,0,7]]}]},{"i":"e1aadbfffc040c2e","q":"A positive integer  $n$  is called *egyptian* if there exists a strictly increasing sequence  $0<a_1<a_2<\\dots<a_k=n$  of integers with last term  $n$  such that\n\\[\\frac{1}{a_1}+\\frac{1}{a_2}+\\dots+\\frac{1}{a_k}=1.\\]\n(a) Determine if  $n=72$  is egyptian.\n(b) Determine if  $n=71$  is egyptian.\n(c) Determine if  $n=72^{71}$  is egyptian.","t":[{"b":2,"e":1.0,"k":"rising","v":0.42857,"x":0.96429,"p":[[0,132,0.0,0.42857,0.13363,0.42857,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,23,0,0,4,0,0,0,0,0,1,0,0],[4,132,0.0303,0.75893,0.23538,0.71429,0.85714,0.85714,0.0,1.0,1,6,1,1,0,1,0,0,1,0,0,0,0,0,3,0,0,7,0,0,13,0,6],[8,132,0.0606,0.85714,0.22304,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,11,0,16],[12,132,0.0909,0.82588,0.20743,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,0,9,0,13],[16,132,0.1212,0.77677,0.19866,0.67857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,9,0,9],[20,132,0.1515,0.72321,0.23941,0.4286,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,8,0,0,2,0,0,5,0,0,8,0,8],[24,132,0.1818,0.82589,0.1504,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,14,0,8],[28,132,0.2121,0.78569,0.19886,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,4,0,0,14,0,7],[32,132,0.2424,0.83928,0.13243,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,10,0,10],[36,132,0.2727,0.81249,0.16919,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,14,0,8],[40,132,0.303,0.82589,0.18808,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,12,0,11],[44,132,0.3333,0.75892,0.20958,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,0,10,0,8],[48,132,0.3636,0.80803,0.19434,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,7,0,12],[52,132,0.3939,0.83928,0.16268,0.85711,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,17,0,9],[56,132,0.4242,0.8125,0.21261,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,0,7,0,13],[60,132,0.4545,0.83929,0.1948,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,14],[64,132,0.4848,0.85714,0.14725,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,10,0,13],[68,132,0.5152,0.83034,0.17657,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,10,0,12],[72,132,0.5455,0.81247,0.20344,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,6,0,14],[76,132,0.5758,0.82143,0.17128,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,9,0,11],[80,132,0.6061,0.76784,0.21652,0.57143,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,5,0,0,4,0,12],[84,132,0.6364,0.78124,0.21721,0.67857,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,6,0,0,7,0,11],[88,132,0.6667,0.87496,0.1418,0.82132,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,9,0,15],[92,132,0.697,0.84822,0.18536,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,3,0,17],[96,132,0.7273,0.81696,0.22084,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,4,0,16],[100,132,0.7576,0.73654,0.26517,0.571,0.78571,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,4,0,0,8,0,0,2,0,0,3,0,13],[104,132,0.7879,0.80803,0.22477,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,0,7,0,14],[108,132,0.8182,0.82142,0.17857,0.71429,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,6,0,13],[112,132,0.8485,0.83928,0.1948,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,3,0,17],[116,132,0.8788,0.8125,0.19045,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,0,9,0,11],[120,132,0.9091,0.83482,0.19597,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,5,0,16],[124,132,0.9394,0.85267,0.17672,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,3,0,17],[128,132,0.9697,0.89285,0.17496,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,6,0,20],[132,132,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]},{"b":4,"e":1.0,"k":"rising","v":0.48214,"x":0.99554,"p":[[0,158,0.0,0.48214,0.17405,0.42857,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,25,0,0,2,0,0,0,0,0,0,0,3],[4,158,0.0253,0.83927,0.21356,0.71429,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,9,0,14],[8,158,0.0506,0.73214,0.30671,0.57143,0.85714,1.0,0.0,1.0,3,10,1,3,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,10,0,10],[12,158,0.0759,0.71875,0.28004,0.57143,0.85714,0.89286,0.0,1.0,1,8,1,1,0,1,0,0,4,0,0,0,0,0,4,0,0,4,0,0,10,0,8],[16,158,0.1013,0.60713,0.29881,0.42857,0.71429,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,3,0,0,3,0,0,5,0,0,6,0,0,7,0,4],[20,158,0.1266,0.58482,0.34323,0.42857,0.57143,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,0,0,0,6,0,0,4,0,0,4,0,0,3,0,8],[24,158,0.1519,0.74112,0.24863,0.67857,0.78571,0.895,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,8,0,0,8,0,8],[28,158,0.1772,0.76339,0.29366,0.53571,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,3,0,0,1,0,0,3,0,0,5,0,15],[32,158,0.2025,0.62045,0.35656,0.28571,0.71429,1.0,0.0,1.0,2,10,0,2,0,5,0,0,2,0,0,5,0,0,0,0,0,3,0,0,5,0,10],[36,158,0.2278,0.60268,0.33452,0.28571,0.57143,1.0,0.0,1.0,2,9,0,2,0,3,0,0,4,0,0,4,0,0,5,0,0,1,0,0,4,0,9],[40,158,0.2532,0.70535,0.33108,0.42857,0.85712,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,3,0,0,2,0,0,4,0,0,4,0,13],[44,158,0.2785,0.74999,0.27895,0.5354,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,3,0,0,2,0,0,3,0,0,6,0,13],[48,158,0.3038,0.70535,0.29437,0.42859,0.78564,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,4,0,0,3,0,0,4,0,0,5,0,11],[52,158,0.3291,0.68301,0.25689,0.4286,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,7,0,0,5,0,0,7,0,0,2,0,9],[56,158,0.3544,0.62498,0.33645,0.49968,0.71429,0.85714,0.0,1.0,5,5,0,5,0,1,0,0,2,0,0,0,0,0,3,0,0,8,0,0,8,0,5],[60,158,0.3797,0.7009,0.29956,0.53572,0.78564,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,0,6,0,10],[64,158,0.4051,0.75446,0.27947,0.57143,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,4,0,13],[68,158,0.4304,0.68303,0.26422,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,8,0,0,4,0,0,7,0,0,1,0,10],[72,158,0.4557,0.82589,0.20743,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,9,0,14],[76,158,0.481,0.67854,0.30723,0.42857,0.71429,1.0,0.0,1.0,2,11,0,2,0,1,0,0,0,0,0,8,0,0,4,0,0,2,0,0,4,0,11],[80,158,0.5063,0.64284,0.3093,0.42857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,5,0,0,3,0,0,4,0,0,7,0,7],[84,158,0.5316,0.69642,0.28515,0.42859,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,1,0,0,6,0,0,3,0,0,5,0,0,6,0,9],[88,158,0.557,0.80802,0.25156,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,7,0,15],[92,158,0.5823,0.79464,0.23941,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,4,0,14],[96,158,0.6076,0.79018,0.2525,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,5,0,14],[100,158,0.6329,0.75446,0.24545,0.57143,0.85707,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,2,0,0,6,0,12],[104,158,0.6582,0.79463,0.23675,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,3,0,14],[108,158,0.6835,0.7723,0.22264,0.57142,0.78564,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,6,0,0,3,0,13],[112,158,0.7089,0.84375,0.21535,0.67857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,3,0,19],[116,158,0.7342,0.79018,0.3214,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,2,0,19],[120,158,0.7595,0.87053,0.21829,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,21],[124,158,0.7848,0.85714,0.19233,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,6,0,17],[128,158,0.8101,0.87052,0.2297,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,21],[132,158,0.8354,0.86161,0.20973,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,3,0,20],[136,158,0.8608,0.87946,0.26513,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,24],[140,158,0.8861,0.76786,0.28291,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,3,0,0,4,0,0,3,0,0,4,0,15],[144,158,0.9114,0.81695,0.25061,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,1,0,19],[148,158,0.9367,0.74105,0.25615,0.57143,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,0,2,0,13],[152,158,0.962,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[156,158,0.9873,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[158,158,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"6e8306f97935f969","q":"A sequence  $(a_n)$  of positive integers is defined by  $a_0=m$  and  $a_{n+1}= a_n^5 +487$  for all  $n\\ge 0$ . \nFind all positive integers  $m$  such that the sequence contains the maximum possible number of perfect squares.","t":[{"b":1,"e":0.57143,"k":"flat","v":0.47765,"x":0.66963,"p":[[0,64,0.0,0.66963,0.18706,0.57143,0.57143,0.85704,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,1,0,0,5,0,5],[4,64,0.0625,0.54463,0.16917,0.42857,0.57121,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,14,0,0,13,0,0,1,0,0,0,0,3],[8,64,0.125,0.56249,0.18877,0.42857,0.49979,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,11,0,0,0,0,0,1,0,4],[12,64,0.1875,0.51786,0.17035,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,16,0,0,10,0,0,0,0,0,3,0,1],[16,64,0.25,0.47765,0.09849,0.42857,0.42857,0.57111,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,21,0,0,9,0,0,0,0,0,1,0,0],[20,64,0.3125,0.5357,0.16366,0.42857,0.42859,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,16,0,0,10,0,0,2,0,0,1,0,2],[24,64,0.375,0.48661,0.08645,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,9,0,0,2,0,0,0,0,0],[28,64,0.4375,0.5,0.08748,0.42857,0.5,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,15,0,0,15,0,0,1,0,0,0,0,0],[32,64,0.5,0.5089,0.12845,0.42857,0.571,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,14,0,0,16,0,0,0,0,0,0,0,1],[36,64,0.5625,0.54463,0.09061,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,21,0,0,1,0,0,1,0,0],[40,64,0.625,0.55804,0.14445,0.42857,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,0,0,0,1,0,2],[44,64,0.6875,0.50893,0.08702,0.42857,0.5,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,14,0,0,2,0,0,0,0,0],[48,64,0.75,0.49109,0.10672,0.42857,0.57121,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,11,0,0,18,0,0,0,0,0,0,0,0],[52,64,0.8125,0.52231,0.13175,0.42857,0.5712,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,14,0,0,15,0,0,0,0,0,1,0,1],[56,64,0.875,0.51336,0.10009,0.42857,0.57121,0.57143,0.28571,0.857,0,0,0,0,0,0,0,0,1,0,0,13,0,0,17,0,0,0,0,0,1,0,0],[60,64,0.9375,0.54018,0.13236,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,14,0,0,2,0,0,1,0,1],[64,64,1.0,0.57585,0.06667,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0,1,0,0]]},{"b":4,"e":0.57143,"k":"flat","v":0.47322,"x":0.68744,"p":[[0,40,0.0,0.68744,0.20962,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,1,0,0,1,0,9],[4,40,0.1,0.5089,0.1181,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,14,0,0,15,0,0,1,0,0,1,0,0],[8,40,0.2,0.5625,0.16727,0.42857,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,13,0,0,15,0,0,0,0,0,1,0,3],[12,40,0.3,0.49105,0.09404,0.42857,0.42857,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,0,0,0,0,1,0,0],[16,40,0.4,0.47322,0.11538,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,19,0,0,10,0,0,0,0,0,1,0,0],[20,40,0.5,0.52676,0.0662,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0],[24,40,0.6,0.56247,0.14698,0.53539,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,7,0,0,20,0,0,1,0,0,2,0,1],[28,40,0.7,0.54018,0.09268,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,20,0,0,1,0,0,1,0,0],[32,40,0.8,0.54907,0.05185,0.57132,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[36,40,0.9,0.58482,0.11495,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,0,0,2],[40,40,1.0,0.54909,0.05186,0.57142,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0]]}]},{"i":"f6d1db2ecff4850c","q":"A sequence $\\left(a_{n}\\right)$ is defined as follows: $a_{1}=\\sqrt{2}, a_{2}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \\geq 2$. Prove that for every $n \\geq 1$ we have\n\n$$\n\\left(1+a_{1}\\right)\\left(1+a_{2}\\right) \\cdots\\left(1+a_{n}\\right)<(2+\\sqrt{2}) a_{1} a_{2} \\cdots a_{n} .\n$$","t":[{"b":1,"e":0.2857,"k":"flat","v":0.2634,"x":0.37054,"p":[[0,36,0.0,0.37054,0.22548,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,2,0,0,24,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[4,36,0.1111,0.35268,0.18893,0.28571,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,1,0,0,25,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[8,36,0.2222,0.31696,0.17399,0.28571,0.28571,0.28571,0.0,1.0,2,1,0,2,0,1,0,0,24,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[12,36,0.3333,0.33482,0.17717,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,2,0,0,23,0,0,1,0,0,3,0,0,1,0,0,0,0,1],[16,36,0.4444,0.2634,0.08073,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,2,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.32143,0.12877,0.28571,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,1,0,0,27,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[28,36,0.7778,0.31696,0.09268,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[32,36,0.8889,0.29464,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.32589,0.10249,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,27,0,0,2,0,0,2,0,0,1,0,0,0,0,0]]},{"b":2,"e":0.71429,"k":"flat","v":0.20089,"x":0.36607,"p":[[0,39,0.0,0.36607,0.20183,0.28571,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,1,0,0,24,0,0,3,0,0,1,0,0,0,0,0,1,0,2],[4,39,0.1026,0.33482,0.20079,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,3,0,0,22,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[8,39,0.2051,0.24106,0.1037,0.14286,0.2857,0.28571,0.0,0.571,1,0,0,1,0,11,0,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,39,0.3077,0.22321,0.08702,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,10,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.24107,0.11538,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,10,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,39,0.5128,0.23661,0.09852,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.28571,0.12877,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,9,0,0,18,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[28,39,0.7179,0.20089,0.09354,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,16,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.23214,0.11152,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,11,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,39,0.9231,0.27232,0.09006,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,7,0,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[39,39,1.0,0.28571,0.06186,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"ec97bee8515728f3","q":"An acute triangle  $ABC$  is inscribed in circle  $\\omega$  centered at  $O$ . Line  $BO$  and side  $AC$  meet at  $B_1$ . Line  $CO$  and side  $AB$  meet at  $C_1$ . Line  $B_1C_1$  meets circle  $\\omega$  at  $P$  and  $Q$ . If  $AP = AQ$ , prove that  $AB = AC$ 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train traveling at  $ 80$  mph begins to cross a  $ 1$  mile long bridge. At this moment, a man begins to walk from the front of the train to the back of the train at a speed of  $5$  mph. The man reaches the back of the train as soon as the train is completely off the bridge. What is the length of the train (as a fraction of a mile)?","t":[{"b":0,"e":0.28571,"k":"flat","v":0.26786,"x":0.28571,"p":[[0,41,0.0,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.23214,"x":0.28572,"p":[[0,37,0.0,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.28572,5e-05,0.28571,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f9cbd6fd3161bc35","q":"3. (UKR) ${ }^{\\mathrm{IMO}}$ Let $I$ be the incenter of triangle $A B C$. Let $K, L$, and $M$ be the points of tangency of the incircle of $A B C$ with $A B, B C$, and $C A$, respectively. The line $t$ passes through $B$ and is parallel to $K L$. The lines $M K$ and $M L$ intersect $t$ at the points $R$ and $S$. Prove that $\\angle R I S$ is acute.","t":[{"b":3,"e":0.42857,"k":"flat","v":0.20536,"x":0.35268,"p":[[0,64,0.0,0.35268,0.19227,0.28571,0.42857,0.42857,0.0,1.0,4,1,1,4,0,3,0,0,3,0,0,21,0,0,0,0,0,0,0,0,0,0,1],[4,64,0.0625,0.28571,0.2369,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,6,0,0,7,0,0,9,0,0,1,0,0,0,0,0,1,0,1],[8,64,0.125,0.25446,0.1665,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,10,0,0,5,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[12,64,0.1875,0.31696,0.15458,0.28571,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,3,0,0,8,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[16,64,0.25,0.24554,0.17215,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,7,0,0,7,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[20,64,0.3125,0.25,0.15972,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,10,0,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.23661,0.15815,0.14286,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,9,0,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.26339,0.15612,0.14286,0.28571,0.42857,0.0,0.42857,5,0,0,5,0,7,0,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.26786,0.15465,0.14286,0.28571,0.42857,0.0,0.42857,5,0,0,5,0,6,0,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.27679,0.19541,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,4,0,0,8,0,0,12,0,0,0,0,0,0,0,0,1,0,0],[40,64,0.625,0.25884,0.14487,0.14286,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,8,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.28124,0.16933,0.14286,0.28571,0.42857,0.0,0.857,4,0,0,4,0,5,0,0,14,0,0,8,0,0,0,0,0,0,0,0,1,0,0],[48,64,0.75,0.26786,0.11152,0.24999,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,6,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.20536,0.16342,0.0,0.2857,0.28571,0.0,0.4286,11,0,0,11,0,2,0,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.25,0.15972,0.10714,0.28571,0.42857,0.0,0.4286,8,0,0,8,0,1,0,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.21429,0.16366,0.0,0.28571,0.28571,0.0,0.42857,10,0,0,10,0,3,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.25,0.16751,0.0,0.28571,0.42857,0.0,0.42857,9,0,0,9,0,0,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.01786,"x":0.39732,"p":[[0,102,0.0,0.39732,0.27137,0.25,0.42857,0.42857,0.0,1.0,5,3,1,5,0,3,0,0,2,0,0,17,0,0,1,0,0,0,0,0,1,0,3],[4,102,0.0392,0.25447,0.1461,0.14286,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,9,0,0,9,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[8,102,0.0784,0.32589,0.17582,0.14286,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,11,0,0,5,0,0,15,0,0,0,0,0,0,0,0,0,0,1],[12,102,0.1176,0.24554,0.15663,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,10,0,0,6,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,102,0.1569,0.24107,0.2126,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,8,0,0,11,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[20,102,0.1961,0.25,0.17128,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,6,0,0,8,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[24,102,0.2353,0.2142,0.1429,0.14214,0.28571,0.28571,0.0,0.42857,7,0,0,7,0,7,0,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,102,0.2745,0.20527,0.16345,0.105,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,10,0,0,7,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[32,102,0.3137,0.21875,0.16746,0.0,0.2857,0.42857,0.0,0.4286,9,0,0,9,0,6,0,0,8,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[36,102,0.3529,0.23665,0.23041,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,6,0,0,2,0,0,12,0,0,0,0,0,0,0,0,0,0,1],[40,102,0.3922,0.26339,0.15612,0.14286,0.28571,0.42857,0.0,0.42857,5,0,0,5,0,7,0,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[44,102,0.4314,0.2767,0.17112,0.14286,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,5,0,0,7,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[48,102,0.4706,0.19196,0.15815,0.0,0.2857,0.28571,0.0,0.4286,11,0,0,11,0,4,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[52,102,0.5098,0.25446,0.18808,0.14286,0.28571,0.28571,0.0,1.0,6,1,0,6,0,4,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[56,102,0.549,0.22304,0.15551,0.105,0.28571,0.28571,0.0,0.4286,8,0,0,8,0,5,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[60,102,0.5882,0.19643,0.15872,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,2,0,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,102,0.6275,0.1875,0.14032,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,1,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,102,0.6667,0.20982,0.15146,0.0,0.28571,0.28571,0.0,0.4286,10,0,0,10,0,1,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[72,102,0.7059,0.15178,0.14698,0.0,0.21428,0.28571,0.0,0.42857,15,0,0,15,0,1,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,102,0.7451,0.16965,0.1448,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,1,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,102,0.7843,0.19643,0.13716,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,1,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,102,0.8235,0.17411,0.13236,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,102,0.8627,0.18751,0.14033,0.0,0.28571,0.28571,0.0,0.42857,11,0,0,11,0,1,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,102,0.902,0.21875,0.12869,0.21427,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,0,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,102,0.9412,0.14732,0.13592,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,102,0.9804,0.12053,0.13882,0.0,0.0,0.28571,0.0,0.28571,18,0,0,18,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[102,102,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ab25b7d9188429e","q":"Adults made up  $\\frac5{12}$  of the crowd of people at a concert. After a bus carrying  $50$  more people arrived, adults made up  $\\frac{11}{25}$  of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.","t":[{"b":4,"e":0.85714,"k":"flat","v":0.92411,"x":0.97768,"p":[[0,35,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,35,0.1143,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[8,35,0.2286,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,35,0.3429,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,35,0.4571,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[20,35,0.5714,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[24,35,0.6857,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[28,35,0.8,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[32,35,0.9143,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[35,35,1.0,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15]]},{"b":7,"e":0.85714,"k":"flat","v":0.91517,"x":0.96429,"p":[[0,19,0.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[4,19,0.2105,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[8,19,0.4211,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[12,19,0.6316,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,19,0.8421,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[19,19,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]}]},{"i":"cb4f40b8ecaec5d5","q":"4. (BUL 4) We are given $n$ points in space. Some pairs of these points are connected by line segments so that the number of segments equals $\\left[n^{2} / 4\\right]$, and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle.","t":[{"b":5,"e":0.28571,"k":"falling","v":0.64731,"x":0.87946,"p":[[0,59,0.0,0.8125,0.27534,0.78561,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,0,6,0,18],[4,59,0.0678,0.87946,0.16409,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,12,0,15],[8,59,0.1356,0.87946,0.22619,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,6,0,21],[12,59,0.2034,0.70536,0.2922,0.42859,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,6,0,0,0,0,0,1,0,0,12,0,8],[16,59,0.2712,0.80357,0.27606,0.78561,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,0,8,0,16],[20,59,0.339,0.82587,0.17401,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,12,0,10],[24,59,0.4068,0.81249,0.25111,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,4,0,0,9,0,14],[28,59,0.4746,0.78571,0.30304,0.82132,0.85714,1.0,0.0,1.0,1,14,0,1,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,10,0,14],[32,59,0.5424,0.81695,0.23484,0.82143,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,12,0,12],[36,59,0.6102,0.80357,0.25443,0.78561,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,0,0,0,10,0,14],[40,59,0.678,0.75446,0.30771,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,2,0,0,1,0,0,3,0,0,8,0,13],[44,59,0.7458,0.73661,0.29904,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,3,0,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,11],[48,59,0.8136,0.74553,0.29175,0.67857,0.85714,1.0,0.0,1.0,2,11,0,2,0,1,0,0,0,0,0,4,0,0,1,0,0,5,0,0,8,0,11],[52,59,0.8814,0.72766,0.27049,0.57132,0.85707,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,3,0,0,5,0,0,2,0,0,8,0,10],[56,59,0.9492,0.64731,0.3072,0.39286,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,5,0,0,1,0,0,5,0,0,4,0,0,7,0,7],[59,59,1.0,0.6607,0.28292,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,5,0,0,5,0,0,3,0,0,5,0,0,5,0,8]]},{"b":6,"e":0.42857,"k":"falling","v":0.42854,"x":0.86607,"p":[[0,44,0.0,0.75446,0.33737,0.42859,1.0,1.0,0.14286,1.0,0,18,0,0,0,5,0,0,2,0,0,2,0,0,0,0,0,2,0,0,3,0,18],[4,44,0.0909,0.86607,0.17105,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,10,0,15],[8,44,0.1818,0.77677,0.26471,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,2,0,0,1,0,0,3,0,0,14,0,9],[12,44,0.2727,0.66072,0.3004,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,2,0,0,2,0,0,7,0,0,1,0,0,7,0,0,2,0,10],[16,44,0.3636,0.75892,0.2683,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,1,0,0,4,0,0,2,0,0,8,0,12],[20,44,0.4545,0.69196,0.31157,0.53571,0.85714,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,2,0,0,2,0,0,3,0,0,3,0,0,10,0,8],[24,44,0.5455,0.62946,0.35867,0.24999,0.71429,1.0,0.0,1.0,1,11,0,1,0,7,0,0,2,0,0,2,0,0,2,0,0,3,0,0,4,0,11],[28,44,0.6364,0.73661,0.28146,0.71429,0.85714,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,0,0,0,0,0,0,3,0,0,6,0,0,12,0,7],[32,44,0.7273,0.74554,0.27371,0.67857,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,3,0,0,1,0,0,4,0,0,11,0,9],[36,44,0.8182,0.55348,0.3089,0.28571,0.42859,0.85714,0.14,1.0,0,5,0,0,0,4,0,0,9,0,0,4,0,0,2,0,0,1,0,0,7,0,5],[40,44,0.9091,0.4464,0.17764,0.28571,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,9,0,0,5,0,0,12,0,0,2,0,0,1,0,0],[44,44,1.0,0.42854,0.1597,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,7,0,0,7,0,0,13,0,0,1,0,0,0,0,0]]}]},{"i":"70ce822e974c5d10","q":"4. (BUL 3) Prove that for every integer $n>1$ the equation  $$ \\frac{x^{n}}{n!}+\\frac{x^{n-1}}{(n-1)!}+\\cdots+\\frac{x^{2}}{2!}+\\frac{x}{1!}+1=0 $$  has no rational roots.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.44643,"x":0.85713,"p":[[0,62,0.0,0.62054,0.36177,0.14286,0.71429,1.0,0.0,1.0,1,11,1,1,0,8,0,0,1,0,0,2,0,0,2,0,0,4,0,0,3,0,11],[4,62,0.0645,0.5,0.34069,0.14286,0.35714,0.85714,0.14286,1.0,0,4,0,0,0,11,0,0,5,0,0,2,0,0,1,0,0,1,0,0,8,0,4],[8,62,0.129,0.49107,0.31731,0.14286,0.35714,0.85714,0.14286,1.0,0,3,0,0,0,9,0,0,7,0,0,2,0,0,2,0,0,2,0,0,7,0,3],[12,62,0.1935,0.44643,0.30878,0.14286,0.42857,0.85714,0.14286,1.0,0,1,0,0,0,13,0,0,2,0,0,6,0,0,0,0,0,2,0,0,8,0,1],[16,62,0.2581,0.46433,0.2945,0.14286,0.42857,0.75,0.14286,1.0,0,1,0,0,0,11,0,0,2,0,0,7,0,0,1,0,0,3,0,0,7,0,1],[20,62,0.3226,0.63839,0.28568,0.42857,0.78571,0.85714,0.14286,1.0,0,1,0,0,0,6,0,0,1,0,0,3,0,0,1,0,0,5,0,0,15,0,1],[24,62,0.3871,0.45089,0.31766,0.14286,0.35714,0.85714,0.0,0.85714,1,0,0,1,0,12,0,0,3,0,0,3,0,0,1,0,0,2,0,0,10,0,0],[28,62,0.4516,0.72319,0.24206,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,2,0,0,1,0,0,4,0,0,20,0,1],[32,62,0.5161,0.67857,0.25754,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,2,0,0,1,0,0,2,0,0,5,0,0,18,0,0],[36,62,0.5806,0.7366,0.25028,0.85711,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,25,0,0],[40,62,0.6452,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[44,62,0.7097,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[48,62,0.7742,0.83929,0.05923,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0],[52,62,0.8387,0.85713,4e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[56,62,0.9032,0.84374,0.04164,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0],[60,62,0.9677,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[62,62,1.0,0.83482,0.05187,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0]]},{"b":2,"e":0.85714,"k":"rising","v":0.52232,"x":0.83482,"p":[[0,35,0.0,0.6517,0.35176,0.2857,0.85714,1.0,0.14,1.0,0,12,0,0,0,7,0,0,2,0,0,4,0,0,1,0,0,1,0,0,5,0,12],[4,35,0.1143,0.52232,0.33237,0.14286,0.49999,0.85714,0.14286,1.0,0,2,0,0,0,10,0,0,5,0,0,1,0,0,2,0,0,0,0,0,12,0,2],[8,35,0.2286,0.74554,0.20434,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,0,23,0,0],[12,35,0.3429,0.58928,0.29397,0.28571,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,4,0,0,5,0,0,2,0,0,3,0,0,2,0,0,15,0,0],[16,35,0.4571,0.59374,0.30328,0.28571,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,5,0,0,5,0,0,0,0,0,2,0,0,4,0,0,15,0,0],[20,35,0.5714,0.74107,0.2126,0.67857,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,2,0,0,21,0,1],[24,35,0.6857,0.81248,0.12595,0.857,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,27,0,0],[28,35,0.8,0.81249,0.10374,0.82132,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,23,0,1],[32,35,0.9143,0.83482,0.05187,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0],[35,35,1.0,0.83482,0.06298,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0]]}]},{"i":"3f8ec6a6fdf505a9","q":"4. (CZS 2) Let $T_{1}$ be a triangle having $a, b, c$ as lengths of its sides and let $T_{2}$ be another triangle having $u, v, w$ as lengths of its sides. If $P, Q$ are the areas of the two triangles, prove that  $$ 16 P Q \\leq a^{2}\\left(-u^{2}+v^{2}+w^{2}\\right)+b^{2}\\left(u^{2}-v^{2}+w^{2}\\right)+c^{2}\\left(u^{2}+v^{2}-w^{2}\\right) $$  When does equality hold?","t":[{"b":3,"e":0.28571,"k":"flat","v":0.20973,"x":0.3125,"p":[[0,57,0.0,0.2142,0.13837,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,57,0.0702,0.28116,0.14063,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,12,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[8,57,0.1404,0.28571,0.17128,0.14286,0.28571,0.32143,0.0,1.0,1,1,0,1,0,10,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[12,57,0.2105,0.27233,0.10926,0.14286,0.28571,0.32164,0.14286,0.4286,0,0,0,0,0,11,0,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.3125,0.10972,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,18,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[20,57,0.3509,0.29465,0.17473,0.14289,0.28571,0.32143,0.0,1.0,1,1,0,1,0,9,0,0,14,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[24,57,0.4211,0.20973,0.1336,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,14,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,57,0.4912,0.27232,0.12037,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,10,0,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.30357,0.07784,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,22,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.27232,0.11495,0.25,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,6,0,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.28125,0.14934,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,10,0,0,9,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[44,57,0.7719,0.28571,0.08748,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,6,0,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.29464,0.11812,0.24999,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,7,0,0,13,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.29022,0.09101,0.2857,0.28571,0.28571,0.14286,0.43,0,0,0,0,0,6,0,0,19,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.27232,0.07457,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,6,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.27232,0.05486,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,4,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.20089,"x":0.32589,"p":[[0,74,0.0,0.20089,0.1551,0.14286,0.14286,0.1786,0.14286,1.0,0,1,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,74,0.0541,0.28571,0.12372,0.24999,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,8,0,0,19,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,74,0.1081,0.24544,0.13,0.14286,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,4,0,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,74,0.1622,0.2767,0.11822,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,9,0,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[16,74,0.2162,0.25884,0.14921,0.14286,0.2857,0.32143,0.0,0.71429,2,0,0,2,0,12,0,0,10,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[20,74,0.2703,0.3125,0.16146,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,7,0,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[24,74,0.3243,0.2679,0.10571,0.14286,0.28571,0.28571,0.14286,0.43,0,0,0,0,0,11,0,0,14,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,74,0.3784,0.28125,0.11564,0.14286,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,10,0,0,14,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[32,74,0.4324,0.2633,0.12939,0.14286,0.28571,0.32143,0.0,0.42857,3,0,0,3,0,7,0,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.30348,0.08581,0.2857,0.28571,0.32143,0.14,0.42857,0,0,0,0,0,4,0,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[40,74,0.5405,0.27678,0.14698,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,9,0,0,12,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[44,74,0.5946,0.29446,0.10089,0.2857,0.28571,0.42857,0.14,0.42857,0,0,0,0,0,7,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[48,74,0.6486,0.29018,0.10403,0.24999,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,8,0,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[52,74,0.7027,0.26339,0.09522,0.25,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,7,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,74,0.7568,0.28571,0.12372,0.14286,0.28571,0.42857,0.0,0.42857,1,0,0,1,0,9,0,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.32589,0.11971,0.28571,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,2,0,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[64,74,0.8649,0.27678,0.11811,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,9,0,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[68,74,0.9189,0.30803,0.08828,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,22,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[72,74,0.973,0.3125,0.0974,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[74,74,1.0,0.25893,0.09061,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"55af2374114cd3f7","q":"17. (ISR 2) In the convex pentagon $A B C D E$, the sides $B C, C D, D E$ have the same length. Moreover, each diagonal of the pentagon is parallel to a side ( $A C$ is parallel to $D E, B D$ is parallel to $A E$, etc.). Prove that $A B C D E$ is a regular pentagon.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.25887,"x":0.42405,"p":[[0,32,0.0,0.30353,0.33642,0.0,0.21429,0.571,0.0,1.0,14,3,0,14,0,2,0,0,4,0,0,2,0,0,4,0,0,3,0,0,0,0,3],[4,32,0.125,0.42405,0.40323,0.0,0.49979,0.67857,0.0,1.0,13,8,0,13,0,0,0,0,2,0,0,1,0,0,8,0,0,0,0,0,0,0,8],[8,32,0.25,0.29459,0.29648,0.0,0.28571,0.57143,0.0,1.0,15,1,0,15,0,0,0,0,2,0,0,1,0,0,13,0,0,0,0,0,0,0,1],[12,32,0.375,0.34368,0.28083,0.0,0.4286,0.57143,0.0,0.857,12,0,0,12,0,0,0,0,1,0,0,4,0,0,13,0,0,1,0,0,1,0,0],[16,32,0.5,0.32135,0.29007,0.0,0.35714,0.57141,0.0,1.0,13,1,0,13,0,0,0,0,3,0,0,1,0,0,14,0,0,0,0,0,0,0,1],[20,32,0.625,0.25887,0.27988,0.0,0.0,0.57143,0.0,0.57143,17,0,0,17,0,0,0,0,1,0,0,0,0,0,14,0,0,0,0,0,0,0,0],[24,32,0.75,0.31246,0.2799,0.0,0.571,0.57143,0.0,0.57143,14,0,0,14,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,0,0,0],[28,32,0.875,0.31696,0.27137,0.0,0.5,0.57143,0.0,0.57143,13,0,0,13,0,0,0,0,2,0,0,1,0,0,16,0,0,0,0,0,0,0,0],[32,32,1.0,0.33034,0.27764,0.0,0.57143,0.57143,0.0,0.57143,13,0,0,13,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.21427,"x":0.47765,"p":[[0,88,0.0,0.45977,0.38086,0.0,0.571,0.74996,0.0,1.0,11,4,2,11,0,0,0,0,3,0,0,1,0,0,3,0,0,6,0,0,4,0,4],[4,88,0.0455,0.43301,0.38045,0.0,0.571,0.60714,0.0,1.0,12,6,0,12,0,0,0,0,1,0,0,2,0,0,9,0,0,1,0,0,1,0,6],[8,88,0.0909,0.47765,0.34554,0.21427,0.57143,0.60714,0.0,1.0,8,6,0,8,0,0,0,0,4,0,0,1,0,0,11,0,0,2,0,0,0,0,6],[12,88,0.1364,0.34372,0.37432,0.0,0.14286,0.57143,0.0,1.0,16,4,0,16,0,0,0,0,1,0,0,0,0,0,9,0,0,1,0,0,1,0,4],[16,88,0.1818,0.45533,0.39998,0.0,0.57121,0.85704,0.0,1.0,12,7,0,12,0,0,0,0,2,0,0,0,0,0,8,0,0,1,0,0,2,0,7],[20,88,0.2273,0.32139,0.32338,0.0,0.28571,0.57143,0.0,1.0,14,2,0,14,0,1,0,0,2,0,0,1,0,0,10,0,0,2,0,0,0,0,2],[24,88,0.2727,0.4062,0.34087,0.0,0.35714,0.57143,0.0,1.0,9,5,0,9,0,1,0,0,6,0,0,2,0,0,8,0,0,1,0,0,0,0,5],[28,88,0.3182,0.37942,0.30846,0.0,0.571,0.57143,0.0,1.0,11,2,0,11,0,0,0,0,3,0,0,0,0,0,15,0,0,1,0,0,0,0,2],[32,88,0.3636,0.37496,0.36725,0.0,0.28571,0.57143,0.0,1.0,12,5,0,12,0,1,0,0,5,0,0,1,0,0,6,0,0,1,0,0,1,0,5],[36,88,0.4091,0.366,0.36928,0.0,0.571,0.57143,0.0,1.0,15,4,0,15,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,4],[40,88,0.4545,0.38391,0.32426,0.0,0.57143,0.57143,0.0,1.0,11,3,0,11,0,1,0,0,2,0,0,0,0,0,15,0,0,0,0,0,0,0,3],[44,88,0.5,0.4687,0.31588,0.21429,0.57143,0.57143,0.0,1.0,8,3,0,8,0,0,0,0,2,0,0,1,0,0,15,0,0,1,0,0,2,0,3],[48,88,0.5455,0.37048,0.32896,0.0,0.42857,0.57111,0.0,1.0,11,4,0,11,0,0,0,0,4,0,0,5,0,0,8,0,0,0,0,0,0,0,4],[52,88,0.5909,0.37049,0.34413,0.0,0.49979,0.57143,0.0,1.0,13,3,0,13,0,0,0,0,2,0,0,1,0,0,11,0,0,1,0,0,1,0,3],[56,88,0.6364,0.35713,0.29666,0.0,0.28571,0.57143,0.0,1.0,10,2,0,10,0,0,0,0,7,0,0,2,0,0,9,0,0,2,0,0,0,0,2],[60,88,0.6818,0.26337,0.32162,0.0,0.0,0.57143,0.0,1.0,18,2,0,18,0,0,0,0,1,0,0,1,0,0,10,0,0,0,0,0,0,0,2],[64,88,0.7273,0.29908,0.29526,0.0,0.28571,0.57143,0.0,1.0,14,1,0,14,0,0,0,0,3,0,0,4,0,0,9,0,0,0,0,0,1,0,1],[68,88,0.7727,0.24102,0.25357,0.0,0.14286,0.571,0.0,0.57143,15,0,0,15,0,2,0,0,3,0,0,2,0,0,10,0,0,0,0,0,0,0,0],[72,88,0.8182,0.32138,0.31739,0.0,0.28571,0.57143,0.0,1.0,13,2,0,13,0,1,0,0,4,0,0,1,0,0,10,0,0,0,0,0,1,0,2],[76,88,0.8636,0.21427,0.30721,0.0,0.0,0.46418,0.0,1.0,20,2,0,20,0,0,0,0,2,0,0,2,0,0,6,0,0,0,0,0,0,0,2],[80,88,0.9091,0.22765,0.292,0.0,0.0,0.46418,0.0,1.0,18,1,0,18,0,0,0,0,4,0,0,2,0,0,6,0,0,0,0,0,1,0,1],[84,88,0.9545,0.27676,0.22848,0.0,0.28571,0.571,0.0,0.57143,11,0,0,11,0,0,0,0,10,0,0,2,0,0,9,0,0,0,0,0,0,0,0],[88,88,1.0,0.26338,0.27917,0.0,0.2143,0.57143,0.0,1.0,14,1,0,14,0,2,0,0,5,0,0,0,0,0,10,0,0,0,0,0,0,0,1]]}]},{"i":"905e02e86afda09e","q":"17. A3 (POL) Let $P$ be a cubic polynomial given by $P(x)=a x^{3}+b x^{2}+c x+$ $d$, where $a, b, c, d$ are integers and $a \\neq 0$. Suppose that $x P(x)=y P(y)$ for infinitely many pairs $x, y$ of integers with $x \\neq y$. Prove that the equation $P(x)=0$ has an integer root.","t":[{"b":1,"e":0.28571,"k":"falling","v":0.55356,"x":0.93302,"p":[[0,39,0.0,0.84821,0.2765,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,22],[4,39,0.1026,0.89286,0.22016,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,2,0,24],[8,39,0.2051,0.93302,0.13359,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[12,39,0.3077,0.91518,0.1551,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,6,0,22],[16,39,0.4103,0.74107,0.3102,0.42859,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,5,0,0,3,0,0,3,0,0,1,0,0,3,0,16],[20,39,0.5128,0.87054,0.22119,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,6,0,20],[24,39,0.6154,0.84374,0.23245,0.67857,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,2,0,0,2,0,20],[28,39,0.7179,0.86161,0.22724,0.78571,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,0,2,0,22],[32,39,0.8205,0.76786,0.23077,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,0,0,0,3,0,14],[36,39,0.9231,0.6473,0.31337,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,2,0,0,2,0,0,4,0,0,5,0,0,4,0,0,4,0,9],[39,39,1.0,0.55356,0.25939,0.42857,0.4998,0.57143,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,11,0,0,9,0,0,0,0,0,1,0,6]]},{"b":7,"e":0.85714,"k":"falling","v":0.62939,"x":0.95535,"p":[[0,80,0.0,0.875,0.20124,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,1,0,22],[4,80,0.05,0.95535,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[8,80,0.1,0.875,0.18123,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,6,0,19],[12,80,0.15,0.88839,0.19475,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,21],[16,80,0.2,0.88393,0.2299,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,4,0,23],[20,80,0.25,0.85714,0.22304,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,4,0,20],[24,80,0.3,0.83035,0.24074,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,7,0,16],[28,80,0.35,0.83482,0.2372,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,4,0,18],[32,80,0.4,0.85713,0.20205,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,4,0,19],[36,80,0.45,0.80799,0.20084,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,2,0,0,7,0,13],[40,80,0.5,0.86606,0.17837,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,3,0,19],[44,80,0.55,0.70982,0.21572,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,5,0,0,4,0,8],[48,80,0.6,0.72768,0.24317,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,4,0,0,5,0,10],[52,80,0.65,0.79462,0.21411,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,1,0,0,5,0,14],[56,80,0.7,0.62939,0.20474,0.571,0.57143,0.85704,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,3,0,0,17,0,0,0,0,0,5,0,4],[60,80,0.75,0.67856,0.18899,0.57143,0.57143,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,3,0,0,5,0,5],[64,80,0.8,0.65624,0.21975,0.57143,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,1,0,0,18,0,0,2,0,0,0,0,8],[68,80,0.85,0.70979,0.20041,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,2,0,0,2,0,9],[72,80,0.9,0.7053,0.19868,0.57143,0.64286,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,14,0,0,4,0,0,6,0,6],[76,80,0.95,0.81696,0.18977,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,3,0,15],[80,80,1.0,0.67411,0.18977,0.57143,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,19,0,0,2,0,0,3,0,6]]}]},{"i":"1bd60b294c9e4690","q":"A perfect square ends with the same two digits. How many possible values\nof this digit are there?","t":[{"b":3,"e":1.0,"k":"flat","v":0.90178,"x":1.0,"p":[[0,44,0.0,0.90624,0.06786,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[4,44,0.0909,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[8,44,0.1818,0.90178,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[12,44,0.2727,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[16,44,0.3636,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,44,0.4545,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,44,0.5455,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,44,0.6364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.86607,"x":0.92857,"p":[[0,76,0.0,0.87499,0.17035,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,10],[4,76,0.0526,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[8,76,0.1053,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[12,76,0.1579,0.88392,0.10374,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,0,9],[16,76,0.2105,0.86607,0.13333,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,20,0,9],[20,76,0.2632,0.87946,0.06298,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,6],[24,76,0.3158,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[28,76,0.3684,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[32,76,0.4211,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[36,76,0.4737,0.88838,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[40,76,0.5263,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[44,76,0.5789,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[48,76,0.6316,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[52,76,0.6842,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[56,76,0.7368,0.88392,0.06622,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,7],[60,76,0.7895,0.87945,0.05188,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[64,76,0.8421,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[68,76,0.8947,0.87945,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[72,76,0.9474,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[76,76,1.0,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2]]}]},{"i":"d684d506df0768d6","q":"A nonempty set  $A$  is called an *$n$ -level-good*set if   $ A \\subseteq \\{1,2,3,\\ldots,n\\}$  and  $|A| \\le \\min_{x\\in A} x$  (where  $|A|$  denotes the number of elements in  $A$  and  $\\min_{x\\in A} x$   denotes the minimum of the elements in  $A$ ). Let  $a_n$  be the number of  $n$ -level-good sets. Prove that for all positive integers  $n$  we have  $a_{n+2}=a_{n+1}+a_{n}+1$ .","t":[{"b":0,"e":0.0,"k":"volatile","v":0.16071,"x":0.875,"p":[[0,17,0.0,0.63393,0.43586,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,17],[4,17,0.2353,0.69196,0.44047,0.10714,1.0,1.0,0.0,1.0,8,21,0,8,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,21],[8,17,0.4706,0.875,0.30671,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,27],[12,17,0.7059,0.20536,0.37447,0.0,0.0,0.10714,0.0,1.0,24,5,0,24,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,5],[16,17,0.9412,0.16071,0.31693,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[17,17,1.0,0.20533,0.32127,0.0,0.0,0.42857,0.0,1.0,21,3,0,21,0,0,0,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,3]]},{"b":2,"e":1.0,"k":"rising","v":0.62054,"x":1.0,"p":[[0,50,0.0,0.83929,0.3004,0.89286,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,24],[4,50,0.08,0.79911,0.36572,0.89286,1.0,1.0,0.0,1.0,4,24,0,4,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,24],[8,50,0.16,0.75893,0.40159,0.53571,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,23],[12,50,0.24,0.76786,0.30252,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,7,0,0,3,0,0,1,0,0,0,0,19],[16,50,0.32,0.83482,0.25281,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,2,0,20],[20,50,0.4,0.62054,0.35644,0.42857,0.57143,1.0,0.0,1.0,5,11,0,5,0,1,0,0,0,0,0,5,0,0,6,0,0,2,0,0,2,0,11],[24,50,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,50,0.56,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,50,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,50,0.72,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,50,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,50,0.88,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,50,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[50,50,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"be8bd1e14070bbfd","q":"7. (NET 1) Given five real numbers $u_{0}, u_{1}, u_{2}, u_{3}, u_{4}$, prove that it is always possible to find five real numbers $v_{0}, v_{1}, v_{2}, v_{3}, v_{4}$ that satisfy the following conditions: (i) $u_{i}-v_{i} \\in \\mathbb{N}$. (ii) $\\sum_{0 \\leq i<j \\leq 4}\\left(v_{i}-v_{j}\\right)^{2}<4$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.59375,"x":1.0,"p":[[0,67,0.0,0.90625,0.16602,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,0,0,24],[4,67,0.0597,0.91071,0.25939,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[8,67,0.1194,0.92411,0.22864,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[12,67,0.1791,0.83929,0.27141,0.67857,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,1,0,0,0,0,23],[16,67,0.2388,0.83036,0.3102,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,3,0,22],[20,67,0.2985,0.82143,0.31542,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,23],[24,67,0.3582,0.86607,0.2878,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,26],[28,67,0.4179,0.76339,0.3722,0.60714,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,21],[32,67,0.4776,0.76786,0.36202,0.5,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,22],[36,67,0.5373,0.83929,0.29395,0.89286,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,24],[40,67,0.597,0.82589,0.34206,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,25],[44,67,0.6567,0.7857,0.34442,0.49968,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,22],[48,67,0.7164,0.70981,0.3737,0.53539,0.92857,1.0,0.0,1.0,5,16,0,5,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,4,0,16],[52,67,0.7761,0.59375,0.4105,0.28571,0.64286,1.0,0.0,1.0,6,15,0,6,0,0,0,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,15],[56,67,0.8358,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,67,0.8955,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,67,0.9552,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[67,67,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.72768,"x":1.0,"p":[[0,50,0.0,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[4,50,0.08,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[8,50,0.16,0.87054,0.26573,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,2,0,24],[12,50,0.24,0.91071,0.24679,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,28],[16,50,0.32,0.80357,0.3458,0.82143,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[20,50,0.4,0.84821,0.30501,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,24],[24,50,0.48,0.72768,0.35956,0.39286,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,5,0,0,1,0,0,3,0,0,1,0,0,0,0,19],[28,50,0.56,0.75446,0.38998,0.5,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,22],[32,50,0.64,0.89286,0.23146,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,26],[36,50,0.72,0.83481,0.29258,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,3,0,22],[40,50,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,50,0.88,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,50,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[50,50,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"39b9ed3f7905074d","q":"6. (USS 4) ${ }^{\\mathrm{IMO}}$ Prove for each triangle $A B C$ the inequality  $$ \\frac{1}{4}<\\frac{I A \\cdot I B \\cdot I C}{l_{A} l_{B} l_{C}} \\leq \\frac{8}{27} $$  where $I$ is the incenter and $l_{A}, l_{B}, l_{C}$ are the lengths of the angle bisectors of $A B C$.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.83035,"x":0.93304,"p":[[0,157,0.0,0.93304,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[4,157,0.0255,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[8,157,0.051,0.90178,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[12,157,0.0764,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[16,157,0.1019,0.91518,0.07016,0.85714,0.85714,1.0,0.8571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[20,157,0.1274,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[24,157,0.1529,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[28,157,0.1783,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[32,157,0.2038,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[36,157,0.2293,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[40,157,0.2548,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[44,157,0.2803,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[48,157,0.3057,0.86607,0.16728,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,8],[52,157,0.3312,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[56,157,0.3567,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[60,157,0.3822,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[64,157,0.4076,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[68,157,0.4331,0.88392,0.09066,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,21,0,9],[72,157,0.4586,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7],[76,157,0.4841,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[80,157,0.5096,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[84,157,0.535,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[88,157,0.5605,0.88392,0.07524,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,8],[92,157,0.586,0.87499,0.07789,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,6],[96,157,0.6115,0.88393,0.08328,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,23,0,8],[100,157,0.6369,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[104,157,0.6624,0.87054,0.07457,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,5],[108,157,0.6879,0.8616,0.09771,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,23,0,6],[112,157,0.7134,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[116,157,0.7389,0.85714,0.06186,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,26,0,3],[120,157,0.7643,0.85714,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[124,157,0.7898,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[128,157,0.8153,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[132,157,0.8408,0.84821,0.06121,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,2],[136,157,0.8662,0.86161,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3],[140,157,0.8917,0.86607,0.06121,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,4],[144,157,0.9172,0.83035,0.14914,0.85714,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,2],[148,157,0.9427,0.85267,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,2],[152,157,0.9682,0.84375,0.06546,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,1],[156,157,0.9936,0.8482,0.07941,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,3],[157,157,1.0,0.86607,0.06121,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,4]]},{"b":4,"e":0.85714,"k":"flat","v":0.87052,"x":0.94196,"p":[[0,83,0.0,0.91964,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,10,0,19],[4,83,0.0482,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[8,83,0.0964,0.9241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(POL 2) ${ }^{\\mathrm{IMO} 1}$ Given a point $M$ on the side $A B$ of the triangle $A B C$, let $r_{1}$ and $r_{2}$ be the radii of the inscribed circles of the triangles $A C M$ and $B C M$ respectively and let $\\rho_{1}$ and $\\rho_{2}$ be the radii of the excircles of the triangles $A C M$ and $B C M$ at the sides $A M$ and $B M$ respectively. Let $r$ and $\\rho$ denote the radii of the inscribed circle and the excircle at the side $A B$ of the triangle $A B C$ respectively. Prove that  $$ \\frac{r_{1}}{\\rho_{1}} \\frac{r_{2}}{\\rho_{2}}=\\frac{r}{\\rho} $$","t":[{"b":5,"e":1.0,"k":"flat","v":0.92854,"x":1.0,"p":[[0,192,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,192,0.0208,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,192,0.0417,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[12,192,0.0625,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,192,0.0833,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,192,0.1042,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,192,0.125,0.95982,0.15662,1.0,1.0,1.0,0.143,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[28,192,0.1458,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,192,0.1667,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,192,0.1875,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[40,192,0.2083,0.92857,0.19562,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[44,192,0.2292,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,192,0.25,0.92854,0.20208,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[52,192,0.2708,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,192,0.2917,0.9598,0.10858,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[60,192,0.3125,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[64,192,0.3333,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,192,0.3542,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,192,0.375,0.97991,0.11188,1.0,1.0,1.0,0.357,1.0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,31],[76,192,0.3958,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,192,0.4167,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,192,0.4375,0.98438,0.04966,1.0,1.0,1.0,0.78571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,29],[88,192,0.4583,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,192,0.4792,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[96,192,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,192,0.5208,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[104,192,0.5417,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[108,192,0.5625,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,192,0.5833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,192,0.6042,0.9732,0.08335,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[120,192,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,192,0.6458,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,192,0.6667,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[132,192,0.6875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[136,192,0.7083,0.97319,0.08336,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[140,192,0.7292,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[144,192,0.75,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[148,192,0.7708,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[152,192,0.7917,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[156,192,0.8125,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[160,192,0.8333,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[164,192,0.8542,0.96873,0.10561,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[168,192,0.875,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[172,192,0.8958,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[176,192,0.9167,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[180,192,0.9375,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[184,192,0.9583,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[188,192,0.9792,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[192,192,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]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C4 (EST) There are $n+1$ fixed positions in a row, labeled 0 to $n$ in increasing order from right to left. Cards numbered 0 to $n$ are shuffled and dealt, one in each position. The object of the game is to have card $i$ in the $i$ th position for $0 \\leq i \\leq n$. If this has not been achieved, the following move is executed. Determine the smallest $k$ such that the $k$ th position is occupied by a card $l>k$. Remove this card, slide all cards from the $(k+1)$ st to the $l$ th position one place to the right, and replace the card $l$ in the $l$ th position. (a) Prove that the game lasts at most $2^{n}-1$ moves. (b) Prove that there exists a unique initial configuration for which the game lasts exactly $2^{n}-1$ moves.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.37936,"x":0.91067,"p":[[0,50,0.0,0.37936,0.21617,0.2857,0.28571,0.42858,0.14,1.0,0,1,0,0,0,6,0,0,14,0,0,5,0,0,2,0,0,3,0,0,1,0,1],[4,50,0.08,0.87054,0.18336,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,2,0,20],[8,50,0.16,0.77675,0.20186,0.57143,0.78564,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,4,0,0,4,0,12],[12,50,0.24,0.76784,0.2918,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,1,0,0,7,0,0,0,0,0,4,0,16],[16,50,0.32,0.72766,0.19352,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,8,0,0,3,0,8],[20,50,0.4,0.79464,0.23673,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,11,0,0,0,0,0,3,0,16],[24,50,0.48,0.86159,0.20357,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,0,0,21],[28,50,0.56,0.79909,0.21976,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,7,0,13],[32,50,0.64,0.79911,0.24964,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,7,0,0,2,0,16],[36,50,0.72,0.91067,0.15474,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,23],[40,50,0.8,0.85714,0.17128,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,16],[44,50,0.88,0.81248,0.19379,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,0,7,0,13],[48,50,0.96,0.62048,0.14556,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,0,0,0,2,0,3],[50,50,1.0,0.64283,0.20826,0.57143,0.57143,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,1,0,0,19,0,0,1,0,0,2,0,6]]},{"b":1,"e":1.0,"k":"rising","v":0.32589,"x":0.98214,"p":[[0,43,0.0,0.32589,0.21497,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,12,0,0,7,0,0,6,0,0,3,0,0,1,0,0,2,0,0],[4,43,0.093,0.81695,0.17943,0.71429,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,2,0,14],[8,43,0.186,0.83036,0.21261,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,0,4,0,16],[12,43,0.2791,0.83925,0.17409,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,4,0,15],[16,43,0.3721,0.85711,0.19237,0.85711,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,9,0,16],[20,43,0.4651,0.91518,0.17445,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,25],[24,43,0.5581,0.81246,0.18711,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,8,0,12],[28,43,0.6512,0.81693,0.20899,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,4,0,15],[32,43,0.7442,0.81691,0.20595,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,5,0,15],[36,43,0.8372,0.90179,0.19377,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,25],[40,43,0.9302,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[43,43,1.0,0.90176,0.18017,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,1,0,24]]}]},{"i":"962428f8d30e0b97","q":"3. (POL 4) ${ }^{\\mathrm{IMO} 4}$ Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle.","t":[{"b":2,"e":1.0,"k":"flat","v":0.93303,"x":1.0,"p":[[0,22,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,22,0.1818,0.97768,0.08828,1.0,1.0,1.0,0.5714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,22,0.3636,0.93303,0.17123,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[12,22,0.5455,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[22,22,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.42857,"k":"falling","v":0.52679,"x":0.98214,"p":[[0,22,0.0,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,22,0.1818,0.92857,0.17496,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,26],[8,22,0.3636,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[12,22,0.5455,0.58929,0.15872,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,19,0,0,0,0,0],[16,22,0.7273,0.54008,0.18485,0.42857,0.42859,0.71429,0.14,1.0,0,2,0,0,0,1,0,0,1,0,0,16,0,0,4,0,0,8,0,0,0,0,2],[20,22,0.9091,0.59804,0.23573,0.42857,0.71429,0.71429,0.14,1.0,0,4,0,0,0,3,0,0,0,0,0,11,0,0,0,0,0,14,0,0,0,0,4],[22,22,1.0,0.52679,0.15335,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,19,0,0,0,0,0,12,0,0,0,0,0]]}]},{"i":"b7562649d7560886","q":"18. G3 (KOR) Let $O$ be the circumcenter of an acute-angled triangle $A B C$ with $\\angle B<\\angle C$. The line $A O$ meets the side $B C$ at $D$. The circumcenters of the triangles $A B D$ and $A C D$ are $E$ and $F$, respectively. Extend the sides $B A$ and $C A$ beyond $A$, and choose on the respective extension points $G$ and $H$ such that $A G=A C$ and $A H=A B$. Prove that the quadrilateral $E F G H$ is a rectangle if and only if $\\angle A C B-\\angle A B C=60^{\\circ}$.","t":[{"b":5,"e":0.14286,"k":"falling","v":0.18304,"x":0.54014,"p":[[0,95,0.0,0.43749,0.25489,0.2857,0.42857,0.71429,0.0,0.71429,3,0,1,3,0,4,0,0,8,0,0,2,0,0,3,0,0,12,0,0,0,0,0],[4,95,0.0421,0.54014,0.21048,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,3,0,0,3,0,0,11,0,0,11,0,0,1,0,0],[8,95,0.0842,0.51338,0.2392,0.39286,0.57143,0.71429,0.0,0.85714,3,0,0,3,0,1,0,0,4,0,0,5,0,0,5,0,0,13,0,0,1,0,0],[12,95,0.1263,0.49998,0.19232,0.42857,0.57121,0.60714,0.0,0.71429,2,0,0,2,0,1,0,0,2,0,0,9,0,0,10,0,0,8,0,0,0,0,0],[16,95,0.1684,0.41517,0.21236,0.25,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,7,0,0,4,0,0,7,0,0,9,0,0,3,0,0,1,0,0],[20,95,0.2105,0.47096,0.22221,0.28571,0.571,0.58929,0.14286,1.0,0,1,0,0,0,6,0,0,5,0,0,4,0,0,9,1,0,6,0,0,0,0,1],[24,95,0.2526,0.41515,0.18679,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,7,0,0,9,0,0,7,0,0,4,0,0,0,0,0],[28,95,0.2947,0.3192,0.20123,0.14286,0.28571,0.42858,0.0,0.71429,4,0,0,4,0,6,0,0,8,0,1,8,0,0,2,0,0,3,0,0,0,0,0],[32,95,0.3368,0.3749,0.22525,0.14286,0.42857,0.57111,0.0,0.71429,3,0,0,3,0,6,0,0,6,0,0,8,0,0,3,0,0,6,0,0,0,0,0],[36,95,0.3789,0.32593,0.21793,0.14286,0.28571,0.57111,0.0,0.71429,3,0,0,3,0,10,0,0,6,0,0,4,0,0,6,0,0,3,0,0,0,0,0],[40,95,0.4211,0.33479,0.24897,0.14286,0.2857,0.57143,0.0,0.71429,5,0,0,5,0,10,0,0,2,0,0,3,0,0,8,0,0,4,0,0,0,0,0],[44,95,0.4632,0.34372,0.21972,0.14286,0.28571,0.46418,0.0,0.71429,3,0,0,3,0,7,0,0,9,0,0,5,0,0,3,0,0,5,0,0,0,0,0],[48,95,0.5053,0.41294,0.22989,0.25,0.42857,0.60714,0.0,0.85714,1,0,0,1,0,7,0,0,6,0,1,6,0,0,3,0,0,7,0,0,1,0,0],[52,95,0.5474,0.34362,0.23924,0.14286,0.28571,0.571,0.0,0.85714,6,0,0,6,0,4,0,0,7,0,0,5,0,0,7,0,0,2,0,0,1,0,0],[56,95,0.5895,0.42857,0.24223,0.2857,0.42859,0.57143,0.0,0.85714,4,0,0,4,0,2,0,0,7,0,0,4,0,0,8,0,0,6,0,0,1,0,0],[60,95,0.6316,0.42853,0.21722,0.24999,0.42859,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,2,0,0,7,0,0,10,0,0,5,0,0,0,0,0],[64,95,0.6737,0.45086,0.22046,0.28571,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,1,0,0,6,0,0,8,0,0,7,0,0,6,0,0,1,0,0],[68,95,0.7158,0.3883,0.21507,0.28571,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,4,0,0,7,0,0,8,0,0,5,0,0,5,0,0,0,0,0],[72,95,0.7579,0.41067,0.1812,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,4,0,0,11,0,0,6,0,0,8,0,0,2,0,0,1,0,0],[76,95,0.8,0.40166,0.20033,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,7,0,0,6,0,0,8,0,0,4,0,0,0,0,0],[80,95,0.8421,0.39498,0.23899,0.25,0.42857,0.57143,0.0,0.85714,4,0,0,4,0,4,0,0,5,0,1,7,0,0,5,0,0,5,0,0,1,0,0],[84,95,0.8842,0.41965,0.2257,0.25002,0.42859,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,4,0,0,7,0,0,6,0,0,7,0,0,0,0,0],[88,95,0.9263,0.30358,0.16656,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,9,0,0,8,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[92,95,0.9684,0.33249,0.19041,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,1,6,0,0,9,0,0,9,0,0,2,0,0,3,0,0,0,0,0],[95,95,1.0,0.18304,0.16065,0.0,0.14286,0.28571,0.0,0.5714,10,0,0,10,0,9,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.40177,"x":0.61158,"p":[[0,138,0.0,0.51337,0.22263,0.42857,0.57143,0.71429,0.0,0.85714,2,0,1,2,0,3,0,0,2,0,0,3,0,0,12,0,0,9,0,0,1,0,0],[4,138,0.029,0.61158,0.14391,0.5713,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,16,0,0,1,0,0],[8,138,0.058,0.51784,0.19804,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,8,0,0,8,0,0,7,0,0,1,0,1],[12,138,0.087,0.46429,0.23419,0.28571,0.4286,0.71429,0.0,0.71429,3,0,0,3,0,2,0,0,5,0,0,7,0,0,4,0,0,11,0,0,0,0,0],[16,138,0.1159,0.48209,0.20436,0.39286,0.571,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,4,0,0,5,0,0,12,0,0,7,0,0,0,0,0],[20,138,0.1449,0.5044,0.21717,0.42857,0.571,0.71429,0.0,0.71429,2,0,0,2,0,3,0,0,1,0,0,7,0,0,8,0,0,11,0,0,0,0,0],[24,138,0.1739,0.50445,0.22582,0.39293,0.57143,0.71429,0.0,0.71429,3,0,0,3,0,1,0,0,4,0,0,3,0,0,10,0,0,11,0,0,0,0,0],[28,138,0.2029,0.44642,0.23076,0.28571,0.42857,0.60714,0.0,0.85714,2,0,0,2,0,3,0,0,7,0,0,7,0,0,5,0,0,6,0,0,2,0,0],[32,138,0.2319,0.45981,0.21938,0.2857,0.57121,0.60714,0.0,0.71429,2,0,0,2,0,3,0,0,6,0,0,4,0,0,9,0,0,8,0,0,0,0,0],[36,138,0.2609,0.54017,0.17029,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,8,0,0,6,0,0,11,0,0,1,0,0],[40,138,0.2899,0.49103,0.21996,0.28571,0.571,0.71429,0.0,0.71429,2,0,0,2,0,2,0,0,5,0,0,5,0,0,7,0,0,11,0,0,0,0,0],[44,138,0.3188,0.49091,0.19878,0.39286,0.57141,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,3,0,0,2,0,0,16,0,0,6,0,0,0,0,0],[48,138,0.3478,0.50887,0.17103,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,0,8,0,0,12,0,0,7,0,0,0,0,0],[52,138,0.3768,0.46866,0.21806,0.28571,0.4286,0.71429,0.0,0.71429,1,0,0,1,0,3,0,0,8,0,0,5,0,0,4,0,0,11,0,0,0,0,0],[56,138,0.4058,0.49105,0.20496,0.39286,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,4,0,0,3,0,0,14,0,0,7,0,0,0,0,0],[60,138,0.4348,0.48437,0.22069,0.33918,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,3,0,0,3,0,1,5,0,0,8,0,0,10,0,0,0,0,0],[64,138,0.4638,0.40177,0.20957,0.2857,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,2,0,0,6,0,0,6,0,0,12,0,0,2,0,0,0,0,0],[68,138,0.4928,0.56251,0.19208,0.42965,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,1,0,0,5,0,0,8,0,0,15,0,0,0,0,0],[72,138,0.5217,0.55357,0.20748,0.4286,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,2,0,0,4,0,0,8,0,0,15,0,0,0,0,0],[76,138,0.5507,0.58929,0.12752,0.4286,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,14,0,0,0,0,0],[80,138,0.5797,0.54909,0.17168,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,5,0,0,8,0,0,13,0,0,0,0,0],[84,138,0.6087,0.54909,0.16793,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,4,0,0,10,0,0,12,0,0,0,0,0],[88,138,0.6377,0.57128,0.1596,0.57132,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,4,0,0,13,0,0,12,0,0,0,0,0],[92,138,0.6667,0.6071,0.13833,0.5713,0.64286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,3,0,0,11,0,0,16,0,0,0,0,0],[96,138,0.6957,0.5714,0.15152,0.42859,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,6,0,0,10,0,0,13,0,0,0,0,0],[100,138,0.7246,0.50443,0.1712,0.42857,0.4286,0.71429,0.14286,0.71429,0,0,0,0,0,3,0,0,1,0,0,13,0,0,6,0,0,9,0,0,0,0,0],[104,138,0.7536,0.52676,0.18363,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,0,8,0,0,8,0,0,11,0,0,0,0,0],[108,138,0.7826,0.54237,0.13815,0.42857,0.57143,0.71407,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,1,8,0,0,11,0,0,9,0,0,0,0,0],[112,138,0.8116,0.49551,0.19879,0.42857,0.57143,0.60714,0.0,0.71429,2,0,0,2,0,1,0,0,4,0,0,6,0,0,11,0,0,8,0,0,0,0,0],[116,138,0.8406,0.54462,0.15745,0.42859,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,8,0,0,9,0,0,11,0,0,0,0,0],[120,138,0.8696,0.53569,0.16751,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,9,0,0,8,0,0,11,0,0,0,0,0],[124,138,0.8986,0.47326,0.20651,0.42857,0.5007,0.57143,0.0,0.71429,3,0,0,3,0,1,0,0,2,0,0,10,0,0,9,0,0,7,0,0,0,0,0],[128,138,0.9275,0.51784,0.16656,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,9,0,0,9,0,0,9,0,0,0,0,0],[132,138,0.9565,0.49999,0.17127,0.42857,0.4998,0.60714,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,10,0,0,8,0,0,8,0,0,0,0,0],[136,138,0.9855,0.49099,0.15143,0.42857,0.4293,0.57143,0.14,0.71429,0,0,0,0,0,1,0,0,5,0,0,11,0,0,9,0,0,6,0,0,0,0,0],[138,138,1.0,0.49772,0.17988,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,2,0,1,8,0,0,11,0,0,7,0,0,0,0,0]]}]},{"i":"7716b5486c28155e","q":"18. 6b.(CAN 5) Let $x_{1}, x_{2}, \\ldots, x_{n}$ be positive numbers. Prove that  $$ \\frac{x_{1}^{2}}{x_{1}^{2}+x_{2} x_{3}}+\\frac{x_{2}^{2}}{x_{2}^{2}+x_{3} x_{4}}+\\cdots+\\frac{x_{n-1}^{2}}{x_{n-1}^{2}+x_{n} x_{1}}+\\frac{x_{n}^{2}}{x_{n}^{2}+x_{1} x_{2}} \\leq n-1 . $$  Supplementary Problems","t":[{"b":0,"e":1.0,"k":"rising","v":0.35268,"x":0.92857,"p":[[0,73,0.0,0.35268,0.09439,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[4,73,0.0548,0.58481,0.3017,0.28571,0.42857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,12,0,0,6,0,0,1,0,0,1,0,0,4,0,8],[8,73,0.1096,0.5625,0.30501,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,8,0,0,2,0,0,0,0,0,0,0,10],[12,73,0.1644,0.57589,0.29556,0.28571,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,9,0,0,9,0,0,0,0,0,3,0,0,2,0,8],[16,73,0.2192,0.54911,0.28596,0.28571,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,10,0,0,9,0,0,1,0,0,1,0,0,4,0,6],[20,73,0.274,0.47768,0.26149,0.28571,0.42857,0.46431,0.14286,1.0,0,5,0,0,0,1,0,0,13,0,0,10,0,0,1,0,0,1,0,0,1,0,5],[24,73,0.3288,0.50446,0.28118,0.28571,0.42857,0.75,0.0,1.0,1,5,0,1,0,0,0,0,12,0,0,9,0,0,1,0,0,1,0,0,3,0,5],[28,73,0.3836,0.50445,0.24739,0.28571,0.42857,0.71407,0.2857,1.0,0,4,0,0,0,0,0,0,11,0,0,12,0,0,0,0,0,3,0,0,2,0,4],[32,73,0.4384,0.52232,0.31055,0.28571,0.42857,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,11,0,0,9,0,0,0,0,0,1,0,0,1,0,8],[36,73,0.4932,0.52678,0.2683,0.28571,0.42857,0.85704,0.1429,1.0,0,4,0,0,0,1,0,0,10,0,0,10,0,0,1,0,0,1,0,0,5,0,4],[40,73,0.5479,0.4375,0.25738,0.28571,0.42857,0.42858,0.0,1.0,2,3,0,2,0,1,0,0,11,0,0,11,0,0,1,0,0,1,0,0,2,0,3],[44,73,0.6027,0.46875,0.27487,0.28571,0.42857,0.50002,0.0,1.0,1,4,0,1,0,1,0,0,13,0,0,9,0,0,0,0,0,1,0,0,3,0,4],[48,73,0.6575,0.41071,0.31693,0.28571,0.28571,0.5,0.0,1.0,5,4,0,5,0,1,0,0,14,0,0,4,0,0,0,0,0,1,0,0,3,0,4],[52,73,0.7123,0.42411,0.31234,0.28571,0.35714,0.46429,0.0,1.0,5,5,0,5,0,1,0,0,10,0,0,8,0,0,1,0,0,1,0,0,1,0,5],[56,73,0.7671,0.53571,0.34441,0.28571,0.42857,1.0,0.0,1.0,2,10,0,2,0,1,0,0,12,0,0,5,0,0,1,0,0,0,0,0,1,0,10],[60,73,0.8219,0.37054,0.25966,0.2857,0.28571,0.42857,0.0,1.0,3,2,0,3,0,2,0,0,17,0,0,5,0,0,0,0,0,0,0,0,3,0,2],[64,73,0.8767,0.47322,0.27301,0.28571,0.28571,0.5,0.2857,1.0,0,5,0,0,0,0,0,0,18,0,0,6,0,0,0,0,0,1,0,0,2,0,5],[68,73,0.9315,0.83034,0.25113,0.82132,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,0,7,0,17],[72,73,0.9863,0.92857,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[73,73,1.0,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,8,0,21]]},{"b":7,"e":0.2857,"k":"flat","v":0.38837,"x":0.64286,"p":[[0,59,0.0,0.41071,0.16656,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,12,0,0,18,0,0,0,0,0,0,0,0,0,0,2],[4,59,0.0678,0.6116,0.29501,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,6,0,0,11,0,0,0,0,0,2,0,0,3,0,9],[8,59,0.1356,0.53125,0.27487,0.28571,0.42857,0.64286,0.14286,1.0,0,7,0,0,0,1,0,0,8,0,0,13,0,0,2,0,0,0,0,0,1,0,7],[12,59,0.2034,0.64286,0.27894,0.42857,0.4286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,14,0,0,2,0,0,1,0,0,1,0,11],[16,59,0.2712,0.49554,0.2575,0.28571,0.42857,0.42858,0.14286,1.0,0,5,0,0,0,1,0,0,9,0,0,15,0,0,0,0,0,0,0,0,2,0,5],[20,59,0.339,0.53571,0.26486,0.39286,0.42857,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,14,0,0,1,0,0,0,0,0,4,0,5],[24,59,0.4068,0.45089,0.20238,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,18,0,0,1,0,0,1,0,0,0,0,3],[28,59,0.4746,0.45982,0.17762,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,7,0,0,20,0,0,0,0,0,3,0,0,0,0,2],[32,59,0.5424,0.48214,0.26426,0.28571,0.42857,0.4286,0.0,1.0,1,4,0,1,0,2,0,0,6,0,0,16,0,0,0,0,0,0,0,0,3,0,4],[36,59,0.6102,0.61161,0.29502,0.42857,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,7,0,0,12,0,0,0,0,0,1,0,0,2,0,10],[40,59,0.678,0.47768,0.23854,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,10,0,0,14,0,0,0,0,0,1,0,0,3,0,3],[44,59,0.7458,0.51784,0.26904,0.28571,0.42857,0.64254,0.2857,1.0,0,6,0,0,0,0,0,0,11,0,0,12,0,0,1,0,0,0,0,0,2,0,6],[48,59,0.8136,0.43302,0.20972,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,12,0,0,14,0,0,1,0,0,1,0,0,0,0,3],[52,59,0.8814,0.45536,0.23266,0.28571,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,13,0,0,14,0,0,0,0,0,0,0,0,1,0,4],[56,59,0.9492,0.4241,0.26118,0.28571,0.28571,0.42858,0.14286,1.0,0,3,0,0,0,2,0,0,20,0,0,3,0,0,0,0,0,1,0,0,3,0,3],[59,59,1.0,0.38837,0.16455,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,17,0,0,9,0,0,2,0,0,1,0,0,2,0,0]]}]},{"i":"ba1256f8e047ca17","q":"2. A2 (IRE) Let $a_{1} \\geq a_{2} \\geq \\cdots \\geq a_{n}$ be real numbers such that  $$ a_{1}^{k}+a_{2}^{k}+\\cdots+a_{n}^{k} \\geq 0 $$  for all integers $k>0$. Let $p=\\max \\left\\{\\left|a_{1}\\right|, \\ldots,\\left|a_{n}\\right|\\right\\}$. Prove that $p=a_{1}$ and that  $$ \\left(x-a_{1}\\right)\\left(x-a_{2}\\right) \\cdots\\left(x-a_{n}\\right) \\leq x^{n}-a_{1}^{n} $$  for all $x>a_{1}$.","t":[{"b":3,"e":0.28571,"k":"falling","v":0.26339,"x":0.45982,"p":[[0,38,0.0,0.45982,0.23347,0.28571,0.42857,0.42857,0.28571,1.0,0,4,0,0,0,0,0,0,13,0,0,13,0,0,1,0,0,0,0,0,1,0,4],[4,38,0.1053,0.45982,0.2618,0.28571,0.28571,0.46431,0.14286,1.0,0,5,0,0,0,1,0,0,16,0,0,7,0,0,1,0,0,2,0,0,0,0,5],[8,38,0.2105,0.39732,0.23347,0.28571,0.28571,0.42857,0.0,1.0,1,3,0,1,0,1,0,0,18,0,0,7,0,0,0,0,0,2,0,0,0,0,3],[12,38,0.3158,0.37053,0.21975,0.28571,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,2,0,0,23,0,0,3,0,0,0,0,0,0,0,0,2,0,2],[16,38,0.4211,0.42857,0.24741,0.28571,0.28571,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,22,0,0,3,0,0,0,0,0,2,0,0,2,0,3],[20,38,0.5263,0.30803,0.13415,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[24,38,0.6316,0.29018,0.07563,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.32143,0.19562,0.28571,0.28571,0.28571,0.0,1.0,2,2,0,2,0,1,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[32,38,0.8421,0.29911,0.06546,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,38,0.9474,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.26339,0.1017,0.28571,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,4,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.28571,"x":0.46875,"p":[[0,69,0.0,0.46875,0.2506,0.28571,0.42857,0.42858,0.2857,1.0,0,4,0,0,0,0,0,0,15,0,0,10,0,0,0,0,0,1,0,0,2,0,4],[4,69,0.058,0.42857,0.26964,0.28571,0.28571,0.42858,0.14286,1.0,0,5,0,0,0,1,0,0,21,0,0,4,0,0,0,0,0,0,0,0,1,0,5],[8,69,0.1159,0.37053,0.19516,0.28571,0.28571,0.32143,0.2857,1.0,0,2,0,0,0,0,0,0,24,0,0,5,0,0,0,0,0,0,0,0,1,0,2],[12,69,0.1739,0.40625,0.26027,0.28571,0.28571,0.32143,0.14286,1.0,0,5,0,0,0,1,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[16,69,0.2319,0.43303,0.27545,0.28571,0.28571,0.42857,0.2857,1.0,0,6,0,0,0,0,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[20,69,0.2899,0.42857,0.25,0.28571,0.28571,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,21,0,0,5,0,0,0,0,0,1,0,0,1,0,4],[24,69,0.3478,0.29911,0.10326,0.28571,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,1,0,0,26,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[28,69,0.4058,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.31696,0.09268,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,25,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[36,69,0.5217,0.35714,0.15972,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,23,0,0,7,0,0,0,0,0,0,0,0,1,0,1],[40,69,0.5797,0.30803,0.06298,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.29911,0.04164,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.29911,0.04164,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.30357,0.06916,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[68,69,0.9855,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.29911,0.04164,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6a979c306efe253f","q":"4. A4 (IRE) ${ }^{\\mathrm{IMO} 5}$ Let $n$ be a positive integer and let $x_{1} \\leq x_{2} \\leq \\cdots \\leq x_{n}$ be real numbers. (a) Prove that  $$ \\left(\\sum_{i, j=1}^{n}\\left|x_{i}-x_{j}\\right|\\right)^{2} \\leq \\frac{2\\left(n^{2}-1\\right)}{3} \\sum_{i, j=1}^{n}\\left(x_{i}-x_{j}\\right)^{2} $$  (b) Show that equality holds if and only if $x_{1}, \\ldots, x_{n}$ is an arithmetic progession.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.4174,"x":0.56693,"p":[[0,144,0.0,0.48214,0.14174,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,7,0,0,17,0,0,0,0,0,1,0,0],[4,144,0.0278,0.50892,0.09406,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,11,0,0,20,0,0,0,0,0,0,0,0],[8,144,0.0556,0.56693,0.14499,0.53539,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,21,0,0,0,0,0,1,0,2],[12,144,0.0833,0.51337,0.07871,0.42857,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,11,0,0,20,0,0,0,0,0,0,0,0],[16,144,0.1111,0.50892,0.12339,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,13,0,0,1,0,0,0,0,1],[20,144,0.1389,0.54018,0.1665,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,10,0,0,17,0,0,0,0,0,1,0,2],[24,144,0.1667,0.52232,0.1411,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,8,0,0,19,0,0,1,0,0,0,0,1],[28,144,0.1944,0.50446,0.11837,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,14,0,0,0,0,0,0,0,1],[32,144,0.2222,0.53125,0.11425,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,11,0,0,18,0,0,0,0,0,2,0,0],[36,144,0.25,0.48219,0.13241,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,17,0,0,11,0,0,1,0,0,1,0,0],[40,144,0.2778,0.52232,0.18074,0.42857,0.50001,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,11,0,0,11,0,0,2,0,0,1,0,2],[44,144,0.3056,0.48659,0.18161,0.42857,0.42859,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,11,0,0,13,0,0,0,0,0,0,0,2],[48,144,0.3333,0.55801,0.1488,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,14,0,0,1,0,0,3,0,1],[52,144,0.3611,0.50893,0.15126,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,12,0,0,15,0,0,0,0,0,1,0,1],[56,144,0.3889,0.48658,0.14663,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,16,0,0,10,0,0,0,0,0,1,0,1],[60,144,0.4167,0.51561,0.18275,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,1,16,0,0,9,0,0,0,0,0,2,0,2],[64,144,0.4444,0.50893,0.15126,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,17,0,0,10,0,0,0,0,0,2,0,1],[68,144,0.4722,0.50446,0.15561,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,14,0,0,12,0,0,1,0,0,1,0,1],[72,144,0.5,0.46874,0.12491,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,19,0,0,10,0,0,0,0,0,1,0,0],[76,144,0.5278,0.4911,0.14696,0.42857,0.42857,0.57111,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,22,0,0,7,0,0,0,0,0,0,0,2],[80,144,0.5556,0.48214,0.09942,0.42857,0.42857,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,20,0,0,10,0,0,0,0,0,1,0,0],[84,144,0.5833,0.49998,0.19231,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,17,0,0,5,0,0,2,0,0,3,0,1],[88,144,0.6111,0.45535,0.0906,0.42857,0.42857,0.4642,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,21,0,0,7,0,0,1,0,0,0,0,0],[92,144,0.6389,0.46426,0.07139,0.42857,0.42857,0.42858,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0],[96,144,0.6667,0.51786,0.17405,0.42857,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,18,0,0,8,0,0,0,0,0,2,0,2],[100,144,0.6944,0.48661,0.12299,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,19,0,0,9,0,0,0,0,0,2,0,0],[104,144,0.7222,0.5134,0.15915,0.42857,0.42859,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,19,0,0,9,0,0,0,0,0,1,0,2],[108,144,0.75,0.51783,0.09941,0.42857,0.571,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,15,0,0,15,0,0,1,0,0,1,0,0],[112,144,0.7778,0.5357,0.16366,0.42857,0.57121,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,13,0,0,13,0,0,1,0,0,1,0,2],[116,144,0.8056,0.45089,0.10779,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,1,0,0,19,0,0,10,0,0,0,0,0,0,0,0],[120,144,0.8333,0.48214,0.13716,0.42857,0.42857,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,14,0,0,11,0,0,0,0,0,2,0,0],[124,144,0.8611,0.52231,0.17717,0.42857,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,18,0,0,7,0,0,1,0,0,2,0,2],[128,144,0.8889,0.5357,0.16751,0.42857,0.42859,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,17,0,0,8,0,0,3,0,0,1,0,2],[132,144,0.9167,0.48214,0.15047,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,16,0,0,9,0,0,2,0,0,0,0,1],[136,144,0.9444,0.47767,0.1365,0.42857,0.42857,0.4642,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,22,0,0,6,0,0,0,0,0,1,0,1],[140,144,0.9722,0.50892,0.16727,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,16,0,0,8,0,0,1,0,0,4,0,0],[144,144,1.0,0.4174,0.10123,0.42857,0.42857,0.42857,0.071,0.57143,0,0,0,0,1,0,0,0,5,0,0,21,0,0,5,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.36161,"x":0.60714,"p":[[0,62,0.0,0.45982,0.1461,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,14,0,0,12,0,0,0,0,0,1,0,0],[4,62,0.0645,0.5357,0.12371,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,17,0,0,0,0,0,1,0,1],[8,62,0.129,0.52232,0.17353,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,7,0,0,19,0,0,0,0,0,0,0,2],[12,62,0.1935,0.51338,0.16311,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,8,0,0,18,0,0,0,0,0,1,0,1],[16,62,0.2581,0.58481,0.18681,0.42859,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,8,0,0,17,0,0,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A5 (UKR) Let $\\mathbb{R}$ be the set of real numbers. Does there exist a function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ that simultaneously satisfies the following three conditions? (a) There is a positive number $M$ such that $-M \\leq f(x) \\leq M$ for all $x$. (b) $f(1)=1$. (c) If $x \\neq 0$, then  $$ f\\left(x+\\frac{1}{x^{2}}\\right)=f(x)+\\left[f\\left(\\frac{1}{x}\\right)\\right]^{2} $$","t":[{"b":1,"e":0.42857,"k":"falling","v":0.18748,"x":0.67857,"p":[[0,100,0.0,0.51785,0.33071,0.14286,0.57143,0.85714,0.0,1.0,3,4,1,3,0,6,0,0,2,0,0,4,0,0,5,0,0,2,0,0,6,0,4],[4,100,0.04,0.674,0.39336,0.14286,0.92857,1.0,0.0,1.0,2,16,0,2,0,8,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,16],[8,100,0.08,0.52679,0.43071,0.14286,0.5,1.0,0.0,1.0,6,12,0,6,0,8,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,12],[12,100,0.12,0.66516,0.34922,0.39286,0.85714,1.0,0.0,1.0,2,11,0,2,0,4,0,0,2,0,0,2,0,0,4,0,0,0,0,0,7,0,11],[16,100,0.16,0.56687,0.38223,0.14286,0.57143,1.0,0.0,1.0,2,11,0,2,0,9,0,0,1,0,0,3,0,0,2,0,0,2,0,0,2,0,11],[20,100,0.2,0.67857,0.34626,0.35714,0.78571,1.0,0.14286,1.0,0,13,0,0,0,8,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,13],[24,100,0.24,0.45087,0.35012,0.14286,0.42857,0.85714,0.0,1.0,5,4,0,5,0,7,0,0,3,0,0,3,0,0,4,0,0,1,0,0,5,0,4],[28,100,0.28,0.53572,0.37796,0.14286,0.57144,0.89286,0.0,1.0,3,8,0,3,0,8,0,0,3,0,0,2,0,0,0,0,0,4,0,0,4,0,8],[32,100,0.32,0.65625,0.33285,0.42857,0.71429,1.0,0.0,1.0,2,11,0,2,0,3,0,0,1,0,0,5,0,0,4,0,0,2,0,0,4,0,11],[36,100,0.36,0.56247,0.37104,0.14286,0.64264,0.89286,0.0,1.0,5,8,0,5,0,4,0,0,1,0,0,4,0,0,2,0,0,4,0,0,4,0,8],[40,100,0.4,0.65624,0.36397,0.39286,0.85714,1.0,0.0,1.0,2,13,0,2,0,5,0,0,1,0,0,4,0,0,2,0,0,1,0,0,4,0,13],[44,100,0.44,0.43746,0.40552,0.0,0.28571,0.85714,0.0,1.0,10,5,0,10,0,5,0,0,2,0,0,0,0,0,3,0,0,0,0,0,7,0,5],[48,100,0.48,0.49107,0.36759,0.14286,0.42857,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,5,0,0,6,0,0,0,0,0,1,0,0,4,0,7],[52,100,0.52,0.54464,0.3787,0.14286,0.5,0.89286,0.0,1.0,3,8,0,3,0,7,0,0,4,0,0,2,0,0,1,0,0,1,0,0,6,0,8],[56,100,0.56,0.47765,0.35999,0.14286,0.35714,0.85714,0.0,1.0,3,5,0,3,0,9,0,0,4,0,0,2,0,0,2,0,0,1,0,0,6,0,5],[60,100,0.6,0.44643,0.37415,0.14286,0.42857,0.85714,0.0,1.0,6,5,0,6,0,8,0,0,1,0,0,5,0,0,0,0,0,2,0,0,5,0,5],[64,100,0.64,0.5625,0.3443,0.28571,0.50001,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,3,0,0,7,0,0,2,0,0,2,0,0,5,0,7],[68,100,0.68,0.53571,0.3677,0.14286,0.42857,1.0,0.0,1.0,2,9,0,2,0,8,0,0,3,0,0,4,0,0,2,0,0,1,0,0,3,0,9],[72,100,0.72,0.53572,0.40721,0.14286,0.57143,1.0,0.0,1.0,4,10,0,4,0,10,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,10],[76,100,0.76,0.43303,0.36155,0.14286,0.35714,0.75,0.0,1.0,4,6,0,4,0,11,0,0,1,0,0,4,0,0,2,0,0,2,0,0,2,0,6],[80,100,0.8,0.34365,0.34047,0.14214,0.2143,0.50002,0.0,1.0,7,3,0,7,0,9,0,0,6,0,0,2,0,0,0,0,0,1,0,0,4,0,3],[84,100,0.84,0.33031,0.30182,0.0,0.28571,0.571,0.0,1.0,9,1,0,9,0,6,0,0,2,0,0,6,0,0,4,0,0,1,0,0,3,0,1],[88,100,0.88,0.34375,0.28763,0.14286,0.2857,0.57143,0.0,1.0,5,2,0,5,0,10,0,0,3,0,0,5,0,0,4,0,0,2,0,0,1,0,2],[92,100,0.92,0.33929,0.2714,0.14286,0.2857,0.57143,0.0,0.85714,4,0,0,4,0,11,0,0,4,0,0,4,0,0,4,0,0,1,0,0,4,0,0],[96,100,0.96,0.23658,0.18761,0.14286,0.14286,0.32143,0.0,0.71429,5,0,0,5,0,14,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[100,100,1.0,0.18748,0.20336,0.0,0.14286,0.32143,0.0,0.714,12,0,0,12,0,10,0,0,2,0,0,5,0,0,2,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.10268,"x":0.74107,"p":[[0,93,0.0,0.50892,0.34057,0.25001,0.42857,0.85714,0.0,1.0,4,6,2,4,0,4,0,0,3,0,0,7,0,0,2,0,0,3,0,0,3,0,6],[4,93,0.043,0.6875,0.35434,0.28571,0.85714,1.0,0.0,1.0,1,15,0,1,0,4,0,0,4,0,0,2,0,0,2,0,0,1,0,0,3,0,15],[8,93,0.086,0.66518,0.38401,0.25,0.92857,1.0,0.0,1.0,1,16,0,1,0,7,0,0,3,0,0,1,0,0,1,0,0,1,0,0,2,0,16],[12,93,0.129,0.58479,0.3542,0.24999,0.57121,1.0,0.0,1.0,1,10,0,1,0,7,0,0,3,0,0,3,0,0,4,0,0,1,0,0,3,0,10],[16,93,0.172,0.53124,0.39807,0.14286,0.42857,1.0,0.0,1.0,3,11,0,3,0,9,0,0,3,0,0,2,0,0,1,0,0,1,0,0,2,0,11],[20,93,0.2151,0.74107,0.34151,0.39286,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,3,0,0,1,0,0,1,0,0,1,0,0,4,0,17],[24,93,0.2581,0.70088,0.34323,0.39286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,6,0,0,2,0,0,2,0,0,1,0,0,4,0,0,2,0,15],[28,93,0.3011,0.54899,0.34657,0.14286,0.49979,0.89286,0.14,1.0,0,8,0,0,0,9,0,0,4,0,0,3,0,0,3,0,0,1,0,0,4,0,8],[32,93,0.3441,0.57143,0.37796,0.24999,0.42857,1.0,0.0,1.0,2,12,0,2,0,6,0,0,4,0,0,6,0,0,0,0,0,0,0,0,2,0,12],[36,93,0.3871,0.60714,0.40406,0.14286,0.78571,1.0,0.0,1.0,1,15,0,1,0,10,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,15],[40,93,0.4301,0.66072,0.4222,0.14286,1.0,1.0,0.0,1.0,2,18,0,2,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,18],[44,93,0.4731,0.59821,0.38868,0.14286,0.64286,1.0,0.0,1.0,2,13,0,2,0,7,0,0,4,0,0,1,0,0,2,0,0,1,0,0,2,0,13],[48,93,0.5161,0.6116,0.39161,0.14286,0.85707,1.0,0.0,1.0,1,12,0,1,0,10,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,12],[52,93,0.5591,0.47758,0.34192,0.14286,0.35714,0.85714,0.0,1.0,1,7,0,1,0,9,0,0,6,0,0,4,0,0,2,0,0,1,0,0,2,0,7],[56,93,0.6022,0.58036,0.35344,0.28571,0.64286,1.0,0.0,1.0,2,9,0,2,0,5,0,0,5,0,0,2,0,0,2,0,0,4,0,0,3,0,9],[60,93,0.6452,0.54911,0.35555,0.25,0.57143,1.0,0.0,1.0,1,9,0,1,0,7,0,0,7,0,0,1,0,0,0,0,0,6,0,0,1,0,9],[64,93,0.6882,0.61606,0.34151,0.2857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,4,0,0,4,0,0,3,0,0,1,0,0,3,0,11],[68,93,0.7312,0.59372,0.39303,0.14286,0.64264,1.0,0.0,1.0,1,13,0,1,0,11,0,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,13],[72,93,0.7742,0.62932,0.38106,0.28571,0.71214,1.0,0.0,1.0,3,14,0,3,0,4,0,0,4,0,0,1,0,0,3,0,0,2,0,0,1,0,14],[76,93,0.8172,0.55804,0.36133,0.24999,0.57143,1.0,0.0,1.0,3,10,0,3,0,5,0,0,3,0,0,4,0,0,4,0,0,2,0,0,1,0,10],[80,93,0.8602,0.58927,0.37923,0.14286,0.71429,1.0,0.0,1.0,3,9,0,3,0,7,0,0,2,0,0,0,0,0,3,0,0,2,0,0,6,0,9],[84,93,0.9032,0.50446,0.33879,0.14286,0.42857,0.85714,0.0,1.0,3,6,0,3,0,6,0,0,3,0,0,5,0,0,4,0,0,2,0,0,3,0,6],[88,93,0.9462,0.49107,0.39113,0.14286,0.35714,0.85714,0.0,1.0,6,7,0,6,0,6,0,0,4,0,0,1,0,0,0,0,0,4,0,0,4,0,7],[92,93,0.9892,0.10268,0.14827,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,13,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[93,93,1.0,0.11161,0.24932,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1]]}]},{"i":"876444a725502f65","q":"A convex polygon on the plane is called wide if the projection of the polygon onto any line in the same plane is a segment with length at least 1 . Prove that a circle of radius $\\frac{1}{3}$ can be placed completely inside any wide polygon.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.11607,"x":0.35268,"p":[[0,26,0.0,0.125,0.13716,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,11,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.29464,0.24468,0.14286,0.2143,0.4286,0.0,0.85714,6,0,0,6,0,10,0,0,3,0,0,7,0,0,3,0,0,1,0,0,2,0,0],[8,26,0.3077,0.26337,0.21457,0.14286,0.2857,0.32143,0.0,0.85714,6,0,0,6,0,9,0,0,9,0,0,3,0,0,3,0,0,1,0,0,1,0,0],[12,26,0.4615,0.30804,0.15198,0.25,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,6,0,0,12,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[16,26,0.6154,0.32588,0.21198,0.14286,0.42857,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,5,0,0,11,0,0,4,0,0,2,0,0,0,0,0],[20,26,0.7692,0.35268,0.18552,0.24999,0.35714,0.42857,0.0,0.85714,1,0,0,1,0,7,0,0,8,0,0,12,0,0,1,0,0,2,0,0,1,0,0],[24,26,0.9231,0.16072,0.16269,0.0,0.14286,0.1786,0.0,0.71429,10,0,0,10,0,14,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[26,26,1.0,0.11607,0.13092,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,15,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.125,"x":0.39731,"p":[[0,22,0.0,0.125,0.14174,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,18,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.3884,0.22371,0.24999,0.42857,0.4286,0.0,0.85714,3,0,0,3,0,5,0,0,4,0,0,13,0,0,1,0,0,5,0,0,1,0,0],[8,22,0.3636,0.39731,0.21048,0.2857,0.42857,0.4642,0.0,0.71429,4,0,0,4,0,2,0,0,4,0,0,14,0,0,3,0,0,5,0,0,0,0,0],[12,22,0.5455,0.36159,0.24216,0.14286,0.42857,0.42858,0.0,0.85714,4,0,0,4,0,7,0,0,3,0,0,11,0,0,2,0,0,3,0,0,2,0,0],[16,22,0.7273,0.22768,0.20159,0.0,0.14288,0.42857,0.0,0.71429,10,0,0,10,0,7,0,0,4,0,0,9,0,0,1,0,0,1,0,0,0,0,0],[20,22,0.9091,0.22322,0.21706,0.0,0.14286,0.42857,0.0,0.71429,9,0,0,9,0,11,0,0,3,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[22,22,1.0,0.14723,0.17672,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,12,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"123d8384d7107fd1","q":"A convex polygon $\\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.55804,"x":0.74552,"p":[[0,31,0.0,0.74552,0.24154,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,3,0,0,7,0,10],[4,31,0.129,0.72767,0.24318,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,3,0,0,10,0,0,1,0,0,6,0,10],[8,31,0.2581,0.66513,0.24383,0.42857,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,6,0,0,10,0,0,1,0,0,4,0,8],[12,31,0.3871,0.65625,0.25218,0.42857,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,1,0,0,4,0,8],[16,31,0.5161,0.57589,0.25123,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,5,0,0,6,0,0,7,0,0,6,0,0,1,0,5],[20,31,0.6452,0.55804,0.24317,0.42857,0.5,0.60714,0.14286,1.0,0,5,0,0,0,1,0,0,5,0,0,10,0,0,8,0,0,1,0,0,2,0,5],[24,31,0.7742,0.63389,0.25986,0.42857,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,7,0,0,8,0,0,1,0,0,3,0,8],[28,31,0.9032,0.64731,0.23416,0.53539,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,3,0,0,9,0,0,6,0,0,3,0,6],[31,31,1.0,0.69634,0.22803,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,10,0,0,8,0,0,2,0,8]]},{"b":6,"e":0.14286,"k":"falling","v":0.30358,"x":0.67409,"p":[[0,41,0.0,0.65179,0.25489,0.42857,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,8,0,0,6,0,0,1,0,0,6,0,7],[4,41,0.0976,0.62946,0.23381,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,9,0,0,10,0,0,0,0,0,6,0,5],[8,41,0.1951,0.62054,0.23854,0.42857,0.57143,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,6,0,0,8,0,0,5,0,0,2,0,6],[12,41,0.2927,0.67409,0.25313,0.42857,0.64286,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,9,0,0,4,0,0,1,0,0,8,0,7],[16,41,0.3902,0.65173,0.20808,0.42857,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,5,0,0,4,0,5],[20,41,0.4878,0.66071,0.21943,0.53572,0.71429,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,9,0,0,2,0,6],[24,41,0.5854,0.59374,0.2055,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,11,0,0,7,0,0,5,0,0,5,0,2],[28,41,0.6829,0.50893,0.24984,0.28571,0.42857,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,6,0,0,9,0,0,4,0,0,6,0,0,1,0,3],[32,41,0.7805,0.47765,0.21608,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,8,0,0,7,0,0,7,0,0,4,0,0,2,0,1],[36,41,0.878,0.41963,0.24983,0.2857,0.35714,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,9,0,0,5,0,0,5,0,0,2,0,0,3,0,1],[40,41,0.9756,0.35266,0.20197,0.14286,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,6,0,0,6,0,0,10,0,0,4,0,0,3,0,0,0,0,0],[41,41,1.0,0.30358,0.15871,0.14289,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,12,0,0,6,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"18a048a24d21d704","q":"A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.\n\n*(5 points)*","t":[{"b":5,"e":0.0,"k":"falling","v":0.04017,"x":0.61161,"p":[[0,36,0.0,0.61161,0.46871,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,18],[4,36,0.1111,0.26786,0.42371,0.0,0.0,0.71429,0.0,1.0,22,7,0,22,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,7],[8,36,0.2222,0.28125,0.39687,0.0,0.0,0.71429,0.0,1.0,19,5,0,19,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,5],[12,36,0.3333,0.33926,0.38917,0.0,0.0,0.71429,0.0,1.0,17,3,0,17,0,1,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,3],[16,36,0.4444,0.11161,0.26422,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[20,36,0.5556,0.21427,0.3642,0.0,0.0,0.24989,0.0,1.0,22,4,0,22,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,4],[24,36,0.6667,0.07579,0.19874,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[28,36,0.7778,0.04017,0.15659,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[32,36,0.8889,0.09821,0.22428,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[36,36,1.0,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.12058,"x":0.38393,"p":[[0,24,0.0,0.37946,0.46786,0.0,0.0,1.0,0.0,1.0,18,11,1,18,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,11],[4,24,0.1667,0.38393,0.44669,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,8],[8,24,0.3333,0.32589,0.45769,0.0,0.0,1.0,0.0,1.0,20,10,0,20,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[12,24,0.5,0.12058,0.29695,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[16,24,0.6667,0.24554,0.37156,0.0,0.0,0.71429,0.0,1.0,20,3,0,20,0,3,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,3],[20,24,0.8333,0.26783,0.35846,0.0,0.0,0.60682,0.0,1.0,18,3,0,18,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,0,0,0,3],[24,24,1.0,0.25,0.32341,0.0,0.0,0.71429,0.0,0.71429,19,0,0,19,0,1,0,0,1,0,0,0,0,0,2,0,0,9,0,0,0,0,0]]}]},{"i":"54f604b9e484dc4d","q":"<span style=\"color:darkred\"> Let  $(R,+,\\cdot)$  be a ring and let  $f$  be a surjective endomorphism of  $R$  such that  $[x,f(x)]=0$  for any  $x\\in R$  , where  $[a,b]=ab-ba$  ,  $a,b\\in R$  . Prove that:\n\n[list]**a)**  $[x,f(y)]=[f(x),y]$  and  $x[x,y]=f(x)[x,y]$  , for any  $x,y\\in R\\ ;$ **b)** If  $R$  is a division ring and  $f$  is different from the identity function, then  $R$  is commutative.  \n[/list]\n</span>","t":[{"b":0,"e":0.71429,"k":"rising","v":0.41947,"x":0.78124,"p":[[0,162,0.0,0.41947,0.24979,0.24999,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,6,0,0,5,0,0,9,0,0,3,0,0,3,0,0,4,0,0],[4,162,0.0247,0.67857,0.27894,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,2,0,0,7,0,0,7,0,0,1,0,10],[8,162,0.0494,0.75444,0.27021,0.57143,0.78571,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,0,0,0,9,0,0,4,0,0,2,0,14],[12,162,0.0741,0.69197,0.23719,0.53572,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,6,0,0,4,0,8],[16,162,0.0988,0.69194,0.24773,0.57143,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,3,0,0,12,0,0,4,0,0,1,0,10],[20,162,0.1235,0.58482,0.31003,0.28571,0.57143,0.89275,0.0,1.0,1,8,0,1,0,4,0,0,4,0,0,3,0,0,7,0,0,4,0,0,1,0,8],[24,162,0.1481,0.72767,0.24318,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,12,0,0,5,0,0,0,0,12],[28,162,0.1728,0.71873,0.23004,0.57143,0.64271,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,14,0,0,5,0,0,0,0,11],[32,162,0.1975,0.73213,0.28065,0.57143,0.71429,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,3,0,0,6,0,0,6,0,0,0,0,14],[36,162,0.2222,0.72322,0.23402,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,2,0,0,14,0,0,3,0,0,0,0,12],[40,162,0.2469,0.74552,0.25688,0.57143,0.78564,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,3,0,0,3,0,13],[44,162,0.2716,0.76338,0.22193,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,5,0,0,3,0,12],[48,162,0.2963,0.75892,0.22142,0.57143,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,5,0,0,1,0,13],[52,162,0.321,0.75004,0.22297,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,5,0,0,2,0,12],[56,162,0.3457,0.74106,0.22991,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,3,0,0,4,0,11],[60,162,0.3704,0.73661,0.22048,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,7,0,0,3,0,10],[64,162,0.3951,0.75444,0.22934,0.57143,0.78564,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,2,0,0,4,0,12],[68,162,0.4198,0.71414,0.23147,0.57132,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,9,0,0,1,0,10],[72,162,0.4444,0.74997,0.22589,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,6,0,0,2,0,12],[76,162,0.4691,0.74103,0.20344,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,8,0,0,0,0,11],[80,162,0.4938,0.67856,0.272,0.57132,0.64286,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,2,0,0,9,0,0,4,0,0,2,0,10],[84,162,0.5185,0.76337,0.21612,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,5,0,0,3,0,12],[88,162,0.5432,0.76339,0.23584,0.57143,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,6,0,0,2,0,13],[92,162,0.5679,0.67411,0.22083,0.57143,0.57143,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,6,0,0,2,0,7],[96,162,0.5926,0.69195,0.29039,0.57132,0.71429,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,0,0,0,3,0,0,8,0,0,4,0,0,1,0,12],[100,162,0.6173,0.71865,0.28023,0.57143,0.71429,1.0,0.14,1.0,0,13,0,0,0,3,0,0,1,0,0,1,0,0,9,0,0,4,0,0,1,0,13],[104,162,0.642,0.78124,0.21125,0.57143,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,0,3,0,13],[108,162,0.6667,0.73659,0.20551,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,6,0,0,4,0,9],[112,162,0.6914,0.71874,0.22725,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,7,0,0,3,0,9],[116,162,0.716,0.70981,0.23552,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,0,0,0,9,0,0,9,0,0,3,0,8],[120,162,0.7407,0.7589,0.24858,0.57143,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,4,0,0,2,0,14],[124,162,0.7654,0.68746,0.29547,0.53539,0.64286,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,2,0,0,3,0,0,8,0,0,3,0,0,0,0,13],[128,162,0.7901,0.77676,0.21708,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,2,0,0,4,0,13],[132,162,0.8148,0.6741,0.2321,0.57143,0.71429,0.85704,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,8,0,0,9,0,0,2,0,7],[136,162,0.8395,0.76339,0.2412,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,1,0,0,3,0,14],[140,162,0.8642,0.70534,0.24985,0.57132,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,4,0,0,5,0,9],[144,162,0.8889,0.60714,0.29233,0.39286,0.64286,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,5,0,0,4,0,0,4,0,0,7,0,0,2,0,7],[148,162,0.9136,0.77232,0.21387,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,7,0,0,3,0,12],[152,162,0.9383,0.64729,0.21425,0.57143,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,13,0,0,8,0,0,0,0,6],[156,162,0.963,0.60719,0.21125,0.42857,0.57143,0.75,0.1429,1.0,0,2,0,0,0,1,0,0,2,0,0,8,0,0,8,0,0,5,0,0,6,0,2],[160,162,0.9877,0.65604,0.24185,0.57075,0.71214,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,0,0,0,4,0,0,8,0,0,7,0,0,5,0,5],[162,162,1.0,0.58036,0.25489,0.42857,0.50001,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,11,0,0,7,0,0,0,0,0,3,0,6]]},{"b":5,"e":0.57143,"k":"flat","v":0.375,"x":0.74557,"p":[[0,222,0.0,0.375,0.21943,0.2857,0.28571,0.46429,0.0,1.0,2,1,0,2,0,5,0,0,10,0,0,7,0,0,4,0,0,3,0,0,0,0,1],[4,222,0.018,0.70534,0.30081,0.57143,0.71429,1.0,0.0,1.0,1,13,0,1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.71429,0.0,1.0,1,6,0,1,0,1,0,0,4,0,0,9,0,0,6,0,0,4,0,0,1,0,6],[188,222,0.8468,0.50435,0.27901,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,6,0,0,6,0,0,6,0,0,3,0,0,2,0,4],[192,222,0.8649,0.46427,0.25,0.28571,0.42859,0.71429,0.0,1.0,3,1,0,3,0,2,0,0,6,0,0,6,0,0,6,0,0,7,0,0,1,0,1],[196,222,0.8829,0.54017,0.28734,0.28571,0.50001,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,4,0,0,6,0,0,2,0,0,7,0,0,3,0,4],[200,222,0.9009,0.5981,0.302,0.28571,0.57143,1.0,0.14,1.0,0,9,0,0,0,3,0,0,7,0,0,3,0,0,5,0,0,5,0,0,0,0,9],[204,222,0.9189,0.57143,0.27893,0.39286,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,3,0,0,7,0,0,0,0,0,10,0,0,3,0,4],[208,222,0.9369,0.56695,0.26362,0.42857,0.57121,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,3,0,0,9,0,0,4,0,0,6,0,0,3,0,4],[212,222,0.955,0.4598,0.28511,0.14289,0.42857,0.71429,0.0,1.0,2,2,0,2,0,7,0,0,3,0,0,6,0,0,5,0,0,4,0,0,3,0,2],[216,222,0.973,0.48656,0.25964,0.25,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,2,0,0,7,0,0,5,0,0,5,0,0,4,0,1],[220,222,0.991,0.46417,0.22314,0.2857,0.42859,0.60714,0.14,1.0,0,1,0,0,0,5,0,0,6,0,0,8,0,0,5,0,0,6,0,0,1,0,1],[222,222,1.0,0.3839,0.28219,0.14286,0.42857,0.57111,0.0,1.0,4,2,0,4,0,8,0,0,3,0,0,7,0,0,4,0,0,3,0,0,1,0,2]]}]},{"i":"2177429eff4dc1d4","q":"Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:\n\n\n- No two numbers are equal.\n- If a number  $m$  is placed on square  $C$ , then at least  $k$  of the squares orthogonally adjacent to  $C$  have a multiple of  $m$  written on them.\n\nWhat is the greatest value of  $k$  for which this is possible?","t":[{"b":0,"e":0.571,"k":"rising","v":0.37051,"x":0.59812,"p":[[0,36,0.0,0.37051,0.2365,0.14286,0.4998,0.57143,0.0,0.85714,3,0,3,3,0,10,0,0,2,0,0,1,0,0,15,0,0,0,0,0,1,0,0],[4,36,0.1111,0.49106,0.23673,0.42857,0.57143,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,0,0,0,3,0,0,18,0,0,1,0,0,2,0,1],[8,36,0.2222,0.58032,0.21998,0.571,0.57143,0.57143,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,3,0,0,19,0,0,3,0,0,0,0,4],[12,36,0.3333,0.52672,0.19044,0.5354,0.57143,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,4,0,0,22,0,0,0,0,0,0,0,2],[16,36,0.4444,0.49097,0.23686,0.42857,0.57143,0.57143,0.0,1.0,3,2,0,3,0,3,0,0,0,0,0,4,0,0,19,0,0,1,0,0,0,0,2],[20,36,0.5556,0.50446,0.1636,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,5,0,0,21,0,0,0,0,0,0,0,1],[24,36,0.6667,0.54014,0.18808,0.5354,0.57143,0.57143,0.0,1.0,2,2,0,2,0,0,0,0,0,0,0,6,0,0,21,0,0,1,0,0,0,0,2],[28,36,0.7778,0.54911,0.1017,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,5,0,0,23,0,0,3,0,0,0,0,0],[32,36,0.8889,0.59812,0.13572,0.5713,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,26,0,0,0,0,0,0,0,3],[36,36,1.0,0.56246,0.0497,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"rising","v":0.35713,"x":0.52677,"p":[[0,36,0.0,0.35713,0.2812,0.14286,0.35714,0.57143,0.0,1.0,7,1,5,7,0,7,0,0,2,0,0,1,0,0,11,0,0,3,0,0,0,0,1],[4,36,0.1111,0.52677,0.23265,0.42859,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,6,0,0,0,0,0,3,0,0,19,0,0,0,0,0,1,0,3],[8,36,0.2222,0.4241,0.26602,0.14286,0.49979,0.57143,0.0,1.0,3,2,0,3,0,8,0,0,0,0,0,5,0,0,12,0,0,2,0,0,0,0,2],[12,36,0.3333,0.45086,0.26026,0.14286,0.5712,0.57143,0.0,1.0,3,2,0,3,0,6,0,0,0,0,0,5,0,0,15,0,0,0,0,0,1,0,2],[16,36,0.4444,0.49103,0.22569,0.42857,0.57143,0.57143,0.0,1.0,2,2,0,2,0,3,0,0,2,0,0,4,0,0,18,0,0,1,0,0,0,0,2],[20,36,0.5556,0.38384,0.2587,0.14286,0.42857,0.57143,0.0,1.0,4,2,0,4,0,7,0,0,2,0,0,7,0,0,10,0,0,0,0,0,0,0,2],[24,36,0.6667,0.50445,0.2624,0.42859,0.57143,0.57143,0.0,1.0,3,3,0,3,0,4,0,0,0,0,0,2,0,0,18,0,0,2,0,0,0,0,3],[28,36,0.7778,0.49551,0.24739,0.42857,0.57143,0.57143,0.0,1.0,4,2,0,4,0,2,0,0,0,0,0,3,0,0,19,0,0,2,0,0,0,0,2],[32,36,0.8889,0.49552,0.19227,0.42857,0.57143,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,1,0,0,6,0,0,21,0,0,0,0,0,0,0,1],[36,36,1.0,0.50889,0.18188,0.53539,0.57143,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,1,0,0,4,0,0,23,0,0,0,0,0,0,0,1]]}]},{"i":"e8394a2a59bc0ad9","q":"Consider a polynomial $P(x)=\\left(x+d_{1}\\right)\\left(x+d_{2}\\right) \\cdot \\ldots \\cdot\\left(x+d_{9}\\right)$, where $d_{1}, d_{2}, \\ldots, d_{9}$ are nine distinct integers. Prove that there exists an integer $N$ such that for all integers $x \\geq N$ the number $P(x)$ is divisible by a prime number greater than 20 .","t":[{"b":6,"e":0.0,"k":"flat","v":0.06688,"x":0.15161,"p":[[0,48,0.0,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.12937,0.05484,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.1384,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.15161,0.13335,0.14214,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,22,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,48,0.5,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.11143,0.06894,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.11598,0.06618,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.06688,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.11161,"x":0.14491,"p":[[0,34,0.0,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.14491,0.05783,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,1,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.12929,0.07454,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.2857,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e4f5bfd0c6200f52","q":"Consider a tree with  $n$  vertices, labeled with  $1,\\ldots,n$  in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action  $n-1$  times, we get another tree with its labeling a permutation of the first graph's labeling.\nProve that this permutation contains exactly one cycle.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.19197,"x":0.54018,"p":[[0,29,0.0,0.36161,0.32924,0.14286,0.2857,0.42858,0.0,1.0,5,5,2,5,0,10,0,0,4,0,0,6,0,0,0,0,0,2,0,0,0,0,5],[4,29,0.1379,0.54018,0.37412,0.14286,0.42857,1.0,0.14286,1.0,0,12,0,0,0,11,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,12],[8,29,0.2759,0.53563,0.39618,0.14286,0.42857,1.0,0.0,1.0,1,13,0,1,0,11,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,13],[12,29,0.4138,0.40625,0.36265,0.14286,0.14286,0.57143,0.0,1.0,2,8,0,2,0,15,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,8],[16,29,0.5517,0.31241,0.26836,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,18,0,0,1,0,0,8,0,0,0,0,0,1,0,0,0,0,3],[20,29,0.6897,0.33928,0.33455,0.14286,0.14286,0.42857,0.0,1.0,2,6,0,2,0,18,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,6],[24,29,0.8276,0.26321,0.26279,0.14286,0.14286,0.32142,0.0,1.0,2,3,0,2,0,21,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[28,29,0.9655,0.34822,0.2765,0.14286,0.28571,0.42857,0.0,1.0,1,4,0,1,0,13,0,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,4],[29,29,1.0,0.19197,0.11633,0.14286,0.14286,0.1786,0.0,0.4286,2,0,0,2,0,22,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"volatile","v":0.35267,"x":0.66076,"p":[[0,15,0.0,0.35267,0.36766,0.14286,0.14286,0.4642,0.0,1.0,6,7,3,6,0,13,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,7],[4,15,0.2667,0.66076,0.3792,0.28571,1.0,1.0,0.0,1.0,1,17,0,1,0,5,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,17],[8,15,0.5333,0.41955,0.36592,0.14286,0.28571,0.78571,0.0,1.0,3,8,0,3,0,12,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,8],[12,15,0.8,0.52232,0.37561,0.14286,0.42857,1.0,0.0,1.0,1,11,0,1,0,9,0,0,5,0,0,5,0,0,0,0,0,0,0,0,1,0,11],[15,15,1.0,0.41518,0.356,0.14286,0.21429,0.57143,0.0,1.0,1,8,0,1,0,15,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,8]]}]},{"i":"562cab32d39c26e8","q":"7. A1 (USA) ${ }^{\\mathrm{IMO} 2}$ Let $a, b, c$ be positive real numbers with product 1. Prove that  $$ \\left(a-1+\\frac{1}{b}\\right)\\left(b-1+\\frac{1}{c}\\right)\\left(c-1+\\frac{1}{a}\\right) \\leq 1 . $$","t":[{"b":3,"e":0.14286,"k":"flat","v":0.11598,"x":0.1875,"p":[[0,32,0.0,0.11598,0.08325,0.105,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.1875,0.13092,0.14286,0.14286,0.17857,0.0,0.42857,4,0,0,4,0,20,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.15616,0.0827,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,27,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.16964,0.13092,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,25,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[16,32,0.5,0.13393,0.11259,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,18,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.11599,0.10372,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,22,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.15625,0.13533,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,21,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.12492,"x":0.17411,"p":[[0,23,0.0,0.12492,0.07782,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.14732,0.12619,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,25,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,23,0.3478,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.125,0.05922,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.17411,0.08552,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.1517,0.09408,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,24,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.15178,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ccc464f6d531d4b3","q":"A box whose shape is a parallelepiped can be completely filled with cubes of side  $1.$  If we put in it the maximum possible number of cubes, each of volume  $2$ , with the sides parallel to those of the box, then exactly  $40$  percent of the volume of the box is occupied. Determine the possible dimensions of the box.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.73213,"x":1.0,"p":[[0,138,0.0,0.87054,0.24836,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[4,138,0.029,0.92857,0.16366,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[8,138,0.058,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[12,138,0.087,0.96427,0.08754,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,138,0.1159,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[20,138,0.1449,0.91963,0.13807,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[24,138,0.1739,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,138,0.2029,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,138,0.2319,0.94196,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[36,138,0.2609,0.91964,0.19212,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,23],[40,138,0.2899,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[44,138,0.3188,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[48,138,0.3478,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,138,0.3768,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,27],[56,138,0.4058,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,138,0.4348,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[64,138,0.4638,0.91518,0.18509,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,24],[68,138,0.4928,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[72,138,0.5217,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,138,0.5507,0.96427,0.12376,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[80,138,0.5797,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[84,138,0.6087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,138,0.6377,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[92,138,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,138,0.6957,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26],[100,138,0.7246,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,138,0.7536,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[108,138,0.7826,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[112,138,0.8116,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[116,138,0.8406,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[120,138,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,138,0.8986,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[128,138,0.9275,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[132,138,0.9565,0.93304,0.17852,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,25],[136,138,0.9855,0.77677,0.23674,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,0,4,0,13],[138,138,1.0,0.73213,0.27375,0.5354,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,5,0,0,3,0,0,6,0,0,2,0,13]]},{"b":5,"e":1.0,"k":"flat","v":0.83032,"x":0.97321,"p":[[0,222,0.0,0.83032,0.2327,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,2,0,0,2,0,19],[4,222,0.018,0.90179,0.18707,0.85714,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,19],[8,222,0.036,0.93749,0.11263,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[12,222,0.0541,0.95089,0.15815,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[16,222,0.0721,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[20,222,0.0901,0.95982,0.12992,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[24,222,0.1081,0.93304,0.14719,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,24],[28,222,0.1261,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[32,222,0.1441,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[36,222,0.1622,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[40,222,0.1802,0.91517,0.17076,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,22],[44,222,0.1982,0.90625,0.1411,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,19],[48,222,0.2162,0.91518,0.14223,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[52,222,0.2342,0.92411,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[56,222,0.2523,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[60,222,0.2703,0.94195,0.11773,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[64,222,0.2883,0.9241,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20],[68,222,0.3063,0.94643,0.1171,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[72,222,0.3243,0.92857,0.14286,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,22],[76,222,0.3423,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[80,222,0.3604,0.92856,0.13836,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[84,222,0.3784,0.91071,0.14174,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[88,222,0.3964,0.9308,0.11776,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,1,22],[92,222,0.4144,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[96,222,0.4324,0.94871,0.09145,0.91179,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,1,23],[100,222,0.4505,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[104,222,0.4685,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21],[108,222,0.4865,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[112,222,0.5045,0.95088,0.12686,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[116,222,0.5225,0.92857,0.10714,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[120,222,0.5405,0.94642,0.09943,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[124,222,0.5586,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[128,222,0.5766,0.92187,0.10619,0.85714,1.0,1.0,0.64286,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,0,0,9,0,19],[132,222,0.5946,0.91964,0.14258,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[136,222,0.6126,0.92856,0.13366,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22],[140,222,0.6306,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[144,222,0.6486,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[148,222,0.6667,0.875,0.15047,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,15],[152,222,0.6847,0.89285,0.13363,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,16],[156,222,0.7027,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[160,222,0.7207,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[164,222,0.7387,0.92411,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20],[168,222,0.7568,0.94197,0.12807,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[172,222,0.7748,0.87945,0.18251,0.85714,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,17],[176,222,0.7928,0.86606,0.1673,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,10,0,15],[180,222,0.8108,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[184,222,0.8288,0.89286,0.13832,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,18],[188,222,0.8468,0.91518,0.14664,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,22],[192,222,0.8649,0.91964,0.14258,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[196,222,0.8829,0.89286,0.15567,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,8,0,18],[200,222,0.9009,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[204,222,0.9189,0.88169,0.14132,0.83936,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,1,0,8,0,16],[208,222,0.9369,0.85713,0.16369,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,7,0,15],[212,222,0.955,0.86607,0.15947,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,8,0,15],[216,222,0.973,0.88392,0.13092,0.85714,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,12,0,14],[220,222,0.991,0.90625,0.14555,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,20],[222,222,1.0,0.85266,0.16937,0.85708,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,13,0,12]]}]},{"i":"bfd61f875734b4ef","q":"Consider an arc  $AB$  of a circle  $C$  and a point  $P$  variable in that arc  $AB$ . Let   $D$  be the midpoint of the arc  $AP$  that doeas not contain  $B$  and let  $E$  be the midpoint of the arc  $BP$  that does not contain  $A$ . Let  $C_1$  be the circle with center  $D$  passing through  $A$  and  $C_2$  be the circle with center  $E$  passing through  $B.$  Prove that the line that contains the intersection points of  $C_1$  and  $C_2$  passes through a fixed point.","t":[{"b":3,"e":1.0,"k":"falling","v":0.51338,"x":0.90178,"p":[[0,76,0.0,0.76338,0.31056,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,2,0,0,2,0,0,2,0,0,4,0,0,2,0,17],[4,76,0.0526,0.90178,0.22429,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,26],[8,76,0.1053,0.88393,0.25364,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,25],[12,76,0.1579,0.89286,0.26487,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[16,76,0.2105,0.88393,0.28221,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[20,76,0.2632,0.82142,0.31744,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,23],[24,76,0.3158,0.86159,0.26121,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,23],[28,76,0.3684,0.88391,0.25113,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,25],[32,76,0.4211,0.82142,0.3093,0.82132,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,0,4,0,20],[36,76,0.4737,0.79911,0.25716,0.67846,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,5,0,0,2,0,17],[40,76,0.5263,0.83034,0.2243,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,17],[44,76,0.5789,0.85714,0.22868,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,5,0,0,1,0,21],[48,76,0.6316,0.81696,0.29066,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,3,0,19],[52,76,0.6842,0.80804,0.31259,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,20],[56,76,0.7368,0.7991,0.31514,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,4,0,19],[60,76,0.7895,0.80356,0.27376,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,2,0,0,2,0,19],[64,76,0.8421,0.71426,0.33504,0.28571,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,7,0,0,0,0,0,4,0,0,2,0,0,0,0,17],[68,76,0.8947,0.79462,0.29654,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,5,0,0,0,0,0,2,0,0,2,0,0,4,0,18],[72,76,0.9474,0.51338,0.34229,0.28571,0.42859,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,9,0,0,3,0,0,5,0,0,1,0,0,1,0,8],[76,76,1.0,0.60268,0.37752,0.28571,0.64286,1.0,0.0,1.0,4,12,0,4,0,1,0,0,7,0,0,3,0,0,1,0,0,1,0,0,3,0,12]]},{"b":7,"e":1.0,"k":"rising","v":0.7455,"x":0.97768,"p":[[0,67,0.0,0.7455,0.34393,0.57132,1.0,1.0,0.0,1.0,3,18,2,3,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,0,0,0,18],[4,67,0.0597,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,67,0.1194,0.88829,0.23915,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,23],[12,67,0.1791,0.94643,0.14616,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[16,67,0.2388,0.8482,0.27881,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,22],[20,67,0.2985,0.88838,0.25188,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,26],[24,67,0.3582,0.90625,0.23854,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,26],[28,67,0.4179,0.89732,0.19638,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,1,0,24],[32,67,0.4776,0.88838,0.21648,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,23],[36,67,0.5373,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[40,67,0.597,0.91964,0.20806,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,26],[44,67,0.6567,0.95981,0.13941,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[48,67,0.7164,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[52,67,0.7761,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[56,67,0.8358,0.93303,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[60,67,0.8955,0.92857,0.18898,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[64,67,0.9552,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[67,67,1.0,0.90177,0.14917,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,22]]}]},{"i":"b0114d799efe3dc7","q":"7. B1 (CAN 2) Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that $2 p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0))$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,29,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,29,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,29,0.5517,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,29,0.6897,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,29,0.8276,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.95535,"x":1.0,"p":[[0,12,0.0,0.95535,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,12,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e84329a1c5257cdd","q":"An infinite set $B$ consisting of positive integers has the following property. For each $a, b \\in B$ with $a>b$ the number $\\frac{a-b}{(a, b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here $(a, b)$ is the greatest common divisor of numbers $a$ and $b$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.3749,"x":0.63836,"p":[[0,59,0.0,0.3749,0.2545,0.14289,0.28571,0.4286,0.0,1.0,2,2,2,2,0,7,0,0,8,0,0,9,0,0,2,0,0,0,0,0,2,0,2],[4,59,0.0678,0.58924,0.24419,0.5354,0.57143,0.74996,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,1,0,0,13,0,0,3,0,0,5,0,3],[8,59,0.1356,0.56688,0.2461,0.42857,0.5712,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,4,0,0,13,0,0,3,0,0,2,0,4],[12,59,0.2034,0.61158,0.212,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,7,0,0,7,0,0,7,0,0,4,0,3],[16,59,0.2712,0.63836,0.23415,0.571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,3,0,0,11,0,0,4,0,0,6,0,4],[20,59,0.339,0.57138,0.19233,0.42857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,9,0,0,10,0,0,6,0,0,2,0,2],[24,59,0.4068,0.47535,0.23944,0.28571,0.4286,0.57143,0.0,1.0,1,2,0,1,1,3,0,0,4,0,0,8,0,0,9,0,0,3,0,0,1,0,2],[28,59,0.4746,0.58929,0.28291,0.42857,0.57143,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,3,0,0,3,0,0,8,0,0,5,0,0,5,0,4],[32,59,0.5424,0.49992,0.21126,0.39286,0.571,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,5,0,0,5,0,0,14,0,0,2,0,0,2,0,1],[36,59,0.6102,0.54015,0.25438,0.28571,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,6,0,0,3,0,0,11,0,0,3,0,0,3,0,3],[40,59,0.678,0.52228,0.19433,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,6,0,0,13,0,0,4,0,0,3,0,0],[44,59,0.7458,0.55349,0.21353,0.5354,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,3,0,0,19,0,0,1,0,0,1,0,3],[48,59,0.8136,0.49995,0.18896,0.42857,0.571,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,9,0,0,12,0,0,2,0,0,3,0,0],[52,59,0.8814,0.44406,0.22859,0.28571,0.4286,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,4,0,1,6,0,0,11,0,0,2,0,0,1,0,1],[56,59,0.9492,0.4821,0.19147,0.42857,0.4286,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,3,0,0,10,0,0,9,0,0,4,0,0,2,0,0],[59,59,1.0,0.44193,0.16885,0.28571,0.42857,0.57143,0.14286,0.857,0,0,0,0,0,2,0,0,10,0,0,7,0,0,10,0,0,2,0,0,1,0,0]]},{"b":1,"e":0.57143,"k":"flat","v":0.49771,"x":0.60267,"p":[[0,30,0.0,0.50891,0.26228,0.28571,0.42857,0.71429,0.0,1.0,1,3,1,1,0,2,0,0,6,0,0,11,0,0,3,0,0,2,0,0,4,0,3],[4,30,0.1333,0.60267,0.22513,0.42859,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,8,0,0,12,0,0,0,0,0,5,0,4],[8,30,0.2667,0.57587,0.2461,0.5354,0.57143,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,2,0,0,3,0,0,14,0,0,4,0,0,3,0,3],[12,30,0.4,0.58926,0.26904,0.5354,0.57143,0.74996,0.0,1.0,2,5,0,2,0,1,0,0,3,0,0,2,0,0,14,0,0,2,0,0,3,0,5],[16,30,0.5333,0.51556,0.20259,0.41068,0.57141,0.60714,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,1,3,0,0,13,0,0,6,0,0,2,0,0],[20,30,0.6667,0.49995,0.1923,0.39286,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,5,0,0,14,0,0,2,0,0,3,0,0],[24,30,0.8,0.58465,0.20619,0.42857,0.57143,0.71429,0.143,1.0,0,2,0,0,0,1,0,0,4,0,0,5,0,0,10,0,0,7,0,0,3,0,2],[28,30,0.9333,0.50888,0.22284,0.42857,0.571,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,2,0,0,6,0,0,13,0,0,3,0,0,1,0,2],[30,30,1.0,0.49771,0.17988,0.33918,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,1,4,0,0,14,0,0,3,0,0,2,0,0]]}]},{"i":"2ea4e25eebb92d79","q":"An eccentric mathematician has a ladder with  $ n$  rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers  $ a$  rungs of the ladder, and when he descends, each step he takes covers  $ b$  rungs of the ladder, where  $ a$  and  $ b$  are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of  $ n,$  expressed in terms of  $ a$  and  $ b.$","t":[{"b":2,"e":0.57143,"k":"flat","v":0.58481,"x":0.7098,"p":[[0,32,0.0,0.64731,0.2172,0.42857,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,9,0,0,5,0,0,6,0,0,6,0,4],[4,32,0.125,0.63838,0.21124,0.57132,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,6,0,0,11,0,0,7,0,0,3,0,4],[8,32,0.25,0.6875,0.19704,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,8,0,0,3,0,6],[12,32,0.375,0.6607,0.17768,0.57143,0.64286,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,10,0,0,2,0,4],[16,32,0.5,0.7098,0.16555,0.57143,0.71429,0.85704,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,10,0,0,4,0,5],[20,32,0.625,0.58926,0.1171,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,7,0,0,2,0,0],[24,32,0.75,0.63391,0.16342,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,7,0,0,5,0,1],[28,32,0.875,0.58481,0.10926,0.57143,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,0,7,0,0,1,0,0],[32,32,1.0,0.59375,0.12428,0.57143,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,18,0,0,8,0,0,0,0,1]]},{"b":6,"e":0.42857,"k":"flat","v":0.60491,"x":0.82143,"p":[[0,53,0.0,0.60491,0.19314,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,8,0,1,6,0,0,7,0,0,6,0,1],[4,53,0.0755,0.73661,0.16793,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,11,0,0,5,0,6],[8,53,0.1509,0.73658,0.21758,0.57132,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,5,0,0,4,0,10],[12,53,0.2264,0.70533,0.17107,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,7,0,0,5,0,5],[16,53,0.3019,0.82143,0.19233,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,4,0,15],[20,53,0.3774,0.79464,0.17473,0.67857,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,10,0,0,2,0,12],[24,53,0.4528,0.79464,0.19212,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,5,0,0,7,0,11],[28,53,0.5283,0.79018,0.1636,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,4,0,10],[32,53,0.6038,0.80804,0.16213,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,2,0,12],[36,53,0.6792,0.7678,0.20444,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,6,0,0,2,0,12],[40,53,0.7547,0.76339,0.23854,0.57143,0.71429,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,3,0,0,5,0,0,8,0,0,3,0,12],[44,53,0.8302,0.74105,0.19704,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,6,0,0,4,0,9],[48,53,0.9057,0.79908,0.17079,0.71429,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,2,0,12],[52,53,0.9811,0.65177,0.10678,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,19,0,0,1,0,0],[53,53,1.0,0.6339,0.1234,0.57143,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,13,0,0,3,0,0]]}]},{"i":"213c7f10150af055","q":"19. C7 (CZS 3) Let $M$ be the set of real numbers of the form $\\frac{m+n}{\\sqrt{m^{2}+n^{2}}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \\in M$, $y \\in M$ with $x<y$, there exists an element $z \\in M$ such that $x<z<y$.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.97768,"x":1.0,"p":[[0,12,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,12,0.3333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"flat","v":0.94643,"x":0.98214,"p":[[0,26,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,26,0.1538,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,26,0.3077,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[12,26,0.4615,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,26,0.6154,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,26,0.7692,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,26,0.9231,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[26,26,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"02b4dd5f5fbb6460","q":"Define a sequence $\\left\\langle f_{0}(x), f_{1}(x), f_{2}(x), \\ldots\\right\\rangle$ of functions by\n\n$$\nf_{0}(x)=1, \\quad f_{1}(x)=x, \\quad\\left(f_{n}(x)\\right)^{2}-1=f_{n+1}(x) f_{n-1}(x), \\text { for } n \\geq 1\n$$\n\nProve that each $f_{n}(x)$ is a polynomial with integer coefficients.","t":[{"b":2,"e":0.0,"k":"volatile","v":0.01786,"x":0.95982,"p":[[0,12,0.0,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,12,0.3333,0.80355,0.31694,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,18],[8,12,0.6667,0.375,0.44571,0.0,0.07143,0.89286,0.0,1.0,16,8,0,16,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,8],[12,12,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.45536,"x":0.95536,"p":[[0,15,0.0,0.94187,0.20816,1.0,1.0,1.0,0.14,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[4,15,0.2667,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[8,15,0.5333,0.84374,0.28428,0.82132,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,0,2,0,22],[12,15,0.8,0.78125,0.34623,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,19],[15,15,1.0,0.45536,0.4141,0.0,0.42857,0.89286,0.0,1.0,10,8,0,10,0,5,0,0,1,0,0,0,0,0,2,0,0,5,0,0,1,0,8]]}]},{"i":"c8e16a310dd62484","q":"An isosceles triangle  $ABC$  is given( $AB=BC$ ). Point  $D$  lies inside of it such that  $\\angle ADC=150$ ,  $E$  lies on  $CD$  such that  $AE=AB$ . It turned out that  $\\angle EBC+\\angle BAE=60$ . Prove that  $\\angle BDC+\\angle CAE=90$ *D. Vasilyev*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.18295,"p":[[0,174,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,174,0.023,0.06696,0.22011,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,174,0.046,0.07143,0.22868,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[12,174,0.069,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,174,0.092,0.06249,0.18874,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[20,174,0.1149,0.11159,0.25686,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[24,174,0.1379,0.05357,0.17035,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[28,174,0.1609,0.07143,0.22304,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[32,174,0.1839,0.06696,0.22583,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[36,174,0.2069,0.04018,0.15663,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,174,0.2299,0.18295,0.34671,0.0,0.0,0.14071,0.0,1.0,23,4,0,23,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,4],[44,174,0.2529,0.0625,0.19541,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[48,174,0.2759,0.09375,0.23854,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[52,174,0.2989,0.05804,0.1984,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[56,174,0.3218,0.11161,0.27833,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[60,174,0.3448,0.07589,0.2448,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[64,174,0.3678,0.10267,0.23482,0.0,0.0,0.0,0.0,0.857,26,0,0,26,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0],[68,174,0.3908,0.05356,0.16264,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[72,174,0.4138,0.05804,0.18509,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[76,174,0.4368,0.06696,0.15966,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[80,174,0.4598,0.06696,0.20198,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[84,174,0.4828,0.07142,0.22865,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[88,174,0.5057,0.07589,0.22011,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[92,174,0.5287,0.16964,0.34706,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,3],[96,174,0.5517,0.06697,0.20198,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[100,174,0.5747,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,174,0.5977,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[108,174,0.6207,0.04463,0.14027,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[112,174,0.6437,0.0558,0.19458,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1],[116,174,0.6667,0.12051,0.25776,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[120,174,0.6897,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,174,0.7126,0.04017,0.15661,0.0,0.0,0.0,0.0,0.857,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[128,174,0.7356,0.07588,0.2258,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0],[132,174,0.7586,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,174,0.7816,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,174,0.8046,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[144,174,0.8276,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[148,174,0.8506,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[152,174,0.8736,0.02679,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,174,0.8966,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[160,174,0.9195,0.08482,0.24448,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[164,174,0.9425,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[168,174,0.9655,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[172,174,0.9885,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[174,174,1.0,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.14731,"p":[[0,71,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.14731,0.31637,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,1],[8,71,0.1127,0.13392,0.28555,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,1],[12,71,0.169,0.08035,0.17101,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,71,0.2254,0.08929,0.23891,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[20,71,0.2817,0.12054,0.30745,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[24,71,0.338,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,71,0.3944,0.07143,0.22868,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[32,71,0.4507,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.06249,0.18532,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[40,71,0.5634,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.03125,0.15038,0.0,0.0,0.0,0.0,0.857,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,71,0.7324,0.04911,0.18074,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,71,0.7887,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,71,0.8451,0.06695,0.20817,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[64,71,0.9014,0.03124,0.11136,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,71,0.9577,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0f79159c12ebdeef","q":"An integer $n \\geqslant 2$ having exactly $s$ positive divisors $1=d_{1}<d_{2}<\\cdots<d_{s}=n$ is said to be good if there exists an integer $k$, with $2 \\leqslant k \\leqslant s$, such that $d_{k}>1+d_{1}+\\cdots+d_{k-1}$. An integer $n \\geqslant 2$ is said to be bad if it is not good.\n\n(a) Show that there are infinitely many bad integers.\n\n(b) Prove that, among any seven consecutive integers all greater than 2, there are always at least four good integers.\n\n(c) Show that there are infinitely many sequences of seven consecutive good integers.\n\n(Gerhard Woeginger, Luxembourg)\n\n#","t":[{"b":1,"e":0.57143,"k":"flat","v":0.70539,"x":0.92857,"p":[[0,30,0.0,0.79017,0.17124,0.67857,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,8,0,9],[4,30,0.1333,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[8,30,0.2667,0.87946,0.17896,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,4,0,19],[12,30,0.4,0.81696,0.19959,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,0,7,0,13],[16,30,0.5333,0.81696,0.2321,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,9,0,13],[20,30,0.6667,0.7946,0.22001,0.571,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,3,0,0,7,0,13],[24,30,0.8,0.81696,0.16065,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,9,0,10],[28,30,0.9333,0.78566,0.15979,0.71429,0.85707,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,10,0,7],[30,30,1.0,0.70539,0.21104,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,8,0,0,8,0,5]]},{"b":6,"e":0.85714,"k":"rising","v":0.68749,"x":0.94642,"p":[[0,34,0.0,0.68749,0.28891,0.53539,0.78571,0.85714,0.0,1.0,3,6,3,3,0,0,0,0,0,0,0,5,0,0,3,0,0,5,0,0,10,0,6],[4,34,0.1176,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[8,34,0.2353,0.88393,0.1357,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,5,0,17],[12,34,0.3529,0.84822,0.20494,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,7,0,16],[16,34,0.4706,0.91071,0.14174,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[20,34,0.5882,0.94642,0.14174,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[24,34,0.7059,0.90175,0.14039,0.82132,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20],[28,34,0.8235,0.89732,0.16065,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,5,0,20],[32,34,0.9412,0.88392,0.14916,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,18],[34,34,1.0,0.90179,0.14913,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,20]]}]},{"i":"f09ba9b160add4c9","q":"Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $|\\sqrt{3} \\cdot d(n)|$ divides $n$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.12947,"x":0.90179,"p":[[0,42,0.0,0.12947,0.2212,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,9,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[4,42,0.0952,0.73213,0.27837,0.5354,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,3,0,0,2,0,0,3,0,0,9,0,10],[8,42,0.1905,0.76349,0.32853,0.67857,0.93,1.0,0.0,1.0,1,16,0,1,0,4,0,0,1,0,0,1,0,0,1,0,0,2,0,0,6,0,16],[12,42,0.2857,0.66513,0.32462,0.53539,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,3,0,0,1,0,0,5,0,0,4,0,0,6,0,9],[16,42,0.381,0.6875,0.29545,0.57143,0.71429,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,3,0,0,0,0,0,2,0,0,8,0,0,8,0,7],[20,42,0.4762,0.6607,0.30878,0.39286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,4,0,0,1,0,0,3,0,0,6,0,0,6,0,8],[24,42,0.5714,0.66973,0.36327,0.2857,0.85714,1.0,0.0,1.0,2,11,0,2,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,9,0,11],[28,42,0.6667,0.79463,0.23129,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,4,0,0,7,0,13],[32,42,0.7619,0.70312,0.3052,0.4286,0.85714,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,0,0,1,3,0,0,1,0,0,5,0,0,7,0,10],[36,42,0.8571,0.76336,0.23856,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,6,0,0,8,0,10],[40,42,0.9524,0.80801,0.26872,0.82132,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,11,0,13],[42,42,1.0,0.90179,0.1729,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,20]]},{"b":4,"e":0.42857,"k":"rising","v":0.18738,"x":0.84372,"p":[[0,31,0.0,0.18738,0.23262,0.0,0.14286,0.2857,0.0,0.85714,12,0,0,12,0,11,0,0,4,0,0,1,0,0,2,0,0,0,0,0,2,0,0],[4,31,0.129,0.70527,0.28797,0.57143,0.78571,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,3,0,0,1,0,0,2,0,0,7,0,0,8,0,8],[8,31,0.2581,0.82143,0.30093,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,19],[12,31,0.3871,0.74991,0.34642,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,15],[16,31,0.5161,0.84372,0.16119,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,11,0,12],[20,31,0.6452,0.73214,0.31693,0.67857,0.85714,1.0,0.0,1.0,4,9,0,4,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,12,0,9],[24,31,0.7742,0.79463,0.23404,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,0,13,0,10],[28,31,0.9032,0.61605,0.34151,0.42857,0.57143,1.0,0.0,1.0,4,9,0,4,0,1,0,0,1,0,0,7,0,0,4,0,0,1,0,0,5,0,9],[31,31,1.0,0.50443,0.28567,0.39286,0.4998,0.60714,0.0,1.0,3,3,0,3,0,3,0,0,2,0,0,8,0,0,8,0,0,1,0,0,4,0,3]]}]},{"i":"8a5b1ad23008ee8f","q":"Define the sequence  $\\{a_n\\}$  in the following manner: $a_1=1$  $a_2=3$  $a_{n+2}=2a_{n+1}a_{n}+1$  ; for all  $n\\geq1$ Prove that the largest power of  $2$  that divides  $a_{4006}-a_{4005}$  is  $2^{2003}.$","t":[{"b":5,"e":0.1429,"k":"falling","v":0.2857,"x":0.61161,"p":[[0,73,0.0,0.5491,0.27689,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,4,0,0,1,0,0,6,0,0,9,0,0,4,0,2],[4,73,0.0548,0.61161,0.23483,0.42857,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,4,0,0,4,0,0,7,0,0,9,0,1],[8,73,0.1096,0.51785,0.24679,0.28571,0.4286,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,6,0,0,7,0,0,3,0,0,5,0,0,7,0,0],[12,73,0.1644,0.49106,0.27649,0.28571,0.42859,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,6,0,0,6,0,0,3,0,0,5,0,0,4,0,2],[16,73,0.2192,0.52232,0.23853,0.28571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,5,0,0,6,0,0,6,0,0,6,0,0,4,0,1],[20,73,0.274,0.55357,0.25692,0.42857,0.42857,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,12,0,0,3,0,0,2,0,0,3,0,5],[24,73,0.3288,0.48658,0.2078,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,11,0,0,5,0,0,2,0,0,5,0,0],[28,73,0.3836,0.48214,0.23623,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,8,0,0,10,0,0,2,0,0,5,0,0,2,0,2],[32,73,0.4384,0.46429,0.23958,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,3,0,0,8,0,0,8,0,0,4,0,0,4,0,0,3,0,1],[36,73,0.4932,0.48658,0.23106,0.39286,0.42857,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,5,0,0,12,0,0,6,0,0,0,0,0,5,0,1],[40,73,0.5479,0.49554,0.23141,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,13,0,0,4,0,0,4,0,0,4,0,0,6,0,0],[44,73,0.6027,0.43749,0.24662,0.2857,0.39286,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,9,0,2,7,0,0,3,0,0,1,0,0,3,0,2],[48,73,0.6575,0.43748,0.2671,0.25,0.42857,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,6,0,0,6,0,0,6,0,0,1,0,0,3,0,2],[52,73,0.7123,0.45979,0.24151,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,9,0,0,6,0,0,7,0,0,1,0,0,4,0,1],[56,73,0.7671,0.40625,0.23449,0.28571,0.35714,0.42857,0.0,1.0,1,2,0,1,0,4,0,0,11,0,0,9,0,0,2,0,0,2,0,0,1,0,2],[60,73,0.8219,0.46873,0.26542,0.2857,0.35714,0.75,0.14286,1.0,0,1,0,0,0,4,0,0,12,0,0,5,0,0,2,0,0,1,0,0,7,0,1],[64,73,0.8767,0.41518,0.18336,0.28571,0.42857,0.46431,0.14286,1.0,0,1,0,0,0,2,0,0,13,0,0,9,0,0,4,0,0,3,0,0,0,0,1],[68,73,0.9315,0.32589,0.23753,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,5,0,0,13,0,0,5,0,0,2,0,0,0,0,0,2,0,1],[72,73,0.9863,0.31919,0.15045,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,15,0,1,6,0,0,2,0,0,0,0,0,1,0,0],[73,73,1.0,0.2857,0.12874,0.25,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.20758,"x":0.66069,"p":[[0,160,0.0,0.48214,0.2519,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,6,0,0,5,0,0,6,0,0,4,0,0,4,0,1],[4,160,0.025,0.64286,0.23958,0.42857,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,5,0,0,4,0,0,7,0,0,11,0,1],[8,160,0.05,0.63393,0.27418,0.42857,0.78571,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,6,0,0,2,0,0,1,0,0,13,0,3],[12,160,0.075,0.62499,0.26905,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,4,0,0,4,0,0,4,0,0,10,0,3],[16,160,0.1,0.56696,0.25123,0.39286,0.5,0.75,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,8,0,0,1,0,0,7,0,0,6,0,2],[20,160,0.125,0.63393,0.24206,0.53571,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,1,0,0,9,0,0,2,0,0,11,0,2],[24,160,0.15,0.57143,0.23146,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,6,0,0,6,0,0,4,0,0,9,0,0],[28,160,0.175,0.58929,0.23623,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,5,0,0,7,0,0,4,0,0,10,0,0],[32,160,0.2,0.62947,0.24185,0.42859,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,5,0,0,4,0,0,5,0,0,13,0,0],[36,160,0.225,0.57589,0.2461,0.39286,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,6,0,0,3,0,0,6,0,0,8,0,1],[40,160,0.25,0.66069,0.2468,0.5354,0.78571,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,1,0,0,14,0,2],[44,160,0.275,0.61159,0.22933,0.42857,0.57143,0.85704,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,8,0,0,6,0,0,5,0,0,6,0,3],[48,160,0.3,0.55802,0.27283,0.28571,0.57121,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,6,0,0,6,0,0,4,0,0,3,0,0,8,0,2],[52,160,0.325,0.49553,0.27195,0.28571,0.42857,0.74996,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,10,0,0,1,0,0,2,0,0,6,0,2],[56,160,0.35,0.41518,0.22968,0.2857,0.28571,0.4286,0.14286,0.85714,0,0,0,0,0,4,0,0,14,0,0,7,0,0,0,0,0,2,0,0,5,0,0],[60,160,0.375,0.55804,0.25843,0.39286,0.50001,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,8,0,0,3,0,0,4,0,0,7,0,2],[64,160,0.4,0.43304,0.21866,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,8,0,0,10,0,0,3,0,0,2,0,0,4,0,0],[68,160,0.425,0.47544,0.24459,0.2857,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,5,0,0,7,0,1,6,0,0,4,0,0,3,0,0,6,0,0],[72,160,0.45,0.44642,0.23622,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,5,0,0,9,0,0,8,0,0,2,0,0,3,0,0,5,0,0],[76,160,0.475,0.43973,0.23563,0.28571,0.42857,0.58929,0.0,1.0,1,1,0,1,0,2,0,0,12,0,0,8,0,0,1,1,0,3,0,0,3,0,1],[80,160,0.5,0.54909,0.2699,0.2857,0.57121,0.75,0.14286,1.0,0,3,0,0,0,3,0,0,8,0,0,4,0,0,4,0,0,5,0,0,5,0,3],[84,160,0.525,0.44197,0.23244,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,7,0,0,8,0,0,4,0,0,4,0,0,2,0,1],[88,160,0.55,0.51561,0.25426,0.28571,0.4643,0.75,0.0,0.85714,1,0,0,1,0,2,0,0,8,0,0,5,0,1,4,0,0,3,0,0,8,0,0],[92,160,0.575,0.51116,0.25823,0.28571,0.42857,0.75,0.14286,1.0,0,1,0,0,0,4,0,0,6,0,1,8,0,0,2,0,0,3,0,0,7,0,1],[96,160,0.6,0.53124,0.26543,0.28571,0.42857,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,6,0,0,10,0,0,3,0,0,1,0,0,6,0,3],[100,160,0.625,0.49993,0.24998,0.28571,0.571,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,8,0,0,3,0,0,10,0,0,1,0,0,4,0,2],[104,160,0.65,0.5357,0.26,0.28571,0.49979,0.75,0.14286,1.0,0,2,0,0,0,4,0,0,5,0,0,7,0,0,5,0,0,3,0,0,6,0,2],[108,160,0.675,0.50893,0.23403,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,6,0,0,4,0,0,4,0,0,5,0,1],[112,160,0.7,0.58482,0.27049,0.39286,0.57143,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,7,0,0,6,0,0,6,0,0,1,0,0,7,0,4],[116,160,0.725,0.5357,0.25504,0.28571,0.49979,0.74996,0.14286,1.0,0,1,0,0,0,4,0,0,5,0,0,7,0,0,4,0,0,4,0,0,7,0,1],[120,160,0.75,0.4375,0.17105,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,9,0,0,12,0,0,4,0,0,4,0,0,1,0,0],[124,160,0.775,0.46875,0.2402,0.28571,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,6,0,0,10,0,0,3,0,0,3,0,0,4,0,1],[128,160,0.8,0.33918,0.2195,0.14286,0.28571,0.42857,0.14,1.0,0,1,0,0,0,12,0,0,8,0,0,6,0,0,3,0,0,1,0,0,1,0,1],[132,160,0.825,0.40179,0.2635,0.1429,0.28571,0.60714,0.0,0.85714,2,0,0,2,0,7,0,0,9,0,0,3,0,0,3,0,0,4,0,0,4,0,0],[136,160,0.85,0.42186,0.21155,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,6,0,0,7,0,0,9,0,1,4,0,0,2,0,0,3,0,0],[140,160,0.875,0.33928,0.24157,0.14286,0.28571,0.46429,0.0,1.0,4,1,0,4,0,6,0,0,10,0,0,4,0,0,5,0,0,1,0,0,1,0,1],[144,160,0.9,0.3125,0.18708,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,8,0,0,11,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[148,160,0.925,0.3125,0.24337,0.14286,0.28571,0.46431,0.0,1.0,4,1,0,4,0,10,0,0,7,0,0,3,0,0,6,0,0,0,0,0,1,0,1],[152,160,0.95,0.21429,0.14725,0.14286,0.14288,0.28571,0.0,0.4286,6,0,0,6,0,11,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[156,160,0.975,0.20758,0.21529,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,7,0,0,8,0,1,2,0,0,1,0,0,1,0,0,1,0,0],[160,160,1.0,0.21874,0.16358,0.14286,0.14288,0.32143,0.0,0.571,7,0,0,7,0,10,0,0,7,0,0,7,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"cd10e6449e2e5310","q":"Determine all integers  $ n > 3$  for which there exist  $ n$  points  $ A_{1},\\cdots ,A_{n}$  in the plane, no three collinear, and real numbers  $ r_{1},\\cdots ,r_{n}$  such that for  $ 1\\leq i < j < k\\leq n$ , the area of  $ \\triangle A_{i}A_{j}A_{k}$  is  $ r_{i} \\plus{} r_{j} \\plus{} r_{k}$ .","t":[{"b":4,"e":0.14,"k":"flat","v":0.27232,"x":0.62053,"p":[[0,62,0.0,0.3125,0.19704,0.14286,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,13,0,0,6,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[4,62,0.0645,0.62053,0.26632,0.4286,0.71429,0.74996,0.14286,1.0,0,4,0,0,0,5,0,0,1,0,0,3,0,0,4,0,0,11,0,0,4,0,4],[8,62,0.129,0.49999,0.30929,0.14286,0.49999,0.71429,0.0,1.0,1,3,0,1,0,9,0,0,2,0,0,4,0,0,3,0,0,6,0,0,4,0,3],[12,62,0.1935,0.61608,0.23804,0.42859,0.71429,0.85704,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,4,0,0,3,0,0,9,0,0,10,0,0],[16,62,0.2581,0.54016,0.25935,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,7,0,0,3,0,0,3,0,0,10,0,0,3,0,2],[20,62,0.3226,0.60268,0.26901,0.39286,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,3,0,0,3,0,0,9,0,0,6,0,3],[24,62,0.3871,0.54464,0.29329,0.2857,0.57143,0.75,0.14286,1.0,0,5,0,0,0,5,0,0,7,0,0,2,0,0,6,0,0,4,0,0,3,0,5],[28,62,0.4516,0.45087,0.27918,0.14286,0.35714,0.71429,0.14286,0.85714,0,0,0,0,0,10,0,0,6,0,0,2,0,0,3,0,0,5,0,0,6,0,0],[32,62,0.5161,0.47768,0.25406,0.28571,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,6,0,0,8,0,0,3,0,0,4,0,0,6,0,0,5,0,0],[36,62,0.5806,0.52223,0.27816,0.2857,0.42859,0.74996,0.0,1.0,1,1,0,1,0,4,0,0,6,0,0,6,0,0,1,0,0,6,0,0,7,0,1],[40,62,0.6452,0.39276,0.28129,0.14286,0.28571,0.60714,0.0,1.0,1,1,0,1,0,11,0,0,7,0,0,3,0,0,2,0,0,3,0,0,4,0,1],[44,62,0.7097,0.49097,0.26959,0.28571,0.4286,0.71429,0.0,0.85714,1,0,0,1,0,5,0,0,6,0,0,6,0,0,2,0,0,5,0,0,7,0,0],[48,62,0.7742,0.38387,0.23534,0.24999,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,6,0,0,10,0,0,4,0,0,4,0,0,4,0,0,2,0,0],[52,62,0.8387,0.39292,0.21723,0.14289,0.35714,0.5711,0.14286,0.85714,0,0,0,0,0,9,0,0,7,0,0,6,0,0,4,0,0,5,0,0,1,0,0],[56,62,0.9032,0.27232,0.17627,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,14,0,0,10,0,0,4,0,0,0,0,0,3,0,0,0,0,0],[60,62,0.9677,0.48213,0.23076,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,6,0,0,4,0,0,6,0,0,4,0,0],[62,62,1.0,0.39283,0.22301,0.14289,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,8,0,0,8,0,0,2,0,0,7,0,0,6,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.26331,"x":0.65176,"p":[[0,94,0.0,0.29902,0.19359,0.14286,0.14286,0.4643,0.14,0.71429,0,0,0,0,0,18,0,0,2,0,0,4,0,0,7,0,0,1,0,0,0,0,0],[4,94,0.0426,0.65176,0.27419,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,1,0,0,5,0,0,2,0,0,9,0,0,5,0,6],[8,94,0.0851,0.56239,0.32926,0.24999,0.57141,0.85714,0.14,1.0,0,7,0,0,0,8,0,0,4,0,0,2,0,0,3,0,0,5,0,0,3,0,7],[12,94,0.1277,0.4866,0.29636,0.14286,0.50001,0.71429,0.0,1.0,2,1,0,2,0,7,0,0,4,0,0,3,0,0,1,0,0,10,0,0,4,0,1],[16,94,0.1702,0.35712,0.2422,0.14286,0.2857,0.571,0.14286,0.85714,0,0,0,0,0,15,0,0,3,0,0,5,0,0,3,0,0,4,0,0,2,0,0],[20,94,0.2128,0.59372,0.30537,0.39285,0.71429,0.74996,0.0,1.0,2,5,0,2,0,4,0,0,2,0,0,3,0,0,2,0,0,11,0,0,3,0,5],[24,94,0.2553,0.56687,0.26376,0.39286,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,4,0,0,6,0,0,0,0,0,11,0,0,6,0,1],[28,94,0.2979,0.46425,0.26485,0.2857,0.4998,0.71429,0.0,1.0,2,1,1,2,0,5,0,0,6,0,0,3,0,0,6,0,0,7,0,0,2,0,1],[32,94,0.3404,0.56695,0.26362,0.28571,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,3,0,0,2,0,0,14,0,0,1,0,3],[36,94,0.383,0.43286,0.25894,0.14286,0.42857,0.71429,0.14,0.85714,0,0,0,0,0,10,0,0,5,0,0,5,0,0,1,0,0,8,0,0,3,0,0],[40,94,0.4255,0.5625,0.30709,0.25,0.71429,0.85704,0.0,1.0,1,3,0,1,0,7,0,0,2,0,0,2,0,0,3,0,0,8,0,0,6,0,3],[44,94,0.4681,0.49552,0.31538,0.14286,0.4998,0.71429,0.0,1.0,1,3,0,1,0,10,0,0,2,0,0,3,0,0,1,0,0,9,0,0,3,0,3],[48,94,0.5106,0.50442,0.23139,0.28571,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,2,0,0,6,0,0,9,0,0,3,0,0],[52,94,0.5532,0.43302,0.2977,0.14286,0.35714,0.71429,0.0,1.0,1,2,0,1,0,12,0,0,3,0,0,2,0,0,2,0,0,9,0,0,1,0,2],[56,94,0.5957,0.37946,0.28258,0.14286,0.21435,0.71429,0.0,1.0,1,1,0,1,0,15,0,0,3,0,0,1,0,0,1,0,0,10,0,0,0,0,1],[60,94,0.6383,0.34803,0.29449,0.14214,0.28571,0.71429,0.0,1.0,6,1,0,6,0,8,0,0,6,0,0,2,0,0,1,0,0,7,0,0,1,0,1],[64,94,0.6809,0.40179,0.29329,0.14286,0.2857,0.71429,0.14286,1.0,0,2,0,0,0,14,0,0,4,0,0,4,0,0,1,0,0,4,0,0,3,0,2],[68,94,0.7234,0.36152,0.24747,0.14286,0.28571,0.71429,0.14,0.71429,0,0,0,0,0,14,0,0,7,0,0,1,0,0,0,0,0,10,0,0,0,0,0],[72,94,0.766,0.37497,0.23889,0.14286,0.28571,0.60682,0.14286,0.85714,0,0,0,0,0,12,0,0,7,0,0,3,0,0,2,0,0,7,0,0,1,0,0],[76,94,0.8085,0.36607,0.23402,0.14286,0.2857,0.57143,0.14286,0.85714,0,0,0,0,0,13,0,0,5,0,0,5,0,0,2,0,0,6,0,0,1,0,0],[80,94,0.8511,0.41518,0.23787,0.14289,0.35714,0.71429,0.14286,0.85714,0,0,0,0,0,9,0,0,7,0,0,5,0,0,1,0,0,9,0,0,1,0,0],[84,94,0.8936,0.26331,0.19274,0.14286,0.14286,0.2857,0.14,0.71429,0,0,0,0,0,20,0,0,5,0,0,3,0,0,0,0,0,4,0,0,0,0,0],[88,94,0.9362,0.3392,0.2363,0.14286,0.2857,0.50002,0.0,0.71429,1,0,0,1,0,13,0,0,7,0,0,3,0,0,0,0,0,8,0,0,0,0,0],[92,94,0.9787,0.36606,0.24726,0.14286,0.28571,0.71429,0.14286,0.71429,0,0,0,0,0,15,0,0,4,0,0,2,0,0,2,0,0,9,0,0,0,0,0],[94,94,1.0,0.38394,0.26591,0.14286,0.2143,0.71429,0.14286,0.71429,0,0,0,0,0,16,0,0,2,0,0,2,0,0,0,0,0,12,0,0,0,0,0]]}]},{"i":"d409d96cbdf06e3e","q":"Determine all positive integers  $n$  such that for every positive devisor  $ d $  of  $n$ ,  $d+1$  is devisor of  $n+1$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.83036,"x":0.94643,"p":[[0,53,0.0,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[4,53,0.0755,0.87054,0.12556,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,10,0,13],[8,53,0.1509,0.86161,0.12103,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,12],[12,53,0.2264,0.86161,0.12103,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,12],[16,53,0.3019,0.87499,0.13719,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,5,0,16],[20,53,0.3774,0.83036,0.1357,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,10,0,9],[24,53,0.4528,0.88393,0.13092,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,17],[28,53,0.5283,0.83929,0.1171,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,9],[32,53,0.6038,0.85714,0.12877,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,6,0,13],[36,53,0.6792,0.87053,0.13997,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,7,0,15],[40,53,0.7547,0.86161,0.13592,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,3,0,15],[44,53,0.8302,0.85268,0.13592,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,3,0,14],[48,53,0.9057,0.8616,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,14],[52,53,0.9811,0.86607,0.1234,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,8,0,13],[53,53,1.0,0.83036,0.13092,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,11]]},{"b":3,"e":0.71429,"k":"falling","v":0.7232,"x":0.89286,"p":[[0,67,0.0,0.89286,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[4,67,0.0597,0.85268,0.12619,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,7,0,12],[8,67,0.1194,0.84375,0.16506,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,4,0,14],[12,67,0.1791,0.88393,0.12078,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,15],[16,67,0.2388,0.80802,0.13178,0.71429,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,4,0,9],[20,67,0.2985,0.79018,0.11285,0.71429,0.71429,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,5,0,6],[24,67,0.3582,0.80804,0.12682,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,3,0,9],[28,67,0.4179,0.80357,0.12242,0.71429,0.71429,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,4,0,8],[32,67,0.4776,0.79018,0.10705,0.71429,0.71429,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,7,0,5],[36,67,0.5373,0.8125,0.11538,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,8,0,7],[40,67,0.597,0.82589,0.12745,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,5,0,10],[44,67,0.6567,0.77679,0.10062,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,6,0,4],[48,67,0.7164,0.80357,0.12242,0.71429,0.71429,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,4,0,8],[52,67,0.7761,0.80803,0.11633,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,7,0,7],[56,67,0.8358,0.85267,0.12619,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,7,0,12],[60,67,0.8955,0.82589,0.13236,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,11],[64,67,0.9552,0.76786,0.10564,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,2,0,5],[67,67,1.0,0.7232,0.0709,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,2,0,1]]}]},{"i":"d8d02ed84204fec2","q":"Determine all pairs  $(k, n)$  of positive integers that satisfy  $$ 1! + 2! + ... + k! = 1 + 2 + ... + n. $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.70982,"x":1.0,"p":[[0,52,0.0,0.70982,0.35081,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[4,52,0.0769,0.90179,0.23808,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[8,52,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,52,0.2308,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,52,0.3077,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,52,0.3846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,52,0.4615,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[28,52,0.5385,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,52,0.6154,0.87946,0.24251,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[36,52,0.6923,0.91518,0.21387,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[40,52,0.7692,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[44,52,0.8462,0.95089,0.14555,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":7,"e":0.28571,"k":"falling","v":0.26786,"x":1.0,"p":[[0,70,0.0,0.86607,0.2788,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[4,70,0.0571,0.91964,0.2111,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[8,70,0.1143,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,70,0.1714,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,70,0.2286,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[20,70,0.2857,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[24,70,0.3429,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[28,70,0.4,0.88839,0.24153,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[32,70,0.4571,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,70,0.5143,0.86161,0.28901,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[40,70,0.5714,0.94643,0.17405,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[44,70,0.6286,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[48,70,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,70,0.7429,0.90603,0.24159,1.0,1.0,1.0,0.14,1.0,0,27,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[56,70,0.8,0.91071,0.22517,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[60,70,0.8571,0.91518,0.21087,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[64,70,0.9143,0.89285,0.27199,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[68,70,0.9714,0.60268,0.37582,0.28571,0.28571,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[70,70,1.0,0.26786,0.04725,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cf2a522580e5e01c","q":"Determine all positive integers  $n$  such that all positive integers less than or equal to  $n$  and relatively prime to  $n$  are pairwise coprime.","t":[{"b":1,"e":0.0,"k":"falling","v":0.33034,"x":0.84375,"p":[[0,56,0.0,0.76339,0.18766,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,13,0,5],[4,56,0.0714,0.84375,0.16115,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,11,0,12],[8,56,0.1429,0.80355,0.13245,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,18,0,4],[12,56,0.2143,0.81696,0.11971,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,19,0,4],[16,56,0.2857,0.7723,0.14224,0.71429,0.71429,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,0,7,0,6],[20,56,0.3571,0.76339,0.15406,0.67857,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,15,0,3],[24,56,0.4286,0.80356,0.14619,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,14,0,6],[28,56,0.5,0.80803,0.15407,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,11,0,8],[32,56,0.5714,0.81695,0.11428,0.71429,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,15,0,5],[36,56,0.6429,0.78125,0.12869,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,9,0,5],[40,56,0.7143,0.79018,0.15561,0.71429,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,8,0,0,14,0,5],[44,56,0.7857,0.75433,0.16844,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,12,0,0,12,0,3],[48,56,0.8571,0.62054,0.26633,0.39286,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,6,0,0,1,0,0,5,0,0,9,0,0,5,0,4],[52,56,0.9286,0.49554,0.2923,0.28571,0.28571,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,14,0,0,1,0,0,4,0,0,1,0,0,6,0,3],[56,56,1.0,0.33034,0.15744,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,4,0,0,22,0,0,1,0,0,3,0,0,1,0,0,1,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.52661,"x":0.87946,"p":[[0,46,0.0,0.8125,0.1448,0.85714,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,23,0,3],[4,46,0.087,0.79464,0.14258,0.71429,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,20,0,2],[8,46,0.1739,0.81245,0.131,0.71429,0.85714,0.85714,0.57,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,14,0,6],[12,46,0.2609,0.75,0.17496,0.71429,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,16,0,2],[16,46,0.3478,0.87053,0.10926,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,16,0,10],[20,46,0.4348,0.86161,0.10403,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,18,0,8],[24,46,0.5217,0.80357,0.14617,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,11,0,7],[28,46,0.6087,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7],[32,46,0.6957,0.86161,0.10999,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,19,0,8],[36,46,0.7826,0.84821,0.1234,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,19,0,7],[40,46,0.8696,0.81696,0.16457,0.85714,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,22,0,4],[44,46,0.9565,0.77656,0.23157,0.71429,0.85714,0.85714,0.14,1.0,0,6,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,5,0,0,16,0,6],[46,46,1.0,0.52661,0.3491,0.14286,0.71429,0.85714,0.0,1.0,3,1,0,3,0,8,0,0,3,0,0,0,0,0,0,0,0,5,0,0,12,0,1]]}]},{"i":"919d4fa067b361e2","q":"Determine all positive integers  $n\\geq 2$  for which there exist integers  $x_1,x_2,\\ldots ,x_{n-1}$  satisfying the condition that if  $0<i<n,0<j<n, i\\neq j$  and  $n$  divides  $2i+j$ , then  $x_i<x_j$ .","t":[{"b":2,"e":0.28571,"k":"falling","v":0.53121,"x":0.7232,"p":[[0,28,0.0,0.7232,0.24728,0.57132,0.85707,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,5,0,0,10,0,7],[4,28,0.1429,0.71874,0.21867,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,4,0,0,6,0,8],[8,28,0.2857,0.56692,0.22724,0.42857,0.571,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,7,0,0,7,0,0,8,0,0,3,0,0,4,0,3],[12,28,0.4286,0.66518,0.22191,0.4286,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,7,0,0,8,0,0,3,0,0,7,0,5],[16,28,0.5714,0.53571,0.21128,0.42857,0.4286,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,14,0,0,6,0,0,0,0,0,5,0,2],[20,28,0.7143,0.56695,0.1906,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,9,0,0,10,0,0,3,0,0,5,0,1],[24,28,0.8571,0.58032,0.16728,0.42859,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,18,0,0,1,0,0,1,0,3],[28,28,1.0,0.53121,0.12491,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,8,0,0,17,0,0,3,0,0,1,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.68304,"x":0.93303,"p":[[0,45,0.0,0.68304,0.21939,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,10,0,0,10,0,2],[4,45,0.0889,0.8281,0.17846,0.82132,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,13,1,10],[8,45,0.1778,0.80802,0.20396,0.57143,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,0,0,0,10,0,12],[12,45,0.2667,0.74105,0.23808,0.57143,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,4,0,0,7,0,0,2,0,0,10,0,8],[16,45,0.3556,0.77679,0.20806,0.57143,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,9,0,10],[20,45,0.4444,0.82589,0.1461,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,12,0,9],[24,45,0.5333,0.85267,0.19063,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,6,0,17],[28,45,0.6222,0.86159,0.18382,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,5,0,18],[32,45,0.7111,0.84375,0.15303,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,10,0,12],[36,45,0.8,0.92855,0.10719,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[40,45,0.8889,0.93303,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[44,45,0.9778,0.87053,0.16115,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,11,0,15],[45,45,1.0,0.89286,0.14286,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,18]]}]},{"i":"6b12beb201af20b5","q":"Consider a right-angled triangle with integer-valued sides  $a<b<c$  where  $a,b,c$  are pairwise co-prime. Let  $d=c-b$  . Suppose  $d$  divides  $a$  . Then **(a)**  Prove that  $d\\leqslant 2$ . **(b)**  Find all such triangles (i.e. all possible triplets  $a,b,c$ ) with perimeter less than  $100$  .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.94196,"x":0.96429,"p":[[0,32,0.0,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[4,32,0.125,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[8,32,0.25,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[12,32,0.375,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,32,0.5,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,32,0.625,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,32,0.75,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,32,0.875,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[32,32,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]},{"b":3,"e":0.57143,"k":"falling","v":0.74107,"x":0.95089,"p":[[0,20,0.0,0.91071,0.12242,0.82143,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[4,20,0.2,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[8,20,0.4,0.84821,0.15947,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,4,0,15],[12,20,0.6,0.77676,0.1595,0.71429,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,0,3,0,9],[16,20,0.8,0.74107,0.14914,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,14,0,0,3,0,6],[20,20,1.0,0.74999,0.16368,0.57143,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,12,0,0,2,0,8]]}]},{"i":"15cd2a6069baa075","q":"Determine functions  $f : \\mathbb{R} \\rightarrow \\mathbb{R},$  with property that \n\n    \\[\n    f(f(x)) + y \\cdot f(x) \\le x + x \\cdot f(f(y)),\n    \\]\n    \nfor every  $x$  and  $y$  are real numbers.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.67856,"x":0.98661,"p":[[0,130,0.0,0.76786,0.27375,0.42857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,3,0,0,1,0,17],[4,130,0.0308,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[8,130,0.0615,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[12,130,0.0923,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[16,130,0.1231,0.91518,0.18161,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,25],[20,130,0.1538,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,130,0.1846,0.90625,0.20395,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,25],[28,130,0.2154,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,130,0.2462,0.91964,0.19541,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,26],[36,130,0.2769,0.93304,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[40,130,0.3077,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,130,0.3385,0.88839,0.18809,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,6,0,20],[48,130,0.3692,0.91964,0.16728,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[52,130,0.4,0.90177,0.15749,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,21],[56,130,0.4308,0.875,0.20124,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,1,0,22],[60,130,0.4615,0.87509,0.17034,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,8,0,17],[64,130,0.4923,0.86607,0.18189,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,7,0,17],[68,130,0.5231,0.85714,0.19233,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,7,0,17],[72,130,0.5538,0.7857,0.22589,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,1,0,0,7,0,13],[76,130,0.5846,0.83034,0.2096,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,5,0,16],[80,130,0.6154,0.80357,0.21053,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,4,0,0,7,0,13],[84,130,0.6462,0.8482,0.22572,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,19],[88,130,0.6769,0.83034,0.2156,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,2,0,18],[92,130,0.7077,0.88839,0.17029,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,7,0,19],[96,130,0.7385,0.90179,0.15335,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,20],[100,130,0.7692,0.86161,0.18029,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,5,0,17],[104,130,0.8,0.76784,0.25693,0.53539,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,3,0,0,3,0,15],[108,130,0.8308,0.80802,0.21611,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,10,0,12],[112,130,0.8615,0.83022,0.18716,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,6,0,14],[116,130,0.8923,0.85268,0.1838,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,15],[120,130,0.9231,0.89286,0.17857,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,20],[124,130,0.9538,0.78571,0.23958,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,7,0,13],[128,130,0.9846,0.79911,0.24448,0.67857,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0,8,0,14],[130,130,1.0,0.67856,0.25755,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,7,0,0,3,0,0,1,0,0,6,0,0,10,0,5]]},{"b":2,"e":1.0,"k":"falling","v":0.60713,"x":0.98661,"p":[[0,130,0.0,0.81696,0.23483,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,6,0,0,1,0,18],[4,130,0.0308,0.96875,0.10555,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[8,130,0.0615,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,130,0.0923,0.94643,0.16269,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[16,130,0.1231,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[20,130,0.1538,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[24,130,0.1846,0.91518,0.17445,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,25],[28,130,0.2154,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,130,0.2462,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[36,130,0.2769,0.91517,0.17807,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[40,130,0.3077,0.9375,0.15947,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,27],[44,130,0.3385,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[48,130,0.3692,0.875,0.20124,0.82132,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,3,0,21],[52,130,0.4,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[56,130,0.4308,0.88393,0.20652,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,3,0,22],[60,130,0.4615,0.87947,0.18249,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,20],[64,130,0.4923,0.83482,0.26027,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,0,4,0,20],[68,130,0.5231,0.89732,0.16457,0.85713,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,21],[72,130,0.5538,0.79909,0.24965,0.67846,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,4,0,0,4,0,16],[76,130,0.5846,0.86604,0.18881,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,4,0,19],[80,130,0.6154,0.84821,0.20497,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,0,2,0,18],[84,130,0.6462,0.78129,0.25495,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,2,0,0,8,0,13],[88,130,0.6769,0.81249,0.22712,0.67857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,4,0,16],[92,130,0.7077,0.86607,0.22286,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,0,0,0,22],[96,130,0.7385,0.82143,0.2342,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,5,0,0,3,0,17],[100,130,0.7692,0.86606,0.21708,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,21],[104,130,0.8,0.88839,0.18466,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,21],[108,130,0.8308,0.77232,0.26452,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,5,0,0,6,0,13],[112,130,0.8615,0.78125,0.27429,0.57143,0.92857,1.0,0.1429,1.0,0,16,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,3,0,0,4,0,16],[116,130,0.8923,0.82589,0.24152,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,4,0,18],[120,130,0.9231,0.73213,0.26905,0.42859,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,7,0,0,4,0,0,4,0,0,1,0,14],[124,130,0.9538,0.71429,0.26,0.42859,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,6,0,0,2,0,0,8,0,0,2,0,11],[128,130,0.9846,0.68304,0.23887,0.4286,0.64286,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,5,0,0,2,0,9],[130,130,1.0,0.60713,0.21428,0.42857,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,12,0,0,5,0,0,6,0,0,3,0,4]]}]},{"i":"07657d8a9e8b569d","q":"Assume that  $X$  is a seperable metric space. Prove that if  $f: X\\longrightarrow\\mathbb R$  is a function that  $\\lim_{x\\rightarrow a}f(x)$  exists for each  $a\\in\\mathbb R$ . Prove that set of points in which  $f$  is not continuous is countable.","t":[{"b":4,"e":1.0,"k":"flat","v":0.875,"x":0.92857,"p":[[0,25,0.0,0.91964,0.18537,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,25],[4,25,0.16,0.91518,0.18851,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[8,25,0.32,0.91972,0.138,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[12,25,0.48,0.91518,0.15916,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,3,0,23],[16,25,0.64,0.92857,0.12877,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,22],[20,25,0.8,0.875,0.20123,0.85714,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,7,0,18],[24,25,0.96,0.91964,0.18536,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,24],[25,25,1.0,0.88839,0.17762,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,5,0,20]]},{"b":5,"e":0.71429,"k":"falling","v":0.70981,"x":0.92857,"p":[[0,24,0.0,0.91964,0.17473,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,2,0,25],[4,24,0.1667,0.92857,0.17128,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,25],[8,24,0.3333,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20],[12,24,0.5,0.86161,0.14054,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,16],[16,24,0.6667,0.72767,0.08271,0.71429,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,1,0,2],[20,24,0.8333,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[24,24,1.0,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0]]}]},{"i":"5757be06d08de6d7","q":"At a meeting of  $ 12k$  people, each person exchanges greetings with exactly  $ 3k\\plus{}6$  others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?","t":[{"b":2,"e":1.0,"k":"flat","v":0.92409,"x":1.0,"p":[[0,150,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,150,0.0267,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,150,0.0533,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,150,0.08,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,150,0.1067,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,150,0.1333,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,150,0.16,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,150,0.1867,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,150,0.2133,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,150,0.24,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,150,0.2667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,150,0.2933,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[48,150,0.32,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[52,150,0.3467,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,150,0.3733,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,150,0.4,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[64,150,0.4267,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[68,150,0.4533,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,150,0.48,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,150,0.5067,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,150,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,150,0.56,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,150,0.5867,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[92,150,0.6133,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[96,150,0.64,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[100,150,0.6667,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[104,150,0.6933,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[108,150,0.72,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[112,150,0.7467,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[116,150,0.7733,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[120,150,0.8,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[124,150,0.8267,0.92411,0.20511,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[128,150,0.8533,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[132,150,0.88,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[136,150,0.9067,0.93302,0.13828,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[140,150,0.9333,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[144,150,0.96,0.92857,0.15152,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,25],[148,150,0.9867,0.92409,0.16363,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,26],[150,150,1.0,0.94195,0.11773,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25]]},{"b":4,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,63,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,63,0.0635,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,63,0.127,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,63,0.1905,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,63,0.254,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,63,0.3175,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,63,0.381,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,63,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,63,0.5079,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[36,63,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,63,0.6349,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,63,0.6984,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,63,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,63,0.8254,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,63,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,63,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[63,63,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"72857baab51a0beb","q":"Consider the grid of points  $X = \\{(m,n) | 0 \\leq m,n \\leq 4 \\}$ . We say a pair of points  $\\{(a,b),(c,d)\\}$  in  $X$  is a knight-move pair if  $( c = a \\pm 2$  and  $d = b \\pm 1)$  or  $( c = a \\pm 1$  and  $d = b \\pm 2)$ . The number of knight-move pairs in  $X$  is:","t":[{"b":4,"e":1.0,"k":"flat","v":0.95535,"x":1.0,"p":[[0,39,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,39,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,39,0.2051,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,39,0.3077,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,39,0.4103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,39,0.5128,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,39,0.6154,0.95535,0.12596,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[28,39,0.7179,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[32,39,0.8205,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[36,39,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[39,39,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]},{"b":7,"e":0.85714,"k":"flat","v":0.93302,"x":1.0,"p":[[0,34,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,34,0.1176,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,34,0.2353,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,34,0.5882,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[24,34,0.7059,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[28,34,0.8235,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[32,34,0.9412,0.93302,0.14722,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,25],[34,34,1.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26]]}]},{"i":"fc90e7c19d8f86a6","q":"Determine the largest natural number  $r$  with the property that among any five subsets with  $500$  elements of the set  $\\{1,2,\\ldots,1000\\}$  there exist two of them which share at least  $r$  elements.","t":[{"b":6,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,75,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,75,0.0533,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,75,0.1067,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,75,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,75,0.2133,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,75,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,75,0.32,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,75,0.3733,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,75,0.4267,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,75,0.48,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,75,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,75,0.5867,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,75,0.64,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,75,0.6933,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,75,0.7467,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,75,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,75,0.8533,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,75,0.9067,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,75,0.96,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[75,75,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,27,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,27,0.1481,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,27,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"369692e11ac7b1a1","q":"An integer is square-free if it is not divisible by  $a^2$  for any integer  $a>1$ . Let  $S$  be the set of positive square-free integers. Determine, with justification, the value of\\[\\sum_{k\\epsilon S}\\left[\\sqrt{\\frac{10^{10}}{k}}\\right]\\]where  $[x]$  denote the greatest integer less than or equal to  $x$","t":[{"b":1,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,52,0.0,0.90625,0.23313,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,27],[4,52,0.0769,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[8,52,0.1538,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[12,52,0.2308,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,52,0.3077,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,52,0.3846,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,52,0.4615,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,52,0.5385,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[32,52,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,52,0.6923,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,37,0.0,0.90625,0.23313,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,27],[4,37,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,37,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,37,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,37,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d7f800cbf73c9fd0","q":"Define the sequence  $x_1, x_2, ...$  inductively by  $x_1 = \\sqrt{5}$  and  $x_{n+1} = x_n^2 - 2$  for each  $n \\geq 1$ . Compute $\\lim_{n \\to \\infty} \\frac{x_1 \\cdot x_2 \\cdot x_3 \\cdot ... \\cdot x_n}{x_{n+1}}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.97766,"x":1.0,"p":[[0,43,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,43,0.093,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,43,0.186,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,43,0.2791,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,43,0.3721,0.97766,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,43,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,43,0.5581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,43,0.6512,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[32,43,0.7442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,43,0.8372,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,43,0.9302,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[43,43,1.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,46,0.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,46,0.087,0.98661,0.07457,1.0,1.0,1.0,0.5714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,46,0.1739,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[12,46,0.2609,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[16,46,0.3478,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,46,0.4348,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[24,46,0.5217,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[28,46,0.6087,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[32,46,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,46,0.7826,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[40,46,0.8696,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,46,0.9565,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[46,46,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"b939e0e6426cd1af","q":"Consider the matrices   $$ A = \\left(\\begin{matrix} 1 & 2  0 & 1 \\end{matrix}\\right)  \\mbox{ and }  B = \\left(\\begin{matrix} 1 & 0  2 & 1 \\end{matrix}\\right). $$  Let  $k\\geq 1$  an integer. Prove that for any nonzero  $i_1,i_2,\\dots,i_{k-1},j_1,j_2,\\dots,j_k$  and any integers  $i_0,i_k$  it holds that   $$ A^{i_0}B^{j_1}A^{i_1}B^{j_2}\\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \\not = I. $$","t":[{"b":0,"e":0.57143,"k":"rising","v":0.30353,"x":0.75,"p":[[0,38,0.0,0.39285,0.36596,0.0,0.35714,0.74996,0.0,1.0,11,3,2,11,0,2,0,0,3,0,0,5,0,0,1,0,0,2,0,0,5,0,3],[4,38,0.1053,0.36607,0.38949,0.0,0.2857,0.71429,0.0,1.0,14,5,0,14,0,1,0,0,4,0,0,1,0,0,1,0,0,5,0,0,1,0,5],[8,38,0.2105,0.30353,0.3209,0.0,0.21428,0.57111,0.0,1.0,14,2,0,14,0,2,0,0,2,0,0,3,0,0,6,0,0,3,0,0,0,0,2],[12,38,0.3158,0.47322,0.38038,0.0,0.57143,0.75,0.0,1.0,9,6,0,9,0,2,0,0,3,0,0,1,0,0,4,0,0,5,0,0,2,0,6],[16,38,0.4211,0.45536,0.35434,0.0,0.57143,0.71429,0.0,1.0,9,3,0,9,0,1,0,0,5,0,0,0,0,0,2,0,0,10,0,0,2,0,3],[20,38,0.5263,0.51785,0.36377,0.25,0.57121,0.85714,0.0,1.0,7,6,0,7,0,1,0,0,4,0,0,3,0,0,2,0,0,6,0,0,3,0,6],[24,38,0.6316,0.47768,0.41743,0.0,0.42857,1.0,0.0,1.0,11,9,0,11,0,0,0,0,5,0,0,0,0,0,3,0,0,2,0,0,2,0,9],[28,38,0.7368,0.51786,0.40367,0.0,0.71429,0.85714,0.0,1.0,10,7,0,10,0,1,0,0,1,0,0,2,0,0,1,0,0,6,0,0,4,0,7],[32,38,0.8421,0.45533,0.34889,0.10714,0.571,0.71429,0.0,1.0,8,4,0,8,0,2,0,0,4,0,0,1,0,0,6,0,0,5,0,0,2,0,4],[36,38,0.9474,0.71427,0.24744,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,3,0,0,1,0,0,4,0,0,11,0,0,4,0,8],[38,38,1.0,0.75,0.23145,0.71429,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,12,0,0,8,0,7]]},{"b":7,"e":0.71429,"k":"rising","v":0.25446,"x":0.67409,"p":[[0,33,0.0,0.29018,0.31639,0.0,0.1429,0.57143,0.0,0.85714,13,0,1,13,0,4,0,0,4,0,0,2,0,0,2,0,0,3,0,0,4,0,0],[4,33,0.1212,0.33928,0.36899,0.0,0.2857,0.60714,0.0,1.0,15,3,0,15,0,0,0,0,4,0,0,1,0,0,4,0,0,2,0,0,3,0,3],[8,33,0.2424,0.44195,0.39018,0.0,0.49979,0.71429,0.0,1.0,12,6,0,12,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,0,1,0,6],[12,33,0.3636,0.33033,0.3736,0.0,0.21429,0.71429,0.0,1.0,15,4,0,15,0,1,0,0,4,0,0,1,0,0,2,0,0,4,0,0,1,0,4],[16,33,0.4848,0.30803,0.35733,0.0,0.14286,0.57143,0.0,1.0,16,3,0,16,0,0,0,0,3,0,0,4,0,0,2,0,0,2,0,0,2,0,3],[20,33,0.6061,0.3482,0.38121,0.0,0.2857,0.71429,0.0,1.0,15,4,0,15,0,0,0,0,5,0,0,0,0,0,1,0,0,6,0,0,1,0,4],[24,33,0.7273,0.35268,0.4103,0.0,0.07143,0.71429,0.0,1.0,16,6,0,16,0,1,0,0,3,0,0,0,0,0,1,0,0,4,0,0,1,0,6],[28,33,0.8485,0.25446,0.31689,0.0,0.14286,0.42857,0.0,1.0,15,2,0,15,0,3,0,0,4,0,0,5,0,0,0,0,0,1,0,0,2,0,2],[32,33,0.9697,0.54016,0.27371,0.28571,0.57143,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,8,0,0,2,0,0,5,0,0,9,0,0,2,0,3],[33,33,1.0,0.67409,0.24805,0.53539,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,13,0,0,2,0,7]]}]},{"i":"fcf93e862ab163ad","q":"Call a real-valued function $f$ very convex if  $$ \\frac{f(x)+f(y)}{2} \\geq f\\left(\\frac{x+y}{2}\\right)+|x-y| $$  holds for all real numbers $x$ and $y$. Prove that no very convex function exists.","t":[{"b":2,"e":1.0,"k":"falling","v":0.62051,"x":0.88839,"p":[[0,172,0.0,0.87946,0.22335,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,21],[4,172,0.0233,0.80357,0.30671,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,21],[8,172,0.0465,0.81247,0.20961,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,4,0,0,4,0,15],[12,172,0.0698,0.7991,0.2942,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,4,0,18],[16,172,0.093,0.88839,0.23347,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,24],[20,172,0.1163,0.84375,0.28203,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,3,0,22],[24,172,0.1395,0.75,0.29233,0.53572,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,0,2,0,15],[28,172,0.1628,0.83036,0.23266,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,3,0,18],[32,172,0.186,0.83482,0.32559,0.92857,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,24],[36,172,0.2093,0.81254,0.26101,0.57143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,3,0,0,0,0,20],[40,172,0.2326,0.83928,0.2468,0.71421,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,1,0,20],[44,172,0.2558,0.83482,0.24772,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,3,0,18],[48,172,0.2791,0.86159,0.22443,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,4,0,20],[52,172,0.3023,0.85268,0.22724,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,5,0,0,4,0,19],[56,172,0.3256,0.76786,0.29179,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,4,0,0,1,0,0,4,0,0,4,0,0,1,0,17],[60,172,0.3488,0.79018,0.29877,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,3,0,0,2,0,0,0,0,0,3,0,0,5,0,17],[64,172,0.3721,0.77679,0.30501,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,1,0,18],[68,172,0.3953,0.83482,0.25781,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,3,0,0,1,0,21],[72,172,0.4186,0.86607,0.2141,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,1,0,21],[76,172,0.4419,0.82586,0.24678,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,1,0,19],[80,172,0.4651,0.87946,0.1992,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,4,0,21],[84,172,0.4884,0.82143,0.23958,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,3,0,17],[88,172,0.5116,0.83929,0.20438,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,9,0,0,4,0,16],[92,172,0.5349,0.75892,0.24857,0.5354,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,5,0,0,4,0,13],[96,172,0.5581,0.7723,0.24186,0.57143,0.78564,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,7,0,0,2,0,14],[100,172,0.5814,0.75445,0.29502,0.53539,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,1,0,0,5,0,0,3,0,0,3,0,0,2,0,16],[104,172,0.6047,0.79018,0.24218,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,3,0,14],[108,172,0.6279,0.83929,0.22799,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,1,0,20],[112,172,0.6512,0.71874,0.29771,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,6,0,0,2,0,0,3,0,0,4,0,0,2,0,14],[116,172,0.6744,0.81687,0.29737,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,0,4,0,20],[120,172,0.6977,0.74995,0.27668,0.571,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,5,0,0,6,0,0,2,0,0,2,0,15],[124,172,0.7209,0.68971,0.31975,0.53539,0.71429,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,1,2,0,0,8,0,0,0,0,0,3,0,13],[128,172,0.7442,0.79909,0.25471,0.57132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,4,0,0,1,0,18],[132,172,0.7674,0.74998,0.27664,0.57143,0.85707,1.0,0.0,1.0,1,13,0,1,0,0,0,0,4,0,0,0,0,0,5,0,0,5,0,0,4,0,13],[136,172,0.7907,0.8125,0.20652,0.67857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,6,0,0,3,0,15],[140,172,0.814,0.70087,0.32998,0.42859,0.78571,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,3,0,0,2,0,0,5,0,0,3,0,13],[144,172,0.8372,0.76783,0.228,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,8,0,0,3,0,12],[148,172,0.8605,0.74998,0.31543,0.57132,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,4,0,0,1,0,0,2,0,0,5,0,0,2,0,16],[152,172,0.8837,0.79017,0.24218,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,4,0,0,4,0,15],[156,172,0.907,0.68301,0.29176,0.42859,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,5,0,0,5,0,0,4,0,0,1,0,12],[160,172,0.9302,0.70533,0.29221,0.53539,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,0,2,0,12],[164,172,0.9535,0.69642,0.26426,0.5357,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,6,0,0,3,0,10],[168,172,0.9767,0.73663,0.30533,0.42964,0.85707,1.0,0.0,1.0,1,15,0,1,0,0,0,0,5,0,0,3,0,0,2,0,0,3,0,0,3,0,15],[172,172,1.0,0.62051,0.2826,0.39286,0.57121,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,8,0,0,6,0,0,5,0,0,2,0,0,2,0,9]]},{"b":3,"e":0.85714,"k":"falling","v":0.59371,"x":0.90179,"p":[[0,68,0.0,0.89284,0.17499,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,21],[4,68,0.0588,0.78125,0.23953,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,6,0,0,2,0,15],[8,68,0.1176,0.77232,0.3131,0.53572,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,2,0,18],[12,68,0.1765,0.85267,0.20042,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,18],[16,68,0.2353,0.82143,0.28121,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,0,2,0,20],[20,68,0.2941,0.90179,0.16145,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,4,0,21],[24,68,0.3529,0.79909,0.21388,0.67857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,8,0,12],[28,68,0.4118,0.77679,0.28557,0.53571,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,5,0,0,1,0,0,4,0,0,2,0,17],[32,68,0.4706,0.76338,0.2615,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,6,0,0,6,0,12],[36,68,0.5294,0.79018,0.24996,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,2,0,0,2,0,17],[40,68,0.5882,0.72321,0.35344,0.53571,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,3,0,16],[44,68,0.6471,0.74554,0.23072,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,6,0,0,2,0,12],[48,68,0.7059,0.74549,0.27836,0.571,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,0,0,0,4,0,0,4,0,0,5,0,0,5,0,12],[52,68,0.7647,0.71873,0.31234,0.4286,0.78564,1.0,0.0,1.0,2,14,0,2,0,0,0,0,3,0,0,4,0,0,2,0,0,5,0,0,2,0,14],[56,68,0.8235,0.83925,0.21654,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,6,0,17],[60,68,0.8824,0.70535,0.30916,0.42859,0.85707,1.0,0.0,1.0,2,10,0,2,0,1,0,0,2,0,0,4,0,0,2,0,0,3,0,0,8,0,10],[64,68,0.9412,0.74106,0.32428,0.57142,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,3,0,0,0,0,0,6,0,0,1,0,0,3,0,16],[68,68,1.0,0.59371,0.27225,0.39286,0.57121,0.78571,0.1429,1.0,0,8,0,0,0,1,0,0,7,0,0,5,0,0,8,0,0,3,0,0,0,0,8]]}]},{"i":"34c30c4dd02daf31","q":"De\ffine the *quasi-primes* as follows. $\\bullet$  The \ffirst quasi-prime is  $q_1 = 2$  $\\bullet$  For  $n \\ge 2$ , the  $n^{th}$  quasi-prime  $q_n$  is the smallest integer greater than  $q_{n_1}$  and not of the form  $q_iq_j$  for some  $1 \\le i \\le j \\le n - 1$ .\nDetermine, with proof, whether or not  $1000$  is a quasi-prime.","t":[{"b":1,"e":1.0,"k":"flat","v":0.84373,"x":1.0,"p":[[0,25,0.0,0.94195,0.15512,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[4,25,0.16,0.84373,0.28651,0.857,1.0,1.0,0.0,1.0,3,18,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,18],[8,25,0.32,0.9375,0.2111,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[12,25,0.48,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,25,0.96,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.71429,"k":"flat","v":0.78571,"x":0.97321,"p":[[0,92,0.0,0.86161,0.26603,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,2,0,23],[4,92,0.0435,0.78571,0.38631,0.71429,1.0,1.0,0.0,1.0,6,23,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,23],[8,92,0.087,0.81695,0.34853,0.85711,1.0,1.0,0.0,1.0,4,22,4,4,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,22],[12,92,0.1304,0.84813,0.31751,0.85714,1.0,1.0,0.0,1.0,3,23,3,3,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,23],[16,92,0.1739,0.81241,0.34907,0.82143,1.0,1.0,0.0,1.0,4,22,4,4,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,22],[20,92,0.2174,0.91964,0.19212,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[24,92,0.2609,0.85701,0.32933,1.0,1.0,1.0,0.0,1.0,4,25,2,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[28,92,0.3043,0.91515,0.15514,0.85711,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,21],[32,92,0.3478,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[36,92,0.3913,0.83482,0.29475,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,21],[40,92,0.4348,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[44,92,0.4783,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[48,92,0.5217,0.92411,0.18893,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[52,92,0.5652,0.9241,0.20198,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[56,92,0.6087,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[60,92,0.6522,0.87945,0.20238,0.857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,5,0,20],[64,92,0.6957,0.94643,0.1171,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[68,92,0.7391,0.86605,0.1673,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,6,0,16],[72,92,0.7826,0.94629,0.12268,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[76,92,0.8261,0.95982,0.10853,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[80,92,0.8696,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[84,92,0.913,0.96874,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[88,92,0.9565,0.91525,0.16695,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,22],[92,92,1.0,0.84807,0.1955,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,7,0,0,4,0,17]]}]},{"i":"fea7bb11255e5a2a","q":"Consider the set  $S$  of lattice points with positive coordinates in the plane. For each point  $P(a,b)$  from  $S$ , we draw a segment between it and each of the points in the set \\[S(P)=\\{(a+b,c)\\mid c\\in\\mathbb{Z}, \\, c>a+b\\}.\\] Show that there is no colouring of the points in  $S$  with a finite number of colours such that every two points joined by a segment are coloured with different colours.\n\n*Ioan Tomescu*","t":[{"b":2,"e":0.1429,"k":"flat","v":0.02232,"x":0.30357,"p":[[0,56,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,27,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.30357,0.26184,0.14286,0.14286,0.42858,0.14286,1.0,0,2,0,0,0,21,0,0,2,0,0,2,0,0,2,0,0,3,0,0,0,0,2],[8,56,0.1429,0.1874,0.13569,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,56,0.2143,0.19634,0.14177,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,25,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[16,56,0.2857,0.18295,0.14393,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,24,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[20,56,0.3571,0.18302,0.14386,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,23,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[24,56,0.4286,0.14732,0.11564,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,26,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,56,0.5,0.15178,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.20534,0.16725,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,23,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[36,56,0.6429,0.19187,0.12686,0.14286,0.14286,0.17857,0.0,0.42857,3,0,0,3,0,21,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.16072,0.20124,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,21,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[44,56,0.7857,0.19634,0.17408,0.14286,0.14286,0.17857,0.0,0.71429,5,0,0,5,0,19,0,0,3,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[48,56,0.8571,0.20972,0.15147,0.14286,0.14286,0.14287,0.0,0.71429,1,0,0,1,0,24,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[52,56,0.9286,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14,"k":"flat","v":0.03572,"x":0.17856,"p":[[0,28,0.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,23,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.16955,0.07527,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.16517,0.11351,0.14286,0.14286,0.14286,0.0,0.571,3,0,0,3,0,25,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,28,0.4286,0.12937,0.07455,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.17856,0.12873,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,23,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,28,0.7143,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,28,0.8571,0.16045,0.11714,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,28,1.0,0.16697,0.09533,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,25,0,1,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"785ee0182c897d4f","q":"Determine the minimum possible amount of distinct prime divisors of  $19^{4n}+4$ , for a positive integer  $n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,56,0.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[4,56,0.0714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,56,0.2857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,56,0.3571,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[24,56,0.4286,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[28,56,0.5,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,56,0.5714,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[36,56,0.6429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,56,0.7143,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[44,56,0.7857,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[48,56,0.8571,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,56,0.9286,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[56,56,1.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22]]},{"b":5,"e":0.57143,"k":"falling","v":0.73212,"x":1.0,"p":[[0,60,0.0,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[4,60,0.0667,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,60,0.1333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,60,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,60,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,60,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,60,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,60,0.4667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,60,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,60,0.6667,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[44,60,0.7333,0.95076,0.111,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[48,60,0.8,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[52,60,0.8667,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[56,60,0.9333,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[60,60,1.0,0.73212,0.20126,0.57143,0.57143,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,1,0,11]]}]},{"i":"c17e6e7b201f43d8","q":"Determine whether there exists a positive integer  $n$  such that it is possible to find at least  $2018$  different quadruples  $(x,y,z,t)$  of positive integers that simultaneously satisfy equations  $$ \\begin{cases}\n      x+y+z=n\n      xyz = 2t^3.\n\\end{cases} $$","t":[{"b":3,"e":1.0,"k":"rising","v":0.51339,"x":0.97768,"p":[[0,28,0.0,0.72767,0.36134,0.39286,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,18],[4,28,0.1429,0.86607,0.28333,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,24],[8,28,0.2857,0.51339,0.45856,0.0,0.35714,1.0,0.0,1.0,10,14,0,10,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,14],[12,28,0.4286,0.64732,0.40718,0.25,1.0,1.0,0.0,1.0,5,17,0,5,0,3,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,17],[16,28,0.5714,0.91071,0.21354,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,23],[20,28,0.7143,0.89732,0.1794,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,6,0,21],[24,28,0.8571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,28,1.0,0.93303,0.22583,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28]]},{"b":4,"e":1.0,"k":"flat","v":0.75447,"x":0.98661,"p":[[0,37,0.0,0.75447,0.35217,0.4286,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,19],[4,37,0.1081,0.96429,0.15567,1.0,1.0,1.0,0.1429,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[8,37,0.2162,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,37,0.3243,0.91071,0.26666,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[16,37,0.4324,0.93749,0.19215,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[20,37,0.5405,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[24,37,0.6486,0.9241,0.22584,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[28,37,0.7568,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[32,37,0.8649,0.90625,0.20395,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,24],[36,37,0.973,0.87937,0.22364,0.85714,1.0,1.0,0.14,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,19],[37,37,1.0,0.81695,0.26544,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,0,3,0,18]]}]},{"i":"49975c517a9095d6","q":"Does there exist a function  $f : \\mathbb N \\to \\mathbb N$ , such that  $f(f(n)) =n + 1987$  for every natural number  $n$ ? *(IMO Problem 4)*\n\n*Proposed by Vietnam.*","t":[{"b":1,"e":0.85714,"k":"flat","v":0.08482,"x":0.35713,"p":[[0,35,0.0,0.24107,0.3223,0.0,0.14286,0.35713,0.0,1.0,14,1,3,14,0,9,0,0,1,0,0,0,0,0,3,0,0,0,0,0,4,0,1],[4,35,0.1143,0.35713,0.37457,0.0,0.14286,0.75,0.0,1.0,11,3,0,11,0,7,0,0,1,0,0,1,0,0,3,0,0,1,0,0,5,0,3],[8,35,0.2286,0.21429,0.33312,0.0,0.0,0.17857,0.0,1.0,18,2,0,18,0,6,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,2],[12,35,0.3429,0.14731,0.24347,0.0,0.0,0.14287,0.0,0.85714,19,0,0,19,0,6,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,0],[16,35,0.4571,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,6,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,35,0.5714,0.17401,0.24931,0.0,0.14143,0.14287,0.0,0.85714,15,0,0,15,0,10,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0],[24,35,0.6857,0.0892,0.19147,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[28,35,0.8,0.10268,0.212,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[32,35,0.9143,0.25893,0.32426,0.0,0.14286,0.32143,0.0,1.0,12,3,0,12,0,8,0,0,4,0,0,3,0,0,0,0,0,0,0,0,2,0,3],[35,35,1.0,0.15177,0.23939,0.0,0.0,0.2857,0.0,0.85714,19,0,0,19,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,0,2,0,0]]},{"b":7,"e":0.57143,"k":"rising","v":0.26784,"x":0.73658,"p":[[0,22,0.0,0.26784,0.34946,0.0,0.14286,0.57111,0.0,1.0,15,4,4,15,0,6,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,4],[4,22,0.1818,0.57143,0.32733,0.39286,0.57143,0.85714,0.0,1.0,3,4,0,3,0,4,0,0,1,0,0,6,0,0,3,0,0,2,0,0,9,0,4],[8,22,0.3636,0.58929,0.30462,0.39285,0.57143,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,2,0,0,4,0,0,6,0,0,1,0,0,9,0,4],[12,22,0.5455,0.64285,0.24485,0.57132,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,1,0,0,3,0,0,10,0,0,4,0,0,7,0,4],[16,22,0.7273,0.73658,0.22049,0.57143,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,1,0,0,8,0,0,3,0,0,13,0,5],[20,22,0.9091,0.65177,0.25238,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,4,0,0,1,0,0,2,0,0,6,0,0,6,0,0,11,0,2],[22,22,1.0,0.71427,0.22017,0.57143,0.78571,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,2,0,0,5,0,0,7,0,0,14,0,2]]}]},{"i":"5b394553da611bf5","q":"Determine continuous functions  $f:\\mathbb{R}\\to \\mathbb{R}$  such that  $\\left( {{a}^{2}}+ab+{{b}^{2}} \\right)\\int\\limits_{a}^{b}{f\\left( x \\right)dx=3\\int\\limits_{a}^{b}{{{x}^{2}}f\\left( x \\right)dx,}}$  for every  $a,b\\in \\mathbb{R}$  .","t":[{"b":3,"e":1.0,"k":"flat","v":0.96875,"x":0.99107,"p":[[0,8,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,8,0.5,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,8,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.77232,"x":0.97768,"p":[[0,22,0.0,0.91071,0.14617,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,21],[4,22,0.1818,0.94196,0.10012,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[8,22,0.3636,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[12,22,0.5455,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,22,0.7273,0.91518,0.12807,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,20],[20,22,0.9091,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[22,22,1.0,0.77232,0.18509,0.57143,0.78571,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,3,0,0,6,0,10]]}]},{"i":"742e012b8be169ec","q":"Each rational point on a real line is assigned an integer.  Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0067,"x":0.20535,"p":[[0,50,0.0,0.07066,0.14623,0.0,0.0,0.05311,0.0,0.57143,23,0,1,24,1,3,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,50,0.08,0.20535,0.33299,0.0,0.0,0.21429,0.0,1.0,19,1,0,19,3,2,0,0,0,0,0,1,0,0,1,0,0,1,1,0,3,0,1],[8,50,0.16,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.15625,0.26752,0.0,0.0,0.17857,0.0,1.0,19,1,0,19,2,3,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,1],[16,50,0.32,0.10263,0.20512,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,1,4,0,0,1,0,1,0,0,0,2,0,0,0,0,0,1,0,0],[20,50,0.4,0.1183,0.22631,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,1,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0],[24,50,0.48,0.08034,0.17831,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[28,50,0.56,0.07589,0.19556,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,50,0.64,0.06025,0.15472,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,1,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[36,50,0.72,0.08929,0.1915,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,50,0.8,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.07143,0.16269,0.0,0.0,0.08929,0.0,0.85714,22,0,0,22,2,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,50,0.96,0.07588,0.15861,0.0,0.0,0.01786,0.0,0.57143,24,0,0,24,1,2,0,0,2,0,1,0,0,0,2,0,0,0,0,0,0,0,0],[50,50,1.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.1607,"p":[[0,43,0.0,0.14955,0.24246,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,1,6,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[4,43,0.093,0.13839,0.21572,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,7,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[8,43,0.186,0.0692,0.16316,0.0,0.0,0.08929,0.0,0.85714,23,0,0,23,1,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,43,0.2791,0.0758,0.15142,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,43,0.3721,0.1607,0.26664,0.0,0.0,0.1429,0.0,1.0,18,2,0,18,0,7,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[20,43,0.4651,0.13391,0.22567,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,2,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[24,43,0.5581,0.09598,0.18351,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,1,4,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[28,43,0.6512,0.10268,0.21498,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[32,43,0.7442,0.02455,0.0524,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.1116,0.23819,0.0,0.0,0.07143,0.0,0.85714,23,0,0,23,2,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[40,43,0.9302,0.03348,0.07359,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b16d28aeb7dd9b5d","q":"Each of the numbers  $1, 2, 3,... , 2006^2$  is placed at random into a cell of a  $2006\u0002\\times 2006$  board. Prove that there exist two cells which share a common side or a common vertex such that the sum of the numbers in them is divisible by  $4$ . (4)","t":[{"b":2,"e":0.14286,"k":"falling","v":0.27232,"x":0.875,"p":[[0,44,0.0,0.84822,0.29219,0.82143,1.0,1.0,0.0,1.0,2,23,2,2,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,23],[4,44,0.0909,0.875,0.27837,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,24],[8,44,0.1818,0.86607,0.23941,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,3,0,21],[12,44,0.2727,0.83927,0.24681,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,5,0,0,1,0,20],[16,44,0.3636,0.77232,0.28316,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,4,0,0,2,0,0,2,0,0,7,0,14],[20,44,0.4545,0.70536,0.37786,0.39286,1.0,1.0,0.0,1.0,4,17,0,4,0,2,0,0,2,0,0,1,0,0,1,0,0,4,0,0,1,0,17],[24,44,0.5455,0.73213,0.31084,0.57132,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,4,0,0,1,0,0,3,0,0,5,0,0,3,0,14],[28,44,0.6364,0.75,0.31944,0.71429,0.85714,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,7,0,13],[32,44,0.7273,0.44195,0.3376,0.14286,0.35714,0.74996,0.0,1.0,4,5,0,4,0,6,0,0,6,0,0,5,0,0,2,0,0,1,0,0,3,0,5],[36,44,0.8182,0.54018,0.3792,0.14286,0.42857,1.0,0.0,1.0,3,10,0,3,0,7,0,0,3,0,0,4,0,0,1,0,0,2,0,0,2,0,10],[40,44,0.9091,0.52678,0.33012,0.28571,0.42857,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,10,0,0,3,0,0,0,0,0,3,0,0,6,0,5],[44,44,1.0,0.27232,0.17985,0.2857,0.28571,0.28571,0.0,1.0,5,1,0,5,0,2,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.73213,"x":0.98214,"p":[[0,39,0.0,0.82589,0.31488,0.71429,1.0,1.0,0.0,1.0,3,22,3,3,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,22],[4,39,0.1026,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[8,39,0.2051,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[12,39,0.3077,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[16,39,0.4103,0.95536,0.13091,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[20,39,0.5128,0.84821,0.23673,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,4,0,19],[24,39,0.6154,0.76337,0.24383,0.57143,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,3,0,0,7,0,0,4,0,0,5,0,12],[28,39,0.7179,0.73213,0.25939,0.57132,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,3,0,0,2,0,13],[32,39,0.8205,0.8616,0.16935,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,3,0,17],[36,39,0.9231,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[39,39,1.0,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27]]}]},{"i":"7a80fc130db1675d","q":"Does there exist an irrational number $\\alpha>1$ such that\n\n$$\n\\left\\lfloor\\alpha^{n}\\right\\rfloor \\equiv 0 \\quad(\\bmod 2017)\n$$\n\nfor all integers $n \\geq 1$ ?","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.77679,"p":[[0,66,0.0,0.77679,0.41178,0.96429,1.0,1.0,0.0,1.0,7,24,2,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[4,66,0.0606,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,66,0.1212,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,66,0.303,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,66,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,66,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,66,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,66,0.7879,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.65625,"p":[[0,73,0.0,0.65625,0.47496,0.0,1.0,1.0,0.0,1.0,11,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[4,73,0.0548,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,73,0.1096,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,73,0.1644,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,73,0.3288,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,73,0.3836,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,73,0.4384,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,73,0.5479,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,73,0.6027,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,73,0.6575,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,73,0.7123,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,73,0.7671,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,73,0.9315,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7f236b32ab3d4c00","q":"Find all  $n$  for which there exists a convex  $n$ -gon all of whose diagonals are equal.","t":[{"b":1,"e":0.57143,"k":"rising","v":0.17857,"x":0.54002,"p":[[0,46,0.0,0.17857,0.09449,0.14286,0.14286,0.1786,0.0,0.42857,2,0,2,2,0,22,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.29911,0.17627,0.14286,0.2857,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,10,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[8,46,0.1739,0.3616,0.22864,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,9,0,0,7,0,0,2,0,0,1,0,0,2,0,1],[12,46,0.2609,0.39285,0.20823,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,14,0,0,8,0,0,2,0,0,1,0,0,2,0,1],[16,46,0.3478,0.36161,0.2082,0.14286,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,10,0,0,5,0,0,13,0,0,1,0,0,1,0,0,1,0,1],[20,46,0.4348,0.25857,0.17679,0.14286,0.14286,0.32143,0.14,0.85714,0,0,0,0,0,19,0,0,5,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[24,46,0.5217,0.22768,0.11214,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,18,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,46,0.6087,0.27659,0.1596,0.14286,0.2857,0.32143,0.14,0.71429,0,0,0,0,0,15,0,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[32,46,0.6957,0.24097,0.13573,0.14286,0.14286,0.2857,0.14,0.57143,0,0,0,0,0,18,0,0,9,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[36,46,0.7826,0.35705,0.22877,0.14286,0.28571,0.57143,0.14,0.85714,0,0,0,0,0,13,0,0,7,0,0,1,0,0,6,0,0,4,0,0,1,0,0],[40,46,0.8696,0.35258,0.21727,0.14286,0.28571,0.57143,0.14,0.85714,0,0,0,0,0,13,0,0,6,0,0,2,0,0,8,0,0,2,0,0,1,0,0],[44,46,0.9565,0.39732,0.19144,0.24999,0.42856,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,8,0,0,0,0,0,15,0,0,1,0,0,0,0,0],[46,46,1.0,0.54002,0.13232,0.57075,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,2,0,0,23,0,0,2,0,0,1,0,0]]},{"b":6,"e":0.28571,"k":"rising","v":0.25442,"x":0.45982,"p":[[0,48,0.0,0.25442,0.15469,0.14286,0.14288,0.28571,0.14,0.71429,0,0,0,0,0,18,0,0,7,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,48,0.0833,0.45978,0.25699,0.28571,0.42857,0.46536,0.14,1.0,0,4,0,0,0,5,0,0,6,0,0,13,0,0,2,0,0,1,0,0,1,0,4],[8,48,0.1667,0.39732,0.16261,0.28571,0.42857,0.4286,0.14286,0.857,0,0,0,0,0,6,0,0,4,0,0,15,0,0,6,0,0,0,0,0,1,0,0],[12,48,0.25,0.4375,0.18189,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,20,0,0,2,0,0,0,0,0,2,0,1],[16,48,0.3333,0.38839,0.10853,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,9,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[20,48,0.4167,0.41962,0.12336,0.42857,0.42857,0.42858,0.0,0.57143,1,0,0,1,0,1,0,0,4,0,0,19,0,0,7,0,0,0,0,0,0,0,0],[24,48,0.5,0.45982,0.17399,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,21,0,0,2,0,0,0,0,0,1,0,2],[28,48,0.5833,0.38396,0.10374,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,9,0,0,18,0,0,3,0,0,0,0,0,0,0,0],[32,48,0.6667,0.4464,0.13714,0.42857,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,21,0,0,4,0,0,0,0,0,2,0,0],[36,48,0.75,0.45533,0.11536,0.42857,0.42857,0.42858,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,25,0,0,4,0,0,0,0,0,0,0,1],[40,48,0.8333,0.45536,0.15746,0.42857,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,25,0,0,2,0,0,0,0,0,0,0,2],[44,48,0.9167,0.43302,0.12618,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,22,0,0,3,0,0,0,0,0,0,0,1],[48,48,1.0,0.43749,0.1181,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,25,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"2ab6cd80686bc279","q":"Find all finite sets of positive integers with at least two elements such that for any two numbers  $ a$ ,  $ b$  ( $ a > b$ ) belonging to the set, the number  $ \\frac {b^2}{a \\minus{} b}$  belongs to the set, too.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.75443,"x":0.94643,"p":[[0,51,0.0,0.77232,0.21387,0.57143,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,6,0,0,4,0,12],[4,51,0.0784,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[8,51,0.1569,0.94196,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[12,51,0.2353,0.88391,0.15749,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,18],[16,51,0.3137,0.82586,0.22515,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,0,3,0,17],[20,51,0.3922,0.93302,0.12873,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[24,51,0.4706,0.89729,0.16846,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,22],[28,51,0.549,0.90625,0.17353,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,1,0,24],[32,51,0.6275,0.88837,0.18811,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,7,0,20],[36,51,0.7059,0.93303,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[40,51,0.7843,0.80357,0.1915,0.67857,0.85707,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,4,0,13],[44,51,0.8627,0.83481,0.17898,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,6,0,14],[48,51,0.9412,0.83033,0.22145,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,1,0,19],[51,51,1.0,0.75443,0.22087,0.571,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,0,5,0,11]]},{"b":7,"e":0.4286,"k":"falling","v":0.52228,"x":0.95536,"p":[[0,39,0.0,0.82139,0.19888,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,2,0,0,7,0,14],[4,39,0.1026,0.91518,0.18851,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[8,39,0.2051,0.91517,0.16314,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[12,39,0.3077,0.90622,0.14117,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,20],[16,39,0.4103,0.87053,0.19678,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,3,0,20],[20,39,0.5128,0.88837,0.13712,0.82132,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,17],[24,39,0.6154,0.875,0.18123,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,5,0,19],[28,39,0.7179,0.95536,0.12078,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[32,39,0.8205,0.85712,0.22018,0.82132,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,19],[36,39,0.9231,0.83035,0.20958,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,6,0,16],[39,39,1.0,0.52228,0.14986,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,20,0,0,7,0,0,2,0,0,2,0,1]]}]},{"i":"27aa52f54707f6bd","q":"Find all functions  $f : R \\to  R$  satisfying the conditions:\n1.  $f (x + 1) \\ge  f (x) + 1$  for all  $x \\in R$   \n2.  $f (x y) \\ge f (x)f (y)$  for all  $x, y \\in R$","t":[{"b":3,"e":0.42857,"k":"rising","v":0.25893,"x":0.58923,"p":[[0,111,0.0,0.25893,0.2126,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,18,0,0,8,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[4,111,0.036,0.58923,0.23077,0.4286,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,4,0,0,13,0,0,5,0,0,0,0,5],[8,111,0.0721,0.49552,0.23685,0.28571,0.57143,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,4,0,0,5,0,0,11,0,0,3,0,0,3,0,1],[12,111,0.1081,0.54014,0.269,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,6,0,0,3,0,0,5,0,0,4,0,0,9,0,0,2,0,3],[16,111,0.1441,0.52677,0.28219,0.28571,0.57121,0.71429,0.0,1.0,1,4,0,1,0,6,0,0,2,0,0,5,0,0,5,0,0,9,0,0,0,0,4],[20,111,0.1802,0.55357,0.23077,0.39286,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,6,0,0,7,0,0,6,0,0,2,0,3],[24,111,0.2162,0.53121,0.19637,0.39286,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,4,0,0,12,0,0,5,0,0,2,0,1],[28,111,0.2523,0.58035,0.24205,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,2,0,0,5,0,0,5,0,0,12,0,0,1,0,3],[32,111,0.2883,0.46429,0.22868,0.28571,0.42859,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,6,0,0,8,0,0,7,0,0,5,0,0,1,0,1],[36,111,0.3243,0.52678,0.21558,0.42857,0.5,0.60714,0.14286,1.0,0,2,0,0,0,3,0,0,2,0,0,11,0,0,8,0,0,4,0,0,2,0,2],[40,111,0.3604,0.55353,0.21053,0.42857,0.57141,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,6,0,0,6,0,0,12,0,0,1,0,1],[44,111,0.3964,0.51782,0.22797,0.42857,0.57143,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,3,0,0,4,0,0,12,0,0,7,0,0,1,0,1],[48,111,0.4324,0.52678,0.25614,0.28571,0.57143,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,4,0,0,5,0,0,5,0,0,8,0,0,4,0,1],[52,111,0.4685,0.50892,0.20805,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,3,0,0,6,0,0,9,0,0,9,0,0,1,0,0],[56,111,0.5045,0.51336,0.2338,0.28571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,3,0,0,2,0,0,12,0,0,6,0,0,2,0,1],[60,111,0.5405,0.53557,0.17876,0.42857,0.57141,0.71429,0.14,0.71429,0,0,0,0,0,3,0,0,2,0,0,6,0,0,10,0,0,11,0,0,0,0,0],[64,111,0.5766,0.47766,0.23584,0.39286,0.57143,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,2,0,0,4,0,0,14,0,0,5,0,0,0,0,1],[68,111,0.6126,0.51335,0.21386,0.39286,0.571,0.60714,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,6,0,0,10,0,0,6,0,0,0,0,2],[72,111,0.6486,0.49106,0.23402,0.39286,0.42859,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,9,0,0,6,0,0,6,0,0,2,0,1],[76,111,0.6847,0.51783,0.19479,0.39286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,4,0,0,13,0,0,5,0,0,1,0,1],[80,111,0.7207,0.55351,0.15871,0.42859,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,9,0,0,9,0,0,10,0,0,1,0,0],[84,111,0.7568,0.43304,0.21865,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,4,0,0,8,0,0,8,0,0,3,0,0,2,0,0],[88,111,0.7928,0.4732,0.19044,0.39286,0.42857,0.57143,0.14286,0.857,0,0,0,0,0,4,0,0,4,0,0,10,0,0,7,0,0,6,0,0,1,0,0],[92,111,0.8288,0.5491,0.23449,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,8,0,0,8,0,0,5,0,0,2,0,3],[96,111,0.8649,0.46426,0.21723,0.2857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,4,0,0,10,0,0,8,0,0,3,0,0,0,0,2],[100,111,0.9009,0.49101,0.24981,0.39286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,6,0,0,2,0,0,11,0,0,6,0,0,1,0,0,4,0,2],[104,111,0.9369,0.53124,0.18976,0.42857,0.4998,0.71429,0.1429,1.0,0,1,0,0,0,2,0,0,2,0,0,12,0,0,6,0,0,8,0,0,1,0,1],[108,111,0.973,0.45981,0.18117,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,2,0,0,12,0,0,7,0,0,6,0,0,0,0,0],[111,111,1.0,0.51783,0.19803,0.42857,0.571,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,11,0,0,12,0,0,2,0,0,1,0,2]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.69642,"p":[[0,208,0.0,0.2499,0.15975,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,19,0,0,7,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[4,208,0.0192,0.53125,0.2506,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,4,0,0,7,0,0,9,0,0,0,0,3],[8,208,0.0385,0.49996,0.20824,0.42857,0.42857,0.60714,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,11,0,0,7,0,0,6,0,0,1,0,1],[12,208,0.0577,0.53123,0.25312,0.42857,0.57143,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,3,0,0,6,0,0,9,0,0,5,0,0,3,0,2],[16,208,0.0769,0.54908,0.24771,0.42857,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,7,0,0,6,0,0,7,0,0,2,0,3],[20,208,0.0962,0.6607,0.28516,0.53539,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,4,0,0,1,0,0,6,0,0,7,0,0,2,0,9],[24,208,0.1154,0.58927,0.23891,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,2,0,0,4,0,0,9,0,0,9,0,0,2,0,3],[28,208,0.1346,0.50438,0.21439,0.42857,0.57143,0.71429,0.14,1.0,0,1,0,0,0,5,0,0,2,0,0,8,0,0,7,0,0,9,0,0,0,0,1],[32,208,0.1538,0.50892,0.23402,0.42857,0.4286,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,0,0,0,12,0,0,6,0,0,6,0,0,1,0,2],[36,208,0.1731,0.50887,0.18876,0.28571,0.571,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,8,0,0,4,0,0,13,0,0,4,0,0,1,0,1],[40,208,0.1923,0.53567,0.23957,0.39286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,3,0,0,4,0,0,9,0,0,6,0,0,4,0,1],[44,208,0.2115,0.57589,0.25625,0.39286,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,2,0,0,9,0,0,6,0,0,4,0,3],[48,208,0.2308,0.42409,0.23002,0.28571,0.42857,0.57111,0.0,1.0,1,1,0,1,0,5,0,0,7,0,0,10,0,0,2,0,0,5,0,0,1,0,1],[52,208,0.25,0.50889,0.2647,0.39286,0.4998,0.57143,0.14286,1.0,0,5,0,0,0,6,0,0,2,0,0,8,0,0,10,0,0,1,0,0,0,0,5],[56,208,0.2692,0.5357,0.23145,0.42857,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,13,0,0,6,0,0,3,0,0,1,0,4],[60,208,0.2885,0.54909,0.22047,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,7,0,0,7,0,0,9,0,0,1,0,2],[64,208,0.3077,0.47322,0.25614,0.28571,0.57143,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,3,0,0,5,0,0,10,0,0,3,0,0,3,0,1],[68,208,0.3269,0.48661,0.26693,0.28571,0.42859,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,4,0,0,7,0,0,4,0,0,6,0,0,1,0,3],[72,208,0.3462,0.55356,0.21354,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,8,0,0,5,0,0,9,0,0,3,0,1],[76,208,0.3654,0.5714,0.19562,0.42857,0.57143,0.60714,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,5,0,0,14,0,0,4,0,0,1,0,3],[80,208,0.3846,0.50893,0.24206,0.42857,0.5,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,3,0,0,9,0,0,9,0,0,3,0,0,1,0,3],[84,208,0.4038,0.50891,0.25489,0.39285,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,6,0,0,2,0,0,9,0,0,5,0,0,6,0,0,1,0,3],[88,208,0.4231,0.58482,0.27283,0.42857,0.57143,0.85704,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,8,0,0,3,0,0,5,0,0,4,0,5],[92,208,0.4423,0.54462,0.30185,0.39286,0.57121,0.71429,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,6,0,0,7,0,0,4,0,0,1,0,6],[96,208,0.4615,0.51328,0.21101,0.42857,0.57121,0.71429,0.14,1.0,0,1,0,0,0,4,0,0,3,0,0,7,0,0,9,0,0,7,0,0,1,0,1],[100,208,0.4808,0.52679,0.32623,0.14289,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,9,0,0,2,0,0,7,0,0,2,0,0,3,0,0,2,0,7],[104,208,0.5,0.49997,0.24222,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,4,0,0,6,0,0,7,0,0,7,0,0,3,0,0,3,0,2],[108,208,0.5192,0.5758,0.26856,0.42857,0.57143,0.71429,0.14,1.0,0,6,0,0,0,4,0,0,1,0,0,10,0,0,5,0,0,5,0,0,1,0,6],[112,208,0.5385,0.62498,0.28291,0.53539,0.71429,0.85704,0.0,1.0,2,5,0,2,0,2,0,0,2,0,0,2,0,0,6,0,0,9,0,0,4,0,5],[116,208,0.5577,0.67854,0.22589,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,0,0,0,3,0,0,10,0,0,7,0,0,4,0,6],[120,208,0.5769,0.63392,0.333,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,4,0,0,2,0,0,2,0,0,3,0,0,6,0,9],[124,208,0.5962,0.62499,0.31492,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,4,0,0,1,0,0,7,0,0,4,0,0,1,0,10],[128,208,0.6154,0.69642,0.30041,0.42857,0.78571,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,2,0,0,5,0,0,4,0,0,2,0,0,4,0,12],[132,208,0.6346,0.63388,0.24206,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,4,0,0,5,0,0,7,0,0,4,0,0,7,0,4],[136,208,0.6538,0.69196,0.29257,0.53571,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,0,0,0,3,0,0,3,0,0,5,0,0,8,0,8],[140,208,0.6731,0.06696,0.21718,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[144,208,0.6923,0.02678,0.14911,0.0,0.0,0.0,0.0,0.857,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[148,208,0.7115,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[152,208,0.7308,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[156,208,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[160,208,0.7692,0.08482,0.26453,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[164,208,0.7885,0.04463,0.17651,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[168,208,0.8077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[172,208,0.8269,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[176,208,0.8462,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[180,208,0.8654,0.08482,0.25719,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[184,208,0.8846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[188,208,0.9038,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[192,208,0.9231,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[196,208,0.9423,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[200,208,0.9615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[204,208,0.9808,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[208,208,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0c16ca3ce5849e93","q":"Find  all nonzero  integers   $a, b, c, d$    with    $a>b>c>d$   that  satisfy   $ab+cd=34$  and  $ac-bd=19.$","t":[{"b":4,"e":0.14286,"k":"flat","v":0.24554,"x":0.81696,"p":[[0,337,0.0,0.43308,0.3754,0.14286,0.14286,1.0,0.14286,1.0,0,9,0,0,0,18,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,9],[4,337,0.0119,0.71429,0.36246,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,2,0,0,1,0,0,0,0,0,4,0,0,0,0,18],[8,337,0.0237,0.57581,0.39857,0.14286,0.71429,1.0,0.14,1.0,0,13,0,0,0,13,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,13],[12,337,0.0356,0.5758,0.3839,0.14286,0.57143,1.0,0.14,1.0,0,13,0,0,0,11,0,0,3,0,0,1,0,0,2,0,0,2,0,0,0,0,13],[16,337,0.0475,0.79911,0.309,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,20],[20,337,0.0593,0.68304,0.3908,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,10,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,18],[24,337,0.0712,0.71875,0.3668,0.4643,1.0,1.0,0.0,1.0,1,18,1,1,0,7,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,18],[28,337,0.0831,0.71429,0.36596,0.25001,1.0,1.0,0.14286,1.0,0,18,0,0,0,8,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,18],[32,337,0.095,0.79018,0.31539,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,20],[36,337,0.1068,0.70089,0.35956,0.46429,0.85714,1.0,0.0,1.0,1,16,1,1,0,7,0,0,0,0,0,0,0,0,2,0,0,6,0,0,0,0,16],[40,337,0.1187,0.66518,0.36177,0.25,0.71429,1.0,0.0,1.0,1,14,1,1,0,7,0,0,1,0,0,1,0,0,2,0,0,5,0,0,1,0,14],[44,337,0.1306,0.76786,0.33072,0.53571,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,3,0,0,1,0,0,1,0,0,3,0,0,0,0,20],[48,337,0.1424,0.66964,0.35792,0.14286,0.71429,1.0,0.14286,1.0,0,14,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,14],[52,337,0.1543,0.81696,0.26543,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,6,0,0,1,0,19],[56,337,0.1662,0.79464,0.29001,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,6,0,0,0,0,19],[60,337,0.178,0.75,0.31339,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,5,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,16],[64,337,0.1899,0.64286,0.33503,0.35714,0.71429,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,0,0,0,3,0,0,0,0,0,10,0,0,0,0,11],[68,337,0.2018,0.63393,0.36235,0.14286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,1,0,0,2,0,0,2,0,0,4,0,0,1,0,13],[72,337,0.2136,0.60268,0.35845,0.24999,0.71429,1.0,0.14286,1.0,0,12,0,0,0,8,0,0,4,0,0,2,0,0,1,0,0,5,0,0,0,0,12],[76,337,0.2255,0.59821,0.37191,0.14286,0.64286,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,4,0,0,1,0,0,2,0,0,3,0,0,0,0,13],[80,337,0.2374,0.69643,0.36202,0.14286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,15],[84,337,0.2493,0.69196,0.3409,0.39286,0.71429,1.0,0.14286,1.0,0,14,0,0,0,7,0,0,1,0,0,1,0,0,1,0,0,7,0,0,1,0,14],[88,337,0.2611,0.62946,0.41011,0.14286,0.85714,1.0,0.0,1.0,2,16,1,2,0,10,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,16],[92,337,0.273,0.73661,0.34462,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,18],[96,337,0.2849,0.7232,0.30502,0.57132,0.71429,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,2,0,0,1,0,0,3,0,0,7,0,0,1,0,14],[100,337,0.2967,0.67411,0.32387,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,0,0,0,2,0,0,1,0,0,9,0,0,2,0,11],[104,337,0.3086,0.60714,0.36943,0.14286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,3,0,0,2,0,0,1,0,0,4,0,0,0,0,13],[108,337,0.3205,0.56241,0.34255,0.14286,0.64286,0.89286,0.14,1.0,0,8,0,0,0,10,0,0,1,0,0,4,0,0,1,0,0,6,0,0,2,0,8],[112,337,0.3323,0.55356,0.37923,0.14286,0.57121,1.0,0.14286,1.0,0,11,0,0,0,12,0,0,3,0,0,0,0,0,2,0,0,3,0,0,1,0,11],[116,337,0.3442,0.41955,0.34437,0.14286,0.14286,0.71429,0.14,1.0,0,6,0,0,0,18,0,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,6],[120,337,0.3561,0.40625,0.30745,0.14286,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,14,0,0,3,0,0,6,0,0,2,0,0,2,0,0,0,0,5],[124,337,0.368,0.50446,0.37455,0.14286,0.35714,1.0,0.14286,1.0,0,10,0,0,0,14,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,10],[128,337,0.3798,0.44643,0.31288,0.14286,0.35714,0.71429,0.14286,1.0,0,5,0,0,0,12,0,0,4,0,0,4,0,0,2,0,0,5,0,0,0,0,5],[132,337,0.3917,0.55795,0.40942,0.14286,0.5,1.0,0.14,1.0,0,14,0,0,0,15,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,14],[136,337,0.4036,0.53115,0.3917,0.14286,0.42836,1.0,0.14,1.0,0,12,0,0,0,14,0,0,2,0,0,0,0,0,3,0,0,1,0,0,0,0,12],[140,337,0.4154,0.67848,0.35006,0.39286,0.71429,1.0,0.14,1.0,0,15,0,0,0,7,0,0,1,0,0,3,0,0,1,0,0,5,0,0,0,0,15],[144,337,0.4273,0.46875,0.37667,0.14286,0.2857,1.0,0.0,1.0,1,9,0,1,0,14,0,0,3,0,0,1,0,0,1,0,0,3,0,0,0,0,9],[148,337,0.4392,0.51339,0.34784,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,11,0,0,3,0,0,4,0,0,1,0,0,4,0,0,1,0,8],[152,337,0.451,0.51784,0.33072,0.14286,0.4998,0.71429,0.14286,1.0,0,7,0,0,0,11,0,0,1,0,0,4,0,0,3,0,0,6,0,0,0,0,7],[156,337,0.4629,0.48214,0.33645,0.14286,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,14,0,0,0,0,0,3,0,0,2,0,0,7,0,0,0,0,6],[160,337,0.4748,0.60266,0.34944,0.24999,0.71429,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,3,0,0,3,0,0,1,0,0,4,0,0,3,0,10],[164,337,0.4866,0.47312,0.35978,0.14286,0.21428,0.71429,0.14,1.0,0,7,0,0,0,16,0,0,1,0,0,0,0,0,1,0,0,7,0,0,0,0,7],[168,337,0.4985,0.54464,0.37362,0.14286,0.50001,1.0,0.14286,1.0,0,11,0,0,0,12,0,0,2,0,0,2,0,0,2,0,0,3,0,0,0,0,11],[172,337,0.5104,0.49097,0.3369,0.14286,0.42857,0.78571,0.14,1.0,0,8,0,0,0,10,0,0,5,0,0,4,0,0,3,0,0,2,0,0,0,0,8],[176,337,0.5223,0.5,0.37796,0.14286,0.35714,1.0,0.0,1.0,1,10,0,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,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[204,273,0.7473,0.92411,0.23686,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[208,273,0.7619,0.91518,0.2392,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[212,273,0.7766,0.94197,0.19515,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[216,273,0.7912,0.90179,0.24598,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[220,273,0.8059,0.93304,0.18893,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[224,273,0.8205,0.73213,0.3421,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,1,0,0,6,0,0,2,0,0,0,0,0,0,0,19],[228,273,0.8352,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[232,273,0.8498,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[236,273,0.8645,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[240,273,0.8791,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[244,273,0.8938,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[248,273,0.9084,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[252,273,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[256,273,0.9377,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[260,273,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[264,273,0.967,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[268,273,0.9817,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[272,273,0.9963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[273,273,1.0,0.27678,0.13333,0.24999,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,6,0,0,19,0,0,2,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"ccb206256f1164dc","q":"Find all integers  $ n\\ge 2$  having the following property: for any  $ k$  integers  $ a_{1},a_{2},\\cdots,a_{k}$  which aren't congruent to each other (modulo  $ n$ ), there exists an integer polynomial  $ f(x)$  such that congruence equation  $ f(x)\\equiv 0 (mod n)$  exactly has  $ k$  roots  $ x\\equiv a_{1},a_{2},\\cdots,a_{k} (mod n).$","t":[{"b":0,"e":0.0,"k":"falling","v":0.02232,"x":0.48661,"p":[[0,20,0.0,0.48661,0.2693,0.39286,0.42857,0.71429,0.0,1.0,3,3,1,3,0,2,0,0,3,0,0,11,0,0,4,0,0,5,0,0,1,0,3],[4,20,0.2,0.2455,0.27713,0.0,0.14286,0.4642,0.0,1.0,15,1,0,15,0,2,0,0,4,0,0,3,0,0,6,0,0,1,0,0,0,0,1],[8,20,0.4,0.19194,0.30429,0.0,0.0,0.35704,0.0,1.0,21,2,0,21,0,1,0,0,2,0,0,0,0,0,6,0,0,0,0,0,0,0,2],[12,20,0.6,0.14286,0.26486,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[16,20,0.8,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,20,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.30785,"x":0.58479,"p":[[0,42,0.0,0.48658,0.35509,0.0,0.57121,0.71429,0.0,1.0,9,4,5,9,0,0,0,0,2,0,0,3,0,0,5,0,0,6,0,0,3,0,4],[4,42,0.0952,0.58479,0.3338,0.42857,0.57143,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,0,0,0,4,0,0,10,0,0,2,0,0,2,0,8],[8,42,0.1905,0.5759,0.27545,0.42857,0.57143,0.71429,0.0,1.0,2,6,0,2,0,1,0,0,2,0,0,8,0,0,8,0,0,4,0,0,1,0,6],[12,42,0.2857,0.56692,0.36853,0.25004,0.64286,0.89286,0.0,1.0,7,8,0,7,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,0,2,0,8],[16,42,0.381,0.55804,0.33381,0.28571,0.57143,0.85714,0.0,1.0,3,7,0,3,0,4,0,0,2,0,0,5,0,0,5,0,0,3,0,0,3,0,7],[20,42,0.4762,0.48214,0.32878,0.28571,0.42857,0.71429,0.0,1.0,5,5,0,5,0,2,0,0,6,0,0,5,0,0,2,0,0,6,0,0,1,0,5],[24,42,0.5714,0.36607,0.33681,0.0,0.42857,0.57143,0.0,1.0,11,4,0,11,0,1,0,0,3,0,0,8,0,0,2,0,0,3,0,0,0,0,4],[28,42,0.6667,0.30785,0.29266,0.0,0.2857,0.46418,0.0,1.0,9,2,0,9,0,6,0,0,5,0,0,4,0,0,3,0,0,3,0,0,0,0,2],[32,42,0.7619,0.35713,0.31541,0.10714,0.42857,0.4286,0.0,1.0,8,4,0,8,0,5,0,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,4],[36,42,0.8571,0.3304,0.26109,0.14286,0.42857,0.42858,0.0,1.0,7,2,0,7,0,5,0,0,2,0,0,14,0,0,1,0,0,1,0,0,0,0,2],[40,42,0.9524,0.32594,0.19962,0.24999,0.42857,0.42857,0.0,0.85714,7,0,0,7,0,1,0,0,3,0,0,20,0,0,0,0,0,0,0,0,1,0,0],[42,42,1.0,0.3125,0.17655,0.25,0.42857,0.42857,0.0,0.4286,7,0,0,7,0,1,0,0,3,0,0,21,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3099b32f6e2ec811","q":"Find all integers  $a$  such that there is a prime number of  $p\\ge  5$  that divides\n  ${p-1 \\choose 2}$   $+ {p-1 \\choose 3} a$   $+{p-1 \\choose 4} a^2$ + ...+ $ {p-1 \\choose p-3} a^{p-5} .$","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":0.99554,"p":[[0,39,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,39,0.1026,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,39,0.2051,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,39,0.3077,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,39,0.4103,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,39,0.5128,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,39,0.6154,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,39,0.7179,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,39,0.8205,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,39,0.9231,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[39,39,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":5,"e":1.0,"k":"flat","v":0.95535,"x":1.0,"p":[[0,45,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,45,0.0889,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,45,0.1778,0.95535,0.11539,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[12,45,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,45,0.3556,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,45,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,45,0.5333,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,45,0.6222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,45,0.7111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,45,0.9778,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[45,45,1.0,0.9866,0.04167,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"14d41b5c9fd168d5","q":"Find all integers  $n \\geq 3$  for which there exist real numbers  $a_1, a_2, \\dots a_{n + 2}$  satisfying  $a_{n + 1} = a_1$ ,  $a_{n + 2} = a_2$  and  $$ a_ia_{i + 1} + 1 = a_{i + 2}, $$  for  $i = 1, 2, \\dots, n$ .\n\n*Proposed by Patrik Bak, Slovakia*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.15616,"x":0.52676,"p":[[0,80,0.0,0.15616,0.09008,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,80,0.05,0.52676,0.33011,0.2857,0.42859,0.85714,0.0,1.0,1,6,0,1,0,6,0,0,6,0,0,5,0,0,2,0,0,1,0,0,5,0,6],[8,80,0.1,0.45981,0.31285,0.2857,0.28571,0.75,0.0,1.0,1,5,0,1,0,6,0,0,10,0,0,5,0,0,1,0,0,1,0,0,3,0,5],[12,80,0.15,0.39722,0.28963,0.14286,0.42857,0.42857,0.0,1.0,3,4,0,3,0,7,0,0,4,0,0,12,0,0,1,0,0,0,0,0,1,0,4],[16,80,0.2,0.30803,0.28596,0.14286,0.2857,0.42857,0.0,1.0,6,3,0,6,0,8,0,0,8,0,0,6,0,0,0,0,0,0,0,0,1,0,3],[20,80,0.25,0.38839,0.30771,0.14286,0.28571,0.42857,0.0,1.0,2,5,0,2,0,9,0,0,8,0,0,7,0,0,0,0,0,0,0,0,1,0,5],[24,80,0.3,0.42409,0.30822,0.14286,0.35714,0.57143,0.0,1.0,2,4,0,2,0,9,0,0,5,0,0,5,0,0,4,0,0,1,0,0,2,0,4],[28,80,0.35,0.32141,0.25753,0.14286,0.21428,0.42857,0.0,1.0,2,2,0,2,0,14,0,0,5,0,0,4,0,0,3,0,0,2,0,0,0,0,2],[32,80,0.4,0.36161,0.29447,0.14286,0.21428,0.46431,0.0,1.0,1,4,0,1,0,15,0,0,3,0,0,5,0,0,3,0,0,1,0,0,0,0,4],[36,80,0.45,0.35713,0.24741,0.14286,0.28571,0.4286,0.0,0.85714,1,0,0,1,0,13,0,0,3,0,0,8,0,0,1,0,0,3,0,0,3,0,0],[40,80,0.5,0.34813,0.25497,0.14286,0.28571,0.42857,0.14,1.0,0,3,0,0,0,14,0,0,4,0,0,10,0,0,0,0,0,1,0,0,0,0,3],[44,80,0.55,0.39285,0.32339,0.14286,0.21428,0.74996,0.0,1.0,2,1,0,2,0,14,0,0,4,0,0,1,0,0,1,0,0,2,0,0,7,0,1],[48,80,0.6,0.44182,0.31196,0.14286,0.28571,0.71429,0.0,1.0,1,3,0,1,0,9,0,0,8,0,0,3,0,0,0,0,0,4,0,0,4,0,3],[52,80,0.65,0.40169,0.33018,0.14286,0.21428,0.74996,0.14,1.0,0,4,0,0,0,16,0,0,4,0,0,3,0,0,0,0,0,1,0,0,4,0,4],[56,80,0.7,0.38818,0.28636,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,14,0,0,2,0,0,3,0,0,6,0,0,2,0,0,2,0,2],[60,80,0.75,0.45089,0.37814,0.14286,0.14286,0.89286,0.14286,1.0,0,8,0,0,0,17,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,8],[64,80,0.8,0.38395,0.31427,0.14286,0.21429,0.57111,0.0,1.0,1,3,0,1,0,15,0,0,3,0,0,4,0,0,2,0,0,0,0,0,4,0,3],[68,80,0.85,0.24107,0.21558,0.14286,0.14286,0.1786,0.14286,1.0,0,1,0,0,0,24,0,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[72,80,0.9,0.20982,0.17122,0.14286,0.14286,0.14287,0.14286,1.0,0,1,0,0,0,25,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[76,80,0.95,0.21429,0.20203,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[80,80,1.0,0.20089,0.13767,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.11607,"x":0.47321,"p":[[0,80,0.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,80,0.05,0.3571,0.28344,0.14286,0.2857,0.42857,0.0,1.0,3,3,0,3,0,11,0,0,3,0,0,8,0,0,3,0,0,0,0,0,1,0,3],[8,80,0.1,0.4107,0.28737,0.14289,0.42857,0.57111,0.0,1.0,3,3,0,3,0,7,0,0,4,0,0,9,0,0,2,0,0,3,0,0,1,0,3],[12,80,0.15,0.46433,0.28121,0.28571,0.42857,0.42895,0.0,1.0,1,5,0,1,0,5,0,0,4,0,0,15,0,0,0,0,0,1,0,0,1,0,5],[16,80,0.2,0.46874,0.34484,0.14286,0.28571,0.85704,0.14286,1.0,0,7,0,0,0,11,0,0,6,0,0,5,0,0,0,0,0,0,0,0,3,0,7],[20,80,0.25,0.41962,0.32913,0.14286,0.28571,0.74996,0.0,1.0,3,3,0,3,0,8,0,0,9,0,0,1,0,0,1,0,0,2,0,0,5,0,3],[24,80,0.3,0.31232,0.25625,0.14286,0.14286,0.42857,0.0,0.85714,1,0,0,1,0,16,0,0,6,0,0,4,0,0,0,0,0,0,0,0,5,0,0],[28,80,0.35,0.35267,0.32336,0.14286,0.2857,0.42858,0.0,1.0,5,4,0,5,0,10,0,0,5,0,0,5,0,0,1,0,0,0,0,0,2,0,4],[32,80,0.4,0.47321,0.32818,0.14286,0.42857,0.85714,0.0,1.0,1,6,0,1,0,8,0,0,6,0,0,6,0,0,2,0,0,0,0,0,3,0,6],[36,80,0.45,0.41518,0.28428,0.14286,0.28571,0.46431,0.14286,1.0,0,3,0,0,0,10,0,0,7,0,0,7,0,0,1,0,0,1,0,0,3,0,3],[40,80,0.5,0.42411,0.31234,0.14286,0.28571,0.53574,0.14286,1.0,0,4,0,0,0,12,0,0,5,0,0,7,0,0,0,0,0,0,0,0,4,0,4],[44,80,0.55,0.4464,0.34946,0.14286,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,14,0,0,4,0,0,3,0,0,2,0,0,0,0,0,2,0,7],[48,80,0.6,0.45981,0.30874,0.14286,0.42857,0.74996,0.14286,1.0,0,4,0,0,0,11,0,0,2,0,0,9,0,0,1,0,0,1,0,0,4,0,4],[52,80,0.65,0.31696,0.269,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,20,0,0,2,0,0,4,0,0,0,0,0,2,0,0,3,0,1],[56,80,0.7,0.37052,0.28539,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,13,0,0,7,0,0,6,0,0,1,0,0,0,0,0,1,0,4],[60,80,0.75,0.28563,0.22593,0.14286,0.14288,0.32143,0.14,1.0,0,1,0,0,0,18,0,0,6,0,0,5,0,0,0,0,0,0,0,0,2,0,1],[64,80,0.8,0.2857,0.26243,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,20,0,0,4,0,0,1,0,0,2,0,0,1,0,0,1,0,2],[68,80,0.85,0.27679,0.22286,0.14286,0.14286,0.32143,0.14286,1.0,0,1,0,0,0,20,0,0,4,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[72,80,0.9,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,80,0.95,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,80,1.0,0.16071,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e2e2ea8713039560","q":"Find all ordered pairs of integers  $x,y$  such that   $$ xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2. $$  *Proposed by Henry Jiang*","t":[{"b":5,"e":0.42857,"k":"flat","v":0.41072,"x":0.42862,"p":[[0,58,0.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.41522,0.04166,0.42857,0.42857,0.42857,0.28571,0.43,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.40179,"x":0.42857,"p":[[0,68,0.0,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.41969,0.03459,0.42857,0.42857,0.42857,0.2857,0.43,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[12,68,0.1765,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.40179,0.07523,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[28,68,0.4118,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,68,0.4706,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[40,68,0.5882,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.41071,0.06916,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"99f58bdaa78569c7","q":"Find all functions  $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ , such that  $f(x+f(x)+f(y))=2f(x)+y$  for all positive reals  $x,y$ .\n\n*Proposed by Athanasios Kontogeorgis, Greece*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.45981,"x":0.82142,"p":[[0,216,0.0,0.53114,0.29077,0.2857,0.42857,0.74996,0.14,1.0,0,6,0,0,0,3,0,0,10,0,0,4,0,0,5,0,0,2,0,0,2,0,6],[4,216,0.0185,0.81695,0.24547,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,0,2,0,18],[8,216,0.037,0.81696,0.23753,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,0,5,0,16],[12,216,0.0556,0.74998,0.25001,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,7,0,0,3,0,12],[16,216,0.0741,0.81249,0.25616,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,1,0,0,7,0,16],[20,216,0.0926,0.76339,0.24643,0.57143,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,7,0,0,3,0,13],[24,216,0.1111,0.75891,0.24599,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,0,0,0,6,0,0,3,0,0,9,0,10],[28,216,0.1296,0.7946,0.2171,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,6,0,0,5,0,13],[32,216,0.1481,0.77229,0.26212,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,1,0,0,7,0,0,1,0,0,7,0,13],[36,216,0.1667,0.78122,0.23144,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,5,0,0,3,0,14],[40,216,0.1852,0.76784,0.2442,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,4,0,0,6,0,12],[44,216,0.2037,0.79241,0.23376,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,1,0,10,0,11],[48,216,0.2222,0.79464,0.24727,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,1,0,0,6,0,15],[52,216,0.2407,0.79464,0.25738,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,0,2,0,17],[56,216,0.2593,0.79462,0.25739,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,0,0,0,5,0,0,1,0,0,11,0,12],[60,216,0.2778,0.75,0.23958,0.57143,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,0,0,0,14],[64,216,0.2963,0.78122,0.23416,0.67836,0.85707,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,0,0,0,5,0,0,6,0,0,6,0,12],[68,216,0.3148,0.78125,0.30926,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,5,0,0,2,0,18],[72,216,0.3333,0.79018,0.23415,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,2,0,15],[76,216,0.3519,0.69643,0.28291,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,4,0,0,1,0,0,8,0,0,1,0,0,6,0,10],[80,216,0.3704,0.72989,0.28557,0.57132,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,2,0,0,3,0,0,4,0,0,7,1,10],[84,216,0.3889,0.77232,0.2547,0.67857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,4,0,0,8,0,12],[88,216,0.4074,0.76785,0.2714,0.53572,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,4,0,0,4,0,0,3,0,0,1,0,0,5,0,15],[92,216,0.4259,0.75536,0.27199,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,3,0,0,5,1,0,2,0,0,4,0,14],[96,216,0.4444,0.73213,0.28516,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,1,0,0,5,0,0,3,0,0,7,0,11],[100,216,0.463,0.81249,0.24339,0.71429,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,0,7,0,15],[104,216,0.4815,0.74999,0.25506,0.57143,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,2,0,0,7,0,0,4,0,0,3,0,13],[108,216,0.5,0.76339,0.27108,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,4,0,0,6,0,13],[112,216,0.5185,0.82142,0.21724,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,0,5,0,15],[116,216,0.537,0.75446,0.24545,0.57143,0.78564,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,0,0,0,7,0,0,6,0,0,5,0,11],[120,216,0.5556,0.76785,0.22232,0.57143,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,3,0,0,7,0,11],[124,216,0.5741,0.70536,0.32721,0.39286,0.85714,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,3,0,0,1,0,0,2,0,0,2,0,0,7,0,12],[128,216,0.5926,0.76785,0.26904,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,1,0,0,2,0,0,3,0,0,6,0,0,5,0,13],[132,216,0.6111,0.68302,0.28061,0.5354,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,2,0,0,7,0,0,2,0,0,6,0,9],[136,216,0.6296,0.64284,0.31944,0.42857,0.57143,1.0,0.0,1.0,1,12,0,1,0,3,0,0,2,0,0,6,0,0,5,0,0,3,0,0,0,0,12],[140,216,0.6481,0.63839,0.30301,0.39286,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,6,0,0,4,0,0,3,0,0,5,0,0,3,0,9],[144,216,0.6667,0.79906,0.21391,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,4,0,0,6,0,13],[148,216,0.6852,0.73213,0.23892,0.5354,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,3,0,0,8,0,9],[152,216,0.7037,0.7589,0.25366,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,2,0,0,5,0,13],[156,216,0.7222,0.76785,0.24679,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,4,0,0,6,0,12],[160,216,0.7407,0.71427,0.30515,0.42859,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,5,0,0,2,0,0,4,0,0,1,0,0,5,0,13],[164,216,0.7593,0.70982,0.31234,0.42857,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,4,0,0,4,0,0,3,0,0,1,0,0,5,0,13],[168,216,0.7778,0.67411,0.32778,0.39286,0.78571,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,2,0,0,1,0,0,4,0,0,3,0,0,5,0,11],[172,216,0.7963,0.65622,0.27633,0.39286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,7,0,0,1,0,0,6,0,0,5,0,0,4,0,8],[176,216,0.8148,0.71427,0.32538,0.49968,0.85707,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,4,0,0,0,0,0,4,0,0,3,0,0,2,0,15],[180,216,0.8333,0.64286,0.30094,0.28571,0.64286,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,7,0,0,2,0,0,5,0,0,4,0,0,2,0,10],[184,216,0.8519,0.65622,0.26931,0.42859,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,2,0,0,4,0,0,6,0,0,4,0,0,7,0,6],[188,216,0.8704,0.72766,0.27284,0.57143,0.78571,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,5,0,0,5,0,11],[192,216,0.8889,0.71875,0.28456,0.57143,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,4,0,0,0,0,0,7,0,0,3,0,0,4,0,12],[196,216,0.9074,0.66517,0.25658,0.57132,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,2,0,0,8,0,0,6,0,0,4,0,7],[200,216,0.9259,0.71875,0.26603,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,5,0,0,1,0,0,10,0,0,0,0,0,4,0,12],[204,216,0.9444,0.45981,0.18117,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,3,0,0,11,0,0,3,0,0,0,0,1],[208,216,0.963,0.52231,0.25657,0.28571,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,1,0,0,13,0,0,0,0,0,10,0,0,2,0,0,2,0,4],[212,216,0.9815,0.5,0.23958,0.28571,0.50001,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,10,0,0,4,0,0,10,0,0,1,0,0,2,0,3],[216,216,1.0,0.46428,0.22016,0.28571,0.57141,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,10,0,0,1,0,0,15,0,0,1,0,0,0,0,2]]},{"b":3,"e":0.42857,"k":"flat","v":0.40177,"x":0.83925,"p":[[0,124,0.0,0.46875,0.29501,0.14286,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,10,0,0,4,0,0,4,0,0,4,0,0,4,0,0,3,0,3],[4,124,0.0323,0.8125,0.21558,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,0,6,0,14],[8,124,0.0645,0.77679,0.25985,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,5,0,0,8,0,12],[12,124,0.0968,0.83034,0.18365,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,8,0,13],[16,124,0.129,0.79016,0.21425,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,5,0,0,8,0,11],[20,124,0.1613,0.82142,0.21429,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,6,0,15],[24,124,0.1935,0.79017,0.25751,0.57132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,2,0,0,6,0,15],[28,124,0.2258,0.81695,0.23485,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,6,0,0,2,0,17],[32,124,0.2581,0.76339,0.25407,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,2,0,0,4,0,14],[36,124,0.2903,0.76786,0.24679,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,5,0,0,6,0,12],[40,124,0.3226,0.76339,0.25657,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,3,0,0,10,0,10],[44,124,0.3548,0.75445,0.24805,0.67857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,5,0,0,10,0,9],[48,124,0.3871,0.81249,0.25365,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,2,0,0,5,0,17],[52,124,0.4194,0.76338,0.27804,0.53571,0.85714,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,4,0,0,3,0,0,1,0,0,2,0,0,8,0,13],[56,124,0.4516,0.71873,0.22725,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,6,0,0,9,0,6],[60,124,0.4839,0.81249,0.18709,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,4,0,13],[64,124,0.5161,0.76336,0.26394,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,4,0,0,7,0,12],[68,124,0.5484,0.75893,0.28221,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,0,2,0,15],[72,124,0.5806,0.82143,0.24484,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,0,4,0,16],[76,124,0.6129,0.76784,0.2442,0.67857,0.85707,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,7,0,0,5,0,12],[80,124,0.6452,0.68304,0.31081,0.42859,0.71429,1.0,0.0,1.0,1,12,0,1,0,2,0,0,3,0,0,3,0,0,6,0,0,2,0,0,3,0,12],[84,124,0.6774,0.74552,0.27372,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,0,4,0,13],[88,124,0.7097,0.77232,0.29636,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,0,9,0,14],[92,124,0.7419,0.77678,0.23941,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,5,0,0,10,0,10],[96,124,0.7742,0.83925,0.19808,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,14],[100,124,0.8065,0.78124,0.25751,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,2,0,0,8,0,13],[104,124,0.8387,0.74999,0.26001,0.67857,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,2,0,0,3,0,0,7,0,0,7,0,10],[108,124,0.871,0.75443,0.28625,0.57132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,3,0,0,1,0,0,6,0,0,0,0,0,6,0,14],[112,124,0.9032,0.75,0.27664,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,4,0,0,10,0,10],[116,124,0.9355,0.56248,0.2878,0.28571,0.57141,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,6,0,0,4,0,0,7,0,0,1,0,0,5,0,5],[120,124,0.9677,0.63392,0.3153,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,8,0,0,3,0,0,4,0,0,2,0,0,2,0,11],[124,124,1.0,0.40177,0.17654,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,15,0,0,2,0,0,10,0,0,1,0,0,1,0,0]]}]},{"i":"5100cfd67c40e9c8","q":"Find all polynomials  $p(x)$  satisfying  $p(x^3+1)=p(x+1)^3$  for all  $x$ .","t":[{"b":3,"e":1.0,"k":"rising","v":0.59373,"x":0.99554,"p":[[0,37,0.0,0.59373,0.22047,0.42859,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,5,0,0,12,0,0,6,0,0,1,0,4],[4,37,0.1081,0.77676,0.26951,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,4,0,0,2,0,16],[8,37,0.2162,0.87945,0.16411,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,7,0,18],[12,37,0.3243,0.84821,0.18189,0.67857,1.0,1.0,0.5714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,4,0,17],[16,37,0.4324,0.83034,0.24599,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,18],[20,37,0.5405,0.92411,0.1636,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,24],[24,37,0.6486,0.91518,0.23107,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[28,37,0.7568,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,37,0.8649,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[37,37,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":5,"e":0.71429,"k":"flat","v":0.63836,"x":0.87946,"p":[[0,31,0.0,0.63836,0.20198,0.5354,0.71429,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,9,0,0,6,0,2],[4,31,0.129,0.78572,0.20825,0.71429,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,9,0,0,4,0,12],[8,31,0.2581,0.87946,0.19597,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,21],[12,31,0.3871,0.80357,0.23891,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,5,0,0,1,0,17],[16,31,0.5161,0.80802,0.20706,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,3,0,15],[20,31,0.6452,0.82142,0.23421,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,3,0,0,0,0,19],[24,31,0.7742,0.71875,0.24086,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,3,0,0,6,0,9],[28,31,0.9032,0.69193,0.22336,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,7,0,0,2,0,8],[31,31,1.0,0.6875,0.24337,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,0,0,0,2,0,0,8,0,0,7,0,0,6,0,6]]}]},{"i":"a390607b2ddabd60","q":"Find all pairs of positive integers  $m, n \\ge 3$  for which there exist infinitely many positive integers  $a$  such that \\[\\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\\] is itself an integer.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.39284,"x":0.68748,"p":[[0,85,0.0,0.45536,0.3163,0.14289,0.28571,0.71429,0.14286,1.0,0,5,0,0,0,9,0,0,10,0,0,1,0,0,2,0,0,3,0,0,2,0,5],[4,85,0.0471,0.52232,0.31055,0.2857,0.42857,0.75,0.0,1.0,1,7,0,1,0,2,0,0,12,0,0,3,0,0,3,0,0,3,0,0,1,0,7],[8,85,0.0941,0.46427,0.28793,0.28571,0.28571,0.71429,0.0,1.0,2,4,0,2,0,2,0,0,13,0,0,2,0,0,4,0,0,4,0,0,1,0,4],[12,85,0.1412,0.47766,0.25657,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,18,0,0,0,0,0,2,0,0,6,0,0,3,0,2],[16,85,0.1882,0.50891,0.26471,0.28571,0.28571,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,17,0,0,1,0,0,3,0,0,4,0,0,4,0,3],[20,85,0.2353,0.41516,0.21829,0.28571,0.28571,0.4642,0.14286,1.0,0,2,0,0,0,1,0,0,19,0,0,4,0,0,3,0,0,2,0,0,1,0,2],[24,85,0.2824,0.53122,0.29066,0.28571,0.28571,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,17,0,0,1,0,0,2,0,0,4,0,0,2,0,6],[28,85,0.3294,0.46428,0.23689,0.28571,0.28586,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,16,0,0,2,0,0,6,0,0,4,0,0,0,0,3],[32,85,0.3765,0.50443,0.23415,0.28571,0.42857,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,2,0,0,3,0,0,9,0,0,1,0,2],[36,85,0.4235,0.47319,0.2635,0.28571,0.35714,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,13,0,0,4,0,0,3,0,0,5,0,0,0,0,4],[40,85,0.4706,0.46871,0.27253,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,5,0,0,10,0,0,5,0,0,4,0,0,3,0,0,1,0,4],[44,85,0.5176,0.46873,0.24545,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,15,0,0,2,0,0,4,0,0,5,0,0,2,0,2],[48,85,0.5647,0.50442,0.24995,0.28571,0.42857,0.60714,0.1429,1.0,0,3,0,0,0,1,0,0,13,0,0,3,0,0,7,0,0,2,0,0,3,0,3],[52,85,0.6118,0.49999,0.21724,0.28571,0.4286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,4,0,0,6,0,0,5,0,0,3,0,1],[56,85,0.6588,0.46872,0.24544,0.2857,0.35714,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,15,0,0,3,0,0,6,0,0,2,0,0,3,0,2],[60,85,0.7059,0.52229,0.23584,0.28571,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,12,0,0,1,0,0,7,0,0,7,0,0,2,0,2],[64,85,0.7529,0.51337,0.24185,0.28571,0.4286,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,4,0,0,2,0,0,7,0,0,3,0,2],[68,85,0.8,0.49553,0.23682,0.28571,0.42859,0.60707,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,5,0,0,5,0,0,2,0,0,4,0,2],[72,85,0.8471,0.39284,0.18897,0.28571,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,18,0,0,1,0,0,4,0,0,6,0,0,0,0,0],[76,85,0.8941,0.52231,0.28707,0.28571,0.42857,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,14,0,0,4,0,0,3,0,0,2,0,0,2,0,6],[80,85,0.9412,0.68748,0.26351,0.42859,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,7,0,0,2,0,0,3,0,0,5,0,0,8,0,7],[84,85,0.9882,0.5491,0.26027,0.28571,0.4286,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,10,0,0,6,0,0,3,0,0,2,0,0,8,0,2],[85,85,1.0,0.59821,0.24337,0.39286,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,4,0,0,3,0,0,10,0,0,4,0,3]]},{"b":6,"e":0.42857,"k":"flat","v":0.37497,"x":0.7187,"p":[[0,58,0.0,0.47318,0.27299,0.28571,0.35714,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,12,0,0,3,0,0,3,0,0,4,0,0,4,0,2],[4,58,0.069,0.46872,0.25059,0.28571,0.28571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,17,0,0,2,0,0,3,0,0,5,0,0,1,0,3],[8,58,0.1379,0.54458,0.24856,0.28571,0.57121,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,13,0,0,1,0,0,6,0,0,6,0,0,3,0,3],[12,58,0.2069,0.50441,0.255,0.28571,0.35714,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,16,0,0,1,0,0,6,0,0,4,0,0,1,0,4],[16,58,0.2759,0.49553,0.25501,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,13,0,0,2,0,0,6,0,0,4,0,0,2,0,3],[20,58,0.3448,0.49106,0.25489,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,16,0,0,1,0,0,4,0,0,4,0,0,4,0,2],[24,58,0.4138,0.5401,0.21645,0.28571,0.571,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,0,0,0,8,0,0,7,0,0,5,0,0],[28,58,0.4828,0.56694,0.25874,0.28571,0.571,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,11,0,0,4,0,0,3,0,0,8,0,0,1,0,5],[32,58,0.5517,0.54907,0.25028,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,11,0,0,2,0,0,4,0,0,9,0,0,2,0,3],[36,58,0.6207,0.6071,0.20517,0.42859,0.64286,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,3,0,0,7,0,0,11,0,0,3,0,2],[40,58,0.6897,0.7187,0.22158,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,0,0,0,7,0,0,8,0,0,6,0,7],[44,58,0.7586,0.54458,0.22989,0.28571,0.5712,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,8,0,0,2,0,0,7,0,0,10,0,0,1,0,2],[48,58,0.8276,0.48212,0.21052,0.28571,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,2,0,0,11,0,0,2,0,0,4,0,0],[52,58,0.8966,0.45977,0.20431,0.28571,0.42836,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,14,0,0,0,0,0,8,0,0,7,0,0,1,0,0],[56,58,0.9655,0.44184,0.2184,0.2857,0.28571,0.60714,0.14,1.0,0,1,0,0,0,1,0,0,18,0,0,1,0,0,4,0,0,6,0,0,1,0,1],[58,58,1.0,0.37497,0.17765,0.28571,0.28571,0.571,0.0,0.71429,1,0,0,1,0,3,0,0,15,0,0,4,0,0,6,0,0,3,0,0,0,0,0]]}]},{"i":"1e421be02647ed99","q":"Find all pairs  $(p,n)$  so that  $p$  is a prime number,  $n$  is a positive integer and \\[p^3-2p^2+p+1=3^n \\] holds.","t":[{"b":5,"e":1.0,"k":"rising","v":0.53571,"x":0.97768,"p":[[0,173,0.0,0.57588,0.37878,0.14289,0.4998,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,6,0,0,1,0,0,1,0,0,2,0,0,0,0,13],[4,173,0.0231,0.67856,0.3407,0.28571,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,8,0,0,1,0,0,1,0,0,2,0,0,3,0,14],[8,173,0.0462,0.70534,0.35882,0.28571,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,17],[12,173,0.0694,0.53571,0.34626,0.28571,0.28571,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,11,0,0,1,0,0,1,0,0,2,0,0,2,0,9],[16,173,0.0925,0.70536,0.35703,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,17],[20,173,0.1156,0.66965,0.36672,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,6,0,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,16],[24,173,0.1387,0.59821,0.31225,0.28571,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,15,0,0,0,0,0,1,0,0,5,0,0,2,0,9],[28,173,0.1618,0.71875,0.33785,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,9,0,0,2,0,0,1,0,0,0,0,0,1,0,18],[32,173,0.185,0.75,0.33881,0.28571,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,20],[36,173,0.2081,0.87053,0.20628,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,2,0,21],[40,173,0.2312,0.91518,0.19516,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,26],[44,173,0.2543,0.89732,0.22654,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,2,0,25],[48,173,0.2775,0.84821,0.26711,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,1,0,0,3,0,22],[52,173,0.3006,0.82589,0.29609,0.78571,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,23],[56,173,0.3237,0.86161,0.25874,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,3,0,23],[60,173,0.3468,0.90179,0.21558,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,1,0,25],[64,173,0.3699,0.90625,0.22477,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[68,173,0.3931,0.91963,0.19867,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,27],[72,173,0.4162,0.9241,0.20512,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[76,173,0.4393,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[80,173,0.4624,0.88393,0.20958,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,3,0,22],[84,173,0.4855,0.94642,0.13717,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[88,173,0.5087,0.9375,0.15947,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[92,173,0.5318,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[96,173,0.5549,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[100,173,0.578,0.94196,0.13767,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[104,173,0.6012,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[108,173,0.6243,0.93304,0.13825,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[112,173,0.6474,0.94642,0.13718,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[116,173,0.6705,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[120,173,0.6936,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[124,173,0.7168,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[128,173,0.7399,0.89285,0.21129,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,5,0,22],[132,173,0.763,0.93312,0.10699,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[136,173,0.7861,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[140,173,0.8092,0.90625,0.19103,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,4,0,23],[144,173,0.8324,0.94642,0.10565,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[148,173,0.8555,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[152,173,0.8786,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[156,173,0.9017,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[160,173,0.9249,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[164,173,0.948,0.92857,0.16367,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[168,173,0.9711,0.92411,0.14719,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,24],[172,173,0.9942,0.91962,0.1285,0.857,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[173,173,1.0,0.88837,0.12748,0.82132,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16]]},{"b":7,"e":0.14286,"k":"falling","v":0.14277,"x":0.83928,"p":[[0,80,0.0,0.66071,0.38091,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,7,0,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,16],[4,80,0.05,0.83928,0.30041,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,24],[8,80,0.1,0.72768,0.34508,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,9,0,0,0,0,0,0,0,0,1,0,0,2,0,18],[12,80,0.15,0.68303,0.36724,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,18],[16,80,0.2,0.74107,0.30813,0.5,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,6,0,0,0,0,0,1,0,0,5,0,0,3,0,15],[20,80,0.25,0.6875,0.33586,0.28571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,9,0,0,0,0,0,3,0,0,1,0,0,2,0,15],[24,80,0.3,0.69643,0.33646,0.28571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,9,0,0,0,0,0,2,0,0,1,0,0,3,0,15],[28,80,0.35,0.80357,0.29613,0.74989,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,5,0,19],[32,80,0.4,0.74106,0.30814,0.39286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,7,0,0,1,0,0,1,0,0,4,0,0,2,0,16],[36,80,0.45,0.65625,0.36746,0.28571,0.85714,1.0,0.0,1.0,1,15,1,1,0,3,0,0,9,0,0,0,0,0,1,0,0,1,0,0,2,0,15],[40,80,0.5,0.65177,0.34615,0.28571,0.78571,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,10,0,0,0,0,0,1,0,0,2,0,0,3,0,13],[44,80,0.55,0.76785,0.28738,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,6,0,0,0,0,0,0,0,0,6,0,0,4,0,15],[48,80,0.6,0.67857,0.3677,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,9,0,0,0,0,0,0,0,0,1,0,0,1,0,17],[52,80,0.65,0.46875,0.35934,0.14286,0.28571,1.0,0.14286,1.0,0,9,0,0,0,11,0,0,9,0,0,1,0,0,0,0,0,2,0,0,0,0,9],[56,80,0.7,0.48205,0.38099,0.14286,0.28571,1.0,0.14,1.0,0,10,0,0,0,13,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,10],[60,80,0.75,0.66518,0.37899,0.14286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,9,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,16],[64,80,0.8,0.56696,0.38711,0.14286,0.35716,1.0,0.14286,1.0,0,12,0,0,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,3,0,12],[68,80,0.85,0.48214,0.34947,0.14286,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,10,0,0,9,0,0,1,0,0,0,0,0,2,0,0,3,0,7],[72,80,0.9,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,80,0.95,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,80,1.0,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63f0bc07e4bf0ee0","q":"Find all positive integers  $a, b$ , and  $c$  such that the numbers  $$ \\frac{a+1}{b}, \\frac{b+1}{c} \\quad \\text{and} \\quad \\frac{c+1}{a} $$  are positive integers.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.59821,"x":0.83482,"p":[[0,136,0.0,0.70089,0.19352,0.53571,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,9,0,0,8,0,4],[4,136,0.0294,0.74106,0.19377,0.67836,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,11,0,0,6,0,7],[8,136,0.0588,0.83482,0.14334,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,7,0,11],[12,136,0.0882,0.73661,0.17169,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,6,0,5],[16,136,0.1176,0.78125,0.17122,0.71429,0.78571,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,12,0,0,9,0,7],[20,136,0.1471,0.75444,0.12996,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,18,0,0,6,0,4],[24,136,0.1765,0.70534,0.14699,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,19,0,0,5,0,2],[28,136,0.2059,0.76338,0.16601,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,15,0,0,7,0,6],[32,136,0.2353,0.72321,0.16342,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,20,0,0,2,0,5],[36,136,0.2647,0.64284,0.15568,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,19,0,0,0,0,2],[40,136,0.2941,0.70982,0.14934,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,21,0,0,3,0,3],[44,136,0.3235,0.66964,0.16917,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,17,0,0,4,0,2],[48,136,0.3529,0.69642,0.19149,0.42857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,13,0,0,6,0,4],[52,136,0.3824,0.69195,0.1641,0.67857,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,19,0,0,1,0,4],[56,136,0.4118,0.67857,0.16751,0.64286,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,19,0,0,2,0,3],[60,136,0.4412,0.62946,0.18161,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,14,0,0,3,0,2],[64,136,0.4706,0.63839,0.13825,0.42859,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,20,0,0,2,0,0],[68,136,0.5,0.65625,0.14223,0.64286,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,22,0,0,1,0,1],[72,136,0.5294,0.66072,0.15047,0.64286,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,22,0,0,0,0,2],[76,136,0.5588,0.63839,0.16746,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,18,0,0,1,0,2],[80,136,0.5882,0.67857,0.17857,0.42859,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,16,0,0,4,0,3],[84,136,0.6176,0.69643,0.14173,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,21,0,0,3,0,2],[88,136,0.6471,0.63393,0.15126,0.4286,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,18,0,0,3,0,0],[92,136,0.6765,0.63839,0.17852,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,15,0,0,3,0,2],[96,136,0.7059,0.62947,0.16311,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,18,0,0,0,0,2],[100,136,0.7353,0.62054,0.15815,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,18,0,0,1,0,1],[104,136,0.7647,0.66517,0.14555,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,19,0,0,3,0,1],[108,136,0.7941,0.66964,0.15335,0.64286,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,19,0,0,4,0,1],[112,136,0.8235,0.62505,0.14167,0.4286,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,18,0,0,2,0,0],[116,136,0.8529,0.65179,0.14698,0.64286,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,21,0,0,3,0,0],[120,136,0.8824,0.70535,0.16342,0.67857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,14,0,0,8,0,2],[124,136,0.9118,0.63396,0.18873,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,9,0,0,1,0,0,15,0,0,3,0,2],[128,136,0.9412,0.65179,0.14258,0.53571,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,21,0,0,1,0,1],[132,136,0.9706,0.65179,0.15947,0.42857,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,18,0,0,5,0,0],[136,136,1.0,0.59821,0.15335,0.42857,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,18,0,0,1,0,0]]},{"b":2,"e":0.57143,"k":"flat","v":0.65625,"x":0.76785,"p":[[0,122,0.0,0.7366,0.17168,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,6,0,5],[4,122,0.0328,0.66964,0.15746,0.67857,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,19,0,0,4,0,1],[8,122,0.0656,0.76785,0.14616,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,15,0,0,10,0,4],[12,122,0.0984,0.69196,0.19597,0.42857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,10,0,0,7,0,4],[16,122,0.1311,0.70088,0.14446,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,20,0,0,4,0,2],[20,122,0.1639,0.73214,0.15465,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,18,0,0,5,0,4],[24,122,0.1967,0.67411,0.14827,0.67857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,19,0,0,4,0,1],[28,122,0.2295,0.72768,0.17985,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,17,0,0,3,0,6],[32,122,0.2623,0.75,0.12372,0.71429,0.71429,0.75,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,0,4,0,4],[36,122,0.2951,0.70981,0.14935,0.71429,0.71429,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,18,0,0,6,0,2],[40,122,0.3279,0.68736,0.13567,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,20,0,0,6,0,0],[44,122,0.3607,0.73647,0.11358,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,23,0,0,5,0,2],[48,122,0.3934,0.66071,0.14617,0.64286,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,21,0,0,2,0,1],[52,122,0.4262,0.72321,0.13803,0.71429,0.71429,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,20,0,0,6,0,2],[56,122,0.459,0.66518,0.12169,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,23,0,0,2,0,0],[60,122,0.4918,0.69643,0.15464,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,21,0,0,2,0,3],[64,122,0.5246,0.7232,0.10064,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0,2,0,2],[68,122,0.5574,0.71875,0.14933,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,21,0,0,2,0,4],[72,122,0.5902,0.66518,0.16602,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,18,0,0,3,0,2],[76,122,0.623,0.74107,0.10374,0.71429,0.71429,0.71429,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,0,2,0,3],[80,122,0.6557,0.73213,0.11712,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,22,0,0,5,0,2],[84,122,0.6885,0.65625,0.12299,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,24,0,0,1,0,0],[88,122,0.7213,0.71875,0.11564,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,23,0,0,5,0,1],[92,122,0.7541,0.66964,0.1357,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,21,0,0,4,0,0],[96,122,0.7869,0.72768,0.09006,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0,2],[100,122,0.8197,0.71429,0.13832,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,23,0,0,2,0,3],[104,122,0.8525,0.68304,0.09932,0.71429,0.71429,0.71429,0.42857,0.8571,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,27,0,0,1,0,0],[108,122,0.8852,0.69196,0.10779,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,25,0,0,1,0,1],[112,122,0.918,0.70982,0.08364,0.71429,0.71429,0.71429,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0,1,0,1],[116,122,0.9508,0.67857,0.11845,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,23,0,0,1,0,1],[120,122,0.9836,0.66518,0.14987,0.67857,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,22,0,0,0,0,2],[122,122,1.0,0.67857,0.11293,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,25,0,0,2,0,0]]}]},{"i":"0260452ac9f395c7","q":"Find all positive integers  $n$  such that  $$ \\gcd(n,1) + \\gcd(n,2) + \\cdots + \\gcd(n,n) = 3n - 3. $$  **Note:** The notation  $\\gcd(a,b)$  denotes the greatest common divisor of  $a$  and  $b$ .\n\n*Proposed by Sergio Vera*","t":[{"b":3,"e":0.28571,"k":"flat","v":0.24999,"x":0.46433,"p":[[0,87,0.0,0.39286,0.14725,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,1,0,0,19,0,0,6,0,0,0,0,0,0,0,0],[4,87,0.046,0.46433,0.1821,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,2,0,0,17,0,0,8,0,0,0,0,0,0,0,2],[8,87,0.092,0.42857,0.17128,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,16,0,0,8,0,0,0,0,0,0,0,1],[12,87,0.1379,0.45536,0.18707,0.42857,0.42857,0.46431,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,18,0,0,5,0,0,1,0,0,0,0,2],[16,87,0.1839,0.41963,0.19864,0.28571,0.42857,0.57111,0.0,1.0,1,1,0,1,0,5,0,0,3,0,0,14,0,0,6,0,0,2,0,0,0,0,1],[20,87,0.2299,0.375,0.13716,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,2,0,0,21,0,0,3,0,0,0,0,0,0,0,0],[24,87,0.2759,0.32579,0.15259,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,2,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[28,87,0.3218,0.33036,0.14914,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,3,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[32,87,0.3678,0.3482,0.20182,0.14286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,4,0,0,15,0,0,3,0,0,0,0,0,0,0,1],[36,87,0.4138,0.3482,0.1921,0.14286,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,1,0,0,14,0,0,7,0,0,0,0,0,0,0,0],[40,87,0.4598,0.32143,0.14725,0.14286,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,5,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[44,87,0.5057,0.3259,0.16065,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,9,0,0,1,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[48,87,0.5517,0.38838,0.14825,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,1,0,0,19,0,0,4,0,0,1,0,0,0,0,0],[52,87,0.5977,0.35714,0.12877,0.24999,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,8,0,0,1,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[56,87,0.6437,0.30804,0.18249,0.14286,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,8,0,0,3,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[60,87,0.6897,0.28572,0.17128,0.14286,0.35714,0.42857,0.0,0.57143,3,0,0,3,0,12,0,0,1,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[64,87,0.7356,0.308,0.16787,0.14286,0.42857,0.42857,0.0,0.571,2,0,0,2,0,11,0,0,2,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[68,87,0.7816,0.34375,0.17445,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,8,0,0,3,0,0,13,0,0,6,0,0,0,0,0,0,0,0],[72,87,0.8276,0.39285,0.11843,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,4,0,0,22,0,0,3,0,0,0,0,0,0,0,0],[76,87,0.8736,0.34375,0.1984,0.14286,0.42857,0.46431,0.0,0.57143,4,0,0,4,0,7,0,0,1,0,0,12,0,0,8,0,0,0,0,0,0,0,0],[80,87,0.9195,0.32142,0.1675,0.14286,0.35714,0.42858,0.0,0.57143,1,0,0,1,0,11,0,0,4,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[84,87,0.9655,0.33481,0.14985,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,2,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[87,87,1.0,0.24999,0.17855,0.14286,0.14286,0.42857,0.0,0.57143,5,0,0,5,0,13,0,0,1,0,0,11,0,0,2,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.25,"x":0.40625,"p":[[0,133,0.0,0.39286,0.14286,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,0,0,0,19,0,0,6,0,0,0,0,0,0,0,0],[4,133,0.0301,0.40625,0.19597,0.28571,0.42857,0.46431,0.0,1.0,1,1,0,1,0,6,0,0,2,0,0,15,0,0,6,0,0,1,0,0,0,0,1],[8,133,0.0602,0.37054,0.11769,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,5,0,0,20,0,0,2,0,0,0,0,0,0,0,0],[12,133,0.0902,0.32589,0.15663,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,5,0,0,15,0,0,1,0,0,1,0,0,0,0,0],[16,133,0.1203,0.35273,0.15968,0.14289,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,3,0,0,15,0,0,5,0,0,0,0,0,0,0,0],[20,133,0.1504,0.3304,0.15338,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,1,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[24,133,0.1805,0.38389,0.15741,0.39286,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,4,0,0,2,0,0,18,0,0,6,0,0,0,0,0,0,0,0],[28,133,0.2105,0.32143,0.17128,0.14286,0.35714,0.42857,0.0,0.57143,2,0,0,2,0,9,0,0,5,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[32,133,0.2406,0.34822,0.14698,0.25,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,7,0,0,4,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[36,133,0.2707,0.33481,0.17715,0.14286,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,6,0,0,6,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[40,133,0.3008,0.27223,0.16895,0.14286,0.21428,0.42857,0.0,0.57143,3,0,0,3,0,13,0,0,2,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[44,133,0.3308,0.34375,0.19516,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,10,0,0,3,0,0,14,0,0,3,0,0,0,0,0,0,0,1],[48,133,0.3609,0.37054,0.13296,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,2,0,0,20,0,0,3,0,0,0,0,0,0,0,0],[52,133,0.391,0.26777,0.16663,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,17,0,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[56,133,0.4211,0.34821,0.17105,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,7,0,0,4,0,0,13,0,0,6,0,0,0,0,0,0,0,0],[60,133,0.4511,0.30802,0.17534,0.14286,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,14,0,0,1,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[64,133,0.4812,0.3125,0.15336,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,0,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[68,133,0.5113,0.27231,0.17624,0.14286,0.21431,0.42857,0.0,0.57143,4,0,0,4,0,12,0,0,1,0,0,13,0,0,2,0,0,0,0,0,0,0,0],[72,133,0.5414,0.29008,0.1617,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,14,0,0,3,0,0,11,0,0,3,0,0,0,0,0,0,0,0],[76,133,0.5714,0.25,0.16751,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,18,0,0,1,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[80,133,0.6015,0.35267,0.21423,0.14286,0.42857,0.4642,0.0,1.0,1,1,0,1,0,12,0,0,1,0,0,10,0,0,7,0,0,0,0,0,0,0,1],[84,133,0.6316,0.35715,0.15567,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,10,0,0,1,0,0,16,0,0,5,0,0,0,0,0,0,0,0],[88,133,0.6617,0.30802,0.23448,0.14286,0.28571,0.42858,0.0,1.0,5,1,0,5,0,10,0,0,2,0,0,8,0,0,6,0,0,0,0,0,0,0,1],[92,133,0.6917,0.28125,0.16554,0.14286,0.35714,0.42857,0.0,0.57143,3,0,0,3,0,12,0,0,1,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[96,133,0.7218,0.3125,0.15746,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,12,0,0,1,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[100,133,0.7519,0.30357,0.16656,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,13,0,0,3,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[104,133,0.782,0.35268,0.15966,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,0,0,0,18,0,0,4,0,0,0,0,0,0,0,0],[108,133,0.812,0.32143,0.18211,0.14286,0.42857,0.42858,0.0,0.57143,3,0,0,3,0,9,0,0,2,0,0,13,0,0,5,0,0,0,0,0,0,0,0],[112,133,0.8421,0.29017,0.16552,0.14286,0.42857,0.42857,0.0,0.571,3,0,0,3,0,11,0,0,1,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[116,133,0.8722,0.34822,0.17105,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,1,0,0,14,0,0,6,0,0,0,0,0,0,0,0],[120,133,0.9023,0.30358,0.15465,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,13,0,0,0,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[124,133,0.9323,0.33032,0.18702,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,11,0,0,1,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[128,133,0.9624,0.33474,0.15826,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,8,0,0,1,0,0,19,0,0,2,0,0,0,0,0,0,0,0],[132,133,0.9925,0.29018,0.17672,0.14286,0.21429,0.42857,0.0,0.57143,2,0,0,2,0,14,0,0,1,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[133,133,1.0,0.35714,0.18558,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,3,0,0,15,0,0,3,0,0,0,0,0,0,0,1]]}]},{"i":"2450e0786ede5359","q":"Find the number of second-degree polynomials  $ f(x)$  with integer coefficients and integer zeros for which  $ f(0)\\equal{}2010$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.86607,"x":0.96875,"p":[[0,54,0.0,0.88393,0.17655,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,0,0,21],[4,54,0.0741,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[8,54,0.1481,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[12,54,0.2222,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[16,54,0.2963,0.92857,0.15567,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,1,0,25],[20,54,0.3704,0.91518,0.14664,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,23],[24,54,0.4444,0.89285,0.13833,0.71429,1.0,1.0,0.714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20],[28,54,0.5185,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22],[32,54,0.5926,0.88393,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,18],[36,54,0.6667,0.91071,0.12753,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[40,54,0.7407,0.87945,0.15201,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,1,0,19],[44,54,0.8148,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20],[48,54,0.8889,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[52,54,0.963,0.88393,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,18],[54,54,1.0,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20]]},{"b":4,"e":1.0,"k":"flat","v":0.87946,"x":0.97321,"p":[[0,48,0.0,0.87946,0.18249,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,0,0,21],[4,48,0.0833,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,48,0.1667,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[12,48,0.25,0.90624,0.12683,0.71429,1.0,1.0,0.714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[16,48,0.3333,0.89286,0.14286,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,1,0,20],[20,48,0.4167,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[24,48,0.5,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[28,48,0.5833,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[32,48,0.6667,0.90625,0.13651,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[36,48,0.75,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[40,48,0.8333,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[44,48,0.9167,0.90177,0.14036,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,21],[48,48,1.0,0.91071,0.12753,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,21]]}]},{"i":"5aa3edbe07bded0b","q":"Find all positive integers  $n$  such that the number  $$ n^6 + 5n^3 + 4n + 116 $$  is the product of two or more consecutive numbers.","t":[{"b":5,"e":1.0,"k":"flat","v":0.83927,"x":0.97768,"p":[[0,154,0.0,0.86607,0.19865,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,18],[4,154,0.026,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,154,0.0519,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,154,0.0779,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[16,154,0.1039,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[20,154,0.1299,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[24,154,0.1558,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,154,0.1818,0.94195,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[32,154,0.2078,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[36,154,0.2338,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[40,154,0.2597,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[44,154,0.2857,0.91071,0.12753,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[48,154,0.3117,0.89284,0.1713,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,21],[52,154,0.3377,0.89732,0.1439,0.82132,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,5,0,19],[56,154,0.3636,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,24],[60,154,0.3896,0.88839,0.14167,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,18],[64,154,0.4156,0.89285,0.13363,0.85711,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,17],[68,154,0.4416,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[72,154,0.4675,0.90177,0.13574,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,3,0,20],[76,154,0.4935,0.875,0.17405,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,4,0,19],[80,154,0.5195,0.88835,0.13242,0.857,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,16],[84,154,0.5455,0.90177,0.14036,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,19],[88,154,0.5714,0.91517,0.12811,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,20],[92,154,0.5974,0.90623,0.14114,0.82132,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,3,0,21],[96,154,0.6234,0.88392,0.13573,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,5,0,17],[100,154,0.6494,0.87946,0.16016,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,18],[104,154,0.6753,0.85714,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,13],[108,154,0.7013,0.88837,0.16652,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,9,0,18],[112,154,0.7273,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,23],[116,154,0.7532,0.87944,0.15615,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,7,0,17],[120,154,0.7792,0.88392,0.13095,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,7,0,16],[124,154,0.8052,0.92856,0.10719,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[128,154,0.8312,0.91069,0.12247,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[132,154,0.8571,0.87053,0.13997,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,9,0,14],[136,154,0.8831,0.87053,0.13997,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,4,0,16],[140,154,0.9091,0.86607,0.19212,0.71429,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,18],[144,154,0.9351,0.88393,0.16917,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,2,0,20],[148,154,0.961,0.88839,0.13236,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,6,0,17],[152,154,0.987,0.92409,0.15969,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,25],[154,154,1.0,0.83927,0.17407,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,0,2,0,15]]},{"b":7,"e":0.857,"k":"flat","v":0.72764,"x":0.99107,"p":[[0,112,0.0,0.80357,0.17767,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,6,0,11],[4,112,0.0357,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,112,0.0714,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[12,112,0.1071,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[16,112,0.1429,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[20,112,0.1786,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[24,112,0.2143,0.95089,0.06785,0.85714,1.0,1.0,0.8571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[28,112,0.25,0.9598,0.08924,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[32,112,0.2857,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[36,112,0.3214,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,112,0.3571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,112,0.3929,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[48,112,0.4286,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[52,112,0.4643,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[56,112,0.5,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[60,112,0.5357,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[64,112,0.5714,0.93303,0.11837,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,21],[68,112,0.6071,0.95535,0.07525,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[72,112,0.6429,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[76,112,0.6786,0.91964,0.14258,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,10,0,20],[80,112,0.7143,0.89283,0.14289,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,12,0,16],[84,112,0.75,0.9107,0.1372,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,19],[88,112,0.7857,0.88839,0.16263,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,8,0,18],[92,112,0.8214,0.86159,0.16937,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,9,0,15],[96,112,0.8571,0.89731,0.15666,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,10,0,18],[100,112,0.8929,0.91517,0.1551,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,22],[104,112,0.9286,0.79906,0.19189,0.71429,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,12,0,9],[108,112,0.9643,0.72764,0.23787,0.57132,0.857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,3,0,0,9,0,8],[112,112,1.0,0.78571,0.22588,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,9,0,11]]}]},{"i":"1d997451c977d015","q":"Find all positive integers  $n$  such that   $$ \\phi(n) + \\sigma(n) = 2n + 8. $$","t":[{"b":2,"e":1.0,"k":"rising","v":0.72317,"x":0.95982,"p":[[0,174,0.0,0.72317,0.27187,0.42857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,7,0,0,5,0,0,0,0,0,4,0,13],[4,174,0.023,0.93304,0.17491,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[8,174,0.046,0.89286,0.22015,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,23],[12,174,0.069,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[16,174,0.092,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[20,174,0.1149,0.94196,0.08646,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[24,174,0.1379,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[28,174,0.1609,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[32,174,0.1839,0.95535,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[36,174,0.2069,0.95088,0.1048,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[40,174,0.2299,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[44,174,0.2529,0.94197,0.12299,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[48,174,0.2759,0.91516,0.12811,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,20],[52,174,0.2989,0.9241,0.15146,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,23],[56,174,0.3218,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[60,174,0.3448,0.875,0.15872,0.82132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,8,0,16],[64,174,0.3678,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[68,174,0.3908,0.87946,0.20858,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,18],[72,174,0.4138,0.91964,0.14258,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[76,174,0.4368,0.89285,0.16754,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,8,0,19],[80,174,0.4598,0.88393,0.15335,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,5,0,18],[84,174,0.4828,0.93749,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[88,174,0.5057,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[92,174,0.5287,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[96,174,0.5517,0.92856,0.11298,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[100,174,0.5747,0.87498,0.1777,0.82132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,6,0,18],[104,174,0.5977,0.94643,0.13243,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[108,174,0.6207,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[112,174,0.6437,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[116,174,0.6667,0.83926,0.21357,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,17],[120,174,0.6897,0.88392,0.25364,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,23],[124,174,0.7126,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[128,174,0.7356,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[132,174,0.7586,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[136,174,0.7816,0.91071,0.13717,0.85711,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,21],[140,174,0.8046,0.90186,0.13092,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,18],[144,174,0.8276,0.90625,0.13175,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,19],[148,174,0.8506,0.9174,0.1506,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,3,0,0,3,0,23],[152,174,0.8736,0.90175,0.14489,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[156,174,0.8966,0.90177,0.12599,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,17],[160,174,0.9195,0.85266,0.16557,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,3,0,16],[164,174,0.9425,0.86159,0.20357,0.82132,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,6,0,18],[168,174,0.9655,0.90177,0.12082,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,17],[172,174,0.9885,0.88839,0.12745,0.82132,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16],[174,174,1.0,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18]]},{"b":6,"e":0.71429,"k":"rising","v":0.65622,"x":0.98214,"p":[[0,180,0.0,0.65622,0.28763,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,2,0,0,6,0,0,3,0,0,3,0,0,9,0,6],[4,180,0.0222,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[8,180,0.0444,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,180,0.0667,0.9375,0.15542,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[16,180,0.0889,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[20,180,0.1111,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[24,180,0.1333,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[28,180,0.1556,0.93525,0.13991,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,1,0,2,0,25],[32,180,0.1778,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[36,180,0.2,0.94196,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[40,180,0.2222,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,27],[44,180,0.2444,0.94643,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[48,180,0.2667,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[52,180,0.2889,0.95087,0.09857,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[56,180,0.3111,0.9375,0.14258,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[60,180,0.3333,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[64,180,0.3556,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[68,180,0.3778,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[72,180,0.4,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[76,180,0.4222,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[80,180,0.4444,0.91515,0.13774,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[84,180,0.4667,0.9375,0.12339,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[88,180,0.4889,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[92,180,0.5111,0.92857,0.13363,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,23],[96,180,0.5333,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[100,180,0.5556,0.93302,0.11289,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[104,180,0.5778,0.95089,0.12169,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[108,180,0.6,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[112,180,0.6222,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[116,180,0.6444,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[120,180,0.6667,0.95535,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[124,180,0.6889,0.92856,0.14726,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[128,180,0.7111,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[132,180,0.7333,0.91961,0.15953,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,24],[136,180,0.7556,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[140,180,0.7778,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[144,180,0.8,0.87053,0.20628,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,2,0,21],[148,180,0.8222,0.92409,0.12873,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[152,180,0.8444,0.93748,0.11263,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[156,180,0.8667,0.93302,0.1129,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[160,180,0.8889,0.88838,0.15041,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,3,0,19],[164,180,0.9111,0.92411,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[168,180,0.9333,0.91514,0.13303,0.85711,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,21],[172,180,0.9556,0.87944,0.14778,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,3,0,18],[176,180,0.9778,0.87946,0.1992,0.82132,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,21],[180,180,1.0,0.82134,0.14736,0.71429,0.85707,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,11,0,9]]}]},{"i":"81d92be1229e15e1","q":"Find the maximum value of real number  $k$  such that \n\\[\\frac{a}{1+9bc+k(b-c)^2}+\\frac{b}{1+9ca+k(c-a)^2}+\\frac{c}{1+9ab+k(a-b)^2}\\geq \\frac{1}{2}\\]\nholds for all non-negative real numbers  $a,\\ b,\\ c$  satisfying  $a+b+c=1$ 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1,0,0,1,0,0,6,0,0,1,0,4],[116,147,0.7891,0.55134,0.3751,0.14286,0.42857,1.0,0.14286,1.0,0,11,0,0,0,11,0,0,2,0,0,5,0,0,1,0,0,0,0,0,1,1,11],[120,147,0.8163,0.48213,0.36377,0.14286,0.35714,1.0,0.14286,1.0,0,9,0,0,0,14,0,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,9],[124,147,0.8435,0.48661,0.3423,0.14286,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,14,0,0,0,0,0,4,0,0,0,0,0,7,0,0,1,0,6],[128,147,0.8707,0.49106,0.32915,0.14286,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,11,0,0,2,0,0,6,0,0,2,0,0,4,0,0,0,0,7],[132,147,0.898,0.41964,0.36759,0.14286,0.21428,0.78571,0.0,1.0,2,8,0,2,0,14,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,8],[136,147,0.9252,0.60268,0.38255,0.14286,0.64286,1.0,0.14286,1.0,0,13,0,0,0,11,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,13],[140,147,0.9524,0.47768,0.34738,0.14286,0.35714,0.75,0.14286,1.0,0,7,0,0,0,13,0,0,3,0,0,3,0,0,1,0,0,4,0,0,1,0,7],[144,147,0.9796,0.47768,0.3773,0.14286,0.28571,0.89286,0.0,1.0,1,8,0,1,0,15,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,8],[147,147,1.0,0.4107,0.3067,0.14286,0.35714,0.60714,0.14286,1.0,0,4,0,0,0,15,0,0,1,0,0,6,0,0,2,0,0,3,0,0,1,0,4]]}]},{"i":"91377f870371f828","q":"For  $ n \\in \\mathbb{N}$ , let  $s(n)$  denote the sum of all positive divisors of  $n$ . Show that for any  $n > 1$ , the product  $s(n - 1)s(n)s(n + 1)$  is an even number.","t":[{"b":3,"e":0.28571,"k":"volatile","v":0.19196,"x":0.80803,"p":[[0,16,0.0,0.70533,0.19867,0.57132,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,6,0,0,6,0,6],[4,16,0.25,0.80803,0.23037,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,1,0,0,8,0,14],[8,16,0.5,0.64286,0.32537,0.28571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,6,0,0,2,0,0,3,0,0,2,0,0,5,0,10],[12,16,0.75,0.41964,0.2765,0.28571,0.28571,0.42857,0.0,1.0,1,5,0,1,0,2,0,0,18,0,0,4,0,0,1,0,0,1,0,0,0,0,5],[16,16,1.0,0.19196,0.13175,0.0,0.2857,0.28571,0.0,0.42857,9,0,0,9,0,4,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.72765,"x":0.86161,"p":[[0,20,0.0,0.72765,0.20318,0.57132,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,6,0,0,7,0,7],[4,20,0.2,0.79902,0.27653,0.67857,1.0,1.0,0.14,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,3,0,0,3,0,18],[8,20,0.4,0.74107,0.31019,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,5,0,0,5,0,0,1,0,0,1,0,0,2,0,17],[12,20,0.6,0.8616,0.10403,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,21,0,7],[16,20,0.8,0.8616,0.14054,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,19,0,9],[20,20,1.0,0.86161,0.08364,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,24,0,5]]}]},{"i":"a76f1e6df44ae0db","q":"For  $n\\geq 1$  let  $M$  be an  $n\\times n$  complex array with distinct eigenvalues  $\\lambda_1,\\lambda_2,\\ldots,\\lambda_k$ , with multiplicities  $m_1,m_2,\\ldots,m_k$  respectively. Consider the linear operator  $L_M$  defined by  $L_MX=MX+XM^T$ , for any complex  $n\\times n$  array  $X$ . Find its eigenvalues and their multiplicities. ( $M^T$  denotes the transpose matrix of  $M$ ).","t":[{"b":4,"e":1.0,"k":"rising","v":0.84375,"x":1.0,"p":[[0,21,0.0,0.84375,0.16888,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,6,0,14],[4,21,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,21,0.381,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,21,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,21,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,21,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[21,21,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.83482,"x":0.97768,"p":[[0,16,0.0,0.83482,0.23176,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,18],[4,16,0.25,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,16,0.5,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[12,16,0.75,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,16,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]}]},{"i":"2a48362b4ca6b48d","q":"Fix positive integers $r>s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r-1$ that shares at least $s$ elements with each set in $F$.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.07134,"x":0.31684,"p":[[0,38,0.0,0.07134,0.10708,0.0,0.0,0.14286,0.0,0.28571,21,0,1,21,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.23657,0.23032,0.10714,0.14286,0.32143,0.0,0.85714,8,0,0,8,0,12,0,0,4,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[8,38,0.2105,0.31684,0.26424,0.14286,0.28571,0.571,0.0,0.857,6,0,0,6,0,9,0,0,5,0,0,3,0,0,4,0,0,3,0,0,2,0,0],[12,38,0.3158,0.25,0.20825,0.14286,0.21428,0.28571,0.0,0.85714,6,0,0,6,0,10,0,0,9,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[16,38,0.4211,0.16518,0.1992,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,13,0,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[20,38,0.5263,0.25447,0.20743,0.14286,0.14288,0.28571,0.0,0.85714,3,0,0,3,0,14,0,0,10,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[24,38,0.6316,0.14285,0.1184,0.10714,0.14286,0.14287,0.0,0.571,8,0,0,8,0,18,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,38,0.7368,0.15625,0.12037,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,12,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.13393,0.10062,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,19,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.15177,0.11807,0.14286,0.14286,0.1429,0.0,0.571,7,0,0,7,0,18,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.14732,0.12103,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,12,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.06696,"x":0.33477,"p":[[0,55,0.0,0.0759,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.20982,0.21124,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,9,0,0,7,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[8,55,0.1455,0.22773,0.22552,0.0,0.1429,0.28571,0.0,0.85714,10,0,0,10,0,7,0,0,8,0,0,4,0,0,0,0,0,2,0,0,1,0,0],[12,55,0.2182,0.15625,0.1488,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,9,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.26339,0.25028,0.14286,0.14286,0.32143,0.0,1.0,6,1,0,6,0,12,0,0,6,0,0,4,0,0,0,0,0,2,0,0,1,0,1],[20,55,0.3636,0.26783,0.24933,0.0,0.28571,0.42858,0.0,0.71429,12,0,0,12,0,2,0,0,6,0,0,5,0,0,4,0,0,3,0,0,0,0,0],[24,55,0.4364,0.17411,0.19144,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,12,0,0,4,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[28,55,0.5091,0.25441,0.22504,0.0,0.2857,0.42857,0.0,0.857,9,0,0,9,0,6,0,0,8,0,0,3,0,0,5,0,0,0,0,0,1,0,0],[32,55,0.5818,0.30801,0.30535,0.0,0.14288,0.60707,0.0,0.85714,11,0,0,11,0,6,0,0,2,0,0,3,0,0,2,0,0,6,0,0,2,0,0],[36,55,0.6545,0.33477,0.30006,0.10714,0.2857,0.571,0.0,1.0,8,1,0,8,0,7,0,0,4,0,0,2,0,0,5,0,0,3,0,0,2,0,1],[40,55,0.7273,0.28572,0.25754,0.10714,0.2857,0.4286,0.0,0.85714,8,0,0,8,0,7,0,0,7,0,0,4,0,0,0,0,0,5,0,0,1,0,0],[44,55,0.8,0.16518,0.17536,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,10,0,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[48,55,0.8727,0.06696,0.08737,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.1339,0.15939,0.0,0.14286,0.2857,0.0,0.571,15,0,0,15,0,8,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[55,55,1.0,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cbd70c011998d6a2","q":"Find all prime  $x,y$  and  $z,$  such that  $x^5 +y^3 -(x+y)^2=3z^3$","t":[{"b":2,"e":0.0,"k":"falling","v":0.19643,"x":0.70533,"p":[[0,128,0.0,0.39728,0.32088,0.14286,0.2857,0.571,0.0,1.0,2,5,0,2,0,12,0,0,4,0,0,4,0,0,4,0,0,0,0,0,1,0,5],[4,128,0.0312,0.70533,0.25491,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,14,0,0,2,0,0,0,0,12],[8,128,0.0625,0.57579,0.33796,0.39286,0.57143,1.0,0.0,1.0,4,10,0,4,0,1,0,0,3,0,0,4,0,0,10,0,0,0,0,0,0,0,10],[12,128,0.0938,0.50892,0.333,0.25,0.57121,0.67857,0.0,1.0,2,8,0,2,0,6,0,0,5,0,0,2,0,0,9,0,0,0,0,0,0,0,8],[16,128,0.125,0.57134,0.32549,0.25,0.57143,0.85714,0.0,1.0,1,7,0,1,0,7,0,0,2,0,0,2,0,0,5,0,0,6,0,0,2,0,7],[20,128,0.1562,0.52679,0.34337,0.14286,0.4286,0.89286,0.0,1.0,1,8,0,1,0,9,0,0,1,0,0,6,0,0,4,0,0,1,0,0,2,0,8],[24,128,0.1875,0.46872,0.34851,0.14286,0.49979,0.74996,0.0,1.0,3,5,0,3,0,10,0,0,2,0,0,1,0,0,5,0,0,3,0,0,3,0,5],[28,128,0.2188,0.48659,0.33854,0.14286,0.4998,0.64282,0.0,1.0,2,7,0,2,0,9,0,0,2,0,0,3,0,0,8,0,0,0,0,0,1,0,7],[32,128,0.25,0.40625,0.32754,0.14286,0.42857,0.57143,0.0,1.0,5,4,0,5,0,8,0,0,2,0,0,6,0,0,4,0,0,1,0,0,2,0,4],[36,128,0.2812,0.46863,0.27263,0.24999,0.4998,0.57143,0.0,1.0,2,3,0,2,0,6,0,0,2,0,0,6,0,0,10,0,0,2,0,0,1,0,3],[40,128,0.3125,0.48658,0.37262,0.14286,0.57121,0.89286,0.0,1.0,5,8,0,5,0,8,0,0,0,0,0,1,0,0,9,0,0,0,0,0,1,0,8],[44,128,0.3438,0.52231,0.36,0.14286,0.5712,1.0,0.0,1.0,2,9,0,2,0,8,0,0,4,0,0,1,0,0,6,0,0,1,0,0,1,0,9],[48,128,0.375,0.41516,0.30169,0.14286,0.42857,0.57143,0.0,1.0,3,4,0,3,0,8,0,0,4,0,0,5,0,0,7,0,0,0,0,0,1,0,4],[52,128,0.4062,0.19643,0.24936,0.0,0.14286,0.2857,0.0,0.85714,15,0,0,15,0,6,0,0,4,0,0,2,0,0,2,0,0,2,0,0,1,0,0],[56,128,0.4375,0.30795,0.31366,0.0,0.14286,0.57143,0.0,1.0,9,2,0,9,0,10,0,0,1,0,0,1,0,0,6,0,0,2,0,0,1,0,2],[60,128,0.4688,0.32142,0.25504,0.14286,0.35714,0.42858,0.0,1.0,7,1,0,7,0,6,0,0,3,0,0,9,0,0,5,0,0,0,0,0,1,0,1],[64,128,0.5,0.27902,0.29253,0.05355,0.14286,0.42857,0.0,1.0,8,2,0,8,1,9,0,0,4,0,0,5,0,0,1,0,0,0,0,0,2,0,2],[68,128,0.5312,0.26777,0.22522,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,12,0,0,2,0,0,7,0,0,3,0,0,1,0,0,1,0,0],[72,128,0.5625,0.28125,0.28231,0.14286,0.14288,0.42857,0.0,1.0,7,2,0,7,0,11,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,2],[76,128,0.5938,0.29909,0.27047,0.14286,0.2857,0.42857,0.0,1.0,7,2,0,7,0,7,0,0,7,0,0,6,0,0,2,0,0,0,0,0,1,0,2],[80,128,0.625,0.24997,0.25997,0.0,0.14286,0.46418,0.0,1.0,11,1,0,11,0,8,0,0,2,0,0,3,0,0,7,0,0,0,0,0,0,0,1],[84,128,0.6562,0.33705,0.26977,0.14286,0.28571,0.57143,0.0,1.0,6,1,0,6,0,8,0,0,3,0,0,5,0,1,5,0,0,2,0,0,1,0,1],[88,128,0.6875,0.29911,0.28203,0.10714,0.21428,0.42858,0.0,1.0,8,2,0,8,0,8,0,0,3,0,0,7,0,0,3,0,0,0,0,0,1,0,2],[92,128,0.7188,0.32589,0.32778,0.0,0.14286,0.57143,0.0,1.0,11,1,0,11,0,6,0,0,0,0,0,6,0,0,2,0,0,2,0,0,4,0,1],[96,128,0.75,0.31693,0.28284,0.14286,0.2143,0.571,0.0,1.0,7,1,0,7,0,9,0,0,3,0,0,4,0,0,5,0,0,1,0,0,2,0,1],[100,128,0.7812,0.38604,0.34762,0.0,0.35714,0.60714,0.0,1.0,9,4,0,9,0,5,0,0,2,0,0,3,0,1,4,0,0,3,0,0,1,0,4],[104,128,0.8125,0.27229,0.24314,0.10714,0.21428,0.42858,0.0,1.0,8,1,0,8,0,8,0,0,5,0,0,4,0,0,6,0,0,0,0,0,0,0,1],[108,128,0.8438,0.28125,0.26119,0.0,0.14288,0.42857,0.0,0.85714,9,0,0,9,0,8,0,0,1,0,0,9,0,0,1,0,0,2,0,0,2,0,0],[112,128,0.875,0.36158,0.30924,0.14286,0.28571,0.57141,0.0,1.0,7,3,0,7,0,6,0,0,5,0,0,3,0,0,5,0,0,3,0,0,0,0,3],[116,128,0.9062,0.29911,0.20316,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,7,0,0,6,0,0,4,0,0,2,0,0,0,0,0],[120,128,0.9375,0.21427,0.22866,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,9,0,0,5,0,0,2,0,0,2,0,0,3,0,0,0,0,0],[124,128,0.9688,0.25893,0.20341,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,4,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[128,128,1.0,0.20088,0.19837,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,7,0,0,4,0,0,6,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.30357,"x":0.79911,"p":[[0,104,0.0,0.49104,0.30291,0.28571,0.42857,0.71429,0.0,1.0,4,4,0,4,0,3,0,0,2,0,0,8,0,0,6,0,0,3,0,0,2,0,4],[4,104,0.0385,0.73661,0.29038,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,4,0,0,7,0,0,1,0,0,2,0,15],[8,104,0.0769,0.61158,0.3159,0.42857,0.57143,1.0,0.0,1.0,1,10,0,1,0,5,0,0,1,0,0,2,0,0,11,0,0,2,0,0,0,0,10],[12,104,0.1154,0.79911,0.27862,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,1,0,0,2,0,19],[16,104,0.1538,0.75893,0.28669,0.53572,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,1,0,0,3,0,16],[20,104,0.1923,0.73214,0.29613,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,1,0,0,3,0,0,7,0,0,1,0,0,2,0,15],[24,104,0.2308,0.79462,0.24209,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,2,0,0,1,0,17],[28,104,0.2692,0.67857,0.33312,0.5,0.78571,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,2,0,0,0,0,0,7,0,0,1,0,0,3,0,13],[32,104,0.3077,0.73661,0.26752,0.57143,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,5,0,0,10,0,0,0,0,0,2,0,14],[36,104,0.3462,0.6875,0.29974,0.57143,0.71429,1.0,0.0,1.0,1,11,0,1,0,3,0,0,1,0,0,1,0,0,9,0,0,3,0,0,3,0,11],[40,104,0.3846,0.62943,0.31311,0.42857,0.57143,1.0,0.0,1.0,1,11,0,1,0,3,0,0,2,0,0,5,0,0,9,0,0,0,0,0,1,0,11],[44,104,0.4231,0.66964,0.30186,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,3,0,0,3,0,0,9,0,0,0,0,0,2,0,12],[48,104,0.4615,0.66517,0.29366,0.57132,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,2,0,0,1,0,0,10,0,0,2,0,0,3,0,10],[52,104,0.5,0.65624,0.27862,0.53539,0.57143,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,2,0,0,3,0,0,10,0,0,2,0,0,3,0,9],[56,104,0.5385,0.53123,0.2993,0.28571,0.49979,0.74996,0.14286,1.0,0,6,0,0,0,6,0,0,5,0,0,5,0,0,6,0,0,2,0,0,2,0,6],[60,104,0.5769,0.56695,0.33784,0.24999,0.57143,1.0,0.0,1.0,1,10,0,1,0,7,0,0,2,0,0,2,0,0,10,0,0,0,0,0,0,0,10],[64,104,0.6154,0.61605,0.29973,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,1,0,0,6,0,0,7,0,0,1,0,0,4,0,8],[68,104,0.6538,0.73212,0.28293,0.57143,0.78571,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,3,0,0,0,0,0,9,0,0,2,0,0,2,0,14],[72,104,0.6923,0.62499,0.31492,0.39286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,4,0,0,5,0,0,5,0,0,1,0,0,3,0,10],[76,104,0.7308,0.60258,0.34032,0.28571,0.57143,1.0,0.0,1.0,1,11,0,1,0,6,0,0,2,0,0,2,0,0,9,0,0,0,0,0,1,0,11],[80,104,0.7692,0.60712,0.31744,0.39286,0.57143,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,1,0,0,2,0,0,9,0,0,2,0,0,2,0,9],[84,104,0.8077,0.70982,0.29339,0.53571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,4,0,0,2,0,0,5,0,0,4,0,0,2,0,13],[88,104,0.8462,0.62945,0.30275,0.5354,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,1,0,0,1,0,0,10,0,0,2,0,0,4,0,8],[92,104,0.8846,0.57141,0.29451,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,5,0,0,4,0,0,3,0,0,10,0,0,1,0,0,2,0,7],[96,104,0.9231,0.52678,0.34337,0.14286,0.5,0.85714,0.14286,1.0,0,7,0,0,0,10,0,0,4,0,0,2,0,0,4,0,0,1,0,0,4,0,7],[100,104,0.9615,0.37052,0.28539,0.14286,0.28571,0.57111,0.14286,1.0,0,3,0,0,0,15,0,0,5,0,0,3,0,0,3,0,0,2,0,0,1,0,3],[104,104,1.0,0.30357,0.25939,0.14286,0.14286,0.42857,0.14286,1.0,0,2,0,0,0,19,0,0,4,0,0,5,0,0,0,0,0,0,0,0,2,0,2]]}]},{"i":"c03cbb271dffad26","q":"Find the sum of all positive integers  $m$  such that  $2^m$  can be expressed as a sum of four factorials (of positive integers).\n\nNote: The factorials do not have to be distinct.  For example,  $2^4=16$  counts, because it equals  $3!+3!+2!+2!$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.1875,"x":0.30798,"p":[[0,92,0.0,0.30798,0.18258,0.14286,0.28571,0.42893,0.0,0.71429,2,0,0,2,0,9,0,0,11,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[4,92,0.0435,0.27677,0.18875,0.14286,0.28571,0.32143,0.0,0.71429,6,0,0,6,0,4,0,0,14,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[8,92,0.087,0.25894,0.14914,0.14286,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,19,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,92,0.1304,0.23195,0.12252,0.14286,0.2857,0.28571,0.0,0.571,3,0,0,3,0,10,0,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,92,0.1739,0.25436,0.18469,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,6,0,0,14,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[20,92,0.2174,0.24535,0.14839,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,9,0,0,13,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[24,92,0.2609,0.1875,0.1448,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,12,0,0,11,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,92,0.3043,0.19187,0.12172,0.14214,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,8,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,92,0.3478,0.27669,0.15955,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,9,0,0,15,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[36,92,0.3913,0.2408,0.11561,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,7,0,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,92,0.4348,0.23213,0.1417,0.14286,0.2857,0.28571,0.0,0.57143,5,0,0,5,0,7,0,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[44,92,0.4783,0.20536,0.11811,0.14286,0.21428,0.28571,0.0,0.57143,4,0,0,4,0,12,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,92,0.5217,0.22321,0.18189,0.10714,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,6,0,0,14,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[52,92,0.5652,0.22767,0.11214,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,9,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,92,0.6087,0.20509,0.13346,0.14,0.21428,0.28571,0.0,0.4286,6,0,0,6,0,10,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,92,0.6522,0.20089,0.15093,0.10714,0.21428,0.28571,0.0,0.57143,8,0,0,8,0,8,0,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[64,92,0.6957,0.22303,0.11271,0.14286,0.2857,0.2857,0.0,0.57143,3,0,0,3,0,10,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,92,0.7391,0.24102,0.1493,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,8,0,0,17,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[72,92,0.7826,0.22334,0.10685,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,7,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,92,0.8261,0.20527,0.11263,0.14286,0.2857,0.28571,0.0,0.42857,5,0,0,5,0,9,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,92,0.8696,0.20535,0.10677,0.14286,0.2857,0.28571,0.0,0.4286,4,0,0,4,0,11,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,92,0.913,0.23215,0.11151,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,11,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[88,92,0.9565,0.25894,0.10374,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,9,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[92,92,1.0,0.22303,0.1069,0.14286,0.2857,0.2857,0.0,0.57143,2,0,0,2,0,12,0,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.19187,"x":0.28571,"p":[[0,64,0.0,0.25438,0.13244,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,8,0,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[4,64,0.0625,0.28127,0.15764,0.14286,0.28571,0.42857,0.0,0.5714,4,0,0,4,0,5,0,0,14,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[8,64,0.125,0.20089,0.16698,0.0,0.2857,0.28571,0.0,0.57143,11,0,1,11,0,3,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,64,0.1875,0.28571,0.14286,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,3,0,0,20,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[16,64,0.25,0.19187,0.11637,0.14214,0.2857,0.28571,0.0,0.28571,7,0,0,7,0,7,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.21411,0.1598,0.105,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,5,0,0,16,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[24,64,0.375,0.24535,0.13953,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,9,0,0,16,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[28,64,0.4375,0.26785,0.1171,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[32,64,0.5,0.2766,0.13821,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,9,0,0,16,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[36,64,0.5625,0.23214,0.12752,0.14286,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,5,0,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,64,0.625,0.23642,0.16612,0.14214,0.2857,0.2857,0.0,0.71429,6,0,0,6,0,6,0,0,17,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[44,64,0.6875,0.24089,0.11554,0.14286,0.2857,0.28571,0.0,0.4286,3,0,0,3,0,8,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.25001,0.11294,0.25,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,5,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,64,0.8125,0.22768,0.11769,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,7,0,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,64,0.875,0.25884,0.08341,0.14286,0.28571,0.28571,0.14,0.4286,0,0,0,0,0,9,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.23652,0.09191,0.14286,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,8,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.22321,0.08702,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,10,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7bcca7fe22b35f46","q":"For a set of real numbers  $A$ , let  $A-A$  be the set of distinct pairwise differences of  $A$ ; that is, \\[A-A:=\\{a-b:a,b\\in A\\}\\] If  $|A-A|=25$ , find the sum of all possible values of  $|A|$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.66067,"x":0.98661,"p":[[0,69,0.0,0.66964,0.15335,0.57143,0.57143,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,4,0,0,7,0,2],[4,69,0.058,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[8,69,0.1159,0.91071,0.1948,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,23],[12,69,0.1739,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,69,0.2319,0.90624,0.14557,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,20],[20,69,0.2899,0.85264,0.26607,0.82143,1.0,1.0,0.0,1.0,2,21,1,2,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,21],[24,69,0.3478,0.91964,0.15542,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,25],[28,69,0.4058,0.91071,0.13716,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,21],[32,69,0.4638,0.89732,0.16457,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,1,0,22],[36,69,0.5217,0.86607,0.16728,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,3,0,18],[40,69,0.5797,0.87946,0.18595,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,3,0,20],[44,69,0.6377,0.85712,0.189,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,1,0,19],[48,69,0.6957,0.79911,0.15093,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,3,0,10],[52,69,0.7536,0.76338,0.19759,0.71429,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,15,0,0,1,0,10],[56,69,0.8116,0.85265,0.1536,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,4,0,15],[60,69,0.8696,0.66963,0.13571,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,15,0,0,0,0,3],[64,69,0.9275,0.72767,0.20001,0.67857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,15,0,0,1,0,8],[68,69,0.9855,0.69196,0.08828,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,25,0,0,2,0,0],[69,69,1.0,0.66067,0.11155,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,20,0,0,0,0,1]]},{"b":7,"e":0.71429,"k":"flat","v":0.6339,"x":0.99107,"p":[[0,84,0.0,0.63391,0.12339,0.57143,0.57143,0.57143,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,1,0,0,5,0,1],[4,84,0.0476,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,84,0.0952,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,84,0.1429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,84,0.1905,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[20,84,0.2381,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22],[24,84,0.2857,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[28,84,0.3333,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[32,84,0.381,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[36,84,0.4286,0.87946,0.21162,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,0,3,0,21],[40,84,0.4762,0.88393,0.1729,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,21],[44,84,0.5238,0.89286,0.14286,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,19],[48,84,0.5714,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[52,84,0.619,0.92857,0.13832,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[56,84,0.6667,0.91964,0.16342,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,24],[60,84,0.7143,0.91071,0.14174,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21],[64,84,0.7619,0.88393,0.1729,0.71429,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,1,0,21],[68,84,0.8095,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[72,84,0.8571,0.90625,0.1411,0.82143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,3,0,21],[76,84,0.9048,0.93303,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23],[80,84,0.9524,0.83927,0.15467,0.71429,0.71429,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,0,0,15],[84,84,1.0,0.6339,0.16343,0.57132,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,18,0,0,1,0,1]]}]},{"i":"9020d2ba10a86c40","q":"Find all prime numbers  $p_1,\u2026,p_n$  (not necessarily different) such that :  $$  \\prod_{i=1}^n p_i=10 \\sum_{i=1}^n p_i $$","t":[{"b":4,"e":0.85714,"k":"falling","v":0.65183,"x":0.89286,"p":[[0,114,0.0,0.89286,0.20203,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,23],[4,114,0.0351,0.85714,0.19562,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,7,0,16],[8,114,0.0702,0.73213,0.24937,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,4,0,0,4,0,11],[12,114,0.1053,0.75893,0.26351,0.57143,0.85714,1.0,0.0,1.0,1,12,1,1,0,1,0,0,1,0,0,1,0,0,5,0,0,6,0,0,5,0,12],[16,114,0.1404,0.79018,0.18893,0.67857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,8,0,10],[20,114,0.1754,0.78571,0.22303,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,6,0,0,8,0,11],[24,114,0.2105,0.7634,0.23312,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,6,0,0,11,0,8],[28,114,0.2456,0.82589,0.17762,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,11,0,11],[32,114,0.2807,0.72768,0.27283,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,4,0,0,9,0,9],[36,114,0.3158,0.7857,0.17498,0.71429,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,9,0,8],[40,114,0.3509,0.70088,0.19679,0.57143,0.71429,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,15,0,0,3,0,5],[44,114,0.386,0.69196,0.24772,0.57142,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,11,0,0,4,0,7],[48,114,0.4211,0.68303,0.23618,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,1,0,0,8,0,0,7,0,0,7,0,5],[52,114,0.4561,0.67854,0.27895,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,3,0,0,5,0,0,3,0,0,7,0,8],[56,114,0.4912,0.69641,0.26184,0.5354,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,5,0,0,6,0,0,4,0,0,5,0,9],[60,114,0.5263,0.74106,0.22991,0.57142,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,3,0,0,8,0,9],[64,114,0.5614,0.86604,0.15547,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,6,0,16],[68,114,0.5965,0.87052,0.17628,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,4,0,18],[72,114,0.6316,0.81247,0.20655,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,0,8,0,13],[76,114,0.6667,0.7321,0.26907,0.571,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,3,0,0,6,0,11],[80,114,0.7018,0.7723,0.20784,0.67857,0.78571,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,8,0,0,6,0,10],[84,114,0.7368,0.70089,0.27048,0.4286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,6,0,0,5,0,0,3,0,0,5,0,10],[88,114,0.7719,0.80357,0.22517,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,5,0,0,7,0,13],[92,114,0.807,0.77231,0.15917,0.71429,0.78571,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,11,0,5],[96,114,0.8421,0.65183,0.22281,0.53605,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,5,0,0,8,0,0,6,0,0,6,0,4],[100,114,0.8772,0.70981,0.23004,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,3,0,0,6,0,8],[104,114,0.9123,0.70085,0.25347,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,7,0,0,6,0,0,2,0,0,6,0,9],[108,114,0.9474,0.77679,0.25738,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,6,0,0,4,0,14],[112,114,0.9825,0.78571,0.25505,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,2,0,0,8,0,13],[114,114,1.0,0.73651,0.22923,0.5354,0.85714,0.85714,0.14,1.0,0,7,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,6,0,0,10,0,7]]},{"b":5,"e":0.71429,"k":"flat","v":0.6607,"x":0.88839,"p":[[0,106,0.0,0.85714,0.24223,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,3,0,21],[4,106,0.0377,0.88839,0.1665,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,20],[8,106,0.0755,0.7098,0.21276,0.57132,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,5,0,0,6,0,7],[12,106,0.1132,0.80357,0.20124,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,7,0,0,7,0,12],[16,106,0.1509,0.77677,0.25239,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,0,10,0,11],[20,106,0.1887,0.68302,0.24931,0.57132,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,0,0,0,4,0,0,6,0,0,5,0,0,9,0,5],[24,106,0.2264,0.6607,0.23892,0.57132,0.71429,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,3,0,0,4,0,0,11,0,0,7,0,3],[28,106,0.2642,0.72313,0.26967,0.57143,0.71429,1.0,0.14,1.0,0,11,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,8,0,0,3,0,11],[32,106,0.3019,0.72771,0.28424,0.42964,0.85714,1.0,0.1429,1.0,0,12,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,0,0,0,7,0,12],[36,106,0.3396,0.77674,0.2171,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,6,0,0,6,0,11],[40,106,0.3774,0.69643,0.17034,0.57143,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,10,0,0,7,0,3],[44,106,0.4151,0.82589,0.19475,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,6,0,14],[48,106,0.4528,0.81694,0.18981,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,4,0,0,12,0,10],[52,106,0.4906,0.83034,0.14482,0.71429,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,10,0,10],[56,106,0.5283,0.77218,0.22267,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,7,0,0,11,0,8],[60,106,0.566,0.81249,0.16148,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,11,0,9],[64,106,0.6038,0.74105,0.20342,0.67857,0.85707,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,7,0,0,13,0,4],[68,106,0.6415,0.79467,0.20799,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,0,10,0,10],[72,106,0.6792,0.79015,0.17126,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,15,0,6],[76,106,0.717,0.75445,0.22372,0.71429,0.85714,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,12,0,6],[80,106,0.7547,0.81247,0.12084,0.71429,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,15,0,5],[84,106,0.7925,0.79462,0.18192,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,12,0,8],[88,106,0.8302,0.81248,0.19047,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,5,0,0,11,0,10],[92,106,0.8679,0.7589,0.18366,0.71429,0.78571,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,11,0,5],[96,106,0.9057,0.74999,0.17498,0.57143,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,13,0,4],[100,106,0.9434,0.81247,0.1938,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,0,14,0,9],[104,106,0.9811,0.80803,0.18423,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,16,0,7],[106,106,1.0,0.76781,0.1666,0.57143,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,13,0,5]]}]},{"i":"64e6c8e95f5ab53d","q":"Find the number of ways to partition a set of  $10$  elements,  $S = \\{1, 2, 3, . . . , 10\\}$  into two parts; that is, the number of unordered pairs  $\\{P, Q\\}$  such that  $P \\cup Q = S$  and  $P \\cap Q = \\emptyset$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,12,0.0,0.89286,0.18558,0.89286,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,24],[4,12,0.3333,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,12,0.6667,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,26,0.0,0.86607,0.19865,0.57143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,22],[4,26,0.1538,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"7acaf619cd5776c7","q":"Find the real numbers  $x, y, z$  such that,   $$ \\frac{1}{x}+\\frac{1}{y+z}=\\frac{1}{2}, \\frac{1}{y}+\\frac{1}{z+x}=\\frac{1}{3}, \\frac{1}{z}+\\frac{1}{x+y}=\\frac{1}{4}. $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.79909,"x":0.96875,"p":[[0,52,0.0,0.79909,0.25967,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,3,0,0,8,0,14],[4,52,0.0769,0.84375,0.20935,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,0,4,0,18],[8,52,0.1538,0.9375,0.13803,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,23],[12,52,0.2308,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[16,52,0.3077,0.96875,0.08553,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,52,0.3846,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,52,0.4615,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,8,0,22],[28,52,0.5385,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[32,52,0.6154,0.94643,0.10565,0.96429,1.0,1.0,0.5714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[36,52,0.6923,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[40,52,0.7692,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,22],[44,52,0.8462,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[48,52,0.9231,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[52,52,1.0,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26]]},{"b":5,"e":1.0,"k":"rising","v":0.68303,"x":0.99107,"p":[[0,69,0.0,0.68303,0.27832,0.57143,0.71429,0.89286,0.0,1.0,1,8,1,1,0,1,0,0,3,0,0,2,0,0,8,0,0,2,0,0,7,0,8],[4,69,0.058,0.83036,0.24856,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,3,0,0,3,0,19],[8,69,0.1159,0.86607,0.20806,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,4,0,20],[12,69,0.1739,0.85266,0.2328,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,21],[16,69,0.2319,0.875,0.1948,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,8,0,18],[20,69,0.2899,0.78571,0.29014,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,0,0,0,5,0,17],[24,69,0.3478,0.90177,0.16148,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,21],[28,69,0.4058,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[32,69,0.4638,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,69,0.5217,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[40,69,0.5797,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,69,0.6377,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[48,69,0.6957,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[52,69,0.7536,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[56,69,0.8116,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,69,0.8696,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,69,0.9275,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[68,69,0.9855,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[69,69,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"b5ef3bb547c36342","q":"For any positive integer  $n$ , define  $f(n)$  to be the smallest positive integer that does not divide  $n$ . For example,  $f(1)=2$ ,  $f(6)=4$ . Prove that for any positive integer  $n$ , either  $f(f(n))$  or  $f(f(f(n)))$  must be equal to  $2$ .","t":[{"b":1,"e":0.1429,"k":"falling","v":0.50437,"x":0.99107,"p":[[0,61,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,61,0.0656,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[8,61,0.1311,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,61,0.1967,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,61,0.2623,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,61,0.3279,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[24,61,0.3934,0.9375,0.19541,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[28,61,0.459,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[32,61,0.5246,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,61,0.5902,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[40,61,0.6557,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[44,61,0.7213,0.92857,0.20516,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[48,61,0.7869,0.91964,0.17105,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[52,61,0.8525,0.80357,0.31084,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,21],[56,61,0.918,0.68304,0.36023,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,5,0,0,3,0,0,0,0,0,2,0,0,0,0,17],[60,61,0.9836,0.62054,0.37899,0.14286,0.78571,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,4,0,0,1,0,0,0,0,0,2,0,0,3,0,13],[61,61,1.0,0.50437,0.36949,0.14286,0.28571,1.0,0.14,1.0,0,11,0,0,0,10,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,11]]},{"b":6,"e":1.0,"k":"flat","v":0.91071,"x":0.99107,"p":[[0,32,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,32,0.125,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[8,32,0.25,0.95088,0.11634,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[12,32,0.375,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[16,32,0.5,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[20,32,0.625,0.91071,0.17405,0.85711,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,0,23],[24,32,0.75,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[28,32,0.875,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,32,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"0baf67759add513e","q":"Find the triplets of natural numbers  $(p,q,r)$  that satisfy the equality   $$ \\frac{1}{p}+\\frac{q}{q^r -1}=1. $$","t":[{"b":3,"e":1.0,"k":"flat","v":0.875,"x":0.98661,"p":[[0,38,0.0,0.875,0.18123,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,7,0,18],[4,38,0.1053,0.89732,0.11971,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,15],[8,38,0.2105,0.91071,0.11152,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,15],[12,38,0.3158,0.91071,0.12242,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,17],[16,38,0.4211,0.92856,0.10106,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[20,38,0.5263,0.90625,0.12682,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,18],[24,38,0.6316,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[28,38,0.7368,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,38,0.8421,0.94196,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[36,38,0.9474,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[38,38,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":6,"e":0.85714,"k":"flat","v":0.80804,"x":0.93304,"p":[[0,40,0.0,0.90625,0.11071,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[4,40,0.1,0.87946,0.17896,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,16],[8,40,0.2,0.89286,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[12,40,0.3,0.80804,0.18423,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,15,0,8],[16,40,0.4,0.87052,0.16508,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,11,0,15],[20,40,0.5,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[24,40,0.6,0.88839,0.12745,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,20,0,11],[28,40,0.7,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[32,40,0.8,0.91517,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[36,40,0.9,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[40,40,1.0,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18]]}]},{"i":"0f298b3474f4fab3","q":"Find the largest real number  $k$ , such that for any positive real numbers  $a,b$ ,  $$ (a+b)(ab+1)(b+1)\\geq kab^2 $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,5,0.0,0.82143,0.3607,0.96429,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,24],[4,5,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[5,5,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"rising","v":0.77232,"x":1.0,"p":[[0,20,0.0,0.77232,0.3551,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,20],[4,20,0.2,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4f1db506e2ca8cfd","q":"For each natural number  $n\\geq 2$ , solve the following system of equations in the integers  $x_1, x_2, ..., x_n$ :  $$ (n^2-n)x_i+\\left(\\prod_{j\\neq i}x_j\\right)S=n^3-n^2,\\qquad \\forall 1\\le i\\le n $$  where  $$ S=x_1^2+x_2^2+\\dots+x_n^2. $$","t":[{"b":4,"e":1.0,"k":"flat","v":0.87724,"x":0.99554,"p":[[0,75,0.0,0.87724,0.14199,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,6,0,0,9,0,15],[4,75,0.0533,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,75,0.1067,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,75,0.16,0.9598,0.09612,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[16,75,0.2133,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[20,75,0.2667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,75,0.32,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,75,0.3733,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,75,0.4267,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,75,0.48,0.96652,0.07781,1.0,1.0,1.0,0.64286,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5,0,26],[40,75,0.5333,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[44,75,0.5867,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[48,75,0.64,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,75,0.6933,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,75,0.7467,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[60,75,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,75,0.8533,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[68,75,0.9067,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,75,0.96,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[75,75,1.0,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26]]},{"b":5,"e":1.0,"k":"flat","v":0.88616,"x":1.0,"p":[[0,115,0.0,0.88616,0.12093,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,13,1,13],[4,115,0.0348,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,115,0.0696,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,115,0.1043,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,115,0.1391,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,115,0.1739,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,115,0.2087,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,115,0.2435,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,115,0.2783,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,115,0.313,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,115,0.3478,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,115,0.3826,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[48,115,0.4174,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[52,115,0.4522,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[56,115,0.487,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[60,115,0.5217,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[64,115,0.5565,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,115,0.5913,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[72,115,0.6261,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[76,115,0.6609,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[80,115,0.6957,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[84,115,0.7304,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[88,115,0.7652,0.95312,0.07908,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,7,0,23],[92,115,0.8,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[96,115,0.8348,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,115,0.8696,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[104,115,0.9043,0.95536,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[108,115,0.9391,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[112,115,0.9739,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[115,115,1.0,0.91518,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16]]}]},{"i":"8712a3d71589904b","q":"For each positive integer n, let  $ f(n) \\equal{} \\sum_{k \\equal{} 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$ . Find the largest value of n for which  $ f(n) \\le 300$ .\r\n\r**Note:**  $ \\lfloor x \\rfloor$  is the greatest integer less than or equal to  $ x$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.71427,"x":0.88839,"p":[[0,74,0.0,0.71427,0.23691,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,7,0,0,3,0,10],[4,74,0.0541,0.82143,0.21429,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,8,0,0,1,0,17],[8,74,0.1081,0.88839,0.17029,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,1,0,21],[12,74,0.1622,0.80804,0.23038,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,9,0,0,0,0,17],[16,74,0.2162,0.83929,0.15047,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,4,0,13],[20,74,0.2703,0.85713,0.19235,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,8,0,0,1,0,19],[24,74,0.3243,0.83036,0.13092,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,11],[28,74,0.3784,0.85268,0.15355,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,3,0,15],[32,74,0.4324,0.85268,0.15765,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,1,0,16],[36,74,0.4865,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[40,74,0.5405,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[44,74,0.5946,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[48,74,0.6486,0.82589,0.15458,0.71429,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,17,0,0,1,0,13],[52,74,0.7027,0.83482,0.13882,0.71429,0.71429,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,13],[56,74,0.7568,0.83482,0.12931,0.71429,0.78564,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,5,0,11],[60,74,0.8108,0.85268,0.14054,0.71429,0.78571,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,15],[64,74,0.8649,0.84821,0.13333,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,13],[68,74,0.9189,0.82143,0.13363,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,2,0,11],[72,74,0.973,0.80357,0.18123,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,12],[74,74,1.0,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16]]},{"b":6,"e":0.71429,"k":"falling","v":0.45536,"x":0.86607,"p":[[0,103,0.0,0.74107,0.21558,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,15,0,0,2,0,9],[4,103,0.0388,0.86607,0.2141,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,22],[8,103,0.0777,0.68304,0.23887,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,7,0,0,2,0,9],[12,103,0.1165,0.70982,0.22442,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,10,0,0,2,0,9],[16,103,0.1553,0.71875,0.25376,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,6,0,0,0,0,13],[20,103,0.1942,0.68304,0.21939,0.42857,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,12,0,0,0,0,8],[24,103,0.233,0.75,0.22016,0.64286,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,11,0,0,2,0,11],[28,103,0.2718,0.71875,0.25874,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,5,0,0,1,0,13],[32,103,0.3107,0.72321,0.25238,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,6,0,0,2,0,12],[36,103,0.3495,0.78125,0.21424,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,2,0,13],[40,103,0.3883,0.71429,0.24744,0.42859,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,7,0,0,1,0,0,10,0,0,2,0,10],[44,103,0.4272,0.74554,0.20743,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,14,0,0,1,0,10],[48,103,0.466,0.74107,0.24074,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,9,0,0,0,0,13],[52,103,0.5049,0.75893,0.24856,0.42857,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,7,0,0,0,0,15],[56,103,0.5437,0.67857,0.22303,0.42857,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,9,0,0,3,0,7],[60,103,0.5825,0.80804,0.19759,0.71429,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,11,0,0,2,0,14],[64,103,0.6214,0.7679,0.25437,0.42857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,4,0,0,1,0,16],[68,103,0.6602,0.66518,0.21902,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,10,0,0,3,0,6],[72,103,0.699,0.65625,0.24707,0.42857,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,6,0,0,1,0,9],[76,103,0.7379,0.70982,0.26362,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,4,0,0,1,0,13],[80,103,0.7767,0.69643,0.22517,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,12,0,0,0,0,9],[84,103,0.8155,0.64284,0.25254,0.42857,0.42859,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,17,0,0,2,0,0,2,0,0,2,0,9],[88,103,0.8544,0.62054,0.22477,0.42857,0.42859,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,8,0,0,1,0,6],[92,103,0.8932,0.62054,0.23585,0.42857,0.42857,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0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an integer $n>0$, denote by $\\mathcal{F}(n)$ the set of integers $m>0$ for which the polynomial $p(x)=x^{2}+m x+n$ has an integer root. (a) Let $S$ denote the set of integers $n>0$ for which $\\mathcal{F}(n)$ contains two consecutive integers. Show that $S$ is infinite but  $$ \\sum_{n \\in S} \\frac{1}{n} \\leq 1 $$  (b) Prove that there are infinitely many positive integers $n$ such that $\\mathcal{F}(n)$ contains three consecutive integers.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.22758,"x":0.74997,"p":[[0,135,0.0,0.74997,0.17499,0.67857,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,12,0,0,5,0,7],[4,135,0.0296,0.62945,0.21387,0.57143,0.71429,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,0,0,0,4,0,0,7,0,0,15,0,0,2,0,2],[8,135,0.0593,0.73214,0.15047,0.71429,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,19,0,0,2,0,5],[12,135,0.0889,0.68304,0.14166,0.67857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,20,0,0,2,0,2],[16,135,0.1185,0.69183,0.1477,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,18,0,0,2,0,3],[20,135,0.1481,0.6875,0.14914,0.67857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,20,0,0,1,0,3],[24,135,0.1778,0.67411,0.09606,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,24,0,0,1,0,0],[28,135,0.2074,0.68747,0.20342,0.57143,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,16,0,0,0,0,6],[32,135,0.237,0.68749,0.11538,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,25,0,0,1,0,1],[36,135,0.2667,0.6607,0.17768,0.57143,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,20,0,0,0,0,3],[40,135,0.2963,0.57588,0.21572,0.42857,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,5,0,0,4,0,0,13,0,0,1,0,2],[44,135,0.3259,0.46428,0.29015,0.2857,0.57143,0.71429,0.0,1.0,6,1,0,6,0,1,0,0,6,0,0,0,0,0,6,0,0,12,0,0,0,0,1],[48,135,0.3556,0.53571,0.24999,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,4,0,0,3,0,0,12,0,0,1,0,2],[52,135,0.3852,0.4286,0.34255,0.0,0.4293,0.71429,0.0,1.0,9,4,0,9,0,2,0,0,2,0,0,4,0,0,5,0,0,6,0,0,0,0,4],[56,135,0.4148,0.49553,0.24996,0.2857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,7,0,0,3,0,0,4,0,0,11,0,0,1,0,1],[60,135,0.4444,0.48661,0.2547,0.28571,0.50001,0.71429,0.0,1.0,3,1,0,3,0,2,0,0,5,0,0,6,0,0,3,0,0,12,0,0,0,0,1],[64,135,0.4741,0.55357,0.25692,0.39286,0.71429,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,5,0,0,4,0,0,2,0,0,15,0,0,2,0,1],[68,135,0.5037,0.44642,0.25938,0.2857,0.42857,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,7,0,0,6,0,0,3,0,0,8,0,0,1,0,1],[72,135,0.5333,0.52232,0.23585,0.39286,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,7,0,0,2,0,0,14,0,0,0,0,1],[76,135,0.563,0.40625,0.26513,0.24999,0.42857,0.71429,0.0,0.71429,6,0,0,6,0,2,0,0,7,0,0,2,0,0,6,0,0,9,0,0,0,0,0],[80,135,0.5926,0.40625,0.26027,0.2857,0.28571,0.71429,0.0,1.0,5,1,0,5,0,0,0,0,12,0,0,4,0,0,2,0,0,8,0,0,0,0,1],[84,135,0.6222,0.43748,0.23402,0.2857,0.42857,0.71429,0.0,0.71429,3,0,0,3,0,2,0,0,8,0,0,6,0,0,3,0,0,10,0,0,0,0,0],[88,135,0.6519,0.42856,0.19561,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,11,0,0,7,0,0,4,0,0,7,0,0,0,0,0],[92,135,0.6815,0.42857,0.24222,0.28571,0.42857,0.71429,0.0,0.71429,4,0,0,4,0,2,0,0,6,0,0,8,0,0,2,0,0,10,0,0,0,0,0],[96,135,0.7111,0.39732,0.23619,0.28571,0.35714,0.60714,0.0,0.71429,4,0,0,4,0,2,0,0,10,0,0,5,0,0,3,0,0,8,0,0,0,0,0],[100,135,0.7407,0.40178,0.27533,0.24999,0.42857,0.71429,0.0,1.0,6,1,0,6,0,2,0,0,7,0,0,5,0,0,3,0,0,8,0,0,0,0,1],[104,135,0.7704,0.34375,0.2854,0.14286,0.2857,0.60714,0.0,1.0,7,1,0,7,0,4,0,0,10,0,0,2,0,0,1,0,0,6,0,0,1,0,1],[108,135,0.8,0.29464,0.24727,0.0,0.28571,0.42858,0.0,0.71429,9,0,0,9,0,3,0,0,9,0,0,4,0,0,2,0,0,5,0,0,0,0,0],[112,135,0.8296,0.33926,0.20121,0.2857,0.28571,0.42858,0.0,0.71429,4,0,0,4,0,3,0,0,12,0,0,6,0,0,4,0,0,3,0,0,0,0,0],[116,135,0.8593,0.30357,0.23891,0.14286,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,8,0,0,6,0,0,5,0,0,2,0,0,5,0,0,0,0,0],[120,135,0.8889,0.27677,0.28106,0.0,0.14288,0.42857,0.0,1.0,9,2,0,9,0,8,0,0,5,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[124,135,0.9185,0.25,0.20516,0.0,0.2857,0.42857,0.0,0.57143,11,0,0,11,0,1,0,0,9,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[128,135,0.9481,0.22758,0.20783,0.0,0.21428,0.42857,0.0,0.71429,11,0,0,11,0,5,0,0,6,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[132,135,0.9778,0.26786,0.21651,0.0,0.28571,0.42857,0.0,0.71429,10,0,0,10,0,1,0,0,10,0,0,7,0,0,2,0,0,2,0,0,0,0,0],[135,135,1.0,0.27679,0.25238,0.0,0.2857,0.42857,0.0,1.0,9,1,0,9,0,4,0,0,10,0,0,4,0,0,1,0,0,3,0,0,0,0,1]]},{"b":4,"e":0.71429,"k":"flat","v":0.60711,"x":0.86161,"p":[[0,165,0.0,0.73646,0.21463,0.57143,0.78564,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,7,0,0,10,0,6],[4,165,0.0242,0.77231,0.12808,0.71429,0.71429,0.75,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,0,1,0,7],[8,165,0.0485,0.79018,0.1636,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,17,0,0,2,0,10],[12,165,0.0727,0.76339,0.17353,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,16,0,0,3,0,8],[16,165,0.097,0.73214,0.20124,0.71429,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,16,0,0,4,0,6],[20,165,0.1212,0.76784,0.17035,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,15,0,0,4,0,8],[24,165,0.1455,0.77232,0.14664,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,7,0,6],[28,165,0.1697,0.81696,0.23753,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,8,0,13],[32,165,0.1939,0.83929,0.17405,0.71429,0.85714,1.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every integer  $n$  not equal to  $1$  or  $-1$ , define  $S(n)$  as the smallest integer greater than  $1$  that divides  $n$ . In particular,  $S(0)=2$ . We also define  $S(1) = S(-1) = 1$ .\n\nLet  $f$  be a non-constant polynomial with integer coefficients such that  $S(f(n)) \\leq S(n)$  for every positive integer  $n$ . Prove that  $f(0)=0$ .**Note:** A non-constant polynomial with integer coefficients is a function of the form  $f(x) = a_0 + a_1 x + a_2 x^2 + \\ldots + a_k x^k$ , where  $k$  is a positive integer and  $a_0,a_1,\\ldots,a_k$  are integers such that  $a_k \\neq 0$ .\n\n*Pitchayut Saengrungkongka, Thailand*","t":[{"b":3,"e":0.71429,"k":"rising","v":0.18972,"x":0.91964,"p":[[0,46,0.0,0.18972,0.31419,0.0,0.0,0.28571,0.0,1.0,21,2,1,21,0,1,0,0,4,0,0,0,0,0,1,0,0,2,1,0,0,0,2],[4,46,0.087,0.75893,0.32031,0.71429,0.85714,1.0,0.0,1.0,3,14,0,3,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,0,6,0,14],[8,46,0.1739,0.80355,0.22233,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,2,0,0,13,0,10],[12,46,0.2609,0.84374,0.16117,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,12,0,11],[16,46,0.3478,0.8482,0.17837,0.85711,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,12,0,13],[20,46,0.4348,0.83928,0.21944,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,6,0,0,6,0,16],[24,46,0.5217,0.875,0.17035,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,9,0,16],[28,46,0.6087,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,21],[32,46,0.6957,0.87499,0.16269,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,13],[36,46,0.7826,0.91963,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[40,46,0.8696,0.84589,0.20243,0.85711,0.85714,1.0,0.14,1.0,0,13,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,4,0,0,12,0,13],[44,46,0.9565,0.82142,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,8,0,0,12,0,9],[46,46,1.0,0.808,0.19762,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,12,0,10]]},{"b":4,"e":0.85714,"k":"rising","v":0.33034,"x":0.92409,"p":[[0,37,0.0,0.33034,0.39518,0.0,0.14286,0.75,0.0,1.0,14,4,0,14,0,6,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,4],[4,37,0.1081,0.66071,0.35848,0.39286,0.85714,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,10,0,8],[8,37,0.2162,0.7098,0.33405,0.57132,0.85714,1.0,0.0,1.0,3,11,0,3,0,2,0,0,1,0,0,1,0,0,3,0,0,3,0,0,8,0,11],[12,37,0.3243,0.78125,0.33117,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,16],[16,37,0.4324,0.78561,0.27221,0.71429,0.85714,1.0,0.14,1.0,0,13,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,3,0,0,9,0,13],[20,37,0.5405,0.8973,0.12996,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,17],[24,37,0.6486,0.92409,0.11289,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[28,37,0.7568,0.87946,0.21162,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,18],[32,37,0.8649,0.91071,0.13717,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[36,37,0.973,0.8616,0.24868,0.85714,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,18],[37,37,1.0,0.83482,0.17168,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,12,0,11]]}]},{"i":"e65e1a5bcc65ca82","q":"Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?","t":[{"b":1,"e":0.0,"k":"volatile","v":0.0,"x":0.83482,"p":[[0,26,0.0,0.77232,0.41011,0.85714,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,23],[4,26,0.1538,0.79018,0.39283,0.96429,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,24],[8,26,0.3077,0.69196,0.43757,0.0,1.0,1.0,0.0,1.0,9,18,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,18],[12,26,0.4615,0.56696,0.48509,0.0,1.0,1.0,0.0,1.0,13,17,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,17],[16,26,0.6154,0.54464,0.47034,0.0,0.78571,1.0,0.0,1.0,13,14,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,14],[20,26,0.7692,0.83482,0.36089,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[24,26,0.9231,0.30357,0.45281,0.0,0.0,1.0,0.0,1.0,22,9,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[26,26,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.37054,"x":0.90179,"p":[[0,27,0.0,0.64732,0.46973,0.0,1.0,1.0,0.0,1.0,11,19,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,19],[4,27,0.1481,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[8,27,0.2963,0.87946,0.30117,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[12,27,0.4444,0.85268,0.33784,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,26],[16,27,0.5926,0.76786,0.40838,0.85714,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,22],[20,27,0.7407,0.6607,0.4385,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,17],[24,27,0.8889,0.49554,0.49614,0.0,0.42857,1.0,0.0,1.0,16,15,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,15],[27,27,1.0,0.37054,0.47897,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,11]]}]},{"i":"09a9669344a092b9","q":"For how many integers  $k$  does the following system of equations has a solution other than  $a=b=c=0$  in the set of real numbers? \\begin{align*} \\begin{cases} a^2+b^2=kc(a+b), b^2+c^2 = ka(b+c), c^2+a^2=kb(c+a).\\end{cases}\\end{align*}","t":[{"b":4,"e":0.857,"k":"rising","v":0.5579,"x":0.76339,"p":[[0,80,0.0,0.5579,0.22959,0.39286,0.64071,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,3,0,0,5,0,0,12,0,0,4,0,0],[4,80,0.05,0.72768,0.24317,0.67857,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,0,0,0,0,0,0,5,0,0,2,0,0,22,0,0],[8,80,0.1,0.71875,0.2435,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,1,0,0,2,0,0,4,0,0,21,0,0],[12,80,0.15,0.66954,0.27782,0.53539,0.85714,0.85714,0.14,0.85714,0,0,0,0,0,5,0,0,2,0,0,1,0,0,2,0,0,2,0,0,20,0,0],[16,80,0.2,0.76339,0.18766,0.85714,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,0,0,0,25,0,0],[20,80,0.25,0.70089,0.22689,0.67857,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,7,0,0,17,0,0],[24,80,0.3,0.74107,0.20652,0.67857,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,22,0,0],[28,80,0.35,0.69194,0.21163,0.57143,0.78569,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,5,0,0,16,0,0],[32,80,0.4,0.69196,0.23176,0.71429,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,10,0,0,15,0,0],[36,80,0.45,0.70088,0.2269,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,0,0,0,4,0,0,5,0,0,18,0,0],[40,80,0.5,0.73214,0.21651,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,8,0,0,19,0,0],[44,80,0.55,0.73214,0.20124,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,9,0,0,18,0,0],[48,80,0.6,0.6875,0.25111,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,2,0,0,2,0,0,0,0,0,8,0,0,17,0,0],[52,80,0.65,0.70536,0.23402,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,0,0,0,1,0,0,9,0,0,17,0,0],[56,80,0.7,0.74107,0.1729,0.71429,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,10,0,0,17,0,0],[60,80,0.75,0.73661,0.19597,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,0,17,0,0],[64,80,0.8,0.65179,0.30291,0.57143,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,5,0,0,0,0,0,0,0,0,2,0,0,5,0,0,18,0,0],[68,80,0.85,0.6875,0.24856,0.71429,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,8,0,0,17,0,0],[72,80,0.9,0.66518,0.2412,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,8,0,0,14,0,0],[76,80,0.95,0.67856,0.22868,0.67836,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,3,0,0,2,0,0,11,0,0,13,0,0],[80,80,1.0,0.72768,0.14445,0.71429,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,17,0,0,11,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.47768,"x":0.77679,"p":[[0,116,0.0,0.55804,0.27049,0.42857,0.64286,0.71429,0.0,0.85714,3,0,0,3,0,2,0,0,2,0,0,4,0,0,5,0,0,9,0,0,7,0,0],[4,116,0.0345,0.74552,0.20745,0.67857,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,0,0,0,6,0,0,2,0,0,22,0,0],[8,116,0.069,0.70536,0.25738,0.67857,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,3,0,0,21,0,0],[12,116,0.1034,0.73213,0.19481,0.57143,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,0,20,0,0],[16,116,0.1379,0.66071,0.24157,0.57143,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,2,0,0,6,0,0,3,0,0,16,0,0],[20,116,0.1724,0.74107,0.14914,0.71429,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,11,0,0,15,0,0],[24,116,0.2069,0.72767,0.19678,0.67857,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,18,0,0],[28,116,0.2414,0.68304,0.18466,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,9,0,0,4,0,0,14,0,0],[32,116,0.2759,0.61606,0.23808,0.53539,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,2,0,0,3,0,0,7,0,0,7,0,0,10,0,0],[36,116,0.3103,0.72321,0.2111,0.57143,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,20,0,0],[40,116,0.3448,0.71428,0.22303,0.57143,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,3,0,0,3,0,0,3,0,0,20,0,0],[44,116,0.3793,0.71874,0.21573,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,7,0,0,18,0,0],[48,116,0.4138,0.64732,0.23954,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,4,0,0,2,0,0,8,0,0,13,0,0],[52,116,0.4483,0.70535,0.20183,0.57143,0.78564,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,0,0,0,5,0,0,7,0,0,16,0,0],[56,116,0.4828,0.62497,0.2468,0.53539,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,3,0,0,1,0,0,3,0,0,4,0,0,10,0,0,10,0,0],[60,116,0.5172,0.70088,0.20317,0.57143,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,6,0,0,16,0,0],[64,116,0.5517,0.73214,0.19149,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,0,20,0,0],[68,116,0.5862,0.68749,0.22143,0.57143,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,0,5,0,0,16,0,0],[72,116,0.6207,0.77679,0.09407,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,17,0,0],[76,116,0.6552,0.73214,0.14617,0.71429,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,12,0,0,14,0,0],[80,116,0.6897,0.64732,0.20511,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,3,0,0,9,0,0,6,0,0,11,0,0],[84,116,0.7241,0.74106,0.16537,0.71429,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,0,16,0,0],[88,116,0.7586,0.65625,0.20781,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,11,0,0,10,0,0],[92,116,0.7931,0.58034,0.28107,0.28571,0.71429,0.85704,0.0,0.85714,1,0,0,1,0,6,0,0,2,0,0,0,0,0,4,0,0,10,0,0,9,0,0],[96,116,0.8276,0.66962,0.19378,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,10,0,0,11,0,0],[100,116,0.8621,0.65175,0.21411,0.571,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,4,0,0,7,0,0,6,0,0,12,0,0],[104,116,0.8966,0.66517,0.21609,0.53569,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,6,0,0,14,0,0],[108,116,0.931,0.67856,0.18558,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,11,0,0,11,0,0],[112,116,0.9655,0.62052,0.18766,0.53539,0.57143,0.74996,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,9,0,0,7,0,0,8,0,0],[116,116,1.0,0.47768,0.17717,0.28571,0.42857,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,11,0,0,2,0,0,8,0,0,1,0,0]]}]},{"i":"ed50efc5bc8649db","q":"For each pair of integers \\( j, k \\geq 2 \\), define the function \\( f_{jk} : \\mathbb{R} \\to \\mathbb{R} \\) given by \n\n\\[\nf_{jk}(x) = 1 - (1 - x^j)^k.\n\\]\n\n(a) Prove that for any integers \\( j, k \\geq 2 \\), there exists a unique real number \\( p_{jk} \\in (0, 1) \\) such that \\( f_{jk}(p_{jk}) = p_{jk} \\). Furthermore, defining \\( \\lambda_{jk} := f'_{jk}(p_{jk}) \\), prove that \\( \\lambda_{jk} > 1 \\).\n\n(b) Prove that \\( p^j_{jk} = 1 - p_{kj} \\) for any integers \\( j, k \\geq 2 \\).\n\n(c) Prove that \\( \\lambda_{jk} = \\lambda_{kj} \\) for any integers \\( j, k \\geq 2 \\).","t":[{"b":3,"e":1.0,"k":"flat","v":0.91964,"x":0.99107,"p":[[0,33,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,33,0.1212,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,33,0.2424,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,33,0.3636,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,33,0.4848,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[20,33,0.6061,0.95981,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[24,33,0.7273,0.91964,0.15947,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,24],[28,33,0.8485,0.9241,0.15146,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,24],[32,33,0.9697,0.93304,0.14279,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,24],[33,33,1.0,0.92857,0.14725,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,25]]},{"b":5,"e":1.0,"k":"flat","v":0.91071,"x":0.99554,"p":[[0,82,0.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[4,82,0.0488,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,82,0.0976,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,82,0.1463,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[16,82,0.1951,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,82,0.2439,0.95089,0.18073,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[24,82,0.2927,0.92857,0.15567,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,24],[28,82,0.3415,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21],[32,82,0.3902,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[36,82,0.439,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,82,0.4878,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,82,0.5366,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[48,82,0.5854,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,82,0.6341,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,82,0.6829,0.9375,0.12846,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,24],[60,82,0.7317,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[64,82,0.7805,0.93304,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,23],[68,82,0.8293,0.91964,0.15126,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[72,82,0.878,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[76,82,0.9268,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[80,82,0.9756,0.93302,0.14282,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[82,82,1.0,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24]]}]},{"i":"952043a8c118e5c2","q":"Given a connected graph with  $n$  edges, where there are no parallel edges. For any two cycles  $C,C'$  in the graph, define its *outer cycle* to be\n\\[C*C'=\\{x|x\\in (C-C')\\cup (C'-C)\\}.\\]\n(1) Let  $r$  be the largest postive integer so that we can choose  $r$  cycles  $C_1,C_2,\\ldots,C_r$  and for all  $1\\leq k\\leq r$  and  $1\\leq i$ ,  $j_1,j_2,\\ldots,j_k\\leq r$ , we have\n\\[C_i\\neq C_{j_1}*C_{j_2}*\\cdots*C_{j_k}.\\]\n(Remark: There should have been an extra condition that either  $j_1\\neq i$  or  $k\\neq 1$ )\n(2) Let  $s$  be the largest positive integer so that we can choose  $s$  edges that do not form a cycle.\n(Remark: A more precise way of saying this is that any nonempty subset of these  $s$  edges does not form a cycle)\nShow that  $r+s=n$ .\n\nNote: A cycle is a set of edges of the form  $\\{A_iA_{i+1},1\\leq i\\leq n\\}$  where  $n\\geq 3$ ,  $A_1,A_2,\\ldots,A_n$  are distinct vertices, and  $A_{n+1}=A_1$ .","t":[{"b":3,"e":0.4286,"k":"volatile","v":0.37053,"x":0.94196,"p":[[0,20,0.0,0.37053,0.21387,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,14,0,0,7,0,0,1,0,0,1,0,0,2,0,1],[4,20,0.2,0.94196,0.19516,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[8,20,0.4,0.81687,0.31406,0.82143,1.0,1.0,0.14,1.0,0,22,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,22],[12,20,0.6,0.71875,0.36155,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,7,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,17],[16,20,0.8,0.67411,0.37327,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,7,0,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,17],[20,20,1.0,0.57579,0.3738,0.14286,0.49979,1.0,0.14,1.0,0,12,0,0,0,9,0,0,5,0,0,2,0,0,2,0,0,0,0,0,2,0,12]]},{"b":6,"e":1.0,"k":"volatile","v":0.41072,"x":0.99554,"p":[[0,15,0.0,0.41072,0.24419,0.25,0.42857,0.46431,0.14286,1.0,0,2,0,0,0,8,0,0,7,0,0,9,0,0,2,0,0,3,0,0,1,0,2],[4,15,0.2667,0.95534,0.12599,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[8,15,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,15,0.8,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[15,15,1.0,0.91518,0.18161,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,25]]}]},{"i":"ed70a6beedaca2c6","q":"For positive integer  $a  \\geq 2$ , denote  $N_a$  as the number of positive integer  $k$  with the following property: the sum of squares of digits of  $k$  in base a representation equals  $k$ . Prove that: \r\n\r\na.)  $N_a$  is odd; \r\n\r\nb.) For every positive integer  $M$ , there exist a positive integer  $a \\geq  2$  such that  $N_a \\geq  M$ .","t":[{"b":0,"e":0.14286,"k":"falling","v":0.13375,"x":0.58029,"p":[[0,34,0.0,0.52227,0.1698,0.42857,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,12,0,0,9,0,0,4,0,0,3,0,0],[4,34,0.1176,0.55353,0.29396,0.42857,0.64286,0.71429,0.0,1.0,3,2,0,3,0,4,0,0,0,0,0,5,0,0,4,0,0,9,0,0,5,0,2],[8,34,0.2353,0.54006,0.33079,0.25,0.57143,0.85714,0.0,1.0,3,6,0,3,0,5,0,0,2,0,0,3,0,0,7,0,0,3,0,0,3,0,6],[12,34,0.3529,0.57588,0.31437,0.42857,0.57143,0.85704,0.0,1.0,3,6,0,3,0,3,0,0,1,0,0,5,0,0,6,0,0,5,0,0,3,0,6],[16,34,0.4706,0.58029,0.26229,0.42857,0.57121,0.74996,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,5,0,0,6,0,0,6,0,0,5,0,3],[20,34,0.5882,0.4107,0.33455,0.14286,0.28571,0.71429,0.0,1.0,6,3,0,6,0,6,0,0,5,0,0,3,0,0,2,0,0,4,0,0,3,0,3],[24,34,0.7059,0.20076,0.15504,0.14286,0.14286,0.28571,0.0,0.571,6,0,0,6,0,13,0,0,10,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[28,34,0.8235,0.14277,0.10714,0.105,0.14286,0.1429,0.0,0.42857,8,0,0,8,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.18742,0.12082,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,15,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.13375,0.12338,0.0,0.14286,0.1429,0.0,0.42857,11,0,0,11,0,14,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.10268,"x":0.61155,"p":[[0,47,0.0,0.49983,0.24246,0.28571,0.4293,0.71429,0.14,1.0,0,1,0,0,0,5,0,0,5,0,0,7,0,0,4,0,0,7,0,0,3,0,1],[4,47,0.0851,0.61155,0.29067,0.49968,0.64286,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,4,0,0,0,0,0,8,0,0,5,0,0,7,0,4],[8,47,0.1702,0.58035,0.27879,0.28571,0.50001,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,7,0,0,7,0,0,2,0,0,4,0,0,5,0,5],[12,47,0.2553,0.53124,0.27254,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,9,0,0,6,0,0,4,0,0,4,0,0,3,0,4],[16,47,0.3404,0.44418,0.30395,0.14286,0.42857,0.60714,0.0,1.0,3,3,0,3,0,6,0,1,4,0,0,5,0,0,5,0,0,2,0,0,3,0,3],[20,47,0.4255,0.46424,0.28345,0.2857,0.42859,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,8,0,0,3,0,0,5,0,0,4,0,0,2,0,3],[24,47,0.5106,0.38839,0.33164,0.14286,0.2857,0.71429,0.0,1.0,7,2,0,7,0,6,0,0,5,0,0,2,0,0,3,0,0,3,0,0,4,0,2],[28,47,0.5957,0.24554,0.20589,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,6,0,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[32,47,0.6809,0.24088,0.22717,0.14,0.14286,0.42858,0.0,0.71429,7,0,0,7,0,14,0,0,2,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[36,47,0.766,0.10268,0.11971,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,47,0.8511,0.13385,0.08702,0.14214,0.14286,0.1429,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.11607,0.14032,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,15,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[47,47,1.0,0.20089,0.1551,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,13,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"c45006244c696cf9","q":"Find all real roots of the equation \\[ \\sqrt{x^2-p}+2\\sqrt{x^2-1}=x  \\] where  $p$  is a real parameter.","t":[{"b":3,"e":1.0,"k":"flat","v":0.86161,"x":0.96429,"p":[[0,85,0.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[4,85,0.0471,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,85,0.0941,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[12,85,0.1412,0.9241,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[16,85,0.1882,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[20,85,0.2353,0.91518,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,18],[24,85,0.2824,0.90178,0.12079,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,17],[28,85,0.3294,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[32,85,0.3765,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[36,85,0.4235,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[40,85,0.4706,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[44,85,0.5176,0.86161,0.14054,0.71429,0.85714,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,10,0,13],[48,85,0.5647,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[52,85,0.6118,0.90177,0.11542,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,16],[56,85,0.6588,0.90179,0.13092,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,19],[60,85,0.7059,0.91517,0.12299,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,20],[64,85,0.7529,0.875,0.13716,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,14],[68,85,0.8,0.92857,0.10714,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[72,85,0.8471,0.90176,0.13577,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,18],[76,85,0.8941,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,18],[80,85,0.9412,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[84,85,0.9882,0.91518,0.11769,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,18],[85,85,1.0,0.89285,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16]]},{"b":5,"e":1.0,"k":"flat","v":0.94196,"x":0.99554,"p":[[0,91,0.0,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,91,0.044,0.94642,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[8,91,0.0879,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[12,91,0.1319,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,91,0.1758,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[20,91,0.2198,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[24,91,0.2637,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,91,0.3077,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,91,0.3516,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,91,0.3956,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,91,0.4396,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[44,91,0.4835,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,91,0.5275,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,91,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,91,0.6154,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,91,0.6593,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,91,0.7033,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,91,0.7473,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[72,91,0.7912,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[76,91,0.8352,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,91,0.8791,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[84,91,0.9231,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[88,91,0.967,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[91,91,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"d93fd1ded4b80fdb","q":"Given a board consists of  $n \\times n$  unit squares ( $n \\ge 3$ ). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any  $2  \\times 2$  square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every  $n$  such that after a finite number of moves, every unit square on the board has a same color.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.35707,"x":1.0,"p":[[0,40,0.0,0.72321,0.29001,0.53572,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,5,0,0,2,0,0,1,0,0,9,0,0,1,0,13],[4,40,0.1,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,40,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,40,0.3,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,40,0.4,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,40,0.5,0.87944,0.25285,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,25],[24,40,0.6,0.86161,0.25626,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,2,0,22],[28,40,0.7,0.86607,0.24728,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,23],[32,40,0.8,0.91071,0.19805,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,24],[36,40,0.9,0.86159,0.25875,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,0,2,0,23],[40,40,1.0,0.35707,0.25995,0.10714,0.28571,0.57143,0.0,1.0,8,1,0,8,0,1,0,0,8,0,0,0,0,0,14,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.0,"k":"volatile","v":0.03125,"x":0.88839,"p":[[0,17,0.0,0.71875,0.29984,0.67857,0.71429,1.0,0.0,1.0,2,12,0,2,0,2,0,0,0,0,0,2,0,0,2,0,0,11,0,0,1,0,12],[4,17,0.2353,0.87946,0.2969,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,26],[8,17,0.4706,0.88839,0.29393,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[12,17,0.7059,0.64732,0.45173,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,18],[16,17,0.9412,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[17,17,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"2e83da267bd7d66f","q":"Given a positive integer  $n,$  let  $s(n)$  denote the sum of the digits of  $n.$  Compute the largest positive integer  $n$  such that  $n = s(n)^2 + 2s(n) - 2.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.74107,"x":0.92857,"p":[[0,74,0.0,0.81696,0.1394,0.82143,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,19,0,5],[4,74,0.0541,0.74552,0.16649,0.71429,0.857,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,9,0,0,18,0,0],[8,74,0.1081,0.74107,0.16917,0.71429,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,12,0,0,11,0,3],[12,74,0.1622,0.79909,0.11217,0.71429,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,19,0,2],[16,74,0.2162,0.79464,0.08702,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,20,0,0],[20,74,0.2703,0.80804,0.15815,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,16,0,6],[24,74,0.3243,0.76339,0.17353,0.71429,0.85707,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,12,0,5],[28,74,0.3784,0.7991,0.10012,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,17,0,2],[32,74,0.4324,0.78124,0.12363,0.71429,0.857,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,16,0,2],[36,74,0.4865,0.78125,0.14718,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,14,0,4],[40,74,0.5405,0.77231,0.13767,0.71429,0.85707,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,11,0,0,17,0,1],[44,74,0.5946,0.7857,0.13834,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,16,0,3],[48,74,0.6486,0.86159,0.10999,0.857,0.85714,1.0,0.5714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,16,0,9],[52,74,0.7027,0.78123,0.12871,0.71429,0.85707,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,14,0,3],[56,74,0.7568,0.83482,0.10779,0.71429,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,13,0,7],[60,74,0.8108,0.85268,0.11564,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,17,0,8],[64,74,0.8649,0.80356,0.16656,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,8,0,0,15,0,6],[68,74,0.9189,0.87054,0.11495,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,11,0,12],[72,74,0.973,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,20],[74,74,1.0,0.90179,0.12078,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,16]]},{"b":7,"e":0.85714,"k":"flat","v":0.77232,"x":0.86607,"p":[[0,62,0.0,0.82589,0.13236,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,22,0,4],[4,62,0.0645,0.83035,0.12078,0.85711,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,20,0,5],[8,62,0.129,0.84375,0.11495,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,22,0,5],[12,62,0.1935,0.8125,0.18363,0.85711,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,20,0,5],[16,62,0.2581,0.83036,0.14032,0.85713,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,19,0,6],[20,62,0.3226,0.81696,0.1439,0.85714,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,23,0,3],[24,62,0.3871,0.85713,0.10714,0.85714,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,22,0,6],[28,62,0.4516,0.82142,0.13362,0.85711,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,24,0,3],[32,62,0.5161,0.82141,0.10717,0.857,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,23,0,2],[36,62,0.5806,0.86607,0.07087,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,24,0,5],[40,62,0.6452,0.83482,0.1017,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,21,0,4],[44,62,0.7097,0.84374,0.09689,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,26,0,3],[48,62,0.7742,0.82142,0.11845,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,23,0,3],[52,62,0.8387,0.82589,0.12234,0.85714,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,25,0,2],[56,62,0.9032,0.84821,0.08702,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,23,0,4],[60,62,0.9677,0.82589,0.09932,0.85714,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,26,0,1],[62,62,1.0,0.77232,0.16311,0.71429,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,20,0,1]]}]},{"i":"2825ccbffbefce4c","q":"Given a rhombus  $ABCD$ , find the locus of the points  $P$  lying inside the rhombus and satisfying  $\\angle APD+\\angle BPC=180^{\\circ}$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.25892,"x":0.77232,"p":[[0,65,0.0,0.34366,0.28096,0.14286,0.21428,0.42857,0.0,1.0,2,1,0,2,0,14,0,0,4,0,0,5,0,0,1,0,0,1,0,0,4,0,1],[4,65,0.0615,0.67411,0.32972,0.78571,0.85714,0.85714,0.0,0.85714,4,0,0,4,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,24,0,0],[8,65,0.1231,0.68304,0.3169,0.71429,0.85714,0.85714,0.0,0.85714,4,0,0,4,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,23,0,0],[12,65,0.1846,0.75892,0.23538,0.85708,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,1,0,0,23,0,2],[16,65,0.2462,0.70301,0.28258,0.7855,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,4,0,0,1,0,0,0,0,1,1,0,0,0,0,0,24,0,0],[20,65,0.3077,0.62043,0.33631,0.35716,0.85714,0.85714,0.0,0.85714,4,0,0,4,0,4,0,0,0,0,0,1,0,0,3,0,0,0,0,0,20,0,0],[24,65,0.3692,0.7723,0.20783,0.85714,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,26,0,0],[28,65,0.4308,0.70982,0.26841,0.67857,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0,23,0,0],[32,65,0.4923,0.77232,0.20473,0.82143,0.85714,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,23,0,1],[36,65,0.5538,0.7366,0.27689,0.85714,0.85714,0.85714,0.0,1.0,2,1,0,2,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,25,0,1],[40,65,0.6154,0.70982,0.31234,0.82132,0.85714,0.85714,0.0,1.0,3,2,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,2],[44,65,0.6769,0.64731,0.32139,0.39286,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,22,0,0],[48,65,0.7385,0.70982,0.29984,0.64286,0.85714,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,19,0,4],[52,65,0.8,0.65633,0.3049,0.5,0.85714,0.85714,0.0,0.86,1,0,0,1,0,6,0,0,1,0,0,0,0,0,1,0,0,3,0,0,20,0,0],[56,65,0.8615,0.5,0.33312,0.14286,0.42857,0.85714,0.0,1.0,2,1,0,2,0,9,0,0,2,0,0,4,0,0,2,0,0,0,0,0,12,0,1],[60,65,0.9231,0.50447,0.2879,0.2857,0.42857,0.85714,0.0,0.85714,1,0,0,1,0,6,0,0,3,0,0,10,0,0,0,0,0,1,0,0,11,0,0],[64,65,0.9846,0.33927,0.26183,0.14286,0.2857,0.57111,0.0,0.85714,3,0,0,3,0,12,0,0,4,0,0,4,0,0,5,0,0,0,0,0,4,0,0],[65,65,1.0,0.25892,0.24597,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,16,0,0,5,0,0,3,0,0,1,0,0,0,0,0,2,0,1]]},{"b":3,"e":0.85714,"k":"rising","v":0.44646,"x":0.75455,"p":[[0,49,0.0,0.44646,0.25937,0.14286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,2,0,0,9,0,0,5,0,0,1,0,0,6,0,0],[4,49,0.0816,0.625,0.36025,0.24999,0.85714,0.85714,0.0,1.0,5,1,0,5,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,21,0,1],[8,49,0.1633,0.70089,0.31615,0.85714,0.85714,0.85714,0.0,1.0,4,1,0,4,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,24,0,1],[12,49,0.2449,0.68304,0.3209,0.82143,0.85714,0.85714,0.0,0.85714,3,0,0,3,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0],[16,49,0.3265,0.625,0.35669,0.14289,0.85714,0.85714,0.0,0.85714,5,0,0,5,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,0],[20,49,0.4082,0.74554,0.23618,0.82143,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,24,0,0],[24,49,0.4898,0.6875,0.3223,0.78571,0.85714,0.85714,0.0,1.0,4,1,0,4,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,23,0,1],[28,49,0.5714,0.62947,0.34042,0.35715,0.85714,0.85714,0.0,0.85714,4,0,0,4,0,4,0,0,0,0,0,2,0,0,0,0,0,1,0,0,21,0,0],[32,49,0.6531,0.6116,0.33547,0.24999,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0,20,0,0],[36,49,0.7347,0.64285,0.32341,0.49967,0.85714,0.85714,0.0,0.85714,3,0,0,3,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,0,21,0,0],[40,49,0.8163,0.61605,0.3489,0.35715,0.85714,0.85714,0.0,1.0,5,1,0,5,0,3,0,0,0,0,0,2,0,0,2,0,0,0,0,0,19,0,1],[44,49,0.898,0.62499,0.34394,0.25001,0.85714,0.85714,0.0,0.85714,4,0,0,4,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,21,0,0],[48,49,0.9796,0.75455,0.27257,0.85714,0.85714,0.85714,0.0,0.86,3,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0],[49,49,1.0,0.71875,0.30615,0.85711,0.85714,0.85714,0.0,1.0,3,2,0,3,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,23,0,2]]}]},{"i":"931318a2661c1c52","q":"Given a triangle  $ ABC$ , let  $ r$  be the external bisector of  $ \\angle ABC$ .  $ P$  and  $ Q$  are the feet of the perpendiculars from  $ A$  and  $ C$  to  $ r$ . If  $ CP \\cap BA \\equal{} M$  and  $ AQ \\cap BC\\equal{}N$ , show that  $ MN$ ,  $ r$  and  $ AC$  concur.","t":[{"b":5,"e":1.0,"k":"falling","v":0.51785,"x":0.95536,"p":[[0,97,0.0,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,97,0.0412,0.95535,0.11539,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[8,97,0.0825,0.92411,0.18552,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[12,97,0.1237,0.89285,0.23419,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,22],[16,97,0.1649,0.9375,0.18189,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[20,97,0.2062,0.93749,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[24,97,0.2474,0.9375,0.17835,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[28,97,0.2887,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[32,97,0.3299,0.9241,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[36,97,0.3711,0.93304,0.18552,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[40,97,0.4124,0.85268,0.27078,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,22],[44,97,0.4536,0.76786,0.29827,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,2,0,0,1,0,0,5,0,0,3,0,16],[48,97,0.4948,0.78125,0.34253,0.71429,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,19],[52,97,0.5361,0.84375,0.27976,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,22],[56,97,0.5773,0.83482,0.32948,0.85714,1.0,1.0,0.0,1.0,4,22,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,22],[60,97,0.6186,0.8482,0.30293,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,22],[64,97,0.6598,0.82142,0.30514,0.82132,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,19],[68,97,0.701,0.85714,0.31135,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,24],[72,97,0.7423,0.86161,0.3164,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,24],[76,97,0.7835,0.84375,0.31003,0.82143,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,23],[80,97,0.8247,0.59375,0.41666,0.0,0.71429,1.0,0.0,1.0,9,10,0,9,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,0,5,0,10],[84,97,0.866,0.69642,0.3973,0.39286,0.92857,1.0,0.0,1.0,6,16,0,6,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,16],[88,97,0.9072,0.74107,0.33204,0.53571,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,3,0,16],[92,97,0.9485,0.51785,0.44284,0.0,0.64286,1.0,0.0,1.0,11,11,0,11,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,11],[96,97,0.9897,0.71875,0.36331,0.53572,0.85714,1.0,0.0,1.0,4,15,0,4,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0,0,6,0,15],[97,97,1.0,0.6428,0.32734,0.571,0.71429,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,2,0,0,0,0,0,6,0,0,7,0,0,4,0,8]]},{"b":6,"e":0.0,"k":"falling","v":0.08482,"x":0.96428,"p":[[0,68,0.0,0.90625,0.27804,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[4,68,0.0588,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[8,68,0.1176,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,68,0.1765,0.94196,0.16312,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[16,68,0.2353,0.91517,0.22829,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,26],[20,68,0.2941,0.89284,0.23421,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,24],[24,68,0.3529,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[28,68,0.4118,0.90178,0.18707,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,19],[32,68,0.4706,0.91964,0.17474,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,24],[36,68,0.5294,0.86159,0.31235,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[40,68,0.5882,0.8482,0.33301,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,25],[44,68,0.6471,0.77219,0.35332,0.71321,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,18],[48,68,0.7059,0.70981,0.3754,0.39286,0.85714,1.0,0.0,1.0,4,15,0,4,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,6,0,15],[52,68,0.7647,0.62946,0.443,0.0,0.85714,1.0,0.0,1.0,10,15,0,10,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,15],[56,68,0.8235,0.51785,0.38918,0.0,0.64286,0.85704,0.0,1.0,9,7,0,9,0,1,0,0,1,0,0,4,0,0,1,0,0,6,0,0,3,0,7],[60,68,0.8824,0.48661,0.46409,0.0,0.42857,1.0,0.0,1.0,14,13,0,14,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,13],[64,68,0.9412,0.50892,0.44883,0.0,0.42857,1.0,0.0,1.0,12,12,0,12,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,12],[68,68,1.0,0.08482,0.20782,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0]]}]},{"i":"334546507fad66e2","q":"Given an infinite sequence of numbers  $a_1, a_2,...$ , in which there are no two equal members.  Segment  $a_i, a_{i+1}, ..., a_{i+m-1}$  of this sequence is called a monotone segment of length  $m$ , if  $a_i < a_{i+1} <...<a_{i+m-1}$  or  $a_i > a_{i+1} >... > a_{i+m-1}$ .  It turned out that for each natural  $k$  the term  $a_k$  is contained in some monotonic segment of length  $k + 1$ . Prove that there exists a natural  $N$  such that the sequence  $a_N , a_{N+1} ,...$  monotonic.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.24107,"x":0.70982,"p":[[0,57,0.0,0.44641,0.33455,0.14286,0.28571,0.71429,0.0,1.0,4,6,0,4,0,5,0,0,8,0,0,2,0,0,4,0,0,3,0,0,0,0,6],[4,57,0.0702,0.70982,0.26362,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,3,0,0,5,0,0,5,0,0,6,0,9],[8,57,0.1404,0.56695,0.23278,0.42857,0.42857,0.64282,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,17,0,0,6,0,0,0,0,0,4,0,4],[12,57,0.2105,0.62498,0.2714,0.42857,0.57143,0.85704,0.0,1.0,1,7,0,1,0,1,0,0,3,0,0,6,0,0,6,0,0,6,0,0,2,0,7],[16,57,0.2807,0.57143,0.31135,0.39286,0.57143,0.85714,0.0,1.0,3,5,0,3,0,2,0,0,3,0,0,5,0,0,5,0,0,4,0,0,5,0,5],[20,57,0.3509,0.51786,0.22799,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,8,0,0,5,0,0,8,0,0,5,0,0,2,0,2],[24,57,0.4211,0.43301,0.21571,0.2857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,10,0,0,6,0,0,6,0,0,5,0,0,0,0,1],[28,57,0.4912,0.49548,0.27659,0.28571,0.42857,0.60714,0.14286,1.0,0,5,0,0,0,3,0,0,12,0,0,3,0,0,6,0,0,2,0,0,1,0,5],[32,57,0.5614,0.49106,0.25489,0.28571,0.4286,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,7,0,0,5,0,0,4,0,0,3,0,2],[36,57,0.6316,0.53125,0.26058,0.28571,0.42857,0.71429,0.1429,1.0,0,5,0,0,0,1,0,0,10,0,0,7,0,0,4,0,0,4,0,0,1,0,5],[40,57,0.7018,0.49107,0.2111,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,12,0,0,6,0,0,1,0,0,1,0,3],[44,57,0.7719,0.49997,0.21128,0.28571,0.571,0.60714,0.0,1.0,1,1,0,1,0,1,0,0,7,0,0,6,0,0,9,0,0,6,0,0,1,0,1],[48,57,0.8421,0.46428,0.22303,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,10,0,0,7,0,0,2,0,0,1,0,2],[52,57,0.9123,0.41047,0.22789,0.2857,0.42857,0.57111,0.0,1.0,2,1,0,2,0,3,0,0,10,0,0,7,0,0,5,0,0,3,0,0,1,0,1],[56,57,0.9825,0.29891,0.22131,0.14286,0.2857,0.42857,0.0,1.0,4,1,0,4,0,8,0,0,11,0,0,3,0,0,4,0,0,1,0,0,0,0,1],[57,57,1.0,0.24107,0.16146,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,7,0,0,12,0,0,5,0,0,2,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.42854,"x":0.63391,"p":[[0,39,0.0,0.4464,0.22515,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,10,0,0,7,0,0,8,0,0,1,0,0,1,0,2],[4,39,0.1026,0.63391,0.2922,0.42857,0.57143,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,4,0,0,3,0,0,10,0,0,1,0,0,4,0,8],[8,39,0.2051,0.54461,0.27301,0.28571,0.4998,0.74996,0.0,1.0,1,4,0,1,0,1,0,0,8,0,0,6,0,0,5,0,0,3,0,0,4,0,4],[12,39,0.3077,0.48659,0.26692,0.28571,0.49979,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,9,0,0,3,0,0,6,0,0,7,0,0,0,0,3],[16,39,0.4103,0.45089,0.27225,0.28571,0.42857,0.57143,0.0,1.0,2,4,0,2,0,3,0,0,9,0,0,6,0,0,6,0,0,2,0,0,0,0,4],[20,39,0.5128,0.5,0.26726,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,13,0,0,2,0,0,5,0,0,3,0,0,5,0,2],[24,39,0.6154,0.56695,0.26842,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,7,0,0,4,0,0,6,0,0,7,0,0,1,0,5],[28,39,0.7179,0.46876,0.30977,0.2857,0.35729,0.71429,0.0,1.0,2,5,0,2,0,4,0,0,10,0,0,5,0,0,0,0,0,5,0,0,1,0,5],[32,39,0.8205,0.44193,0.22404,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,9,0,0,9,0,0,6,0,0,2,0,0,2,0,1],[36,39,0.9231,0.43303,0.18723,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,10,0,0,5,0,0,1,0,0,3,0,0],[39,39,1.0,0.42854,0.20823,0.28571,0.42857,0.4286,0.0,1.0,1,2,0,1,0,1,0,0,10,0,0,13,0,0,4,0,0,0,0,0,1,0,2]]}]},{"i":"f96a766d02252189","q":"For any two positive integers  $n$  and  $p$ , prove that there are exactly  ${{(p+1)}^{n+1}}-{{p}^{n+1}}$  functions \n\t\t $f:\\left\\{ 1,2,...,n \\right\\}\\to \\left\\{ -p,-p+1,-p+2,....,p-1,p \\right\\}$ such that  $\\left| f(i)-f(j) \\right|\\le p$  for all  $i,j\\in \\left\\{ 1,2,...,n \\right\\}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,14,0.0,0.92857,0.22304,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[4,14,0.2857,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,14,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,14,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[14,14,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"rising","v":0.84375,"x":1.0,"p":[[0,23,0.0,0.84375,0.29093,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,24],[4,23,0.1739,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,23,0.3478,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ce636ea8aa8cb30a","q":"Given a triangle $A B C$, let $P$ lie on the circumcircle of the triangle and be the midpoint of the arc $B C$ which does not contain $A$. Draw a straight line $l$ through $P$ so that $l$ is parallel to $A B$. Denote by $k$ the circle which passes through $B$, and is tangent to $l$ at the point $P$. Let $Q$ be the second point of intersection of $k$ and the line $A B$ (if there is no second point of intersection, choose $Q=B)$. Prove that $A Q=A C$.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.31696,"x":0.50447,"p":[[0,95,0.0,0.46429,0.29667,0.14286,0.42857,0.71429,0.0,1.0,1,2,1,1,0,10,0,0,1,0,0,6,0,0,5,0,0,2,0,0,5,0,2],[4,95,0.0421,0.40179,0.25364,0.24999,0.42857,0.57143,0.0,1.0,5,1,0,5,0,3,0,0,4,0,0,9,0,0,5,0,0,5,0,0,0,0,1],[8,95,0.0842,0.42411,0.29121,0.14289,0.35714,0.71429,0.0,1.0,3,1,0,3,0,6,0,0,7,0,0,5,0,0,2,0,0,3,0,0,5,0,1],[12,95,0.1263,0.45536,0.30813,0.24999,0.42857,0.75,0.0,1.0,4,1,0,4,0,4,0,0,6,0,0,6,0,0,1,0,0,3,0,0,7,0,1],[16,95,0.1684,0.42857,0.27664,0.25,0.42857,0.60714,0.0,1.0,4,2,0,4,0,4,0,0,4,0,0,9,0,0,3,0,0,5,0,0,1,0,2],[20,95,0.2105,0.50447,0.22299,0.42857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,2,0,0,14,0,0,2,0,0,6,0,0,3,0,1],[24,95,0.2526,0.36607,0.29653,0.14286,0.35714,0.46431,0.0,1.0,7,2,0,7,0,4,0,0,5,0,0,8,0,0,2,0,0,2,0,0,2,0,2],[28,95,0.2947,0.32588,0.24803,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,8,0,0,4,0,0,9,0,0,3,0,0,1,0,0,1,0,1],[32,95,0.3368,0.45536,0.27765,0.28571,0.42857,0.57143,0.0,1.0,3,3,0,3,0,4,0,0,4,0,0,8,0,0,6,0,0,3,0,0,1,0,3],[36,95,0.3789,0.36611,0.29438,0.14286,0.28571,0.46536,0.0,1.0,7,2,0,7,0,3,0,0,7,0,0,7,0,0,2,0,0,2,0,0,2,0,2],[40,95,0.4211,0.38839,0.30563,0.14286,0.35714,0.46429,0.0,1.0,5,3,0,5,0,6,0,0,5,0,0,8,0,0,1,0,0,2,0,0,2,0,3],[44,95,0.4632,0.4017,0.20663,0.14289,0.42857,0.46431,0.14,0.85714,0,0,0,0,0,9,0,0,2,0,0,13,0,0,4,0,0,2,0,0,2,0,0],[48,95,0.5053,0.41071,0.20437,0.28571,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,6,0,0,2,0,0,16,0,0,4,0,0,0,0,0,3,0,0],[52,95,0.5474,0.39286,0.29233,0.14286,0.42857,0.42857,0.0,1.0,4,2,0,4,0,7,0,0,3,0,0,11,0,0,1,0,0,0,0,0,4,0,2],[56,95,0.5895,0.33036,0.1357,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,10,0,0,3,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[60,95,0.6316,0.41518,0.15714,0.42857,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,5,0,0,2,0,0,19,0,0,4,0,0,1,0,0,1,0,0],[64,95,0.6737,0.33917,0.15054,0.14286,0.42857,0.42857,0.14,0.57143,0,0,0,0,0,11,0,0,1,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[68,95,0.7158,0.37947,0.1411,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,1,0,0,23,0,0,1,0,0,1,0,0,0,0,0],[72,95,0.7579,0.4107,0.15045,0.42857,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,3,0,0,2,0,0,21,0,0,4,0,0,0,0,0,1,0,0],[76,95,0.8,0.36598,0.16354,0.35714,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,6,0,0,0,0,0,21,0,0,2,0,0,1,0,0,0,0,0],[80,95,0.8421,0.42857,0.23958,0.35714,0.42857,0.42857,0.0,1.0,1,1,0,1,0,7,0,0,0,0,0,18,0,0,1,0,0,0,0,0,4,0,1],[84,95,0.8842,0.34813,0.14271,0.25001,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,7,0,0,3,0,0,19,0,0,2,0,0,0,0,0,0,0,0],[88,95,0.9263,0.31696,0.15458,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,11,0,0,2,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[92,95,0.9684,0.33487,0.14558,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,1,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[95,95,1.0,0.35715,0.11845,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.21875,"x":0.51339,"p":[[0,98,0.0,0.51339,0.28316,0.28571,0.42857,0.71429,0.0,1.0,2,4,1,2,0,3,0,0,4,0,0,8,0,0,5,0,0,4,0,0,2,0,4],[4,98,0.0408,0.39286,0.23419,0.28571,0.42857,0.42858,0.0,1.0,3,2,0,3,0,4,0,0,5,0,0,13,0,0,4,0,0,1,0,0,0,0,2],[8,98,0.0816,0.375,0.25442,0.25,0.28571,0.42857,0.0,1.0,3,2,0,3,0,5,0,0,9,0,0,9,0,0,1,0,0,2,0,0,1,0,2],[12,98,0.1224,0.38839,0.2237,0.25,0.42857,0.46431,0.0,0.85714,2,0,0,2,0,6,0,0,6,0,0,10,0,0,3,0,0,3,0,0,2,0,0],[16,98,0.1633,0.36608,0.2765,0.1429,0.35714,0.4286,0.0,1.0,6,1,0,6,0,3,0,0,7,0,0,10,0,0,1,0,0,0,0,0,4,0,1],[20,98,0.2041,0.44642,0.24935,0.28571,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,3,0,0,12,0,0,5,0,0,2,0,0,3,0,1],[24,98,0.2449,0.43748,0.23941,0.2857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,7,0,0,10,0,0,4,0,0,1,0,0,5,0,0],[28,98,0.2857,0.47768,0.2412,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,11,0,0,10,0,0,3,0,0,1,0,0,4,0,2],[32,98,0.3265,0.38839,0.26058,0.14286,0.42857,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,4,0,0,10,0,0,3,0,0,4,0,0,1,0,1],[36,98,0.3673,0.30802,0.24512,0.14286,0.28571,0.42858,0.0,1.0,7,1,0,7,0,4,0,0,10,0,0,4,0,0,4,0,0,2,0,0,0,0,1],[40,98,0.4082,0.36606,0.24726,0.14286,0.42857,0.4642,0.0,0.85714,5,0,0,5,0,4,0,0,6,0,0,9,0,0,4,0,0,1,0,0,3,0,0],[44,98,0.449,0.29,0.21878,0.14,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,5,0,0,7,0,0,9,0,0,2,0,0,1,0,0,1,0,0],[48,98,0.4898,0.30357,0.22516,0.14286,0.28571,0.42857,0.0,0.857,6,0,0,6,0,7,0,0,4,0,0,11,0,0,1,0,0,2,0,0,1,0,0],[52,98,0.5306,0.33482,0.2412,0.14286,0.28571,0.46431,0.0,0.85714,5,0,0,5,0,5,0,0,10,0,0,4,0,0,4,0,0,2,0,0,2,0,0],[56,98,0.5714,0.28575,0.22869,0.10714,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,4,0,0,8,0,0,8,0,0,1,0,0,2,0,0,1,0,0],[60,98,0.6122,0.35713,0.26485,0.14286,0.28571,0.57111,0.0,1.0,6,1,0,6,0,4,0,0,7,0,0,6,0,0,4,0,0,3,0,0,1,0,1],[64,98,0.6531,0.28571,0.23419,0.14286,0.2857,0.42857,0.0,1.0,7,1,0,7,0,4,0,0,12,0,0,5,0,0,2,0,0,0,0,0,1,0,1],[68,98,0.6939,0.29018,0.27313,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,3,0,0,9,0,0,4,0,0,1,0,0,2,0,0,3,0,0],[72,98,0.7347,0.2232,0.24726,0.0,0.14286,0.42857,0.0,1.0,12,1,0,12,0,7,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[76,98,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are  $50$  points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least  $130$  scalene triangles with vertices of that color.","t":[{"b":0,"e":0.71429,"k":"rising","v":0.63392,"x":0.98214,"p":[[0,41,0.0,0.63392,0.34615,0.28571,0.49999,1.0,0.0,1.0,1,14,0,1,0,0,0,0,11,0,0,4,0,0,1,0,0,0,0,0,1,0,14],[4,41,0.0976,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,41,0.1951,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[12,41,0.2927,0.92857,0.19562,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,27],[16,41,0.3902,0.77678,0.3153,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,3,0,0,4,0,0,0,0,0,2,0,0,2,0,19],[20,41,0.4878,0.87053,0.26812,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,3,0,24],[24,41,0.5854,0.87054,0.25345,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,5,0,22],[28,41,0.6829,0.85701,0.26515,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,2,0,23],[32,41,0.7805,0.91518,0.17076,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,24],[36,41,0.878,0.9241,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[40,41,0.9756,0.87945,0.14337,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,17],[41,41,1.0,0.88839,0.1461,0.82132,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,6,0,18]]},{"b":4,"e":0.14286,"k":"volatile","v":0.375,"x":0.88393,"p":[[0,20,0.0,0.60714,0.32143,0.28571,0.42857,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,11,0,0,8,0,0,0,0,0,0,0,0,1,0,12],[4,20,0.2,0.88393,0.25614,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,26],[8,20,0.4,0.8661,0.26943,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,23],[12,20,0.6,0.65178,0.40711,0.2857,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0,3,0,16],[16,20,0.8,0.66508,0.41139,0.24999,1.0,1.0,0.0,1.0,4,18,0,4,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,18],[20,20,1.0,0.375,0.40681,0.0,0.14286,1.0,0.0,1.0,9,9,0,9,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,9]]}]},{"i":"7c162d1a0d0986e9","q":"Given an integer \\( n \\geq 1 \\), Jo-An\u00e9 alternately writes crosses (\\( \\mathcal{X} \\)) and circles (\\( \\mathcal{O}\\)) in the cells of a square grid with \\( 2n + 1 \\) rows and \\( 2n + 1 \\) columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum \\( \\mathcal{X}+ \\mathcal{O} \\), where \\( \\mathcal{X} \\) is the number of rows containing more crosses than circles and \\( \\mathcal{O} \\) is the number of columns containing more circles than crosses.\n\nDetermine, in terms of \\( n \\), the highest possible score that Jo-An\u00e9 can obtain..","t":[{"b":6,"e":0.42857,"k":"falling","v":0.23661,"x":0.95089,"p":[[0,64,0.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[4,64,0.0625,0.875,0.25442,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,24],[8,64,0.125,0.80804,0.33996,0.85714,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[12,64,0.1875,0.82143,0.29667,0.75,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,2,0,22],[16,64,0.25,0.72768,0.33571,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0,3,0,17],[20,64,0.3125,0.71875,0.34531,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,18],[24,64,0.375,0.47768,0.34921,0.14286,0.35714,0.89286,0.14286,1.0,0,8,0,0,0,11,0,0,5,0,0,6,0,0,0,0,0,0,0,0,2,0,8],[28,64,0.4375,0.52678,0.34523,0.28571,0.42857,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,9,0,0,6,0,0,0,0,0,0,0,0,1,0,10],[32,64,0.5,0.52679,0.34523,0.28571,0.42857,1.0,0.0,1.0,2,9,0,2,0,4,0,0,6,0,0,9,0,0,0,0,0,0,0,0,2,0,9],[36,64,0.5625,0.32589,0.28846,0.14286,0.14288,0.42857,0.0,1.0,1,4,0,1,0,16,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,4],[40,64,0.625,0.23661,0.18073,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,20,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[44,64,0.6875,0.30357,0.23891,0.14286,0.14286,0.42857,0.14286,1.0,0,2,0,0,0,17,0,0,5,0,0,7,0,0,0,0,0,0,0,0,1,0,2],[48,64,0.75,0.29464,0.24468,0.14286,0.14286,0.42857,0.0,1.0,1,2,0,1,0,17,0,0,4,0,0,7,0,0,0,0,0,0,0,0,1,0,2],[52,64,0.8125,0.41518,0.31003,0.14286,0.42857,0.42857,0.0,1.0,2,5,0,2,0,9,0,0,4,0,0,10,0,0,0,0,0,1,0,0,1,0,5],[56,64,0.875,0.35714,0.31542,0.14286,0.21428,0.42857,0.0,1.0,2,5,0,2,0,14,0,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,5],[60,64,0.9375,0.41518,0.34508,0.14286,0.28571,0.53571,0.0,1.0,1,7,0,1,0,13,0,0,5,0,0,5,0,0,0,0,0,0,0,0,1,0,7],[64,64,1.0,0.25,0.18898,0.14286,0.14286,0.42857,0.0,1.0,2,1,0,2,0,17,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":1.0,"k":"flat","v":0.80803,"x":0.99554,"p":[[0,73,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,73,0.0548,0.86161,0.29121,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[8,73,0.1096,0.85268,0.28901,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,24],[12,73,0.1644,0.89732,0.2402,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,26],[16,73,0.2192,0.83929,0.30878,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,25],[20,73,0.274,0.85268,0.30406,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,24],[24,73,0.3288,0.87946,0.28372,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[28,73,0.3836,0.80803,0.32657,0.75,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,23],[32,73,0.4384,0.9375,0.17105,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[36,73,0.4932,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[40,73,0.5479,0.95088,0.13179,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[44,73,0.6027,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[48,73,0.6575,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,73,0.7123,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[56,73,0.7671,0.94643,0.17768,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[60,73,0.8219,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[64,73,0.8767,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[68,73,0.9315,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[72,73,0.9863,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[73,73,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"0dff58f27c300c42","q":"For each positive integer  $n$ , find the number of  $n$ -digit positive integers that satisfy both of the following conditions:\n\n\n- no two consecutive digits are equal, and\n- the last digit is a prime.","t":[{"b":5,"e":1.0,"k":"flat","v":0.9375,"x":0.97768,"p":[[0,52,0.0,0.94196,0.09354,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[4,52,0.0769,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,52,0.1538,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[12,52,0.2308,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[16,52,0.3077,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,52,0.3846,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,52,0.4615,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,52,0.5385,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[32,52,0.6154,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,52,0.6923,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[40,52,0.7692,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[44,52,0.8462,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[48,52,0.9231,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[52,52,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.91518,"x":0.97768,"p":[[0,48,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,48,0.0833,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,48,0.1667,0.94196,0.14664,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[12,48,0.25,0.9375,0.14699,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,24],[16,48,0.3333,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,48,0.4167,0.91518,0.13296,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,19],[24,48,0.5,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[28,48,0.5833,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[32,48,0.6667,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[36,48,0.75,0.93304,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[40,48,0.8333,0.93304,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,23],[44,48,0.9167,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,25],[48,48,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]}]},{"i":"77b36cf58e39177c","q":"Find all triples  $(m,p,q)$  where  $ m $  is a positive integer and  $ p , q $  are primes.\n\\[ 2^m p^2 + 1 = q^5 \\]","t":[{"b":1,"e":0.57143,"k":"falling","v":0.61159,"x":0.91518,"p":[[0,48,0.0,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,24],[4,48,0.0833,0.79464,0.20497,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,2,0,15],[8,48,0.1667,0.74554,0.21349,0.57143,0.57143,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,1,0,0,0,0,13],[12,48,0.25,0.69195,0.19598,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,1,0,0,2,0,8],[16,48,0.3333,0.66964,0.16536,0.57143,0.57143,0.75,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,1,0,0,3,0,5],[20,48,0.4167,0.67411,0.16457,0.57143,0.57143,0.75,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,2,0,0,3,0,5],[24,48,0.5,0.69463,0.19026,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,1,0,1,1,0,8],[28,48,0.5833,0.66964,0.19704,0.57143,0.57143,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,18,0,0,3,0,0,1,0,7],[32,48,0.6667,0.69196,0.19269,0.57143,0.57143,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,9],[36,48,0.75,0.61159,0.15664,0.57143,0.57143,0.57143,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,0,0,0,0,0,4],[40,48,0.8333,0.69196,0.18595,0.57143,0.57143,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,0,1,0,8],[44,48,0.9167,0.64283,0.14727,0.57143,0.57143,0.57143,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,2,0,0,1,0,4],[48,48,1.0,0.61161,0.12492,0.57143,0.57143,0.57143,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,0,0,0,3]]},{"b":3,"e":0.14286,"k":"falling","v":0.11152,"x":0.91517,"p":[[0,53,0.0,0.91517,0.19186,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,25],[4,53,0.0755,0.75893,0.20341,0.57143,0.64286,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,3,0,0,0,0,13],[8,53,0.1509,0.87052,0.20002,0.57143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,22],[12,53,0.2264,0.58026,0.35355,0.14289,0.57143,1.0,0.0,1.0,2,10,0,2,0,7,0,0,0,0,0,3,0,0,8,0,0,0,0,0,2,0,10],[16,53,0.3019,0.57589,0.32238,0.5,0.57143,0.85714,0.0,1.0,3,7,0,3,0,4,0,0,1,0,0,0,0,0,14,0,0,0,0,0,3,0,7],[20,53,0.3774,0.49999,0.3481,0.14286,0.57143,0.75,0.0,1.0,3,7,0,3,0,9,0,0,0,0,0,1,0,0,10,0,0,1,0,0,1,0,7],[24,53,0.4528,0.44196,0.31614,0.14286,0.35714,0.60714,0.14286,1.0,0,5,0,0,0,14,0,0,2,0,0,1,0,0,7,0,0,3,0,0,0,0,5],[28,53,0.5283,0.38829,0.26065,0.14286,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,2,0,0,5,0,0,6,0,0,3,0,0,3,0,0],[32,53,0.6038,0.25447,0.28287,0.0,0.14286,0.42857,0.0,1.0,9,2,0,9,0,12,0,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,2],[36,53,0.6792,0.42402,0.33792,0.14286,0.35716,0.57143,0.0,1.0,4,5,0,4,0,10,0,0,2,0,0,1,0,0,8,0,0,1,0,0,1,0,5],[40,53,0.7547,0.44634,0.28074,0.14286,0.57143,0.57143,0.0,1.0,2,2,0,2,0,9,0,0,1,0,0,3,0,0,11,0,0,2,0,0,2,0,2],[44,53,0.8302,0.23213,0.2223,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,22,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,1],[48,53,0.9057,0.24554,0.2402,0.14286,0.14286,0.32143,0.0,1.0,5,1,0,5,0,18,0,0,1,0,0,1,0,0,5,0,0,1,0,0,0,0,1],[52,53,0.9811,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.13393,0.11811,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,19,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a62d08dda2bd9a6b","q":"Find all triplets  $ (x,y,z) $  of real numbers such that\n\\[ 2x^3 + 1 = 3zx \\]\\[ 2y^3 + 1 = 3xy \\]\\[ 2z^3 + 1 =  3yz \\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,60,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,60,0.0667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,60,0.1333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,60,0.2,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,60,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,60,0.3333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,60,0.4,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[28,60,0.4667,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,60,0.5333,0.97545,0.06341,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,1,27],[36,60,0.6,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,60,0.6667,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[44,60,0.7333,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[48,60,0.8,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[52,60,0.8667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,60,0.9333,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,60,1.0,0.97098,0.12609,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,29]]},{"b":1,"e":1.0,"k":"flat","v":0.9375,"x":0.99554,"p":[[0,76,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,76,0.0526,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,76,0.1053,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,76,0.1579,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[16,76,0.2105,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,76,0.2632,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,76,0.3158,0.97098,0.06648,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,1,26],[28,76,0.3684,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,76,0.4211,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,76,0.4737,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[40,76,0.5263,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,76,0.5789,0.96427,0.12376,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[48,76,0.6316,0.9375,0.19212,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[52,76,0.6842,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,76,0.7368,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[60,76,0.7895,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,76,0.8421,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,76,0.8947,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,76,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,76,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"f34edb0104b960b3","q":"Find the biggest natural number  $m$  that has the following property: among any five 500-element subsets of  $\\{ 1,2,\\dots, 1000\\}$  there exist two sets, whose intersection contains at least  $m$  numbers.","t":[{"b":1,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,54,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,54,0.0741,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,54,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,54,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,54,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,54,0.3704,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,54,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,54,0.5185,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,54,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,54,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,54,0.7407,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,54,0.8148,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,54,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,54,0.963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[54,54,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,47,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,47,0.0851,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,47,0.1702,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,47,0.2553,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,47,0.3404,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,47,0.4255,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,47,0.5106,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,47,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,47,0.6809,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[47,47,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"fbf3e7b79fbdcac3","q":"Find all sets of four real numbers  $x_1, x_2, x_3, x_4$  such that the sum of any one and the product of the other three is equal to 2.","t":[{"b":0,"e":1.0,"k":"flat","v":0.96429,"x":0.99107,"p":[[0,36,0.0,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,36,0.1111,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,36,0.2222,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,36,0.3333,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,36,0.4444,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,36,0.5556,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,36,0.6667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,36,0.7778,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,36,0.8889,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,36,1.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,57,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,57,0.0702,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[8,57,0.1404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,57,0.2105,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,57,0.2807,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,57,0.3509,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[24,57,0.4211,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,57,0.4912,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,57,0.5614,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,57,0.6316,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,57,0.7018,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,57,0.7719,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,57,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,57,0.9123,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[56,57,0.9825,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[57,57,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"debf6d1436457c78","q":"For a non-empty finite set  $A$  of positive integers, let  $\\text{lcm}(A)$  denote the least common multiple of elements in  $A$ , and let  $d(A)$  denote the number of prime factors of  $\\text{lcm}(A)$  (counting multiplicity). Given a finite set  $S$  of positive integers, and   $$ f_S(x)=\\sum_{\\emptyset \\neq A \\subset S} \\frac{(-1)^{|A|} x^{d(A)}}{\\text{lcm}(A)}. $$  Prove that, if  $0 \\le x \\le 2$ , then  $-1 \\le f_S(x) \\le 0$ .","t":[{"b":4,"e":0.0,"k":"falling","v":0.02232,"x":0.57143,"p":[[0,47,0.0,0.27232,0.25843,0.14286,0.2143,0.28571,0.0,1.0,4,3,1,4,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[4,47,0.0851,0.4508,0.35563,0.14286,0.28571,0.75,0.0,1.0,1,7,0,1,0,14,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,7],[8,47,0.1702,0.57143,0.37796,0.14286,0.64286,1.0,0.0,1.0,1,11,0,1,0,9,0,0,4,0,0,1,0,0,1,0,0,3,0,0,2,0,11],[12,47,0.2553,0.42409,0.3508,0.14286,0.28571,0.71429,0.0,1.0,2,6,0,2,0,13,0,0,4,0,0,1,0,0,2,0,0,3,0,0,1,0,6],[16,47,0.3404,0.50446,0.39283,0.14286,0.5,1.0,0.0,1.0,6,9,0,6,0,6,0,0,2,0,0,2,0,0,2,0,0,4,0,0,1,0,9],[20,47,0.4255,0.4732,0.34522,0.14286,0.49979,0.71429,0.0,1.0,4,5,0,4,0,7,0,0,4,0,0,1,0,0,4,0,0,5,0,0,2,0,5],[24,47,0.5106,0.36606,0.36058,0.10714,0.2143,0.57143,0.0,1.0,8,6,0,8,0,8,0,0,3,0,0,3,0,0,3,0,0,1,0,0,0,0,6],[28,47,0.5957,0.26786,0.3531,0.0,0.14286,0.32143,0.0,1.0,13,5,0,13,0,8,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,5],[32,47,0.6809,0.35268,0.36767,0.0,0.14286,0.60714,0.0,1.0,9,5,0,9,0,9,0,0,1,0,0,4,0,0,1,0,0,1,0,0,2,0,5],[36,47,0.766,0.24552,0.29929,0.0,0.14286,0.32143,0.0,1.0,10,3,0,10,0,12,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[40,47,0.8511,0.08036,0.16342,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[44,47,0.9362,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[47,47,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.22322,"x":0.70536,"p":[[0,52,0.0,0.22322,0.12339,0.14286,0.14288,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,52,0.0769,0.40179,0.35434,0.14286,0.14286,0.71429,0.0,1.0,1,6,0,1,0,17,0,0,3,0,0,0,0,0,1,0,0,3,0,0,1,0,6],[8,52,0.1538,0.49552,0.35889,0.14289,0.42857,0.89286,0.0,1.0,4,8,0,4,0,5,0,0,5,0,0,4,0,0,3,0,0,2,0,0,1,0,8],[12,52,0.2308,0.41964,0.3387,0.14286,0.28571,0.71429,0.0,1.0,2,5,0,2,0,13,0,0,2,0,0,4,0,0,2,0,0,2,0,0,2,0,5],[16,52,0.3077,0.52231,0.39867,0.14286,0.42857,1.0,0.0,1.0,3,12,0,3,0,9,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,12],[20,52,0.3846,0.70536,0.37276,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,18],[24,52,0.4615,0.50446,0.37112,0.14286,0.42857,1.0,0.0,1.0,1,9,0,1,0,12,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,9],[28,52,0.5385,0.46428,0.37796,0.14286,0.28571,1.0,0.0,1.0,2,9,0,2,0,12,0,0,4,0,0,2,0,0,0,0,0,3,0,0,0,0,9],[32,52,0.6154,0.625,0.3989,0.14286,0.78571,1.0,0.0,1.0,3,14,0,3,0,7,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,14],[36,52,0.6923,0.47768,0.38067,0.14286,0.42857,1.0,0.0,1.0,3,9,0,3,0,11,0,0,1,0,0,4,0,0,2,0,0,1,0,0,1,0,9],[40,52,0.7692,0.33929,0.28516,0.14286,0.28571,0.46429,0.0,1.0,3,2,0,3,0,12,0,0,7,0,0,2,0,0,1,0,0,4,0,0,1,0,2],[44,52,0.8462,0.51785,0.3549,0.14289,0.49979,1.0,0.0,1.0,2,9,0,2,0,8,0,0,3,0,0,3,0,0,5,0,0,2,0,0,0,0,9],[48,52,0.9231,0.49106,0.29867,0.24999,0.49979,0.60714,0.0,1.0,1,5,0,1,0,7,0,0,4,0,0,4,0,0,8,0,0,2,0,0,1,0,5],[52,52,1.0,0.50443,0.26721,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,3,0,0,5,0,0,8,0,0,4,0,0,2,0,3]]}]},{"i":"d696c3c89dbe0ea9","q":"If  $x$  is a positive rational number show that  $x$  can be uniquely expressed in the form  $x = \\sum^n_{k=1} \\frac{a_k}{k!}$  where  $a_1, a_2, \\ldots$  are integers,  $0 \\leq a_n \\leq n - 1$ , for  $n > 1,$  and the series terminates. Show that  $x$  can be expressed as the sum of reciprocals of different integers, each of which is greater than  $10^6.$","t":[{"b":0,"e":0.14286,"k":"volatile","v":0.16518,"x":0.65624,"p":[[0,13,0.0,0.16518,0.06298,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.57588,0.27312,0.42857,0.57143,0.75,0.0,1.0,1,3,0,1,0,4,0,0,2,0,0,4,0,0,7,0,0,6,0,0,5,0,3],[8,13,0.6154,0.65624,0.23653,0.57132,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,3,0,0,11,0,0,2,0,0,7,0,5],[12,13,0.9231,0.56249,0.31326,0.28571,0.57143,0.85714,0.0,1.0,3,3,0,3,0,2,0,0,6,0,0,1,0,0,5,0,0,4,0,0,8,0,3],[13,13,1.0,0.62497,0.25693,0.42859,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,3,0,0,8,0,0,3,0,0,9,0,3]]},{"b":5,"e":0.57143,"k":"volatile","v":0.16072,"x":0.69639,"p":[[0,12,0.0,0.16072,0.07784,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,12,0.3333,0.64726,0.17853,0.57132,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,16,0,0,2,0,0,9,0,1],[8,12,0.6667,0.69639,0.20441,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,4,0,0,13,0,2],[12,12,1.0,0.54461,0.18011,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,9,0,0,12,0,0,2,0,0,5,0,0]]}]},{"i":"5a5226c5b3d5146b","q":"Given any integer  $n\\geq 3$ . A finite series is called  $n$ -series if it satisfies the following two conditions $1)$  It has at least  $3$  terms and each term of it belongs to  $\\{ 1,2,...,n\\}$  $2)$  If series has  $m$  terms  $a_1,a_2,...,a_m$  then  $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$  for all  $k=1,2,...,m-2$ How many  $n$ -series are there  $?$","t":[{"b":2,"e":0.42857,"k":"falling","v":0.23661,"x":0.87946,"p":[[0,160,0.0,0.87946,0.28146,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,26],[4,160,0.025,0.75446,0.35934,0.2857,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,21],[8,160,0.05,0.69643,0.3989,0.28571,1.0,1.0,0.0,1.0,3,20,0,3,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[12,160,0.075,0.72768,0.37005,0.28571,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,20],[16,160,0.1,0.5625,0.44311,0.14286,0.71429,1.0,0.0,1.0,7,15,0,7,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,15],[20,160,0.125,0.7142,0.37641,0.28571,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,5,0,0,0,0,0,2,0,0,0,0,0,1,0,19],[24,160,0.15,0.43304,0.4064,0.14286,0.28571,1.0,0.0,1.0,7,10,0,7,0,7,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,10],[28,160,0.175,0.49554,0.41952,0.14286,0.35714,1.0,0.0,1.0,7,12,0,7,0,5,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,12],[32,160,0.2,0.58034,0.40867,0.14289,0.49979,1.0,0.0,1.0,3,15,0,3,0,6,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,15],[36,160,0.225,0.54902,0.42882,0.14286,0.57143,1.0,0.0,1.0,6,14,0,6,0,6,0,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,14],[40,160,0.25,0.5,0.4475,0.0,0.28571,1.0,0.0,1.0,9,13,0,9,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,13],[44,160,0.275,0.51786,0.40208,0.24999,0.28571,1.0,0.0,1.0,6,12,0,6,0,2,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,12],[48,160,0.3,0.47321,0.4,0.14286,0.28571,1.0,0.0,1.0,5,11,0,5,0,6,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,11],[52,160,0.325,0.38839,0.36811,0.14286,0.2857,0.60714,0.0,1.0,7,7,0,7,0,7,0,0,6,0,0,2,0,0,2,0,0,1,0,0,0,0,7],[56,160,0.35,0.49107,0.43292,0.10714,0.28571,1.0,0.0,1.0,8,12,0,8,0,6,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,12],[60,160,0.375,0.28572,0.32927,0.0,0.14286,0.32144,0.0,1.0,9,4,0,9,0,11,0,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[64,160,0.4,0.35268,0.37113,0.10714,0.14288,0.60714,0.0,1.0,8,6,0,8,0,10,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[68,160,0.425,0.26786,0.3004,0.14286,0.14286,0.28571,0.0,1.0,6,4,0,6,0,15,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[72,160,0.45,0.23661,0.26871,0.0,0.14286,0.32143,0.0,1.0,10,2,0,10,0,10,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,2],[76,160,0.475,0.29465,0.30917,0.10714,0.14286,0.4286,0.0,1.0,8,2,0,8,0,11,0,0,2,0,0,5,0,0,1,0,0,0,0,0,3,0,2],[80,160,0.5,0.375,0.3531,0.14286,0.14286,0.57143,0.0,1.0,5,5,0,5,0,13,0,0,1,0,0,2,0,0,4,0,0,0,0,0,2,0,5],[84,160,0.525,0.41964,0.31931,0.14286,0.28571,0.71429,0.0,1.0,3,4,0,3,0,9,0,0,5,0,0,3,0,0,3,0,0,4,0,0,1,0,4],[88,160,0.55,0.48213,0.32878,0.14289,0.4286,0.85714,0.0,1.0,3,4,0,3,0,7,0,0,2,0,0,6,0,0,4,0,0,1,0,0,5,0,4],[92,160,0.575,0.30354,0.26181,0.14286,0.2857,0.46418,0.0,1.0,6,1,0,6,0,9,0,0,7,0,0,2,0,0,3,0,0,4,0,0,0,0,1],[96,160,0.6,0.34366,0.30699,0.14214,0.28571,0.57143,0.0,1.0,7,2,0,7,0,8,0,0,3,0,0,5,0,0,3,0,0,2,0,0,2,0,2],[100,160,0.625,0.28571,0.29451,0.14286,0.14288,0.28571,0.0,1.0,7,3,0,7,0,10,0,0,8,0,0,1,0,0,1,0,0,2,0,0,0,0,3],[104,160,0.65,0.32143,0.35714,0.0,0.14286,0.57143,0.0,1.0,10,5,0,10,0,9,0,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,5],[108,160,0.675,0.30338,0.36032,0.0,0.14286,0.46418,0.0,1.0,11,5,0,11,0,9,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,5],[112,160,0.7,0.31249,0.28219,0.14286,0.14288,0.42858,0.0,1.0,4,2,0,4,0,13,0,0,5,0,0,3,0,0,3,0,0,0,0,0,2,0,2],[116,160,0.725,0.28122,0.29336,0.0,0.14286,0.571,0.0,1.0,10,2,0,10,0,9,0,0,1,0,0,3,0,0,6,0,0,1,0,0,0,0,2],[120,160,0.75,0.433,0.35621,0.14286,0.28571,0.74996,0.0,1.0,6,6,0,6,0,4,0,0,8,0,0,2,0,0,3,0,0,1,0,0,2,0,6],[124,160,0.775,0.33929,0.31693,0.14286,0.14288,0.57143,0.0,1.0,3,4,0,3,0,15,0,0,5,0,0,0,0,0,2,0,0,3,0,0,0,0,4],[128,160,0.8,0.34811,0.34435,0.0,0.28571,0.57111,0.0,1.0,9,5,0,9,0,6,0,0,4,0,0,4,0,0,3,0,0,1,0,0,0,0,5],[132,160,0.825,0.39732,0.31285,0.14286,0.28571,0.57143,0.0,1.0,4,3,0,4,0,9,0,0,4,0,0,3,0,0,5,0,0,2,0,0,2,0,3],[136,160,0.85,0.42405,0.28677,0.2857,0.35714,0.57143,0.0,1.0,4,3,0,4,0,3,0,0,9,0,0,3,0,0,7,0,0,2,0,0,1,0,3],[140,160,0.875,0.43293,0.20678,0.2857,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,4,0,0,11,0,0,4,0,0,9,0,0,1,0,0,3,0,0],[144,160,0.9,0.42857,0.25254,0.2857,0.35714,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,10,0,0,5,0,0,5,0,0,1,0,0,4,0,1],[148,160,0.925,0.45089,0.22899,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,11,0,0,4,0,0,6,0,0,4,0,0,2,0,1],[152,160,0.95,0.42858,0.202,0.28571,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,10,0,0,3,0,0,9,0,0,4,0,0,1,0,0],[156,160,0.975,0.45534,0.25363,0.2857,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,13,0,0,2,0,0,6,0,0,2,0,0,3,0,2],[160,160,1.0,0.48661,0.2693,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,10,0,0,6,0,0,2,0,0,4,0,0,3,0,3]]},{"b":6,"e":0.28571,"k":"falling","v":0.26778,"x":0.85268,"p":[[0,149,0.0,0.82142,0.29882,0.78561,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0,2,0,22],[4,149,0.0268,0.69197,0.35733,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,18],[8,149,0.0537,0.8125,0.31831,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,4,0,21],[12,149,0.0805,0.85268,0.29984,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[16,149,0.1074,0.82143,0.32143,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[20,149,0.1342,0.76786,0.37585,0.28571,1.0,1.0,0.0,1.0,2,23,0,2,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[24,149,0.1611,0.63393,0.40238,0.24999,1.0,1.0,0.0,1.0,2,17,0,2,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,17],[28,149,0.1879,0.71428,0.3677,0.28571,1.0,1.0,0.0,1.0,1,19,0,1,0,4,0,0,5,0,0,0,0,0,2,0,0,1,0,0,0,0,19],[32,149,0.2148,0.6875,0.39679,0.24999,1.0,1.0,0.0,1.0,2,18,0,2,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,18],[36,149,0.2416,0.69643,0.40367,0.14286,1.0,1.0,0.0,1.0,2,20,0,2,0,7,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,20],[40,149,0.2685,0.70536,0.37104,0.42857,1.0,1.0,0.0,1.0,3,17,0,3,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,0,2,0,17],[44,149,0.2953,0.48652,0.41636,0.14286,0.28571,1.0,0.0,1.0,4,12,0,4,0,11,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,12],[48,149,0.3221,0.58482,0.41088,0.14286,0.57141,1.0,0.0,1.0,4,15,0,4,0,5,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,15],[52,149,0.349,0.72322,0.38454,0.28571,1.0,1.0,0.0,1.0,2,20,0,2,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,20],[56,149,0.3758,0.49107,0.39275,0.14286,0.28571,1.0,0.0,1.0,2,11,0,2,0,11,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,11],[60,149,0.4027,0.66964,0.41255,0.25,1.0,1.0,0.0,1.0,4,19,0,4,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,19],[64,149,0.4295,0.78572,0.36596,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[68,149,0.4564,0.59813,0.41726,0.14286,0.85714,1.0,0.0,1.0,1,16,0,1,0,12,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,16],[72,149,0.4832,0.66518,0.41127,0.14286,1.0,1.0,0.0,1.0,3,18,0,3,0,7,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,18],[76,149,0.5101,0.44195,0.4062,0.14286,0.28571,1.0,0.0,1.0,7,10,0,7,0,8,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,10],[80,149,0.5369,0.64286,0.43448,0.14286,1.0,1.0,0.0,1.0,3,19,0,3,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[84,149,0.5638,0.41518,0.44228,0.0,0.14286,1.0,0.0,1.0,11,10,0,11,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,10],[88,149,0.5906,0.74545,0.39256,0.25,1.0,1.0,0.0,1.0,2,22,0,2,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,22],[92,149,0.6174,0.65625,0.443,0.14286,1.0,1.0,0.0,1.0,7,18,0,7,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,18],[96,149,0.6443,0.70089,0.42009,0.14286,1.0,1.0,0.0,1.0,4,20,0,4,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,20],[100,149,0.6711,0.56696,0.44102,0.14286,0.7143,1.0,0.0,1.0,5,16,0,5,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,16],[104,149,0.698,0.36161,0.31336,0.14286,0.28571,0.46431,0.0,1.0,4,4,0,4,0,10,0,0,6,0,0,4,0,0,2,0,0,1,0,0,1,0,4],[108,149,0.7248,0.31248,0.30604,0.10714,0.2857,0.42857,0.0,1.0,8,3,0,8,0,6,0,0,9,0,0,2,0,0,2,0,0,1,0,0,1,0,3],[112,149,0.7517,0.26778,0.25696,0.14214,0.14286,0.32164,0.0,1.0,7,1,0,7,0,10,0,0,7,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[116,149,0.7785,0.33476,0.2332,0.14286,0.28571,0.42893,0.0,0.85714,3,0,0,3,0,6,0,0,14,0,0,2,0,0,3,0,0,1,0,0,3,0,0],[120,149,0.8054,0.30348,0.28743,0.14286,0.21428,0.42857,0.0,1.0,6,2,0,6,0,10,0,0,7,0,0,3,0,0,0,0,0,3,0,0,1,0,2],[124,149,0.8322,0.30803,0.26989,0.14286,0.2857,0.28571,0.0,1.0,5,2,0,5,0,7,0,0,13,0,0,2,0,0,1,0,0,0,0,0,2,0,2],[128,149,0.8591,0.38839,0.31183,0.14286,0.28571,0.57143,0.0,1.0,4,4,0,4,0,6,0,0,10,0,0,3,0,0,3,0,0,0,0,0,2,0,4],[132,149,0.8859,0.29018,0.21572,0.14286,0.28571,0.28571,0.0,1.0,3,2,0,3,0,7,0,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[136,149,0.9128,0.35265,0.15962,0.2857,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,2,0,0,19,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[140,149,0.9396,0.41072,0.21351,0.2857,0.28571,0.571,0.14286,1.0,0,1,0,0,0,4,0,0,13,0,0,6,0,0,5,0,0,1,0,0,2,0,1],[144,149,0.9664,0.35265,0.18202,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,3,0,0,18,0,0,4,0,0,4,0,0,0,0,0,2,0,0],[148,149,0.9933,0.41522,0.22119,0.28571,0.35714,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,11,0,0,7,0,0,4,0,0,2,0,0,2,0,1],[149,149,1.0,0.37053,0.18509,0.2857,0.28571,0.46428,0.14286,0.85714,0,0,0,0,0,6,0,0,13,0,0,5,0,0,5,0,0,2,0,0,1,0,0]]}]},{"i":"0e53feb9123953d0","q":"If  $k$  is an integer, let  $\\mathrm{c}(k)$  denote the largest cube that is less than or equal to  $k$ . Find all positive integers  $p$  for which the following sequence is bounded: $a_0 = p$  and  $a_{n+1} = 3a_n-2\\mathrm{c}(a_n)$  for  $n \\geqslant 0$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.93304,"x":0.99554,"p":[[0,51,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,51,0.0784,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,51,0.1569,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,51,0.2353,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[16,51,0.3137,0.93304,0.15966,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,24],[20,51,0.3922,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[24,51,0.4706,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,51,0.549,0.95089,0.12682,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[32,51,0.6275,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[36,51,0.7059,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[40,51,0.7843,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,51,0.8627,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,51,0.9412,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[51,51,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.95981,"x":0.98214,"p":[[0,44,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,44,0.0909,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,44,0.1818,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,44,0.2727,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,44,0.3636,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[20,44,0.4545,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[24,44,0.5455,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,44,0.6364,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,44,0.7273,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[36,44,0.8182,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,44,0.9091,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,44,1.0,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28]]}]},{"i":"f0e6e1445397da8e","q":"If  $n$  is a positive integer and  $n+1$  is divisible with  $24$ , prove that sum of all positive divisors of  $n$  is divisible with  $24$","t":[{"b":2,"e":0.14286,"k":"falling","v":0.23214,"x":0.58036,"p":[[0,12,0.0,0.58036,0.44455,0.0,0.71429,1.0,0.0,1.0,10,14,0,10,0,2,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,14],[4,12,0.3333,0.54018,0.45978,0.0,0.71429,1.0,0.0,1.0,11,14,0,11,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,14],[8,12,0.6667,0.33482,0.41895,0.0,0.14286,0.78571,0.0,1.0,13,8,0,13,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,8],[12,12,1.0,0.23214,0.37415,0.0,0.07143,0.14286,0.0,1.0,16,6,0,16,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6]]},{"b":6,"e":0.14286,"k":"falling","v":0.07143,"x":0.9375,"p":[[0,50,0.0,0.41964,0.4683,0.0,0.14286,1.0,0.0,1.0,15,12,0,15,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12],[4,50,0.08,0.88839,0.26901,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[8,50,0.16,0.85714,0.33312,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[12,50,0.24,0.9375,0.21109,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[16,50,0.32,0.9375,0.17105,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[20,50,0.4,0.86607,0.33108,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[24,50,0.48,0.84822,0.333,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[28,50,0.56,0.87946,0.30328,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[32,50,0.64,0.88839,0.26901,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[36,50,0.72,0.74554,0.40679,0.57144,1.0,1.0,0.0,1.0,5,22,0,5,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,22],[40,50,0.8,0.31241,0.43369,0.0,0.07,1.0,0.0,1.0,16,9,0,16,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[44,50,0.88,0.16518,0.32165,0.0,0.0,0.14286,0.0,1.0,19,4,0,19,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[48,50,0.96,0.20536,0.35344,0.0,0.0,0.14286,0.0,1.0,18,5,0,18,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[50,50,1.0,0.07143,0.17857,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"42114323e192d888","q":"For positive integers  $x,y$ , \ffind all pairs  $(x,y)$  such that  $x^2y + x$  is a multiple of  $xy^2 + 7$ .","t":[{"b":1,"e":0.571,"k":"flat","v":0.53125,"x":0.83032,"p":[[0,152,0.0,0.58032,0.20497,0.42857,0.57143,0.71429,0.0,1.0,1,3,1,1,0,0,0,0,2,0,0,6,0,0,14,0,0,5,0,0,1,0,3],[4,152,0.0263,0.77677,0.22852,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,1,0,0,2,0,15],[8,152,0.0526,0.83032,0.21857,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,2,0,0,1,0,19],[12,152,0.0789,0.73658,0.21758,0.57143,0.64286,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,2,0,0,3,0,11],[16,152,0.1053,0.78122,0.22586,0.57143,0.85707,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,2,0,0,2,0,15],[20,152,0.1316,0.71428,0.24484,0.42859,0.64286,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,1,0,0,4,0,11],[24,152,0.1579,0.73657,0.2399,0.57132,0.57143,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,0,0,0,2,0,13],[28,152,0.1842,0.69192,0.21463,0.57132,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,1,0,0,2,0,9],[32,152,0.2105,0.7321,0.23893,0.571,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,1,0,0,5,0,11],[36,152,0.2368,0.77676,0.22572,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,3,0,0,3,0,14],[40,152,0.2632,0.79911,0.23652,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,1,0,0,2,0,17],[44,152,0.2895,0.73211,0.24158,0.571,0.57143,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,0,0,0,2,0,13],[48,152,0.3158,0.67408,0.21794,0.57143,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,16,0,0,2,0,0,0,0,9],[52,152,0.3421,0.63837,0.21125,0.4286,0.57143,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,9,0,0,14,0,0,1,0,0,1,0,7],[56,152,0.3684,0.68302,0.21049,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,0,0,0,3,0,8],[60,152,0.3947,0.63392,0.2141,0.42857,0.57143,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,10,0,0,13,0,0,1,0,0,1,0,7],[64,152,0.4211,0.65624,0.22264,0.53539,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,7,0,0,13,0,0,2,0,0,1,0,8],[68,152,0.4474,0.59371,0.19269,0.42859,0.57143,0.57143,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,10,0,0,14,0,0,1,0,0,2,0,4],[72,152,0.4737,0.63839,0.22583,0.42857,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,0,0,0,3,0,7],[76,152,0.5,0.61603,0.18364,0.53539,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,17,0,0,0,0,0,3,0,4],[80,152,0.5263,0.65623,0.22261,0.42964,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,13,0,0,0,0,0,2,0,8],[84,152,0.5526,0.62498,0.20746,0.42859,0.57143,0.64286,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,14,0,0,0,0,0,2,0,6],[88,152,0.5789,0.6339,0.21706,0.42857,0.57143,0.74996,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,11,0,0,2,0,0,1,0,7],[92,152,0.6053,0.58033,0.14698,0.5354,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,20,0,0,0,0,0,2,0,2],[96,152,0.6316,0.6116,0.1996,0.4286,0.57143,0.57143,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,15,0,0,1,0,0,0,0,6],[100,152,0.6579,0.57143,0.16366,0.42859,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,14,0,0,3,0,0,0,0,3],[104,152,0.6842,0.55799,0.15714,0.42857,0.5712,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,15,0,0,0,0,0,2,0,2],[108,152,0.7105,0.58035,0.18536,0.42857,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,13,0,0,1,0,0,1,0,4],[112,152,0.7368,0.56243,0.14257,0.42857,0.57141,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,19,0,0,0,0,0,1,0,2],[116,152,0.7632,0.66067,0.21355,0.571,0.57143,0.89275,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,7,0,0,15,0,0,1,0,0,1,0,8],[120,152,0.7895,0.61157,0.20897,0.42859,0.57143,0.57143,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,9,0,0,15,0,0,0,0,0,1,0,6],[124,152,0.8158,0.5803,0.19541,0.42857,0.571,0.57143,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,14,0,0,12,0,0,1,0,0,0,0,5],[128,152,0.8421,0.60708,0.18209,0.571,0.57143,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,19,0,0,0,0,0,2,0,4],[132,152,0.8684,0.58032,0.15126,0.5354,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,20,0,0,1,0,0,0,0,3],[136,152,0.8947,0.558,0.1488,0.42857,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,16,0,0,1,0,0,1,0,2],[140,152,0.9211,0.55351,0.17034,0.42857,0.5705,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,15,0,0,13,0,0,0,0,0,1,0,3],[144,152,0.9474,0.54464,0.1448,0.42857,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,11,0,0,17,0,0,1,0,0,0,0,2],[148,152,0.9737,0.53125,0.12988,0.42857,0.571,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,14,0,0,1,0,0,1,0,1],[152,152,1.0,0.56692,0.11564,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,17,0,0,4,0,0,2,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.66069,"x":1.0,"p":[[0,133,0.0,0.66069,0.21355,0.57143,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,3,0,0,16,0,0,2,0,0,2,0,7],[4,133,0.0301,0.77677,0.22571,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,0,0,0,4,0,14],[8,133,0.0602,0.76339,0.23585,0.57143,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,1,0,0,1,0,15],[12,133,0.0902,0.79909,0.21684,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,1,0,0,2,0,16],[16,133,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a  $10\\times 10$  table, positive numbers are written. It is known that, looking left-right, the numbers in each row form an arithmetic progression and, looking up-down, the numbers is each column form a geometric progression. Prove that all the ratios of the geometric progressions are equal.","t":[{"b":5,"e":0.2857,"k":"flat","v":0.67857,"x":0.88616,"p":[[0,20,0.0,0.79464,0.21335,0.78561,0.85714,0.94643,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,1,1,0,0,4,0,0,0,0,0,15,1,8],[4,20,0.2,0.86607,0.13803,0.85714,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,20,0,9],[8,20,0.4,0.88616,0.13105,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,1,11],[12,20,0.6,0.76338,0.18424,0.71429,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,2,0,0,22,0,1],[16,20,0.8,0.67857,0.26726,0.57143,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,0,0,0,4,0,0,1,0,0,4,0,0,1,0,0,20,0,0],[20,20,1.0,0.74553,0.22794,0.78561,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,23,0,1]]},{"b":7,"e":0.28571,"k":"falling","v":0.40625,"x":0.86161,"p":[[0,38,0.0,0.69196,0.29258,0.53571,0.85714,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,5,0,0,1,0,0,3,0,0,0,0,0,17,0,4],[4,38,0.1053,0.77232,0.274,0.85711,0.85714,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,17,0,8],[8,38,0.2105,0.78793,0.21242,0.85714,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,1,0,0,0,3,0,0,0,0,0,19,0,6],[12,38,0.3158,0.84375,0.16115,0.85711,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,15,0,10],[16,38,0.4211,0.86161,0.19061,0.85714,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,16,0,12],[20,38,0.5263,0.83034,0.15748,0.85714,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,21,0,6],[24,38,0.6316,0.77679,0.26229,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,17,0,8],[28,38,0.7368,0.83036,0.21558,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,17,0,10],[32,38,0.8421,0.81248,0.17657,0.85714,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,21,0,5],[36,38,0.9474,0.41972,0.33502,0.10714,0.28571,0.85714,0.0,1.0,8,1,0,8,0,1,0,0,8,0,0,2,0,0,4,0,0,0,0,0,8,0,1],[38,38,1.0,0.40625,0.32165,0.10714,0.42857,0.60714,0.0,1.0,8,1,0,8,0,3,0,0,4,0,0,3,0,0,6,0,0,2,0,0,5,0,1]]}]},{"i":"24127f9164184d62","q":"In a country there are 2014 airports, no three of them lying on a line. Two airports are connected by a direct flight if and only if the line passing through them divides the country in two parts, each with 1006 airports in it. Show that there are no two airports such that one can travel from the first to the second, visiting each of the 2014 airports exactly once.","t":[{"b":2,"e":0.42857,"k":"falling","v":0.13839,"x":0.40615,"p":[[0,21,0.0,0.40615,0.40899,0.0,0.2857,0.75,0.0,1.0,13,7,0,13,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,7],[4,21,0.1905,0.24554,0.36811,0.0,0.0,0.46429,0.0,1.0,20,4,0,20,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,4],[8,21,0.381,0.32589,0.39161,0.0,0.0,0.71429,0.0,1.0,17,5,0,17,0,0,0,0,3,0,0,1,0,0,1,0,0,5,0,0,0,0,5],[12,21,0.5714,0.24106,0.30184,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,1,0,0,6,0,0,3,0,0,3,0,0,0,0,0,1,0,2],[16,21,0.7619,0.28571,0.38299,0.0,0.07143,0.46429,0.0,1.0,16,5,0,16,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,5],[20,21,0.9524,0.23661,0.32656,0.0,0.0,0.42857,0.0,1.0,18,3,0,18,0,0,0,0,5,0,0,4,0,0,1,0,0,0,0,0,1,0,3],[21,21,1.0,0.13839,0.16935,0.0,0.07143,0.17857,0.0,0.42857,16,0,0,16,0,8,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.40179,"x":0.91518,"p":[[0,33,0.0,0.40179,0.41869,0.0,0.28571,0.78571,0.0,1.0,14,8,0,14,0,1,0,0,3,0,0,0,0,0,3,0,0,3,0,0,0,0,8],[4,33,0.1212,0.46429,0.43154,0.0,0.57143,1.0,0.0,1.0,13,10,0,13,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,0,0,0,10],[8,33,0.2424,0.51344,0.4132,0.0,0.50071,1.0,0.0,1.0,9,11,0,9,0,2,0,0,1,0,0,4,0,0,4,0,0,0,0,0,1,0,11],[12,33,0.3636,0.46875,0.4392,0.0,0.42857,1.0,0.0,1.0,13,11,0,13,0,0,0,0,1,0,0,4,0,0,2,0,0,0,0,0,1,0,11],[16,33,0.4848,0.45088,0.35911,0.0,0.57121,0.60714,0.0,1.0,9,6,0,9,0,2,0,0,1,0,0,3,0,0,9,0,0,2,0,0,0,0,6],[20,33,0.6061,0.45982,0.39243,0.0,0.42859,0.85714,0.0,1.0,10,6,0,10,0,2,0,0,2,0,0,3,0,0,3,0,0,2,0,0,4,0,6],[24,33,0.7273,0.57142,0.35535,0.28571,0.57143,1.0,0.0,1.0,4,9,0,4,0,2,0,0,4,0,0,5,0,0,3,0,0,2,0,0,3,0,9],[28,33,0.8485,0.79018,0.30301,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,3,0,0,4,0,0,1,0,0,2,0,19],[32,33,0.9697,0.91518,0.14664,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,5,0,22],[33,33,1.0,0.86161,0.19393,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,8,0,17]]}]},{"i":"0bbce82128e9abc5","q":"In  $\\triangle$ ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.","t":[{"b":0,"e":0.0,"k":"flat","v":0.03571,"x":0.15625,"p":[[0,212,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,212,0.0189,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,212,0.0377,0.07589,0.12869,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,212,0.0566,0.10714,0.15153,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,212,0.0755,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,212,0.0943,0.12058,0.15621,0.0,0.0,0.2857,0.0,0.43,19,0,0,19,0,2,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,212,0.1132,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,212,0.1321,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,212,0.1509,0.09821,0.15335,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,212,0.1698,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,212,0.1887,0.10268,0.15251,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,212,0.2075,0.10268,0.15251,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,5,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,212,0.2264,0.12053,0.15198,0.0,0.0,0.17857,0.0,0.4286,17,0,0,17,0,7,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,212,0.2453,0.10715,0.15568,0.0,0.0,0.1786,0.0,0.4286,20,0,0,20,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,212,0.2642,0.09375,0.13651,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,212,0.283,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,212,0.3019,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,212,0.3208,0.09821,0.15335,0.0,0.0,0.2857,0.0,0.4286,22,0,0,22,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[72,212,0.3396,0.10705,0.1515,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,212,0.3585,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[80,212,0.3774,0.08036,0.11812,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,212,0.3962,0.09822,0.14481,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,2,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[88,212,0.4151,0.12054,0.15198,0.0,0.0,0.1786,0.0,0.4286,17,0,0,17,0,7,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[92,212,0.434,0.11161,0.13709,0.0,0.0,0.1786,0.0,0.42857,17,0,0,17,0,7,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[96,212,0.4528,0.10714,0.15152,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,3,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[100,212,0.4717,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[104,212,0.4906,0.13393,0.17105,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,4,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[108,212,0.5094,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[112,212,0.5283,0.11161,0.1504,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[116,212,0.5472,0.07589,0.09438,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,212,0.566,0.09822,0.1357,0.0,0.0,0.14287,0.0,0.42857,19,0,0,19,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[124,212,0.5849,0.09821,0.14032,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[128,212,0.6038,0.09822,0.15336,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[132,212,0.6226,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[136,212,0.6415,0.05804,0.12299,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[140,212,0.6604,0.03572,0.09449,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[144,212,0.6792,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[148,212,0.6981,0.0625,0.13803,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[152,212,0.717,0.09152,0.13391,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,1,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[156,212,0.7358,0.11161,0.15458,0.0,0.0,0.17857,0.0,0.42857,19,0,0,19,0,5,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[160,212,0.7547,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[164,212,0.7736,0.11161,0.1461,0.0,0.0,0.1786,0.0,0.4286,18,0,0,18,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[168,212,0.7925,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[172,212,0.8113,0.10268,0.16458,0.0,0.0,0.17857,0.0,0.4286,22,0,0,22,0,2,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[176,212,0.8302,0.09375,0.1411,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[180,212,0.8491,0.13839,0.16554,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[184,212,0.867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,0,1,0,0,0,0,0,0,0,0,0,0,0],[156,169,0.9231,0.06696,0.12869,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[160,169,0.9467,0.07143,0.14726,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[164,169,0.9704,0.07589,0.11836,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,169,0.9941,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[169,169,1.0,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce1c7fb4649e5c5c","q":"How many ways are there to choose distinct positive integers  $a, b, c, d$  dividing  $15^6$  such that none of  $a, b, c,$  or  $d$  divide each other? (Order does not matter.)\n\n*Proposed by Miles Yamner and Andrew Wu*\n\n(Note: wording changed from original to clarify)","t":[{"b":5,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,18,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5df48adb34097fab","q":"I have an $n \\times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that  $$ \\frac{1}{7} n^{2}-c n \\leq b(n) \\leq \\frac{1}{5} n^{2}+d n $$  for all $n>0$.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.28125,"x":0.85267,"p":[[0,43,0.0,0.28125,0.2977,0.0,0.14286,0.46429,0.0,1.0,14,1,4,14,0,3,0,0,0,0,0,7,0,0,3,0,0,4,0,0,0,0,1],[4,43,0.093,0.72767,0.26574,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,2,0,0,4,0,0,8,0,0,6,0,9],[8,43,0.186,0.73657,0.24515,0.5713,0.85707,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,2,0,0,7,0,10],[12,43,0.2791,0.67857,0.26964,0.53571,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,4,0,0,5,0,0,7,0,0,4,0,8],[16,43,0.3721,0.72768,0.29529,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,3,0,0,2,0,0,5,0,0,3,0,0,4,0,13],[20,43,0.4651,0.78125,0.1988,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,7,0,0,11,0,8],[24,43,0.5581,0.77232,0.21683,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,6,0,0,11,0,8],[28,43,0.6512,0.66961,0.28891,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,4,0,0,3,0,0,5,0,0,5,0,0,4,0,9],[32,43,0.7442,0.77679,0.27649,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,0,4,0,15],[36,43,0.8372,0.85267,0.17672,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,5,0,16],[40,43,0.9302,0.8125,0.2126,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,10,0,11],[43,43,1.0,0.80357,0.1948,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,11,0,10]]},{"b":4,"e":1.0,"k":"rising","v":0.38839,"x":0.93302,"p":[[0,26,0.0,0.38839,0.36811,0.0,0.28571,0.71429,0.0,1.0,10,6,5,10,0,3,0,0,4,0,0,5,0,0,1,0,0,3,0,0,0,0,6],[4,26,0.1538,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[8,26,0.3077,0.8125,0.20652,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,5,0,0,7,0,13],[12,26,0.4615,0.78124,0.22013,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,7,0,0,7,0,11],[16,26,0.6154,0.75893,0.2911,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,4,0,0,1,0,0,1,0,0,6,0,0,3,0,15],[20,26,0.7692,0.75222,0.24223,0.57143,0.75,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,6,1,0,4,0,11],[24,26,0.9231,0.62946,0.31514,0.42857,0.71429,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,2,0,0,2,0,0,4,0,0,7,0,0,6,0,6],[26,26,1.0,0.63393,0.30916,0.39286,0.71429,0.85714,0.0,1.0,1,5,0,1,0,5,0,0,2,0,0,2,0,0,1,0,0,8,0,0,8,0,5]]}]},{"i":"12b4e4ea2afe47a4","q":"In a city of gnomes there are  $1000$  identical towers, each of which has  $1000$  stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of  $1000$  possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the  $k$ -th story of his tower and the other one in the  $(k+1)$ -st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends.\n\n*Authored by Nikola Velov*","t":[{"b":2,"e":0.0,"k":"falling","v":0.08482,"x":0.5625,"p":[[0,91,0.0,0.33929,0.26666,0.0,0.42857,0.57143,0.0,0.71429,9,0,3,9,0,3,0,0,3,0,0,7,0,0,4,0,0,6,0,0,0,0,0],[4,91,0.044,0.49554,0.3589,0.14286,0.71429,0.71429,0.0,1.0,5,5,0,5,0,7,0,0,2,0,0,0,0,0,1,0,0,11,0,0,1,0,5],[8,91,0.0879,0.5625,0.34058,0.39286,0.71429,0.85714,0.0,1.0,6,5,0,6,0,1,0,0,1,0,0,4,0,0,3,0,0,8,0,0,4,0,5],[12,91,0.1319,0.38393,0.3597,0.14286,0.14286,0.71429,0.0,1.0,7,4,0,7,0,10,0,0,1,0,0,3,0,0,0,0,0,5,0,0,2,0,4],[16,91,0.1758,0.46875,0.33926,0.14286,0.57143,0.71429,0.0,1.0,6,2,0,6,0,6,0,0,0,0,0,3,0,0,4,0,0,6,0,0,5,0,2],[20,91,0.2198,0.39732,0.35845,0.10714,0.14286,0.71429,0.0,1.0,8,3,0,8,0,9,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,3],[24,91,0.2637,0.34375,0.36397,0.0,0.14286,0.71429,0.0,1.0,11,4,0,11,0,7,0,0,2,0,0,0,0,0,2,0,0,6,0,0,0,0,4],[28,91,0.3077,0.45536,0.36323,0.0,0.5,0.71429,0.0,1.0,9,4,0,9,0,3,0,0,1,0,0,3,0,0,1,0,0,10,0,0,1,0,4],[32,91,0.3516,0.29018,0.33595,0.0,0.14286,0.60714,0.0,1.0,12,1,0,12,0,8,0,0,2,0,0,1,0,0,1,0,0,3,0,0,4,0,1],[36,91,0.3956,0.33482,0.31665,0.14286,0.14286,0.71429,0.0,1.0,5,2,0,5,0,15,0,0,0,0,0,2,0,0,1,0,0,6,0,0,1,0,2],[40,91,0.4396,0.40179,0.37019,0.0,0.14286,0.71429,0.0,1.0,9,3,0,9,0,8,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,3],[44,91,0.4835,0.29018,0.32632,0.0,0.14286,0.60714,0.0,1.0,10,2,0,10,0,11,0,0,1,0,0,1,0,0,1,0,0,5,0,0,1,0,2],[48,91,0.5275,0.40625,0.35912,0.0,0.35714,0.71429,0.0,1.0,9,4,0,9,0,5,0,0,2,0,0,2,0,0,3,0,0,6,0,0,1,0,4],[52,91,0.5714,0.24107,0.30606,0.0,0.14286,0.32143,0.0,1.0,12,2,0,12,0,11,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,2],[56,91,0.6154,0.14286,0.17128,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,15,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[60,91,0.6593,0.16964,0.26351,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,10,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,1],[64,91,0.7033,0.17857,0.21724,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,18,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[68,91,0.7473,0.12947,0.20316,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[72,91,0.7912,0.15625,0.22689,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,16,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[76,91,0.8352,0.20536,0.26471,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,14,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,1],[80,91,0.8791,0.08482,0.09354,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,91,0.9231,0.13393,0.14258,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[88,91,0.967,0.16071,0.17768,0.0,0.14286,0.42857,0.0,0.42857,14,0,0,14,0,9,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[91,91,1.0,0.09822,0.12596,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.09821,"x":0.64732,"p":[[0,75,0.0,0.28125,0.27545,0.0,0.28571,0.42857,0.0,1.0,12,1,4,12,0,2,0,0,6,0,0,5,0,0,3,0,0,3,0,0,0,0,1],[4,75,0.0533,0.50447,0.36067,0.14286,0.71429,0.71429,0.0,1.0,6,5,0,6,0,6,0,0,0,0,0,1,0,0,2,0,0,11,0,0,1,0,5],[8,75,0.1067,0.64732,0.28344,0.71429,0.71429,0.85714,0.0,1.0,2,3,0,2,0,4,0,0,0,0,0,0,0,0,1,0,0,16,0,0,6,0,3],[12,75,0.16,0.40625,0.36615,0.0,0.28573,0.71429,0.0,1.0,9,3,0,9,0,7,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,3],[16,75,0.2133,0.34375,0.33285,0.0,0.14286,0.71429,0.0,1.0,9,1,0,9,0,9,0,0,0,0,0,3,0,0,1,0,0,6,0,0,3,0,1],[20,75,0.2667,0.45982,0.34392,0.14286,0.64286,0.71429,0.0,1.0,7,2,0,7,0,6,0,0,0,0,0,1,0,0,2,0,0,12,0,0,2,0,2],[24,75,0.32,0.51786,0.37244,0.10714,0.71429,0.75,0.0,1.0,8,4,0,8,0,3,0,0,1,0,0,0,0,0,1,0,0,11,0,0,4,0,4],[28,75,0.3733,0.46875,0.35398,0.14286,0.50001,0.71429,0.0,1.0,6,3,0,6,0,7,0,0,0,0,0,3,0,0,1,0,0,8,0,0,4,0,3],[32,75,0.4267,0.32589,0.33926,0.0,0.14286,0.71429,0.0,1.0,10,2,0,10,0,9,0,0,1,0,0,1,0,0,1,0,0,7,0,0,1,0,2],[36,75,0.48,0.29464,0.31731,0.0,0.14286,0.57143,0.0,1.0,13,1,0,13,0,6,0,0,0,0,0,1,0,0,5,0,0,6,0,0,0,0,1],[40,75,0.5333,0.36161,0.33309,0.10714,0.14286,0.71429,0.0,1.0,8,2,0,8,0,9,0,0,0,0,0,4,0,0,0,0,0,8,0,0,1,0,2],[44,75,0.5867,0.24107,0.31831,0.0,0.14286,0.1429,0.0,1.0,9,4,0,9,0,16,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[48,75,0.64,0.23214,0.28516,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,9,0,0,1,0,0,3,0,0,0,0,0,4,0,0,2,0,0],[52,75,0.6933,0.21875,0.25501,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,9,0,0,0,0,0,4,0,0,2,0,0,4,0,0,0,0,0],[56,75,0.7467,0.29911,0.33761,0.0,0.14286,0.71429,0.0,1.0,10,2,0,10,0,11,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,2],[60,75,0.8,0.30356,0.33263,0.0,0.14286,0.60682,0.0,1.0,10,2,0,10,0,10,0,0,1,0,0,2,0,0,1,0,0,4,0,0,2,0,2],[64,75,0.8533,0.17857,0.16751,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,22,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[68,75,0.9067,0.18304,0.1931,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,18,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[72,75,0.96,0.125,0.15047,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,17,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[75,75,1.0,0.09821,0.10972,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bfadc5887a741d09","q":"Given is a square $A B C D$ with circumcircle $\\Gamma_{1}$. Let $P$ be a point on arc $A C$ where $B$ also lies. A circle $\\Gamma_{2}$ is internally tangent to $\\Gamma_{1}$ at $P$ and also tangent to diagonal $A C$ at $Q$. Let $R$ be a point on $\\Gamma_{2}$ such that the line $D R$ is tangent to $\\Gamma_{2}$. Prove that $|D R|=|D A|$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.09813,"p":[[0,23,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.09813,0.24336,0.0,0.0,0.14071,0.0,1.0,23,2,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,23,0.3478,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.03125,"x":0.08929,"p":[[0,28,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.08929,0.24157,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,28,0.2857,0.06697,0.18552,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,28,0.4286,0.07143,0.18211,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,28,0.5714,0.03572,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.03126,0.05907,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ffef9dff1d06ce54","q":"In a triangle  $ABC$ , let  $P$  be a point on the bisector of  $\\angle BAC$  and let  $A',B'$  and  $C'$  be points on lines  $BC,CA$  and  $AB$  respectively such that  $PA'$  is perpendicular to  $BC,PB'\\perp AC$ , and  $PC'\\perp AB$ . Prove that  $PA'$  and  $B'C'$  intersect on the median  $AM$ , where  $M$  is the midpoint of  $BC$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.01339,"x":0.08036,"p":[[0,135,0.0,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,135,0.0296,0.07589,0.10091,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,135,0.0593,0.0625,0.08702,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,135,0.0889,0.0758,0.11832,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,135,0.1185,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,135,0.1481,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,135,0.1778,0.04455,0.09731,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,135,0.2074,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,135,0.237,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,135,0.2667,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,135,0.2963,0.06696,0.13356,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,135,0.3259,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,135,0.3556,0.05803,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,135,0.3852,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,135,0.4148,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,135,0.4444,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,135,0.4741,0.04465,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,135,0.5037,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,135,0.5333,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,135,0.563,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,135,0.5926,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,135,0.6222,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,135,0.6519,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,135,0.6815,0.05795,0.08636,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,135,0.7111,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,135,0.7407,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,135,0.7704,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,135,0.8,0.05357,0.1171,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,135,0.8296,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,135,0.8593,0.06697,0.11837,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[120,135,0.8889,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,135,0.9185,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,135,0.9481,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,135,0.9778,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[135,135,1.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.02679,"x":0.10714,"p":[[0,128,0.0,0.09821,0.15335,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,128,0.0312,0.09812,0.09736,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,128,0.0625,0.08929,0.09942,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,128,0.0938,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,128,0.125,0.10713,0.13358,0.0,0.14286,0.14286,0.0,0.571,15,0,0,15,0,13,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,128,0.1562,0.06697,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,128,0.1875,0.08928,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,128,0.2188,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,128,0.25,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,128,0.2812,0.10714,0.12877,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,10,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,128,0.3125,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,11,0,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an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.41955,"x":0.69194,"p":[[0,42,0.0,0.50892,0.22285,0.28571,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,7,0,0,5,0,0,7,0,0,7,0,0,2,0,1],[4,42,0.0952,0.51339,0.29203,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,8,0,0,4,0,0,2,0,0,7,0,0,2,0,4],[8,42,0.1905,0.41955,0.26481,0.14289,0.35714,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,7,0,0,4,0,0,5,0,0,4,0,0,1,0,2],[12,42,0.2857,0.49106,0.23673,0.28571,0.42859,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,4,0,0,9,0,0,6,0,0,3,0,0,4,0,1],[16,42,0.381,0.56696,0.27078,0.28571,0.57143,0.75,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,2,0,0,6,0,0,6,0,0,5,0,3],[20,42,0.4762,0.54463,0.25614,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,5,0,0,5,0,0,5,0,0,8,0,0,2,0,3],[24,42,0.5714,0.62946,0.17445,0.57143,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,4,0,0,10,0,0,11,0,0,4,0,1],[28,42,0.6667,0.6696,0.20343,0.57143,0.71429,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,11,0,0,4,0,4],[32,42,0.7619,0.68304,0.17029,0.57143,0.71429,0.75,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,11,0,0,5,0,3],[36,42,0.8571,0.69194,0.152,0.57143,0.71429,0.75,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,10,0,0,5,0,3],[40,42,0.9524,0.47321,0.17655,0.28571,0.5,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,4,0,0,12,0,0,3,0,0,0,0,1],[42,42,1.0,0.47762,0.19431,0.28571,0.5705,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,8,0,0,4,0,0,10,0,0,6,0,0,1,0,0]]},{"b":6,"e":0.2857,"k":"falling","v":0.36607,"x":0.67855,"p":[[0,75,0.0,0.5424,0.22718,0.39286,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,1,5,0,0,5,0,0,5,0,0,10,0,0,4,0,0],[4,75,0.0533,0.6071,0.22304,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,2,0,0,5,0,0,10,0,0,6,0,0,4,0,3],[8,75,0.1067,0.51338,0.25218,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,8,0,0,7,0,0,3,0,0,5,0,0,4,0,2],[12,75,0.16,0.43303,0.2435,0.2857,0.42857,0.46429,0.0,1.0,2,1,0,2,0,3,0,0,7,0,0,12,0,0,1,0,0,3,0,0,3,0,1],[16,75,0.2133,0.49553,0.30719,0.24999,0.42857,0.71429,0.0,1.0,2,3,0,2,0,6,0,0,5,0,0,4,0,0,1,0,0,8,0,0,3,0,3],[20,75,0.2667,0.51339,0.30064,0.28571,0.42857,0.71429,0.0,1.0,1,6,0,1,0,5,0,0,4,0,0,9,0,0,3,0,0,3,0,0,1,0,6],[24,75,0.32,0.67855,0.28794,0.42857,0.85707,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,5,0,0,3,0,0,1,0,0,9,0,8],[28,75,0.3733,0.55808,0.25089,0.39286,0.57143,0.75,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,6,0,0,6,0,0,4,0,0,6,0,2],[32,75,0.4267,0.5625,0.25738,0.42857,0.42857,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,11,0,0,1,0,0,4,0,0,6,0,3],[36,75,0.48,0.55356,0.27141,0.28571,0.64286,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,1,0,0,4,0,0,10,0,0,4,0,2],[40,75,0.5333,0.52233,0.24119,0.28571,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,5,0,0,6,0,0,3,0,0,10,0,0,4,0,0],[44,75,0.5867,0.42411,0.20973,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,8,0,0,10,0,0,5,0,0,3,0,0,2,0,0],[48,75,0.64,0.49106,0.15541,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,17,0,0,4,0,0,6,0,0,1,0,0],[52,75,0.6933,0.48214,0.16267,0.42857,0.42857,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,18,0,0,5,0,0,0,0,0,4,0,0],[56,75,0.7467,0.48213,0.14173,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,16,0,0,6,0,0,4,0,0,1,0,0],[60,75,0.8,0.5,0.20203,0.28571,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,9,0,0,4,0,0,6,0,0,2,0,1],[64,75,0.8533,0.51338,0.13767,0.42857,0.42859,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,14,0,0,9,0,0,5,0,0,1,0,0],[68,75,0.9067,0.49548,0.1855,0.39286,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,11,0,0,7,0,0,3,0,0,2,0,1],[72,75,0.96,0.42853,0.16748,0.28571,0.42857,0.46418,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,11,0,0,6,0,0,0,0,0,1,0,1],[75,75,1.0,0.36607,0.1234,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,17,0,0,11,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"22c2bbab461ed565","q":"Find all triples  $(x,y,z)$  of positive integers satisfying the system of equations\n\\[\\begin{cases} x^2=2(y+z) x^6=y^6+z^6+31(y^2+z^2)\\end{cases}\\]","t":[{"b":0,"e":0.42857,"k":"flat","v":0.64732,"x":0.69195,"p":[[0,4,0.0,0.69195,0.17169,0.67857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,17,0,0,4,0,3],[4,4,1.0,0.64732,0.11285,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,23,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.64732,"x":0.69643,"p":[[0,10,0.0,0.69643,0.15465,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,18,0,0,5,0,2],[4,10,0.4,0.64732,0.18205,0.42857,0.71429,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,13,0,0,8,0,0],[8,10,0.8,0.66964,0.10374,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,27,0,0,0,0,0],[10,10,1.0,0.68304,0.08552,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,28,0,0,0,0,0]]}]},{"i":"34d66b0e7e1912d6","q":"In a quadrilateral  $ABCD$  sides  $AB$  and  $CD$  are equal,  $\\angle A=150^\\circ,$   $\\angle B=44^\\circ,$   $\\angle C=72^\\circ.$  \nPerpendicular bisector of the segment  $AD$  meets the side  $BC$  at point  $P.$  \nFind  $\\angle APD.$ *Proposed by F. Bakharev*","t":[{"b":2,"e":0.42857,"k":"falling","v":0.35268,"x":0.86606,"p":[[0,154,0.0,0.75443,0.31792,0.67857,0.9285,1.0,0.0,1.0,3,16,1,3,0,0,0,0,2,0,0,0,0,0,3,0,0,7,0,0,1,0,16],[4,154,0.026,0.86162,0.20664,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,0,0,0,21],[8,154,0.0519,0.86606,0.21412,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,5,0,0,1,0,21],[12,154,0.0779,0.67846,0.34084,0.4286,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,0,0,0,4,0,0,2,0,0,7,0,0,2,0,12],[16,154,0.1039,0.67408,0.36288,0.42857,0.71429,1.0,0.0,1.0,4,15,0,4,0,2,0,0,0,0,0,3,0,0,5,0,0,3,0,0,0,0,15],[20,154,0.1299,0.81694,0.32585,0.71429,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,22],[24,154,0.1558,0.68747,0.3223,0.53539,0.71429,1.0,0.0,1.0,2,12,0,2,0,3,0,0,0,0,0,3,0,0,4,0,0,6,0,0,2,0,12],[28,154,0.1818,0.73214,0.34764,0.67857,0.92857,1.0,0.0,1.0,4,16,0,4,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,1,0,16],[32,154,0.2078,0.63826,0.37622,0.42857,0.71429,1.0,0.0,1.0,7,11,0,7,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,2,0,11],[36,154,0.2338,0.6116,0.39647,0.21427,0.71429,1.0,0.0,1.0,8,11,0,8,0,0,0,0,1,0,0,2,0,0,0,0,0,8,0,0,2,0,11],[40,154,0.2597,0.58036,0.41023,0.0,0.71429,1.0,0.0,1.0,9,11,0,9,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,3,0,11],[44,154,0.2857,0.56249,0.35163,0.14286,0.71429,0.75,0.0,1.0,5,7,0,5,0,4,0,0,0,0,0,2,0,0,4,0,0,9,0,0,1,0,7],[48,154,0.3117,0.58482,0.41243,0.0,0.71429,1.0,0.0,1.0,9,12,0,9,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,1,0,12],[52,154,0.3377,0.73661,0.32754,0.57143,0.85714,1.0,0.0,1.0,3,16,0,3,0,1,0,0,0,0,0,3,0,0,2,0,0,7,0,0,0,0,16],[56,154,0.3636,0.55802,0.37857,0.25,0.64286,1.0,0.0,1.0,6,10,0,6,0,2,0,0,3,0,0,3,0,0,2,0,0,6,0,0,0,0,10],[60,154,0.3896,0.67847,0.412,0.25,1.0,1.0,0.0,1.0,6,18,0,6,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,18],[64,154,0.4156,0.607,0.33689,0.28571,0.71429,0.89286,0.0,1.0,3,8,0,3,0,4,0,0,2,0,0,1,0,0,2,0,0,11,0,0,1,0,8],[68,154,0.4416,0.62499,0.37243,0.39286,0.71429,1.0,0.0,1.0,6,10,0,6,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,4,0,10],[72,154,0.4675,0.54909,0.39465,0.10714,0.71429,1.0,0.0,1.0,8,10,0,8,0,1,0,0,2,0,0,3,0,0,1,0,0,7,0,0,0,0,10],[76,154,0.4935,0.59821,0.43365,0.0,0.78564,1.0,0.0,1.0,9,13,0,9,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,13],[80,154,0.5195,0.58034,0.41487,0.10714,0.71429,1.0,0.0,1.0,8,11,0,8,0,3,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,11],[84,154,0.5455,0.57589,0.37709,0.14289,0.71429,1.0,0.0,1.0,6,9,0,6,0,3,0,0,1,0,0,2,0,0,1,0,0,9,0,0,1,0,9],[88,154,0.5714,0.4933,0.40381,0.0,0.5,0.89286,0.0,1.0,10,8,0,10,0,2,0,0,0,0,0,4,0,0,1,0,0,5,1,0,1,0,8],[92,154,0.5974,0.58929,0.42371,0.14286,0.71429,1.0,0.0,1.0,7,14,0,7,0,4,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,14],[96,154,0.6234,0.64282,0.34442,0.42857,0.64286,1.0,0.0,1.0,2,12,0,2,0,4,0,0,1,0,0,4,0,0,5,0,0,2,0,0,2,0,12],[100,154,0.6494,0.64286,0.38132,0.35714,0.71429,1.0,0.0,1.0,6,12,0,6,0,2,0,0,0,0,0,1,0,0,1,0,0,9,0,0,1,0,12],[104,154,0.6753,0.61605,0.34151,0.42857,0.71429,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,1,0,0,3,0,0,2,0,0,10,0,0,2,0,8],[108,154,0.7013,0.64734,0.34064,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,5,0,0,1,0,0,5,0,0,3,0,0,4,0,0,0,0,13],[112,154,0.7273,0.63391,0.42097,0.0,0.78564,1.0,0.0,1.0,9,14,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,14],[116,154,0.7532,0.5714,0.35892,0.25001,0.714,0.89286,0.0,1.0,5,8,0,5,0,3,0,0,1,0,0,4,0,0,2,0,0,7,0,0,2,0,8],[120,154,0.7792,0.49999,0.35892,0.14286,0.49979,0.75,0.0,1.0,6,5,0,6,0,5,0,0,0,0,0,5,0,0,1,0,0,7,0,0,3,0,5],[124,154,0.8052,0.55804,0.37857,0.14286,0.64286,1.0,0.0,1.0,6,10,0,6,0,3,0,0,0,0,0,6,0,0,1,0,0,6,0,0,0,0,10],[128,154,0.8312,0.53124,0.34669,0.14286,0.71429,0.71429,0.0,1.0,5,5,0,5,0,5,0,0,0,0,0,3,0,0,2,0,0,10,0,0,2,0,5],[132,154,0.8571,0.53571,0.38299,0.14286,0.64286,0.89286,0.0,1.0,7,8,0,7,0,3,0,0,1,0,0,3,0,0,2,0,0,6,0,0,2,0,8],[136,154,0.8831,0.54464,0.36845,0.14286,0.71429,0.85704,0.0,1.0,5,6,0,5,0,6,0,0,0,0,0,2,0,0,1,0,0,8,0,0,4,0,6],[140,154,0.9091,0.58036,0.38122,0.14286,0.71429,1.0,0.0,1.0,4,11,0,4,0,6,0,0,0,0,0,4,0,0,0,0,0,7,0,0,0,0,11],[144,154,0.9351,0.57589,0.38545,0.10714,0.71429,1.0,0.0,1.0,8,9,0,8,0,1,0,0,0,0,0,2,0,0,2,0,0,9,0,0,1,0,9],[148,154,0.961,0.41518,0.39343,0.0,0.4286,0.74996,0.0,1.0,13,4,0,13,0,1,0,0,1,0,0,3,0,0,1,0,0,5,0,0,4,0,4],[152,154,0.987,0.40619,0.36085,0.0,0.42857,0.71429,0.0,1.0,10,4,0,10,0,3,0,0,2,0,0,4,0,0,3,0,0,4,0,0,2,0,4],[154,154,1.0,0.35268,0.33117,0.10714,0.2857,0.71429,0.0,1.0,8,3,0,8,0,7,0,0,5,0,0,2,0,0,0,0,0,7,0,0,0,0,3]]},{"b":6,"e":0.0,"k":"falling","v":0.05804,"x":0.85714,"p":[[0,265,0.0,0.74554,0.30459,0.67857,0.78571,1.0,0.0,1.0,2,14,0,2,0,2,0,0,0,0,0,1,0,0,3,0,0,8,0,0,2,0,14],[4,265,0.0151,0.85714,0.22304,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,8,0,0,1,0,20],[8,265,0.0302,0.85268,0.26362,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,21],[12,265,0.0453,0.76337,0.27806,0.571,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,5,0,0,2,0,0,6,0,0,2,0,15],[16,265,0.0604,0.7991,0.32115,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,19],[20,265,0.0755,0.7991,0.30485,0.71429,1.0,1.0,0.0,1.0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an acute triangle  $ABC,$  let  $O,G,H$  be its circumcentre, centroid and orthocenter. Let  $D\\in BC, E\\in CA$  and  $OD\\perp BC, HE\\perp CA.$  Let  $F$  be the midpoint of  $AB.$  If the triangles  $ODC, HEA, GFB$  have the same area, find all the possible values of  $\\angle 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 $ M$  be a point on  $ BC$  and  $ N$  be a point on  $ AB$  such that  $ AM$  and  $ CN$  are angle bisectors of the triangle  $ ABC$ . Given that  $ \\frac {\\angle BNM}{\\angle MNC} \\equal{} \\frac {\\angle BMN}{\\angle NMA}$ , prove that the triangle  $ ABC$  is isosceles.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.0625,"x":0.22097,"p":[[0,144,0.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,144,0.0278,0.16277,0.12785,0.14214,0.14286,0.14287,0.0,0.57143,6,0,0,6,0,19,0,1,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,144,0.0556,0.17402,0.11145,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,19,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,144,0.0833,0.20069,0.14665,0.14286,0.14286,0.28571,0.0,0.571,5,0,0,5,0,15,0,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,144,0.1111,0.22097,0.13755,0.14286,0.14286,0.2857,0.0,0.571,3,0,0,3,0,15,0,1,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,144,0.1389,0.17634,0.11149,0.14286,0.14286,0.16075,0.0,0.57143,3,0,0,3,0,21,0,1,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,144,0.1667,0.15625,0.10326,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,23,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,144,0.1944,0.18304,0.10248,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,144,0.2222,0.18741,0.09744,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,22,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,144,0.25,0.17857,0.10714,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,20,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,144,0.2778,0.16964,0.14032,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,19,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[44,144,0.3056,0.16286,0.10602,0.14286,0.14286,0.2857,0.0,0.4286,6,0,0,6,0,16,0,1,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,144,0.3333,0.17857,0.11845,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,14,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,144,0.3611,0.20089,0.13767,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[56,144,0.3889,0.14286,0.07986,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,144,0.4167,0.16964,0.12078,0.14286,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,16,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,144,0.4444,0.14733,0.12619,0.10714,0.14286,0.1429,0.0,0.57143,8,0,0,8,0,18,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[68,144,0.4722,0.12946,0.09006,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,144,0.5,0.15402,0.11309,0.14286,0.14286,0.14287,0.0,0.57143,6,0,0,6,0,19,0,1,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,144,0.5278,0.1317,0.07599,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,22,0,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,144,0.5556,0.14951,0.09513,0.14286,0.14286,0.1786,0.0,0.28571,6,0,0,6,1,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,144,0.5833,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,144,0.6111,0.13615,0.11344,0.0,0.14286,0.2857,0.0,0.28571,11,0,0,11,0,11,0,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,144,0.6389,0.11607,0.10972,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,144,0.6667,0.14286,0.08748,0.14286,0.14286,0.14287,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,144,0.6944,0.14062,0.09859,0.10714,0.14286,0.16075,0.0,0.28571,8,0,0,8,0,16,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,144,0.7222,0.15402,0.09798,0.14286,0.14286,0.14289,0.0,0.42857,5,0,0,5,1,19,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,144,0.75,0.13839,0.08364,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,144,0.7778,0.16509,0.09525,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,144,0.8056,0.17857,0.08748,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,18,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,144,0.8333,0.1875,0.09585,0.14286,0.14286,0.2857,0.0,0.29,4,0,0,4,0,13,0,2,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,144,0.8611,0.15402,0.06707,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,1,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,144,0.8889,0.14723,0.09771,0.14214,0.14286,0.17857,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,144,0.9167,0.14286,0.10101,0.10714,0.14286,0.1786,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,144,0.9444,0.18527,0.07332,0.14286,0.14286,0.23214,0.14286,0.42857,0,0,0,0,0,23,0,1,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[140,144,0.9722,0.13607,0.08813,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,19,0,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,144,1.0,0.1875,0.10971,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,13,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.09464,"x":0.20312,"p":[[0,158,0.0,0.09464,0.07741,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,18,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,158,0.0253,0.16072,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,158,0.0506,0.18304,0.09606,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,158,0.0759,0.17847,0.13361,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,18,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,158,0.1013,0.20089,0.11769,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,19,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,158,0.1266,0.15839,0.11537,0.14286,0.14286,0.16075,0.0,0.57143,6,0,0,6,0,18,0,1,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,158,0.1519,0.18527,0.11415,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,19,0,1,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,158,0.1772,0.17857,0.10102,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,20,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,158,0.2025,0.12723,0.09737,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,1,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,158,0.2278,0.19186,0.12681,0.14286,0.14286,0.1786,0.0,0.57143,2,0,0,2,0,22,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,158,0.2532,0.16741,0.1245,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,20,0,0,5,0,0,0,0,1,1,0,0,0,0,0,0,0,0],[44,158,0.2785,0.17179,0.1099,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,17,0,1,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,158,0.3038,0.18062,0.10106,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,19,0,1,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,158,0.3291,0.13839,0.10999,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,21,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,158,0.3544,0.17857,0.07986,0.14286,0.14286,0.28571,0.0,0.28571,2,0,0,2,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,158,0.3797,0.16518,0.11904,0.14286,0.14286,0.14287,0.0,0.57143,5,0,0,5,0,20,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,158,0.4051,0.18079,0.11287,0.14286,0.14286,0.28571,0.0,0.571,4,0,0,4,0,17,0,1,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,158,0.4304,0.17187,0.10693,0.14286,0.14286,0.17857,0.0,0.57143,3,0,0,3,1,20,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,158,0.4557,0.16507,0.12425,0.14214,0.14286,0.2857,0.0,0.571,7,0,0,7,0,15,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,158,0.481,0.20312,0.11725,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,15,0,1,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[80,158,0.5063,0.12947,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,158,0.5316,0.18303,0.16457,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,14,0,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[88,158,0.557,0.11607,0.08328,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,158,0.5823,0.17848,0.10717,0.14286,0.14286,0.1786,0.0,0.42857,3,0,0,3,0,21,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[96,158,0.6076,0.18072,0.136,0.14286,0.14286,0.17868,0.0,0.57143,4,0,0,4,1,19,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[100,158,0.6329,0.15625,0.11904,0.14286,0.14286,0.14287,0.0,0.57143,6,0,0,6,0,19,0,2,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[104,158,0.6582,0.14732,0.0977,0.14286,0.14286,0.17868,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,158,0.6835,0.16741,0.08858,0.14286,0.14286,0.16071,0.0,0.42857,3,0,0,3,0,21,0,1,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[112,158,0.7089,0.17411,0.14167,0.14286,0.14286,0.1786,0.0,0.57143,6,0,0,6,0,18,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[116,158,0.7342,0.17857,0.12372,0.14286,0.14286,0.1786,0.0,0.57143,4,0,0,4,0,20,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[120,158,0.7595,0.15625,0.11495,0.14286,0.14286,0.14287,0.0,0.57143,6,0,0,6,0,19,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[124,158,0.7848,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[128,158,0.8101,0.15178,0.11258,0.10714,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,158,0.8354,0.16964,0.10374,0.14286,0.14286,0.17857,0.0,0.42857,4,0,0,4,0,20,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[136,158,0.8608,0.18302,0.14386,0.14286,0.14286,0.2857,0.0,0.571,7,0,0,7,0,14,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[140,158,0.8861,0.19196,0.12682,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,17,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[144,158,0.9114,0.18746,0.12064,0.14286,0.14286,0.2857,0.0,0.57,4,0,0,4,0,17,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[148,158,0.9367,0.13839,0.10403,0.10714,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,18,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[152,158,0.962,0.14286,0.10714,0.14286,0.14286,0.14286,0.0,0.5714,6,0,0,6,0,22,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[156,158,0.9873,0.12947,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[158,158,1.0,0.18304,0.08171,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f5c38fbf0244d792","q":"Let  $ a_i, b_i$  be positive real numbers,  $ i\\equal{}1,2,\\ldots,n$ ,  $ n\\geq 2$ , such that  $ a_i<b_i$ , for all  $ i$ , and also \\[ b_1\\plus{}b_2\\plus{}\\cdots \\plus{} b_n < 1 \\plus{} a_1\\plus{}\\cdots \\plus{} a_n.\\] Prove that there exists a  $ c\\in\\mathbb R$  such that for all  $ i\\equal{}1,2,\\ldots,n$ , and  $ k\\in\\mathbb Z$  we have \\[ (a_i\\plus{}c\\plus{}k)(b_i\\plus{}c\\plus{}k) > 0.\\]","t":[{"b":6,"e":1.0,"k":"flat","v":0.98213,"x":1.0,"p":[[0,21,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,21,0.1905,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,21,0.381,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,21,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,21,0.7619,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,21,0.9524,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[21,21,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,16,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,16,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,16,0.5,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,16,0.75,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,16,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"bb2bc5556323444a","q":"Let  $ N$  denote the set of natural numbers. Let  $ \\phi: N\\rightarrow N$  be a bijective function and assume that there exists a finite limit\r\n\\[ \\lim_{n\\rightarrow\\infty}\\frac{\\phi(n)}{n}\\equal{}L.\r\n\\] What are the possible values of  $ L$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.88393,"x":0.94196,"p":[[0,11,0.0,0.92857,0.13832,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[4,11,0.3636,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[8,11,0.7273,0.88393,0.19377,0.78571,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,1,0,23],[11,11,1.0,0.90624,0.1772,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,24]]},{"b":5,"e":0.71429,"k":"falling","v":0.70088,"x":0.95088,"p":[[0,28,0.0,0.95088,0.12686,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[4,28,0.1429,0.89286,0.19885,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,22],[8,28,0.2857,0.86161,0.27078,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,23],[12,28,0.4286,0.84374,0.24055,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,19],[16,28,0.5714,0.87052,0.24319,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,22],[20,28,0.7143,0.87054,0.1488,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,3,0,17],[24,28,0.8571,0.70088,0.3038,0.57132,0.85707,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,1,0,0,1,0,0,2,0,0,6,0,0,9,0,8],[28,28,1.0,0.7098,0.29339,0.57143,0.78564,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,2,0,0,6,0,0,4,0,0,6,0,10]]}]},{"i":"2dce414411a35ec5","q":"Let  $ S$  be a set of all positive integers which can be represented as  $ a^2 \\plus{} 5b^2$  for some integers  $ a,b$  such that  $ a\\bot b$ . Let  $ p$  be a prime number such that  $ p \\equal{} 4n \\plus{} 3$  for some integer  $ n$ . Show that if for some positive integer  $ k$  the number  $ kp$  is in  $ S$ , then  $ 2p$  is in  $ S$  as well.\r\n\r\nHere, the notation  $ a\\bot b$  means that the integers  $ a$  and  $ b$  are coprime.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13393,"x":0.23652,"p":[[0,19,0.0,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.21429,0.12877,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,19,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.23652,0.1164,0.14286,0.14286,0.28571,0.14,0.42857,0,0,0,0,0,18,0,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.09821,"x":0.1383,"p":[[0,24,0.0,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.1383,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bc8c51d2ad15067b","q":"Let  $ P $  be a point in the interior of a triangle  $ ABC $ , and let  $ D, E, F $  be the point of intersection of the line  $ AP $  and the side  $ BC $  of the triangle, of the line  $ BP $  and the side  $ CA $ , and of the line  $ CP $  and the side  $ AB $ , respectively. Prove that the area of the triangle  $ ABC $  must be  $ 6  $  if the area of each of the triangles  $ PFA, PDB $  and  $ PEC $  is  $ 1 $ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.49107,"x":0.82588,"p":[[0,147,0.0,0.61158,0.22084,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,10,0,0,4,0,0,6,0,0,7,0,2],[4,147,0.0272,0.72321,0.26949,0.57143,0.78571,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,7,0,0,7,0,9],[8,147,0.0544,0.54017,0.24675,0.39286,0.42857,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,7,0,0,9,0,0,4,0,0,5,0,0,3,0,3],[12,147,0.0816,0.49551,0.23953,0.28571,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,9,0,0,5,0,0,4,0,0,1,0,3],[16,147,0.1088,0.59821,0.27067,0.39286,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,6,0,0,3,0,0,4,0,0,7,0,4],[20,147,0.1361,0.57141,0.26963,0.42857,0.57121,0.74996,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,8,0,0,3,0,0,6,0,0,4,0,4],[24,147,0.1633,0.4955,0.26482,0.28571,0.42857,0.64286,0.14286,1.0,0,2,0,0,0,5,0,0,6,0,0,8,0,0,5,0,0,0,0,0,6,0,2],[28,147,0.1905,0.49107,0.28557,0.24999,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,8,0,0,3,0,0,8,0,0,3,0,0,3,0,0,4,0,3],[32,147,0.2177,0.6607,0.2594,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,9,0,0,4,0,0,4,0,0,4,0,8],[36,147,0.2449,0.55803,0.27516,0.39286,0.42857,0.74996,0.14286,1.0,0,6,0,0,0,2,0,0,6,0,0,11,0,0,1,0,0,4,0,0,2,0,6],[40,147,0.2721,0.62946,0.25217,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,9,0,0,0,0,0,7,0,0,10,0,2],[44,147,0.2993,0.54025,0.26433,0.39286,0.4998,0.75,0.0,1.0,1,3,0,1,0,2,0,0,5,0,0,8,0,0,6,0,0,2,0,0,5,0,3],[48,147,0.3265,0.53125,0.25313,0.42857,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,11,0,0,3,0,0,5,0,0,3,0,3],[52,147,0.3537,0.61607,0.27765,0.39286,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,5,0,0,3,0,0,4,0,0,7,0,5],[56,147,0.381,0.54464,0.29329,0.28571,0.57143,0.75,0.14286,1.0,0,4,0,0,0,6,0,0,5,0,0,4,0,0,3,0,0,6,0,0,4,0,4],[60,147,0.4082,0.63836,0.28569,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,3,0,0,6,0,0,3,0,0,6,0,0,3,0,8],[64,147,0.4354,0.60713,0.30723,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,7,0,0,4,0,0,3,0,0,3,0,0,4,0,8],[68,147,0.4626,0.58924,0.27606,0.39286,0.57143,0.85704,0.14286,1.0,0,4,0,0,0,4,0,0,4,0,0,4,0,0,6,0,0,4,0,0,6,0,4],[72,147,0.4898,0.62945,0.2854,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,3,0,0,4,0,0,4,0,0,4,0,0,8,0,5],[76,147,0.517,0.5982,0.29329,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,3,0,0,3,0,0,7,0,0,3,0,0,2,0,0,9,0,4],[80,147,0.5442,0.65622,0.20783,0.571,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,8,0,0,8,0,2],[84,147,0.5714,0.66964,0.2822,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,5,0,0,4,0,0,4,0,0,6,0,8],[88,147,0.5986,0.58929,0.31288,0.28571,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,5,0,0,2,0,0,3,0,0,2,0,0,10,0,4],[92,147,0.6259,0.65625,0.2547,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,5,0,0,3,0,0,6,0,0,9,0,4],[96,147,0.6531,0.70982,0.2382,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,10,0,0,4,0,8],[100,147,0.6803,0.64731,0.27661,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,3,0,0,5,0,0,2,0,0,7,0,0,6,0,6],[104,147,0.7075,0.70982,0.2461,0.57143,0.78571,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,7,0,0,11,0,5],[108,147,0.7347,0.74998,0.2113,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,7,0,0,6,0,9],[112,147,0.7619,0.62052,0.249,0.42859,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,3,0,0,6,0,0,8,0,0,6,0,3],[116,147,0.7891,0.71875,0.21572,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,10,0,0,12,0,3],[120,147,0.8163,0.78347,0.17632,0.71429,0.85714,0.85714,0.357,1.0,0,6,0,0,0,0,0,0,0,0,1,3,0,0,1,0,0,8,0,0,13,0,6],[124,147,0.8435,0.82588,0.15868,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,11,0,10],[128,147,0.8707,0.72765,0.23519,0.53539,0.85707,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,5,0,0,10,0,7],[132,147,0.898,0.68748,0.23266,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,6,0,0,8,0,5],[136,147,0.9252,0.60268,0.24675,0.42857,0.64286,0.75,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,4,0,0,5,0,0,8,0,0,5,0,3],[140,147,0.9524,0.73659,0.2262,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,8,0,0,8,0,7],[144,147,0.9796,0.7052,0.24231,0.571,0.78571,0.85714,0.14,1.0,0,5,0,0,0,2,0,0,2,0,0,1,0,0,7,0,0,4,0,0,11,0,5],[147,147,1.0,0.63392,0.23942,0.4286,0.71429,0.75,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,4,0,0,5,0,0,10,0,0,4,0,4]]},{"b":1,"e":0.71429,"k":"flat","v":0.52679,"x":0.78569,"p":[[0,119,0.0,0.62488,0.22536,0.42857,0.64286,0.85714,0.14,1.0,0,1,0,0,0,2,0,0,1,0,0,8,0,0,5,0,0,5,0,0,10,0,1],[4,119,0.0336,0.52679,0.27993,0.28571,0.42857,0.75,0.14286,1.0,0,4,0,0,0,4,0,0,6,0,0,10,0,0,0,0,0,4,0,0,4,0,4],[8,119,0.0672,0.58929,0.30878,0.28571,0.50001,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,6,0,0,6,0,0,1,0,0,2,0,0,7,0,6],[12,119,0.1008,0.63839,0.24218,0.42859,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,6,0,0,5,0,0,7,0,0,6,0,4],[16,119,0.1345,0.64732,0.28118,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,9,0,0,1,0,0,3,0,0,7,0,7],[20,119,0.1681,0.64729,0.28119,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,1,0,0,5,0,0,5,0,0,5,0,0,5,0,7],[24,119,0.2017,0.58929,0.28515,0.39286,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,6,0,0,2,0,0,4,0,0,3,0,0,7,0,0,7,0,3],[28,119,0.2353,0.64731,0.28344,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,4,0,0,4,0,0,5,0,0,4,0,8],[32,119,0.2689,0.66518,0.30011,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,6,0,0,0,0,0,4,0,0,4,0,0,7,0,8],[36,119,0.3025,0.70534,0.25239,0.5354,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,3,0,0,4,0,0,5,0,0,7,0,8],[40,119,0.3361,0.66072,0.28959,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,7,0,0,0,0,0,6,0,0,4,0,9],[44,119,0.3697,0.70088,0.24054,0.42857,0.78571,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,7,0,0,2,0,0,4,0,0,10,0,6],[48,119,0.4034,0.56696,0.29771,0.28571,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,5,0,0,8,0,0,2,0,0,4,0,0,2,0,7],[52,119,0.437,0.61158,0.27255,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,4,0,0,4,0,0,7,0,0,6,0,4],[56,119,0.4706,0.63839,0.27661,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,5,0,0,2,0,0,6,0,0,6,0,6],[60,119,0.5042,0.64285,0.26726,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,3,0,0,3,0,0,9,0,0,4,0,6],[64,119,0.5378,0.56696,0.31841,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,4,0,0,9,0,0,3,0,0,0,0,0,2,0,9],[68,119,0.5714,0.625,0.28064,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,7,0,0,1,0,0,5,0,0,9,0,4],[72,119,0.605,0.5714,0.26486,0.42857,0.57143,0.75,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,6,0,0,7,0,0,4,0,0,4,0,4],[76,119,0.6387,0.69643,0.27141,0.53571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,0,0,0,4,0,0,1,0,0,9,0,0,7,0,7],[80,119,0.6723,0.6964,0.25693,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,4,0,0,5,0,0,6,0,0,7,0,7],[84,119,0.7059,0.68304,0.26901,0.53572,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,2,0,0,3,0,0,5,0,0,2,0,0,12,0,5],[88,119,0.7395,0.78569,0.26488,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,5,0,0,6,0,14],[92,119,0.7731,0.78124,0.22585,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,0,10,0,10],[96,119,0.8067,0.58929,0.28516,0.28571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,6,0,0,6,0,0,2,0,0,3,0,0,8,0,4],[100,119,0.8403,0.69643,0.28738,0.42857,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,1,0,0,4,0,0,2,0,0,3,0,0,11,0,7],[104,119,0.8739,0.72321,0.24984,0.42859,0.78564,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,7,0,0,2,0,0,5,0,0,7,0,9],[108,119,0.9076,0.76332,0.19107,0.57143,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,6,0,0,8,0,8],[112,119,0.9412,0.69642,0.25939,0.57143,0.78564,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,0,0,0,3,0,0,7,0,0,12,0,4],[116,119,0.9748,0.69642,0.23622,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,7,0,0,9,0,5],[119,119,1.0,0.70981,0.21867,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,8,0,0,10,0,4]]}]},{"i":"c0d82856dd0eb0f8","q":"Let  $ f$  be a function from the set of real numbers  $ \\mathbb{R}$  into itself such for all  $ x \\in \\mathbb{R},$  we have  $ |f(x)| \\leq 1$  and \r\n\r\n\\[ f \\left( x \\plus{} \\frac{13}{42} \\right) \\plus{} f(x) \\equal{} f \\left( x \\plus{} \\frac{1}{6} \\right) \\plus{} f \\left( x \\plus{} \\frac{1}{7} \\right).\\]\r\n\r\nProve that  $ f$  is a periodic function (that is, there exists a non-zero real number  $ c$  such  $ f(x\\plus{}c) \\equal{} f(x)$  for all  $ x \\in \\mathbb{R}$ ).","t":[{"b":1,"e":0.42857,"k":"volatile","v":0.40625,"x":0.93304,"p":[[0,17,0.0,0.58928,0.14174,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,22,0,0,3,0,0,2,0,1],[4,17,0.2353,0.86607,0.22286,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,2,0,22],[8,17,0.4706,0.93304,0.2172,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[12,17,0.7059,0.54464,0.24338,0.42857,0.42857,0.42857,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,23,0,0,0,0,0,0,0,0,0,0,7],[16,17,0.9412,0.45536,0.1448,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,2],[17,17,1.0,0.40625,0.05187,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.42857,"k":"flat","v":0.53571,"x":0.87054,"p":[[0,22,0.0,0.59374,0.26271,0.53539,0.57143,0.71429,0.0,1.0,2,5,1,2,0,1,0,0,2,0,0,3,0,0,13,0,0,4,0,0,2,0,5],[4,22,0.1818,0.70982,0.26119,0.57143,0.78571,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,2,0,0,7,0,9],[8,22,0.3636,0.78125,0.21424,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,3,0,0,6,0,12],[12,22,0.5455,0.87054,0.21535,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,4,0,21],[16,22,0.7273,0.84375,0.20935,0.82132,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,8,0,16],[20,22,0.9091,0.78125,0.25997,0.42857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,5,0,16],[22,22,1.0,0.53571,0.22868,0.42857,0.42857,0.46431,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,22,0,0,2,0,0,0,0,0,0,0,6]]}]},{"i":"e4aae4e01e7d0e33","q":"Let  $(R,+,\\cdot)$  be a ring with center  $Z=\\{a\\in\\mathbb{R}:ar=ra,\\forall r\\in\\mathbb{R}\\}$  with the property that the group  $U=U(R)$  of its invertible elements is finite. Given that  $G$  is the group of automorphisms of the additive group  $(R,+),$  prove that \\[|G|\\geq\\frac{|U|^2}{|Z\\cap U|}.\\]*Drago\u0219 Cri\u0219an*","t":[{"b":1,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,15,0.0,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[4,15,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,15,0.5333,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.571,"k":"flat","v":0.89732,"x":0.96875,"p":[[0,12,0.0,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[4,12,0.3333,0.89732,0.18977,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,1,0,24],[8,12,0.6667,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[12,12,1.0,0.92856,0.16754,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,26]]}]},{"i":"a589996d42e2a768","q":"In a convex quadrilateral  $ABCD$  it is given that  $\\angle{CAB} = 40^{\\circ}, \\angle{CAD} = 30^{\\circ}, \\angle{DBA} = 75^{\\circ}$ , and  $\\angle{DBC}=25^{\\circ}$ . Find  $\\angle{BDC}$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.40179,"x":0.41974,"p":[[0,21,0.0,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.41518,0.07457,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[8,21,0.381,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.41974,0.04973,0.42857,0.42857,0.4286,0.14286,0.43,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.40179,0.08328,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.40616,"x":0.42862,"p":[[0,23,0.0,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.41076,0.06917,0.42857,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.41965,0.04971,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.41964,0.04971,0.42857,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.40616,0.07273,0.42857,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"583715e91b13d581","q":"In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. \nProve that each participant has lost a different number of games.","t":[{"b":5,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,20,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,20,0.2,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.88393,"x":1.0,"p":[[0,13,0.0,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[4,13,0.3077,0.88393,0.18707,0.67857,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,0,0,23],[8,13,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"17df199a90482750","q":"Let  $ f(x)$  be a polynomial of second degree the roots of which are contained in the interval  $ [\\minus{}1,\\plus{}1]$  and let there be a point  $ x_0\\in [\\minus{}1.\\plus{}1]$  such that  $ |f(x_0)|\\equal{}1$ . Prove that for every  $ \\alpha \\in [0,1]$ , there exists a  $ \\zeta \\in [\\minus{}1,\\plus{}1]$  such that  $ |f'(\\zeta)|\\equal{}\\alpha$  and that this statement is not true if  $ \\alpha>1$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.80133,"x":0.91071,"p":[[0,42,0.0,0.80133,0.15124,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,3,2,0,2,0,0,18,1,4],[4,42,0.0952,0.91071,0.09942,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,15],[8,42,0.1905,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[12,42,0.2857,0.89955,0.08904,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,19,1,11],[16,42,0.381,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[20,42,0.4762,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[24,42,0.5714,0.89733,0.06665,0.85714,0.85714,1.0,0.786,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,21,1,9],[28,42,0.6667,0.87498,0.07789,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,6],[32,42,0.7619,0.87053,0.04164,0.85714,0.85714,0.85714,0.8571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[36,42,0.8571,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[40,42,0.9524,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[42,42,1.0,0.89289,0.05926,0.85714,0.85714,0.92893,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,2,7]]},{"b":6,"e":0.57143,"k":"flat","v":0.8058,"x":0.91741,"p":[[0,27,0.0,0.82143,0.08747,0.83918,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,4,1,0,22,1,1],[4,27,0.1481,0.88393,0.08884,0.85714,0.85714,0.94643,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,20,2,8],[8,27,0.2963,0.8817,0.08107,0.85714,0.85714,0.875,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,23,1,7],[12,27,0.4444,0.91741,0.07386,0.85714,0.85714,1.0,0.78571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,17,0,14],[16,27,0.5926,0.87499,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[20,27,0.7407,0.89284,0.08753,0.85714,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10],[24,27,0.8889,0.8415,0.11539,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,1,0,21,0,5],[27,27,1.0,0.8058,0.16193,0.67857,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,1,0,10,0,9]]}]},{"i":"d1ae52180087be68","q":"Let  $A$  be a set of  $N$  residues  $\\pmod{N^2}$ . Prove that there exists a set  $B$  of  $N$  residues  $\\pmod{N^2}$  such that the set  $A+B=\\{a+b \\vert a \\in A, b \\in B \\}$  contains at least half of all the residues  $\\pmod{N^2}$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.81688,"x":0.99554,"p":[[0,12,0.0,0.81688,0.34687,0.96429,1.0,1.0,0.0,1.0,3,24,2,3,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,24],[4,12,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,12,0.6667,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[12,12,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,14,0.0,0.89732,0.26302,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[4,14,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,14,0.5714,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,14,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[14,14,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"0f5ce06e86a03fe3","q":"Let  $A,B\\in\\mathcal{M}_{2}(\\mathbb{R})$  (real  $2\\times 2$  matrices), that satisfy  $A^{2}+B^{2}=AB$ . Prove that  $(AB-BA)^{2}=O_{2}$ .","t":[{"b":5,"e":0.42857,"k":"rising","v":0.34374,"x":0.74105,"p":[[0,112,0.0,0.45086,0.32752,0.14286,0.42857,0.57143,0.0,1.0,2,5,0,2,0,11,0,0,1,0,0,3,0,0,8,0,0,0,0,0,2,0,5],[4,112,0.0357,0.71866,0.29788,0.57143,0.85714,1.0,0.14,1.0,0,11,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,1,0,0,7,0,11],[8,112,0.0714,0.74105,0.24338,0.57143,0.857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,3,0,0,7,0,10],[12,112,0.1071,0.69195,0.32165,0.42857,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,5,0,0,5,0,0,0,0,0,4,0,13],[16,112,0.1429,0.63838,0.29878,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,3,0,0,1,0,0,5,0,0,8,0,0,2,0,0,3,0,9],[20,112,0.1786,0.61159,0.30143,0.42857,0.57143,0.85714,0.0,1.0,1,6,0,1,0,4,0,0,1,0,0,6,0,0,6,0,0,1,0,0,7,0,6],[24,112,0.2143,0.65177,0.31122,0.57132,0.57143,1.0,0.0,1.0,2,9,0,2,0,3,0,0,0,0,0,2,0,0,11,0,0,0,0,0,5,0,9],[28,112,0.25,0.66067,0.29828,0.42857,0.57143,1.0,0.0,1.0,1,10,0,1,0,2,0,0,2,0,0,4,0,0,9,0,0,0,0,0,4,0,10],[32,112,0.2857,0.58033,0.33491,0.28571,0.57143,0.89286,0.0,1.0,2,8,0,2,0,5,0,0,2,0,0,3,0,0,8,0,0,0,0,0,4,0,8],[36,112,0.3214,0.47322,0.3489,0.14286,0.42857,0.85714,0.0,1.0,2,6,0,2,0,11,0,0,1,0,0,4,0,0,4,0,0,1,0,0,3,0,6],[40,112,0.3571,0.54015,0.33069,0.2857,0.571,0.85714,0.0,1.0,3,7,0,3,0,4,0,0,2,0,0,6,0,0,7,0,0,0,0,0,3,0,7],[44,112,0.3929,0.49999,0.30722,0.14286,0.57121,0.57143,0.14286,1.0,0,6,0,0,0,10,0,0,1,0,0,4,0,0,10,0,0,0,0,0,1,0,6],[48,112,0.4286,0.34374,0.27165,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,8,0,0,4,0,0,8,0,0,3,0,0,0,0,0,3,0,1],[52,112,0.4643,0.46873,0.32777,0.14286,0.4998,0.71429,0.0,1.0,5,3,0,5,0,6,0,0,0,0,0,5,0,0,6,0,0,3,0,0,4,0,3],[56,112,0.5,0.6339,0.29868,0.42857,0.64286,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,2,0,0,7,0,0,4,0,0,3,0,0,6,0,7],[60,112,0.5357,0.42855,0.31338,0.14286,0.42857,0.57143,0.0,1.0,6,3,0,6,0,4,0,0,2,0,0,7,0,0,7,0,0,0,0,0,3,0,3],[64,112,0.5714,0.41962,0.35523,0.10714,0.42857,0.57143,0.0,1.0,8,5,0,8,0,5,0,0,1,0,0,4,0,0,7,0,0,0,0,0,2,0,5],[68,112,0.6071,0.375,0.33834,0.10714,0.42857,0.60714,0.0,1.0,8,3,0,8,0,7,0,0,0,0,0,8,0,0,1,0,0,2,0,0,3,0,3],[72,112,0.6429,0.44643,0.32291,0.14286,0.42857,0.71429,0.0,1.0,5,4,0,5,0,6,0,0,0,0,0,9,0,0,2,0,0,5,0,0,1,0,4],[76,112,0.6786,0.54908,0.36962,0.14286,0.57121,0.85714,0.0,1.0,6,6,0,6,0,3,0,0,1,0,0,4,0,0,3,0,0,2,0,0,7,0,6],[80,112,0.7143,0.59819,0.37701,0.25,0.57143,1.0,0.0,1.0,5,10,0,5,0,3,0,0,1,0,0,3,0,0,5,0,0,0,0,0,5,0,10],[84,112,0.75,0.48212,0.36201,0.14286,0.42857,0.85714,0.0,1.0,5,6,0,5,0,7,0,0,0,0,0,5,0,0,5,0,0,0,0,0,4,0,6],[88,112,0.7857,0.52233,0.33428,0.28571,0.42859,0.85714,0.0,1.0,4,7,0,4,0,3,0,0,2,0,0,8,0,0,5,0,0,1,0,0,2,0,7],[92,112,0.8214,0.47768,0.31259,0.14289,0.42857,0.64286,0.0,1.0,2,4,0,2,0,8,0,0,1,0,0,7,0,0,6,0,0,0,0,0,4,0,4],[96,112,0.8571,0.44643,0.33455,0.14286,0.42857,0.75,0.0,1.0,6,3,0,6,0,5,0,0,1,0,0,7,0,0,4,0,0,1,0,0,5,0,3],[100,112,0.8929,0.60267,0.26662,0.42857,0.5,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,13,0,0,3,0,0,2,0,0,6,0,5],[104,112,0.9286,0.68304,0.23619,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,13,0,0,3,0,0,0,0,0,10,0,6],[108,112,0.9643,0.66072,0.21354,0.42857,0.78571,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,1,0,0,15,0,1],[112,112,1.0,0.63392,0.22285,0.42857,0.4286,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,1,0,0,12,0,2]]},{"b":7,"e":0.71429,"k":"flat","v":0.33929,"x":0.76786,"p":[[0,134,0.0,0.44642,0.32683,0.14286,0.42857,0.57143,0.0,1.0,3,6,0,3,0,8,0,0,3,0,0,5,0,0,6,0,0,1,0,0,0,0,6],[4,134,0.0299,0.63839,0.3174,0.42857,0.57143,1.0,0.0,1.0,2,9,0,2,0,3,0,0,1,0,0,3,0,0,8,0,0,2,0,0,4,0,9],[8,134,0.0597,0.72319,0.2574,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,3,0,0,11,0,0,0,0,0,5,0,11],[12,134,0.0896,0.62052,0.33045,0.42857,0.64286,0.89286,0.0,1.0,3,8,0,3,0,3,0,0,0,0,0,5,0,0,5,0,0,3,0,0,5,0,8],[16,134,0.1194,0.65625,0.2693,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,10,0,0,5,0,0,1,0,0,4,0,9],[20,134,0.1493,0.76786,0.22232,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,1,0,0,4,0,13],[24,134,0.1791,0.74107,0.32818,0.42859,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,0,0,0,5,0,0,2,0,0,0,0,0,6,0,15],[28,134,0.209,0.6607,0.35491,0.42859,0.85714,1.0,0.0,1.0,3,12,0,3,0,3,0,0,1,0,0,3,0,0,5,0,0,0,0,0,5,0,12],[32,134,0.2388,0.62052,0.33045,0.42857,0.57143,1.0,0.0,1.0,3,10,0,3,0,2,0,0,0,0,0,7,0,0,7,0,0,0,0,0,3,0,10],[36,134,0.2687,0.71874,0.28005,0.57142,0.78571,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,3,0,0,9,0,0,1,0,0,4,0,12],[40,134,0.2985,0.63838,0.29663,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,1,0,0,7,0,0,6,0,0,1,0,0,4,0,9],[44,134,0.3284,0.61604,0.28669,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,1,0,0,6,0,0,10,0,0,1,0,0,1,0,9],[48,134,0.3582,0.68302,0.25936,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,9,0,0,7,0,0,0,0,0,7,0,8],[52,134,0.3881,0.75892,0.26831,0.57132,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,6,0,0,6,0,0,0,0,0,5,0,14],[56,134,0.4179,0.625,0.31288,0.42857,0.57143,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,0,0,0,11,0,0,2,0,0,1,0,0,6,0,8],[60,134,0.4478,0.70982,0.24868,0.53572,0.78571,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,6,0,0,4,0,0,4,0,0,9,0,7],[64,134,0.4776,0.74997,0.23421,0.57143,0.857,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,0,6,0,11],[68,134,0.5075,0.75,0.31542,0.53571,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,6,0,14],[72,134,0.5373,0.73212,0.27376,0.571,0.78571,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,5,0,0,8,0,0,1,0,0,2,0,14],[76,134,0.5672,0.74106,0.26351,0.57132,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,5,0,0,5,0,0,2,0,0,7,0,11],[80,134,0.597,0.6875,0.32623,0.42857,0.85714,1.0,0.0,1.0,2,11,0,2,0,2,0,0,1,0,0,6,0,0,2,0,0,1,0,0,7,0,11],[84,134,0.6269,0.71874,0.23552,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,0,5,0,10],[88,134,0.6567,0.62054,0.36,0.42857,0.57143,1.0,0.0,1.0,5,10,0,5,0,1,0,0,1,0,0,4,0,0,6,0,0,0,0,0,5,0,10],[92,134,0.6866,0.70982,0.29984,0.42859,0.78571,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,8,0,0,4,0,0,2,0,0,3,0,13],[96,134,0.7164,0.57588,0.3204,0.42857,0.49979,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,2,0,0,9,0,0,4,0,0,1,0,0,3,0,8],[100,134,0.7463,0.65179,0.36059,0.42859,0.85714,1.0,0.0,1.0,5,10,0,5,0,1,0,0,1,0,0,2,0,0,5,0,0,1,0,0,7,0,10],[104,134,0.7761,0.60712,0.36421,0.42857,0.57143,1.0,0.0,1.0,5,10,0,5,0,1,0,0,1,0,0,7,0,0,3,0,0,0,0,0,5,0,10],[108,134,0.806,0.62053,0.30848,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,0,0,0,7,0,0,4,0,0,1,0,0,7,0,7],[112,134,0.8358,0.58034,0.33108,0.39286,0.4998,0.85714,0.0,1.0,2,7,0,2,0,4,0,0,2,0,0,8,0,0,2,0,0,1,0,0,6,0,7],[116,134,0.8657,0.64286,0.34256,0.39286,0.71429,1.0,0.0,1.0,1,11,0,1,0,5,0,0,2,0,0,4,0,0,4,0,0,0,0,0,5,0,11],[120,134,0.8955,0.47756,0.34377,0.14286,0.42857,0.85704,0.0,1.0,5,5,0,5,0,5,0,0,1,0,0,8,0,0,3,0,0,1,0,0,4,0,5],[124,134,0.9254,0.50892,0.31529,0.39285,0.57143,0.74996,0.0,1.0,6,2,0,6,0,1,0,0,1,0,0,7,0,0,5,0,0,4,0,0,6,0,2],[128,134,0.9552,0.43292,0.31647,0.14286,0.42857,0.57143,0.0,1.0,5,5,0,5,0,6,0,0,0,0,0,8,0,0,8,0,0,0,0,0,0,0,5],[132,134,0.9851,0.48661,0.33476,0.14286,0.42857,0.85714,0.0,1.0,5,4,0,5,0,4,0,0,0,0,0,11,0,0,2,0,0,0,0,0,6,0,4],[134,134,1.0,0.33929,0.29179,0.0,0.35714,0.42858,0.0,1.0,9,1,0,9,0,3,0,0,4,0,0,9,0,0,2,0,0,1,0,0,3,0,1]]}]},{"i":"6e4eda95d4ac60f6","q":"Let  $0!!=1!!=1$  and  $n!!=n\\cdot (n-2)!!$  for all integers  $n\\geq 2$ . Find all positive integers  $n$  such that \n\\[\\dfrac{(2^n+1)!!-1}{2^{n+1}}\\]\nis an integer.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.4732,"x":0.75446,"p":[[0,79,0.0,0.4732,0.36147,0.14286,0.35714,0.78571,0.0,1.0,2,8,0,2,0,11,0,0,3,0,0,2,0,0,3,0,0,3,0,0,0,0,8],[4,79,0.0506,0.72321,0.32525,0.39286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,6,0,0,3,0,0,1,0,0,1,0,0,3,0,16],[8,79,0.1013,0.7232,0.30502,0.42857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,5,0,0,5,0,0,2,0,0,1,0,0,3,0,15],[12,79,0.1519,0.70982,0.31438,0.39286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,7,0,0,3,0,0,2,0,0,2,0,0,2,0,15],[16,79,0.2025,0.62052,0.30433,0.28571,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,6,0,0,5,0,0,2,0,0,2,0,0,7,0,7],[20,79,0.2532,0.59813,0.29559,0.42857,0.57143,0.89286,0.14,1.0,0,8,0,0,0,4,0,0,3,0,0,7,0,0,5,0,0,3,0,0,2,0,8],[24,79,0.3038,0.75446,0.30563,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,4,0,0,5,0,0,0,0,0,2,0,0,4,0,16],[28,79,0.3544,0.71874,0.25124,0.4286,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,3,0,0,4,0,11],[32,79,0.4051,0.56696,0.31841,0.28571,0.5,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,9,0,0,3,0,0,2,0,0,4,0,0,2,0,8],[36,79,0.4557,0.62051,0.28033,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,4,0,0,4,0,0,7,0,0,2,0,0,6,0,6],[40,79,0.5063,0.65622,0.30276,0.42859,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,3,0,0,3,0,0,5,0,0,3,0,0,5,0,9],[44,79,0.557,0.63838,0.23415,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,7,0,0,8,0,0,4,0,0,5,0,5],[48,79,0.6076,0.58929,0.3004,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,5,0,0,4,0,0,4,0,0,4,0,0,6,0,0,2,0,7],[52,79,0.6582,0.6205,0.27804,0.39286,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,5,0,0,2,0,0,7,0,0,3,0,0,7,0,5],[56,79,0.7089,0.52232,0.29148,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,4,0,0,10,0,0,3,0,0,3,0,0,5,0,0,2,0,5],[60,79,0.7595,0.57142,0.26726,0.39286,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,5,0,0,4,0,0,10,0,0,3,0,0,1,0,6],[64,79,0.8101,0.64286,0.29667,0.39286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,4,0,0,2,0,0,4,0,0,5,0,0,6,0,7],[68,79,0.8608,0.65174,0.26713,0.53539,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,2,0,0,9,0,0,3,0,0,5,0,7],[72,79,0.9114,0.5,0.26964,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,8,0,0,7,0,0,4,0,0,3,0,0,2,0,4],[76,79,0.962,0.5089,0.25984,0.28571,0.4998,0.60714,0.14286,1.0,0,3,0,0,0,5,0,0,5,0,0,6,0,0,8,0,0,2,0,0,3,0,3],[79,79,1.0,0.48214,0.22232,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,9,0,0,8,0,0,7,0,0,2,0,0,2,0,2]]},{"b":4,"e":0.14286,"k":"flat","v":0.30804,"x":0.72768,"p":[[0,76,0.0,0.43302,0.33972,0.14286,0.28571,0.74996,0.0,1.0,2,4,0,2,0,12,0,0,4,0,0,1,0,0,3,0,0,2,0,0,4,0,4],[4,76,0.0526,0.69643,0.28959,0.42857,0.78571,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,2,0,0,5,0,11],[8,76,0.1053,0.72768,0.31412,0.39286,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,6,0,0,2,0,0,0,0,0,3,0,0,5,0,14],[12,76,0.1579,0.6741,0.30978,0.28571,0.64286,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,9,0,0,3,0,0,4,0,0,2,0,0,0,0,14],[16,76,0.2105,0.55803,0.29957,0.28571,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,9,0,0,5,0,0,2,0,0,3,0,0,4,0,6],[20,76,0.2632,0.5982,0.2911,0.28571,0.57121,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,7,0,0,6,0,0,3,0,0,3,0,0,4,0,7],[24,76,0.3158,0.55802,0.27049,0.28571,0.42857,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,9,0,0,8,0,0,1,0,0,5,0,0,3,0,5],[28,76,0.3684,0.55357,0.27141,0.28571,0.42857,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,8,0,0,7,0,0,3,0,0,3,0,0,5,0,4],[32,76,0.4211,0.52231,0.29797,0.2857,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,5,0,0,8,0,0,4,0,0,4,0,0,4,0,0,1,0,6],[36,76,0.4737,0.56247,0.32328,0.2857,0.49979,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,6,0,0,5,0,0,3,0,0,3,0,0,2,0,8],[40,76,0.5263,0.49552,0.24218,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,9,0,0,6,0,0,2,0,0,2,0,3],[44,76,0.5789,0.53125,0.27254,0.28571,0.42857,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,10,0,0,10,0,0,1,0,0,0,0,0,6,0,4],[48,76,0.6316,0.47326,0.24073,0.28571,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,3,0,0,10,0,0,7,0,0,4,0,0,5,0,0,0,0,3],[52,76,0.6842,0.3616,0.19557,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,13,0,0,9,0,0,1,0,0,1,0,0,1,0,1],[56,76,0.7368,0.50446,0.27429,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,10,0,0,7,0,0,2,0,0,3,0,0,3,0,4],[60,76,0.7895,0.48218,0.2569,0.28571,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,2,0,0,11,0,0,10,0,0,1,0,0,1,0,0,4,0,3],[64,76,0.8421,0.51784,0.27374,0.28571,0.42857,0.75,0.14286,1.0,0,4,0,0,0,2,0,0,11,0,0,6,0,0,3,0,0,2,0,0,4,0,4],[68,76,0.8947,0.48659,0.30485,0.2857,0.42857,0.71429,0.0,1.0,1,5,1,1,0,6,0,0,6,0,0,7,0,0,2,0,0,3,0,0,2,0,5],[72,76,0.9474,0.55357,0.31084,0.28571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,7,0,0,2,0,0,2,0,0,4,0,0,7,0,4],[76,76,1.0,0.30804,0.18595,0.14286,0.2857,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,9,0,0,9,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"4539be577a80e5d3","q":"Let  $ABC$  be a triangle and  $AD,BE,CF$  be cevians concurrent at a point  $P$ . Suppose each of the quadrilaterals  $PDCE,PEAF$  and  $PFBD$  has both circumcircle and incircle. Prove that  $ABC$  is equilateral and  $P$  coincides with the center of the triangle.","t":[{"b":1,"e":0.14,"k":"falling","v":0.33929,"x":0.67409,"p":[[0,72,0.0,0.67409,0.29717,0.57132,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,3,0,0,0,0,0,4,0,0,8,0,0,5,0,8],[4,72,0.0556,0.6339,0.33682,0.28571,0.57143,1.0,0.0,1.0,1,13,0,1,0,3,0,0,5,0,0,3,0,0,6,0,0,1,0,0,0,0,13],[8,72,0.1111,0.56693,0.32632,0.39286,0.57143,0.78571,0.0,1.0,3,8,0,3,0,4,0,0,1,0,0,3,0,0,9,0,0,4,0,0,0,0,8],[12,72,0.1667,0.62944,0.274,0.42857,0.57143,0.78571,0.0,1.0,1,8,0,1,0,2,0,0,1,0,0,6,0,0,7,0,0,7,0,0,0,0,8],[16,72,0.2222,0.51337,0.27859,0.39285,0.571,0.57143,0.0,1.0,2,5,0,2,0,3,0,0,3,0,0,7,0,0,11,0,0,0,0,0,1,0,5],[20,72,0.2778,0.66515,0.28033,0.571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,0,0,0,3,0,0,10,0,0,4,0,0,1,0,10],[24,72,0.3333,0.57142,0.30514,0.28571,0.57143,0.71429,0.0,1.0,1,7,0,1,0,5,0,0,3,0,0,3,0,0,6,0,0,7,0,0,0,0,7],[28,72,0.3889,0.62484,0.21938,0.42859,0.57143,0.75,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,8,0,0,11,0,0,4,0,0,4,0,4],[32,72,0.4444,0.58925,0.24157,0.42857,0.57143,0.57143,0.14286,1.0,0,6,0,0,0,2,0,0,3,0,0,4,0,0,16,0,0,0,0,0,1,0,6],[36,72,0.5,0.6205,0.25407,0.42859,0.57143,0.85714,0.1429,1.0,0,6,0,0,0,3,0,0,1,0,0,5,0,0,11,0,0,3,0,0,3,0,6],[40,72,0.5556,0.51784,0.26666,0.28571,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,6,0,0,4,0,0,3,0,0,11,0,0,3,0,0,1,0,4],[44,72,0.6111,0.64729,0.20199,0.5354,0.64286,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,10,0,0,1,0,5],[48,72,0.6667,0.58035,0.25985,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,2,0,0,7,0,0,9,0,0,4,0,0,2,0,5],[52,72,0.7222,0.51335,0.20157,0.42857,0.57121,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,1,0,0,8,0,0,14,0,0,3,0,0,0,0,2],[56,72,0.7778,0.55802,0.22967,0.42859,0.57143,0.60714,0.0,1.0,1,3,0,1,0,2,0,0,1,0,0,7,0,0,13,0,0,3,0,0,2,0,3],[60,72,0.8333,0.5714,0.18898,0.4286,0.57143,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,5,0,0,15,0,0,4,0,0,2,0,2],[64,72,0.8889,0.37945,0.14553,0.2857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,6,0,0,6,0,0,13,0,0,7,0,0,0,0,0,0,0,0],[68,72,0.9444,0.33929,0.15464,0.1429,0.42857,0.42858,0.0,0.57143,1,0,0,1,0,8,0,0,5,0,0,14,0,0,4,0,0,0,0,0,0,0,0],[72,72,1.0,0.41964,0.15124,0.28592,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,4,0,0,11,0,0,12,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"falling","v":0.33481,"x":0.66513,"p":[[0,76,0.0,0.66513,0.35105,0.2857,0.85707,1.0,0.0,1.0,1,12,0,1,0,5,0,0,4,0,0,0,0,0,3,0,0,2,0,0,5,0,12],[4,76,0.0526,0.64281,0.31543,0.53539,0.57143,1.0,0.0,1.0,2,10,0,2,0,2,0,0,3,0,0,1,0,0,9,0,0,3,0,0,2,0,10],[8,76,0.1053,0.54005,0.3047,0.28571,0.57121,0.71429,0.0,1.0,1,6,0,1,0,6,0,0,3,0,0,3,0,0,8,0,0,4,0,0,1,0,6],[12,76,0.1579,0.54461,0.25861,0.39286,0.57121,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,4,0,0,6,0,0,8,0,0,4,0,0,2,0,4],[16,76,0.2105,0.58032,0.2878,0.39286,0.57143,0.75,0.0,1.0,2,6,0,2,0,1,0,0,5,0,0,3,0,0,9,0,0,4,0,0,2,0,6],[20,76,0.2632,0.50441,0.25248,0.42857,0.571,0.60714,0.0,1.0,3,2,0,3,0,2,0,0,2,0,0,6,0,0,11,0,0,5,0,0,1,0,2],[24,76,0.3158,0.49103,0.26228,0.2857,0.571,0.57143,0.0,1.0,1,3,0,1,0,5,0,0,5,0,0,3,0,0,11,0,0,3,0,0,1,0,3],[28,76,0.3684,0.46874,0.24283,0.28571,0.4286,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,4,0,0,7,0,0,9,0,0,3,0,0,1,0,2],[32,76,0.4211,0.49551,0.26721,0.2857,0.571,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,4,0,0,5,0,0,7,0,0,6,0,0,2,0,2],[36,76,0.4737,0.52675,0.30604,0.28571,0.57143,0.60714,0.0,1.0,4,5,0,4,0,2,0,0,3,0,0,2,0,0,13,0,0,1,0,0,2,0,5],[40,76,0.5263,0.43304,0.25121,0.24999,0.42859,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,5,0,0,6,0,0,8,0,0,2,0,0,1,0,2],[44,76,0.5789,0.50894,0.27647,0.28571,0.571,0.71407,0.0,1.0,1,4,0,1,0,5,0,0,4,0,0,5,0,0,8,0,0,4,0,0,1,0,4],[48,76,0.6316,0.41962,0.25983,0.2857,0.42857,0.571,0.0,1.0,2,3,0,2,0,5,0,0,6,0,0,10,0,0,5,0,0,0,0,0,1,0,3],[52,76,0.6842,0.42846,0.24493,0.25,0.4998,0.57143,0.0,1.0,4,1,0,4,0,4,0,0,2,0,0,6,0,0,12,0,0,3,0,0,0,0,1],[56,76,0.7368,0.41515,0.23243,0.2857,0.42859,0.57143,0.0,1.0,4,1,0,4,0,3,0,0,3,0,0,9,0,0,10,0,0,2,0,0,0,0,1],[60,76,0.7895,0.49101,0.234,0.39286,0.571,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,5,0,0,13,0,0,4,0,0,0,0,2],[64,76,0.8421,0.40169,0.27309,0.14286,0.42857,0.57143,0.0,1.0,4,3,0,4,0,6,0,0,2,0,0,9,0,0,8,0,0,0,0,0,0,0,3],[68,76,0.8947,0.36596,0.22004,0.1429,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,6,0,0,4,0,0,6,0,0,10,0,0,2,0,0,0,0,0],[72,76,0.9474,0.39283,0.26243,0.25,0.42857,0.4642,0.0,1.0,4,3,0,4,0,4,0,0,5,0,0,11,0,0,5,0,0,0,0,0,0,0,3],[76,76,1.0,0.33481,0.19757,0.1429,0.42857,0.4642,0.0,0.57143,5,0,0,5,0,4,0,0,6,0,0,9,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"7d77787fcbdb0743","q":"Let  $ABC$  be a triangle and  $H$  and  $D$  be the feet of the height and bisector relative to  $A$  in  $BC$ , respectively. Let  $E$  be the intersection of the tangent to the circumcircle of  $ABC$  by  $A$  with  $BC$  and  $M$  be the midpoint of  $AD$ . Finally, let  $r$  be the line perpendicular to  $BC$  that passes through  $M$ . Show that  $r$  is tangent to the circumcircle of  $AHE$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.25893,"x":0.53123,"p":[[0,61,0.0,0.40177,0.31427,0.14286,0.2857,0.60714,0.0,1.0,3,3,0,3,0,9,0,0,7,0,0,2,0,0,3,0,0,2,0,0,3,0,3],[4,61,0.0656,0.29464,0.26471,0.10714,0.2857,0.4286,0.0,0.85714,8,0,0,8,0,6,0,0,8,0,0,3,0,0,3,0,0,1,0,0,3,0,0],[8,61,0.1311,0.40148,0.23794,0.14289,0.42857,0.57111,0.0,0.85714,3,0,0,3,0,6,0,0,5,0,0,4,0,0,9,0,0,4,0,0,1,0,0],[12,61,0.1967,0.25893,0.22711,0.0,0.21428,0.42857,0.0,0.71429,9,0,0,9,0,7,0,0,4,0,0,8,0,0,1,0,0,3,0,0,0,0,0],[16,61,0.2623,0.42852,0.25997,0.25,0.42857,0.60714,0.0,0.85714,3,0,0,3,0,5,0,0,6,0,0,4,0,0,6,0,0,5,0,0,3,0,0],[20,61,0.3279,0.46872,0.1758,0.28571,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,6,0,0,8,0,0,9,0,0,6,0,0,0,0,0],[24,61,0.3934,0.51781,0.20747,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,6,0,0,9,0,0,8,0,0,2,0,0],[28,61,0.459,0.4955,0.22298,0.28571,0.57121,0.71429,0.0,0.85714,2,0,0,2,0,2,0,0,5,0,0,4,0,0,9,0,0,9,0,0,1,0,0],[32,61,0.5246,0.53123,0.20896,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,6,0,0,8,0,0,8,0,0,3,0,0],[36,61,0.5902,0.43746,0.19209,0.28571,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,0,0,0,6,0,0,10,0,0,9,0,0,4,0,0,0,0,0],[40,61,0.6557,0.46871,0.21198,0.42857,0.4998,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,1,0,0,10,0,0,9,0,0,7,0,0,0,0,0],[44,61,0.7213,0.46426,0.17126,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,6,0,0,8,0,0,10,0,0,5,0,0,0,0,0],[48,61,0.7869,0.48658,0.17074,0.39286,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,7,0,0,12,0,0,4,0,0,1,0,0],[52,61,0.8525,0.5133,0.18506,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,7,0,0,12,0,0,5,0,0,2,0,0],[56,61,0.918,0.45978,0.18115,0.28571,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,5,0,0,9,0,0,8,0,0,6,0,0,0,0,0],[60,61,0.9836,0.45973,0.17756,0.28571,0.4286,0.571,0.14286,0.85714,0,0,0,0,0,3,0,0,6,0,0,10,0,0,8,0,0,4,0,0,1,0,0],[61,61,1.0,0.49551,0.18892,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,6,0,0,13,0,0,5,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.11152,"x":0.61157,"p":[[0,92,0.0,0.48214,0.33455,0.24999,0.42857,0.85714,0.0,1.0,4,4,0,4,0,4,0,0,6,0,0,5,0,0,1,0,0,3,0,0,5,0,4],[4,92,0.0435,0.61157,0.27487,0.53539,0.64286,0.857,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,1,0,0,8,0,0,7,0,0,5,0,4],[8,92,0.087,0.45982,0.28955,0.2857,0.42859,0.71429,0.0,1.0,4,1,0,4,0,3,0,0,6,0,0,5,0,0,3,0,0,6,0,0,4,0,1],[12,92,0.1304,0.43299,0.24347,0.25,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,7,0,0,5,0,0,6,0,0,7,0,0,2,0,0,4,0,0],[16,92,0.1739,0.3571,0.2474,0.14289,0.28571,0.57036,0.0,0.85714,4,0,0,4,0,6,0,0,8,0,0,5,0,0,5,0,0,1,0,0,3,0,0],[20,92,0.2174,0.3348,0.28926,0.10714,0.2857,0.57143,0.0,1.0,8,1,0,8,0,5,0,0,6,0,0,3,0,0,5,0,0,2,0,0,2,0,1],[24,92,0.2609,0.3973,0.25185,0.14289,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,5,0,0,1,0,0,9,0,0,6,0,0,5,0,0,1,0,0],[28,92,0.3043,0.45972,0.27147,0.2857,0.42859,0.71429,0.0,0.85714,4,0,0,4,0,3,0,0,4,0,0,6,0,0,5,0,0,6,0,0,4,0,0],[32,92,0.3478,0.29006,0.23823,0.14214,0.28571,0.4642,0.0,0.85714,7,0,0,7,0,8,0,0,5,0,0,4,0,0,6,0,0,1,0,0,1,0,0],[36,92,0.3913,0.22321,0.22851,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,5,0,0,6,0,0,6,0,0,1,0,0,1,0,0,1,0,0],[40,92,0.4348,0.23659,0.21009,0.0,0.2857,0.32143,0.0,0.71429,10,0,0,10,0,5,0,0,9,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[44,92,0.4783,0.20069,0.22264,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,11,0,0,3,0,0,2,0,0,4,0,0,0,0,0,1,0,0],[48,92,0.5217,0.25,0.26,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,4,0,0,3,0,0,6,0,0,3,0,0,2,0,0,1,0,0],[52,92,0.5652,0.17409,0.19142,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,11,0,0,3,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[56,92,0.6087,0.27222,0.20004,0.14286,0.2143,0.42857,0.0,0.71429,4,0,0,4,0,12,0,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[60,92,0.6522,0.26339,0.19269,0.14286,0.21428,0.42857,0.0,0.71429,4,0,0,4,0,12,0,0,7,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[64,92,0.6957,0.1383,0.0977,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,20,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,92,0.7391,0.22322,0.16728,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,13,0,0,8,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[72,92,0.7826,0.17411,0.12745,0.14286,0.14286,0.2857,0.0,0.4286,7,0,0,7,0,14,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,92,0.8261,0.19643,0.17768,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,13,0,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[80,92,0.8696,0.17409,0.18806,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,15,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[84,92,0.913,0.21426,0.21124,0.0,0.14286,0.42857,0.0,0.57143,11,0,0,11,0,9,0,0,2,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[88,92,0.9565,0.16963,0.17287,0.0,0.14286,0.1786,0.0,0.57143,10,0,0,10,0,14,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[92,92,1.0,0.11152,0.12743,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"4e5189577e7b1a8c","q":"Let  $ABC$  be a triangle with  $\\angle BAC \\neq 90^{\\circ}.$  Let  $O$  be the circumcenter of the triangle  $ABC$  and  $\\Gamma$  be the circumcircle of the triangle  $BOC.$  Suppose that  $\\Gamma$  intersects the line segment  $AB$  at  $P$  different from  $B$ , and the line segment  $AC$  at  $Q$  different from  $C.$  Let  $ON$  be the diameter of the circle  $\\Gamma.$  Prove that the quadrilateral  $APNQ$  is a parallelogram.","t":[{"b":3,"e":0.2857,"k":"rising","v":0.29464,"x":0.59821,"p":[[0,91,0.0,0.35267,0.33498,0.14286,0.14295,0.60682,0.0,1.0,6,3,1,6,0,11,0,0,3,0,0,3,0,0,1,0,0,2,0,0,3,0,3],[4,91,0.044,0.33035,0.29973,0.14286,0.2857,0.28571,0.0,1.0,2,3,0,2,0,13,0,0,11,0,0,0,0,0,0,0,0,0,0,0,3,0,3],[8,91,0.0879,0.29893,0.24065,0.14286,0.2857,0.28571,0.0,1.0,1,2,0,1,0,13,0,0,13,0,0,0,0,0,2,0,0,0,0,0,1,0,2],[12,91,0.1319,0.38391,0.32622,0.14286,0.28571,0.60682,0.0,1.0,3,4,0,3,0,9,0,0,11,0,0,0,0,0,1,0,0,1,0,0,3,0,4],[16,91,0.1758,0.38393,0.33964,0.14286,0.28571,0.32143,0.0,1.0,3,7,0,3,0,8,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[20,91,0.2198,0.53124,0.33925,0.2857,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,11,0,0,1,0,0,1,0,0,1,0,0,5,0,7],[24,91,0.2637,0.39286,0.34069,0.14286,0.2857,0.85714,0.0,1.0,2,4,0,2,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0,5,0,4],[28,91,0.3077,0.39732,0.35667,0.14286,0.2857,0.85714,0.0,1.0,2,6,0,2,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,3,0,6],[32,91,0.3516,0.33929,0.28291,0.14286,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,14,0,0,12,0,0,0,0,0,0,0,0,1,0,0,2,0,3],[36,91,0.3956,0.51339,0.35689,0.24999,0.28571,0.89286,0.14286,1.0,0,8,0,0,0,8,0,0,11,0,0,0,0,0,0,0,0,1,0,0,4,0,8],[40,91,0.4396,0.33034,0.31629,0.14286,0.14288,0.28571,0.0,1.0,2,4,0,2,0,16,0,0,7,0,0,0,0,0,1,0,0,0,0,0,2,0,4],[44,91,0.4835,0.40168,0.36852,0.14286,0.21429,0.85714,0.0,1.0,4,6,0,4,0,12,0,0,5,0,0,0,0,0,2,0,0,0,0,0,3,0,6],[48,91,0.5275,0.40177,0.34706,0.14286,0.28571,0.75,0.0,1.0,4,5,0,4,0,8,0,0,10,0,0,0,0,0,1,0,0,1,0,0,3,0,5],[52,91,0.5714,0.42848,0.35182,0.14286,0.28571,0.85714,0.0,1.0,2,5,0,2,0,11,0,0,8,0,0,0,0,0,0,0,0,2,0,0,4,0,5],[56,91,0.6154,0.41509,0.35065,0.14286,0.28571,0.75,0.0,1.0,2,6,0,2,0,11,0,0,9,0,0,0,0,0,0,0,0,2,0,0,2,0,6],[60,91,0.6593,0.41516,0.31614,0.14286,0.28571,0.64286,0.14286,1.0,0,4,0,0,0,11,0,0,11,0,0,0,0,0,2,0,0,0,0,0,4,0,4],[64,91,0.7033,0.38839,0.32972,0.14286,0.28571,0.64286,0.0,1.0,2,4,0,2,0,11,0,0,10,0,0,0,0,0,1,0,0,0,0,0,4,0,4],[68,91,0.7473,0.38838,0.35216,0.14286,0.2143,0.75,0.0,1.0,3,5,0,3,0,13,0,0,6,0,0,0,0,0,1,0,0,1,0,0,3,0,5],[72,91,0.7912,0.35715,0.28793,0.14286,0.28571,0.32143,0.14286,1.0,0,3,0,0,0,13,0,0,11,0,0,2,0,0,0,0,0,0,0,0,3,0,3],[76,91,0.8352,0.33482,0.28928,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,13,0,0,10,0,0,0,0,0,0,0,0,1,0,0,5,0,1],[80,91,0.8791,0.30357,0.27374,0.14286,0.14288,0.28571,0.0,1.0,1,2,0,1,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,0,3,0,2],[84,91,0.9231,0.29464,0.30291,0.14286,0.14286,0.28571,0.0,1.0,3,4,0,3,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[88,91,0.967,0.49553,0.38793,0.14286,0.28571,0.89286,0.0,1.0,4,8,0,4,0,7,0,0,7,0,0,0,0,0,0,0,0,2,0,0,4,0,8],[91,91,1.0,0.59821,0.39357,0.14286,0.85714,1.0,0.0,1.0,3,10,0,3,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,8,0,10]]},{"b":4,"e":0.14286,"k":"falling","v":0.09811,"x":0.56695,"p":[[0,81,0.0,0.36159,0.3517,0.0,0.28571,0.60714,0.0,1.0,9,3,1,9,0,6,0,0,5,0,0,1,0,0,3,0,0,1,0,0,4,0,3],[4,81,0.0494,0.37053,0.36396,0.14286,0.14286,0.53568,0.0,1.0,3,7,0,3,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[8,81,0.0988,0.45536,0.3597,0.14286,0.28571,0.75,0.0,1.0,3,7,0,3,0,8,0,0,8,0,0,1,0,0,0,0,0,4,0,0,1,0,7],[12,81,0.1481,0.39731,0.32484,0.14286,0.2857,0.71407,0.0,1.0,2,4,0,2,0,10,0,0,10,0,0,1,0,0,0,0,0,2,0,0,3,0,4],[16,81,0.1975,0.33473,0.29588,0.14286,0.2857,0.28571,0.0,1.0,2,4,0,2,0,12,0,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,4],[20,81,0.2469,0.44195,0.36658,0.14286,0.28571,0.85714,0.0,1.0,3,7,0,3,0,9,0,0,8,0,0,1,0,0,1,0,0,0,0,0,3,0,7],[24,81,0.2963,0.32589,0.31791,0.14286,0.14288,0.28571,0.0,1.0,4,3,0,4,0,13,0,0,8,0,0,0,0,0,0,0,0,1,0,0,3,0,3],[28,81,0.3457,0.56695,0.36155,0.24999,0.64264,0.89286,0.0,1.0,1,8,0,1,0,7,0,0,7,0,0,0,0,0,1,0,0,2,0,0,6,0,8],[32,81,0.3951,0.45982,0.3908,0.14286,0.28571,0.89286,0.0,1.0,3,8,0,3,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,4,0,8],[36,81,0.4444,0.24554,0.29501,0.0,0.14286,0.28571,0.0,1.0,9,3,0,9,0,11,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[40,81,0.4938,0.37937,0.36881,0.14286,0.2857,0.46429,0.0,1.0,4,8,0,4,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[44,81,0.5432,0.49778,0.37944,0.14286,0.28571,1.0,0.0,1.0,2,9,0,2,0,9,0,0,7,0,0,1,0,0,0,0,0,1,1,0,2,0,9],[48,81,0.5926,0.50444,0.37793,0.14286,0.28571,1.0,0.0,1.0,3,9,0,3,0,6,0,0,9,0,0,0,0,0,2,0,0,0,0,0,3,0,9],[52,81,0.642,0.45089,0.37134,0.14286,0.2857,0.85714,0.0,1.0,2,6,0,2,0,12,0,0,6,0,0,0,0,0,0,0,0,1,0,0,5,0,6],[56,81,0.6914,0.3124,0.34155,0.105,0.14286,0.32143,0.0,1.0,8,4,0,8,0,10,0,0,6,0,0,1,0,0,0,0,0,1,0,0,2,0,4],[60,81,0.7407,0.2946,0.31727,0.10714,0.14286,0.46418,0.0,1.0,8,3,0,8,0,12,0,0,2,0,0,2,0,0,3,0,0,1,0,0,1,0,3],[64,81,0.7901,0.35715,0.33881,0.14286,0.2143,0.46431,0.0,1.0,5,5,0,5,0,11,0,0,5,0,0,3,0,0,1,0,0,1,0,0,1,0,5],[68,81,0.8395,0.35713,0.35174,0.14286,0.2143,0.60682,0.0,1.0,7,5,0,7,0,9,0,0,5,0,0,2,0,0,1,0,0,2,0,0,1,0,5],[72,81,0.8889,0.24107,0.32818,0.0,0.14286,0.14287,0.0,1.0,11,2,0,11,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,2],[76,81,0.9383,0.33482,0.32264,0.14286,0.2857,0.39286,0.0,1.0,6,3,0,6,0,9,0,0,9,0,0,0,0,0,0,0,0,3,0,0,2,0,3],[80,81,0.9877,0.16964,0.2822,0.0,0.0,0.14286,0.0,1.0,17,2,0,17,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[81,81,1.0,0.09811,0.1904,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,7,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0]]}]},{"i":"8eaadd8ab2e92c45","q":"Let  $ABC$  be a triangle with sides  $51, 52, 53$ . Let  $\\Omega$  denote the incircle of  $\\bigtriangleup ABC$ . Draw tangents to  $\\Omega$  which are parallel to the sides of  $ABC$ . Let  $r_1, r_2, r_3$  be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed  $r_1 + r_2 + r_3$ .","t":[{"b":1,"e":0.28571,"k":"falling","v":0.48214,"x":1.0,"p":[[0,40,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,40,0.1,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,40,0.2,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,40,0.3,0.95982,0.1439,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[16,40,0.4,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[20,40,0.5,0.90178,0.23538,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,26],[24,40,0.6,0.64286,0.35714,0.28571,0.64286,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[28,40,0.7,0.57589,0.35081,0.28571,0.28571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[32,40,0.8,0.48214,0.32488,0.28571,0.28571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[36,40,0.9,0.50893,0.33108,0.28571,0.28571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[40,40,1.0,0.48661,0.32115,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,9]]},{"b":5,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,26,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,26,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"46d666b94ccc8407","q":"Let  $ABC$  be an acute triangle with  $AB\\neq AC$ ,  $M$  be the median of  $BC$ , and  $H$  be the orthocenter of  $\\triangle ABC$ . The circumcircle of  $B$ ,  $H$ , and  $C$  intersects the median  $AM$  at  $N$ . Show that  $\\angle ANH=90^\\circ$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.02679,"x":0.16518,"p":[[0,63,0.0,0.09819,0.19039,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,6,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,63,0.0635,0.10268,0.20277,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,4,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[8,63,0.127,0.16518,0.2372,0.0,0.0,0.17857,0.0,0.71429,17,0,0,17,0,7,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[12,63,0.1905,0.09375,0.19103,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,3,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[16,63,0.254,0.10268,0.18638,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[20,63,0.3175,0.15179,0.24206,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[24,63,0.381,0.0982,0.18704,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[28,63,0.4444,0.12499,0.21941,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[32,63,0.5079,0.09812,0.18011,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,9,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[36,63,0.5714,0.08929,0.17768,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[40,63,0.6349,0.08036,0.17105,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[44,63,0.6984,0.04911,0.13175,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,63,0.7619,0.08929,0.20124,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[52,63,0.8254,0.07587,0.18546,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[56,63,0.8889,0.0357,0.10708,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,63,0.9524,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[63,63,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.01339,"x":0.21873,"p":[[0,76,0.0,0.21873,0.2834,0.0,0.0,0.5711,0.0,0.71429,17,0,0,17,0,5,0,0,0,0,0,0,0,0,6,0,0,4,0,0,0,0,0],[4,76,0.0526,0.06697,0.15561,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,76,0.1053,0.08929,0.1948,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[12,76,0.1579,0.16518,0.26752,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,8,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,1],[16,76,0.2105,0.08482,0.1551,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,76,0.2632,0.05804,0.13767,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,76,0.3158,0.04018,0.10853,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,76,0.3684,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,76,0.4211,0.08036,0.17835,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[36,76,0.4737,0.0759,0.14719,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,76,0.5263,0.08482,0.16698,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[44,76,0.5789,0.04911,0.09853,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.04911,0.11071,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b63a8235615dbac1","q":"Let  $ABC$  be an acute-angled triangle with  $AB < AC$  and  $\\Gamma$  the circumference that passes through  $A,\\ B$  and  $C$ . Let  $D$  be the point diametrically opposite  $A$  on  $\\Gamma$  and  $\\ell$  the tangent through  $D$  to  $\\Gamma$ . Let  $P, Q$  and  $R$  be the intersection points of  $B C$  with  $\\ell$ , of  $A P$  with  $\\Gamma$  such that  $Q \\neq A$  and of  $Q D$  with the  $A$ -altitude of the triangle  $ABC$ , respectively. Define  $S$  to be the intersection of  $AB$  with  $\\ell$  and  $T$  to be the intersection of  $A C$  with  $\\ell$ . Show that  $S$  and  $T$  lie on the circumference that passes through  $A, Q$  and  $R$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.06241,"x":0.35713,"p":[[0,97,0.0,0.31473,0.27541,0.0,0.28571,0.60714,0.0,0.71429,9,0,0,9,1,3,0,0,7,0,0,3,0,0,1,0,0,8,0,0,0,0,0],[4,97,0.0412,0.33482,0.23038,0.14286,0.42857,0.42857,0.0,0.71429,7,0,0,7,0,2,0,0,6,0,0,12,0,0,0,0,0,5,0,0,0,0,0],[8,97,0.0825,0.32142,0.22586,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,5,0,0,8,0,0,1,0,0,5,0,0,0,0,0],[12,97,0.1237,0.29017,0.21571,0.14286,0.28571,0.4286,0.0,0.71429,7,0,0,7,0,5,0,0,7,0,0,9,0,0,1,0,0,3,0,0,0,0,0],[16,97,0.1649,0.35713,0.23689,0.14286,0.42857,0.4286,0.0,0.71429,6,0,0,6,0,3,0,0,5,0,0,11,0,0,1,0,0,6,0,0,0,0,0],[20,97,0.2062,0.24992,0.21729,0.105,0.1429,0.42857,0.0,0.71429,8,0,0,8,0,9,0,0,4,0,0,8,0,0,0,0,0,3,0,0,0,0,0],[24,97,0.2474,0.18304,0.22654,0.0,0.14286,0.1786,0.0,0.71429,13,0,0,13,0,11,0,0,1,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[28,97,0.2887,0.20536,0.23128,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,5,0,0,4,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[32,97,0.3299,0.26339,0.25028,0.0,0.14286,0.42857,0.0,0.71429,10,0,0,10,0,7,0,0,3,0,0,7,0,0,0,0,0,5,0,0,0,0,0],[36,97,0.3711,0.2009,0.2547,0.0,0.0,0.42857,0.0,0.71429,17,0,0,17,0,3,0,0,2,0,0,6,0,0,0,0,0,4,0,0,0,0,0],[40,97,0.4124,0.22768,0.22263,0.0,0.14288,0.42857,0.0,0.71429,12,0,0,12,0,5,0,0,4,0,0,8,0,0,1,0,0,2,0,0,0,0,0],[44,97,0.4536,0.18741,0.2096,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,5,0,0,6,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[48,97,0.4948,0.16963,0.20956,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,3,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[52,97,0.5361,0.25892,0.24597,0.0,0.21431,0.42858,0.0,0.71429,11,0,0,11,0,5,0,0,4,0,0,7,0,0,1,0,0,4,0,0,0,0,0],[56,97,0.5773,0.18295,0.23212,0.0,0.14143,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,3,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[60,97,0.6186,0.25446,0.24415,0.0,0.14286,0.42857,0.0,0.71429,9,0,0,9,0,9,0,0,4,0,0,5,0,0,0,0,0,5,0,0,0,0,0],[64,97,0.6598,0.1829,0.21465,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,11,0,0,3,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[68,97,0.701,0.1875,0.19377,0.0,0.14286,0.32143,0.0,0.71429,13,0,0,13,0,6,0,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[72,97,0.7423,0.17402,0.23888,0.0,0.14286,0.1429,0.0,1.0,14,1,0,14,0,11,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[76,97,0.7835,0.16962,0.21847,0.0,0.0,0.32143,0.0,0.71429,18,0,0,18,0,2,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[80,97,0.8247,0.20525,0.22286,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,10,0,0,4,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[84,97,0.866,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[88,97,0.9072,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,97,0.9485,0.08929,0.17405,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,4,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[96,97,0.9897,0.07571,0.13347,0.0,0.0,0.14071,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[97,97,1.0,0.06241,0.1284,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.04911,"x":0.25872,"p":[[0,73,0.0,0.20982,0.25501,0.0,0.14286,0.32143,0.0,0.71429,14,0,0,14,0,7,0,0,3,0,0,3,0,0,0,0,0,5,0,0,0,0,0],[4,73,0.0548,0.22772,0.27169,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,4,0,0,4,0,0,3,0,0,0,0,0,6,0,0,0,0,0],[8,73,0.1096,0.25872,0.24603,0.0,0.21428,0.42857,0.0,0.71429,10,0,0,10,0,6,0,0,7,0,0,2,0,0,3,0,0,4,0,0,0,0,0],[12,73,0.1644,0.20981,0.23684,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,5,0,0,4,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[16,73,0.2192,0.14732,0.16554,0.0,0.07143,0.2857,0.0,0.4286,16,0,0,16,0,4,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.07589,0.16742,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[24,73,0.3288,0.16072,0.17768,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,9,0,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[28,73,0.3836,0.16072,0.18123,0.0,0.14286,0.2857,0.0,0.71429,14,0,0,14,0,7,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[32,73,0.4384,0.13393,0.14698,0.0,0.14286,0.1429,0.0,0.57143,13,0,0,13,0,12,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,73,0.4932,0.18293,0.22083,0.0,0.14143,0.32143,0.0,0.71429,15,0,0,15,0,6,0,0,3,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[40,73,0.5479,0.24107,0.27067,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,4,0,0,4,0,0,4,0,0,0,0,0,6,0,0,0,0,0],[44,73,0.6027,0.16294,0.21815,0.0,0.10693,0.2857,0.0,0.71429,15,0,0,15,1,7,0,0,4,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[48,73,0.6575,0.16071,0.24157,0.0,0.0,0.17857,0.0,0.71429,18,0,0,18,0,6,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,0],[52,73,0.7123,0.11607,0.21261,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,8,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[56,73,0.7671,0.24558,0.24288,0.0,0.2143,0.42857,0.0,0.71429,12,0,0,12,0,4,0,0,5,0,0,7,0,0,0,0,0,4,0,0,0,0,0],[60,73,0.8219,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.0758,0.16357,0.0,0.0,0.035,0.0,0.71429,24,0,0,24,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[68,73,0.9315,0.08034,0.14694,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[72,73,0.9863,0.06688,0.14714,0.0,0.0,0.14071,0.0,0.71429,23,0,0,23,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[73,73,1.0,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d2b17f23f470c400","q":"Let  $ABC$  denote a triangle. The point  $X$  lies on the extension of  $AC$  beyond  $A$ , such that  $AX = AB$ . Similarly, the point  $Y$  lies on the extension of  $BC$  beyond  $B$  such that  $BY = AB$ . Prove that the circumcircles of  $ACY$  and  $BCX$  intersect a second time in a point different from  $C$  that lies on the bisector of the angle  $\\angle BCA$ .\n\n(Theresia Eisenk\u00f6lbl)","t":[{"b":3,"e":0.14,"k":"flat","v":0.0,"x":0.06696,"p":[[0,57,0.0,0.06696,0.19556,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,57,0.0702,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,57,0.2807,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.04009,0.17576,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,57,0.4912,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,67,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,67,0.2985,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,67,0.7164,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,67,0.8955,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a09840619c13e295","q":"Let  $ABCD$  be a cyclic quadrilateral with  $|AB|=26$ ,  $|BC|=10$ ,  $m(\\widehat{ABD})=45^\\circ$ , $m(\\widehat{ACB})=90^\\circ$ . What is the area of  $\\triangle DAC$  ? $ \\textbf{(A)}\\ 120\n\\qquad\\textbf{(B)}\\ 108\n\\qquad\\textbf{(C)}\\ 90\n\\qquad\\textbf{(D)}\\ 84\n\\qquad\\textbf{(E)}\\ 80\n$","t":[{"b":6,"e":0.571,"k":"falling","v":0.56692,"x":0.95089,"p":[[0,47,0.0,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[4,47,0.0851,0.86604,0.2421,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,24],[8,47,0.1702,0.8214,0.23148,0.67846,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,0,0,19],[12,47,0.2553,0.87946,0.23449,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,24],[16,47,0.3404,0.83927,0.21945,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,0,0,19],[20,47,0.4255,0.8482,0.20498,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,1,0,19],[24,47,0.5106,0.87945,0.19271,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,1,0,22],[28,47,0.5957,0.75441,0.24288,0.571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,6,0,0,0,0,14],[32,47,0.6809,0.79462,0.23676,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,1,0,16],[36,47,0.766,0.82589,0.25438,0.57143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,2,0,0,0,0,21],[40,47,0.8511,0.88392,0.20654,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,23],[44,47,0.9362,0.62052,0.23313,0.42857,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,8,0,0,1,0,0,12,0,0,1,0,5],[47,47,1.0,0.56692,0.2004,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,5,0,0,8,0,0,11,0,0,0,0,2]]},{"b":7,"e":1.0,"k":"falling","v":0.53558,"x":0.96429,"p":[[0,52,0.0,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[4,52,0.0769,0.90624,0.1772,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,0,0,24],[8,52,0.1538,0.7991,0.22548,0.71421,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,0,0,16],[12,52,0.2308,0.82589,0.2618,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,7,0,0,0,0,20],[16,52,0.3077,0.8839,0.22146,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,0,0,24],[20,52,0.3846,0.80348,0.24703,0.71429,1.0,1.0,0.14,1.0,0,17,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,7,0,0,1,0,17],[24,52,0.4615,0.81249,0.24857,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,2,0,0,1,0,19],[28,52,0.5385,0.77231,0.20783,0.57143,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,10,0,0,0,0,13],[32,52,0.6154,0.80804,0.21312,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,10,0,0,0,0,16],[36,52,0.6923,0.79909,0.27167,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,4,0,0,0,0,19],[40,52,0.7692,0.84821,0.22851,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,5,0,0,0,0,21],[44,52,0.8462,0.79909,0.26454,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,3,0,0,0,0,19],[48,52,0.9231,0.79908,0.26455,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,5,0,0,1,0,18],[52,52,1.0,0.53558,0.17876,0.39286,0.57143,0.71429,0.14,0.85714,0,0,0,0,0,1,0,0,7,0,0,2,0,0,12,0,0,9,0,0,1,0,0]]}]},{"i":"9a85777ea8bcaccd","q":"Let  $ABC$  be a triangle inscribed in circle  $\\Gamma$ , centered at  $O$  with radius  $333.$  Let  $M$  be the midpoint of  $AB$ ,  $N$  be the midpoint of  $AC$ , and  $D$  be the point where line  $AO$  intersects  $BC$ . Given that lines  $MN$  and  $BO$  concur on  $\\Gamma$  and that  $BC = 665$ , find the length of segment  $AD$ .\n\n*Author: Alex Zhu*","t":[{"b":0,"e":1.0,"k":"flat","v":0.52679,"x":0.73661,"p":[[0,187,0.0,0.66964,0.32818,0.28571,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,8,0,0,0,0,0,1,0,0,8,0,0,1,0,12],[4,187,0.0214,0.66518,0.31866,0.28571,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,4,0,0,1,0,13],[8,187,0.0428,0.69196,0.30328,0.28571,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,8,0,0,0,0,0,0,0,0,11,0,0,0,0,12],[12,187,0.0642,0.62054,0.29366,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,12,0,0,1,0,0,1,0,0,9,0,0,0,0,9],[16,187,0.0856,0.66071,0.28516,0.28571,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,10,0,0,0,0,10],[20,187,0.107,0.70087,0.26574,0.571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,4,0,0,10,0,0,0,0,11],[24,187,0.1283,0.65626,0.30063,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,3,0,0,6,0,0,1,0,11],[28,187,0.1497,0.71429,0.27664,0.57143,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,0,0,0,5,0,0,7,0,0,0,0,13],[32,187,0.1711,0.625,0.31693,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,7,0,0,0,0,11],[36,187,0.1925,0.73661,0.31157,0.5,0.85714,1.0,0.0,1.0,1,16,0,1,0,0,0,0,7,0,0,0,0,0,1,0,0,7,0,0,0,0,16],[40,187,0.2139,0.64286,0.32143,0.28571,0.71429,1.0,0.0,1.0,2,11,0,2,0,0,0,0,8,0,0,0,0,0,4,0,0,7,0,0,0,0,11],[44,187,0.2353,0.66071,0.3004,0.28571,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,2,0,0,7,0,0,1,0,11],[48,187,0.2567,0.58036,0.31326,0.28571,0.57143,1.0,0.0,1.0,1,9,0,1,0,0,0,0,13,0,0,0,0,0,4,0,0,5,0,0,0,0,9],[52,187,0.2781,0.65178,0.32328,0.28571,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,1,0,0,5,0,0,0,0,13],[56,187,0.2995,0.62053,0.3126,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,11,0,0,0,0,0,3,0,0,7,0,0,0,0,10],[60,187,0.3209,0.68303,0.34761,0.28571,0.71429,1.0,0.0,1.0,2,15,0,2,0,0,0,0,9,0,0,0,0,0,0,0,0,6,0,0,0,0,15],[64,187,0.3422,0.62945,0.27862,0.28571,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,10,0,0,0,0,0,7,0,0,6,0,0,0,0,9],[68,187,0.3636,0.66964,0.3223,0.28571,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,4,0,0,0,0,14],[72,187,0.385,0.68304,0.30037,0.39286,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,7,0,0,1,0,0,3,0,0,8,0,0,0,0,12],[76,187,0.4064,0.61159,0.29285,0.28571,0.64286,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,12,0,0,1,0,0,3,0,0,7,0,0,0,0,9],[80,187,0.4278,0.61161,0.29717,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,8,0,0,0,0,9],[84,187,0.4492,0.57143,0.27664,0.28571,0.57143,0.71429,0.28571,1.0,0,7,0,0,0,0,0,0,13,0,0,0,0,0,7,0,0,5,0,0,0,0,7],[88,187,0.4706,0.63393,0.2878,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,11,0,0,0,0,0,4,0,0,7,0,0,1,0,9],[92,187,0.492,0.64284,0.27433,0.28571,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,10,0,0,0,0,0,3,0,0,10,0,0,1,0,8],[96,187,0.5134,0.58034,0.30501,0.28571,0.57121,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,15,0,0,0,0,0,3,0,0,5,0,0,0,0,9],[100,187,0.5348,0.69641,0.30671,0.28571,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,6,0,0,0,0,14],[104,187,0.5561,0.56696,0.30406,0.28571,0.57143,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,13,0,0,0,0,0,5,0,0,5,0,0,0,0,8],[108,187,0.5775,0.62946,0.31106,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,6,0,0,0,0,11],[112,187,0.5989,0.65178,0.31731,0.28571,0.71429,1.0,0.0,1.0,1,11,1,1,0,0,0,0,10,0,0,0,0,0,3,0,0,5,0,0,2,0,11],[116,187,0.6203,0.65625,0.27166,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,10,0,0,0,0,9],[120,187,0.6417,0.66964,0.2911,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,9,0,0,0,0,11],[124,187,0.6631,0.65625,0.33093,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,11,0,0,0,0,0,1,0,0,6,0,0,0,0,13],[128,187,0.6845,0.7232,0.24469,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,11,0,0,0,0,11],[132,187,0.7059,0.64732,0.30719,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,6,0,0,1,0,11],[136,187,0.7273,0.68304,0.3169,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,9,0,0,0,0,0,1,0,0,8,0,0,0,0,13],[140,187,0.7487,0.58036,0.30501,0.28571,0.57143,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,15,0,0,0,0,0,3,0,0,5,0,0,0,0,9],[144,187,0.7701,0.58481,0.29311,0.28571,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,14,0,0,0,0,0,3,0,0,7,0,0,0,0,8],[148,187,0.7914,0.625,0.28516,0.28571,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,11,0,0,0,0,0,5,0,0,7,0,0,0,0,9],[152,187,0.8128,0.72768,0.30797,0.28571,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,9,0,0,0,0,0,3,0,0,3,0,0,1,0,16],[156,187,0.8342,0.69643,0.27375,0.5,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,8,0,0,0,0,0,2,0,0,11,0,0,0,0,11],[160,187,0.8556,0.71429,0.32733,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,3,0,0,0,0,17],[164,187,0.877,0.58036,0.3071,0.28571,0.64286,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,13,0,0,0,0,0,2,0,0,8,0,0,0,0,8],[168,187,0.8984,0.64732,0.2923,0.28571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,11,0,0,0,0,0,2,0,0,9,0,0,0,0,10],[172,187,0.9198,0.54464,0.29545,0.28571,0.28571,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,17,0,0,0,0,0,1,0,0,7,0,0,0,0,7],[176,187,0.9412,0.57141,0.29233,0.28571,0.57121,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,15,0,0,0,0,0,2,0,0,7,0,0,1,0,7],[180,187,0.9626,0.52679,0.3102,0.28571,0.28571,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,17,0,0,0,0,0,0,0,0,7,0,0,0,0,7],[184,187,0.984,0.5357,0.27664,0.28571,0.35714,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,16,0,0,1,0,0,1,0,0,8,0,0,1,0,5],[187,187,1.0,0.70088,0.26573,0.57132,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,4,0,0,10,0,0,0,0,11]]},{"b":7,"e":0.71429,"k":"flat","v":0.51786,"x":0.74107,"p":[[0,88,0.0,0.62946,0.29636,0.28571,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,10,0,0,0,0,0,1,0,0,11,0,0,1,0,8],[4,88,0.0455,0.63393,0.30292,0.28571,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,7,0,0,0,0,0,5,0,0,9,0,0,0,0,9],[8,88,0.0909,0.62945,0.30275,0.28571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,1,0,0,2,0,0,6,0,0,1,0,10],[12,88,0.1364,0.74107,0.33964,0.28571,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,19],[16,88,0.1818,0.73661,0.31157,0.5,0.85714,1.0,0.0,1.0,1,16,1,1,0,0,0,0,7,0,0,0,0,0,1,0,0,7,0,0,0,0,16],[20,88,0.2273,0.52232,0.35465,0.28571,0.28571,1.0,0.0,1.0,4,9,0,4,0,0,0,0,13,0,0,0,0,0,2,0,0,4,0,0,0,0,9],[24,88,0.2727,0.56696,0.32436,0.28571,0.64286,0.78571,0.0,1.0,2,8,0,2,0,1,0,0,11,0,0,0,0,0,2,0,0,8,0,0,0,0,8],[28,88,0.3182,0.70536,0.2922,0.57143,0.71429,1.0,0.0,1.0,2,11,0,2,0,0,0,0,4,0,0,0,0,0,3,0,0,11,0,0,1,0,11],[32,88,0.3636,0.61161,0.31991,0.28571,0.64286,1.0,0.0,1.0,1,10,0,1,0,0,0,0,12,0,0,0,0,0,3,0,0,5,0,0,1,0,10],[36,88,0.4091,0.62946,0.34416,0.28571,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,12],[40,88,0.4545,0.6339,0.3071,0.28571,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,12,0,0,0,0,0,5,0,0,3,0,0,1,0,11],[44,88,0.5,0.72768,0.27976,0.57143,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,7,0,0,0,0,0,4,0,0,7,0,0,0,0,14],[48,88,0.5455,0.56249,0.25238,0.28571,0.64286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,3,0,0,12,0,0,0,0,4],[52,88,0.5909,0.57143,0.29234,0.28571,0.57143,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,12,0,0,0,0,0,5,0,0,7,0,0,0,0,7],[56,88,0.6364,0.61161,0.27718,0.28571,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,11,0,0,0,0,0,6,0,0,7,0,0,0,0,8],[60,88,0.6818,0.62052,0.28708,0.28571,0.71429,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,9,0,0,0,0,0,5,0,0,9,0,0,0,0,8],[64,88,0.7273,0.52679,0.2683,0.28571,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,16,0,0,0,0,0,4,0,0,7,0,0,0,0,5],[68,88,0.7727,0.55357,0.29827,0.28571,0.42857,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,16,0,0,0,0,0,4,0,0,4,0,0,0,0,8],[72,88,0.8182,0.51786,0.31894,0.28571,0.28571,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,18,0,0,0,0,0,1,0,0,4,0,0,0,0,8],[76,88,0.8636,0.57589,0.26362,0.28571,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,12,0,0,0,0,5],[80,88,0.9091,0.61607,0.33586,0.28571,0.64286,1.0,0.0,1.0,1,12,0,1,0,0,0,0,13,0,0,0,0,0,2,0,0,4,0,0,0,0,12],[84,88,0.9545,0.56696,0.30196,0.28571,0.42857,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,7,0,0,0,0,8],[88,88,1.0,0.64286,0.30514,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,7,0,0,0,0,11]]}]},{"i":"4284f9740a6f78bd","q":"Let  $ABCD$  be a cyclic quadrilateral, so that  $|AB| + |CD|  = |BC|$ . Show that the intersection of the bisector of  $\\angle DAB$  and   $\\angle CDA$  lies on the side  $BC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.1741,"p":[[0,64,0.0,0.08482,0.20473,0.0,0.0,0.03571,0.0,1.0,24,1,3,24,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,64,0.0625,0.1741,0.19798,0.0,0.14286,0.28571,0.0,0.857,15,0,0,15,0,2,0,0,11,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[8,64,0.125,0.10714,0.14725,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,2,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.13839,0.19719,0.0,0.0,0.2857,0.0,0.85714,18,0,0,18,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[16,64,0.25,0.08482,0.13296,0.0,0.0,0.17857,0.0,0.4286,22,0,0,22,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.08928,0.12752,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.11161,0.1461,0.0,0.0,0.2857,0.0,0.4286,20,0,0,20,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.09821,0.14032,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,1,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.08036,0.12846,0.0,0.0,0.2857,0.0,0.28571,23,0,0,23,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.08482,0.15093,0.0,0.0,0.07143,0.0,0.42857,24,0,0,24,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.09821,0.18363,0.0,0.0,0.2857,0.0,0.85714,23,0,0,23,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,64,0.6875,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.03572,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04464,"x":0.29911,"p":[[0,74,0.0,0.06696,0.10705,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,74,0.0541,0.15625,0.22968,0.0,0.0,0.2857,0.0,0.85714,18,0,0,18,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[8,74,0.1081,0.08927,0.15042,0.0,0.0,0.2857,0.0,0.571,23,0,0,23,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,74,0.1622,0.17857,0.20516,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,5,0,0,8,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[16,74,0.2162,0.16964,0.20652,0.0,0.07143,0.28571,0.0,0.85714,16,0,0,16,0,3,0,0,7,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[20,74,0.2703,0.16517,0.20548,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,3,0,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[24,74,0.3243,0.16955,0.22143,0.0,0.14286,0.2857,0.0,0.85714,14,0,0,14,0,8,0,0,6,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[28,74,0.3784,0.22768,0.274,0.0,0.14286,0.32143,0.0,0.85714,15,0,0,15,0,2,0,0,7,0,0,4,0,0,0,0,0,1,0,0,3,0,0],[32,74,0.4324,0.15625,0.2212,0.0,0.0,0.2857,0.0,0.85714,18,0,0,18,0,3,0,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[36,74,0.4865,0.1875,0.1729,0.0,0.14286,0.32143,0.0,0.42857,12,0,0,12,0,6,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[40,74,0.5405,0.16072,0.16269,0.0,0.14286,0.28571,0.0,0.4286,14,0,0,14,0,5,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,74,0.5946,0.18303,0.2321,0.0,0.0,0.32143,0.0,0.85714,17,0,0,17,0,2,0,0,5,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[48,74,0.6486,0.29911,0.25595,0.10714,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,5,0,0,7,0,0,7,0,0,0,0,0,3,0,0,2,0,0],[52,74,0.7027,0.15625,0.20935,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,5,0,0,3,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[56,74,0.7568,0.21429,0.23419,0.0,0.14286,0.32143,0.0,0.85714,13,0,0,13,0,4,0,0,7,0,0,6,0,0,0,0,0,0,0,0,2,0,0],[60,74,0.8108,0.22767,0.23651,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,5,0,0,5,0,0,8,0,0,0,0,0,0,0,0,2,0,0],[64,74,0.8649,0.20089,0.16698,0.0,0.21428,0.28571,0.0,0.57143,10,0,0,10,0,6,0,0,10,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[68,74,0.9189,0.125,0.17767,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,3,0,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[72,74,0.973,0.12946,0.1488,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,3,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[74,74,1.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"09da89c8789fa063","q":"Let  $ABCD$  be a cyclic quadrilateral whose side  $AB$  is at the same time the diameter of the circle. The lines  $AC$  and  $BD$  intersect at point  $E$  and the extensions of lines  $AD$  and  $BC$  intersect at point  $F$ .  Segment  $EF$  intersects the circle at  $G$  and the extension of segment  $EF$  intersects  $AB$  at  $H$ . Show that if  $G$  is the midpoint of  $FH$ , then  $E$  is the midpoint of  $GH$ .","t":[{"b":4,"e":0.0,"k":"falling","v":0.02232,"x":0.45087,"p":[[0,118,0.0,0.26784,0.31081,0.0,0.14286,0.46418,0.0,1.0,12,2,0,12,0,7,0,0,4,0,0,1,0,0,3,0,0,2,0,0,1,0,2],[4,118,0.0339,0.33258,0.3142,0.14286,0.2143,0.46418,0.0,1.0,7,2,0,7,0,9,0,0,4,0,0,4,0,0,1,1,0,1,0,0,3,0,2],[8,118,0.0678,0.26339,0.20858,0.14286,0.14286,0.32143,0.0,0.85714,4,0,0,4,0,13,0,0,7,0,0,3,0,0,3,0,0,1,0,0,1,0,0],[12,118,0.1017,0.29463,0.28999,0.14286,0.1429,0.42857,0.0,1.0,7,2,0,7,0,10,0,0,5,0,0,4,0,0,2,0,0,0,0,0,2,0,2],[16,118,0.1356,0.3482,0.3213,0.14286,0.28571,0.57143,0.0,1.0,7,3,1,7,0,8,0,0,5,0,0,2,0,0,3,0,0,3,0,0,1,0,3],[20,118,0.1695,0.29464,0.23402,0.14286,0.28571,0.32143,0.0,1.0,6,1,0,6,0,5,0,0,13,0,0,2,0,0,3,0,0,2,0,0,0,0,1],[24,118,0.2034,0.27231,0.22966,0.14286,0.14288,0.42857,0.0,0.71429,6,0,0,6,0,11,0,0,5,0,0,4,0,0,2,0,0,4,0,0,0,0,0],[28,118,0.2373,0.29463,0.26948,0.10714,0.2857,0.4286,0.0,0.85714,8,0,0,8,0,7,0,0,7,0,0,3,0,0,1,0,0,4,0,0,2,0,0],[32,118,0.2712,0.35268,0.3009,0.14286,0.35714,0.46431,0.0,1.0,7,2,0,7,0,7,0,0,2,0,0,8,0,0,1,0,0,4,0,0,1,0,2],[36,118,0.3051,0.41506,0.31419,0.14286,0.35714,0.71429,0.0,1.0,5,2,0,5,0,7,0,0,4,0,0,2,0,0,4,0,0,6,0,0,2,0,2],[40,118,0.339,0.25445,0.24413,0.0,0.14288,0.42857,0.0,0.71429,10,0,0,10,0,7,0,0,5,0,0,4,0,0,2,0,0,4,0,0,0,0,0],[44,118,0.3729,0.33035,0.29328,0.14286,0.2857,0.57143,0.0,0.85714,7,0,0,7,0,8,0,0,5,0,0,3,0,0,2,0,0,3,0,0,4,0,0],[48,118,0.4068,0.26786,0.26666,0.0,0.14288,0.42857,0.0,1.0,9,1,0,9,0,8,0,0,6,0,0,3,0,0,1,0,0,4,0,0,0,0,1],[52,118,0.4407,0.3549,0.33478,0.0,0.28571,0.57143,0.0,1.0,9,3,0,9,0,6,0,0,2,0,1,4,0,0,3,0,0,2,0,0,2,0,3],[56,118,0.4746,0.30803,0.27457,0.10714,0.2857,0.42858,0.0,1.0,8,1,0,8,0,6,0,0,6,0,0,5,0,0,2,0,0,3,0,0,1,0,1],[60,118,0.5085,0.29015,0.26359,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,8,0,0,8,0,0,2,0,0,3,0,0,2,0,0,1,0,1],[64,118,0.5424,0.29911,0.28203,0.0,0.2857,0.57143,0.0,1.0,11,1,0,11,0,4,0,0,3,0,0,5,0,0,5,0,0,3,0,0,0,0,1],[68,118,0.5763,0.25892,0.28667,0.14286,0.14286,0.28571,0.0,1.0,7,2,0,7,0,15,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,2],[72,118,0.6102,0.28572,0.27894,0.10714,0.14286,0.57143,0.0,0.85714,8,0,0,8,0,11,0,0,3,0,0,1,0,0,2,0,0,6,0,0,1,0,0],[76,118,0.6441,0.45087,0.30115,0.14286,0.42857,0.71429,0.0,1.0,2,2,0,2,0,9,0,0,3,0,0,4,0,0,3,0,0,6,0,0,3,0,2],[80,118,0.678,0.2856,0.29667,0.0,0.14286,0.57111,0.0,1.0,10,1,0,10,0,9,0,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,1],[84,118,0.7119,0.23652,0.25411,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,8,0,0,8,0,0,1,0,0,1,0,0,3,0,0,0,0,1],[88,118,0.7458,0.24554,0.29501,0.0,0.14286,0.42857,0.0,1.0,12,1,0,12,0,9,0,0,2,0,0,3,0,0,1,0,0,2,0,0,2,0,1],[92,118,0.7797,0.2857,0.29013,0.0,0.2143,0.42858,0.0,1.0,10,2,0,10,0,6,0,0,4,0,0,7,0,0,1,0,0,1,0,0,1,0,2],[96,118,0.8136,0.23203,0.22798,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,6,0,0,7,0,0,2,0,0,4,0,0,2,0,0,0,0,0],[100,118,0.8475,0.27677,0.29218,0.0,0.2857,0.42857,0.0,1.0,12,2,0,12,0,2,0,0,8,0,0,5,0,0,1,0,0,1,0,0,1,0,2],[104,118,0.8814,0.1875,0.27067,0.0,0.0,0.2857,0.0,1.0,17,2,0,17,0,3,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[108,118,0.9153,0.16964,0.22711,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,8,0,0,4,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[112,118,0.9492,0.183,0.22927,0.0,0.07143,0.42857,0.0,0.71429,16,0,0,16,0,6,0,0,1,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[116,118,0.9831,0.09374,0.21309,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[118,118,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.08483,"x":0.42185,"p":[[0,63,0.0,0.27676,0.29865,0.0,0.14286,0.42857,0.0,1.0,11,2,0,11,0,6,0,0,5,0,0,3,0,0,3,0,0,1,0,0,1,0,2],[4,63,0.0635,0.39733,0.31488,0.14289,0.28571,0.71429,0.0,1.0,5,4,0,5,0,5,0,0,8,0,0,5,0,0,0,0,0,5,0,0,0,0,4],[8,63,0.127,0.26786,0.26426,0.10714,0.14286,0.42857,0.0,1.0,8,1,0,8,0,10,0,0,5,0,0,3,0,0,1,0,0,4,0,0,0,0,1],[12,63,0.1905,0.375,0.31894,0.14286,0.28571,0.57143,0.0,1.0,7,4,0,7,0,5,0,0,5,0,0,5,0,0,4,0,0,2,0,0,0,0,4],[16,63,0.254,0.21865,0.22584,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,5,0,0,2,0,0,2,0,0,3,0,0,0,0,0],[20,63,0.3175,0.22322,0.26229,0.0,0.14286,0.2857,0.0,1.0,9,2,0,9,0,13,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[24,63,0.381,0.30348,0.28296,0.105,0.28571,0.42858,0.0,1.0,8,2,0,8,0,6,0,0,8,0,0,3,0,0,2,0,0,3,0,0,0,0,2],[28,63,0.4444,0.42185,0.33423,0.14286,0.28571,0.66071,0.0,1.0,5,4,0,5,0,6,0,0,6,0,0,3,0,0,3,1,0,1,0,0,3,0,4],[32,63,0.5079,0.35045,0.33281,0.14286,0.21431,0.71429,0.0,1.0,6,3,0,6,1,9,0,0,5,0,0,2,0,0,0,0,0,4,0,0,2,0,3],[36,63,0.5714,0.27228,0.29953,0.0,0.14286,0.42858,0.0,1.0,11,2,0,11,0,7,0,0,4,0,0,3,0,0,3,0,0,1,0,0,1,0,2],[40,63,0.6349,0.28572,0.31339,0.10714,0.14288,0.32143,0.0,1.0,8,3,0,8,0,11,0,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,3],[44,63,0.6984,0.36159,0.33116,0.14286,0.2857,0.60714,0.0,1.0,7,3,0,7,0,8,0,0,4,0,0,3,0,0,2,0,0,3,0,0,2,0,3],[48,63,0.7619,0.31472,0.31229,0.10714,0.2143,0.42857,0.0,1.0,8,3,0,8,0,8,0,0,4,0,1,5,0,0,0,0,0,2,0,0,1,0,3],[52,63,0.8254,0.25443,0.29172,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,5,0,0,8,0,0,1,0,0,2,0,0,0,0,0,3,0,1],[56,63,0.8889,0.35257,0.35893,0.14214,0.14288,0.60682,0.0,1.0,7,5,0,7,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,5],[60,63,0.9524,0.30355,0.25936,0.10714,0.28571,0.4642,0.0,0.85714,8,0,0,8,0,6,0,0,6,0,0,4,0,0,3,0,0,4,0,0,1,0,0],[63,63,1.0,0.08483,0.17445,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"7345a7e7d0bbd5a1","q":"Let  $(G,\\cdot)$  be a group and  $H\\neq G$  be a subgroup so that  $x^2=y^2$  for all  $x,y\\in G\\setminus H.$  Show that  $(H,\\cdot)$  is an Abelian group.","t":[{"b":1,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,32,0.0,0.92411,0.15966,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,25],[4,32,0.125,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[8,32,0.25,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,38,0.0,0.91964,0.15541,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,23],[4,38,0.1053,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,26],[8,38,0.2105,0.91071,0.16656,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0,0,24],[12,38,0.3158,0.91071,0.16656,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0,0,24],[16,38,0.4211,0.875,0.21053,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,2,0,22],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,38,0.7368,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,38,0.8421,0.93304,0.17122,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,26],[36,38,0.9474,0.90625,0.14987,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,22],[38,38,1.0,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26]]}]},{"i":"417f5c348c1b28dd","q":"Let  $A_1$ ,  $A_2$ ,  $A_3$  be three points in the plane, and for convenience, let  $A_4= A_1$ ,  $A_5 = A_2$ .  For  $n = 1$ ,  $2$ , and  $3$ , suppose that  $B_n$  is the midpoint of  $A_n A_{n+1}$ , and suppose that  $C_n$  is the midpoint of  $A_n B_n$ . Suppose that  $A_n C_{n+1}$  and  $B_n A_{n+2}$  meet at  $D_n$ , and that  $A_n B_{n+1}$  and  $C_n A_{n+2}$  meet at  $E_n$ .  \r\nCalculate the ratio of the area of triangle  $D_1 D_2 D_3$  to the area of triangle  $E_1 E_2 E_3$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,83,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,83,0.0482,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,83,0.0964,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[12,83,0.1446,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[16,83,0.1928,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,83,0.241,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,83,0.2892,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,83,0.3373,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,83,0.3855,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,83,0.4337,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,83,0.4819,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,83,0.5301,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[48,83,0.5783,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,83,0.6265,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,83,0.6747,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,83,0.7229,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,83,0.7711,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,83,0.8193,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,83,0.8675,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[76,83,0.9157,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,83,0.9639,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[83,83,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,65,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,65,0.0615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,65,0.1231,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,65,0.1846,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,65,0.2462,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,65,0.3077,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,65,0.3692,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,65,0.4308,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,65,0.4923,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,65,0.5538,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,65,0.6154,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,65,0.6769,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[48,65,0.7385,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,65,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,65,0.8615,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,65,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,65,0.9846,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[65,65,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"48368e2a66665839","q":"Let  $ABCDE$  be a convex pentagon with  $CD= DE$  and  $\\angle EDC \\ne 2 \\cdot \\angle ADB$ . \nSuppose that a point  $P$  is located in the interior of the pentagon such that  $AP =AE$  and  $BP= BC$ . \nProve that  $P$  lies on the diagonal  $CE$  if and only if area  $(BCD)$  + area  $(ADE)$  = area  $(ABD)$  + area  $(ABP)$ .\n\n(Hungary)","t":[{"b":1,"e":0.42857,"k":"rising","v":0.33039,"x":0.88837,"p":[[0,88,0.0,0.33039,0.22429,0.14286,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,7,0,0,7,0,0,8,0,0,2,0,0,3,0,0,1,0,0],[4,88,0.0455,0.88837,0.10557,0.85711,0.85714,1.0,0.714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,13],[8,88,0.0909,0.81691,0.16848,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,11,0,10],[12,88,0.1364,0.85713,0.18213,0.82143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,9,0,15],[16,88,0.1818,0.79005,0.22588,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,9,0,0,10,0,9],[20,88,0.2273,0.83031,0.13094,0.82132,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,18,0,6],[24,88,0.2727,0.74986,0.25255,0.71321,0.857,1.0,0.0,1.0,1,9,0,1,0,0,0,0,3,0,0,1,0,0,2,0,0,8,0,0,8,0,9],[28,88,0.3182,0.65175,0.20496,0.42857,0.71429,0.857,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,8,0,0,8,0,2],[32,88,0.3636,0.71873,0.23279,0.57132,0.857,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,5,0,0,2,0,0,6,0,0,13,0,4],[36,88,0.4091,0.63391,0.28107,0.42859,0.71429,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,3,0,0,4,0,0,3,0,0,6,0,0,10,0,3],[40,88,0.4545,0.69196,0.25028,0.53572,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,4,0,0,2,0,0,8,0,0,9,0,5],[44,88,0.5,0.63835,0.30927,0.5354,0.71429,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,2,0,0,1,0,0,4,0,0,6,0,0,10,0,4],[48,88,0.5455,0.66515,0.23854,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,2,0,0,1,0,0,5,0,0,9,0,0,10,0,2],[52,88,0.5909,0.64729,0.22864,0.42859,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,6,0,0,6,0,0,5,0,0,10,0,2],[56,88,0.6364,0.69196,0.24771,0.4286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,7,0,0,3,0,0,7,0,0,6,0,7],[60,88,0.6818,0.56251,0.30914,0.42857,0.57143,0.75,0.0,1.0,4,5,0,4,0,2,0,0,0,0,0,6,0,0,7,0,0,5,0,0,3,0,5],[64,88,0.7273,0.67854,0.26965,0.57132,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,2,0,0,0,0,0,5,0,0,7,0,0,9,0,5],[68,88,0.7727,0.65622,0.25219,0.42859,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,2,0,0,5,0,0,3,0,0,9,0,0,7,0,4],[72,88,0.8182,0.68301,0.2194,0.57132,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,4,0,0,6,0,0,8,0,0,9,0,3],[76,88,0.8636,0.62141,0.28917,0.42857,0.71429,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,1,0,0,4,0,0,5,1,0,6,0,0,7,0,4],[80,88,0.9091,0.64284,0.25753,0.42857,0.71429,0.85714,0.1429,1.0,0,5,0,0,0,2,0,0,4,0,0,3,0,0,6,0,0,6,0,0,6,0,5],[84,88,0.9545,0.61159,0.18637,0.4286,0.64286,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,8,0,0,5,0,0,9,0,0,7,0,0],[88,88,1.0,0.5402,0.23343,0.42857,0.57121,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,4,0,0,9,0,0,5,0,0,6,0,0,6,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.33033,"x":0.80354,"p":[[0,50,0.0,0.33033,0.18704,0.1429,0.28571,0.4642,0.0,0.7143,1,0,0,1,0,9,0,0,11,0,0,3,0,0,6,0,0,2,0,0,0,0,0],[4,50,0.08,0.76328,0.20107,0.71429,0.85714,0.85714,0.14,1.0,0,3,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,7,0,0,17,0,3],[8,50,0.16,0.7857,0.20517,0.71429,0.85714,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,10,0,8],[12,50,0.24,0.80354,0.17038,0.71429,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,13,0,8],[16,50,0.32,0.78123,0.18894,0.71429,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,8,0,0,13,0,6],[20,50,0.4,0.72319,0.19865,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,9,0,0,10,0,4],[24,50,0.48,0.78123,0.16361,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,14,0,5],[28,50,0.56,0.71413,0.1893,0.57132,0.71429,0.85714,0.14,1.0,0,4,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,9,0,0,8,0,4],[32,50,0.64,0.76331,0.1944,0.57143,0.857,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,5,0,0,12,0,6],[36,50,0.72,0.67406,0.24547,0.571,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,5,0,0,8,0,0,4,0,0,8,0,5],[40,50,0.8,0.7857,0.17857,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,6,0,0,16,0,5],[44,50,0.88,0.7142,0.18561,0.571,0.78564,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,4,0,0,14,0,2],[48,50,0.96,0.6741,0.19957,0.57143,0.71429,0.85704,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,10,0,0,10,0,1],[50,50,1.0,0.69195,0.21756,0.53571,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,5,0,0,1,0,0,9,0,0,12,0,2]]}]},{"i":"619201d1823f1c44","q":"Let  $D$  be a point on side  $[BC]$  of triangle  $ABC$  such that  $[AD]$  is an angle bisector,  $|BD|=4$ , and  $|DC|=3$ . Let  $E$  be a point on side  $[AB]$  and different than  $A$  such that  $m(\\widehat{BED})=m(\\widehat{DEC})$ . If the perpendicular bisector of segment  $[AE]$  meets the line  $BC$  at  $M$ , what is  $|CM|$ ? $ \n\\textbf{(A)}\\ 12\n\\qquad\\textbf{(B)}\\ 9\n\\qquad\\textbf{(C)}\\ 7\n\\qquad\\textbf{(D)}\\ 5\n\\qquad\\textbf{(E)}\\ \\text { None of above}\n$","t":[{"b":4,"e":0.14286,"k":"falling","v":0.30355,"x":0.90179,"p":[[0,101,0.0,0.90179,0.19704,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,24],[4,101,0.0396,0.61159,0.27254,0.53539,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,7,0,0,0,0,0,1,0,0,1,0,0,19,0,0,0,0,4],[8,101,0.0792,0.60268,0.24932,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,23,0,0,1,0,1],[12,101,0.1188,0.66964,0.1729,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,24,0,0,0,0,2],[16,101,0.1584,0.69197,0.17536,0.71429,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,25,0,0,0,0,3],[20,101,0.198,0.65625,0.20473,0.67857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,21,0,0,0,0,3],[24,101,0.2376,0.6607,0.17768,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,26,0,0,0,0,1],[28,101,0.2772,0.58481,0.28428,0.35714,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,0,0,0,2,0,0,1,0,0,17,0,0,0,0,4],[32,101,0.3168,0.66964,0.24856,0.71429,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,5],[36,101,0.3564,0.51339,0.30485,0.14286,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,12,0,0,0,0,0,1,0,0,1,0,0,15,0,0,0,0,3],[40,101,0.396,0.63839,0.20198,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,25,0,0,0,0,1],[44,101,0.4356,0.48659,0.30485,0.14286,0.64264,0.71429,0.14286,1.0,0,3,0,0,0,12,0,0,2,0,0,1,0,0,1,0,0,13,0,0,0,0,3],[48,101,0.4752,0.54018,0.28956,0.14286,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,10,0,0,0,0,0,1,0,0,3,0,0,15,0,0,0,0,3],[52,101,0.5149,0.36607,0.27418,0.14286,0.14286,0.71429,0.14286,1.0,0,1,0,0,0,18,0,0,0,0,0,4,0,0,0,0,0,9,0,0,0,0,1],[56,101,0.5545,0.53572,0.28347,0.14286,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,9,0,0,0,0,0,5,0,0,0,0,0,15,0,0,0,0,3],[60,101,0.5941,0.58036,0.2549,0.39286,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,2,0,0,2,0,0,0,0,0,20,0,0,0,0,2],[64,101,0.6337,0.34821,0.24984,0.14286,0.14288,0.60714,0.14286,0.85714,0,0,0,0,0,17,0,0,2,0,0,4,0,0,1,0,0,7,0,0,1,0,0],[68,101,0.6733,0.47322,0.3223,0.14286,0.50001,0.71429,0.14286,1.0,0,4,0,0,0,14,0,0,1,0,0,1,0,0,1,0,0,11,0,0,0,0,4],[72,101,0.7129,0.50893,0.25738,0.14286,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,9,0,0,0,0,0,5,0,0,2,0,0,15,0,0,0,0,1],[76,101,0.7525,0.49107,0.2922,0.14286,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,15,0,0,0,0,2],[80,101,0.7921,0.56697,0.26118,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,6,0,0,1,0,0,6,0,0,0,0,0,16,0,0,0,0,3],[84,101,0.8317,0.45088,0.27689,0.14286,0.42859,0.71429,0.14286,1.0,0,2,0,0,0,12,0,0,1,0,0,4,0,0,4,0,0,9,0,0,0,0,2],[88,101,0.8713,0.43302,0.28679,0.14286,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,15,0,0,0,0,0,2,0,0,1,0,0,13,0,0,0,0,1],[92,101,0.9109,0.4642,0.28581,0.14286,0.42857,0.71429,0.14,1.0,0,2,0,0,0,12,0,0,1,0,0,4,0,0,1,0,0,12,0,0,0,0,2],[96,101,0.9505,0.37945,0.26148,0.14286,0.28571,0.60714,0.14286,1.0,0,1,0,0,0,16,0,0,0,0,0,5,0,0,3,0,0,7,0,0,0,0,1],[100,101,0.9901,0.30355,0.22514,0.14286,0.14286,0.4642,0.14286,0.71429,0,0,0,0,0,20,0,0,1,0,0,3,0,0,3,0,0,5,0,0,0,0,0],[101,101,1.0,0.34372,0.23104,0.14286,0.14286,0.571,0.14286,0.71429,0,0,0,0,0,17,0,0,0,0,0,6,0,0,3,0,0,6,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.22321,"x":0.83482,"p":[[0,92,0.0,0.83482,0.26513,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,20],[4,92,0.0435,0.57589,0.27545,0.35714,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,8,0,0,0,0,0,2,0,0,1,0,0,18,0,0,0,0,3],[8,92,0.087,0.55804,0.24317,0.39286,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,2,0,0,3,0,0,1,0,0,19,0,0,0,0,1],[12,92,0.1304,0.57133,0.245,0.42857,0.71429,0.71429,0.14,1.0,0,1,0,0,0,7,0,0,0,0,0,2,0,0,2,0,0,20,0,0,0,0,1],[16,92,0.1739,0.44643,0.25692,0.14286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,12,0,0,0,0,0,6,0,0,0,0,0,14,0,0,0,0,0],[20,92,0.2174,0.52232,0.28928,0.14286,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,11,0,0,0,0,0,1,0,0,1,0,0,17,0,0,0,0,2],[24,92,0.2609,0.46875,0.24545,0.14286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,10,0,0,0,0,0,7,0,0,1,0,0,14,0,0,0,0,0],[28,92,0.3043,0.41069,0.27605,0.14286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,15,0,0,0,0,0,4,0,0,3,0,0,8,0,0,1,0,1],[32,92,0.3478,0.33927,0.2335,0.14286,0.14286,0.46418,0.14286,0.71429,0,0,0,0,0,17,0,0,1,0,0,6,0,0,1,0,0,7,0,0,0,0,0],[36,92,0.3913,0.40179,0.30606,0.14286,0.14286,0.71429,0.14286,1.0,0,3,0,0,0,17,0,0,0,0,0,4,0,0,0,0,0,8,0,0,0,0,3],[40,92,0.4348,0.46875,0.29284,0.14286,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,13,0,0,0,0,0,3,0,0,1,0,0,13,0,0,0,0,2],[44,92,0.4783,0.35714,0.25754,0.14286,0.14286,0.71429,0.14286,0.71429,0,0,0,0,0,18,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0],[48,92,0.5217,0.45089,0.27458,0.14286,0.5,0.71429,0.14286,1.0,0,1,0,0,0,13,0,0,0,0,0,3,0,0,3,0,0,12,0,0,0,0,1],[52,92,0.5652,0.23214,0.1948,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0],[56,92,0.6087,0.41963,0.26948,0.14286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,15,0,0,0,0,0,2,0,0,2,0,0,13,0,0,0,0,0],[60,92,0.6522,0.38393,0.30397,0.14286,0.14286,0.71429,0.14286,1.0,0,3,0,0,0,18,0,0,0,0,0,4,0,0,0,0,0,7,0,0,0,0,3],[64,92,0.6957,0.4241,0.25375,0.14286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,12,0,0,0,0,0,9,0,0,1,0,0,9,0,0,0,0,1],[68,92,0.7391,0.37946,0.3001,0.14286,0.14286,0.46429,0.14286,1.0,0,4,0,0,0,17,0,0,0,0,0,7,0,0,1,0,0,3,0,0,0,0,4],[72,92,0.7826,0.33482,0.2412,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,17,0,0,1,0,0,8,0,0,0,0,0,5,0,0,0,0,1],[76,92,0.8261,0.34375,0.28316,0.14286,0.14286,0.60714,0.14286,1.0,0,2,0,0,0,20,0,0,0,0,0,3,0,0,1,0,0,6,0,0,0,0,2],[80,92,0.8696,0.28571,0.22588,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,22,0,0,0,0,0,4,0,0,0,0,0,6,0,0,0,0,0],[84,92,0.913,0.32143,0.26486,0.14286,0.14286,0.5,0.14286,1.0,0,1,0,0,0,21,0,0,0,0,0,3,0,0,0,0,0,7,0,0,0,0,1],[88,92,0.9565,0.24991,0.21133,0.14286,0.14286,0.21429,0.14,1.0,0,1,0,0,0,24,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,1],[92,92,1.0,0.22321,0.14698,0.14286,0.14286,0.21429,0.14286,0.71429,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"517a20ddc0d1806a","q":"Let  $G = (V, E)$  be a finite simple graph on  $n$  vertices.  An edge  $e$  of  $G$  is called a *bottleneck* if one can partition  $V$  into two disjoint sets  $A$  and  $B$  such that \n\n\n- at most  $100$  edges of  $G$  have one endpoint in  $A$  and one endpoint in  $B$ ; and\n- the edge  $e$  is one such edge (meaning the edge  $e$  also has one endpoint in  $A$  and one endpoint in  $B$ ).\n\nProve that at most  $100n$  edges of  $G$  are bottlenecks.\n\n*Proposed by Yang Liu*","t":[{"b":3,"e":0.28571,"k":"rising","v":0.22768,"x":0.78571,"p":[[0,26,0.0,0.22768,0.27166,0.0,0.14286,0.28571,0.0,1.0,12,2,1,12,0,5,0,0,11,0,0,0,0,0,0,0,0,2,0,0,0,0,2],[4,26,0.1538,0.78571,0.27894,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0,6,0,0,0,0,18],[8,26,0.3077,0.59375,0.27919,0.42857,0.42857,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,4,0,0,13,0,0,0,0,0,6,0,0,0,0,8],[12,26,0.4615,0.62947,0.26692,0.42857,0.50001,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,13,0,0,1,0,0,6,0,0,0,0,9],[16,26,0.6154,0.54018,0.28061,0.42857,0.42857,0.71429,0.0,1.0,2,6,0,2,0,0,0,0,5,0,0,13,0,0,0,0,0,6,0,0,0,0,6],[20,26,0.7692,0.44643,0.26184,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,10,0,0,9,0,0,0,0,0,6,0,0,0,0,3],[24,26,0.9231,0.46429,0.20516,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,10,0,0,12,0,0,2,0,0,5,0,0,0,0,2],[26,26,1.0,0.41518,0.16506,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,16,0,0,0,0,0,3,0,0,0,0,1]]},{"b":7,"e":0.42857,"k":"rising","v":0.30356,"x":0.62054,"p":[[0,25,0.0,0.30356,0.27835,0.0,0.2857,0.4642,0.0,1.0,9,1,2,9,0,5,0,0,7,0,0,3,0,0,2,0,0,5,0,0,0,0,1],[4,25,0.16,0.45982,0.1461,0.42857,0.42857,0.4286,0.28571,0.71429,0,0,0,0,0,0,0,0,7,0,0,18,0,0,0,0,0,7,0,0,0,0,0],[8,25,0.32,0.50893,0.18877,0.42857,0.42859,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,17,0,0,1,0,0,7,0,0,0,0,2],[12,25,0.48,0.46429,0.14286,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,19,0,0,0,0,0,7,0,0,0,0,0],[16,25,0.64,0.47768,0.17717,0.42857,0.42857,0.42858,0.28571,1.0,0,2,0,0,0,0,0,0,5,0,0,21,0,0,0,0,0,4,0,0,0,0,2],[20,25,0.8,0.47768,0.14987,0.42857,0.42857,0.42858,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,23,0,0,0,0,0,5,0,0,0,0,1],[24,25,0.96,0.62054,0.20705,0.42857,0.64286,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,11,0,0,0,0,5],[25,25,1.0,0.5759,0.18379,0.42857,0.50001,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,14,0,0,1,0,0,13,0,0,0,0,2]]}]},{"i":"aad139c5e50128b2","q":"Let  $ABC$  be a right angled triangle with  $\\angle B=90^{\\circ}$ . Let  $I$  be the incentre of triangle  $ABC$ . Suppose  $AI$  is extended to meet  $BC$  at  $F$  . The perpendicular on  $AI$  at  $I$  is extended to meet  $AC$  at  $E$  . Prove that  $IE = IF$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.81696,"x":0.97767,"p":[[0,28,0.0,0.81696,0.31182,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,4,0,21],[4,28,0.1429,0.9107,0.14617,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,10,0,19],[8,28,0.2857,0.94642,0.08565,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[12,28,0.4286,0.91517,0.12299,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,18],[16,28,0.5714,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,28,0.7143,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[24,28,0.8571,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,28,1.0,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.86159,"x":1.0,"p":[[0,27,0.0,0.86159,0.22444,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,3,0,21],[4,27,0.1481,0.91516,0.19187,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,0,22],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,27,0.5926,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[27,27,1.0,0.96427,0.14289,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30]]}]},{"i":"5bc5e54fefa9550d","q":"Let  $\\mathbb{N}$  be the set of all positive integers and  $S=\\left\\{(a, b, c, d) \\in \\mathbb{N}^4: a^2+b^2+c^2=d^2\\right\\}$ . Find the largest positive integer  $m$  such that  $m$  divides abcd for all  $(a, b, c, d) \\in S$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,47,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,47,0.0851,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,47,0.1702,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,47,0.2553,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,47,0.3404,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,47,0.4255,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,47,0.5106,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,47,0.5957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,47,0.6809,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[36,47,0.766,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,47,0.8511,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[44,47,0.9362,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[47,47,1.0,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28]]},{"b":3,"e":0.57143,"k":"flat","v":0.94196,"x":1.0,"p":[[0,48,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,48,0.0833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,48,0.1667,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,48,0.25,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,48,0.3333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,48,0.4167,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,48,0.5,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[28,48,0.5833,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,48,0.6667,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[36,48,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,48,0.8333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,48,0.9167,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[48,48,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"624fc57af16d4ead","q":"Let  $\\epsilon$  be an  $n$ -th root of the unity and suppose  $z=p(\\epsilon)$  is a real number where  $p$  is some polinomial with integer coefficients. Prove there exists a polinomial  $q$  with integer coefficients such that  $z=q(2\\cos(2\\pi/n))$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.52678,"x":0.77679,"p":[[0,14,0.0,0.52678,0.2271,0.42857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,15,0,0,2,0,0,2,0,0,5,0,2],[4,14,0.2857,0.77679,0.2111,0.71429,0.78571,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,5,0,11],[8,14,0.5714,0.70088,0.24837,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,7,0,0,5,0,8],[12,14,0.8571,0.68747,0.24339,0.53539,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,6,0,0,9,0,5],[14,14,1.0,0.62498,0.20438,0.42857,0.64286,0.85702,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,9,0,0,5,0,0,7,0,0,8,0,1]]},{"b":4,"e":0.42857,"k":"volatile","v":0.45534,"x":0.82142,"p":[[0,9,0.0,0.45534,0.16916,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,16,0,0,2,0,0,2,0,0,3,0,0],[4,9,0.4444,0.82142,0.22868,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,8,0,14],[8,9,0.8889,0.7098,0.22156,0.5354,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,3,0,0,6,0,8],[9,9,1.0,0.72766,0.2269,0.57132,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,6,0,0,7,0,8]]}]},{"i":"52bf984e3c35235d","q":"Let  $\\mathcal{F}$  be a family of subsets of  $\\{1,2,\\ldots, 2017\\}$  with the following property: if  $S_1$  and  $S_2$  are two elements of  $\\mathcal{F}$  with  $S_1\\subsetneq S_2$ , then  $|S_2\\setminus S_1|$  is odd. Compute the largest number of subsets  $\\mathcal{F}$  may contain.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.78571,"x":0.92857,"p":[[0,19,0.0,0.78571,0.17496,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,0,4,0,10],[4,19,0.2105,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20],[8,19,0.4211,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[12,19,0.6316,0.91071,0.12753,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,21],[16,19,0.8421,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[19,19,1.0,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20]]},{"b":4,"e":0.71429,"k":"flat","v":0.79911,"x":0.9308,"p":[[0,25,0.0,0.79911,0.17445,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,0,2,0,11],[4,25,0.16,0.9308,0.12813,0.98214,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,1,24],[8,25,0.32,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[12,25,0.48,0.89732,0.12993,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,19],[16,25,0.64,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[20,25,0.8,0.88616,0.13583,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,1,18],[24,25,0.96,0.87946,0.13882,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,1,0,18],[25,25,1.0,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20]]}]},{"i":"bc2473e2144dbf56","q":"Let  $\\mathbb N = {1, 2, 3, . . .}$ . For real  $x, y$ , set  $S(x, y) = \\{s | s = [nx+y], n \\in \\mathbb N\\}$ . Prove that if  $r > 1$  is a rational number, there exist real numbers  $u$  and  $v$  such that\n\\[S(r, 0) \\cap S(u, v) = \\emptyset, S(r, 0) \\cup S(u, v) = \\mathbb N.\\]","t":[{"b":4,"e":0.71429,"k":"falling","v":0.19642,"x":0.59819,"p":[[0,39,0.0,0.47321,0.3102,0.24999,0.42857,0.71429,0.0,1.0,3,3,1,3,0,5,0,0,6,0,0,4,0,0,2,0,0,6,0,0,3,0,3],[4,39,0.1026,0.59819,0.28891,0.39286,0.57143,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,5,0,0,3,0,0,8,0,0,4,0,0,2,0,7],[8,39,0.2051,0.53122,0.30771,0.28571,0.57121,0.71429,0.0,1.0,3,4,0,3,0,2,0,0,7,0,0,2,0,0,4,0,0,7,0,0,3,0,4],[12,39,0.3077,0.56695,0.31841,0.28571,0.64286,0.85714,0.0,1.0,3,5,0,3,0,3,0,0,3,0,0,4,0,0,3,0,0,7,0,0,4,0,5],[16,39,0.4103,0.58926,0.25692,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,2,0,0,3,0,0,5,0,0,10,0,0,5,0,2],[20,39,0.5128,0.55801,0.28203,0.39286,0.571,0.85714,0.0,1.0,1,4,0,1,0,3,0,0,4,0,0,7,0,0,5,0,0,3,0,0,5,0,4],[24,39,0.6154,0.47765,0.26631,0.28571,0.4998,0.71429,0.0,1.0,2,1,0,2,0,5,0,0,4,0,0,5,0,0,6,0,0,6,0,0,3,0,1],[28,39,0.7179,0.48658,0.32114,0.14286,0.42859,0.75,0.0,1.0,3,3,0,3,0,6,0,0,4,0,0,4,0,0,3,0,0,4,0,0,5,0,3],[32,39,0.8205,0.39283,0.27892,0.14286,0.28571,0.57143,0.0,1.0,3,2,0,3,0,7,0,0,8,0,0,3,0,0,4,0,0,4,0,0,1,0,2],[36,39,0.9231,0.29017,0.2244,0.14286,0.28571,0.4642,0.0,0.71429,5,0,0,5,0,10,0,0,7,0,0,2,0,0,5,0,0,3,0,0,0,0,0],[39,39,1.0,0.19642,0.17402,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,14,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.35267,"x":0.6205,"p":[[0,45,0.0,0.42856,0.34069,0.14286,0.42857,0.75,0.0,1.0,7,2,1,7,0,5,0,0,2,0,0,5,0,0,3,0,0,2,0,0,6,0,2],[4,45,0.0889,0.6205,0.25408,0.42857,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,5,0,0,8,0,3],[8,45,0.1778,0.56695,0.29121,0.28571,0.57143,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,5,0,0,3,0,0,6,0,0,7,0,0,2,0,5],[12,45,0.2667,0.45532,0.2586,0.14289,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,9,0,0,3,0,0,6,0,0,6,0,0,5,0,0,1,0,2],[16,45,0.3556,0.51786,0.29179,0.2857,0.57143,0.71429,0.0,1.0,3,2,0,3,0,2,0,0,6,0,0,4,0,0,4,0,0,6,0,0,5,0,2],[20,45,0.4444,0.45969,0.30882,0.14286,0.571,0.71429,0.0,1.0,4,1,0,4,0,7,0,0,2,0,0,2,0,0,5,0,0,7,0,0,4,0,1],[24,45,0.5333,0.47322,0.25614,0.25,0.42857,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,1,0,0,9,0,0,5,0,0,5,0,0,3,0,1],[28,45,0.6222,0.59371,0.21756,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,7,0,0,5,0,0,9,0,0,4,0,2],[32,45,0.7111,0.56247,0.24468,0.42857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,2,0,0,8,0,0,4,0,0,8,0,0,4,0,2],[36,45,0.8,0.43746,0.24982,0.25,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,6,0,0,5,0,0,3,0,0,3,0,1],[40,45,0.8889,0.37943,0.22756,0.14286,0.42857,0.571,0.0,0.85714,2,0,0,2,0,8,0,0,4,0,0,9,0,0,5,0,0,2,0,0,2,0,0],[44,45,0.9778,0.38391,0.22139,0.14286,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,11,0,0,5,0,0,5,0,0,6,0,0,4,0,0,1,0,0],[45,45,1.0,0.35267,0.20819,0.28571,0.28571,0.42858,0.0,0.71429,3,0,0,3,0,4,0,0,12,0,0,6,0,0,2,0,0,5,0,0,0,0,0]]}]},{"i":"5cfc61b62c83501b","q":"Let  $H{}$  be the orthocenter of the triangle  $ABC{}$  and  $X{}$  be the midpoint of the side  $BC.$  The perpendicular at  $H{}$  to  $HX{}$  intersects the sides  $(AB)$  and  $(AC)$  at  $Y{}$  and  $Z{}$  respectively. Let  $O{}$  be the circumcenter of  $ABC{}$  and  $O'$  be the circumcenter of  $BHC.$  [list=a]\n[*]Prove that  $HY=HZ.$ [*]Prove that  $\\overrightarrow{AY}+\\overrightarrow{AZ}=2\\overrightarrow{OO'}.$ [/list]","t":[{"b":3,"e":0.85714,"k":"falling","v":0.47765,"x":0.99554,"p":[[0,86,0.0,0.83927,0.25693,0.67857,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,1,0,21],[4,86,0.0465,0.92857,0.21129,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[8,86,0.093,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,86,0.1395,0.90625,0.21312,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,25],[16,86,0.186,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,86,0.2326,0.93308,0.19545,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[24,86,0.2791,0.94643,0.17405,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[28,86,0.3256,0.94196,0.17076,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[32,86,0.3721,0.92856,0.1786,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[36,86,0.4186,0.92856,0.17499,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[40,86,0.4651,0.90178,0.22142,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,6,0,23],[44,86,0.5116,0.96428,0.12878,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[48,86,0.5581,0.95535,0.13092,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[52,86,0.6047,0.9375,0.15126,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[56,86,0.6512,0.86158,0.23005,0.857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,6,0,19],[60,86,0.6977,0.84811,0.26736,0.85711,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,19],[64,86,0.7442,0.75891,0.24599,0.57143,0.85707,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,0,7,0,11],[68,86,0.7907,0.76784,0.26905,0.5713,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,4,0,0,5,0,14],[72,86,0.8372,0.58033,0.2878,0.28571,0.57143,0.85704,0.0,1.0,2,4,0,2,0,1,0,0,6,0,0,3,0,0,5,0,0,6,0,0,5,0,4],[76,86,0.8837,0.66512,0.29367,0.39286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,5,0,0,1,0,0,4,0,0,5,0,0,6,0,8],[80,86,0.9302,0.68302,0.27604,0.49968,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,7,0,0,0,0,0,3,0,0,9,0,0,3,0,9],[84,86,0.9767,0.65624,0.30275,0.39286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,5,0,0,2,0,0,2,0,0,7,0,0,5,0,8],[86,86,1.0,0.47765,0.25904,0.2857,0.42859,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,11,0,0,2,0,0,5,0,0,5,0,0,4,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.61599,"x":0.9732,"p":[[0,90,0.0,0.7991,0.24448,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,1,0,0,5,0,16],[4,90,0.0444,0.88839,0.23072,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,22],[8,90,0.0889,0.90179,0.22711,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,25],[12,90,0.1333,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[16,90,0.1778,0.9732,0.05579,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,90,0.2222,0.86158,0.23822,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,3,0,21],[24,90,0.2667,0.86606,0.22851,0.85708,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,7,0,19],[28,90,0.3111,0.90178,0.17655,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,6,0,21],[32,90,0.3556,0.83035,0.26351,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,0,5,0,18],[36,90,0.4,0.85714,0.25254,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,20],[40,90,0.4444,0.85712,0.23147,0.82132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,4,0,20],[44,90,0.4889,0.82588,0.25688,0.82132,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,0,7,0,17],[48,90,0.5333,0.88392,0.1692,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,4,0,19],[52,90,0.5778,0.84373,0.26574,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,21],[56,90,0.6222,0.83035,0.23266,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,5,0,0,6,0,16],[60,90,0.6667,0.90178,0.21559,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,4,0,24],[64,90,0.7111,0.84808,0.22859,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,6,0,17],[68,90,0.7556,0.79906,0.25723,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,4,0,0,4,0,16],[72,90,0.8,0.6741,0.33357,0.2857,0.78564,1.0,0.0,1.0,1,11,0,1,0,3,0,0,6,0,0,0,0,0,1,0,0,5,0,0,5,0,11],[76,90,0.8444,0.70527,0.26004,0.57143,0.71429,0.85714,0.14,1.0,0,7,0,0,0,2,0,0,4,0,0,0,0,0,3,0,0,9,0,0,7,0,7],[80,90,0.8889,0.70981,0.23551,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,9,0,0,8,0,6],[84,90,0.9333,0.61599,0.29123,0.28571,0.71429,0.85714,0.14,1.0,0,5,0,0,0,4,0,0,5,0,0,2,0,0,4,0,0,5,0,0,7,0,5],[88,90,0.9778,0.63392,0.23674,0.42859,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,2,0,0,4,0,0,9,0,0,8,0,2],[90,90,1.0,0.71427,0.23957,0.57142,0.78564,0.85714,0.143,1.0,0,5,0,0,0,1,0,0,4,0,0,1,0,0,3,0,0,7,0,0,11,0,5]]}]},{"i":"04d521715ccd4bbc","q":"Let  $a < b < c < d < e$  be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values \u200b\u200bthat  $e$  can take.","t":[{"b":6,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,46,0.0,0.86607,0.17835,0.78571,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,6,0,18],[4,46,0.087,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[8,46,0.1739,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[12,46,0.2609,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[16,46,0.3478,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[20,46,0.4348,0.92857,0.12877,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,23],[24,46,0.5217,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[28,46,0.6087,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,46,0.6957,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,46,0.7826,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,46,0.8696,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.84374,"x":0.99554,"p":[[0,87,0.0,0.84374,0.20002,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,1,0,19],[4,87,0.046,0.92411,0.13825,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,23],[8,87,0.092,0.93304,0.14279,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[12,87,0.1379,0.90179,0.16917,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,23],[16,87,0.1839,0.92857,0.15567,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[20,87,0.2299,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[24,87,0.2759,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[28,87,0.3218,0.94196,0.10631,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[32,87,0.3678,0.87946,0.19269,0.57143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,23],[36,87,0.4138,0.90625,0.14987,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,21],[40,87,0.4598,0.91964,0.15126,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24],[44,87,0.5057,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[48,87,0.5517,0.92857,0.14725,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,25],[52,87,0.5977,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[56,87,0.6437,0.91518,0.16698,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,25],[60,87,0.6897,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[64,87,0.7356,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[68,87,0.7816,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[72,87,0.8276,0.94643,0.12242,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[76,87,0.8736,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[80,87,0.9195,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,87,0.9655,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[87,87,1.0,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26]]}]},{"i":"1c6af154b8706642","q":"Let  $\\tau (n)$  denote the number of positive integer divisors of  $n$ . Find the sum of the six least positive integers  $n$  that are solutions to  $\\tau (n) + \\tau (n+1) = 7$ .","t":[{"b":0,"e":0.857,"k":"flat","v":0.88839,"x":1.0,"p":[[0,125,0.0,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[4,125,0.032,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,125,0.064,0.92411,0.2448,1.0,1.0,1.0,0.0,1.0,2,28,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[12,125,0.096,0.95089,0.17717,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[16,125,0.128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,125,0.16,0.88839,0.29175,1.0,1.0,1.0,0.0,1.0,3,26,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[24,125,0.192,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[28,125,0.224,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,125,0.256,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,125,0.288,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,125,0.32,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,125,0.352,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[48,125,0.384,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,125,0.416,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[56,125,0.448,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[60,125,0.48,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[64,125,0.512,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[68,125,0.544,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[72,125,0.576,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[76,125,0.608,0.9375,0.18536,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[80,125,0.64,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[84,125,0.672,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[88,125,0.704,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[92,125,0.736,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[96,125,0.768,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[100,125,0.8,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[104,125,0.832,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[108,125,0.864,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[112,125,0.896,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[116,125,0.928,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[120,125,0.96,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[124,125,0.992,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[125,125,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,85,0.0,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[4,85,0.0471,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,85,0.0941,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,85,0.1412,0.92411,0.24218,1.0,1.0,1.0,0.0,1.0,2,27,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[16,85,0.1882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,85,0.2353,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,85,0.2824,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,85,0.3294,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,85,0.3765,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,85,0.4235,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,85,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,85,0.5176,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,85,0.5647,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[52,85,0.6118,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,85,0.6588,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[60,85,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,85,0.7529,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,85,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,85,0.8471,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[76,85,0.8941,0.96652,0.07781,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,1,26],[80,85,0.9412,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[84,85,0.9882,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[85,85,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"984480fdf9b64214","q":"Let  $a$  and  $b$  be relatively prime positive integers such that  $a/b$  is the maximum possible value of \\[\\sin^2x_1+\\sin^2x_2+\\sin^2x_3+\\cdots+\\sin^2x_{2007},\\] where, for  $1\\leq i\\leq 2007$ ,  $x_i$  is a nonnegative real number, and \\[x_1+x_2+x_3+\\cdots+x_{2007}=\\pi.\\] Find the value of  $a+b$ .","t":[{"b":1,"e":0.0,"k":"falling","v":0.12054,"x":0.88393,"p":[[0,40,0.0,0.62946,0.19186,0.53571,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,6,0,0,8,0,0,10,0,0,4,0,2],[4,40,0.1,0.88393,0.12595,0.85714,0.85714,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,17,0,12],[8,40,0.2,0.82143,0.17496,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,13,0,10],[12,40,0.3,0.8125,0.16146,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,13,0,8],[16,40,0.4,0.84375,0.15714,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,15,0,10],[20,40,0.5,0.80803,0.20079,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,14,0,9],[24,40,0.6,0.78571,0.21429,0.71429,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,0,17,0,6],[28,40,0.7,0.78124,0.29664,0.85714,0.85714,1.0,0.0,1.0,3,9,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,9],[32,40,0.8,0.58036,0.33491,0.25,0.71429,0.85714,0.0,1.0,4,2,0,4,0,4,0,0,1,0,0,2,0,0,3,0,0,4,0,0,12,0,2],[36,40,0.9,0.47321,0.40632,0.0,0.57143,0.85714,0.0,1.0,10,4,0,10,0,4,0,0,1,0,0,0,0,0,2,0,0,2,0,0,9,0,4],[40,40,1.0,0.12054,0.27458,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0]]},{"b":2,"e":0.85714,"k":"rising","v":0.60714,"x":0.88393,"p":[[0,61,0.0,0.60714,0.14725,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,4,0,0,4,0,1],[4,61,0.0656,0.85268,0.16554,0.82143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,11,0,13],[8,61,0.1311,0.78571,0.22304,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,0,13,0,9],[12,61,0.1967,0.79018,0.22011,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,9,0,11],[16,61,0.2623,0.86594,0.17485,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,13],[20,61,0.3279,0.82141,0.23421,0.82132,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,12,0,12],[24,61,0.3934,0.83929,0.1948,0.85714,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,13,0,12],[28,61,0.459,0.87054,0.12556,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,18,0,10],[32,61,0.5246,0.85714,0.16366,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,14,0,12],[36,61,0.5902,0.8125,0.19045,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,14,0,9],[40,61,0.6557,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,12],[44,61,0.7213,0.7857,0.20518,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,0,17,0,6],[48,61,0.7869,0.77679,0.20183,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,11,0,7],[52,61,0.8525,0.85268,0.1838,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,11,0,14],[56,61,0.918,0.79464,0.18877,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,0,18,0,5],[60,61,0.9836,0.78571,0.15567,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,13,0,5],[61,61,1.0,0.80804,0.16982,0.82143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,18,0,6]]}]},{"i":"53e415cc9b51892c","q":"Let  $a,b \\in \\mathbb R,f(x)=ax+b+\\frac{9}{x}.$  Prove that there exists  $x_0 \\in \\left[1,9 \\right],$  such that  $|f(x_0)| \\ge 2.$","t":[{"b":2,"e":0.0,"k":"flat","v":0.04464,"x":0.17857,"p":[[0,68,0.0,0.10714,0.21429,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[4,68,0.0588,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.10714,0.17496,0.0,0.0,0.17857,0.0,0.71429,21,0,0,21,0,3,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,68,0.1765,0.17857,0.24223,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,3,0,0,5,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[16,68,0.2353,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.14732,0.22442,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,0,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[28,68,0.4118,0.09821,0.20024,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[32,68,0.4706,0.11161,0.1665,0.0,0.0,0.17857,0.0,0.71429,19,0,0,19,0,5,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,68,0.5294,0.12054,0.2055,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,68,0.5882,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.11607,0.15335,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,68,0.7059,0.10268,0.18638,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[52,68,0.7647,0.06688,0.11279,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.07589,0.12869,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.0758,0.107,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.04464,0.1357,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,36,0.0,0.03116,0.08541,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,36,0.7778,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"423c3985ab1aa490","q":"Let  $ABCD$  be a trapezium with bases  $AB$  and  $CD$  in which  $AB + CD = AD$ . Diagonals  $AC$  and  $BD$  intersect in point  $E$ . Line passing through point  $E$  and parallel to bases of trapezium cuts  $AD$  in point  $F$ . Prove that  $\\sphericalangle BFC = 90 ^{\\circ}$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.93303,"x":1.0,"p":[[0,57,0.0,0.93303,0.14719,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,25],[4,57,0.0702,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,57,0.1404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,57,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,57,0.2807,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,57,0.3509,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,57,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,57,0.4912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,57,0.5614,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,57,0.6316,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[40,57,0.7018,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,57,0.7719,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,57,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,57,0.9123,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,57,0.9825,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[57,57,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.71429,"k":"falling","v":0.64734,"x":0.95089,"p":[[0,24,0.0,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[4,24,0.1667,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[8,24,0.3333,0.87058,0.19009,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,5,0,19],[12,24,0.5,0.75,0.16366,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,9,0,0,7,0,6],[16,24,0.6667,0.75445,0.17942,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,5,0,8],[20,24,0.8333,0.64734,0.19876,0.57132,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,4,0,0,13,0,0,6,0,0,2,0,5],[24,24,1.0,0.73213,0.18815,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,9,0,0,5,0,7]]}]},{"i":"bda63257d19cfb34","q":"Let  $M=\\{1,2,\\dots,49\\}$  be the set of the first  $49$  positive integers. Determine the maximum integer  $k$  such that the set  $M$  has a subset of  $k$  elements such that there is no  $6$  consecutive integers in such subset. For this value of  $k$ , find the number of subsets of  $M$  with  $k$  elements with the given property.","t":[{"b":2,"e":0.71429,"k":"rising","v":0.6875,"x":0.96875,"p":[[0,26,0.0,0.6875,0.18707,0.57143,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,3,0,0,4,0,6],[4,26,0.1538,0.90179,0.17655,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,22],[8,26,0.3077,0.85267,0.14936,0.82143,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,12,0,12],[12,26,0.4615,0.86161,0.17307,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,6,0,17],[16,26,0.6154,0.88393,0.16146,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,8,0,18],[20,26,0.7692,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,26,0.9231,0.93304,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[26,26,1.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25]]},{"b":4,"e":1.0,"k":"rising","v":0.65177,"x":0.96429,"p":[[0,29,0.0,0.65177,0.18537,0.57143,0.57143,0.71429,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,1,0,0,16,0,0,8,0,0,3,0,3],[4,29,0.1379,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[8,29,0.2759,0.90179,0.14032,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,19],[12,29,0.4138,0.91964,0.15947,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,25],[16,29,0.5517,0.89732,0.1394,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,18],[20,29,0.6897,0.87054,0.16888,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,5,0,18],[24,29,0.8276,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[28,29,0.9655,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[29,29,1.0,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26]]}]},{"i":"961514e6f4120fb6","q":"Let  $a,b,c$  be integer numbers such that  $(a+b+c) \\mid (a^{2}+b^{2}+c^{2})$ . Show that there exist infinitely many positive integers  $n$  such that  $(a+b+c) \\mid (a^{n}+b^{n}+c^{n})$ .\r\n\r\n*Laurentiu Panaitopol*","t":[{"b":5,"e":0.28571,"k":"flat","v":0.35266,"x":0.60266,"p":[[0,33,0.0,0.46427,0.31943,0.24999,0.42859,0.71429,0.0,1.0,5,3,0,5,0,3,0,0,6,0,0,3,0,0,4,0,0,5,0,0,3,0,3],[4,33,0.1212,0.42854,0.35713,0.14286,0.35714,0.74996,0.0,1.0,6,5,0,6,0,7,0,0,3,0,0,3,0,0,4,0,0,1,0,0,3,0,5],[8,33,0.2424,0.58926,0.31894,0.28571,0.57143,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,5,0,0,3,0,0,5,0,0,5,0,0,3,0,7],[12,33,0.3636,0.35266,0.29662,0.14286,0.28571,0.4642,0.0,1.0,7,1,0,7,0,5,0,0,6,0,0,6,0,0,2,0,0,1,0,0,4,0,1],[16,33,0.4848,0.59372,0.25532,0.39286,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,6,0,0,3,0,0,3,0,0,12,0,0,3,0,3],[20,33,0.6061,0.45534,0.32031,0.24999,0.42859,0.71429,0.0,1.0,6,3,0,6,0,2,0,0,6,0,0,3,0,0,4,0,0,6,0,0,2,0,3],[24,33,0.7273,0.58033,0.28557,0.42857,0.57143,0.71429,0.0,1.0,1,6,0,1,0,4,0,0,2,0,0,4,0,0,8,0,0,6,0,0,1,0,6],[28,33,0.8485,0.60266,0.31285,0.39286,0.71429,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,2,0,0,5,0,0,7,0,0,4,0,6],[32,33,0.9697,0.49986,0.20217,0.42857,0.57143,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,4,0,0,4,0,0,16,0,0,3,0,0,2,0,0],[33,33,1.0,0.45083,0.14767,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,6,0,0,13,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.44641,"x":0.85711,"p":[[0,38,0.0,0.55802,0.28202,0.28571,0.57121,0.85704,0.14286,1.0,0,4,0,0,0,4,0,0,6,0,0,5,0,0,4,0,0,4,0,0,5,0,4],[4,38,0.1053,0.62054,0.34738,0.39288,0.71429,0.85714,0.0,1.0,5,6,0,5,0,2,0,0,1,0,0,1,0,0,2,0,0,8,0,0,7,0,6],[8,38,0.2105,0.62946,0.33476,0.28571,0.71429,0.85714,0.0,1.0,3,7,0,3,0,2,0,0,4,0,0,1,0,0,4,0,0,3,0,0,8,0,7],[12,38,0.3158,0.60266,0.3655,0.14286,0.71429,0.89286,0.0,1.0,3,8,0,3,0,6,0,0,2,0,0,0,0,0,2,0,0,5,0,0,6,0,8],[16,38,0.4211,0.47768,0.32264,0.24999,0.42857,0.71429,0.0,1.0,2,6,0,2,0,6,0,0,6,0,0,6,0,0,2,0,0,3,0,0,1,0,6],[20,38,0.5263,0.65177,0.3071,0.42857,0.71429,0.85714,0.0,1.0,3,6,0,3,0,0,0,0,4,0,0,2,0,0,3,0,0,6,0,0,8,0,6],[24,38,0.6316,0.44643,0.35848,0.14286,0.28571,0.75,0.0,1.0,5,4,0,5,0,8,0,0,4,0,0,1,0,0,1,0,0,5,0,0,4,0,4],[28,38,0.7368,0.44641,0.32877,0.14286,0.42859,0.71429,0.0,1.0,7,2,0,7,0,3,0,0,3,0,0,4,0,0,4,0,0,5,0,0,4,0,2],[32,38,0.8421,0.71429,0.29014,0.57143,0.85714,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,2,0,0,1,0,0,3,0,0,6,0,0,9,0,8],[36,38,0.9474,0.75893,0.21852,0.71429,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,14,0,0,9,0,6],[38,38,1.0,0.85711,0.14291,0.82132,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,12,0,12]]}]},{"i":"304e124b760dfc27","q":"Let  $D$  be a point on the inside of triangle  $ABC$  such that  $AD=CD$ ,  $\\angle DAB=70^{\\circ}$ ,  $\\angle DBA=30^{\\circ}$  and  $\\angle DBC=20^{\\circ}$ . Find the measure of angle  $\\angle DCB$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.1026,"x":0.15152,"p":[[0,44,0.0,0.12499,0.09936,0.14286,0.14286,0.14286,0.0,0.571,7,0,0,7,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,44,0.0909,0.12947,0.09689,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,44,0.1818,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.1026,0.06418,0.0,0.14286,0.14286,0.0,0.143,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.12491,0.09941,0.14214,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,44,0.8182,0.15152,0.07939,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,44,0.9091,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.12054,"x":0.17856,"p":[[0,14,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.16955,0.16147,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[8,14,0.5714,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.17856,0.15564,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,24,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[14,14,1.0,0.125,0.09942,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"d9d3992e766fb432","q":"Let  $O$  be the circumcenter of acute  $\\triangle ABC$ ( $AB<AC$ ), the angle bisector of  $\\angle BAC$  meets  $BC$  at  $T$  and  $M$  is the midpoint of  $AT$ . Point  $P$  lies inside  $\\triangle ABC$  such that  $PB\\perp PC$ .  $D,E$  distinct from  $P$  lies on the perpendicular to  $AP$  through  $P$  such that  $BD=BP, CE=CP$ . If  $AO$  bisects segment  $DE$ , prove that  $AO$  is tangent to the circumcircle of  $\\triangle AMP$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.19187,"x":0.45086,"p":[[0,103,0.0,0.21866,0.1515,0.14286,0.14286,0.28571,0.0,0.71429,3,0,1,3,0,17,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[4,103,0.0388,0.37947,0.19434,0.14286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,2,0,0,12,0,0,7,0,0,2,0,0,0,0,0],[8,103,0.0777,0.40179,0.19045,0.39285,0.42857,0.42858,0.0,1.0,2,1,0,2,0,4,0,0,2,0,0,17,0,0,6,0,0,0,0,0,0,0,1],[12,103,0.1165,0.39286,0.23958,0.14289,0.35714,0.42858,0.14286,1.0,0,2,0,0,0,9,0,0,7,0,0,9,0,0,1,0,0,4,0,0,0,0,2],[16,103,0.1553,0.41072,0.15046,0.28571,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,3,0,0,5,0,0,13,0,0,10,0,0,0,0,0,0,0,0],[20,103,0.1942,0.35268,0.20511,0.14286,0.35714,0.57141,0.0,0.71429,1,0,0,1,0,11,0,0,4,0,0,7,0,0,6,0,0,3,0,0,0,0,0],[24,103,0.233,0.30804,0.17168,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,10,0,0,5,0,0,12,0,0,2,0,0,1,0,0,0,0,0],[28,103,0.2718,0.45086,0.16013,0.42857,0.42857,0.57141,0.0,0.71429,1,0,0,1,0,2,0,0,3,0,0,14,0,0,9,0,0,3,0,0,0,0,0],[32,103,0.3107,0.39283,0.15969,0.2857,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,7,0,0,3,0,0,14,0,0,7,0,0,1,0,0,0,0,0],[36,103,0.3495,0.35268,0.15561,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,5,0,0,12,0,0,6,0,0,0,0,0,0,0,0],[40,103,0.3883,0.40177,0.17654,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,5,0,0,11,0,0,8,0,0,2,0,0,0,0,0],[44,103,0.4272,0.45084,0.17165,0.39286,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,12,0,0,10,0,0,1,0,0,0,0,1],[48,103,0.466,0.41514,0.17982,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,6,0,0,9,0,0,11,0,0,0,0,0,1,0,0],[52,103,0.5049,0.39732,0.20434,0.2857,0.42857,0.46431,0.0,1.0,1,1,0,1,0,6,0,0,5,0,0,12,0,0,5,0,0,2,0,0,0,0,1],[56,103,0.5437,0.38838,0.21198,0.25,0.42857,0.57143,0.0,1.0,2,1,0,2,0,6,0,0,5,0,0,8,0,0,10,0,0,0,0,0,0,0,1],[60,103,0.5825,0.39286,0.20203,0.2857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,6,0,0,10,0,0,7,0,0,1,0,0,0,0,1],[64,103,0.6214,0.3125,0.23808,0.14286,0.28574,0.42857,0.0,1.0,4,1,0,4,0,12,0,0,0,0,0,11,0,0,2,0,0,2,0,0,0,0,1],[68,103,0.6602,0.35268,0.18893,0.25,0.42857,0.4286,0.0,0.71429,3,0,0,3,0,5,0,0,6,0,0,12,0,0,4,0,0,2,0,0,0,0,0],[72,103,0.699,0.33036,0.2034,0.14286,0.35714,0.42858,0.0,0.71429,3,0,0,3,0,9,0,0,4,0,0,9,0,0,5,0,0,2,0,0,0,0,0],[76,103,0.7379,0.36161,0.1636,0.25,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,4,0,0,15,0,0,4,0,0,1,0,0,0,0,0],[80,103,0.7767,0.31694,0.17025,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,9,0,0,6,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[84,103,0.8155,0.37947,0.18423,0.2857,0.42857,0.42858,0.0,0.71429,2,0,0,2,0,5,0,0,4,0,0,15,0,0,3,0,0,3,0,0,0,0,0],[88,103,0.8544,0.37947,0.19759,0.1429,0.42857,0.42857,0.0,1.0,1,1,0,1,0,8,0,0,2,0,0,14,0,0,6,0,0,0,0,0,0,0,1],[92,103,0.8932,0.33928,0.16269,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,12,0,0,7,0,0,5,0,0,1,0,0,0,0,0],[96,103,0.932,0.2857,0.18209,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,9,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[100,103,0.9709,0.29464,0.15947,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,10,0,0,7,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[103,103,1.0,0.19187,0.11632,0.14286,0.14288,0.2857,0.0,0.571,3,0,0,3,0,18,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.15616,"x":0.33036,"p":[[0,68,0.0,0.21875,0.17851,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,15,0,0,6,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[4,68,0.0588,0.25447,0.20743,0.10714,0.1429,0.42857,0.0,0.71429,8,0,0,8,0,9,0,0,1,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[8,68,0.1176,0.33033,0.17286,0.14286,0.35714,0.4286,0.0,0.57143,2,0,0,2,0,8,0,0,6,0,0,10,0,0,6,0,0,0,0,0,0,0,0],[12,68,0.1765,0.27676,0.19861,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,10,0,0,5,0,0,7,0,0,4,0,0,1,0,0,0,0,0],[16,68,0.2353,0.23661,0.17353,0.14286,0.1429,0.42857,0.0,0.57143,6,0,0,6,0,11,0,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[20,68,0.2941,0.33036,0.21558,0.14286,0.35714,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,4,0,0,10,0,0,5,0,0,0,0,0,0,0,1],[24,68,0.3529,0.24545,0.15257,0.14286,0.2143,0.32143,0.0,0.57143,3,0,0,3,0,13,0,0,8,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[28,68,0.4118,0.24107,0.1729,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,14,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[32,68,0.4706,0.24554,0.1439,0.14286,0.2143,0.28571,0.0,0.57143,2,0,0,2,0,14,0,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[36,68,0.5294,0.19179,0.14116,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,17,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[40,68,0.5882,0.25893,0.20341,0.14286,0.2143,0.42857,0.0,0.57143,7,0,0,7,0,9,0,0,5,0,0,5,0,0,6,0,0,0,0,0,0,0,0],[44,68,0.6471,0.17858,0.15152,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,14,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[48,68,0.7059,0.2009,0.15093,0.14286,0.1429,0.28571,0.0,0.57143,6,0,0,6,0,14,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[52,68,0.7647,0.30346,0.21357,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,10,0,0,8,0,0,5,0,0,5,0,0,0,0,0,0,0,1],[56,68,0.8235,0.1875,0.1357,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,17,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[60,68,0.8824,0.19188,0.14113,0.14286,0.14286,0.2857,0.0,0.4286,6,0,0,6,0,15,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.20972,0.16746,0.14286,0.14286,0.32143,0.0,0.57143,6,0,0,6,0,15,0,0,3,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[68,68,1.0,0.15616,0.17628,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,10,0,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"180f355aaf6a8037","q":"Let  $a, b, c,$  and  $d$  be real numbers such that  $a \\geq b \\geq c \\geq d$  and  $$ a+b+c+d = 13 $$    $$ a^2+b^2+c^2+d^2=43. $$  Show that  $ab \\geq 3 + cd$ 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 $P$  be a point inside the equilateral triangle  $ABC$  such that  $6\\angle  PBC = 3\\angle  PAC = 2\\angle  PCA$ . Find the measure of the angle  $\\angle PBC$  .","t":[{"b":0,"e":0.0,"k":"flat","v":0.63838,"x":0.87945,"p":[[0,94,0.0,0.63838,0.32924,0.28571,0.64264,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,13,0,0,2,0,0,1,0,0,2,0,0,1,0,13],[4,94,0.0426,0.75891,0.27535,0.57132,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,6,0,0,2,0,15],[8,94,0.0851,0.79911,0.24186,0.67857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,7,0,0,0,0,17],[12,94,0.1277,0.82141,0.24745,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,2,0,0,2,0,19],[16,94,0.1702,0.81249,0.26832,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0,4,0,18],[20,94,0.2128,0.85713,0.23959,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,0,1,0,22],[24,94,0.2553,0.72768,0.28203,0.42857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,7,0,0,2,0,0,1,0,0,5,0,0,5,0,12],[28,94,0.2979,0.6875,0.31019,0.42857,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,7,0,0,7,0,0,1,0,0,2,0,0,0,0,15],[32,94,0.3404,0.8348,0.23177,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,2,0,19],[36,94,0.383,0.82142,0.24745,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,2,0,0,0,0,20],[40,94,0.4255,0.83927,0.22234,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,4,0,18],[44,94,0.4681,0.87945,0.21758,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,24],[48,94,0.5106,0.82143,0.2369,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,7,0,0,1,0,18],[52,94,0.5532,0.72765,0.26574,0.4286,0.78564,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,2,0,0,3,0,13],[56,94,0.5957,0.7991,0.26453,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,2,0,0,4,0,17],[60,94,0.6383,0.85267,0.21275,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,4,0,0,3,0,19],[64,94,0.6809,0.80802,0.25156,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,2,0,18],[68,94,0.7234,0.82589,0.2618,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,3,0,0,2,0,20],[72,94,0.766,0.75892,0.2822,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,6,0,0,4,0,14],[76,94,0.8085,0.72767,0.26332,0.42859,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,5,0,0,1,0,0,9,0,0,0,0,13],[80,94,0.8511,0.77232,0.29203,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,4,0,0,2,0,0,1,0,0,6,0,0,1,0,17],[84,94,0.8936,0.83481,0.22336,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,1,0,19],[88,94,0.9362,0.78122,0.25752,0.57132,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,4,0,0,4,0,15],[92,94,0.9787,0.73659,0.26753,0.53539,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,7,0,0,2,0,13],[94,94,1.0,0.72767,0.32015,0.42857,0.9285,1.0,0.0,1.0,2,16,0,2,0,0,0,0,3,0,0,4,0,0,3,0,0,3,0,0,1,0,16]]},{"b":1,"e":0.57143,"k":"rising","v":0.61159,"x":0.93302,"p":[[0,80,0.0,0.61159,0.31991,0.28571,0.49979,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,13,0,0,3,0,0,2,0,0,1,0,0,2,0,11],[4,80,0.05,0.88839,0.20119,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,1,0,23],[8,80,0.1,0.86159,0.24351,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,23],[12,80,0.15,0.83928,0.22798,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,3,0,19],[16,80,0.2,0.85266,0.2328,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,4,0,0,1,0,21],[20,80,0.25,0.79909,0.26212,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,17],[24,80,0.3,0.82589,0.24675,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,2,0,0,1,0,20],[28,80,0.35,0.77228,0.28986,0.571,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,2,0,0,1,0,18],[32,80,0.4,0.86607,0.2549,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,24],[36,80,0.45,0.80803,0.27108,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,3,0,0,4,0,18],[40,80,0.5,0.83036,0.26351,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,3,0,0,1,0,21],[44,80,0.55,0.78125,0.28344,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,4,0,0,1,0,18],[48,80,0.6,0.83929,0.25692,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,2,0,0,2,0,21],[52,80,0.65,0.88389,0.17662,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,0,0,22],[56,80,0.7,0.90179,0.19704,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,1,0,24],[60,80,0.75,0.83482,0.23449,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,7,0,0,1,0,19],[64,80,0.8,0.93302,0.15149,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,26],[68,80,0.85,0.85712,0.23961,0.67857,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,0,0,23],[72,80,0.9,0.83036,0.25614,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,4,0,0,0,0,21],[76,80,0.95,0.84821,0.26471,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,23],[80,80,1.0,0.84822,0.26229,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,23]]}]},{"i":"a31d0435682c34ce","q":"Let  $Q(x)$  be a quadratic trinomial. Given that the function  $P(x)=x^{2}Q(x)$  is increasing in the interval  $(0,\\infty )$ , prove that:\n\\[P(x) + P(y) + P(z) > 0\\]\nfor all real numbers  $x,y,z$  such that  $x+y+z>0$  and  $xyz>0$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.07143,"x":0.56249,"p":[[0,76,0.0,0.56249,0.2141,0.42857,0.57121,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,9,0,0,5,0,0,8,0,0,1,0,3],[4,76,0.0526,0.28125,0.2382,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,5,0,0,8,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[8,76,0.1053,0.39731,0.30036,0.14286,0.28571,0.60682,0.0,1.0,3,3,0,3,0,7,0,0,9,0,0,4,0,0,1,0,0,3,0,0,2,0,3],[12,76,0.1579,0.37947,0.32851,0.14286,0.28571,0.57143,0.0,1.0,6,3,0,6,0,7,0,0,5,0,0,5,0,0,2,0,0,0,0,0,4,0,3],[16,76,0.2105,0.31241,0.22435,0.14286,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,7,0,0,11,0,0,4,0,0,3,0,0,1,0,0,2,0,0],[20,76,0.2632,0.29911,0.26088,0.14286,0.21428,0.42857,0.0,1.0,5,1,0,5,0,11,0,0,5,0,0,6,0,0,1,0,0,1,0,0,2,0,1],[24,76,0.3158,0.25,0.20203,0.14286,0.14286,0.32143,0.0,0.85714,3,0,0,3,0,17,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[28,76,0.3684,0.29902,0.26578,0.14286,0.2857,0.42857,0.0,1.0,5,1,0,5,0,10,0,0,8,0,0,4,0,0,1,0,0,0,0,0,3,0,1],[32,76,0.4211,0.36161,0.2696,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,9,0,0,5,0,0,3,0,0,5,0,0,3,0,0,3,0,0],[36,76,0.4737,0.2767,0.21416,0.14286,0.14286,0.42857,0.0,0.71429,5,0,0,5,0,12,0,0,3,0,0,6,0,0,4,0,0,2,0,0,0,0,0],[40,76,0.5263,0.2009,0.14223,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,15,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[44,76,0.5789,0.22767,0.18849,0.14286,0.14286,0.28571,0.0,0.857,6,0,0,6,0,11,0,0,10,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[48,76,0.6316,0.21866,0.23144,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,16,0,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[52,76,0.6842,0.28125,0.25626,0.14286,0.14286,0.32143,0.0,0.85714,6,0,0,6,0,11,0,0,7,0,0,1,0,0,2,0,0,3,0,0,2,0,0],[56,76,0.7368,0.15179,0.13333,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,16,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.10705,0.10098,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.10268,0.08171,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.10259,0.1085,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.30804,"x":0.58036,"p":[[0,71,0.0,0.58034,0.20496,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,9,0,0,5,0,0,4,0,2],[4,71,0.0563,0.45534,0.25862,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,5,0,0,7,0,0,6,0,0,6,0,0,4,0,0,0,0,3],[8,71,0.1127,0.42855,0.30513,0.14286,0.42857,0.57143,0.0,1.0,4,3,0,4,0,6,0,0,4,0,0,5,0,0,7,0,0,0,0,0,3,0,3],[12,71,0.169,0.30804,0.23176,0.14286,0.28571,0.57143,0.0,0.71429,5,0,0,5,0,10,0,0,4,0,0,4,0,0,6,0,0,3,0,0,0,0,0],[16,71,0.2254,0.33482,0.249,0.14286,0.28571,0.46429,0.0,1.0,4,1,0,4,0,10,0,0,3,0,0,7,0,0,4,0,0,3,0,0,0,0,1],[20,71,0.2817,0.38393,0.29329,0.14286,0.28571,0.71429,0.0,1.0,3,1,0,3,0,11,0,0,3,0,0,5,0,0,1,0,0,5,0,0,3,0,1],[24,71,0.338,0.37947,0.25657,0.14286,0.35714,0.57143,0.0,0.85714,2,0,0,2,0,10,0,0,4,0,0,7,0,0,2,0,0,4,0,0,3,0,0],[28,71,0.3944,0.36159,0.23414,0.25,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,4,0,0,9,0,0,8,0,0,2,0,0,3,0,0,2,0,0],[32,71,0.4507,0.31696,0.21049,0.14286,0.28571,0.42858,0.0,0.71429,4,0,0,4,0,9,0,0,4,0,0,8,0,0,5,0,0,2,0,0,0,0,0],[36,71,0.507,0.36607,0.29653,0.14286,0.28571,0.46429,0.0,1.0,2,3,0,2,0,13,0,0,4,0,0,5,0,0,1,0,0,3,0,0,1,0,3],[40,71,0.5634,0.51786,0.26426,0.39286,0.57141,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,7,0,0,7,0,0,4,0,0,4,0,2],[44,71,0.6197,0.58036,0.25238,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,6,0,0,4,0,0,8,0,0,4,0,3],[48,71,0.6761,0.50446,0.24219,0.39286,0.42859,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,9,0,0,3,0,0,9,0,0,2,0,1],[52,71,0.7324,0.54463,0.27765,0.28571,0.57121,0.75,0.0,1.0,1,3,0,1,0,2,0,0,8,0,0,4,0,0,4,0,0,5,0,0,5,0,3],[56,71,0.7887,0.54447,0.18,0.42857,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,9,0,0,8,0,0,9,0,0,2,0,0],[60,71,0.8451,0.46429,0.21429,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,5,0,0,6,0,0,7,0,0,6,0,0,6,0,0,2,0,0],[64,71,0.9014,0.54908,0.2055,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,8,0,0,11,0,0,6,0,0,1,0,2],[68,71,0.9577,0.39286,0.19562,0.25,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,4,0,0,11,0,0,5,0,0,4,0,0,0,0,0],[71,71,1.0,0.48212,0.18122,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,3,0,0,9,0,0,9,0,0,7,0,0,0,0,0]]}]},{"i":"d340bf5adacc7bb4","q":"Let  $\\alpha$  be a root of the equation  $x^3-5x+3=0$  and let  $f(x)$  be a polynomial with rational coefficients. Prove that if  $f(\\alpha)$  be the root of equation above, then  $f(f(\\alpha))$  is a root, too.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.54464,"x":0.75,"p":[[0,44,0.0,0.75,0.27199,0.67857,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,0,0,0,1,0,0,1,0,0,15,0,8],[4,44,0.0909,0.65614,0.305,0.28571,0.85714,0.85714,0.14,1.0,0,4,0,0,0,4,0,0,6,0,0,0,0,0,2,0,0,1,0,0,15,0,4],[8,44,0.1818,0.65175,0.24469,0.28571,0.78564,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,0,0,0,3,0,0,4,0,0,16,0,0],[12,44,0.2727,0.62499,0.26183,0.28571,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,0,0,0,2,0,0,6,0,0,11,0,2],[16,44,0.3636,0.65846,0.2629,0.28571,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,2,0,0,1,1,0,17,0,1],[20,44,0.4545,0.54464,0.27993,0.28571,0.42857,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,15,0,0,0,0,0,1,0,0,5,0,0,8,0,2],[24,44,0.5455,0.58481,0.27515,0.28571,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,0,0,0,1,0,0,5,0,0,10,0,2],[28,44,0.6364,0.59375,0.27225,0.28571,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,0,0,0,1,0,0,4,0,0,14,0,0],[32,44,0.7273,0.6607,0.24937,0.28571,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,0,0,0,3,0,0,2,0,0,18,0,0],[36,44,0.8182,0.58036,0.2549,0.28571,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,6,0,0,11,0,0],[40,44,0.9091,0.6473,0.2575,0.28571,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,8,0,0,0,0,0,3,0,0,6,0,0,12,0,2],[44,44,1.0,0.69642,0.23891,0.71429,0.857,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,0,0,0,0,0,0,8,0,0,16,0,1]]},{"b":7,"e":0.57143,"k":"flat","v":0.66065,"x":0.80354,"p":[[0,43,0.0,0.74106,0.24856,0.67857,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,0,0,0,2,0,0,1,0,0,19,0,4],[4,43,0.093,0.7991,0.22263,0.85708,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,18,0,7],[8,43,0.186,0.66963,0.24857,0.39288,0.78571,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,1,0,0,2,0,0,5,0,0,14,0,2],[12,43,0.2791,0.70532,0.23129,0.57132,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,0,0,0,3,0,0,3,0,0,20,0,0],[16,43,0.3721,0.73658,0.21162,0.71429,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,4,0,0,20,0,1],[20,43,0.4651,0.67632,0.22728,0.57132,0.78571,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,0,0,1,4,0,0,5,0,0,16,0,0],[24,43,0.5581,0.73211,0.19804,0.67857,0.857,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,5,0,0,18,0,1],[28,43,0.6512,0.74105,0.18013,0.67857,0.85707,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,6,0,0,17,0,1],[32,43,0.7442,0.80354,0.16272,0.85711,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,22,0,3],[36,43,0.8372,0.75891,0.20342,0.82132,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,3,0,0,24,0,0],[40,43,0.9302,0.77678,0.17105,0.82132,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,0,24,0,0],[43,43,1.0,0.66065,0.25192,0.28571,0.85707,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,0,0,0,19,0,0]]}]},{"i":"ff14324fdbbad5fc","q":"Let  $S_n$  be the sum of the first  $n$  prime numbers. For example,\n\\[ S_5 = 2 + 3 + 5 + 7 + 11 = 28.\\]\nDoes there exist an integer  $k$  such that  $S_{2023} < k^2 < S_{2024}$ ?","t":[{"b":6,"e":0.85714,"k":"flat","v":0.93303,"x":0.97321,"p":[[0,26,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,26,0.1538,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,26,0.3077,0.94642,0.11156,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[12,26,0.4615,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,26,0.6154,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,26,0.7692,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[24,26,0.9231,0.93303,0.17852,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,23],[26,26,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":7,"e":1.0,"k":"flat","v":0.94196,"x":0.98214,"p":[[0,33,0.0,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[4,33,0.1212,0.95535,0.13092,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[8,33,0.2424,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,33,0.3636,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,33,0.4848,0.94196,0.17807,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,25],[20,33,0.6061,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,33,0.7273,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,33,0.8485,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,33,0.9697,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[33,33,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"d7851dad59ab8131","q":"Let  $N$  be a positive integer with  $2k$  digits. Its chunks are defined by the two numbers formed by the digits from  $1$  to  $k$  and  $k+1$  to  $2k$  (e.g. the chunks of 142856 are 142 and 856). We define the  $N$ -*reverse* as the number formed by switching its chunks (e.g. the reverse of 142856 is 856142 and for 1401 it is 114). We call a number *cearense* is it satisfies the following conditions:\n[list=i]\n[*] Has an even number of digits\n[*] Its chunks are relatively prime\n[*]Divides its reverse\n[/list]\nFind the two smallest cearense 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238,0.86611,0.17824,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,1,0,19],[136,252,0.5397,0.87945,0.15615,0.71429,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,4,0,18],[140,252,0.5556,0.86607,0.17835,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,4,0,18],[144,252,0.5714,0.79463,0.21707,0.71429,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,9,0,0,4,0,13],[148,252,0.5873,0.86606,0.15542,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,16],[152,252,0.6032,0.82589,0.17762,0.71429,0.78564,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,1,0,15],[156,252,0.619,0.85268,0.15355,0.71429,0.92857,1.0,0.5714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,1,0,16],[160,252,0.6349,0.82142,0.19562,0.71429,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,14,0,0,2,0,14],[164,252,0.6508,0.80802,0.20081,0.71429,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,12,0,0,3,0,13],[168,252,0.6667,0.77231,0.17446,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,0,1,0,10],[172,252,0.6825,0.81695,0.17583,0.71429,0.78564,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,3,0,13],[176,252,0.6984,0.82143,0.19885,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,9,0,0,4,0,14],[180,252,0.7143,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[184,252,0.7302,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[188,252,0.746,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[192,252,0.7619,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[196,252,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[200,252,0.7937,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[204,252,0.8095,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[208,252,0.8254,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[212,252,0.8413,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[216,252,0.8571,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[220,252,0.873,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[224,252,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[228,252,0.9048,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[232,252,0.9206,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[236,252,0.9365,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[240,252,0.9524,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[244,252,0.9683,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[248,252,0.9841,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[252,252,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c451f21c25bb9458","q":"Let  $a$ ,  $b$ ,  $c$  be real numbers greater than or equal to  $1$ . Prove that\n\\[ \\min \\left(\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )\\leq abc. \\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,43,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,43,0.093,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,43,0.186,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,43,0.2791,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,43,0.3721,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,43,0.4651,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,43,0.5581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,43,0.6512,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,43,0.7442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,43,0.8372,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,43,0.9302,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[43,43,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,18,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,18,0.2222,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,18,0.4444,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,18,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"89b1e9ad193214ac","q":"Let  $S(n)$  be the sum of digits of  $n$ . Determine all the pairs  $(a, b)$  of positive integers, such that the expression $S(an + b) - S(n)$  has a finite number of values, where  $n$  is varying in the positive integers.","t":[{"b":0,"e":0.2857,"k":"falling","v":0.35273,"x":0.89285,"p":[[0,43,0.0,0.82142,0.18898,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,6,0,13],[4,43,0.093,0.89285,0.24223,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,23],[8,43,0.186,0.83482,0.24513,0.82143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,7,0,17],[12,43,0.2791,0.89284,0.22306,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,22],[16,43,0.3721,0.79018,0.2766,0.53572,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,7,0,0,2,0,0,2,0,0,2,0,18],[20,43,0.4651,0.74107,0.32427,0.42857,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,1,0,0,5,0,0,0,0,0,3,0,0,6,0,14],[24,43,0.5581,0.75,0.25754,0.57143,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,5,0,0,3,0,13],[28,43,0.6512,0.60268,0.30249,0.42857,0.42859,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,4,0,0,11,0,0,0,0,0,3,0,0,5,0,7],[32,43,0.7442,0.56696,0.3164,0.28571,0.42857,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,6,0,0,9,0,0,1,0,0,3,0,0,2,0,8],[36,43,0.8372,0.51337,0.28315,0.28571,0.42857,0.74996,0.0,1.0,1,5,0,1,0,1,0,0,10,0,0,8,0,0,3,0,0,1,0,0,3,0,5],[40,43,0.9302,0.48661,0.31106,0.28571,0.42857,0.75,0.0,1.0,3,6,0,3,0,1,0,0,8,0,0,11,0,0,0,0,0,1,0,0,2,0,6],[43,43,1.0,0.35273,0.18207,0.2857,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,4,0,0,9,0,0,15,0,0,0,0,0,0,0,0,2,0,0]]},{"b":7,"e":0.2857,"k":"falling","v":0.40625,"x":0.85714,"p":[[0,40,0.0,0.84374,0.13997,0.71429,0.85705,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,7,0,12],[4,40,0.1,0.85714,0.32927,1.0,1.0,1.0,0.0,1.0,4,25,2,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[8,40,0.2,0.84806,0.29445,0.85711,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,21],[12,40,0.3,0.58482,0.33571,0.42857,0.42857,1.0,0.0,1.0,4,9,1,4,0,0,0,0,2,0,0,11,0,0,2,0,0,1,0,0,3,0,9],[16,40,0.4,0.58036,0.31931,0.42857,0.42857,1.0,0.0,1.0,3,9,1,3,0,0,0,0,3,0,0,12,0,0,1,0,0,3,0,0,1,0,9],[20,40,0.5,0.46429,0.33882,0.25,0.42857,0.75,0.0,1.0,6,6,0,6,0,2,0,0,2,0,0,13,0,0,0,0,0,1,0,0,2,0,6],[24,40,0.6,0.54465,0.34522,0.42857,0.42857,1.0,0.0,1.0,5,9,0,5,0,0,0,0,2,0,0,13,0,0,0,0,0,2,0,0,1,0,9],[28,40,0.7,0.40625,0.26271,0.2857,0.42857,0.4286,0.0,1.0,6,2,0,6,0,0,0,0,4,0,0,15,0,0,2,0,0,2,0,0,1,0,2],[32,40,0.8,0.46429,0.31542,0.28571,0.42857,0.75,0.0,1.0,5,4,0,5,0,1,0,0,5,0,0,12,0,0,0,0,0,1,0,0,4,0,4],[36,40,0.9,0.42857,0.27433,0.2857,0.42857,0.42858,0.0,1.0,3,4,0,3,0,1,0,0,10,0,0,12,0,0,0,0,0,1,0,0,1,0,4],[40,40,1.0,0.41518,0.20628,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,5,0,0,5,0,0,18,0,0,1,0,0,0,0,0,1,0,2]]}]},{"i":"b3820c616d0b8373","q":"Let  $S={1,2,\\ldots,100}$ . Consider a partition of  $S$  into  $S_1,S_2,\\ldots,S_n$  for some  $n$ , i.e.  $S_i$  are nonempty, pairwise disjoint and  $\\displaystyle S=\\bigcup_{i=1}^n S_i$ . Let  $a_i$  be the average of elements of the set  $S_i$ . Define the score of this partition by \n\\[\\dfrac{a_1+a_2+\\ldots+a_n}{n}.\\]\n\nAmong all  $n$  and partitions of  $S$ , determine the minimum possible score.","t":[{"b":0,"e":1.0,"k":"rising","v":0.75,"x":1.0,"p":[[0,144,0.0,0.75,0.38631,0.53571,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,21],[4,144,0.0278,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,144,0.0556,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,144,0.0833,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,144,0.1111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,144,0.1389,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,144,0.1667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,144,0.1944,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[32,144,0.2222,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,144,0.25,0.9375,0.16728,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,27],[40,144,0.2778,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[44,144,0.3056,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[48,144,0.3333,0.97545,0.06341,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,1,27],[52,144,0.3611,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[56,144,0.3889,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,144,0.4167,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,144,0.4444,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,144,0.4722,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[72,144,0.5,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[76,144,0.5278,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,144,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,144,0.5833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,144,0.6111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,144,0.6389,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,144,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,144,0.6944,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,144,0.7222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[108,144,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,144,0.7778,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[116,144,0.8056,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[120,144,0.8333,0.98438,0.04966,1.0,1.0,1.0,0.78571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,29],[124,144,0.8611,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[128,144,0.8889,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[132,144,0.9167,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[136,144,0.9444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[140,144,0.9722,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[144,144,1.0,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26]]},{"b":1,"e":1.0,"k":"rising","v":0.69196,"x":1.0,"p":[[0,84,0.0,0.69196,0.37306,0.28571,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,6,0,0,1,0,0,1,0,0,2,0,0,1,0,17],[4,84,0.0476,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,84,0.0952,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,84,0.1429,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[16,84,0.1905,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,84,0.2381,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[24,84,0.2857,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[28,84,0.3333,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,84,0.381,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,84,0.4286,0.93304,0.19228,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[40,84,0.4762,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[44,84,0.5238,0.96429,0.11293,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[48,84,0.5714,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[52,84,0.619,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,84,0.6667,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[60,84,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,84,0.7619,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,84,0.8095,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,84,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,84,0.9048,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,84,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,84,1.0,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]}]},{"i":"3275eb2beb873ec7","q":"Let  $\\alpha$  and  $\\beta$  be real numbers with  $\\beta \\ne 0$ . Determine all functions  $f:\\mathbb{R} \\to \\mathbb{R}$  such that\n\\[f(\\alpha f(x)+f(y))=\\beta x+f(y)\\]\nholds for all real  $x$  and  $y$ .\n\n*(Walther Janous)*","t":[{"b":2,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,109,0.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[4,109,0.0367,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,109,0.0734,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,109,0.1101,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[16,109,0.1468,0.90625,0.16213,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,23],[20,109,0.1835,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[24,109,0.2202,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[28,109,0.2569,0.92857,0.15152,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,25],[32,109,0.2936,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,109,0.3303,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,109,0.367,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,109,0.4037,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[48,109,0.4404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,109,0.4771,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,109,0.5138,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,109,0.5505,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,109,0.5872,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,109,0.6239,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,109,0.6606,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,109,0.6972,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[80,109,0.7339,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,109,0.7706,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[88,109,0.8073,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,109,0.844,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,109,0.8807,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,109,0.9174,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[104,109,0.9541,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[108,109,0.9908,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[109,109,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.88838,"x":0.98661,"p":[[0,71,0.0,0.92856,0.14289,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,25],[4,71,0.0563,0.92411,0.14719,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,25],[8,71,0.1127,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,71,0.169,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[16,71,0.2254,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[20,71,0.2817,0.91964,0.15126,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24],[24,71,0.338,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[28,71,0.3944,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[32,71,0.4507,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[36,71,0.507,0.92856,0.14289,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,25],[40,71,0.5634,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[44,71,0.6197,0.93749,0.15545,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,27],[48,71,0.6761,0.91071,0.16269,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,1,0,24],[52,71,0.7324,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[56,71,0.7887,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[60,71,0.8451,0.88838,0.19802,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,0,0,24],[64,71,0.9014,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[68,71,0.9577,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[71,71,1.0,0.91518,0.1551,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,24]]}]},{"i":"a445e258a0a31091","q":"Let  $[AB]$  be a chord of the circle  $\\Gamma$  not passing through its center and let  $M$  be the midpoint of  $[AB].$  Let  $C$  be a variable point on  $\\Gamma$  different from  $A$  and  $B$  and  $P$  be the point of intersection of the tangent lines at  $A$  of circumcircle of  $CAM$  and at  $B$  of circumcircle of  $CBM.$  Show that all  $CP$  lines pass through a fixed point.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.07143,"x":0.25893,"p":[[0,52,0.0,0.17411,0.18808,0.0,0.21428,0.28571,0.0,0.71429,15,0,5,15,0,1,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,52,0.0769,0.22768,0.20781,0.0,0.2857,0.2857,0.0,1.0,9,1,0,9,0,4,0,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[8,52,0.1538,0.25893,0.21852,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,5,0,0,15,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[12,52,0.2308,0.23214,0.1171,0.14286,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,7,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.19643,0.14617,0.0,0.2857,0.28571,0.0,0.57143,9,0,0,9,0,5,0,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,52,0.3846,0.20088,0.16695,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,5,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[24,52,0.4615,0.21429,0.15568,0.14286,0.2857,0.28571,0.0,0.71429,7,0,0,7,0,7,0,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,52,0.5385,0.23661,0.19759,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,6,0,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[32,52,0.6154,0.11607,0.1357,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,5,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.17411,0.13709,0.0,0.14295,0.28571,0.0,0.4286,10,0,0,10,0,7,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.18749,0.1836,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,8,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[44,52,0.8462,0.16072,0.13716,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.11374,0.14486,0.0,0.0,0.2857,0.0,0.571,17,0,0,17,1,5,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.09143,"x":0.36606,"p":[[0,51,0.0,0.24107,0.16146,0.14286,0.28571,0.28571,0.0,0.57143,7,0,3,7,0,3,0,0,18,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,51,0.0784,0.32589,0.21793,0.2857,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,2,0,0,16,0,0,2,0,0,2,0,0,5,0,0,0,0,0],[8,51,0.1569,0.31696,0.25439,0.24999,0.28571,0.28571,0.0,1.0,7,1,0,7,0,1,0,0,17,0,0,0,0,0,1,0,0,5,0,0,0,0,1],[12,51,0.2353,0.36606,0.31121,0.10714,0.28571,0.57143,0.0,1.0,8,3,0,8,0,3,0,0,8,0,0,1,0,0,6,0,0,3,0,0,0,0,3],[16,51,0.3137,0.24554,0.19638,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,4,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[20,51,0.3922,0.16072,0.16269,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,51,0.4706,0.24107,0.2299,0.0,0.28571,0.28571,0.0,1.0,10,1,0,10,0,2,0,0,16,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[28,51,0.549,0.19643,0.16656,0.0,0.28571,0.28571,0.0,0.57143,11,0,0,11,0,3,0,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,51,0.6275,0.15625,0.14445,0.0,0.21428,0.28571,0.0,0.42857,14,0,0,14,0,2,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.14732,0.145,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,7,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,51,0.7843,0.17415,0.17035,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,9,0,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[44,51,0.8627,0.14277,0.16752,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,9,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[48,51,0.9412,0.09143,0.12398,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,11,0,1,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[51,51,1.0,0.10259,0.1085,0.0,0.14143,0.1429,0.0,0.28571,15,0,0,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"72a363cf138a3834","q":"Let  $a,b,c,d$  be positive real numbers with  $a+b+c+d=2$ . Prove the following inequality:  $$ \\frac{(a+c)^{2}}{ad+bc}+\\frac{(b+d)^{2}}{ac+bd}+4\\geq 4\\left ( \\frac{a+b+1}{c+d+1}+\\frac{c+d+1}{a+b+1} \\right). $$  *Proposed by Mohammad Jafari*","t":[{"b":1,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,74,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,74,0.0541,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,74,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,74,0.1622,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,74,0.2162,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,74,0.2703,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,74,0.3243,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,74,0.3784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,74,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,74,0.4865,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,74,0.5405,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,74,0.5946,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,74,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,74,0.7027,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,74,0.7568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,74,0.8108,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,74,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,74,0.9189,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,74,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[74,74,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.96428,"x":1.0,"p":[[0,118,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,118,0.0339,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,118,0.0678,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,118,0.1017,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,118,0.1356,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,118,0.1695,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,118,0.2034,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,118,0.2373,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,118,0.2712,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[36,118,0.3051,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,118,0.339,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,118,0.3729,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,118,0.4068,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,118,0.4407,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,118,0.4746,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,118,0.5085,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,118,0.5424,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,118,0.5763,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,118,0.6102,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,118,0.6441,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[80,118,0.678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,118,0.7119,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,118,0.7458,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[92,118,0.7797,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,118,0.8136,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,118,0.8475,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[104,118,0.8814,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[108,118,0.9153,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[112,118,0.9492,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[116,118,0.9831,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[118,118,1.0,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]}]},{"i":"9d89fafac4774948","q":"Let  $a_0,a_1,a_2,\\ldots$  be a sequence of integers and  $b_0,b_1,b_2,\\ldots$  be a sequence of *positive* integers such that  $a_0=0,a_1=1$ , and \n\\[\na_{n+1} =\n        \\begin{cases}\n            a_nb_n+a_{n-1} & \\text{if  $b_{n-1}=1$ } \n            a_nb_n-a_{n-1} & \\text{if  $b_{n-1}>1$ }\n        \\end{cases}\\qquad\\text{for }n=1,2,\\ldots.\n\\]\nfor  $n=1,2,\\ldots.$  Prove that at least one of the two numbers  $a_{2017}$  and  $a_{2018}$  must be greater than or equal to  $2017$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.08036,"x":0.33704,"p":[[0,115,0.0,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,115,0.0348,0.31248,0.26828,0.0,0.28571,0.4642,0.0,0.85714,9,0,0,9,0,4,0,0,6,0,0,5,0,0,2,0,0,5,0,0,1,0,0],[8,115,0.0696,0.2455,0.2454,0.0,0.14286,0.42858,0.0,0.85714,10,0,0,10,0,8,0,0,5,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[12,115,0.1043,0.25892,0.22709,0.0,0.2857,0.42857,0.0,0.85714,9,0,0,9,0,6,0,0,6,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[16,115,0.1391,0.25446,0.21646,0.0,0.2857,0.42857,0.0,0.71429,9,0,0,9,0,6,0,0,6,0,0,7,0,0,2,0,0,2,0,0,0,0,0],[20,115,0.1739,0.33704,0.26258,0.14286,0.28571,0.51775,0.0,1.0,6,1,0,6,0,7,0,0,4,0,0,6,0,1,3,0,0,4,0,0,0,0,1],[24,115,0.2087,0.25438,0.23351,0.0,0.2143,0.42857,0.0,0.71429,11,0,0,11,0,5,0,0,2,0,0,11,0,0,0,0,0,3,0,0,0,0,0],[28,115,0.2435,0.29902,0.24583,0.0,0.28571,0.42858,0.0,0.71429,10,0,0,10,0,2,0,0,5,0,0,9,0,0,2,0,0,4,0,0,0,0,0],[32,115,0.2783,0.25,0.22588,0.0,0.14288,0.42857,0.0,0.71429,10,0,0,10,0,7,0,0,2,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[36,115,0.313,0.31695,0.22511,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,4,0,0,9,0,0,7,0,0,3,0,0,2,0,0,1,0,0],[40,115,0.3478,0.30794,0.23454,0.14286,0.2857,0.4286,0.0,0.85714,6,0,0,6,0,7,0,0,6,0,0,6,0,0,4,0,0,2,0,0,1,0,0],[44,115,0.3826,0.22767,0.21973,0.0,0.1429,0.42857,0.0,0.71429,11,0,0,11,0,6,0,0,6,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[48,115,0.4174,0.16518,0.15612,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,5,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,115,0.4522,0.20533,0.2498,0.0,0.14286,0.32143,0.0,1.0,14,1,0,14,0,6,0,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[56,115,0.487,0.20085,0.24192,0.0,0.14143,0.32143,0.0,0.85714,15,0,0,15,0,4,0,0,5,0,0,5,0,0,0,0,0,2,0,0,1,0,0],[60,115,0.5217,0.20088,0.19513,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,7,0,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[64,115,0.5565,0.23212,0.20743,0.0,0.21435,0.32143,0.0,0.85714,9,0,0,9,0,7,0,0,8,0,0,5,0,0,2,0,0,0,0,0,1,0,0],[68,115,0.5913,0.20981,0.19877,0.0,0.1429,0.42857,0.0,0.71429,11,0,0,11,0,7,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[72,115,0.6261,0.23212,0.20119,0.10714,0.14288,0.42857,0.0,0.71429,8,0,0,8,0,10,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[76,115,0.6609,0.18304,0.18293,0.0,0.14288,0.28571,0.0,0.71429,12,0,0,12,0,7,0,0,7,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[80,115,0.6957,0.2945,0.24468,0.105,0.28571,0.4642,0.0,0.85714,8,0,0,8,0,7,0,0,3,0,0,6,0,0,6,0,0,1,0,0,1,0,0],[84,115,0.7304,0.22322,0.20806,0.0,0.1429,0.32143,0.0,0.71429,10,0,0,10,0,7,0,0,7,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[88,115,0.7652,0.19197,0.18766,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,6,0,0,7,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[92,115,0.8,0.19634,0.18817,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,8,0,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[96,115,0.8348,0.22769,0.1851,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,5,0,0,6,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[100,115,0.8696,0.19188,0.16605,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,11,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[104,115,0.9043,0.17855,0.17122,0.0,0.14286,0.28571,0.0,0.571,10,0,0,10,0,12,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[108,115,0.9391,0.16072,0.15872,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,10,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[112,115,0.9739,0.2008,0.18164,0.0,0.14286,0.42857,0.0,0.57143,11,0,0,11,0,7,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[115,115,1.0,0.16516,0.1679,0.0,0.14286,0.28571,0.0,0.5714,12,0,0,12,0,10,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.07141,"x":0.25444,"p":[[0,44,0.0,0.07141,0.14281,0.0,0.0,0.03571,0.0,0.571,24,0,0,24,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,44,0.0909,0.24989,0.27895,0.0,0.14286,0.42858,0.0,0.857,14,0,0,14,0,4,0,0,3,0,0,4,0,0,2,0,0,4,0,0,1,0,0],[8,44,0.1818,0.25444,0.28509,0.0,0.14286,0.42858,0.0,1.0,13,1,0,13,0,4,0,0,5,0,0,4,0,0,3,0,0,0,0,0,2,0,1],[12,44,0.2727,0.17856,0.1923,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,3,0,0,7,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,44,0.3636,0.10714,0.16366,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,6,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,44,0.4545,0.20972,0.24997,0.0,0.14143,0.42857,0.0,0.85714,15,0,0,15,0,4,0,0,4,0,0,5,0,0,1,0,0,2,0,0,1,0,0],[24,44,0.5455,0.14285,0.19882,0.0,0.0,0.42857,0.0,0.571,20,0,0,20,0,2,0,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[28,44,0.6364,0.14732,0.17673,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,4,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.17856,0.21722,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,4,0,0,5,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[36,44,0.8182,0.13831,0.17307,0.0,0.07,0.2857,0.0,0.71429,16,0,0,16,0,6,0,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[40,44,0.9091,0.14283,0.21717,0.0,0.0,0.2857,0.0,0.857,19,0,0,19,0,4,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[44,44,1.0,0.11606,0.15332,0.0,0.0,0.14287,0.0,0.571,17,0,0,17,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"61dc951d99851b87","q":"Let  $a_1,a_2,\\dots$  be a sequence of positive numbers satisfying, for any positive integers  $k,l,m,n$  such that  $k+n=m+l$ ,   $$ \\frac{a_k+a_n}{1+a_ka_n}=\\frac{a_m+a_l}{1+a_ma_l}. $$  Show that there exist positive numbers  $b,c$  so that  $b\\le a_n\\le c$  for any positive integer  $n$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.19634,"x":0.6429,"p":[[0,55,0.0,0.55802,0.28651,0.42857,0.71429,0.71429,0.0,1.0,4,2,0,4,0,2,0,0,1,0,0,3,0,0,4,0,0,14,0,0,2,0,2],[4,55,0.0727,0.6429,0.27429,0.39286,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,3,0,0,2,0,0,5,0,0,10,0,4],[8,55,0.1455,0.62946,0.27862,0.28571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,5,0,0,0,0,0,3,0,0,8,0,0,9,0,3],[12,55,0.2182,0.625,0.29827,0.28571,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,3,0,0,0,0,0,4,0,0,6,0,0,9,0,4],[16,55,0.2909,0.49545,0.29674,0.14286,0.57141,0.75,0.0,0.85714,1,0,0,1,0,8,0,0,5,0,0,1,0,0,3,0,0,6,0,0,8,0,0],[20,55,0.3636,0.46875,0.34853,0.14286,0.35714,0.75,0.0,1.0,2,4,0,2,0,12,0,0,2,0,0,1,0,0,1,0,0,6,0,0,4,0,4],[24,55,0.4364,0.59821,0.32623,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,7,0,0,2,0,0,1,0,0,5,0,0,4,0,8],[28,55,0.5091,0.55804,0.30797,0.28571,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,7,0,0,1,0,0,2,0,0,6,0,0,6,0,4],[32,55,0.5818,0.55804,0.28428,0.28571,0.64286,0.75,0.14286,1.0,0,3,0,0,0,5,0,0,6,0,0,3,0,0,2,0,0,8,0,0,5,0,3],[36,55,0.6545,0.55804,0.30797,0.28571,0.64286,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,6,0,0,1,0,0,3,0,0,7,0,0,5,0,4],[40,55,0.7273,0.43295,0.29348,0.14289,0.28571,0.75,0.0,0.85714,1,0,0,1,0,8,0,0,10,0,0,2,0,0,0,0,0,3,0,0,8,0,0],[44,55,0.8,0.50882,0.32141,0.2857,0.28571,0.85714,0.14,1.0,0,6,0,0,0,6,0,0,11,0,0,1,0,0,2,0,0,3,0,0,3,0,6],[48,55,0.8727,0.46875,0.35577,0.14286,0.28571,0.85714,0.0,1.0,2,5,0,2,0,10,0,0,6,0,0,1,0,0,0,0,0,3,0,0,5,0,5],[52,55,0.9455,0.26786,0.22798,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,14,0,0,9,0,0,2,0,0,1,0,0,1,0,0,1,0,1],[55,55,1.0,0.19634,0.14177,0.14286,0.14286,0.2857,0.0,0.85714,2,0,0,2,0,20,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":1,"e":0.14286,"k":"falling","v":0.14733,"x":0.75,"p":[[0,73,0.0,0.60712,0.24484,0.57132,0.71429,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,2,0,0,2,0,0,8,0,0,11,0,0,4,0,2],[4,73,0.0548,0.62499,0.23077,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,7,0,0,5,0,0,8,0,0,7,0,2],[8,73,0.1096,0.68304,0.27137,0.67857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,1,0,0,1,0,0,11,0,0,8,0,5],[12,73,0.1644,0.65625,0.3251,0.42857,0.78571,0.85714,0.0,1.0,1,7,0,1,0,6,0,0,0,0,0,3,0,0,1,0,0,5,0,0,9,0,7],[16,73,0.2192,0.67411,0.29717,0.39286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,5,0,0,2,0,0,1,0,0,6,0,0,7,0,8],[20,73,0.274,0.75,0.21129,0.71429,0.78571,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,10,0,0,10,0,6],[24,73,0.3288,0.68304,0.3209,0.42857,0.85714,1.0,0.0,1.0,2,9,0,2,0,2,0,0,2,0,0,4,0,0,2,0,0,2,0,0,9,0,9],[28,73,0.3836,0.58036,0.34981,0.28571,0.71429,0.89286,0.0,1.0,2,8,0,2,0,5,0,0,5,0,0,2,0,0,1,0,0,5,0,0,4,0,8],[32,73,0.4384,0.65624,0.31514,0.39286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,2,0,0,1,0,0,3,0,0,7,0,0,4,0,9],[36,73,0.4932,0.58036,0.29001,0.28571,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,5,0,0,5,0,0,3,0,0,2,0,0,8,0,0,5,0,4],[40,73,0.5479,0.50892,0.3443,0.14286,0.42857,0.85714,0.0,1.0,3,4,0,3,0,6,0,0,5,0,0,3,0,0,1,0,0,3,0,0,7,0,4],[44,73,0.6027,0.33482,0.33045,0.10714,0.28571,0.42857,0.0,1.0,8,5,0,8,0,6,0,0,7,0,0,4,0,0,2,0,0,0,0,0,0,0,5],[48,73,0.6575,0.14733,0.08365,0.14286,0.14286,0.14286,0.0,0.286,5,0,0,5,0,21,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,73,0.7123,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,73,0.7671,0.17411,0.09268,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,17,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.19643,0.06916,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.16518,0.09523,0.14286,0.14286,0.17857,0.0,0.42857,4,0,0,4,0,20,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,73,0.9315,0.20536,0.07936,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,16,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.15608,0.0746,0.14286,0.14286,0.14289,0.0,0.28571,3,0,0,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.17411,0.06902,0.14286,0.14286,0.17857,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a36141cbb8041f7","q":"Let  $f:[0,1]\\rightarrow [0,1]$  a continuous function in  $0$  and in  $1$ , which has one-side limits in any point and  $f(x-0)\\le f(x)\\le f(x+0),\\ (\\forall)x\\in (0,1)$ . Prove that:\n\na)for the set  $A=\\{x\\in [0,1]\\ |\\ f(x)\\ge x\\}$ , we have  $\\sup A\\in A$ .\nb)there is  $x_0\\in [0,1]$  such that  $f(x_0)=x_0$ .\n\n*Mihai Piticari*","t":[{"b":3,"e":0.57143,"k":"falling","v":0.52229,"x":0.94196,"p":[[0,12,0.0,0.92411,0.1636,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,25],[4,12,0.3333,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[8,12,0.6667,0.62947,0.21975,0.42857,0.57143,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,12,0,0,5,0,0,7,0,0,1,0,6],[12,12,1.0,0.52229,0.11632,0.42857,0.57121,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,12,0,0,13,0,0,5,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.52679,"x":0.95982,"p":[[0,13,0.0,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[4,13,0.3077,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[8,13,0.6154,0.81249,0.23809,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,3,0,0,3,0,17],[12,13,0.9231,0.54013,0.19799,0.42857,0.4286,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,14,0,0,8,0,0,2,0,0,3,0,2],[13,13,1.0,0.52679,0.21261,0.42857,0.42859,0.60714,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,12,0,0,7,0,0,5,0,0,0,0,3]]}]},{"i":"b213fb7d8eb4ab4d","q":"Let  $d$  be a real number such that  $d^2=r^2+s^2$ , where  $r$  and  $s$  are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance  $d$  have different colors.","t":[{"b":4,"e":0.42857,"k":"flat","v":0.19643,"x":0.53567,"p":[[0,37,0.0,0.29017,0.09771,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,0,0,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,37,0.1081,0.47768,0.31461,0.2857,0.4293,0.85704,0.0,1.0,5,1,0,5,0,2,0,0,6,0,0,4,0,0,4,0,0,2,0,0,8,0,1],[8,37,0.2162,0.47318,0.24596,0.2857,0.42859,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,10,0,0,4,0,0,8,0,0,1,0,0,6,0,0],[12,37,0.3243,0.53567,0.30303,0.2857,0.57121,0.857,0.0,1.0,2,1,0,2,0,5,0,0,4,0,0,3,0,0,3,0,0,5,0,0,9,0,1],[16,37,0.4324,0.308,0.27914,0.0,0.28571,0.571,0.0,0.85714,9,0,0,9,0,6,0,0,5,0,0,2,0,0,5,0,0,3,0,0,2,0,0],[20,37,0.5405,0.45536,0.2683,0.24999,0.42857,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,6,0,0,4,0,0,4,0,0,6,0,0,3,0,1],[24,37,0.6486,0.23204,0.23893,0.0,0.14286,0.28571,0.0,0.85714,9,0,0,9,0,10,0,0,6,0,0,3,0,0,1,0,0,1,0,0,2,0,0],[28,37,0.7568,0.31249,0.25613,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,9,0,0,8,0,0,3,0,0,1,0,0,4,0,0,2,0,0],[32,37,0.8649,0.27231,0.24315,0.0,0.2857,0.42857,0.0,0.85714,10,0,0,10,0,4,0,0,6,0,0,7,0,0,2,0,0,2,0,0,1,0,0],[36,37,0.973,0.22767,0.26929,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,8,0,0,5,0,0,2,0,0,1,0,0,1,0,0,3,0,0],[37,37,1.0,0.19643,0.17034,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,15,0,0,8,0,0,1,0,0,0,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.11152,"x":0.53571,"p":[[0,54,0.0,0.32588,0.13938,0.2857,0.28571,0.28571,0.0,0.857,1,0,1,1,0,0,0,0,25,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[4,54,0.0741,0.53571,0.28347,0.28571,0.57143,0.85714,0.0,0.85714,2,0,0,2,0,3,0,0,5,0,0,4,0,0,6,0,0,1,0,0,11,0,0],[8,54,0.1481,0.48659,0.28539,0.2857,0.5712,0.71429,0.0,0.85714,3,0,0,3,0,4,0,0,5,0,0,3,0,0,6,0,0,4,0,0,7,0,0],[12,54,0.2222,0.43299,0.25871,0.2857,0.4998,0.57143,0.0,0.85714,5,0,0,5,0,2,0,0,4,0,0,5,0,0,12,0,0,0,0,0,4,0,0],[16,54,0.2963,0.4598,0.28286,0.2857,0.42859,0.64282,0.0,1.0,2,1,0,2,0,4,0,0,9,0,0,4,0,0,5,0,0,0,0,0,7,0,1],[20,54,0.3704,0.42411,0.29984,0.14289,0.42857,0.71429,0.0,0.85714,7,0,0,7,0,2,0,0,4,0,0,5,0,0,5,0,0,4,0,0,5,0,0],[24,54,0.4444,0.47758,0.3065,0.14286,0.42859,0.85704,0.0,0.85714,3,0,0,3,0,6,0,0,3,0,0,6,0,0,3,0,0,1,0,0,10,0,0],[28,54,0.5185,0.37499,0.27139,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,10,0,0,7,0,0,3,0,0,4,0,0,2,0,0,3,0,1],[32,54,0.5926,0.30357,0.28064,0.10714,0.14288,0.57143,0.0,0.85714,8,0,0,8,0,9,0,0,3,0,0,3,0,0,3,0,0,4,0,0,2,0,0],[36,54,0.6667,0.36607,0.30291,0.14286,0.28571,0.60714,0.0,1.0,6,1,0,6,0,6,0,0,7,0,0,4,0,0,1,0,0,3,0,0,4,0,1],[40,54,0.7407,0.30356,0.26665,0.10714,0.2857,0.42857,0.0,0.85714,8,0,0,8,0,6,0,0,6,0,0,5,0,0,3,0,0,1,0,0,3,0,0],[44,54,0.8148,0.12946,0.19351,0.0,0.0,0.1786,0.0,0.71429,18,0,0,18,0,6,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[48,54,0.8889,0.16964,0.15335,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,6,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.16071,0.15465,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,11,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[54,54,1.0,0.11152,0.1114,0.0,0.14286,0.1429,0.0,0.28571,14,0,0,14,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5cfe17d9c528a7a6","q":"Let  $d_1,d_2,\\dots,d_{12}$  be real numbers in the open interval  $(1,12).$  Show that there exist distinct indices  $i,j,k$  such that  $d_i,d_j,d_k$  are the side lengths of an acute triangle.","t":[{"b":3,"e":0.28571,"k":"falling","v":0.26339,"x":0.95536,"p":[[0,51,0.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[4,51,0.0784,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[8,51,0.1569,0.91964,0.16342,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,7,0,22],[12,51,0.2353,0.91071,0.12243,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,17],[16,51,0.3137,0.94196,0.07874,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[20,51,0.3922,0.87054,0.27516,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,21],[24,51,0.4706,0.90625,0.18423,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,21],[28,51,0.549,0.8125,0.27534,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,12,0,14],[32,51,0.6275,0.75891,0.34152,0.53539,1.0,1.0,0.0,1.0,1,17,0,1,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0,6,0,17],[36,51,0.7059,0.79464,0.2878,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,13,0,13],[40,51,0.7843,0.76339,0.31463,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,3,0,0,2,0,0,0,0,0,2,0,0,3,0,0,6,0,15],[44,51,0.8627,0.87946,0.22048,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,9,0,19],[48,51,0.9412,0.82142,0.29233,0.85711,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,9,0,17],[51,51,1.0,0.26339,0.16016,0.14286,0.2857,0.32143,0.0,0.71429,3,0,0,3,0,10,0,0,11,0,0,6,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.90179,"x":0.96875,"p":[[0,35,0.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,35,0.1143,0.91964,0.16342,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,21],[8,35,0.2286,0.90624,0.11071,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[12,35,0.3429,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[16,35,0.4571,0.90179,0.19377,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,20],[20,35,0.5714,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[24,35,0.6857,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[28,35,0.8,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,35,0.9143,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[35,35,1.0,0.92411,0.16746,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,24]]}]},{"i":"f3fa34bdf4b91a4a","q":"Let  $M=\\{1,2,\\cdots,n\\}$ , each element of  $M$  is colored in either red, blue or yellow. Set $A=\\{(x,y,z)\\in M\\times M\\times M|x+y+z\\equiv 0\\mod n$ ,  $x,y,z$  are of same color $\\},$  $B=\\{(x,y,z)\\in M\\times M\\times M|x+y+z\\equiv 0\\mod n,$   $x,y,z$  are of pairwise distinct color $\\}.$ Prove that  $2|A|\\geq |B|$ .","t":[{"b":0,"e":0.71429,"k":"falling","v":0.68749,"x":0.97321,"p":[[0,9,0.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,9,0.4444,0.87946,0.28372,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,26],[8,9,0.8889,0.69643,0.16269,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,13,0,0,7,0,2],[9,9,1.0,0.68749,0.18364,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,11,0,0,8,0,2]]},{"b":6,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,35,0.0,0.90625,0.27804,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[4,35,0.1143,0.98219,0.09918,1.0,1.0,1.0,0.43,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,35,0.2286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d65a66c38543c4b7","q":"Let  $d$  be a positive integer. \nThe seqeunce  $a_1, a_2, a_3,...$  of positive integers is defined by   $a_1 = 1$  and  $a_{n + 1} = n\\left \\lfloor \\frac{a_n}{n}  \\right \\rfloor+ d$  for  $n = 1,2,3, ...$  .\nProve that there exists a positive integer  $N$  so that the terms  $a_N,a_{N + 1}, a_{N + 2},...$  form an arithmetic progression.\n\nNote: If  $x$  is a real number,  $\\left \\lfloor  x \\right \\rfloor $  denotes the largest integer that is less than or equal to  $x$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.83035,"x":0.95982,"p":[[0,49,0.0,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,49,0.0816,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,49,0.1633,0.94196,0.18161,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[12,49,0.2449,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[16,49,0.3265,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[20,49,0.4082,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[24,49,0.4898,0.95759,0.10236,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,1,26],[28,49,0.5714,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[32,49,0.6531,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[36,49,0.7347,0.90178,0.12596,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,12,0,16],[40,49,0.8163,0.86159,0.15357,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,17,0,11],[44,49,0.898,0.86606,0.07936,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,6],[48,49,0.9796,0.83035,0.1357,0.85711,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,21,0,5],[49,49,1.0,0.83479,0.12434,0.85711,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,19,0,6]]},{"b":4,"e":0.85714,"k":"flat","v":0.86161,"x":0.97768,"p":[[0,75,0.0,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[4,75,0.0533,0.9375,0.16342,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[8,75,0.1067,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,75,0.16,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,75,0.2133,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,75,0.2667,0.91964,0.18536,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[24,75,0.32,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[28,75,0.3733,0.93527,0.08631,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,1,19],[32,75,0.4267,0.94643,0.1171,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[36,75,0.48,0.91516,0.15096,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,9,0,20],[40,75,0.5333,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[44,75,0.5867,0.91964,0.14698,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,20],[48,75,0.64,0.88836,0.15871,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,12,0,16],[52,75,0.6933,0.92409,0.10097,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[56,75,0.7467,0.88616,0.09425,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,16,0,11],[60,75,0.8,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[64,75,0.8533,0.92411,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,17],[68,75,0.9067,0.87053,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[72,75,0.96,0.87052,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[75,75,1.0,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1]]}]},{"i":"53fecaaf27ed808a","q":"Let  $g:[2013,2014]\\to\\mathbb{R}$  a function that satisfy the following two conditions:\n\ni)   $g(2013)=g(2014) = 0,$ ii) for any  $a,b \\in [2013,2014]$  it hold that  $g\\left(\\frac{a+b}{2}\\right) \\leq g(a) + g(b).$ Prove that  $g$  has zeros in any open subinterval  $(c,d) \\subset[2013,2014].$","t":[{"b":2,"e":0.57143,"k":"flat","v":0.33036,"x":0.56697,"p":[[0,30,0.0,0.53125,0.07349,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0],[4,30,0.1333,0.46427,0.20824,0.42857,0.57143,0.57143,0.0,0.71429,5,0,1,5,0,0,0,0,0,0,0,5,0,0,21,0,0,1,0,0,0,0,0],[8,30,0.2667,0.43304,0.25376,0.25001,0.57143,0.57143,0.0,1.0,6,1,0,6,0,2,0,0,1,0,0,3,0,0,18,0,0,1,0,0,0,0,1],[12,30,0.4,0.39284,0.21723,0.28571,0.42857,0.57143,0.0,0.57143,6,0,0,6,0,1,0,0,3,0,0,7,0,0,15,0,0,0,0,0,0,0,0],[16,30,0.5333,0.33036,0.21261,0.10714,0.42857,0.42858,0.0,0.57143,8,0,0,8,0,1,0,0,3,0,0,13,0,0,7,0,0,0,0,0,0,0,0],[20,30,0.6667,0.53124,0.07348,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0],[24,30,0.8,0.56697,0.09094,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,0,0,1],[28,30,0.9333,0.53124,0.06422,0.42859,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0],[30,30,1.0,0.55356,0.04724,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.13393,"x":0.54911,"p":[[0,60,0.0,0.54911,0.10779,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,21,0,0,1,0,0,0,0,1],[4,60,0.0667,0.39284,0.26725,0.10714,0.4998,0.57143,0.0,1.0,8,1,0,8,0,2,0,0,0,0,0,6,0,0,14,0,0,1,0,0,0,0,1],[8,60,0.1333,0.45536,0.24338,0.39286,0.57143,0.57143,0.0,1.0,4,2,0,4,0,2,0,0,2,0,0,6,0,0,16,0,0,0,0,0,0,0,2],[12,60,0.2,0.3482,0.23672,0.0,0.42857,0.57143,0.0,0.57143,9,0,0,9,0,1,0,0,1,0,0,9,0,0,12,0,0,0,0,0,0,0,0],[16,60,0.2667,0.37946,0.20079,0.28571,0.42857,0.57143,0.0,0.57143,5,0,0,5,0,2,0,0,3,0,0,11,0,0,11,0,0,0,0,0,0,0,0],[20,60,0.3333,0.36161,0.2172,0.24999,0.42857,0.57143,0.0,0.57143,7,0,0,7,0,1,0,0,3,0,0,10,0,0,11,0,0,0,0,0,0,0,0],[24,60,0.4,0.35268,0.23415,0.10714,0.42857,0.46429,0.0,0.71429,8,0,0,8,0,1,0,0,2,0,0,13,0,0,5,0,0,3,0,0,0,0,0],[28,60,0.4667,0.33929,0.23622,0.10714,0.42857,0.57143,0.0,0.57143,8,0,0,8,0,4,0,0,0,0,0,8,0,0,12,0,0,0,0,0,0,0,0],[32,60,0.5333,0.35268,0.23686,0.10714,0.42857,0.57143,0.0,0.71429,8,0,0,8,0,2,0,0,2,0,0,8,0,0,11,0,0,1,0,0,0,0,0],[36,60,0.6,0.29018,0.22442,0.0,0.35714,0.42858,0.0,0.57143,10,0,0,10,0,2,0,0,4,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[40,60,0.6667,0.36161,0.2082,0.24999,0.42857,0.57143,0.0,0.57143,6,0,0,6,0,2,0,0,3,0,0,11,0,0,10,0,0,0,0,0,0,0,0],[44,60,0.7333,0.35713,0.18556,0.28571,0.42857,0.4286,0.0,0.57143,5,0,0,5,0,1,0,0,6,0,0,13,0,0,7,0,0,0,0,0,0,0,0],[48,60,0.8,0.29909,0.21235,0.0,0.42857,0.42857,0.0,0.57143,10,0,0,10,0,0,0,0,3,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[52,60,0.8667,0.3616,0.18551,0.39285,0.42857,0.42858,0.0,0.57143,6,0,0,6,0,0,0,0,2,0,0,19,0,0,5,0,0,0,0,0,0,0,0],[56,60,0.9333,0.29909,0.21534,0.0,0.42857,0.42857,0.0,0.71429,9,0,0,9,0,2,0,0,3,0,0,14,0,0,3,0,0,1,0,0,0,0,0],[60,60,1.0,0.13393,0.18189,0.0,0.0,0.32142,0.0,0.4286,19,0,0,19,0,4,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dc178a61dd8bda42","q":"Let  $S$  be a circle with radius  $2$ , let  $S_1$  be a circle,with radius  $1$  and tangent, internally to  $S$  in  $B$  and let  $S_2$  be a circle, with radius  $1$  and tangent to  $S_1$  in  $A$ , but  $S_2$  isn't tangent to  $S$ . If  $K$  is the point of intersection of the line  $AB$  and the circle  $S$ , prove that  $K$  is in the circle  $S_2$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,39,0.0,0.91071,0.25443,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[4,39,0.1026,0.94196,0.18161,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[8,39,0.2051,0.93304,0.2172,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[12,39,0.3077,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,39,0.4103,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,39,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,39,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,39,0.7179,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,39,0.8205,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,39,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,36,0.0,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[4,36,0.1111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,36,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"aabd2b12d7f87084","q":"Let  $k\\geqslant 2$  be an integer, and  $a,b$  be real numbers. prove that  $a-b$  is an integer divisible by  $k$  if and only if for every positive integer  $n$   $$ \\lfloor an \\rfloor \\equiv \\lfloor bn \\rfloor \\ (mod \\ k) $$  Proposed by Navid Safaei","t":[{"b":0,"e":0.571,"k":"rising","v":0.29009,"x":0.79464,"p":[[0,36,0.0,0.29009,0.27317,0.14286,0.14286,0.2857,0.14,1.0,0,3,0,0,0,22,0,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,3],[4,36,0.1111,0.65179,0.35881,0.28571,0.78571,1.0,0.14286,1.0,0,14,0,0,0,7,0,0,3,0,0,2,0,0,3,0,0,1,0,0,2,0,14],[8,36,0.2222,0.66516,0.36353,0.24999,0.85707,1.0,0.14286,1.0,0,14,0,0,0,8,0,0,2,0,0,1,0,0,2,0,0,2,0,0,3,0,14],[12,36,0.3333,0.79464,0.3213,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,21],[16,36,0.4444,0.60489,0.3531,0.14286,0.64286,1.0,0.14286,1.0,0,10,0,0,0,9,0,0,1,0,1,2,0,0,3,0,0,2,0,0,4,0,10],[20,36,0.5556,0.62499,0.36026,0.24999,0.71429,1.0,0.14286,1.0,0,12,0,0,0,8,0,0,3,0,0,2,0,0,2,0,0,2,0,0,3,0,12],[24,36,0.6667,0.61161,0.39807,0.14286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,11,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,15],[28,36,0.7778,0.68525,0.28786,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,2,0,0,4,0,0,5,0,0,3,1,0,4,0,10],[32,36,0.8889,0.66964,0.34151,0.42857,0.78571,1.0,0.14286,1.0,0,13,0,0,0,7,0,0,0,0,0,4,0,0,3,0,0,2,0,0,3,0,13],[36,36,1.0,0.69195,0.33902,0.28571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,3,0,0,0,0,0,2,0,0,4,0,0,4,0,13]]},{"b":4,"e":1.0,"k":"rising","v":0.33481,"x":0.625,"p":[[0,32,0.0,0.33481,0.32655,0.14286,0.14286,0.42857,0.0,1.0,3,4,2,3,0,16,0,0,4,0,0,2,0,0,0,0,0,1,0,0,2,0,4],[4,32,0.125,0.625,0.35669,0.24999,0.85707,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,3,0,0,2,0,0,2,0,0,0,0,0,7,0,10],[8,32,0.25,0.54463,0.38205,0.14286,0.42836,1.0,0.14286,1.0,0,10,0,0,0,12,0,0,4,0,0,0,0,0,2,0,0,0,0,0,4,0,10],[12,32,0.375,0.42857,0.34626,0.14286,0.28571,0.71429,0.14286,1.0,0,7,0,0,0,15,0,0,5,0,0,0,0,0,3,0,0,2,0,0,0,0,7],[16,32,0.5,0.53571,0.34626,0.24999,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,8,0,0,7,0,0,2,0,0,3,0,0,0,0,0,4,0,8],[20,32,0.625,0.52677,0.35434,0.14286,0.42859,1.0,0.14286,1.0,0,9,0,0,0,10,0,0,5,0,0,2,0,0,2,0,0,3,0,0,1,0,9],[24,32,0.75,0.56474,0.35191,0.24999,0.5,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,5,0,0,3,0,0,2,1,0,1,0,0,2,0,10],[28,32,0.875,0.49094,0.34622,0.14286,0.35714,0.857,0.14,1.0,0,7,0,0,0,11,0,0,5,0,0,3,0,0,2,0,0,1,0,0,3,0,7],[32,32,1.0,0.59371,0.26752,0.42857,0.57143,0.85704,0.14286,1.0,0,4,0,0,0,3,0,0,4,0,0,6,0,0,5,0,0,4,0,0,6,0,4]]}]},{"i":"376eaa5066f1456e","q":"Let  $p{}$  be a prime number and  $x_1,x_2,\\ldots,x_p$  be integers for which  $x_1^n+x_2^n+\\cdots+x_p^n$  is divisible by  $p{}$  for any positive integer  $n{}$ . Prove that  $x_1-x_2$  is divisible by  $p{}.$","t":[{"b":2,"e":0.42857,"k":"flat","v":0.26339,"x":0.36607,"p":[[0,12,0.0,0.26339,0.17896,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,13,0,0,1,0,0,13,0,0,0,0,0,1,0,0,0,0,0],[4,12,0.3333,0.36161,0.20511,0.14286,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,11,0,0,0,0,0,14,0,0,3,0,0,2,0,0,1,0,0],[8,12,0.6667,0.33034,0.14912,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,3,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[12,12,1.0,0.36607,0.13333,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,2,0,0,22,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.1429,"k":"flat","v":0.1875,"x":0.33927,"p":[[0,12,0.0,0.33473,0.22199,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,12,0,0,4,0,0,11,0,0,1,0,0,1,0,0,1,0,1],[4,12,0.3333,0.33927,0.21942,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,13,0,0,1,0,0,12,0,0,2,0,0,2,0,0,0,0,1],[8,12,0.6667,0.1875,0.14913,0.14286,0.14286,0.17857,0.0,0.71429,4,0,0,4,0,20,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,12,1.0,0.20089,0.13767,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,21,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"35c6c43905e26240","q":"Let  $n \\ge 1$  and  $x_1, \\ldots, x_n \\ge 0$ . Prove that  $$   (x_1 + \\frac{x_2}{2} + \\ldots + \\frac{x_n}{n}) (x_1 + 2x_2 + \\ldots + nx_n) \\le \\frac{(n+1)^2}{4n} (x_1 + x_2 + \\ldots + x_n)^2 . $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.04464,"x":0.22768,"p":[[0,109,0.0,0.11149,0.17391,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,8,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[4,109,0.0367,0.12052,0.16012,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,13,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,109,0.0734,0.16963,0.26349,0.0,0.0,0.1429,0.0,1.0,17,1,0,17,0,8,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,1],[12,109,0.1101,0.14732,0.20972,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,9,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[16,109,0.1468,0.10714,0.14726,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,10,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,109,0.1835,0.17411,0.27137,0.0,0.14286,0.14286,0.0,1.0,15,2,0,15,0,11,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[24,109,0.2202,0.08482,0.09354,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,109,0.2569,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,109,0.2936,0.15625,0.21237,0.0,0.07143,0.14287,0.0,0.71429,16,0,0,16,0,9,0,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[36,109,0.3303,0.20982,0.28118,0.0,0.14286,0.42857,0.0,1.0,14,2,0,14,0,9,0,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[40,109,0.367,0.14286,0.22868,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,7,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[44,109,0.4037,0.2232,0.27416,0.0,0.14286,0.46418,0.0,1.0,14,1,0,14,0,8,0,0,0,0,0,2,0,0,6,0,0,1,0,0,0,0,1],[48,109,0.4404,0.17409,0.2415,0.0,0.14286,0.1786,0.0,0.85714,15,0,0,15,0,9,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,0],[52,109,0.4771,0.13384,0.18536,0.0,0.14143,0.14286,0.0,0.71429,15,0,0,15,0,12,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[56,109,0.5138,0.19197,0.19103,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,14,0,0,2,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[60,109,0.5505,0.13839,0.22724,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,8,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[64,109,0.5872,0.10714,0.17128,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[68,109,0.6239,0.12491,0.2165,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[72,109,0.6606,0.13393,0.2141,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[76,109,0.6972,0.20088,0.2786,0.0,0.14286,0.1429,0.0,1.0,13,2,0,13,0,12,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,2],[80,109,0.7339,0.15625,0.21535,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,11,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[84,109,0.7706,0.16965,0.25614,0.0,0.14286,0.14287,0.0,1.0,14,2,0,14,0,12,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[88,109,0.8073,0.10714,0.16366,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,10,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[92,109,0.844,0.22768,0.274,0.0,0.14286,0.42857,0.0,1.0,12,2,0,12,0,9,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,2],[96,109,0.8807,0.10714,0.16751,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,12,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[100,109,0.9174,0.07143,0.12877,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,9,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[104,109,0.9541,0.125,0.21354,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,8,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[108,109,0.9908,0.06696,0.12869,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[109,109,1.0,0.04464,0.11538,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.06696,"x":0.18749,"p":[[0,47,0.0,0.11607,0.26107,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[4,47,0.0851,0.18749,0.2586,0.0,0.14286,0.1786,0.0,1.0,15,1,0,15,0,9,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,1],[8,47,0.1702,0.15177,0.18533,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,13,0,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,47,0.2553,0.16963,0.22988,0.0,0.14286,0.14287,0.0,1.0,14,1,0,14,0,11,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[16,47,0.3404,0.08929,0.12753,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,11,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.12054,0.16409,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,15,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[24,47,0.5106,0.13393,0.21706,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,12,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[28,47,0.5957,0.09822,0.17655,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[32,47,0.6809,0.10714,0.13832,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,47,0.766,0.08482,0.12299,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,9,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.08036,0.15542,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,47,0.9362,0.0758,0.12359,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.06696,0.15146,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,5,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"977ee26c3bd22223","q":"Let  $r_1,r_2,r$ , with  $r_1 < r_2 < r$ , be the radii of three circles  $\\Gamma_1,\\Gamma_2,\\Gamma$ , respectively. The circles  $\\Gamma_1,\\Gamma_2$  are internally tangent to  $\\Gamma$  at two distinct points  $A,B$  and intersect in two distinct points. Prove that the segment  $AB$  contains an intersection point of  $\\Gamma_1$  and  $\\Gamma_2$  if and only if  $r_1 +r_2 = r$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.43745,"x":0.65619,"p":[[0,65,0.0,0.56246,0.2141,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,7,0,0,9,0,0,8,0,0,0,0,3],[4,65,0.0615,0.61159,0.15251,0.57142,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,15,0,0,8,0,0,1,0,2],[8,65,0.1231,0.60266,0.15458,0.5354,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,8,0,0,15,0,0,5,0,0,2,0,2],[12,65,0.1846,0.54903,0.1561,0.42859,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,7,0,0,15,0,0,6,0,0,0,0,1],[16,65,0.2462,0.58472,0.1488,0.571,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,19,0,0,3,0,0,1,0,2],[20,65,0.3077,0.65176,0.17105,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,7,0,0,2,0,4],[24,65,0.3692,0.65619,0.16313,0.571,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,9,0,0,3,0,3],[28,65,0.4308,0.59364,0.14774,0.4286,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,13,0,0,8,0,0,0,0,2],[32,65,0.4923,0.5803,0.20184,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,12,0,0,5,0,0,0,0,4],[36,65,0.5538,0.58927,0.23623,0.42857,0.57121,0.60714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,12,0,0,9,0,0,1,0,0,0,0,7],[40,65,0.6154,0.60259,0.2012,0.42857,0.571,0.60714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,10,0,0,13,0,0,2,0,0,1,0,5],[44,65,0.6769,0.58478,0.23517,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,5,0,0,11,0,0,7,0,0,2,0,3],[48,65,0.7385,0.64726,0.17492,0.57132,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,18,0,0,4,0,0,1,0,5],[52,65,0.8,0.58482,0.16883,0.42965,0.57143,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,7,0,0,15,0,0,4,0,0,2,0,2],[56,65,0.8615,0.58475,0.16506,0.42857,0.5712,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,13,0,0,5,0,0,0,0,3],[60,65,0.9231,0.58918,0.20125,0.42859,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,5,0,0,12,0,0,7,0,0,1,0,3],[64,65,0.9846,0.49107,0.23941,0.39286,0.4286,0.57143,0.0,1.0,2,3,0,2,0,0,0,0,6,0,0,11,0,0,7,0,0,2,0,0,1,0,3],[65,65,1.0,0.43745,0.20493,0.2857,0.42857,0.571,0.0,1.0,1,1,0,1,0,2,0,0,9,0,0,9,0,0,7,0,0,2,0,0,1,0,1]]},{"b":5,"e":0.57143,"k":"flat","v":0.51335,"x":0.65172,"p":[[0,116,0.0,0.51335,0.19513,0.42857,0.4998,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,11,0,0,8,0,0,5,0,0,3,0,0],[4,116,0.0345,0.62926,0.15505,0.571,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,7,0,0,1,0,3],[8,116,0.069,0.60264,0.17399,0.4286,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,9,0,0,11,0,0,6,0,0,3,0,2],[12,116,0.1034,0.65172,0.1673,0.571,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,6,0,0,2,0,4],[16,116,0.1379,0.57136,0.14286,0.5354,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,6,0,0,17,0,0,5,0,0,1,0,1],[20,116,0.1724,0.60259,0.16263,0.571,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,2,0,0,18,0,0,4,0,0,4,0,1],[24,116,0.2069,0.59817,0.15746,0.5354,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,7,0,0,15,0,0,4,0,0,4,0,1],[28,116,0.2414,0.54904,0.17895,0.42857,0.571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,9,0,0,12,0,0,4,0,0,1,0,2],[32,116,0.2759,0.61605,0.23266,0.42857,0.57143,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,9,0,0,9,0,0,4,0,0,0,0,7],[36,116,0.3103,0.58034,0.18189,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,7,0,0,13,0,0,6,0,0,0,0,3],[40,116,0.3448,0.61151,0.17943,0.571,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,4,0,0,17,0,0,5,0,0,0,0,4],[44,116,0.3793,0.58922,0.15465,0.5354,0.57143,0.60714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,16,0,0,5,0,0,1,0,2],[48,116,0.4138,0.62046,0.17355,0.5354,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,6,0,0,4,0,2],[52,116,0.4483,0.58928,0.23077,0.42857,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,10,0,0,8,0,0,4,0,0,0,0,6],[56,116,0.4828,0.58925,0.15047,0.42859,0.57143,0.60714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,15,0,0,5,0,0,1,0,2],[60,116,0.5172,0.63388,0.18191,0.571,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,14,0,0,6,0,0,0,0,5],[64,116,0.5517,0.64283,0.18557,0.5713,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,8,0,0,2,0,4],[68,116,0.5862,0.5892,0.11152,0.571,0.57143,0.71429,0.28571,0.857,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,0,8,0,0,1,0,0],[72,116,0.6207,0.58472,0.10926,0.571,0.57143,0.71407,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,8,0,0,1,0,0],[76,116,0.6552,0.61601,0.16917,0.571,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,15,0,0,6,0,0,3,0,2],[80,116,0.6897,0.57141,0.12366,0.571,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,6,0,0,20,0,0,2,0,0,3,0,0],[84,116,0.7241,0.63388,0.19542,0.5354,0.57143,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,6,0,0,11,0,0,5,0,0,5,0,3],[88,116,0.7586,0.62049,0.21011,0.42859,0.57143,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,10,0,0,11,0,0,2,0,0,3,0,5],[92,116,0.7931,0.58476,0.13534,0.571,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,19,0,0,4,0,0,0,0,2],[96,116,0.8276,0.58031,0.11811,0.571,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,0,3,0,0,3,0,0],[100,116,0.8621,0.5892,0.15048,0.571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,18,0,0,4,0,0,1,0,2],[104,116,0.8966,0.59367,0.19597,0.42859,0.57141,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,7,0,0,13,0,0,3,0,0,3,0,3],[108,116,0.931,0.64732,0.22299,0.4286,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,12,0,0,3,0,0,0,0,8],[112,116,0.9655,0.64726,0.18554,0.5354,0.64286,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,9,0,0,4,0,3],[116,116,1.0,0.59819,0.19378,0.5354,0.57143,0.60714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,5,0,0,16,0,0,3,0,0,1,0,4]]}]},{"i":"ed916a5fbfc3add6","q":"Let  $n \\in \\mathbb N$  and  $A_n$  set of all permutations  $(a_1, \\ldots, a_n)$  of the set  $\\{1, 2, \\ldots , n\\}$  for which\n\\[k|2(a_1 + \\cdots+ a_k), \\text{ for all } 1 \\leq k \\leq n.\\]\nFind the number of elements of the set  $A_n$ .\n\n*Proposed by Vidan Govedarica, Serbia*","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.98661,"p":[[0,92,0.0,0.29911,0.31412,0.14286,0.14286,0.32144,0.0,1.0,4,3,4,4,0,18,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,3],[4,92,0.0435,0.91964,0.24206,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[8,92,0.087,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,92,0.1304,0.84375,0.31209,0.85714,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,20],[16,92,0.1739,0.87054,0.28428,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[20,92,0.2174,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,92,0.2609,0.91518,0.1551,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,22],[28,92,0.3043,0.87054,0.26812,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,22],[32,92,0.3478,0.92856,0.11849,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[36,92,0.3913,0.91071,0.14174,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[40,92,0.4348,0.90179,0.18363,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,21],[44,92,0.4783,0.79911,0.2693,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,10,0,13],[48,92,0.5217,0.72321,0.30501,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,2,0,0,1,0,0,5,0,0,4,0,0,5,0,12],[52,92,0.5652,0.63393,0.35881,0.2857,0.78571,1.0,0.0,1.0,2,10,0,2,0,5,0,0,3,0,0,2,0,0,1,0,0,3,0,0,6,0,10],[56,92,0.6087,0.53572,0.3993,0.10714,0.71429,0.85714,0.0,1.0,8,6,0,8,0,4,0,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,6],[60,92,0.6522,0.51786,0.3567,0.24999,0.64286,0.75,0.0,1.0,7,5,0,7,0,1,0,0,4,0,0,2,0,0,2,0,0,8,0,0,3,0,5],[64,92,0.6957,0.46429,0.35174,0.14286,0.42857,0.75,0.0,1.0,7,3,0,7,0,4,0,0,2,0,0,4,0,0,2,0,0,5,0,0,5,0,3],[68,92,0.7391,0.51338,0.36572,0.14286,0.57121,0.85714,0.0,1.0,6,5,0,6,0,3,0,0,4,0,0,2,0,0,4,0,0,1,0,0,7,0,5],[72,92,0.7826,0.45522,0.37692,0.0,0.57143,0.85714,0.0,1.0,9,3,0,9,0,4,0,0,2,0,0,0,0,0,3,0,0,5,0,0,6,0,3],[76,92,0.8261,0.48659,0.39747,0.0,0.64264,0.85714,0.0,1.0,9,6,0,9,0,3,0,0,3,0,0,0,0,0,1,0,0,6,0,0,4,0,6],[80,92,0.8696,0.03571,0.11294,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[84,92,0.913,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,92,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,92,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.26785,"x":0.94642,"p":[[0,174,0.0,0.375,0.36727,0.14286,0.14286,0.75,0.0,1.0,6,4,4,6,0,13,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,4],[4,174,0.023,0.94642,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[8,174,0.046,0.91518,0.20472,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,24],[12,174,0.069,0.87946,0.26752,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,23],[16,174,0.092,0.80795,0.32869,0.85714,1.0,1.0,0.0,1.0,2,19,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,7,0,19],[20,174,0.1149,0.78571,0.32537,0.85714,0.92857,1.0,0.0,1.0,2,16,0,2,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,9,0,16],[24,174,0.1379,0.74107,0.31224,0.67857,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,0,8,0,12],[28,174,0.1609,0.80357,0.2714,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,4,0,0,9,0,14],[32,174,0.1839,0.79463,0.28333,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,3,0,0,0,0,0,1,0,0,4,0,0,7,0,15],[36,174,0.2069,0.72322,0.32915,0.53571,0.85714,1.0,0.0,1.0,3,10,0,3,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,12,0,10],[40,174,0.2299,0.6607,0.31288,0.39286,0.78564,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,4,0,0,1,0,0,2,0,0,5,0,0,10,0,6],[44,174,0.2529,0.75,0.3312,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,3,0,0,1,0,0,1,0,0,0,0,0,4,0,0,7,0,14],[48,174,0.2759,0.54455,0.39203,0.14286,0.42857,1.0,0.0,1.0,3,11,0,3,0,7,0,0,5,0,0,2,0,0,0,0,0,2,0,0,2,0,11],[52,174,0.2989,0.5625,0.36759,0.24999,0.71429,0.85714,0.0,1.0,5,5,0,5,0,3,0,0,5,0,0,0,0,0,1,0,0,4,0,0,9,0,5],[56,174,0.3218,0.58035,0.37954,0.14286,0.71429,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,3,0,0,0,0,0,2,0,0,3,0,0,8,0,7],[60,174,0.3448,0.41964,0.39275,0.0,0.28574,0.85714,0.0,1.0,11,3,0,11,0,5,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,3],[64,174,0.3678,0.54017,0.33069,0.28571,0.57143,0.85714,0.0,1.0,3,4,0,3,0,4,0,0,5,0,0,2,0,0,4,0,0,3,0,0,7,0,4],[68,174,0.3908,0.62053,0.35285,0.28571,0.85714,0.85714,0.0,1.0,3,6,0,3,0,3,0,0,5,0,0,1,0,0,1,0,0,1,0,0,12,0,6],[72,174,0.4138,0.57143,0.33312,0.28571,0.71429,0.85714,0.0,1.0,3,3,0,3,0,4,0,0,4,0,0,2,0,0,1,0,0,5,0,0,10,0,3],[76,174,0.4368,0.54464,0.34522,0.14289,0.57143,0.85714,0.0,1.0,3,5,0,3,0,6,0,0,3,0,0,1,0,0,4,0,0,4,0,0,6,0,5],[80,174,0.4598,0.49105,0.36058,0.2857,0.42857,0.85714,0.0,1.0,6,6,0,6,0,1,0,0,8,0,0,3,0,0,3,0,0,0,0,0,5,0,6],[84,174,0.4828,0.55795,0.36495,0.24999,0.5,1.0,0.0,1.0,3,9,0,3,0,5,0,0,4,0,0,4,0,0,1,0,0,3,0,0,3,0,9],[88,174,0.5057,0.58482,0.32803,0.28571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,5,0,0,3,0,0,2,0,0,3,0,0,5,0,0,7,0,5],[92,174,0.5287,0.65625,0.35149,0.28571,0.78571,1.0,0.0,1.0,2,11,0,2,0,4,0,0,3,0,0,2,0,0,1,0,0,4,0,0,5,0,11],[96,174,0.5517,0.57588,0.35262,0.2857,0.57143,1.0,0.0,1.0,3,9,0,3,0,2,0,0,8,0,0,1,0,0,3,0,0,3,0,0,3,0,9],[100,174,0.5747,0.62499,0.28291,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,5,0,0,2,0,0,5,0,0,9,0,4],[104,174,0.5977,0.55357,0.30251,0.28571,0.50001,0.85714,0.0,1.0,2,3,0,2,0,3,0,0,4,0,0,7,0,0,2,0,0,3,0,0,8,0,3],[108,174,0.6207,0.76339,0.25657,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,4,0,0,10,0,10],[112,174,0.6437,0.64732,0.33309,0.28571,0.71429,1.0,0.0,1.0,1,9,0,1,0,5,0,0,3,0,0,2,0,0,0,0,0,7,0,0,5,0,9],[116,174,0.6667,0.50446,0.3273,0.28571,0.42857,0.85714,0.0,1.0,4,3,0,4,0,3,0,0,6,0,0,4,0,0,1,0,0,5,0,0,6,0,3],[120,174,0.6897,0.52677,0.38703,0.14286,0.71429,0.85714,0.0,1.0,6,6,0,6,0,6,0,0,1,0,0,1,0,0,1,0,0,5,0,0,6,0,6],[124,174,0.7126,0.45089,0.23449,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,9,0,0,8,0,0,4,0,0,2,0,0,5,0,0],[128,174,0.7356,0.5625,0.30917,0.28571,0.42857,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,10,0,0,5,0,0,0,0,0,4,0,0,6,0,5],[132,174,0.7586,0.41518,0.24836,0.2857,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,9,0,0,6,0,0,4,0,0,5,0,0,1,0,1],[136,174,0.7816,0.50893,0.26711,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,7,0,0,7,0,0,2,0,0,6,0,0,4,0,2],[140,174,0.8046,0.46874,0.30143,0.2857,0.28571,0.74996,0.0,1.0,1,3,0,1,0,5,0,0,12,0,0,1,0,0,3,0,0,2,0,0,5,0,3],[144,174,0.8276,0.48214,0.29179,0.28571,0.42857,0.71429,0.0,1.0,3,4,0,3,0,2,0,0,6,0,0,9,0,0,3,0,0,3,0,0,2,0,4],[148,174,0.8506,0.33928,0.24157,0.2857,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,1,0,0,14,0,0,4,0,0,2,0,0,3,0,0,2,0,0],[152,174,0.8736,0.30357,0.22232,0.24999,0.28571,0.32143,0.0,1.0,5,1,0,5,0,3,0,0,16,0,0,5,0,0,0,0,0,1,0,0,1,0,1],[156,174,0.8966,0.28571,0.17857,0.24999,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,3,0,0,15,0,0,7,0,0,1,0,0,0,0,0,1,0,0],[160,174,0.9195,0.33036,0.21558,0.2857,0.28571,0.4287,0.0,0.85714,6,0,0,6,0,1,0,0,12,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[164,174,0.9425,0.2991,0.22406,0.24999,0.28571,0.28571,0.0,1.0,6,1,0,6,0,2,0,0,17,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[168,174,0.9655,0.43304,0.25123,0.28571,0.42857,0.4286,0.0,1.0,2,1,0,2,0,1,0,0,12,0,0,10,0,0,0,0,0,1,0,0,5,0,1],[172,174,0.9885,0.26785,0.05922,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[174,174,1.0,0.28572,0.03571,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"981bde5fc9d2c82b","q":"Let  $p_1,p_2,p_3,p_4$  be four distinct primes, and let  $1=d_1<d_2<\\ldots<d_{16}=n$  be the divisors of  $n=p_1p_2p_3p_4$ . Determine all  $n<2001$  with the property that $d_9-d_8=22$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.75,"x":0.9866,"p":[[0,203,0.0,0.87054,0.19352,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,2,0,20],[4,203,0.0197,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,203,0.0394,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,203,0.0591,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[16,203,0.0788,0.92857,0.18558,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[20,203,0.0985,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[24,203,0.1182,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,203,0.1379,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[32,203,0.1576,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[36,203,0.1773,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[40,203,0.197,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[44,203,0.2167,0.93311,0.10703,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[48,203,0.2365,0.91963,0.08703,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[52,203,0.2562,0.92408,0.11291,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[56,203,0.2759,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[60,203,0.2956,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[64,203,0.3153,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[68,203,0.335,0.92857,0.10715,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[72,203,0.3547,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[76,203,0.3744,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[80,203,0.3941,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[84,203,0.4138,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[88,203,0.4335,0.93304,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[92,203,0.4532,0.92411,0.18205,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,22],[96,203,0.4729,0.9241,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[100,203,0.4926,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[104,203,0.5123,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[108,203,0.532,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[112,203,0.5517,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[116,203,0.5714,0.9866,0.04167,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[120,203,0.5911,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[124,203,0.6108,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[128,203,0.6305,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[132,203,0.6502,0.94195,0.10635,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[136,203,0.67,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[140,203,0.6897,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[144,203,0.7094,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[148,203,0.7291,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[152,203,0.7488,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[156,203,0.7685,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[160,203,0.7882,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[164,203,0.8079,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[168,203,0.8276,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[172,203,0.8473,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[176,203,0.867,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[180,203,0.8867,0.94643,0.12753,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[184,203,0.9064,0.91069,0.13248,0.82143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,21],[188,203,0.9261,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[192,203,0.9458,0.90179,0.1448,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,21],[196,203,0.9655,0.89732,0.15251,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,1,0,21],[200,203,0.9852,0.75445,0.10854,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[203,203,1.0,0.75,0.10102,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,1,0,4]]},{"b":3,"e":1.0,"k":"flat","v":0.88391,"x":1.0,"p":[[0,69,0.0,0.88391,0.18015,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,20],[4,69,0.058,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,69,0.1159,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[12,69,0.1739,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,69,0.2319,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,69,0.2899,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,69,0.3478,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,69,0.4058,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,69,0.4638,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,69,0.5217,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[40,69,0.5797,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,69,0.6377,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,69,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,69,0.7536,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,69,0.8116,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,69,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,69,0.9275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,69,0.9855,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[69,69,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ba580f39002d7e2b","q":"Let  $x_1, x_2 \\ldots , x_5$  be real numbers. Find the least positive integer  $n$  with the following property: if some  $n$  distinct sums of the form  $x_p+x_q+x_r$  (with  $1\\le p<q<r\\le 5$ ) are equal to  $0$ , then  $x_1=x_2=\\cdots=x_5=0$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.44644,"x":0.9375,"p":[[0,79,0.0,0.44644,0.26183,0.2857,0.28571,0.4286,0.2857,1.0,0,5,0,0,0,0,0,0,20,0,0,5,0,0,0,0,0,2,0,0,0,0,5],[4,79,0.0506,0.83481,0.27226,0.57143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,23],[8,79,0.1013,0.81247,0.25367,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,2,0,0,5,0,17],[12,79,0.1519,0.88838,0.21051,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,23],[16,79,0.2025,0.84374,0.24319,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,19],[20,79,0.2532,0.9375,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[24,79,0.3038,0.71426,0.29016,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,4,0,0,6,0,0,1,0,0,2,0,14],[28,79,0.3544,0.88839,0.21939,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,1,0,24],[32,79,0.4051,0.83926,0.24939,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,19],[36,79,0.4557,0.69196,0.40738,0.42857,1.0,1.0,0.0,1.0,7,18,0,7,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,18],[40,79,0.5063,0.64731,0.35712,0.42857,0.64286,1.0,0.0,1.0,5,13,0,5,0,0,0,0,0,0,0,6,0,0,5,0,0,2,0,0,1,0,13],[44,79,0.557,0.70534,0.27185,0.42857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,5,0,0,0,0,13],[48,79,0.6076,0.70088,0.33381,0.42857,0.71429,1.0,0.0,1.0,3,15,0,3,0,0,0,0,2,0,0,5,0,0,2,0,0,5,0,0,0,0,15],[52,79,0.6582,0.81249,0.22991,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,3,0,0,3,0,17],[56,79,0.7089,0.60712,0.31944,0.42857,0.57143,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,2,0,0,5,0,0,6,0,0,5,0,0,2,0,8],[60,79,0.7595,0.75447,0.26058,0.53572,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,7,0,0,0,0,15],[64,79,0.8101,0.65176,0.28108,0.53539,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,4,0,0,2,0,0,5,0,0,9,0,0,3,0,7],[68,79,0.8608,0.71426,0.20825,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,10,0,0,4,0,7],[72,79,0.9114,0.75,0.25254,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,9,0,0,1,0,13],[76,79,0.962,0.70533,0.23404,0.57132,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,13,0,0,2,0,8],[79,79,1.0,0.71424,0.25508,0.571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,4,0,0,3,0,11]]},{"b":3,"e":1.0,"k":"rising","v":0.43749,"x":0.90622,"p":[[0,128,0.0,0.43749,0.22569,0.28571,0.28571,0.4642,0.2857,1.0,0,2,0,0,0,0,0,0,18,0,0,6,0,0,3,0,0,0,0,0,3,0,2],[4,128,0.0312,0.90622,0.20399,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,26],[8,128,0.0625,0.79464,0.24468,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,1,0,17],[12,128,0.0938,0.77678,0.24727,0.67857,0.78564,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,8,0,0,1,0,15],[16,128,0.125,0.89286,0.20825,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,24],[20,128,0.1562,0.85267,0.21868,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,2,0,20],[24,128,0.1875,0.83927,0.24938,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,5,0,19],[28,128,0.2188,0.85714,0.24485,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,22],[32,128,0.25,0.7857,0.3134,0.57132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,1,0,20],[36,128,0.2812,0.89732,0.20897,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,24],[40,128,0.3125,0.86606,0.21111,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,6,0,19],[44,128,0.3438,0.87945,0.23989,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,25],[48,128,0.375,0.86159,0.24351,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,0,3,0,22],[52,128,0.4062,0.86606,0.24208,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,21],[56,128,0.4375,0.85714,0.22016,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,5,0,19],[60,128,0.4688,0.89732,0.2237,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,25],[64,128,0.5,0.83929,0.27836,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,22],[68,128,0.5312,0.89284,0.20518,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,24],[72,128,0.5625,0.88838,0.21648,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,22],[76,128,0.5938,0.85714,0.27432,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,23],[80,128,0.625,0.73659,0.26514,0.57132,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,4,0,0,1,0,14],[84,128,0.6562,0.75,0.28121,0.42857,0.9285,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,4,0,0,1,0,16],[88,128,0.6875,0.72767,0.28204,0.42857,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,2,0,0,1,0,15],[92,128,0.7188,0.66517,0.28929,0.42857,0.64286,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,7,0,0,5,0,0,4,0,0,4,0,0,0,0,12],[96,128,0.75,0.72768,0.29957,0.42857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,9,0,0,1,0,0,1,0,0,0,0,17],[100,128,0.7812,0.69642,0.31084,0.42857,0.78571,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,16],[104,128,0.8125,0.69641,0.31491,0.42857,0.78571,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,8,0,0,2,0,0,1,0,0,1,0,15],[108,128,0.8438,0.74106,0.30187,0.42857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,6,0,0,5,0,0,1,0,0,2,0,0,1,0,17],[112,128,0.875,0.73213,0.30253,0.42857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,6,0,0,5,0,0,2,0,0,2,0,0,0,0,17],[116,128,0.9062,0.78124,0.28345,0.42859,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,7,0,0,1,0,0,1,0,0,1,0,19],[120,128,0.9375,0.69642,0.31289,0.42857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,7,0,0,7,0,0,1,0,0,0,0,0,2,0,15],[124,128,0.9688,0.74106,0.30814,0.39286,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,8,0,0,2,0,0,1,0,0,3,0,0,1,0,17],[128,128,1.0,0.71872,0.29772,0.42857,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,7,0,0,2,0,0,2,0,0,0,0,16]]}]},{"i":"bf256ea866c39b25","q":"Let  $z_1 = 18 + 83i$ ,  $z_2 = 18 + 39i, $  and  $z_3 = 78 + 99i,$  where  $i = \\sqrt{-1}$ . Let  $z$  be the unique complex number with the properties that  $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$  is a real number and the imaginary part of  $z$  is the greatest possible. Find the real part of  $z$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.77232,"x":1.0,"p":[[0,208,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,208,0.0192,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,208,0.0385,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[12,208,0.0577,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[16,208,0.0769,0.94196,0.13767,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,27],[20,208,0.0962,0.93304,0.14718,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,26],[24,208,0.1154,0.91071,0.18123,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,0,0,25],[28,208,0.1346,0.875,0.21943,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,1,0,23],[32,208,0.1538,0.90625,0.16602,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,0,0,24],[36,208,0.1731,0.92857,0.15567,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,26],[40,208,0.1923,0.88839,0.24416,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,26],[44,208,0.2115,0.87946,0.22047,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,24],[48,208,0.2308,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[52,208,0.25,0.95089,0.16213,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[56,208,0.2692,0.84375,0.27976,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,23],[60,208,0.2885,0.81696,0.22654,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,6,0,0,0,0,18],[64,208,0.3077,0.93302,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[68,208,0.3269,0.87053,0.2212,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,2,0,22],[72,208,0.3462,0.91518,0.1885,1.0,1.0,1.0,0.1429,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[76,208,0.3654,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[80,208,0.3846,0.90177,0.17293,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,2,0,23],[84,208,0.4038,0.89732,0.20277,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,24],[88,208,0.4231,0.87052,0.21537,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,21],[92,208,0.4423,0.82589,0.24415,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,0,0,0,20],[96,208,0.4615,0.83929,0.23077,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,20],[100,208,0.4808,0.88393,0.20958,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,22],[104,208,0.5,0.9375,0.14699,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,26],[108,208,0.5192,0.81697,0.27487,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,1,0,0,2,0,20],[112,208,0.5385,0.77232,0.29851,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,19],[116,208,0.5577,0.86161,0.19393,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,1,0,20],[120,208,0.5769,0.94643,0.13243,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[124,208,0.5962,0.85267,0.2328,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,21],[128,208,0.6154,0.88839,0.19475,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,1,0,23],[132,208,0.6346,0.89286,0.18898,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,1,0,23],[136,208,0.6538,0.91071,0.19805,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,26],[140,208,0.6731,0.89284,0.18901,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,1,0,23],[144,208,0.6923,0.90625,0.23854,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[148,208,0.7115,0.85714,0.26,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,23],[152,208,0.7308,0.90625,0.18423,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,25],[156,208,0.75,0.9107,0.21356,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,27],[160,208,0.7692,0.875,0.21943,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,0,0,23],[164,208,0.7885,0.92411,0.17851,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,26],[168,208,0.8077,0.83482,0.23449,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,0,1,0,19],[172,208,0.8269,0.85714,0.20825,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,2,0,20],[176,208,0.8462,0.89732,0.17215,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,0,0,23],[180,208,0.8654,0.91518,0.16311,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[184,208,0.8846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[188,208,0.9038,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[192,208,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[196,208,0.9423,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[200,208,0.9615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[204,208,0.9808,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[208,208,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,55,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,55,0.0727,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,55,0.1455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,55,0.2182,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,55,0.2909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,55,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,55,0.4364,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,55,0.5818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,55,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,55,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,55,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,55,0.9455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"73b43561441f5884","q":"Let $(a_n)$ be a strictly increasing sequence of positive integers such that $a_1 = 1$ and for all $n \\geq 1, a_{n+1} \\leq 2n$. Show that for every integer $n \\geq 1$, there exist two indices $p$ and $q$ such that $a_p - a_q = n$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.01339,"x":0.90179,"p":[[0,76,0.0,0.90179,0.2299,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,26],[4,76,0.0526,0.72768,0.42612,0.53571,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,21],[8,76,0.1053,0.49554,0.47378,0.0,0.71429,1.0,0.0,1.0,15,13,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,13],[12,76,0.1579,0.39732,0.47479,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,12],[16,76,0.2105,0.23214,0.41304,0.0,0.0,0.10714,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[20,76,0.2632,0.59821,0.47034,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,17],[24,76,0.3158,0.45089,0.45612,0.0,0.21429,1.0,0.0,1.0,14,11,0,14,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,11],[28,76,0.3684,0.375,0.48412,0.0,0.0,1.0,0.0,1.0,20,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[32,76,0.4211,0.35268,0.45032,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,10],[36,76,0.4737,0.4375,0.47102,0.0,0.07143,1.0,0.0,1.0,16,12,0,16,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,12],[40,76,0.5263,0.33482,0.44979,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,9],[44,76,0.5789,0.34375,0.45995,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,10],[48,76,0.6316,0.32143,0.4475,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,9],[52,76,0.6842,0.26339,0.42873,0.0,0.0,0.46429,0.0,1.0,22,8,0,22,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[56,76,0.7368,0.42411,0.48245,0.0,0.0,1.0,0.0,1.0,17,13,0,17,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13],[60,76,0.7895,0.32143,0.45034,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[64,76,0.8421,0.20536,0.38455,0.0,0.0,0.14286,0.0,1.0,22,6,0,22,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[68,76,0.8947,0.07589,0.24218,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[72,76,0.9474,0.08036,0.2141,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[76,76,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"volatile","v":0.00446,"x":0.88393,"p":[[0,43,0.0,0.88393,0.3102,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,43,0.093,0.36607,0.46966,0.0,0.0,1.0,0.0,1.0,19,11,0,19,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11],[8,43,0.186,0.36607,0.46694,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,10],[12,43,0.2791,0.625,0.47077,0.0,1.0,1.0,0.0,1.0,11,19,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,19],[16,43,0.3721,0.43304,0.45943,0.0,0.21428,1.0,0.0,1.0,15,11,0,15,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,11],[20,43,0.4651,0.37946,0.45402,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,10],[24,43,0.5581,0.48659,0.47361,0.0,0.42835,1.0,0.0,1.0,14,14,0,14,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,14],[28,43,0.6512,0.16964,0.36147,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[32,43,0.7442,0.22768,0.41166,0.0,0.0,0.07143,0.0,1.0,24,7,0,24,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[36,43,0.8372,0.19643,0.36378,0.0,0.0,0.14287,0.0,1.0,23,4,0,23,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[40,43,0.9302,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"5b479910425e86bd","q":"Let $a, b, c$ be positive real numbers. Prove the inequality\n\n$$\n\\left(a^{2}+a c+c^{2}\\right)\\left(\\frac{1}{a+b+c}+\\frac{1}{a+c}\\right)+b^{2}\\left(\\frac{1}{b+c}+\\frac{1}{a+b}\\right)>a+b+c\n$$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.07143,"x":0.45982,"p":[[0,104,0.0,0.13832,0.08365,0.14286,0.14286,0.14286,0.0,0.286,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,104,0.0385,0.27679,0.24984,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,3,0,0,6,0,0,1,0,0,3,0,0,0,0,1],[8,104,0.0769,0.3259,0.24804,0.14286,0.2143,0.46431,0.0,0.85714,3,0,0,3,0,13,0,0,3,0,0,5,0,0,2,0,0,5,0,0,1,0,0],[12,104,0.1154,0.36161,0.32139,0.14286,0.21435,0.71429,0.0,1.0,7,2,0,7,0,9,0,0,2,0,0,3,0,0,0,0,0,9,0,0,0,0,2],[16,104,0.1538,0.39286,0.28121,0.14286,0.42857,0.57143,0.0,1.0,5,2,0,5,0,5,0,0,5,0,0,6,0,0,4,0,0,5,0,0,0,0,2],[20,104,0.1923,0.3525,0.26497,0.14286,0.28571,0.57143,0.0,1.0,3,1,0,3,0,11,0,0,5,0,0,3,0,0,3,0,0,6,0,0,0,0,1],[24,104,0.2308,0.3125,0.2635,0.14286,0.2143,0.42857,0.0,1.0,4,1,0,4,0,12,0,0,5,0,0,5,0,0,0,0,0,4,0,0,1,0,1],[28,104,0.2692,0.375,0.3004,0.14286,0.28571,0.71429,0.0,1.0,7,1,0,7,0,5,0,0,5,0,0,4,0,0,1,0,0,8,0,0,1,0,1],[32,104,0.3077,0.37946,0.27574,0.14286,0.28571,0.60714,0.0,1.0,4,1,0,4,0,7,0,0,7,0,0,3,0,0,3,0,0,6,0,0,1,0,1],[36,104,0.3462,0.33036,0.31428,0.14286,0.14286,0.60714,0.0,1.0,7,2,0,7,0,11,0,0,1,0,0,4,0,0,1,0,0,5,0,0,1,0,2],[40,104,0.3846,0.38392,0.30812,0.14286,0.35714,0.71429,0.0,1.0,6,1,0,6,0,8,0,0,2,0,0,4,0,0,2,0,0,7,0,0,2,0,1],[44,104,0.4231,0.39286,0.24223,0.14286,0.42857,0.71429,0.0,0.71429,3,0,0,3,0,7,0,0,3,0,0,10,0,0,0,0,0,9,0,0,0,0,0],[48,104,0.4615,0.41063,0.29187,0.14286,0.42857,0.71429,0.0,1.0,3,2,0,3,0,9,0,0,2,0,0,8,0,0,0,0,0,7,0,0,1,0,2],[52,104,0.5,0.41518,0.28203,0.14286,0.42857,0.71429,0.0,1.0,2,2,0,2,0,10,0,0,2,0,0,7,0,0,1,0,0,8,0,0,0,0,2],[56,104,0.5385,0.45982,0.26663,0.28571,0.42857,0.71429,0.0,1.0,2,2,0,2,0,5,0,0,5,0,0,7,0,0,2,0,0,9,0,0,0,0,2],[60,104,0.5769,0.30804,0.24251,0.14286,0.21428,0.42857,0.0,0.85714,5,0,0,5,0,11,0,0,2,0,0,7,0,0,3,0,0,3,0,0,1,0,0],[64,104,0.6154,0.38393,0.31428,0.14286,0.42857,0.71429,0.0,1.0,5,3,0,5,0,10,0,0,0,0,0,7,0,0,1,0,0,6,0,0,0,0,3],[68,104,0.6538,0.35714,0.26726,0.14286,0.42857,0.60714,0.0,0.85714,5,0,0,5,0,9,0,0,1,0,0,8,0,0,1,0,0,7,0,0,1,0,0],[72,104,0.6923,0.32125,0.27676,0.14214,0.21431,0.42857,0.0,1.0,6,1,0,6,0,10,0,0,2,0,0,7,0,0,1,0,0,4,0,0,1,0,1],[76,104,0.7308,0.30804,0.29904,0.14286,0.14286,0.42858,0.0,1.0,6,3,0,6,0,12,0,0,3,0,0,4,0,0,2,0,0,2,0,0,0,0,3],[80,104,0.7692,0.28129,0.29558,0.0,0.14286,0.42893,0.0,1.0,10,1,0,10,0,9,0,0,1,0,0,5,0,0,2,0,0,2,0,0,2,0,1],[84,104,0.8077,0.15179,0.17835,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,18,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[88,104,0.8462,0.10714,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,104,0.8846,0.09375,0.09182,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,104,0.9231,0.16054,0.13719,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,23,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[100,104,0.9615,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,104,1.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.14732,"x":0.41964,"p":[[0,107,0.0,0.20536,0.17474,0.14286,0.14286,0.42857,0.0,0.57143,7,0,0,7,0,15,0,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[4,107,0.0374,0.3125,0.28669,0.10714,0.14288,0.46429,0.0,1.0,8,1,0,8,0,9,0,0,1,0,0,6,0,0,1,0,0,6,0,0,0,0,1],[8,107,0.0748,0.41964,0.23128,0.14286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,10,0,0,1,0,0,10,0,0,6,0,0,3,0,0,1,0,1],[12,107,0.1121,0.34821,0.2141,0.14286,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,12,0,0,2,0,0,11,0,0,2,0,0,3,0,0,1,0,0],[16,107,0.1495,0.30804,0.19597,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,14,0,0,4,0,0,8,0,0,2,0,0,3,0,0,0,0,0],[20,107,0.1869,0.36161,0.26241,0.14286,0.35714,0.42858,0.0,1.0,1,2,0,1,0,14,0,0,1,0,0,9,0,0,1,0,0,4,0,0,0,0,2],[24,107,0.2243,0.36159,0.1785,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,1,0,0,14,0,0,4,0,0,2,0,0,0,0,0],[28,107,0.2617,0.28571,0.21724,0.14286,0.14286,0.42857,0.0,1.0,1,1,0,1,0,18,0,0,1,0,0,9,0,0,0,0,0,2,0,0,0,0,1],[32,107,0.2991,0.34375,0.23381,0.14286,0.35714,0.42857,0.0,1.0,1,1,0,1,0,13,0,0,2,0,0,11,0,0,1,0,0,2,0,0,1,0,1],[36,107,0.3364,0.27677,0.18187,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,17,0,0,2,0,0,9,0,0,1,0,0,2,0,0,0,0,0],[40,107,0.3738,0.35714,0.21724,0.14286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,10,0,0,2,0,0,11,0,0,2,0,0,5,0,0,0,0,0],[44,107,0.4112,0.29464,0.2141,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,16,0,0,1,0,0,8,0,0,1,0,0,4,0,0,0,0,0],[48,107,0.4486,0.28125,0.1838,0.14286,0.21428,0.42857,0.0,0.85714,2,0,0,2,0,14,0,0,3,0,0,11,0,0,1,0,0,0,0,0,1,0,0],[52,107,0.486,0.3125,0.1448,0.14286,0.42857,0.42857,0.0,0.42857,1,0,0,1,0,11,0,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[56,107,0.5234,0.3125,0.14914,0.14286,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,5,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[60,107,0.5607,0.30357,0.22232,0.14286,0.21428,0.42857,0.0,0.85714,3,0,0,3,0,13,0,0,2,0,0,10,0,0,0,0,0,3,0,0,1,0,0],[64,107,0.5981,0.28571,0.23958,0.14286,0.14286,0.42857,0.0,0.85714,3,0,0,3,0,17,0,0,1,0,0,5,0,0,1,0,0,4,0,0,1,0,0],[68,107,0.6355,0.27679,0.2257,0.14286,0.14286,0.42857,0.0,0.85714,4,0,0,4,0,15,0,0,1,0,0,7,0,0,2,0,0,2,0,0,1,0,0],[72,107,0.6729,0.25447,0.19475,0.14286,0.14286,0.42857,0.0,0.85714,4,0,0,4,0,15,0,0,1,0,0,10,0,0,1,0,0,0,0,0,1,0,0],[76,107,0.7103,0.21875,0.16746,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,12,0,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[80,107,0.7477,0.25446,0.23347,0.14286,0.14286,0.32143,0.0,1.0,3,1,0,3,0,19,0,0,2,0,0,4,0,0,0,0,0,3,0,0,0,0,1],[84,107,0.785,0.29464,0.2111,0.14286,0.21428,0.42857,0.0,0.71429,3,0,0,3,0,13,0,0,3,0,0,9,0,0,0,0,0,4,0,0,0,0,0],[88,107,0.8224,0.23661,0.14987,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,20,0,0,2,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[92,107,0.8598,0.27677,0.16725,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,15,0,0,0,0,0,13,0,0,2,0,0,0,0,0,0,0,0],[96,107,0.8972,0.25438,0.16656,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,17,0,0,1,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[100,107,0.9346,0.3125,0.15746,0.14286,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,13,0,0,3,0,0,14,0,0,1,0,0,1,0,0,0,0,0],[104,107,0.972,0.25893,0.16146,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,15,0,0,4,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[107,107,1.0,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,24,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"924632dd3a8898bb","q":"Let $a, b, c$ be positive real numbers such that $a b c=\\frac{2}{3}$. Prove that\n\n$$\n\\frac{a b}{a+b}+\\frac{b c}{b+c}+\\frac{c a}{c+a} \\geqslant \\frac{a+b+c}{a^{3}+b^{3}+c^{3}}\n$$\n\n(FYR Macedonia)","t":[{"b":1,"e":0.42857,"k":"flat","v":0.08928,"x":0.36161,"p":[[0,134,0.0,0.18286,0.27257,0.0,0.14286,0.14286,0.0,1.0,10,3,0,10,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,134,0.0299,0.26338,0.27688,0.14286,0.14286,0.28571,0.0,1.0,6,3,0,6,0,13,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,3],[8,134,0.0597,0.27231,0.2509,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,6,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[12,134,0.0896,0.22312,0.23944,0.14286,0.14286,0.2857,0.0,1.0,6,2,0,6,0,16,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[16,134,0.1194,0.36161,0.34253,0.14286,0.2857,0.5,0.0,1.0,6,4,0,6,0,9,0,0,6,0,0,3,0,0,0,0,0,1,0,0,3,0,4],[20,134,0.1493,0.27232,0.25344,0.14286,0.2857,0.42857,0.0,1.0,7,2,0,7,0,8,0,0,8,0,0,6,0,0,0,0,0,1,0,0,0,0,2],[24,134,0.1791,0.26339,0.27458,0.10714,0.14286,0.32143,0.0,1.0,8,2,0,8,0,10,0,0,6,0,0,3,0,0,2,0,0,0,0,0,1,0,2],[28,134,0.209,0.30804,0.29904,0.14286,0.14286,0.42857,0.0,1.0,6,3,0,6,0,12,0,0,3,0,0,4,0,0,2,0,0,2,0,0,0,0,3],[32,134,0.2388,0.27235,0.21238,0.14286,0.14288,0.42857,0.0,0.71429,5,0,0,5,0,12,0,0,4,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[36,134,0.2687,0.31701,0.25937,0.14286,0.28571,0.42857,0.0,1.0,6,2,0,6,0,8,0,0,3,0,0,9,0,0,4,0,0,0,0,0,0,0,2],[40,134,0.2985,0.30357,0.2519,0.14286,0.1429,0.42857,0.0,1.0,3,2,0,3,0,14,0,0,3,0,0,7,0,0,2,0,0,1,0,0,0,0,2],[44,134,0.3284,0.35268,0.23954,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,10,0,0,5,0,0,8,0,0,3,0,0,2,0,0,1,0,1],[48,134,0.3582,0.29006,0.23553,0.14286,0.14286,0.42858,0.0,1.0,5,1,0,5,0,12,0,0,2,0,0,6,0,0,6,0,0,0,0,0,0,0,1],[52,134,0.3881,0.22321,0.25238,0.0,0.14286,0.42857,0.0,1.0,12,1,0,12,0,8,0,0,3,0,0,4,0,0,3,0,0,1,0,0,0,0,1],[56,134,0.4179,0.26339,0.27689,0.0,0.14286,0.32143,0.0,1.0,9,2,0,9,0,8,0,0,7,0,0,3,0,0,2,0,0,0,0,0,1,0,2],[60,134,0.4478,0.30348,0.32688,0.14214,0.14286,0.42857,0.0,1.0,7,4,0,7,0,12,0,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,4],[64,134,0.4776,0.20088,0.21084,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,9,0,0,2,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[68,134,0.5075,0.30357,0.32684,0.0,0.14286,0.46429,0.0,1.0,9,4,0,9,0,9,0,0,4,0,0,2,0,0,3,0,0,1,0,0,0,0,4],[72,134,0.5373,0.25,0.22016,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,14,0,0,1,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[76,134,0.5672,0.28572,0.26964,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,2,0,0,6,0,0,3,0,0,0,0,0,2,0,1],[80,134,0.597,0.27679,0.23673,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,7,0,0,4,0,0,8,0,0,4,0,0,0,0,0,0,0,1],[84,134,0.6269,0.3259,0.284,0.14286,0.2857,0.57143,0.0,1.0,6,1,0,6,0,9,0,0,6,0,0,2,0,0,2,0,0,5,0,0,1,0,1],[88,134,0.6567,0.28116,0.24873,0.14214,0.14288,0.42857,0.0,1.0,7,1,0,7,0,10,0,0,3,0,0,5,0,0,5,0,0,1,0,0,0,0,1],[92,134,0.6866,0.22321,0.2765,0.0,0.14286,0.2857,0.0,1.0,11,2,0,11,0,10,0,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,2],[96,134,0.7164,0.16954,0.15744,0.0,0.14286,0.2857,0.0,0.571,10,0,0,10,0,12,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[100,134,0.7463,0.16518,0.18249,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,7,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[104,134,0.7761,0.16963,0.16143,0.0,0.14286,0.28571,0.0,0.571,11,0,0,11,0,10,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[108,134,0.806,0.09821,0.14032,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[112,134,0.8358,0.12946,0.21829,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[116,134,0.8657,0.16518,0.18249,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,9,0,0,4,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[120,134,0.8955,0.20089,0.22263,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,9,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[124,134,0.9254,0.29017,0.23549,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,5,0,0,5,0,0,9,0,0,4,0,0,0,0,0,0,0,1],[128,134,0.9552,0.08928,0.12242,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,134,0.9851,0.19194,0.19428,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,14,0,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[134,134,1.0,0.13393,0.18189,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,5,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.25429,"x":1.0,"p":[[0,60,0.0,0.25429,0.34583,0.0,0.14286,0.14286,0.0,1.0,10,5,0,10,0,15,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[4,60,0.0667,0.51783,0.36899,0.14286,0.42857,1.0,0.0,1.0,2,10,0,2,0,8,0,0,4,0,0,4,0,0,3,0,0,0,0,0,1,0,10],[8,60,0.1333,0.36608,0.34798,0.14286,0.14288,0.46429,0.0,1.0,3,6,0,3,0,15,0,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,6],[12,60,0.2,0.3482,0.29436,0.14286,0.28571,0.42857,0.0,1.0,4,4,0,4,0,10,0,0,4,0,0,8,0,0,2,0,0,0,0,0,0,0,4],[16,60,0.2667,0.39731,0.32484,0.14286,0.35714,0.57143,0.0,1.0,4,5,0,4,0,10,0,0,2,0,0,5,0,0,5,0,0,1,0,0,0,0,5],[20,60,0.3333,0.54911,0.38978,0.14286,0.57143,1.0,0.0,1.0,4,12,0,4,0,5,0,0,5,0,0,1,0,0,4,0,0,1,0,0,0,0,12],[24,60,0.4,0.36159,0.34252,0.14286,0.21428,0.46418,0.0,1.0,5,6,0,5,0,11,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,6],[28,60,0.4667,0.77232,0.36746,0.39285,1.0,1.0,0.14286,1.0,0,23,0,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,23],[32,60,0.5333,0.92411,0.23686,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[36,60,0.6,0.94643,0.17405,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[40,60,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,60,0.7333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,60,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,60,0.8667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3dc3c00b8649e3d8","q":"Let  $n, k$  be natural numbers,  $1 \\leq k < n$ . In each vertex of a regular polygon with  $n$  sides is written  $1$  or  $-1$ . At each step we choose  $k$  consecutive vertices and change their signs. Is it possible that, starting from a certain configuration and by doing the operation a few times to obtain any other configuration?","t":[{"b":1,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,19,0.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,19,0.2105,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,19,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,19,0.6316,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,19,0.8421,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[19,19,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":4,"e":0.571,"k":"falling","v":0.70979,"x":1.0,"p":[[0,14,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,14,0.2857,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[8,14,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,14,0.8571,0.75889,0.20345,0.57143,0.64286,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,3,0,0,0,0,13],[14,14,1.0,0.70979,0.20975,0.57143,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,4,0,0,0,0,10]]}]},{"i":"693531409aba068a","q":"Let $a, b, c$ be real numbers, such that $0 \\leq a \\leq b \\leq c$ and $a+b+c=a b+b c+c a>0$. Prove that $\\sqrt{b c}(a+1) \\geq 2$. Find all triples $(a, b, c)$ for which equality holds.\n\n## Proposed by Romania","t":[{"b":0,"e":0.14286,"k":"flat","v":0.1517,"x":0.38393,"p":[[0,102,0.0,0.23214,0.17032,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,23,0,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[4,102,0.0392,0.30803,0.19595,0.14286,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,15,0,0,7,0,0,3,0,0,4,0,0,3,0,0,0,0,0],[8,102,0.0784,0.32143,0.21724,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,14,0,0,7,0,0,6,0,0,2,0,0,0,0,0,3,0,0],[12,102,0.1176,0.38393,0.23265,0.14286,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,10,0,0,8,0,0,4,0,0,5,0,0,2,0,0,3,0,0],[16,102,0.1569,0.3348,0.20076,0.14286,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,12,0,0,6,0,0,2,0,0,10,0,0,1,0,0,0,0,0],[20,102,0.1961,0.29007,0.18726,0.14286,0.14288,0.42857,0.14,0.71429,0,0,0,0,0,18,0,0,3,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[24,102,0.2353,0.30357,0.19804,0.14286,0.21431,0.42857,0.14286,0.85714,0,0,0,0,0,16,0,0,4,0,0,8,0,0,1,0,0,2,0,0,1,0,0],[28,102,0.2745,0.32588,0.25058,0.14286,0.14286,0.46418,0.14286,1.0,0,1,0,0,0,17,0,0,5,0,0,2,0,0,4,0,0,1,0,0,2,0,1],[32,102,0.3137,0.36596,0.22004,0.14286,0.28571,0.57143,0.14,0.85714,0,0,0,0,0,12,0,0,6,0,0,3,0,0,7,0,0,3,0,0,1,0,0],[36,102,0.3529,0.30356,0.18812,0.14286,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,15,0,0,7,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[40,102,0.3922,0.22762,0.15518,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,23,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[44,102,0.4314,0.25891,0.16915,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,20,0,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[48,102,0.4706,0.2589,0.1614,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,19,0,0,5,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[52,102,0.5098,0.24554,0.13474,0.14286,0.1429,0.28571,0.14286,0.57143,0,0,0,0,0,17,0,0,10,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[56,102,0.549,0.27223,0.19684,0.14286,0.14286,0.42857,0.14,0.71429,0,0,0,0,0,21,0,0,1,0,0,5,0,0,2,0,0,3,0,0,0,0,0],[60,102,0.5882,0.21874,0.13821,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,23,0,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[64,102,0.6275,0.27231,0.19998,0.14286,0.14286,0.35704,0.14286,0.71429,0,0,0,0,0,21,0,0,3,0,0,0,0,0,6,0,0,2,0,0,0,0,0],[68,102,0.6667,0.21872,0.16354,0.14286,0.14286,0.1786,0.14286,0.85714,0,0,0,0,0,24,0,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[72,102,0.7059,0.20982,0.11837,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,23,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[76,102,0.7451,0.22765,0.14658,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,21,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[80,102,0.7843,0.19196,0.11628,0.14286,0.14286,0.14287,0.14286,0.57143,0,0,0,0,0,26,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[84,102,0.8235,0.21875,0.17851,0.14286,0.14286,0.1429,0.14286,1.0,0,1,0,0,0,25,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[88,102,0.8627,0.22767,0.17073,0.14286,0.14286,0.14287,0.14286,0.71429,0,0,0,0,0,25,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[92,102,0.902,0.18302,0.10848,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,27,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[96,102,0.9412,0.20088,0.13292,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,26,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[100,102,0.9804,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[102,102,1.0,0.17411,0.09268,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.19197,"x":0.38839,"p":[[0,118,0.0,0.19197,0.12682,0.14286,0.14286,0.1429,0.14286,0.71429,0,0,0,0,0,27,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,118,0.0339,0.29464,0.18536,0.14286,0.21429,0.42858,0.14286,0.71429,0,0,0,0,0,16,0,0,6,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[8,118,0.0678,0.27677,0.15123,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,14,0,0,11,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[12,118,0.1017,0.38839,0.22084,0.14286,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,9,0,0,8,0,0,5,0,0,5,0,0,3,0,0,2,0,0],[16,118,0.1356,0.3125,0.20958,0.14286,0.2857,0.42858,0.14286,0.85714,0,0,0,0,0,14,0,0,9,0,0,3,0,0,3,0,0,1,0,0,2,0,0],[20,118,0.1695,0.32143,0.24744,0.14286,0.14286,0.42858,0.14286,1.0,0,1,0,0,0,17,0,0,5,0,0,3,0,0,3,0,0,1,0,0,2,0,1],[24,118,0.2034,0.32589,0.1931,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,13,0,0,7,0,0,5,0,0,4,0,0,3,0,0,0,0,0],[28,118,0.2373,0.35268,0.22864,0.14286,0.28571,0.46429,0.14286,1.0,0,1,0,0,0,12,0,0,8,0,0,4,0,0,3,0,0,4,0,0,0,0,1],[32,118,0.2712,0.34371,0.20154,0.14286,0.28571,0.571,0.14286,0.71429,0,0,0,0,0,13,0,0,5,0,0,5,0,0,6,0,0,3,0,0,0,0,0],[36,118,0.3051,0.21875,0.13355,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,21,0,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[40,118,0.339,0.34374,0.21681,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,13,0,0,8,0,0,1,0,0,5,0,0,5,0,0,0,0,0],[44,118,0.3729,0.33926,0.23074,0.14286,0.28571,0.46418,0.14286,0.85714,0,0,0,0,0,14,0,0,7,0,0,3,0,0,3,0,0,3,0,0,2,0,0],[48,118,0.4068,0.30345,0.19484,0.14286,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,15,0,0,7,0,0,4,0,0,4,0,0,1,0,0,1,0,0],[52,118,0.4407,0.31247,0.23804,0.14286,0.14286,0.571,0.14286,0.85714,0,0,0,0,0,18,0,0,5,0,0,0,0,0,6,0,0,0,0,0,3,0,0],[56,118,0.4746,0.25,0.19233,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,22,0,0,3,0,0,4,0,0,0,0,0,2,0,0,1,0,0],[60,118,0.5085,0.30355,0.2194,0.14286,0.14288,0.571,0.14286,0.85714,0,0,0,0,0,18,0,0,5,0,0,0,0,0,6,0,0,2,0,0,1,0,0],[64,118,0.5424,0.21426,0.12364,0.14286,0.14286,0.28571,0.14286,0.571,0,0,0,0,0,22,0,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[68,118,0.5763,0.23213,0.1812,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,22,0,0,3,0,0,2,0,0,3,0,0,0,0,0,1,0,0],[72,118,0.6102,0.27675,0.21104,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,21,0,0,2,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[76,118,0.6441,0.26784,0.21051,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,21,0,0,4,0,0,2,0,0,1,0,0,3,0,0,1,0,0],[80,118,0.678,0.21875,0.14279,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,22,0,0,7,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[84,118,0.7119,0.29463,0.20804,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,16,0,0,5,0,0,4,0,0,2,0,0,4,0,0,0,0,0],[88,118,0.7458,0.31694,0.21643,0.14286,0.14286,0.571,0.14286,0.71429,0,0,0,0,0,17,0,0,4,0,0,2,0,0,5,0,0,4,0,0,0,0,0],[92,118,0.7797,0.2946,0.20491,0.14286,0.14286,0.46418,0.14286,0.71429,0,0,0,0,0,18,0,0,5,0,0,1,0,0,5,0,0,3,0,0,0,0,0],[96,118,0.8136,0.22759,0.1467,0.14286,0.14286,0.2857,0.14,0.57143,0,0,0,0,0,23,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[100,118,0.8475,0.23205,0.18127,0.14286,0.14286,0.17857,0.14,0.85714,0,0,0,0,0,24,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[104,118,0.8814,0.26786,0.16656,0.14286,0.14288,0.28571,0.14286,0.71429,0,0,0,0,0,17,0,0,8,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[108,118,0.9153,0.26787,0.18814,0.14286,0.14288,0.28571,0.14286,0.85714,0,0,0,0,0,18,0,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,0],[112,118,0.9492,0.29911,0.17261,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,8,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[116,118,0.9831,0.26339,0.17896,0.14286,0.14286,0.32143,0.14286,0.71429,0,0,0,0,0,20,0,0,4,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[118,118,1.0,0.24105,0.18702,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,22,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"0c8f65eb539e8b05","q":"Let  $p$  and  $q$  be given prime numbers and  $S$  be a subset of  ${1,2,3,\\dots ,p-2,p-1}$ . Prove that the number of elements in the set $A=\\{ (x_1,x_2,\u2026,x_q ):x_i\\in S,\\sum_{i=1}^q x_i \\equiv 0(mod\\: p)\\}$ is multiple of  $q$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.09822,"x":0.14286,"p":[[0,10,0.0,0.14277,0.12877,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,9,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.14286,0.15152,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,5,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.09822,0.10374,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.07134,"x":0.125,"p":[[0,8,0.0,0.07134,0.11288,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.125,0.15872,0.0,0.0,0.17857,0.0,0.42857,17,0,0,17,0,7,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.10714,0.11845,0.0,0.07143,0.17857,0.0,0.28571,16,0,0,16,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"87b0c466e988ff15","q":"Let  $n>2$  be a positive integer. Masha writes down  $n$  natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers  $a$  and  $b$ , she writes a divisor of the number  $a+b$  greater than  $1$ , then Taya erases the original numbers and obtains a new set of  $n$  numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal? \n*Proposed by T. Korotchenko*","t":[{"b":4,"e":0.14286,"k":"flat","v":0.00446,"x":0.13366,"p":[[0,57,0.0,0.13366,0.087,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.12053,0.08827,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.10697,0.08741,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.10259,0.09602,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.08929,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.09821,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.07572,0.07113,0.0,0.14,0.14286,0.0,0.143,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.13822,"p":[[0,48,0.0,0.13822,0.06665,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.13357,0.09404,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.12938,0.10325,0.0,0.14286,0.14286,0.0,0.42857,9,0,1,9,0,18,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.12928,0.09686,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.11161,0.11701,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.09821,0.12078,0.0,0.0,0.17857,0.0,0.28571,18,0,0,18,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.06687,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.08027,0.11254,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.07116,0.08727,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"18ec6482a830a03d","q":"Let  $n \\ge 2$  be a positive integer and let  $A \\in \\mathcal{M}_n(\\mathbb{R})$  be a matrix such that  $A^2=-I_n$ . If  $B \\in \\mathcal{M}_n(\\mathbb{R})$  and  $AB = BA$ , prove that  $\\det B \\ge 0$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.21875,"x":0.25893,"p":[[0,44,0.0,0.23661,0.12169,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,10,0,0,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,44,0.0909,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.24107,0.07523,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.24553,0.06423,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.25446,0.06901,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.2366,0.08458,0.14286,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,7,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.22769,0.07873,0.14286,0.28571,0.28571,0.0,0.286,1,0,0,1,0,11,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.23214,0.06916,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.24554,0.08171,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,5,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.24107,0.07523,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.25893,0.05576,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.2366,0.06785,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.20982,"x":0.24107,"p":[[0,33,0.0,0.22335,0.09416,0.14286,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,11,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.24107,0.1729,0.14286,0.28571,0.28571,0.0,1.0,5,1,0,5,0,5,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,33,0.2424,0.23214,0.08564,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.2366,0.07667,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,9,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.2366,0.06785,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.20982,0.08736,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,13,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.23652,0.06798,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.23214,0.08564,0.14286,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.24107,0.06622,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.21875,0.07129,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a78d2f8f5b619eb4","q":"Let  $n\\in \\Bbb{N}, n \\geq 4.$  Determine all sets  $ A = \\{a_1, a_2, . . . , a_n\\} \\subset \\Bbb{N}$  containing  $2015$  and having the property that  $ |a_i - a_j|$  is prime, for all distinct   $i, j\\in \\{1, 2, . . . , n\\}.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,56,0.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,56,0.0714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,56,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,56,0.3571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,56,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,56,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,56,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,56,0.6429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,56,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,56,0.7857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,56,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.82588,"x":1.0,"p":[[0,47,0.0,0.96429,0.11293,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[4,47,0.0851,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,47,0.1702,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[12,47,0.2553,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,47,0.3404,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,47,0.4255,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,47,0.5106,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,47,0.5957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,47,0.6809,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,47,0.8511,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,47,0.9362,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[47,47,1.0,0.82588,0.32486,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,22]]}]},{"i":"13879c6986fc00c7","q":"Let  $n\\ge2$  be an integer.  A regular  $(2n+1)-gon$  is divided in to  $2n-1$  triangles by diagonals which do not meet except at the vertices.  Prove that at least three of these triangles are isosceles.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.3839,"x":0.64732,"p":[[0,63,0.0,0.40624,0.26989,0.14286,0.42857,0.57111,0.0,1.0,3,2,1,3,0,6,0,0,5,0,0,9,0,0,2,0,0,4,0,0,1,0,2],[4,63,0.0635,0.64732,0.2448,0.57143,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,3,0,0,12,0,0,1,0,0,7,0,5],[8,63,0.127,0.60713,0.23146,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,5,0,0,12,0,0,2,0,0,6,0,3],[12,63,0.1905,0.57143,0.26726,0.42857,0.57143,0.71429,0.0,1.0,3,5,0,3,0,0,0,0,2,0,0,4,0,0,14,0,0,3,0,0,1,0,5],[16,63,0.254,0.5848,0.22119,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,4,0,0,13,0,0,3,0,0,5,0,2],[20,63,0.3175,0.64285,0.24743,0.57143,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,2,0,0,11,0,0,3,0,0,6,0,5],[24,63,0.381,0.56249,0.20806,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,8,0,0,12,0,0,1,0,0,6,0,1],[28,63,0.4444,0.54018,0.27833,0.39286,0.57143,0.75,0.0,1.0,2,3,0,2,0,2,0,0,4,0,0,7,0,0,6,0,0,3,0,0,5,0,3],[32,63,0.5079,0.55357,0.22799,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,6,0,0,9,0,0,4,0,0,4,0,2],[36,63,0.5714,0.49996,0.23957,0.39286,0.571,0.71429,0.0,0.85714,3,0,0,3,0,1,0,0,4,0,0,6,0,0,8,0,0,7,0,0,3,0,0],[40,63,0.6349,0.51337,0.21086,0.39286,0.57121,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,7,0,0,10,0,0,3,0,0,3,0,1],[44,63,0.6984,0.62499,0.24679,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,4,0,0,8,0,0,3,0,0,7,0,4],[48,63,0.7619,0.57589,0.27078,0.28571,0.57143,0.74996,0.0,1.0,1,5,0,1,0,1,0,0,7,0,0,2,0,0,10,0,0,3,0,0,3,0,5],[52,63,0.8254,0.42411,0.27545,0.2857,0.42857,0.57143,0.0,1.0,4,3,0,4,0,2,0,0,9,0,0,5,0,0,6,0,0,3,0,0,0,0,3],[56,63,0.8889,0.51785,0.25691,0.39286,0.57143,0.71429,0.0,1.0,3,1,0,3,0,1,0,0,4,0,0,5,0,0,9,0,0,5,0,0,4,0,1],[60,63,0.9524,0.3839,0.23806,0.2857,0.28571,0.57111,0.0,1.0,4,1,0,4,0,2,0,0,11,0,0,5,0,0,5,0,0,4,0,0,0,0,1],[63,63,1.0,0.47319,0.17289,0.28571,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,9,0,0,5,0,0,8,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.30356,"x":0.56696,"p":[[0,42,0.0,0.41518,0.21237,0.39286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,2,0,0,15,0,0,6,0,0,2,0,0,0,0,1],[4,42,0.0952,0.46874,0.29284,0.28571,0.57143,0.57143,0.0,1.0,5,4,0,5,0,1,0,0,6,0,0,1,0,0,14,0,0,1,0,0,0,0,4],[8,42,0.1905,0.54016,0.25438,0.42857,0.57143,0.60714,0.0,1.0,3,2,0,3,0,0,0,0,4,0,0,3,0,0,14,0,0,2,0,0,4,0,2],[12,42,0.2857,0.56696,0.24868,0.42857,0.57143,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,2,0,0,6,0,0,10,0,0,4,0,0,5,0,2],[16,42,0.381,0.49551,0.21423,0.39286,0.57143,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,4,0,0,15,0,0,3,0,0,1,0,1],[20,42,0.4762,0.48213,0.21053,0.28571,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,7,0,0,5,0,0,11,0,0,3,0,0,3,0,0],[24,42,0.5714,0.52677,0.27301,0.28571,0.49979,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,6,0,0,7,0,0,6,0,0,4,0,0,2,0,4],[28,42,0.6667,0.40177,0.27764,0.24999,0.42857,0.57143,0.0,1.0,6,2,0,6,0,2,0,0,6,0,0,5,0,0,9,0,0,1,0,0,1,0,2],[32,42,0.7619,0.30356,0.20746,0.14286,0.28571,0.4642,0.0,0.57143,7,0,0,7,0,3,0,0,9,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[36,42,0.8571,0.40179,0.23266,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,8,0,0,7,0,0,4,0,0,5,0,0,0,0,1],[40,42,0.9524,0.51338,0.22829,0.39286,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,6,0,0,6,0,0,13,0,0,1,0,0,1,0,3],[42,42,1.0,0.48659,0.18161,0.39286,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,7,0,0,14,0,0,1,0,0,1,0,1]]}]},{"i":"66adfc58f7343e58","q":"Let  $p$  be a positive prime integer,  $S(p)$  be the number of triples  $(x,y,z)$  such that  $x,y,z\\in\\{0,1,..., p-1\\}$  and  $x^2+y^2+z^2$  is divided by  $p$ . Prove that  $S(p) \\ge 2p- 1$ .\n\n(I. Bliznets)","t":[{"b":2,"e":0.14286,"k":"flat","v":0.21428,"x":0.36159,"p":[[0,18,0.0,0.36159,0.24217,0.14286,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,13,0,0,6,0,0,4,0,0,4,0,0,3,0,0,1,0,1],[4,18,0.2222,0.34375,0.2854,0.14286,0.14286,0.46429,0.14286,1.0,0,2,0,0,0,18,0,0,4,0,0,2,0,0,1,0,0,3,0,0,2,0,2],[8,18,0.4444,0.28571,0.23146,0.14286,0.14286,0.32143,0.14286,1.0,0,1,0,0,0,20,0,0,4,0,0,2,0,0,3,0,0,1,0,0,1,0,1],[12,18,0.6667,0.23661,0.14987,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,18,0,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[16,18,0.8889,0.23661,0.16982,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,19,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[18,18,1.0,0.21428,0.11845,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,20,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.16071,"x":0.46429,"p":[[0,27,0.0,0.38392,0.24597,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,11,0,0,8,0,0,1,0,0,8,0,0,1,0,0,2,0,1],[4,27,0.1481,0.46429,0.33882,0.14286,0.28571,0.85714,0.14286,1.0,0,5,0,0,0,13,0,0,4,0,0,2,0,0,2,0,0,2,0,0,4,0,5],[8,27,0.2963,0.34366,0.26937,0.14286,0.14288,0.57143,0.14,1.0,0,2,0,0,0,17,0,0,5,0,0,0,0,0,4,0,0,4,0,0,0,0,2],[12,27,0.4444,0.38838,0.30142,0.14286,0.28571,0.57143,0.0,1.0,1,4,0,1,0,12,0,0,7,0,0,2,0,0,4,0,0,1,0,0,1,0,4],[16,27,0.5926,0.37946,0.28928,0.14286,0.14286,0.57143,0.14286,1.0,0,1,0,0,0,17,0,0,2,0,0,1,0,0,5,0,0,2,0,0,4,0,1],[20,27,0.7407,0.22759,0.18513,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[24,27,0.8889,0.16071,0.04725,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.17857,0.08748,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,26,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b6210a818255fce9","q":"Let  $q(n)$  denote the sum of the digits of a natural number  $n$ . Determine  $q(q(q(2000^{2000})))$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,56,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,56,0.0714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,56,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,56,0.3571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,56,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,56,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,56,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,56,0.6429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,56,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,56,0.7857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,56,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,24,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,24,0.1667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,24,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9c8330ac42f44e1e","q":"Let $\\mathcal{K}$ be an integer lattice. Does there exist a bijection $f: \\mathbb{N} \\rightarrow \\mathcal{K}$ such that for all mutually distinct $a, b, c \\in \\mathbb{N}$ we have\n\n$$\n\\text { GCD }(a, b, c)>1 \\quad \\Longrightarrow \\quad f(a), f(b), f(c) \\text { are not collinear? }\n$$\n\n(An integer lattice is the set of points in the plane with integer coordinates in the Cartesian coordinate system.)\n\n(Stevan Gajovi\u0107)","t":[{"b":5,"e":1.0,"k":"flat","v":0.22768,"x":0.70533,"p":[[0,42,0.0,0.22768,0.2693,0.0,0.0,0.57143,0.0,0.85714,17,0,2,17,0,1,0,0,3,0,0,2,0,0,8,0,0,0,0,0,1,0,0],[4,42,0.0952,0.70533,0.31732,0.57143,0.78571,1.0,0.0,1.0,3,13,0,3,0,0,0,0,2,0,0,0,0,0,10,0,0,1,0,0,3,0,13],[8,42,0.1905,0.67405,0.34854,0.571,0.64286,1.0,0.0,1.0,5,13,0,5,0,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,2,0,13],[12,42,0.2857,0.66958,0.31833,0.571,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,1,0,0,0,0,0,10,0,0,4,0,0,3,0,10],[16,42,0.381,0.70089,0.3141,0.571,0.78564,1.0,0.0,1.0,3,11,0,3,0,0,0,0,2,0,0,2,0,0,5,0,0,4,0,0,5,0,11],[20,42,0.4762,0.60709,0.34993,0.5354,0.57143,1.0,0.0,1.0,6,10,0,6,0,0,0,0,0,0,0,2,0,0,11,0,0,2,0,0,1,0,10],[24,42,0.5714,0.61604,0.33012,0.42857,0.57143,1.0,0.0,1.0,4,9,0,4,0,0,0,0,3,0,0,3,0,0,8,0,0,2,0,0,3,0,9],[28,42,0.6667,0.67412,0.27016,0.57143,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,2,0,0,1,0,0,10,0,0,6,0,0,3,0,8],[32,42,0.7619,0.54013,0.35307,0.28571,0.57143,0.85714,0.0,1.0,7,7,0,7,0,0,0,0,2,0,0,2,0,0,11,0,0,0,0,0,3,0,7],[36,42,0.8571,0.39729,0.27601,0.10714,0.5712,0.57143,0.0,1.0,8,1,0,8,0,2,0,0,2,0,0,2,0,0,15,0,0,2,0,0,0,0,1],[40,42,0.9524,0.36152,0.24486,0.14286,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,4,0,0,5,0,0,5,0,0,9,0,0,2,0,0,1,0,0],[42,42,1.0,0.34374,0.26929,0.14286,0.28571,0.57111,0.0,1.0,6,2,0,6,0,6,0,0,5,0,0,6,0,0,6,0,0,1,0,0,0,0,2]]},{"b":6,"e":1.0,"k":"rising","v":0.17408,"x":0.97768,"p":[[0,32,0.0,0.17408,0.24928,0.0,0.0,0.42858,0.0,0.71429,20,0,4,20,0,2,0,0,1,0,0,2,0,0,6,0,0,1,0,0,0,0,0],[4,32,0.125,0.75445,0.26059,0.57143,0.78571,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,3,0,0,1,0,15],[8,32,0.25,0.70088,0.3395,0.57143,0.85714,1.0,0.0,1.0,4,13,0,4,0,0,0,0,2,0,0,0,0,0,7,0,0,2,0,0,4,0,13],[12,32,0.375,0.77678,0.2878,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,2,0,17],[16,32,0.5,0.808,0.25661,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,2,0,18],[20,32,0.625,0.88825,0.15881,0.857,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,6,0,19],[24,32,0.75,0.93748,0.1595,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[28,32,0.875,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,32,1.0,0.95089,0.15815,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,29]]}]},{"i":"7c770e6866543517","q":"Let  $p$  be a prime number and let  $m, n$  be integers greater than  $1$  such that  $n | m^{p(n-1)} - 1$ . Prove that  $gcd(m^{n-1} - 1, n) > 1$ .","t":[{"b":2,"e":0.42857,"k":"falling","v":0.16063,"x":0.71875,"p":[[0,35,0.0,0.71875,0.20666,0.71429,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,12,0,0,10,0,3],[4,35,0.1143,0.63393,0.29437,0.42857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,1,0,0,4,0,0,4,0,0,5,0,0,8,0,5],[8,35,0.2286,0.46429,0.28121,0.25,0.35714,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,8,0,0,3,0,0,1,0,0,6,0,0,5,0,1],[12,35,0.3429,0.35713,0.25253,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,7,0,0,8,0,0,2,0,0,6,0,0,3,0,0,2,0,0],[16,35,0.4571,0.27678,0.18877,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,12,0,0,11,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[20,35,0.5714,0.32588,0.26057,0.14286,0.14286,0.4642,0.0,1.0,1,1,0,1,0,16,0,0,5,0,0,2,0,0,3,0,0,2,0,0,2,0,1],[24,35,0.6857,0.27232,0.23244,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,15,0,0,7,0,0,2,0,0,1,0,0,3,0,0,0,0,1],[28,35,0.8,0.36607,0.26229,0.14286,0.2857,0.60714,0.0,0.85714,2,0,0,2,0,10,0,0,8,0,0,3,0,0,1,0,0,5,0,0,3,0,0],[32,35,0.9143,0.20536,0.18877,0.14286,0.14286,0.17857,0.0,0.71429,5,0,0,5,0,19,0,0,3,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[35,35,1.0,0.16063,0.11711,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,21,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.25437,"x":0.82589,"p":[[0,55,0.0,0.68302,0.12745,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,19,0,0,3,0,1],[4,55,0.0727,0.82589,0.17029,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,9,0,11],[8,55,0.1455,0.70979,0.24351,0.53539,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,3,0,0,13,0,5],[12,55,0.2182,0.78125,0.2474,0.71429,0.85712,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,8,0,0,3,0,14],[16,55,0.2909,0.70088,0.24579,0.53539,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,3,0,0,12,0,5],[20,55,0.3636,0.65623,0.29203,0.39286,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,6,0,0,2,0,0,6,0,0,1,0,0,7,0,8],[24,55,0.4364,0.56249,0.28557,0.28571,0.4998,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,11,0,0,4,0,0,3,0,0,1,0,0,9,0,3],[28,55,0.5091,0.57143,0.27894,0.28571,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,10,0,0,3,0,0,1,0,0,6,0,0,7,0,3],[32,55,0.5818,0.63393,0.26949,0.42857,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,4,0,0,5,0,0,2,0,0,10,0,4],[36,55,0.6545,0.51339,0.2809,0.28571,0.5,0.75,0.0,1.0,1,2,0,1,0,2,0,0,12,0,0,1,0,0,4,0,0,4,0,0,6,0,2],[40,55,0.7273,0.52677,0.31019,0.28571,0.42859,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,12,0,0,2,0,0,2,0,0,3,0,0,6,0,4],[44,55,0.8,0.55357,0.30041,0.28571,0.57143,0.85714,0.0,1.0,1,2,0,1,0,3,0,0,9,0,0,2,0,0,2,0,0,3,0,0,10,0,2],[48,55,0.8727,0.41071,0.24157,0.28571,0.28571,0.42858,0.14286,1.0,0,2,0,0,0,3,0,0,18,0,0,4,0,0,0,0,0,3,0,0,2,0,2],[52,55,0.9455,0.32142,0.15152,0.2857,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,2,0,0,23,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[55,55,1.0,0.25437,0.07784,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"406898093811e4d6","q":"Let $a, b, c$ be positive real numbers such that abc $=1$. Show that\n\n$$\n\\frac{1}{a^{3}+b c}+\\frac{1}{b^{3}+c a}+\\frac{1}{c^{3}+a b} \\leq \\frac{(a b+b c+c a)^{2}}{6}\n$$\n\nso","t":[{"b":3,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,35,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,35,0.3429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,35,0.4571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,60,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,60,0.0667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,60,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,60,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,60,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,60,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,60,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,60,0.4667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,60,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,60,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,60,0.7333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,60,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,60,0.8667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,60,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,60,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"0d22c48edc39493b","q":"Let $a \\in [0 ; 1]$. We define the sequence $\\left(x_{n}\\right)$ by\n$x_{0}=a$ and $x_{n+1}=1-\\left|1-2 x_{n}\\right|$, for all $n \\geq 0$.\nProve that the sequence $\\left(x_{n}\\right)$ is periodic from a certain rank if and only if $a$ is a rational number.\n(Note: we say that $\\left(x_{n}\\right)$ is periodic from a certain rank if there exist integers $T>0$ and $n \\geqslant 0$ such that $x_{k+T}=x_{k}$ for all $k \\geqslant n$.)","t":[{"b":1,"e":0.42857,"k":"falling","v":0.45982,"x":0.94642,"p":[[0,37,0.0,0.94642,0.14177,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[4,37,0.1081,0.89286,0.22304,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,2,0,23],[8,37,0.2162,0.68304,0.34206,0.42857,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,0,0,0,4,0,0,2,0,0,6,0,0,3,0,12],[12,37,0.3243,0.5982,0.33964,0.42857,0.71429,0.85714,0.0,1.0,6,7,0,6,0,0,0,0,1,0,0,2,0,0,5,0,0,9,0,0,2,0,7],[16,37,0.4324,0.62946,0.35148,0.39286,0.71429,1.0,0.0,1.0,4,9,0,4,0,2,0,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,9],[20,37,0.5405,0.60264,0.34762,0.39286,0.64286,1.0,0.0,1.0,4,9,0,4,0,2,0,0,2,0,0,4,0,0,4,0,0,4,0,0,3,0,9],[24,37,0.6486,0.46874,0.4304,0.0,0.42857,1.0,0.0,1.0,12,10,0,12,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,10],[28,37,0.7568,0.45982,0.39405,0.0,0.42857,0.89286,0.0,1.0,11,8,0,11,0,0,0,0,1,0,0,6,0,0,4,0,0,1,0,0,1,0,8],[32,37,0.8649,0.54472,0.19372,0.42857,0.42857,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,21,0,0,4,0,0,3,0,0,0,0,4],[36,37,0.973,0.48661,0.15093,0.42857,0.42857,0.4286,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,27,0,0,1,0,0,2,0,0,0,0,2],[37,37,1.0,0.50447,0.15966,0.42857,0.42857,0.4286,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,25,0,0,1,0,0,4,0,0,0,0,2]]},{"b":3,"e":0.57143,"k":"falling","v":0.71427,"x":0.95982,"p":[[0,18,0.0,0.91516,0.20475,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,25],[4,18,0.2222,0.76785,0.29179,0.71429,0.85707,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,3,0,14],[8,18,0.4444,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[12,18,0.6667,0.88839,0.19475,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,0,0,23],[16,18,0.8889,0.74999,0.20204,0.57143,0.71429,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,5,0,0,0,0,12],[18,18,1.0,0.71427,0.2113,0.57143,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,3,0,0,1,0,10]]}]},{"i":"864968dd6f7a021d","q":"Let $\\omega_{1}$ and $\\omega_{2}$ be two circles with centers $\\mathrm{O}_{1}$ and $\\mathrm{O}_{2}$, respectively. Suppose that $\\omega_{1}$ and $\\omega_{2}$ intersect at points $A$ and $B$. The line $\\left(O_{1} A\\right)$ intersects the circle $\\omega_{2}$ again at $C$, while the line $\\left(O_{2} A\\right)$ intersects the circle $\\omega_{1}$ again at $D$. Show that the points $\\mathrm{D}, \\mathrm{O}_{1}, \\mathrm{~B}, \\mathrm{O}_{2}$, and $C$ lie on the same circle.","t":[{"b":1,"e":0.4286,"k":"flat","v":0.23205,"x":0.42409,"p":[[0,74,0.0,0.28106,0.20676,0.14286,0.2857,0.42857,0.0,1.0,4,1,0,4,0,10,0,0,7,0,0,8,0,0,2,0,0,0,0,0,0,0,1],[4,74,0.0541,0.29911,0.09007,0.2857,0.28571,0.32143,0.14286,0.4286,0,0,0,0,0,5,0,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,74,0.1081,0.32589,0.18293,0.2857,0.28571,0.42857,0.0,1.0,3,1,0,3,0,2,0,0,16,0,0,8,0,0,2,0,0,0,0,0,0,0,1],[12,74,0.1622,0.3125,0.15746,0.24999,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,6,0,0,12,0,0,9,0,0,2,0,0,1,0,0,0,0,0],[16,74,0.2162,0.27232,0.12037,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,15,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[20,74,0.2703,0.31696,0.1504,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[24,74,0.3243,0.25,0.13832,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,6,0,0,18,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[28,74,0.3784,0.29018,0.12619,0.2857,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,4,0,0,19,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[32,74,0.4324,0.26786,0.09279,0.2857,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,3,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.32589,0.10853,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,23,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[40,74,0.5405,0.29909,0.10322,0.2857,0.28571,0.28571,0.0,0.571,1,0,0,1,0,3,0,0,21,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[44,74,0.5946,0.25004,0.12378,0.14289,0.2857,0.28571,0.0,0.43,4,0,0,4,0,5,0,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,74,0.6486,0.24998,0.1336,0.24999,0.2857,0.28571,0.0,0.571,5,0,0,5,0,3,0,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[52,74,0.7027,0.27232,0.15714,0.25,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,3,0,0,17,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[56,74,0.7568,0.25,0.10101,0.2857,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,4,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.23205,0.12759,0.14286,0.2857,0.2857,0.0,0.57143,4,0,0,4,0,8,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,74,0.8649,0.3303,0.14471,0.2857,0.2857,0.42858,0.0,0.57143,1,0,0,1,0,4,0,0,17,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[68,74,0.9189,0.42409,0.20665,0.2857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,9,0,0,8,0,0,4,0,0,7,0,0,0,0,0],[72,74,0.973,0.37042,0.23113,0.25,0.28571,0.57111,0.0,0.71429,4,0,0,4,0,4,0,0,9,0,0,5,0,0,4,0,0,6,0,0,0,0,0],[74,74,1.0,0.36157,0.18889,0.2857,0.28571,0.571,0.0,0.71429,3,0,0,3,0,3,0,0,11,0,0,5,0,0,9,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.20518,"x":0.36605,"p":[[0,38,0.0,0.20518,0.20189,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,9,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[4,38,0.1053,0.36605,0.17471,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,22,0,0,4,0,0,3,0,0,0,0,0,1,0,1],[8,38,0.2105,0.32147,0.10107,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,4,0,0,17,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[12,38,0.3158,0.29911,0.10926,0.2857,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,16,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[16,38,0.4211,0.30356,0.1224,0.2857,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,2,0,0,20,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[20,38,0.5263,0.2857,0.12874,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,5,0,0,18,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[24,38,0.6316,0.2991,0.10326,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,23,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[28,38,0.7368,0.27678,0.06121,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,4,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.25892,0.08328,0.2857,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.25893,0.07523,0.25,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,8,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.24107,0.06621,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3521fd579dcffa31","q":"Let $a, b, c$ be real numbers. Show that:\n\n$$\n2 a^{2}+20 b^{2}+5 c^{2}+8 a b-4 b c-4 a c \\geqslant 0\n$$\n\nand find the cases of equality.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.61607,"x":0.72768,"p":[[0,29,0.0,0.61607,0.2976,0.42857,0.42857,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,16,0,0,0,0,0,4,0,0,0,0,10],[4,29,0.1379,0.69643,0.27606,0.42857,0.57144,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,2,0,0,0,0,14],[8,29,0.2759,0.70982,0.3204,0.42857,0.85714,1.0,0.0,1.0,1,16,0,1,0,2,0,0,0,0,0,10,0,0,0,0,0,3,0,0,0,0,16],[12,29,0.4138,0.72768,0.28202,0.42857,0.71429,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,1,0,0,10,0,0,0,0,0,5,0,0,0,0,15],[16,29,0.5517,0.69643,0.27606,0.42857,0.57144,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,2,0,0,0,0,14],[20,29,0.6897,0.66964,0.31831,0.42857,0.57143,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,14,0,0,0,0,0,2,0,0,0,0,14],[24,29,0.8276,0.62946,0.31916,0.42857,0.42857,1.0,0.0,1.0,1,11,0,1,0,3,0,0,0,0,0,13,0,0,0,0,0,2,0,0,2,0,11],[28,29,0.9655,0.70536,0.3008,0.42857,0.71429,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,12,0,0,0,0,0,3,0,0,0,0,15],[29,29,1.0,0.66964,0.25862,0.42857,0.57144,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,5,0,0,0,0,11]]},{"b":3,"e":1.0,"k":"flat","v":0.625,"x":0.70089,"p":[[0,10,0.0,0.625,0.3004,0.42857,0.42859,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,12],[4,10,0.4,0.67411,0.33926,0.42857,0.71429,1.0,0.0,1.0,3,15,0,3,0,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,0,0,15],[8,10,0.8,0.70089,0.32607,0.42857,0.78571,1.0,0.0,1.0,2,15,0,2,0,1,0,0,0,0,0,10,0,0,0,0,0,3,0,0,1,0,15],[10,10,1.0,0.66071,0.27141,0.42857,0.42857,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,2,0,0,0,0,12]]}]},{"i":"f529ab8ad3b344a3","q":"Let  $p,q$  and  $s{}$  be prime numbers such that  $2^sq =p^y-1$  where  $y > 1.$  Find all possible values of  $p.$","t":[{"b":1,"e":0.0,"k":"falling","v":0.04018,"x":0.62923,"p":[[0,91,0.0,0.45981,0.27135,0.2857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,8,0,0,3,0,0,4,0,0,5,0,0,4,0,1],[4,91,0.044,0.62923,0.23934,0.42859,0.71429,0.75,0.14,1.0,0,2,0,0,0,2,0,0,5,0,0,2,0,0,2,0,0,13,0,0,6,0,2],[8,91,0.0879,0.47321,0.2976,0.2857,0.42857,0.71429,0.0,1.0,2,3,1,2,0,5,0,0,8,0,0,3,0,0,2,0,0,7,0,0,2,0,3],[12,91,0.1319,0.43283,0.31249,0.14286,0.28571,0.71429,0.0,1.0,3,2,0,3,0,8,0,0,6,0,0,1,0,0,4,0,0,4,0,0,4,0,2],[16,91,0.1758,0.38838,0.25562,0.14286,0.28571,0.60714,0.0,0.85714,2,0,0,2,0,9,0,0,6,0,0,4,0,0,3,0,0,6,0,0,2,0,0],[20,91,0.2198,0.38839,0.32583,0.14286,0.28571,0.71429,0.0,1.0,7,2,0,7,0,7,0,0,3,0,0,1,0,0,5,0,0,5,0,0,2,0,2],[24,91,0.2637,0.38372,0.23553,0.14286,0.28571,0.5714,0.14,1.0,0,1,0,0,0,11,0,0,6,0,0,4,0,0,7,0,0,2,0,0,1,0,1],[28,91,0.3077,0.33908,0.21664,0.14286,0.28571,0.4286,0.0,0.85714,4,0,0,4,0,5,0,0,9,0,0,7,0,0,4,0,0,2,0,0,1,0,0],[32,91,0.3516,0.34825,0.26712,0.14286,0.28571,0.4286,0.0,1.0,4,1,0,4,0,8,0,0,6,0,0,8,0,0,1,0,0,1,0,0,3,0,1],[36,91,0.3956,0.34366,0.22836,0.14286,0.28571,0.46431,0.0,0.857,2,0,0,2,0,10,0,0,7,0,0,5,0,0,3,0,0,4,0,0,1,0,0],[40,91,0.4396,0.33932,0.24157,0.14289,0.28571,0.42895,0.0,1.0,4,1,0,4,0,5,0,0,12,0,0,4,0,0,3,0,0,2,0,0,1,0,1],[44,91,0.4835,0.33482,0.21609,0.14286,0.28571,0.4286,0.0,1.0,2,1,0,2,0,7,0,0,11,0,0,8,0,0,1,0,0,1,0,0,1,0,1],[48,91,0.5275,0.29017,0.27544,0.0,0.21429,0.42857,0.0,0.85714,9,0,0,9,0,7,0,0,5,0,0,4,0,0,1,0,0,4,0,0,2,0,0],[52,91,0.5714,0.31247,0.22987,0.14286,0.2857,0.46418,0.0,0.85714,3,0,0,3,0,11,0,0,8,0,0,2,0,0,4,0,0,3,0,0,1,0,0],[56,91,0.6154,0.32134,0.2475,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,8,0,0,7,0,0,5,0,0,1,0,0,5,0,0,1,0,0],[60,91,0.6593,0.27231,0.22966,0.0,0.2857,0.42857,0.0,0.71429,9,0,0,9,0,6,0,0,4,0,0,7,0,0,4,0,0,2,0,0,0,0,0],[64,91,0.7033,0.32586,0.2556,0.14286,0.28571,0.571,0.0,0.85714,6,0,0,6,0,7,0,0,7,0,0,2,0,0,5,0,0,4,0,0,1,0,0],[68,91,0.7473,0.28556,0.21398,0.14286,0.2857,0.42858,0.0,0.71429,5,0,0,5,0,10,0,0,6,0,0,4,0,0,5,0,0,2,0,0,0,0,0],[72,91,0.7912,0.20527,0.14702,0.14286,0.1429,0.28571,0.0,0.57143,6,0,0,6,0,11,0,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[76,91,0.8352,0.16954,0.15333,0.0,0.14286,0.2857,0.0,0.57143,9,0,0,9,0,13,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[80,91,0.8791,0.2188,0.22587,0.0,0.14288,0.28571,0.0,0.71429,12,0,0,12,0,5,0,0,8,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[84,91,0.9231,0.17393,0.21051,0.0,0.14286,0.2857,0.0,1.0,12,1,0,12,0,10,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[88,91,0.967,0.12499,0.17764,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,8,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[91,91,1.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.16965,"x":0.61158,"p":[[0,70,0.0,0.46412,0.21998,0.28571,0.571,0.711,0.0,0.71429,1,0,0,1,0,5,0,0,5,0,0,4,0,0,8,0,0,9,0,0,0,0,0],[4,70,0.0571,0.61158,0.23754,0.4286,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,3,0,0,9,0,0,4,0,0,8,0,2],[8,70,0.1143,0.48661,0.22548,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,7,0,0,10,0,0,3,0,0,5,0,0,3,0,1],[12,70,0.1714,0.41508,0.19991,0.2857,0.42857,0.571,0.0,0.85714,1,0,1,1,0,3,0,0,11,0,0,5,0,0,9,0,0,1,0,0,2,0,0],[16,70,0.2286,0.45982,0.25439,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,9,0,0,6,0,0,2,0,0,6,0,0,3,0,1],[20,70,0.2857,0.37043,0.22837,0.14286,0.28571,0.4642,0.14,1.0,0,1,0,0,0,9,0,0,11,0,0,4,0,0,3,0,0,3,0,0,1,0,1],[24,70,0.3429,0.37499,0.23075,0.14286,0.28571,0.46418,0.14286,1.0,0,1,0,0,0,10,0,0,8,0,0,6,0,0,3,0,0,3,0,0,1,0,1],[28,70,0.4,0.3973,0.22225,0.25,0.42857,0.5711,0.0,0.85714,1,0,1,1,0,7,0,0,7,0,0,7,0,0,5,0,0,3,0,0,2,0,0],[32,70,0.4571,0.45981,0.26899,0.24999,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,5,0,0,4,0,0,3,0,0,5,0,1],[36,70,0.5143,0.40616,0.2579,0.14286,0.28571,0.60714,0.14,1.0,0,2,0,0,0,10,0,0,7,0,0,5,0,0,2,0,0,6,0,0,0,0,2],[40,70,0.5714,0.43749,0.24983,0.25,0.35714,0.71429,0.14286,0.85714,0,0,0,0,0,8,0,0,8,0,0,3,0,0,3,0,0,7,0,0,3,0,0],[44,70,0.6286,0.3839,0.2214,0.14286,0.35714,0.57141,0.14286,0.85714,0,0,0,0,0,11,0,0,5,0,0,5,0,0,6,0,0,4,0,0,1,0,0],[48,70,0.6857,0.38838,0.23482,0.14286,0.35714,0.57111,0.14286,0.85714,0,0,0,0,0,11,0,0,5,0,0,7,0,0,2,0,0,5,0,0,2,0,0],[52,70,0.7429,0.33929,0.21354,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,11,0,0,9,0,0,7,0,0,2,0,0,1,0,0,1,0,1],[56,70,0.8,0.32143,0.19562,0.14286,0.2857,0.32143,0.14286,0.85714,0,0,0,0,0,10,0,0,14,0,0,3,0,0,2,0,0,1,0,0,2,0,0],[60,70,0.8571,0.28563,0.19239,0.14286,0.1429,0.32143,0.14,0.71429,0,0,0,0,0,17,0,0,7,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[64,70,0.9143,0.30802,0.17894,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,13,0,0,8,0,0,7,0,0,1,0,0,3,0,0,0,0,0],[68,70,0.9714,0.24553,0.18975,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,17,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[70,70,1.0,0.16965,0.10971,0.14286,0.14286,0.14287,0.0,0.71429,1,0,0,1,0,27,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"20924b262d4572ab","q":"Let $a, b$ and $c$ be three strictly positive real numbers. Prove that\n\n$$\n4\\left(a^{3}+b^{3}+c^{3}+3\\right) \\geqslant 3(a+1)(b+1)(c+1) .\n$$","t":[{"b":1,"e":0.14286,"k":"falling","v":0.14732,"x":0.35714,"p":[[0,47,0.0,0.33929,0.30671,0.14286,0.2143,0.42857,0.0,1.0,4,4,0,4,0,12,0,0,3,0,0,8,0,0,0,0,0,0,0,0,1,0,4],[4,47,0.0851,0.35713,0.2857,0.10714,0.42857,0.57143,0.0,1.0,8,2,0,8,0,3,0,0,4,0,0,7,0,0,7,0,0,0,0,0,1,0,2],[8,47,0.1702,0.2857,0.23689,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,8,0,0,4,0,0,7,0,0,5,0,0,0,0,0,0,0,1],[12,47,0.2553,0.35259,0.29669,0.14286,0.28571,0.57143,0.0,1.0,5,3,0,5,0,9,0,0,4,0,0,5,0,0,5,0,0,0,0,0,1,0,3],[16,47,0.3404,0.35714,0.27894,0.14286,0.35714,0.46431,0.0,1.0,5,3,0,5,0,7,0,0,4,0,0,8,0,0,5,0,0,0,0,0,0,0,3],[20,47,0.4255,0.24999,0.23418,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,15,0,0,1,0,0,4,0,0,5,0,0,0,0,0,0,0,1],[24,47,0.5106,0.31686,0.28739,0.14286,0.14286,0.57111,0.0,1.0,3,3,0,3,0,17,0,0,0,0,0,3,0,0,6,0,0,0,0,0,0,0,3],[28,47,0.5957,0.23651,0.20707,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,17,0,0,0,0,0,6,0,0,3,0,0,0,0,0,1,0,0],[32,47,0.6809,0.26786,0.28065,0.14286,0.14286,0.42857,0.0,1.0,6,2,0,6,0,16,0,0,1,0,0,2,0,0,4,0,0,0,0,0,1,0,2],[36,47,0.766,0.15179,0.13333,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,23,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,47,0.8511,0.19196,0.25407,0.0,0.14286,0.14286,0.0,1.0,10,2,0,10,0,16,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[44,47,0.9362,0.14732,0.10999,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[47,47,1.0,0.1875,0.1729,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1]]},{"b":4,"e":0.14286,"k":"falling","v":0.14732,"x":0.47321,"p":[[0,129,0.0,0.37499,0.31692,0.14286,0.35714,0.4286,0.0,1.0,5,5,0,5,0,8,0,0,3,0,0,9,0,0,2,0,0,0,0,0,0,0,5],[4,129,0.031,0.36607,0.30291,0.0,0.42857,0.57143,0.0,1.0,9,3,0,9,0,3,0,0,1,0,0,8,0,0,8,0,0,0,0,0,0,0,3],[8,129,0.062,0.25446,0.17762,0.14286,0.14286,0.42857,0.0,0.57143,4,0,0,4,0,14,0,0,2,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[12,129,0.093,0.47321,0.28668,0.28571,0.42857,0.57143,0.0,1.0,2,4,0,2,0,5,0,0,3,0,0,10,0,0,5,0,0,1,0,0,2,0,4],[16,129,0.124,0.27677,0.21996,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,3,0,0,8,0,0,5,0,0,1,0,0,0,0,0],[20,129,0.155,0.39286,0.30514,0.14286,0.42857,0.4286,0.0,1.0,6,4,0,6,0,4,0,0,3,0,0,12,0,0,2,0,0,0,0,0,1,0,4],[24,129,0.186,0.45981,0.27136,0.28571,0.42859,0.57143,0.0,1.0,5,3,0,5,0,0,0,0,4,0,0,8,0,0,11,0,0,0,0,0,1,0,3],[28,129,0.2171,0.42857,0.30723,0.14286,0.42857,0.57143,0.0,1.0,2,5,0,2,0,8,0,0,5,0,0,7,0,0,4,0,0,0,0,0,1,0,5],[32,129,0.2481,0.41072,0.31693,0.14286,0.42857,0.57143,0.0,1.0,5,5,1,5,0,7,0,0,0,0,0,10,0,0,5,0,0,0,0,0,0,0,5],[36,129,0.2791,0.45534,0.31019,0.24999,0.42857,0.57143,0.0,1.0,3,6,0,3,0,5,0,0,4,0,0,9,0,0,5,0,0,0,0,0,0,0,6],[40,129,0.3101,0.35267,0.29229,0.14286,0.28571,0.57143,0.0,1.0,7,3,0,7,0,5,0,0,5,0,0,5,0,0,7,0,0,0,0,0,0,0,3],[44,129,0.3411,0.46427,0.27664,0.28571,0.42857,0.57143,0.0,1.0,3,4,0,3,0,3,0,0,3,0,0,11,0,0,7,0,0,0,0,0,1,0,4],[48,129,0.3721,0.35268,0.29663,0.14286,0.42857,0.57143,0.0,1.0,7,2,0,7,0,8,0,0,0,0,0,6,0,0,7,0,0,1,0,0,1,0,2],[52,129,0.4031,0.32143,0.26726,0.14286,0.28571,0.42857,0.0,1.0,6,2,0,6,0,7,0,0,5,0,0,9,0,0,2,0,0,0,0,0,1,0,2],[56,129,0.4341,0.35713,0.30513,0.14286,0.2143,0.57143,0.0,1.0,5,3,0,5,0,11,0,0,1,0,0,4,0,0,7,0,0,0,0,0,1,0,3],[60,129,0.4651,0.33927,0.28737,0.14286,0.21428,0.57143,0.0,1.0,4,3,0,4,0,12,0,0,3,0,0,3,0,0,7,0,0,0,0,0,0,0,3],[64,129,0.4961,0.35713,0.2812,0.14286,0.42857,0.57143,0.0,1.0,6,2,0,6,0,7,0,0,2,0,0,7,0,0,7,0,0,0,0,0,1,0,2],[68,129,0.5271,0.2857,0.26962,0.10714,0.14288,0.42857,0.0,1.0,8,2,0,8,0,9,0,0,2,0,0,7,0,0,4,0,0,0,0,0,0,0,2],[72,129,0.5581,0.29464,0.31326,0.0,0.14286,0.57143,0.0,1.0,13,1,0,13,0,4,0,0,1,0,0,5,0,0,5,0,0,0,0,0,3,0,1],[76,129,0.5891,0.26338,0.2425,0.0,0.21429,0.42857,0.0,1.0,10,1,0,10,0,6,0,0,2,0,0,10,0,0,3,0,0,0,0,0,0,0,1],[80,129,0.6202,0.26786,0.29827,0.0,0.14286,0.42857,0.0,1.0,11,2,0,11,0,8,0,0,2,0,0,5,0,0,2,0,0,1,0,0,1,0,2],[84,129,0.6512,0.32589,0.28846,0.10714,0.28571,0.57143,0.0,1.0,8,2,0,8,0,6,0,0,4,0,0,5,0,0,6,0,0,0,0,0,1,0,2],[88,129,0.6822,0.29463,0.29436,0.10714,0.14286,0.57143,0.0,1.0,8,2,0,8,0,10,0,0,4,0,0,0,0,0,7,0,0,0,0,0,1,0,2],[92,129,0.7132,0.33926,0.20745,0.14286,0.42857,0.571,0.0,0.57143,5,0,0,5,0,6,0,0,2,0,0,10,0,0,9,0,0,0,0,0,0,0,0],[96,129,0.7442,0.37945,0.37048,0.10714,0.28571,0.57143,0.0,1.0,8,7,0,8,0,7,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,7],[100,129,0.7752,0.39731,0.34577,0.14286,0.35714,0.57143,0.0,1.0,7,5,0,7,0,7,0,0,2,0,0,3,0,0,7,0,0,0,0,0,1,0,5],[104,129,0.8062,0.36152,0.33315,0.14286,0.14286,0.57143,0.0,1.0,5,5,0,5,0,12,0,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,5],[108,129,0.8372,0.29464,0.22851,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,10,0,0,4,0,0,8,0,0,3,0,0,0,0,0,2,0,0],[112,129,0.8682,0.32589,0.31387,0.14286,0.14286,0.46429,0.0,1.0,6,4,0,6,0,12,0,0,1,0,0,5,0,0,4,0,0,0,0,0,0,0,4],[116,129,0.8992,0.30357,0.29827,0.14286,0.14286,0.46429,0.0,1.0,7,3,0,7,0,12,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,3],[120,129,0.9302,0.20079,0.22829,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,8,0,0,2,0,0,5,0,0,3,0,0,0,0,0,1,0,0],[124,129,0.9612,0.34822,0.31122,0.14286,0.2143,0.57143,0.0,1.0,7,3,0,7,0,9,0,0,1,0,0,4,0,0,6,0,0,2,0,0,0,0,3],[128,129,0.9922,0.30802,0.25027,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,12,0,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,2],[129,129,1.0,0.14732,0.09771,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,20,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d9184b363f81d80a","q":"Let  $p$  be a prime number and  $f\\in \\mathbb{Z}[X]$  given by\n\\[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \\cdots + a_2x+ a_1 , \\]\nwhere  $a_i = \\left( \\tfrac ip\\right)$  is the Legendre symbol of  $i$  with respect to  $p$  (i.e.  $a_i=1$  if  $ i^{\\frac {p-1}2} \\equiv 1 \\pmod p$  and  $a_i=-1$  otherwise, for all  $i=1,2,\\ldots,p-1$ ).\n\na) Prove that  $f(x)$  is divisible with  $(x-1)$ , but not with  $(x-1)^2$  iff  $p \\equiv 3 \\pmod 4$ ;\nb) Prove that if  $p\\equiv 5 \\pmod 8$  then  $f(x)$  is divisible with  $(x-1)^2$  but not with  $(x-1)^3$ .\n\n*Sugested by Calin Popescu.*","t":[{"b":3,"e":1.0,"k":"rising","v":0.54911,"x":0.90624,"p":[[0,35,0.0,0.54911,0.11904,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,17,0,0,3,0,0,0,0,1],[4,35,0.1143,0.84372,0.19682,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,4,0,17],[8,35,0.2286,0.83924,0.19485,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,7,0,15],[12,35,0.3429,0.87053,0.18681,0.82143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,18],[16,35,0.4571,0.90622,0.15413,0.85711,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,22],[20,35,0.5714,0.90624,0.1499,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,21],[24,35,0.6857,0.8214,0.23693,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,2,0,0,5,0,17],[28,35,0.8,0.82137,0.19239,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,3,0,15],[32,35,0.9143,0.72767,0.23518,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,4,0,0,9,0,0,5,0,0,2,0,11],[35,35,1.0,0.86604,0.20187,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,3,0,20]]},{"b":6,"e":1.0,"k":"rising","v":0.52679,"x":0.89729,"p":[[0,47,0.0,0.52679,0.07523,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,20,0,0,1,0,0,0,0,0],[4,47,0.0851,0.83036,0.18013,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,5,0,14],[8,47,0.1702,0.88388,0.15754,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,4,0,19],[12,47,0.2553,0.84374,0.17987,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,4,0,16],[16,47,0.3404,0.80801,0.19106,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,5,0,13],[20,47,0.4255,0.85714,0.17857,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,6,0,16],[24,47,0.5106,0.7946,0.1784,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,0,3,0,12],[28,47,0.5957,0.741,0.21857,0.571,0.78564,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,3,0,0,7,0,9],[32,47,0.6809,0.74997,0.2259,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,5,0,0,4,0,11],[36,47,0.766,0.89729,0.14833,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,19],[40,47,0.8511,0.7723,0.20473,0.67857,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,6,0,10],[44,47,0.9362,0.82589,0.17029,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,8,0,12],[47,47,1.0,0.81246,0.21563,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,1,0,0,2,0,17]]}]},{"i":"f2b553622d583cfc","q":"Let $a, b, c, d$ be natural numbers such that $0<|a d-b c|<\\min (c, d)$.\nProve that for all integers $x, y>1$ that are coprime, the number $x^{a}+y^{b}$ is not divisible by $x^{c}+y^{d}$.","t":[{"b":1,"e":0.4286,"k":"flat","v":0.37498,"x":0.61607,"p":[[0,24,0.0,0.5446,0.24598,0.42857,0.57143,0.71429,0.0,0.85714,3,0,1,3,0,0,0,0,3,0,0,6,0,0,8,0,0,6,0,0,6,0,0],[4,24,0.1667,0.39728,0.21346,0.28571,0.42857,0.571,0.0,0.71429,5,0,0,5,0,1,0,0,3,0,0,14,0,0,5,0,0,4,0,0,0,0,0],[8,24,0.3333,0.37498,0.26424,0.14286,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,3,0,0,3,0,0,8,0,0,6,0,0,3,0,0,2,0,0],[12,24,0.5,0.51782,0.25691,0.39286,0.57143,0.71429,0.0,0.85714,3,0,0,3,0,2,0,0,3,0,0,4,0,0,9,0,0,6,0,0,5,0,0],[16,24,0.6667,0.54458,0.14031,0.42857,0.57121,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,13,0,0,12,0,0,3,0,0,3,0,0],[20,24,0.8333,0.61607,0.18707,0.42859,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,8,0,0,10,0,0,7,0,0,4,0,2],[24,24,1.0,0.50888,0.18534,0.42857,0.4286,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,0,0,0,15,0,0,8,0,0,5,0,0,2,0,0]]},{"b":6,"e":0.0,"k":"rising","v":0.42399,"x":0.83928,"p":[[0,68,0.0,0.42399,0.23012,0.39286,0.42857,0.57111,0.0,0.85714,4,0,0,4,0,2,0,0,2,0,0,14,0,0,6,0,0,1,0,0,3,0,0],[4,68,0.0588,0.6875,0.21852,0.64286,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,6,0,0,0,0,0,12,0,0,10,0,2],[8,68,0.1176,0.82143,0.14286,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,13,0,8],[12,68,0.1765,0.83491,0.1641,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,11,0,11],[16,68,0.2353,0.76785,0.33455,0.67857,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,16],[20,68,0.2941,0.64283,0.34626,0.53539,0.71429,0.85714,0.0,1.0,6,7,0,6,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,7,0,7],[24,68,0.3529,0.71427,0.31135,0.67857,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,5,0,10],[28,68,0.4118,0.73201,0.31289,0.67536,0.85714,1.0,0.0,1.0,4,9,0,4,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,11,0,9],[32,68,0.4706,0.6875,0.35072,0.71429,0.78571,1.0,0.0,1.0,6,9,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,7,0,9],[36,68,0.5294,0.65622,0.3551,0.571,0.78564,0.89286,0.0,1.0,6,8,0,6,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,8,0,8],[40,68,0.5882,0.78571,0.28794,0.71429,0.85714,1.0,0.0,1.0,3,12,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,10,0,12],[44,68,0.6471,0.65179,0.3387,0.57143,0.71429,0.85714,0.0,1.0,6,6,0,6,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,8,0,6],[48,68,0.7059,0.72767,0.31003,0.71429,0.85712,1.0,0.0,1.0,4,9,0,4,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,9,0,9],[52,68,0.7647,0.78571,0.31744,0.71429,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,15],[56,68,0.8235,0.78124,0.28119,0.71429,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,10,0,11],[60,68,0.8824,0.77677,0.31932,0.71429,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,15],[64,68,0.9412,0.63838,0.40087,0.32143,0.85714,1.0,0.0,1.0,8,12,0,8,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,5,0,12],[68,68,1.0,0.83928,0.16269,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,7,0,13]]}]},{"i":"d6dbfd74c566e12e","q":"Let $a, b, c$ be the side lengths of a triangle. Prove that\n\n$$\n\\sqrt[3]{\\left(a^{2}+b c\\right)\\left(b^{2}+c a\\right)\\left(c^{2}+a b\\right)}>\\frac{a^{2}+b^{2}+c^{2}}{2} .\n$$","t":[{"b":1,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,12,0.0,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,12,0.3333,0.94642,0.19152,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[8,12,0.6667,0.875,0.31894,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,8,0.0,0.89286,0.27199,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,26],[4,8,0.5,0.91518,0.25218,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e894697349d6448a","q":"Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.15616,"x":0.48214,"p":[[0,38,0.0,0.16964,0.17655,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,38,0.1053,0.23643,0.21019,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,24,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,1],[8,38,0.2105,0.25446,0.24932,0.14286,0.14286,0.17857,0.0,1.0,2,2,0,2,0,22,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,2],[12,38,0.3158,0.15616,0.0969,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,28,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,38,0.4211,0.25,0.17496,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,21,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[20,38,0.5263,0.25,0.15152,0.14286,0.14286,0.42857,0.14286,0.57143,0,0,0,0,0,21,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[24,38,0.6316,0.21866,0.1515,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,25,0,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[28,38,0.7368,0.19643,0.12753,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,25,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[32,38,0.8421,0.25438,0.15872,0.14286,0.14286,0.42857,0.14,0.57143,0,0,0,0,0,21,0,0,0,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[36,38,0.9474,0.48214,0.06916,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0],[38,38,1.0,0.47321,0.06622,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.17839,"x":0.60266,"p":[[0,72,0.0,0.17839,0.15572,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,22,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[4,72,0.0556,0.24545,0.25317,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,24,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,2],[8,72,0.1111,0.27232,0.19678,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,20,0,0,0,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[12,72,0.1667,0.23205,0.22802,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,24,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,1],[16,72,0.2222,0.27232,0.18681,0.14286,0.14286,0.42858,0.0,0.57143,1,0,0,1,0,19,0,0,1,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[20,72,0.2778,0.22768,0.18509,0.14286,0.14286,0.21429,0.0,0.57143,3,0,0,3,0,21,0,0,0,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[24,72,0.3333,0.25435,0.19476,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,22,0,0,0,0,0,2,0,0,6,0,0,1,0,0,0,0,0],[28,72,0.3889,0.41964,0.23402,0.14286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,12,0,0,0,0,0,3,0,0,14,0,0,2,0,0,0,0,1],[32,72,0.4444,0.32134,0.21436,0.14286,0.14286,0.57143,0.14,0.71429,0,0,0,0,0,18,0,0,0,0,0,5,0,0,6,0,0,3,0,0,0,0,0],[36,72,0.5,0.33927,0.2389,0.14286,0.14286,0.57143,0.0,0.85714,2,0,0,2,0,15,0,0,0,0,0,2,0,0,11,0,0,1,0,0,1,0,0],[40,72,0.5556,0.44195,0.1868,0.35714,0.57121,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,0,0,0,7,0,0,15,0,0,2,0,0,0,0,0],[44,72,0.6111,0.43304,0.23003,0.14286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,11,0,0,0,0,0,3,0,0,15,0,0,2,0,0,0,0,1],[48,72,0.6667,0.36158,0.20817,0.14286,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,13,0,0,0,0,0,4,0,0,14,0,0,0,0,0,0,0,0],[52,72,0.7222,0.35259,0.21133,0.14286,0.42859,0.57143,0.0,0.57143,1,0,0,1,0,14,0,0,0,0,0,3,0,0,14,0,0,0,0,0,0,0,0],[56,72,0.7778,0.36161,0.22299,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,16,0,0,0,0,0,1,0,0,13,0,0,2,0,0,0,0,0],[60,72,0.8333,0.44196,0.21829,0.14286,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,10,0,0,0,0,0,4,0,0,14,0,0,3,0,0,1,0,0],[64,72,0.8889,0.39284,0.20824,0.14286,0.49979,0.57143,0.0,0.57143,2,0,0,2,0,9,0,0,0,0,0,5,0,0,16,0,0,0,0,0,0,0,0],[68,72,0.9444,0.58927,0.06916,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,6,0,0,0,0,0],[72,72,1.0,0.60266,0.09933,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,6,0,0,0,0,1]]}]},{"i":"7e32d54e67496225","q":"Let $O$ be an interior point in the equilateral triangle $A B C$, of side length $a$. The lines $A O, B O$, and $C O$ intersect the sides of the triangle in the points $A_{1}, B_{1}$, and $C_{1}$. Show that\n\n$$\n\\left|O A_{1}\\right|+\\left|O B_{1}\\right|+\\left|O C_{1}\\right|<a\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.15625,"x":1.0,"p":[[0,55,0.0,0.27232,0.27283,0.14286,0.14286,0.42857,0.0,1.0,4,3,0,4,0,17,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[4,55,0.0727,0.17857,0.17857,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,24,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[8,55,0.1455,0.24554,0.27487,0.14286,0.14286,0.14286,0.0,1.0,3,3,0,3,0,23,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[12,55,0.2182,0.24107,0.30812,0.0,0.14286,0.14287,0.0,1.0,9,2,0,9,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,2],[16,55,0.2909,0.23661,0.32851,0.0,0.14286,0.14286,0.0,1.0,10,4,0,10,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,4],[20,55,0.3636,0.3125,0.35072,0.14286,0.14286,0.42857,0.0,1.0,7,5,0,7,0,14,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,5],[24,55,0.4364,0.26339,0.27689,0.14286,0.14286,0.21429,0.0,1.0,3,3,0,3,0,21,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,3],[28,55,0.5091,0.24554,0.33165,0.0,0.14286,0.17857,0.0,1.0,11,4,0,11,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,4],[32,55,0.5818,0.21866,0.25252,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,20,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,1],[36,55,0.6545,0.15625,0.17985,0.14286,0.14286,0.14286,0.0,1.0,7,1,0,7,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,55,0.7273,0.17857,0.22868,0.14286,0.14286,0.14286,0.0,1.0,7,2,0,7,0,21,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[44,55,0.8,0.15625,0.18681,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,18,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,55,0.9455,0.97321,0.07525,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[55,55,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.42857,"k":"rising","v":0.2679,"x":0.5625,"p":[[0,47,0.0,0.2679,0.20442,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,15,0,0,2,0,0,10,0,0,1,0,0,0,0,0,0,0,1],[4,47,0.0851,0.3259,0.32387,0.14286,0.14286,0.42857,0.0,1.0,5,5,0,5,0,14,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5],[8,47,0.1702,0.43304,0.35801,0.14286,0.28574,0.78571,0.0,1.0,1,8,0,1,0,15,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,8],[12,47,0.2553,0.34375,0.34415,0.14286,0.14286,0.42857,0.0,1.0,5,6,0,5,0,14,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,6],[16,47,0.3404,0.5625,0.41178,0.14286,0.42857,1.0,0.0,1.0,1,14,0,1,0,13,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,14],[20,47,0.4255,0.51339,0.40855,0.14286,0.21431,1.0,0.14286,1.0,0,13,0,0,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,13],[24,47,0.5106,0.45536,0.35434,0.14286,0.42857,0.67857,0.0,1.0,4,8,0,4,0,8,0,0,0,0,0,10,0,0,2,0,0,0,0,0,0,0,8],[28,47,0.5957,0.48661,0.40384,0.14286,0.42857,1.0,0.0,1.0,5,11,0,5,0,9,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,11],[32,47,0.6809,0.45089,0.38978,0.14286,0.14286,1.0,0.0,1.0,2,10,0,2,0,15,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,10],[36,47,0.766,0.42411,0.38546,0.14286,0.14288,1.0,0.0,1.0,4,9,0,4,0,13,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,9],[40,47,0.8511,0.36161,0.34066,0.10714,0.28571,0.57143,0.0,1.0,8,4,0,8,0,8,0,0,0,0,0,7,0,0,3,0,0,0,0,0,2,0,4],[44,47,0.9362,0.34375,0.3251,0.14286,0.14286,0.42857,0.0,1.0,5,5,0,5,0,12,0,0,2,0,0,6,0,0,2,0,0,0,0,0,0,0,5],[47,47,1.0,0.49997,0.13362,0.42857,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,12,0,0,17,0,0,0,0,0,1,0,0]]}]},{"i":"882f09561e5a5411","q":"Let $a, b, c, d$ be positive integers such that $a d \\neq b c$ and $\\operatorname{gcd}(a, b, c, d)=1$. Prove that, as $n$ runs through the positive integers, the values $\\operatorname{gcd}(a n+b, c n+d)$ may achieve form the set of all positive divisors of some integer.\n\nRaul Alcantara, Peru","t":[{"b":2,"e":0.28571,"k":"flat","v":0.16071,"x":0.39731,"p":[[0,24,0.0,0.16071,0.21943,0.0,0.0,0.32143,0.0,0.57143,20,0,0,20,0,0,0,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[4,24,0.1667,0.39731,0.21047,0.28571,0.57121,0.57143,0.0,0.57143,5,0,0,5,0,0,0,0,9,0,0,1,0,0,17,0,0,0,0,0,0,0,0],[8,24,0.3333,0.26784,0.26425,0.0,0.28571,0.57143,0.0,0.71429,14,0,0,14,0,0,0,0,7,0,0,0,0,0,9,0,0,2,0,0,0,0,0],[12,24,0.5,0.38838,0.20895,0.2857,0.49979,0.57143,0.0,0.57143,5,0,0,5,0,0,0,0,10,0,0,1,0,0,16,0,0,0,0,0,0,0,0],[16,24,0.6667,0.25447,0.26901,0.0,0.14286,0.57143,0.0,0.71429,16,0,0,16,0,0,0,0,3,0,0,2,0,0,10,0,0,1,0,0,0,0,0],[20,24,0.8333,0.24106,0.24596,0.0,0.28571,0.57111,0.0,0.57143,15,0,0,15,0,0,0,0,6,0,0,2,0,0,9,0,0,0,0,0,0,0,0],[24,24,1.0,0.30356,0.22797,0.0,0.28571,0.57111,0.0,0.57143,10,0,0,10,0,0,0,0,7,0,0,6,0,0,9,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.15179,"x":0.39729,"p":[[0,45,0.0,0.15179,0.20497,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,4,0,0,4,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[4,45,0.0889,0.39729,0.22225,0.28571,0.49979,0.57143,0.0,0.71429,5,0,0,5,0,1,0,0,8,0,0,2,0,0,14,0,0,2,0,0,0,0,0],[8,45,0.1778,0.37945,0.19433,0.28571,0.28571,0.57143,0.0,0.57143,4,0,0,4,0,0,0,0,13,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[12,45,0.2667,0.33929,0.1948,0.28571,0.28571,0.57143,0.0,0.57143,5,0,0,5,0,1,0,0,13,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[16,45,0.3556,0.33481,0.19758,0.2857,0.28571,0.4642,0.0,0.71429,5,0,0,5,0,2,0,0,11,0,0,6,0,0,7,0,0,1,0,0,0,0,0],[20,45,0.4444,0.33926,0.1881,0.2857,0.28571,0.571,0.0,0.71429,4,0,0,4,0,1,0,0,16,0,0,2,0,0,8,0,0,1,0,0,0,0,0],[24,45,0.5333,0.29909,0.18679,0.2857,0.28571,0.4286,0.0,0.57143,6,0,0,6,0,1,0,0,16,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[28,45,0.6222,0.30357,0.14617,0.2857,0.28571,0.32143,0.0,0.57143,3,0,0,3,0,2,0,0,19,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[32,45,0.7111,0.30802,0.17167,0.2857,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,1,0,0,15,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[36,45,0.8,0.29007,0.1694,0.2857,0.28571,0.32143,0.0,0.57143,5,0,0,5,0,2,0,0,17,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[40,45,0.8889,0.30349,0.13254,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,4,0,0,22,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[44,45,0.9778,0.30803,0.11355,0.28571,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,2,0,0,23,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[45,45,1.0,0.30134,0.07933,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,1,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"0592dc4443c2936f","q":"Let $a, b, c$ be real numbers. Prove that\n\n$$\na b+b c+c a+\\max \\{|a-b|,|b-c|,|c-a|\\} \\leq 1+\\frac{1}{3}(a+b+c)^{2} .\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.74107,"x":0.86161,"p":[[0,11,0.0,0.74107,0.28669,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,4,0,0,1,0,0,9,0,0,0,0,0,1,0,16],[4,11,0.3636,0.78124,0.23143,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,2,0,0,2,0,15],[8,11,0.7273,0.83929,0.23077,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,20],[11,11,1.0,0.86161,0.23279,0.67857,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,0,0,0,23]]},{"b":2,"e":1.0,"k":"rising","v":0.69642,"x":0.86607,"p":[[0,16,0.0,0.69642,0.26905,0.57132,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,3,0,0,10,0,0,2,0,0,1,0,12],[4,16,0.25,0.70982,0.2435,0.57143,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,3,0,0,14,0,0,0,0,0,1,0,12],[8,16,0.5,0.75893,0.27067,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,2,0,0,8,0,0,0,0,0,2,0,16],[12,16,0.75,0.80357,0.25442,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,4,0,0,4,0,16],[16,16,1.0,0.86607,0.18536,0.82143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,7,0,17]]}]},{"i":"db0f2cc420fd35d3","q":"Let $a_{0}$ be an arbitrary positive integer. Consider the infinite sequence $\\left(a_{n}\\right)_{n \\geq 1}$, defined inductively as follows: given $a_{0}, a_{1}, \\ldots, a_{n-1}$ define the term $a_{n}$ as the smallest positive integer such that $a_{0}+a_{1}+\\ldots+a_{n}$ is divisible by $n$. Prove that there exists a positive integer $M$ such that $a_{n+1}=a_{n}$ for all $n \\geq M$.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.41071,"x":0.83481,"p":[[0,85,0.0,0.83481,0.18937,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,7,0,15],[4,85,0.0471,0.75445,0.21201,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,3,0,0,4,0,11],[8,85,0.0941,0.81249,0.16538,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,11,0,9],[12,85,0.1412,0.81694,0.19641,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,0,6,0,14],[16,85,0.1882,0.79911,0.18161,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,11,0,9],[20,85,0.2353,0.77678,0.18189,0.57143,0.85707,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,6,0,0,9,0,8],[24,85,0.2824,0.80357,0.21053,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,3,0,0,5,0,14],[28,85,0.3294,0.79909,0.19187,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,6,0,12],[32,85,0.3765,0.80804,0.18423,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,9,0,11],[36,85,0.4235,0.7946,0.20188,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,0,7,0,12],[40,85,0.4706,0.75,0.24223,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,3,0,0,9,0,0,1,0,0,8,0,10],[44,85,0.5176,0.76339,0.21011,0.57143,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,5,0,11],[48,85,0.5647,0.68302,0.2135,0.57143,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,10,0,0,5,0,0,5,0,6],[52,85,0.6118,0.68289,0.20118,0.57143,0.64071,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,5,0,0,6,0,5],[56,85,0.6588,0.79911,0.18161,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,13,0,8],[60,85,0.7059,0.70536,0.17105,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,10,0,0,6,0,4],[64,85,0.7529,0.70534,0.19213,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,6,0,0,9,0,4],[68,85,0.8,0.66518,0.20705,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,10,0,0,8,0,0,5,0,4],[72,85,0.8471,0.78125,0.20512,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,10,0,9],[76,85,0.8941,0.72767,0.2,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,7,0,0,6,0,7],[80,85,0.9412,0.65178,0.23403,0.53571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,5,0,0,7,0,0,7,0,0,6,0,4],[84,85,0.9882,0.45979,0.25934,0.28571,0.571,0.71429,0.0,0.85714,5,0,0,5,0,1,0,0,4,0,0,5,0,0,8,0,0,7,0,0,2,0,0],[85,85,1.0,0.41071,0.29397,0.14286,0.35714,0.71429,0.0,0.85714,7,0,0,7,0,2,0,0,7,0,0,2,0,0,1,0,0,12,0,0,1,0,0]]},{"b":6,"e":0.57143,"k":"falling","v":0.59371,"x":0.84374,"p":[[0,52,0.0,0.84374,0.16508,0.82143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,12,0,12],[4,52,0.0769,0.78571,0.18898,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,12,0,8],[8,52,0.1538,0.6741,0.18637,0.57143,0.57143,0.85702,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,5,0,0,4,0,5],[12,52,0.2308,0.67856,0.16367,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,12,0,0,2,0,4],[16,52,0.3077,0.64731,0.20821,0.42857,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,10,0,0,2,0,0,5,0,5],[20,52,0.3846,0.63393,0.20497,0.42857,0.57143,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,11,0,0,5,0,0,6,0,0,6,0,3],[24,52,0.4615,0.66964,0.18013,0.57143,0.57143,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,7,0,0,4,0,4],[28,52,0.5385,0.61157,0.08919,0.57143,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,6,0,0,2,0,0],[32,52,0.6154,0.62945,0.1177,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,5,0,0,5,0,0],[36,52,0.6923,0.60268,0.11143,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,4,0,0,3,0,0],[40,52,0.7692,0.59371,0.12429,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,25,0,0,1,0,0,4,0,0],[44,52,0.8462,0.62499,0.1171,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,4,0,0,5,0,0],[48,52,0.9231,0.63837,0.14279,0.57143,0.57143,0.85704,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,0,0,0,9,0,0],[52,52,1.0,0.6339,0.11813,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,3,0,0,6,0,0]]}]},{"i":"99e87d5c28112881","q":"Let $k \\geqslant 1$ be an integer. A telephone operator offers each customer $k$ free call numbers (if person $A$ chooses the number of $B$, then calls from $A$ to $B$ and from $B$ to $A$ are free). Consider a group of $n$ people.\n\n1) If $n \\geqslant 2 k+2$, show that there exist two people who will not be able to communicate for free.\n2) If $n=2 k+1$, show that the $n$ people can arrange it so that any person can communicate for free with any other.","t":[{"b":0,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,6,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,6,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[6,6,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,17,0.0,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,17,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,17,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,17,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,17,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[17,17,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a0832983ebf77457","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}, \\ldots$ be a sequence of positive integers such that  $$ \\frac{a_{1}}{a_{2}}+\\frac{a_{2}}{a_{3}}+\\cdots+\\frac{a_{n-1}}{a_{n}}+\\frac{a_{n}}{a_{1}} $$  is an integer for all $n \\geqslant k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{n}=a_{n+1}$ for all $n \\geqslant m$.  (Mongolia)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08036,"x":0.30357,"p":[[0,89,0.0,0.16509,0.09525,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,26,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,89,0.0449,0.24999,0.16749,0.14286,0.14286,0.42857,0.0,0.57143,3,0,0,3,0,16,0,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[8,89,0.0899,0.21429,0.18558,0.10714,0.14286,0.32143,0.0,0.71429,8,0,0,8,0,11,0,0,5,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[12,89,0.1348,0.29455,0.18543,0.14286,0.2143,0.42857,0.0,0.71429,1,0,0,1,0,15,0,0,3,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[16,89,0.1798,0.30357,0.16269,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,10,0,0,8,0,0,12,0,0,0,0,0,0,0,0,1,0,0],[20,89,0.2247,0.26786,0.15872,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,18,0,0,3,0,0,9,0,0,1,0,0,1,0,0,0,0,0],[24,89,0.2697,0.28123,0.18721,0.14286,0.21428,0.42857,0.0,0.71429,3,0,0,3,0,13,0,0,3,0,0,9,0,0,3,0,0,1,0,0,0,0,0],[28,89,0.3146,0.24991,0.13839,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[32,89,0.3596,0.26777,0.15879,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,16,0,0,3,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[36,89,0.4045,0.26339,0.14335,0.14286,0.21428,0.42857,0.0,0.57143,1,0,0,1,0,15,0,0,5,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[40,89,0.4494,0.25,0.17497,0.14286,0.14286,0.42857,0.0,0.85714,2,0,0,2,0,17,0,0,3,0,0,9,0,0,0,0,0,0,0,0,1,0,0],[44,89,0.4944,0.20535,0.12842,0.14286,0.14286,0.2857,0.0,0.571,2,0,0,2,0,20,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[48,89,0.5393,0.23215,0.18814,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,13,0,0,4,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[52,89,0.5843,0.24999,0.17494,0.14286,0.14286,0.42857,0.0,0.57143,3,0,0,3,0,16,0,0,3,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[56,89,0.6292,0.23205,0.14622,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,16,0,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[60,89,0.6742,0.17857,0.17128,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,14,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[64,89,0.7191,0.20087,0.15086,0.14286,0.14286,0.28571,0.0,0.571,5,0,0,5,0,16,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[68,89,0.764,0.16072,0.15872,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,12,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[72,89,0.809,0.16509,0.1243,0.105,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.20536,0.1234,0.14286,0.14286,0.2857,0.0,0.4286,2,0,0,2,0,20,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.12936,0.13528,0.0,0.14286,0.14286,0.0,0.571,11,0,0,11,0,17,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[84,89,0.9438,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.13393,0.08703,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.11607,0.06621,0.14286,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.16518,"x":0.27679,"p":[[0,55,0.0,0.16518,0.0724,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.20982,0.14279,0.14286,0.14288,0.28571,0.0,0.4286,6,0,0,6,0,11,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.25893,0.20652,0.14286,0.14286,0.32143,0.0,1.0,2,1,0,2,0,17,0,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[12,55,0.2182,0.25446,0.12745,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,14,0,0,8,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.22768,0.13296,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,19,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[20,55,0.3636,0.20982,0.12364,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,21,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,55,0.4364,0.20536,0.14698,0.14286,0.14288,0.28571,0.0,0.57143,6,0,0,6,0,12,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[28,55,0.5091,0.19634,0.12757,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,20,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.27679,0.17105,0.14286,0.21428,0.42857,0.0,0.71429,2,0,0,2,0,14,0,0,3,0,0,11,0,0,1,0,0,1,0,0,0,0,0],[36,55,0.6545,0.2633,0.14342,0.14286,0.2143,0.42857,0.0,0.57143,1,0,0,1,0,15,0,0,5,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[40,55,0.7273,0.24545,0.13946,0.14286,0.14286,0.42857,0.0,0.42857,1,0,0,1,0,18,0,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.26786,0.15047,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[48,55,0.8727,0.20089,0.13767,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,21,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[52,55,0.9455,0.21875,0.10705,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,20,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.17411,0.07771,0.14286,0.14286,0.1429,0.0,0.42857,1,0,0,1,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c00623510607ddd0","q":"Let $f(x)=\\frac{ax+b}{cx+d}$  $F_n(x)=f(f(f...f(x)...))$  (with  $n\\ f's$ )\n\nSuppose that  $f(0) \\not =0$ ,  $f(f(0)) \\not = 0$ , and for some  $n$  we have  $F_n(0)=0$ ,\nshow that  $F_n(x)=x$  (for any valid x).","t":[{"b":4,"e":0.85714,"k":"flat","v":0.74552,"x":0.86607,"p":[[0,21,0.0,0.74552,0.20744,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,8,0,0,6,0,8],[4,21,0.1905,0.81248,0.14915,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,13,0,7],[8,21,0.381,0.76339,0.13651,0.71429,0.71429,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,0,12,0,3],[12,21,0.5714,0.78125,0.15561,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,11,0,0,12,0,5],[16,21,0.7619,0.77232,0.21086,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,12,0,6],[20,21,0.9524,0.86607,0.09407,0.85713,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,8],[21,21,1.0,0.78125,0.13825,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,10,0,5]]},{"b":5,"e":0.85714,"k":"flat","v":0.75892,"x":0.81696,"p":[[0,14,0.0,0.75892,0.18707,0.57143,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,3,0,0,14,0,5],[4,14,0.2857,0.81696,0.1394,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,10,0,8],[8,14,0.5714,0.80803,0.11633,0.71429,0.78564,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,10,0,6],[12,14,0.8571,0.80357,0.16656,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,9,0,9],[14,14,1.0,0.78125,0.12869,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,9,0,5]]}]},{"i":"08d366669307d89a","q":"Let $n \\geqslant 3$ be a positive integer and let $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\\{1,2, \\ldots, n\\}$ such that the value of  $$ \\left|1-\\sum_{i \\in X} a_{i}\\right| $$  is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\\left(b_{1}, b_{2}, \\ldots, b_{n}\\right)$ with sum equal to 2 such that  $$ \\sum_{i \\in X} b_{i}=1 $$  (New Zealand) Common remarks. In all solutions, we say an index set $X$ is $\\left(a_{i}\\right)$-minimising if it has the property in the problem for the given sequence $\\left(a_{i}\\right)$. Write $X^{c}$ for the complement of $X$, and $[a, b]$ for the interval of integers $k$ such that $a \\leqslant k \\leqslant b$. Note that  $$ \\left|1-\\sum_{i \\in X} a_{i}\\right|=\\left|1-\\sum_{i \\in X^{c}} a_{i}\\right|, $$  so we may exchange $X$ and $X^{c}$ where convenient. Let  $$ \\Delta=\\sum_{i \\in X^{c}} a_{i}-\\sum_{i \\in X} a_{i} $$  and note that $X$ is $\\left(a_{i}\\right)$-minimising if and only if it minimises $|\\Delta|$, and that $\\sum_{i \\in X} a_{i}=1$ if and only if $\\Delta=0$.  In some solutions, a scaling process is used. If we have a strictly increasing sequence of positive real numbers $c_{i}$ (typically obtained by perturbing the $a_{i}$ in some way) such that  $$ \\sum_{i \\in X} c_{i}=\\sum_{i \\in X^{c}} c_{i} $$  then we may put $b_{i}=2 c_{i} / \\sum_{j=1}^{n} c_{j}$. So it suffices to construct such a sequence without needing its sum to be 2 .  The solutions below show various possible approaches to the problem. Solutions 1 and 2 perturb a few of the $a_{i}$ to form the $b_{i}$ (with scaling in the case of Solution 1, without scaling in the case of Solution 2). Solutions 3 and 4 look at properties of the index set $X$. Solution 3 then perturbs many of the $a_{i}$ to form the $b_{i}$, together with scaling. Rather than using such perturbations, Solution 4 constructs a sequence $\\left(b_{i}\\right)$ directly from the set $X$ with the required properties. Solution 4 can be used to give a complete description of sets $X$ that are $\\left(a_{i}\\right)$-minimising for some $\\left(a_{i}\\right)$.","t":[{"b":0,"e":0.14286,"k":"rising","v":0.08036,"x":0.35719,"p":[[0,49,0.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,1,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.35712,0.26962,0.14286,0.21428,0.60714,0.0,0.85714,2,0,0,2,0,14,0,0,3,0,0,2,0,0,3,0,0,6,0,0,2,0,0],[8,49,0.1633,0.29911,0.21237,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,12,0,0,6,0,0,4,0,0,4,0,0,3,0,0,0,0,0],[12,49,0.2449,0.28572,0.18558,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,12,0,0,9,0,0,4,0,0,3,0,0,2,0,0,0,0,0],[16,49,0.3265,0.25893,0.20341,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,14,0,0,9,0,0,2,0,0,1,0,0,2,0,0,1,0,0],[20,49,0.4082,0.20534,0.138,0.14286,0.14286,0.17857,0.0,0.57143,1,0,0,1,0,23,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[24,49,0.4898,0.23643,0.19442,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,15,0,0,7,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[28,49,0.5714,0.2767,0.1748,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,13,0,0,8,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[32,49,0.6531,0.3125,0.20025,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,7,0,0,11,0,0,7,0,0,1,0,0,2,0,0,1,0,0],[36,49,0.7347,0.34362,0.239,0.14286,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,7,0,0,3,0,0,3,0,0,5,0,0,1,0,0],[40,49,0.8163,0.2857,0.21722,0.14286,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,9,0,0,12,0,0,2,0,0,1,0,0,3,0,0,1,0,0],[44,49,0.898,0.29465,0.23128,0.14286,0.28571,0.32143,0.0,0.85714,3,0,0,3,0,12,0,0,9,0,0,2,0,0,2,0,0,2,0,0,2,0,0],[48,49,0.9796,0.35714,0.22304,0.14286,0.35714,0.46431,0.0,0.85714,3,0,0,3,0,7,0,0,6,0,0,8,0,0,4,0,0,3,0,0,1,0,0],[49,49,1.0,0.35719,0.23147,0.14286,0.28571,0.57143,0.0,0.85714,3,0,0,3,0,8,0,0,6,0,0,5,0,0,6,0,0,3,0,0,1,0,0]]},{"b":1,"e":0.71429,"k":"flat","v":0.08482,"x":0.36155,"p":[[0,30,0.0,0.08482,0.11214,0.0,0.07143,0.14286,0.0,0.57143,16,0,1,16,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,30,0.1333,0.25436,0.22797,0.14214,0.14286,0.42857,0.0,0.85714,7,0,0,7,0,11,0,0,5,0,0,3,0,0,4,0,0,1,0,0,1,0,0],[8,30,0.2667,0.34821,0.24468,0.14286,0.21428,0.57143,0.14286,0.85714,0,0,0,0,0,16,0,0,3,0,0,4,0,0,3,0,0,4,0,0,2,0,0],[12,30,0.4,0.31249,0.25861,0.14286,0.14286,0.5711,0.0,0.85714,4,0,0,4,0,14,0,0,2,0,0,3,0,0,3,0,0,5,0,0,1,0,0],[16,30,0.5333,0.3124,0.24342,0.14286,0.14286,0.46418,0.0,0.85714,2,0,0,2,0,15,0,0,5,0,0,2,0,0,2,0,0,5,0,0,1,0,0],[20,30,0.6667,0.30795,0.20557,0.14286,0.2143,0.42857,0.14,0.85714,0,0,0,0,0,16,0,0,5,0,0,5,0,0,3,0,0,2,0,0,1,0,0],[24,30,0.8,0.32129,0.24201,0.14286,0.14286,0.57143,0.14286,0.85714,0,0,0,0,0,19,0,0,2,0,0,2,0,0,3,0,0,5,0,0,1,0,0],[28,30,0.9333,0.36155,0.2673,0.14286,0.14286,0.60714,0.0,1.0,1,1,0,1,0,16,0,0,0,0,0,5,0,0,2,0,0,7,0,0,0,0,1],[30,30,1.0,0.17411,0.09268,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"93306ca8f6b58c9d","q":"Let $n \\geqslant 1$ be an integer. Bosphore has written the number 2 on the board $n$ times. He then performs, $n-1$ times in a row, the following operation: he chooses two numbers written on the board, which he calls $a$ and $b$, then erases them and writes the number $\\sqrt{(a b+1) / 2}$ in their place. Finally, he calls $x$ the number written on the board after these $n-1$ operations, and $y$ the number $\\sqrt{(n+3) / n}$.\na) Prove that $x \\geqslant y$.\nb) Prove that there are infinitely many integers $n \\geqslant 1$ for which $x > y$ necessarily.\nc) Prove that there are infinitely many integers $n \\geqslant 1$ for which Bosphore can ensure that $x = y$.","t":[{"b":2,"e":0.1429,"k":"falling","v":0.19641,"x":0.9508,"p":[[0,61,0.0,0.9508,0.15861,1.0,1.0,1.0,0.14,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,61,0.0656,0.72768,0.28651,0.42857,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,3,0,0,6,0,0,3,0,0,2,0,0,3,0,14],[8,61,0.1311,0.73661,0.29904,0.57143,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,1,0,0,1,0,0,6,0,0,4,0,0,4,0,13],[12,61,0.1967,0.71426,0.36771,0.42859,1.0,1.0,0.0,1.0,3,17,0,3,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,17],[16,61,0.2623,0.68747,0.29112,0.5354,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,3,0,0,7,0,0,3,0,0,3,0,11],[20,61,0.3279,0.66518,0.34554,0.5,0.71429,1.0,0.0,1.0,2,13,0,2,0,4,0,0,2,0,0,0,0,0,6,0,0,4,0,0,1,0,13],[24,61,0.3934,0.69195,0.31158,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,2,0,0,2,0,0,4,0,0,6,0,0,3,0,0,0,0,14],[28,61,0.459,0.59822,0.36846,0.28571,0.64286,1.0,0.0,1.0,3,12,0,3,0,3,0,0,5,0,0,4,0,0,1,0,0,3,0,0,1,0,12],[32,61,0.5246,0.62947,0.35149,0.39286,0.71429,1.0,0.0,1.0,3,12,0,3,0,2,0,0,3,0,0,5,0,0,2,0,0,4,0,0,1,0,12],[36,61,0.5902,0.50442,0.3694,0.14286,0.4998,0.89286,0.0,1.0,5,8,0,5,0,5,0,0,3,0,0,3,0,0,5,0,0,1,0,0,2,0,8],[40,61,0.6557,0.41071,0.35129,0.14286,0.28571,0.60714,0.0,1.0,6,6,0,6,0,6,0,0,5,0,0,5,0,0,2,0,0,1,0,0,1,0,6],[44,61,0.7213,0.36607,0.27418,0.14286,0.42857,0.42858,0.0,1.0,4,3,0,4,0,8,0,0,2,0,0,12,0,0,2,0,0,1,0,0,0,0,3],[48,61,0.7869,0.42857,0.24223,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,7,0,0,10,0,0,3,0,0,3,0,0,1,0,2],[52,61,0.8525,0.24107,0.22142,0.0,0.2143,0.42857,0.0,0.71429,11,0,0,11,0,5,0,0,5,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[56,61,0.918,0.20087,0.15909,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,17,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[60,61,0.9836,0.19641,0.19475,0.0,0.14286,0.32143,0.0,0.71429,9,0,0,9,0,14,0,0,1,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[61,61,1.0,0.2009,0.17445,0.0,0.2143,0.28571,0.0,0.57143,11,0,0,11,0,5,0,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.15179,"x":0.91517,"p":[[0,35,0.0,0.91517,0.18851,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,25],[4,35,0.1143,0.76561,0.30839,0.57132,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,1,1,0,0,3,0,0,5,0,0,0,0,0,2,0,18],[8,35,0.2286,0.70089,0.29093,0.42857,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,7,0,0,3,0,0,2,0,0,4,0,12],[12,35,0.3429,0.6741,0.29931,0.42857,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,4,0,0,6,0,0,3,0,0,4,0,10],[16,35,0.4571,0.63829,0.29027,0.42859,0.64286,1.0,0.0,1.0,1,9,0,1,0,3,0,0,1,0,0,4,0,0,7,0,0,7,0,0,0,0,9],[20,35,0.5714,0.55357,0.32488,0.39286,0.57143,0.85704,0.0,1.0,4,6,0,4,0,2,0,0,2,0,0,6,0,0,5,0,0,4,0,0,3,0,6],[24,35,0.6857,0.60268,0.3169,0.42857,0.57143,1.0,0.0,1.0,1,10,0,1,0,3,0,0,3,0,0,8,0,0,4,0,0,2,0,0,1,0,10],[28,35,0.8,0.44643,0.36025,0.14286,0.35714,0.75,0.0,1.0,6,7,0,6,0,4,0,0,6,0,0,4,0,0,3,0,0,1,0,0,1,0,7],[32,35,0.9143,0.25893,0.33775,0.0,0.07143,0.42858,0.0,1.0,16,3,0,16,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,0,1,0,3],[35,35,1.0,0.15179,0.20183,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,4,0,0,6,0,0,4,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"a692308ace9d1a80","q":"Let $n \\geqslant 2$ and $p$ be an odd prime number. Let $u$ be the set of positive integers less than or equal to $p^{n}$ and coprime with $p$, and let $N=|U|$. Show that there exists a permutation $a_{1}, \\ldots, a_{N}$ of the elements of $U$ such that $\\sum_{k=1}^{N} a_{k} a_{k+1}$ (with $a_{N+1}=a_{1}$) is divisible by $p^{n-1}$ but not by $p^{n}$.","t":[{"b":3,"e":0.571,"k":"rising","v":0.40848,"x":0.82143,"p":[[0,37,0.0,0.40848,0.29409,0.28571,0.28571,0.57143,0.0,1.0,4,4,3,4,0,2,0,0,12,0,1,4,0,0,3,0,0,1,0,0,1,0,4],[4,37,0.1081,0.80802,0.26151,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,2,0,0,5,0,17],[8,37,0.2162,0.62499,0.30252,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,2,0,0,7,0,0,2,0,0,6,0,0,3,0,8],[12,37,0.3243,0.71872,0.2487,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,5,0,0,3,0,11],[16,37,0.4324,0.74106,0.30187,0.53539,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,1,0,0,4,0,0,2,0,0,4,0,0,4,0,14],[20,37,0.5405,0.82143,0.22868,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,5,0,16],[24,37,0.6486,0.72765,0.26813,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,7,0,0,4,0,11],[28,37,0.7568,0.72767,0.27977,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,7,0,0,3,0,0,3,0,0,2,0,14],[32,37,0.8649,0.73659,0.26753,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,2,0,0,5,0,0,3,0,0,8,0,10],[36,37,0.973,0.67857,0.26,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,6,0,0,3,0,0,3,0,0,8,0,7],[37,37,1.0,0.56692,0.2461,0.28571,0.49979,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,7,0,0,3,0,0,5,0,0,5,0,3]]},{"b":6,"e":1.0,"k":"rising","v":0.55802,"x":0.90625,"p":[[0,44,0.0,0.55802,0.34876,0.28571,0.42857,1.0,0.0,1.0,1,9,0,1,0,4,0,0,10,0,0,2,0,0,2,0,0,0,0,0,4,0,9],[4,44,0.0909,0.74105,0.30186,0.42859,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,5,0,0,2,0,0,3,0,0,3,0,15],[8,44,0.1818,0.69643,0.28064,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,9,0,0,0,0,0,5,0,0,5,0,10],[12,44,0.2727,0.77678,0.26229,0.53572,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,1,0,0,9,0,13],[16,44,0.3636,0.60714,0.28793,0.42857,0.64286,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,7,0,0,2,0,0,3,0,0,9,0,4],[20,44,0.4545,0.6875,0.27067,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,3,0,0,5,0,0,4,0,0,6,0,0,4,0,9],[24,44,0.5455,0.66963,0.26108,0.42857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,5,0,0,3,0,0,7,0,0,8,0,5],[28,44,0.6364,0.64286,0.3312,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,5,0,0,5,0,0,1,0,0,3,0,0,2,0,12],[32,44,0.7273,0.66964,0.27302,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,7,0,0,2,0,0,4,0,0,6,0,8],[36,44,0.8182,0.79464,0.31529,0.42857,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,22],[40,44,0.9091,0.875,0.23623,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,25],[44,44,1.0,0.90625,0.20705,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,26]]}]},{"i":"ee6fe5a50bfe6ea5","q":"Let $n$ be a positive integer. Define a chameleon to be any sequence of $3 n$ letters, with exactly $n$ occurrences of each of the letters $a, b$, and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$, there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3 n^{2} / 2$ swaps. (Australia)","t":[{"b":2,"e":1.0,"k":"rising","v":0.67857,"x":1.0,"p":[[0,32,0.0,0.82589,0.26901,0.67857,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,2,0,0,1,0,21],[4,32,0.125,0.67857,0.32733,0.28571,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,3,0,0,0,0,15],[8,32,0.25,0.86607,0.2788,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[12,32,0.375,0.875,0.25191,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,25],[16,32,0.5,0.89286,0.23958,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,26],[20,32,0.625,0.75893,0.33012,0.39286,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,20],[24,32,0.75,0.75893,0.33964,0.28571,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,7,0,0,0,0,0,0,0,0,3,0,0,0,0,20],[28,32,0.875,0.87946,0.26027,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.375,"x":1.0,"p":[[0,31,0.0,0.78125,0.34439,0.5,1.0,1.0,0.0,1.0,2,22,1,2,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,22],[4,31,0.129,0.375,0.23077,0.28571,0.28571,0.28571,0.0,1.0,1,3,0,1,0,0,0,0,25,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[8,31,0.2581,0.83929,0.26426,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,4,0,0,0,0,22],[12,31,0.3871,0.95089,0.14555,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[16,31,0.5161,0.91071,0.20438,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,0,0,26],[20,31,0.6452,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"f0eaa1f243e69ebe","q":"Let $n$ be a positive integer and $b$ the greatest integer less than $(\\sqrt[3]{28}-3)^{-n}$. Prove that $b$ cannot be divisible by 6.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.17402,"x":1.0,"p":[[0,62,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,62,0.0645,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,62,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,62,0.1935,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,62,0.2581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,62,0.3226,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,62,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,62,0.4516,0.95089,0.17354,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[32,62,0.5161,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,62,0.5806,0.84375,0.32015,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[40,62,0.6452,0.74554,0.36375,0.42857,1.0,1.0,0.0,1.0,2,20,0,2,0,3,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,20],[44,62,0.7097,0.55795,0.41407,0.14289,0.42857,1.0,0.0,1.0,5,14,0,5,0,4,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,14],[48,62,0.7742,0.44196,0.38525,0.14286,0.28571,1.0,0.0,1.0,4,9,0,4,0,9,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,9],[52,62,0.8387,0.35259,0.32343,0.14286,0.28571,0.42857,0.0,1.0,3,5,0,3,0,12,0,0,8,0,0,2,0,0,1,0,0,0,0,0,1,0,5],[56,62,0.9032,0.34822,0.333,0.14286,0.2143,0.46431,0.0,1.0,5,5,0,5,0,11,0,0,6,0,0,2,0,0,1,0,0,2,0,0,0,0,5],[60,62,0.9677,0.22313,0.20187,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,13,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[62,62,1.0,0.17402,0.10558,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,16,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"falling","v":0.6875,"x":1.0,"p":[[0,33,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,33,0.1212,0.95536,0.1448,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[8,33,0.2424,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,33,0.3636,0.97768,0.08828,1.0,1.0,1.0,0.5714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,33,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,33,0.6061,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,33,0.7273,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,33,0.8485,0.89731,0.17943,0.71429,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,0,0,23],[32,33,0.9697,0.69196,0.25028,0.57142,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,2,0,0,7,0,0,6,0,0,3,0,9],[33,33,1.0,0.6875,0.19045,0.57142,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,10,0,0,4,0,5]]}]},{"i":"e64b873f1f89501a","q":"One hundred circles of radius one are positioned in the plane so that the area of any triangle formed by the centres of three of these circles is at most 2017. Prove that there is a line intersecting at least three of these circles.","t":[{"b":2,"e":0.0,"k":"flat","v":0.04903,"x":0.36156,"p":[[0,38,0.0,0.04903,0.09846,0.0,0.0,0.0,0.0,0.286,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.08927,0.16265,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,38,0.2105,0.16963,0.23805,0.0,0.0,0.42857,0.0,0.71429,19,0,0,19,0,3,0,0,1,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[12,38,0.3158,0.24106,0.29759,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,5,0,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,1],[16,38,0.4211,0.14286,0.18211,0.0,0.0,0.28571,0.0,0.5714,18,0,0,18,0,3,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,38,0.5263,0.29016,0.22722,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,5,0,0,9,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[24,38,0.6316,0.32141,0.29879,0.10714,0.2857,0.57111,0.0,1.0,8,2,0,8,0,6,0,0,7,0,0,2,0,0,5,0,0,0,0,0,2,0,2],[28,38,0.7368,0.29018,0.25123,0.0,0.2857,0.57143,0.0,0.71429,10,0,0,10,0,4,0,0,5,0,0,4,0,0,6,0,0,3,0,0,0,0,0],[32,38,0.8421,0.36156,0.33497,0.0,0.28571,0.57143,0.0,1.0,9,4,0,9,0,4,0,0,5,0,0,4,0,0,4,0,0,1,0,0,1,0,4],[36,38,0.9474,0.22759,0.25473,0.0,0.21428,0.28571,0.0,1.0,12,2,0,12,0,4,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[38,38,1.0,0.18737,0.18718,0.0,0.14286,0.21432,0.0,0.71429,9,0,0,9,0,15,0,0,0,0,0,6,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"rising","v":0.05357,"x":0.36605,"p":[[0,39,0.0,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,25,0,2,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.28123,0.31232,0.0,0.14286,0.57143,0.0,1.0,14,1,0,14,0,4,0,0,1,0,0,3,0,0,6,0,0,1,0,0,2,0,1],[8,39,0.2051,0.24104,0.2799,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,1,0,0,2,0,0,8,0,0,3,0,0,0,0,0,1,0,1],[12,39,0.3077,0.2232,0.27183,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,2,0,0,4,0,0,5,0,0,2,0,0,2,0,0,0,0,1],[16,39,0.4103,0.21429,0.26486,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,3,0,0,3,0,0,5,0,0,3,0,0,1,0,0,0,0,1],[20,39,0.5128,0.2857,0.24998,0.0,0.28571,0.42858,0.0,0.85714,11,0,0,11,0,2,0,0,5,0,0,7,0,0,5,0,0,1,0,0,1,0,0],[24,39,0.6154,0.19196,0.29797,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,4,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,1],[28,39,0.7179,0.33036,0.27994,0.0,0.35714,0.4286,0.0,1.0,9,1,0,9,0,3,0,0,4,0,0,9,0,0,3,0,0,1,0,0,2,0,1],[32,39,0.8205,0.19187,0.23856,0.0,0.07,0.28571,0.0,0.85714,16,0,0,16,0,3,0,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[36,39,0.9231,0.36605,0.24466,0.24999,0.42857,0.42858,0.0,1.0,7,1,0,7,0,1,0,0,3,0,0,14,0,0,5,0,0,0,0,0,1,0,1],[39,39,1.0,0.20536,0.19212,0.0,0.21428,0.42857,0.0,0.57143,13,0,0,13,0,3,0,0,6,0,0,9,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"16ce78e51395dd49","q":"Let $a, b, c$ and $d$ be four real numbers. Suppose there exists a permutation $(x, y, z, t)$ of the numbers $a, b, c$ and $d$ such that\n\n$$\nx \\leqslant 2 a-b, y \\leqslant 2 b-c, z \\leqslant 2 c-d \\text { and } t \\leqslant 2 d-a .\n$$\n\nProve that $a=b=c=d$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,79,0.0,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,79,0.0506,0.95534,0.12083,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[8,79,0.1013,0.88839,0.23887,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,24],[12,79,0.1519,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,79,0.2025,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,79,0.2532,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,79,0.3038,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,79,0.3544,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,79,0.4051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,79,0.4557,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,79,0.5063,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,79,0.557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,79,0.6076,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,79,0.6582,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,79,0.7089,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,79,0.7595,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,79,0.8101,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,79,0.8608,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,79,0.9114,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,79,0.962,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[79,79,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.94641,"x":0.99107,"p":[[0,9,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,9,0.4444,0.94641,0.11715,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[8,9,0.8889,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[9,9,1.0,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25]]}]},{"i":"bd097f4245d00ac9","q":"Prove that a group  $G$  in which exactly two elements other than the identity commute with each other is isomorphic to  $\\mathbb{Z}/3 \\mathbb{Z}$  or  $S_3.$","t":[{"b":6,"e":0.85714,"k":"flat","v":0.95087,"x":0.98661,"p":[[0,29,0.0,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,29,0.1379,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[8,29,0.2759,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,29,0.4138,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,29,0.5517,0.98213,0.05924,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,29,0.6897,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,29,0.8276,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,29,0.9655,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[29,29,1.0,0.95087,0.09013,0.91071,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,2,22]]},{"b":7,"e":1.0,"k":"flat","v":0.95981,"x":0.99554,"p":[[0,28,0.0,0.9665,0.08568,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,1,26],[4,28,0.1429,0.9866,0.04166,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,28,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,28,0.4286,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,28,0.5714,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,28,0.7143,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,28,0.8571,0.95981,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[28,28,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"6bd237c8e04b0f61","q":"Positive real numbers $a_{1}, a_{2}, \\ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\\frac{x^{2}+6 x y+y^{2}}{x+y}$ on the blackboard. After 2023 moves, only one number $c$ will remain on the blackboard. Prove that\n\n$$\nc<2024\\left(a_{1}+a_{2}+\\ldots+a_{2024}\\right)\n$$","t":[{"b":0,"e":0.0,"k":"falling","v":0.05357,"x":0.66072,"p":[[0,70,0.0,0.51786,0.30252,0.39286,0.42857,0.67857,0.0,1.0,2,8,0,2,0,1,0,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,8],[4,70,0.0571,0.66072,0.33834,0.42857,0.64286,1.0,0.0,1.0,3,14,0,3,0,0,0,0,2,0,0,8,0,0,3,0,0,2,0,0,0,0,14],[8,70,0.1143,0.54469,0.31018,0.42857,0.42859,1.0,0.0,1.0,2,9,0,2,0,1,0,0,4,0,0,14,0,0,2,0,0,0,0,0,0,0,9],[12,70,0.1714,0.625,0.31288,0.42857,0.42857,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,15,0,0,1,0,0,0,0,0,2,0,11],[16,70,0.2286,0.4375,0.31529,0.24999,0.42857,0.42858,0.0,1.0,5,6,0,5,0,3,0,0,2,0,0,15,0,0,1,0,0,0,0,0,0,0,6],[20,70,0.2857,0.49554,0.30719,0.42857,0.42857,0.50002,0.0,1.0,3,7,0,3,0,3,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,7],[24,70,0.3429,0.55357,0.34209,0.39286,0.42857,1.0,0.0,1.0,3,10,0,3,0,3,0,0,2,0,0,10,0,0,3,0,0,1,0,0,0,0,10],[28,70,0.4,0.45536,0.24074,0.42857,0.42857,0.4286,0.0,1.0,3,3,0,3,0,0,0,0,3,0,0,20,0,0,1,0,0,1,0,0,1,0,3],[32,70,0.4571,0.53125,0.27487,0.42857,0.42857,0.71429,0.0,1.0,2,6,0,2,0,0,0,0,3,0,0,17,0,0,1,0,0,2,0,0,1,0,6],[36,70,0.5143,0.45089,0.27458,0.28571,0.42857,0.42858,0.0,1.0,3,5,0,3,0,1,0,0,6,0,0,15,0,0,2,0,0,0,0,0,0,0,5],[40,70,0.5714,0.39732,0.33261,0.0,0.42857,0.42857,0.0,1.0,9,5,0,9,0,0,0,0,4,0,0,12,0,0,1,0,0,0,0,0,1,0,5],[44,70,0.6286,0.44196,0.28873,0.42857,0.42857,0.42857,0.0,1.0,5,5,0,5,0,1,0,0,1,0,0,19,0,0,1,0,0,0,0,0,0,0,5],[48,70,0.6857,0.47768,0.34921,0.28571,0.42857,0.78571,0.0,1.0,6,8,0,6,0,1,0,0,4,0,0,11,0,0,1,0,0,1,0,0,0,0,8],[52,70,0.7429,0.37946,0.37899,0.0,0.35714,0.57143,0.0,1.0,12,7,0,12,0,1,0,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,7],[56,70,0.8,0.32589,0.29066,0.0,0.42857,0.42857,0.0,1.0,10,3,0,10,0,2,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,3],[60,70,0.8571,0.16072,0.26905,0.0,0.0,0.2857,0.0,1.0,20,2,0,20,0,3,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[64,70,0.9143,0.05804,0.19186,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[68,70,0.9714,0.05357,0.1915,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[70,70,1.0,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.00893,"x":0.54912,"p":[[0,73,0.0,0.54912,0.30327,0.42857,0.42857,1.0,0.0,1.0,1,9,0,1,0,2,0,0,4,0,0,14,0,0,2,0,0,0,0,0,0,0,9],[4,73,0.0548,0.47322,0.27302,0.39286,0.42857,0.4286,0.0,1.0,2,5,0,2,0,2,0,0,4,0,0,17,0,0,1,0,0,0,0,0,1,0,5],[8,73,0.1096,0.41518,0.32608,0.14286,0.42857,0.46431,0.0,1.0,7,5,0,7,0,3,0,0,1,0,0,13,0,0,2,0,0,0,0,0,1,0,5],[12,73,0.1644,0.28125,0.26119,0.0,0.28571,0.42857,0.0,1.0,10,2,0,10,0,4,0,0,3,0,0,13,0,0,0,0,0,0,0,0,0,0,2],[16,73,0.2192,0.52677,0.3489,0.28571,0.42857,1.0,0.0,1.0,3,10,0,3,0,3,0,0,5,0,0,9,0,0,2,0,0,0,0,0,0,0,10],[20,73,0.274,0.30804,0.27225,0.0,0.42857,0.42857,0.0,1.0,11,2,0,11,0,0,0,0,4,0,0,14,0,0,0,0,0,1,0,0,0,0,2],[24,73,0.3288,0.47321,0.30397,0.28571,0.42857,0.60714,0.0,1.0,4,6,0,4,0,0,0,0,7,0,0,12,0,0,1,0,0,2,0,0,0,0,6],[28,73,0.3836,0.32589,0.30563,0.0,0.42857,0.42857,0.0,1.0,10,3,0,10,0,4,0,0,0,0,0,14,0,0,0,0,0,0,0,0,1,0,3],[32,73,0.4384,0.44197,0.34508,0.14286,0.42857,0.5,0.0,1.0,6,7,0,6,0,4,0,0,1,0,0,13,0,0,0,0,0,1,0,0,0,0,7],[36,73,0.4932,0.36606,0.31326,0.0,0.42857,0.42858,0.0,1.0,10,4,0,10,0,0,0,0,3,0,0,12,0,0,3,0,0,0,0,0,0,0,4],[40,73,0.5479,0.41518,0.36832,0.0,0.42857,0.60714,0.0,1.0,9,7,0,9,0,3,0,0,2,0,0,8,0,0,2,0,0,1,0,0,0,0,7],[44,73,0.6027,0.26339,0.24251,0.0,0.35714,0.42857,0.0,1.0,12,1,0,12,0,2,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,1],[48,73,0.6575,0.44643,0.36553,0.10714,0.42857,0.67857,0.0,1.0,8,8,0,8,0,1,0,0,4,0,0,9,0,0,2,0,0,0,0,0,0,0,8],[52,73,0.7123,0.23214,0.27606,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,1,0,0,4,0,0,6,0,0,3,0,0,0,0,0,1,0,1],[56,73,0.7671,0.28125,0.2004,0.0,0.42857,0.42857,0.0,0.57143,9,0,0,9,0,3,0,0,2,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[60,73,0.8219,0.22768,0.21086,0.0,0.21428,0.42857,0.0,0.71429,12,0,0,12,0,4,0,0,4,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[64,73,0.8767,0.16964,0.19377,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,4,0,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[68,73,0.9315,0.14284,0.22013,0.0,0.0,0.32142,0.0,0.71429,21,0,0,21,0,2,0,0,1,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[72,73,0.9863,0.08482,0.24964,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[73,73,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9831c23086ab2214","q":"Points  $D,E,F$  are midpoints of the sides  $AB,BC,CA$  of triangle  $ABC$ . Angle bisectors of the angles  $BDC$  and  $ADC$  intersect the lines  $BC$  and  $AC$  respectively at the points  $M$  and  $N$ , and the line  $MN$  intersects the line  $CD$  at the point  $O$ . Let the lines  $EO$  and  $FO$  intersect respectively the lines  $AC$  and  $BC$  at the points  $P$  and  $Q$ . Prove that  $CD=PQ$ . *(Plamen Koshlukov)*","t":[{"b":3,"e":0.42857,"k":"flat","v":0.55804,"x":0.70982,"p":[[0,92,0.0,0.65179,0.24468,0.42859,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,6,0,0,6,0,0,8,0,0,3,0,6],[4,92,0.0435,0.70982,0.17672,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,7,0,0,6,0,5],[8,92,0.087,0.58034,0.19541,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,8,0,0,9,0,0,6,0,0,3,0,2],[12,92,0.1304,0.67855,0.20826,0.42859,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,7,0,0,4,0,6],[16,92,0.1739,0.69643,0.17768,0.57143,0.64286,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,6,0,0,5,0,5],[20,92,0.2174,0.61161,0.1931,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,9,0,0,2,0,3],[24,92,0.2609,0.64284,0.21129,0.53539,0.57143,0.85704,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,12,0,0,3,0,0,4,0,5],[28,92,0.3043,0.57588,0.20356,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,8,0,0,8,0,0,9,0,0,0,0,3],[32,92,0.3478,0.60713,0.21429,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,12,0,0,7,0,0,5,0,0,3,0,4],[36,92,0.3913,0.61607,0.20341,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,9,0,0,3,0,3],[40,92,0.4348,0.63837,0.19881,0.42857,0.64286,0.75,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,10,0,0,5,0,0,8,0,0,5,0,3],[44,92,0.4783,0.65177,0.21998,0.53539,0.64286,0.71429,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,6,0,0,8,0,0,9,0,0,1,0,6],[48,92,0.5217,0.6607,0.22233,0.53539,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,7,0,0,10,0,0,4,0,0,4,0,6],[52,92,0.5652,0.67411,0.21199,0.4286,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,7,0,0,4,0,6],[56,92,0.6087,0.64728,0.23687,0.42857,0.57143,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,9,0,0,7,0,0,6,0,0,3,0,6],[60,92,0.6522,0.64284,0.22016,0.42857,0.64286,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,8,0,0,3,0,5],[64,92,0.6957,0.625,0.17404,0.4286,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,5,0,0,5,0,2],[68,92,0.7391,0.56696,0.23278,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,8,0,0,6,0,0,5,0,0,5,0,2],[72,92,0.7826,0.625,0.21053,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,10,0,0,5,0,0,8,0,0,3,0,4],[76,92,0.8261,0.66963,0.22429,0.57132,0.71429,0.75,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,0,3,0,5],[80,92,0.8696,0.66515,0.2161,0.42857,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,5,0,0,8,0,4],[84,92,0.913,0.60713,0.22868,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,9,0,0,10,0,0,4,0,0,2,0,5],[88,92,0.9565,0.58929,0.21943,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,10,0,0,8,0,0,5,0,0,2,0,4],[92,92,1.0,0.55804,0.14445,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,10,0,0,11,0,0,7,0,0,2,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.45089,"x":0.67409,"p":[[0,187,0.0,0.67409,0.19961,0.57143,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,3,0,0,12,0,0,4,0,0,7,0,4],[4,187,0.0214,0.62052,0.21609,0.4286,0.57143,0.64286,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,7,0,0,15,0,0,0,0,0,2,0,6],[8,187,0.0428,0.59821,0.16917,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,13,0,0,7,0,0,0,0,3],[12,187,0.0642,0.5625,0.18189,0.42857,0.57143,0.60714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,12,0,0,10,0,0,5,0,0,0,0,3],[16,187,0.0856,0.53124,0.15663,0.42857,0.4998,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,14,0,0,11,0,0,2,0,0,2,0,1],[20,187,0.107,0.56249,0.17104,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,11,0,0,7,0,0,7,0,0,4,0,0],[24,187,0.1283,0.5803,0.17474,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,7,0,0,14,0,0,6,0,0,1,0,2],[28,187,0.1497,0.5223,0.23037,0.42857,0.4998,0.60714,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,9,0,0,8,0,0,3,0,0,3,0,2],[32,187,0.1711,0.61607,0.20958,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,6,0,0,10,0,0,6,0,0,4,0,3],[36,187,0.1925,0.54907,0.14772,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,8,0,0,15,0,0,3,0,0,3,0,0],[40,187,0.2139,0.53122,0.16065,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,9,0,0,13,0,0,6,0,0,1,0,0],[44,187,0.2353,0.50892,0.12845,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,14,0,0,11,0,0,5,0,0,0,0,0],[48,187,0.2567,0.53572,0.15972,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,12,0,0,12,0,0,4,0,0,1,0,1],[52,187,0.2781,0.58036,0.19212,0.42857,0.57143,0.60714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,8,0,0,13,0,0,3,0,0,2,0,3],[56,187,0.2995,0.54015,0.19474,0.42857,0.57121,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,10,0,0,10,0,0,4,0,0,3,0,1],[60,187,0.3209,0.62499,0.1915,0.5354,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,14,0,0,3,0,0,3,0,4],[64,187,0.3422,0.49997,0.1821,0.39286,0.42859,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,9,0,0,7,0,0,6,0,0,2,0,0],[68,187,0.3636,0.50893,0.21998,0.42857,0.57143,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,2,0,0,9,0,0,13,0,0,2,0,0,1,0,2],[72,187,0.385,0.5089,0.14257,0.42857,0.49979,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,13,0,0,11,0,0,4,0,0,1,0,0],[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real numbers  $x,y$  satisfy   $$ \\left \\lfloor xy \\right \\rfloor - \\lfloor x \\rfloor \\lfloor y \\rfloor = 8. $$   Find the sum of all possible values of the quantity  $\\left \\lfloor 2xy \\right \\rfloor - \\lfloor 2x \\rfloor \\lfloor y 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634,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[84,142,0.5915,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[88,142,0.6197,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[92,142,0.6479,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[96,142,0.6761,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[100,142,0.7042,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[104,142,0.7324,0.93303,0.11285,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,20],[108,142,0.7606,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[112,142,0.7887,0.90625,0.15407,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,16],[116,142,0.8169,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[120,142,0.8451,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[124,142,0.8732,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[128,142,0.9014,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[132,142,0.9296,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[136,142,0.9577,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[140,142,0.9859,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[142,142,1.0,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16]]}]},{"i":"13eac012a9783e78","q":"Prove that for all non-negative real numbers $x, y, z$, not all equal to 0 , the following inequality holds\n\n$$\n\\frac{2 x^{2}-x+y+z}{x+y^{2}+z^{2}}+\\frac{2 y^{2}+x-y+z}{x^{2}+y+z^{2}}+\\frac{2 z^{2}+x+y-z}{x^{2}+y^{2}+z} \\geqslant 3\n$$\n\nDetermine all the triples $(x, y, z)$ for which the equality holds.","t":[{"b":5,"e":0.14286,"k":"falling","v":0.10268,"x":0.62946,"p":[[0,61,0.0,0.47768,0.24383,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,4,0,0,11,0,0,3,0,0,6,0,0,1,0,2],[4,61,0.0656,0.62946,0.2854,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,9,0,0,0,0,0,6,0,0,3,0,8],[8,61,0.1311,0.52231,0.23853,0.42857,0.4286,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,4,0,0,10,0,0,4,0,0,7,0,0,2,0,2],[12,61,0.1967,0.55803,0.23244,0.42857,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,12,0,0,3,0,0,5,0,0,5,0,2],[16,61,0.2623,0.54018,0.22794,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,4,0,0,8,0,0,4,0,0,10,0,0,1,0,2],[20,61,0.3279,0.50446,0.28568,0.28571,0.42857,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,3,0,0,8,0,0,2,0,0,6,0,0,3,0,3],[24,61,0.3934,0.46429,0.30513,0.14296,0.42857,0.71429,0.0,1.0,1,5,0,1,0,8,0,0,4,0,0,8,0,0,2,0,0,3,0,0,1,0,5],[28,61,0.459,0.45081,0.20564,0.28571,0.42857,0.60714,0.14,0.85714,0,0,0,0,0,5,0,0,6,0,0,9,0,0,4,0,0,7,0,0,1,0,0],[32,61,0.5246,0.48214,0.26666,0.28571,0.42859,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,5,0,0,5,0,0,6,0,0,5,0,0,1,0,3],[36,61,0.5902,0.35259,0.26729,0.14286,0.21429,0.46429,0.0,1.0,1,2,0,1,0,15,0,0,2,0,0,6,0,0,2,0,0,4,0,0,0,0,2],[40,61,0.6557,0.34821,0.23941,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,11,0,0,5,0,0,7,0,0,0,0,0,6,0,0,1,0,0],[44,61,0.7213,0.25,0.25754,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,18,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[48,61,0.7869,0.26768,0.20449,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,18,0,0,3,0,0,2,0,0,5,0,0,2,0,0,0,0,0],[52,61,0.8525,0.25893,0.23266,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,18,0,0,2,0,0,6,0,0,0,0,0,1,0,0,1,0,1],[56,61,0.918,0.13384,0.10061,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,24,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,61,0.9836,0.12946,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.15616,"x":0.51338,"p":[[0,74,0.0,0.49106,0.26949,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,4,0,0,10,0,0,4,0,0,4,0,0,2,0,3],[4,74,0.0541,0.51338,0.25469,0.42857,0.4286,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,2,0,0,10,0,0,6,0,0,3,0,0,4,0,2],[8,74,0.1081,0.41517,0.26331,0.14286,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,13,0,0,0,0,0,8,0,0,3,0,0,5,0,0,2,0,1],[12,74,0.1622,0.39732,0.22794,0.14286,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,9,0,0,3,0,0,15,0,0,0,0,0,3,0,0,0,0,2],[16,74,0.2162,0.30795,0.23456,0.14286,0.21429,0.42857,0.0,1.0,2,1,0,2,0,14,0,0,5,0,0,5,0,0,2,0,0,3,0,0,0,0,1],[20,74,0.2703,0.41517,0.31003,0.14286,0.35714,0.60714,0.0,1.0,2,3,0,2,0,10,0,0,4,0,0,7,0,0,1,0,0,1,0,0,4,0,3],[24,74,0.3243,0.40625,0.2969,0.14286,0.42857,0.71429,0.0,1.0,1,3,0,1,0,13,0,0,1,0,0,8,0,0,0,0,0,5,0,0,1,0,3],[28,74,0.3784,0.39731,0.19474,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,5,0,0,15,0,0,3,0,0,2,0,0,0,0,1],[32,74,0.4324,0.34375,0.20782,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,5,0,0,10,0,0,2,0,0,2,0,0,0,0,1],[36,74,0.4865,0.30804,0.21756,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,17,0,0,2,0,0,9,0,0,2,0,0,0,0,0,1,0,1],[40,74,0.5405,0.29465,0.20805,0.14286,0.21435,0.42857,0.0,0.85714,1,0,0,1,0,15,0,0,5,0,0,8,0,0,0,0,0,1,0,0,2,0,0],[44,74,0.5946,0.25893,0.1729,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,17,0,0,1,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[48,74,0.6486,0.27232,0.20628,0.14286,0.14286,0.32143,0.0,1.0,1,1,0,1,0,17,0,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[52,74,0.7027,0.25447,0.21349,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,19,0,0,1,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[56,74,0.7568,0.28562,0.26005,0.14286,0.14286,0.32143,0.0,1.0,1,1,0,1,0,21,0,0,2,0,0,2,0,0,1,0,0,2,0,0,2,0,1],[60,74,0.8108,0.23205,0.1741,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,19,0,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[64,74,0.8649,0.25,0.17857,0.14286,0.14286,0.32142,0.14286,0.71429,0,0,0,0,0,22,0,0,2,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[68,74,0.9189,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[72,74,0.973,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[74,74,1.0,0.15616,0.07459,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1a582c51a1639e0c","q":"Problem. Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold:\n(i) $a^{2021}$ divides $n$,\n(ii) $b^{2021}$ divides $n$,\n(iii) 2022 divides $a-b$.\n\nProve that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^{2}$.","t":[{"b":4,"e":0.43,"k":"rising","v":0.15625,"x":0.82589,"p":[[0,63,0.0,0.15625,0.14445,0.0,0.14286,0.2857,0.0,0.42857,12,0,4,12,0,8,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.81696,0.22654,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,0,10,0,13],[8,63,0.127,0.81695,0.19962,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,10,0,11],[12,63,0.1905,0.82589,0.23072,0.82132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,10,0,14],[16,63,0.254,0.80803,0.28707,0.71429,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,16],[20,63,0.3175,0.70089,0.27515,0.42859,0.78564,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,4,0,0,8,0,0,2,0,0,2,0,0,5,0,11],[24,63,0.381,0.76338,0.2826,0.5354,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,3,0,0,4,0,0,1,0,0,2,0,0,8,0,13],[28,63,0.4444,0.71425,0.28795,0.42857,0.85707,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,4,0,0,3,0,0,2,0,0,3,0,0,8,0,10],[32,63,0.5079,0.77232,0.24448,0.53572,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,3,0,0,8,0,12],[36,63,0.5714,0.79017,0.2082,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,7,0,0,9,0,10],[40,63,0.6349,0.72767,0.24578,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,7,0,8],[44,63,0.6984,0.70535,0.2878,0.42859,0.78564,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,6,0,0,2,0,0,3,0,0,4,0,0,5,0,11],[48,63,0.7619,0.70982,0.2461,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,7,0,0,1,0,0,6,0,0,7,0,8],[52,63,0.8254,0.70089,0.25595,0.42857,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,8,0,0,3,0,0,1,0,0,9,0,8],[56,63,0.8889,0.62499,0.25692,0.42857,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,4,0,0,9,0,0,2,0,0,5,0,0,6,0,5],[60,63,0.9524,0.58481,0.23244,0.42857,0.5712,0.75,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,12,0,0,5,0,0,4,0,0,5,0,3],[63,63,1.0,0.50891,0.25236,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,13,0,0,5,0,0,0,0,0,4,0,3]]},{"b":5,"e":0.0,"k":"falling","v":0.01786,"x":0.67409,"p":[[0,40,0.0,0.2232,0.22284,0.0,0.14286,0.32143,0.0,0.85714,9,0,3,9,0,11,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[4,40,0.1,0.67409,0.31387,0.42857,0.78571,1.0,0.0,1.0,2,10,0,2,0,1,0,0,2,0,0,6,0,0,3,0,0,2,0,0,6,0,10],[8,40,0.2,0.56696,0.29554,0.28593,0.42859,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,7,0,0,9,0,0,2,0,0,1,0,0,5,0,6],[12,40,0.3,0.50444,0.31335,0.28571,0.42857,0.85711,0.0,1.0,4,3,0,4,0,1,0,0,6,0,0,8,0,0,2,0,0,1,0,0,7,0,3],[16,40,0.4,0.47768,0.32264,0.2857,0.42857,0.75,0.0,1.0,5,4,0,5,0,2,0,0,4,0,0,9,0,0,2,0,0,2,0,0,4,0,4],[20,40,0.5,0.39286,0.33312,0.10714,0.28571,0.71429,0.0,1.0,8,4,0,8,0,2,0,0,7,0,0,6,0,0,0,0,0,4,0,0,1,0,4],[24,40,0.6,0.51785,0.34022,0.2857,0.49999,0.85714,0.0,1.0,4,5,0,4,0,3,0,0,6,0,0,3,0,0,3,0,0,3,0,0,5,0,5],[28,40,0.7,0.30356,0.29611,0.14286,0.2143,0.42857,0.0,1.0,7,3,0,7,0,9,0,0,5,0,0,6,0,0,1,0,0,0,0,0,1,0,3],[32,40,0.8,0.23213,0.34021,0.0,0.0,0.42857,0.0,1.0,18,3,0,18,0,4,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,3],[36,40,0.9,0.30357,0.31083,0.10714,0.14286,0.42857,0.0,1.0,8,2,0,8,0,9,0,0,5,0,0,4,0,0,0,0,0,1,0,0,3,0,2],[40,40,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"50c58f52588298b2","q":"Prove that for any positive integer $k$, there exists an arithmetic sequence\n\n$$\n\\frac{a_{1}}{b_{1}}, \\quad \\frac{a_{2}}{b_{2}}, \\ldots, \\quad \\frac{a_{k}}{b_{k}}\n$$\n\nof rational numbers, where $a_{i}, b_{i}$ are relatively prime positive integers for each $i=1,2, \\ldots, k$, such that the positive integers $a_{1}, b_{1}, a_{2}, b_{2}, \\ldots, a_{k}, b_{k}$ are all distinct.","t":[{"b":0,"e":1.0,"k":"falling","v":0.35265,"x":0.99554,"p":[[0,65,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,65,0.0615,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,65,0.1231,0.91058,0.18828,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,2,0,24],[12,65,0.1846,0.86606,0.2788,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,23],[16,65,0.2462,0.85268,0.29984,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,23],[20,65,0.3077,0.92857,0.18557,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[24,65,0.3692,0.93749,0.11812,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[28,65,0.4308,0.92857,0.21129,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[32,65,0.4923,0.82589,0.25688,0.71429,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,4,0,0,1,0,20],[36,65,0.5538,0.83034,0.23809,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,2,0,19],[40,65,0.6154,0.79464,0.27879,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,4,0,17],[44,65,0.6769,0.8482,0.31932,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,25],[48,65,0.7385,0.81696,0.30563,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,20],[52,65,0.8,0.4732,0.37361,0.10717,0.42859,0.85714,0.0,1.0,8,7,0,8,0,1,0,0,5,0,0,4,0,0,3,0,0,2,0,0,2,0,7],[56,65,0.8615,0.49979,0.36087,0.24999,0.42857,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,6,0,0,4,0,0,2,0,0,2,0,0,3,0,7],[60,65,0.9231,0.43298,0.29981,0.14289,0.4286,0.71429,0.0,1.0,6,2,0,6,0,4,0,0,1,0,0,7,0,0,5,0,0,6,0,0,1,0,2],[64,65,0.9846,0.51775,0.33465,0.2857,0.42857,0.85714,0.0,1.0,4,5,0,4,0,2,0,0,6,0,0,6,0,0,1,0,0,3,0,0,5,0,5],[65,65,1.0,0.35265,0.25748,0.14286,0.28571,0.571,0.0,1.0,5,1,0,5,0,5,0,0,9,0,0,4,0,0,3,0,0,5,0,0,0,0,1]]},{"b":6,"e":0.2857,"k":"flat","v":0.85267,"x":0.99554,"p":[[0,58,0.0,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,58,0.069,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[8,58,0.1379,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,58,0.2069,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,58,0.2759,0.9464,0.11718,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[20,58,0.3448,0.97326,0.10949,1.0,1.0,1.0,0.43,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[24,58,0.4138,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[28,58,0.4828,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,58,0.5517,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[36,58,0.6207,0.95982,0.1439,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[40,58,0.6897,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,58,0.7586,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[48,58,0.8276,0.93304,0.17852,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[52,58,0.8966,0.89719,0.18306,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,4,0,22],[56,58,0.9655,0.85267,0.23822,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,0,1,0,22],[58,58,1.0,0.89732,0.22934,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,26]]}]},{"i":"1e7bc2f7b5310faa","q":"Show that any number in row $n$ (for $n>0$ ) is at most $2^{n-1}$.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.23661,"x":0.95089,"p":[[0,6,0.0,0.95089,0.19103,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,6,0.6667,0.53571,0.46839,0.0,0.64286,1.0,0.0,1.0,9,16,0,9,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[6,6,1.0,0.23661,0.3722,0.0,0.14286,0.14286,0.0,1.0,15,6,0,15,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6]]},{"b":7,"e":1.0,"k":"volatile","v":0.46429,"x":0.97768,"p":[[0,8,0.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[4,8,0.5,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,8,1.0,0.46429,0.45457,0.0,0.14286,1.0,0.0,1.0,10,13,0,10,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,13]]}]},{"i":"bf1356530828cd33","q":"Prove the following assertion: The four altitudes of a tetrahedron  $ABCD$  intersect in a point if and only if\n\\[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\\]","t":[{"b":3,"e":1.0,"k":"flat","v":0.70089,"x":0.95981,"p":[[0,65,0.0,0.80343,0.18477,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,14,0,8],[4,65,0.0615,0.83929,0.22798,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,7,0,0,6,0,16],[8,65,0.1231,0.78124,0.30091,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,16],[12,65,0.1846,0.75893,0.29545,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,6,0,0,8,0,12],[16,65,0.2462,0.70089,0.32802,0.42857,0.85714,1.0,0.0,1.0,2,11,0,2,0,3,0,0,1,0,0,3,0,0,0,0,0,6,0,0,6,0,11],[20,65,0.3077,0.91518,0.1885,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,25],[24,65,0.3692,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[28,65,0.4308,0.95981,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[32,65,0.4923,0.87053,0.20628,0.857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,6,0,19],[36,65,0.5538,0.91518,0.15093,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,21],[40,65,0.6154,0.86608,0.17467,0.85711,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,9,0,16],[44,65,0.6769,0.90624,0.1499,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,20],[48,65,0.7385,0.93303,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[52,65,0.8,0.81696,0.20589,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,5,0,0,10,0,12],[56,65,0.8615,0.87499,0.15874,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,11,0,15],[60,65,0.9231,0.87945,0.16411,0.85711,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,8,0,17],[64,65,0.9846,0.86161,0.16935,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,11,0,14],[65,65,1.0,0.88839,0.15458,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,18]]},{"b":5,"e":0.4286,"k":"flat","v":0.61157,"x":0.8616,"p":[[0,100,0.0,0.68293,0.27851,0.42857,0.71429,1.0,0.14,1.0,0,10,0,0,0,2,0,0,2,0,0,7,0,0,2,0,0,6,0,0,3,0,10],[4,100,0.04,0.70981,0.33784,0.67836,0.85714,1.0,0.0,1.0,3,12,0,3,0,2,0,0,2,0,0,0,0,0,1,0,0,7,0,0,5,0,12],[8,100,0.08,0.74105,0.30186,0.57143,0.85707,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,13],[12,100,0.12,0.71862,0.31438,0.71321,0.85714,1.0,0.0,1.0,2,10,0,2,0,3,0,0,1,0,0,0,0,0,1,0,0,8,0,0,7,0,10],[16,100,0.16,0.79017,0.27195,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,3,0,0,6,0,15],[20,100,0.2,0.61157,0.33166,0.39286,0.71429,1.0,0.0,1.0,2,9,0,2,0,4,0,0,2,0,0,4,0,0,3,0,0,6,0,0,2,0,9],[24,100,0.24,0.64285,0.34069,0.28571,0.71429,0.89286,0.0,1.0,2,8,0,2,0,5,0,0,2,0,0,1,0,0,0,0,0,8,0,0,6,0,8],[28,100,0.28,0.7723,0.30064,0.67857,0.85714,1.0,0.0,1.0,2,15,0,2,0,1,0,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,15],[32,100,0.32,0.7723,0.27167,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,2,0,0,1,0,0,0,0,0,2,0,0,6,0,0,9,0,11],[36,100,0.36,0.7187,0.31032,0.571,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,2,0,0,5,0,0,3,0,0,4,0,13],[40,100,0.4,0.74998,0.24745,0.57132,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,6,0,0,7,0,10],[44,100,0.44,0.7366,0.27689,0.57142,0.85707,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,6,0,0,6,0,11],[48,100,0.48,0.85714,0.22588,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,7,0,18],[52,100,0.52,0.63391,0.35344,0.25,0.71429,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,3,0,0,0,0,0,3,0,0,3,0,0,4,0,11],[56,100,0.56,0.69191,0.30953,0.42857,0.857,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,2,0,0,4,0,0,3,0,0,2,0,0,6,0,11],[60,100,0.6,0.82134,0.25529,0.82143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,0,9,0,15],[64,100,0.64,0.71414,0.30723,0.42857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,2,0,0,3,0,0,2,0,0,2,0,0,8,0,11],[68,100,0.68,0.75446,0.30771,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,0,0,0,2,0,0,1,0,0,4,0,0,6,0,14],[72,100,0.72,0.73652,0.26772,0.57143,0.78571,1.0,0.14,1.0,0,11,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,7,0,0,5,0,11],[76,100,0.76,0.82597,0.23621,0.71429,0.93,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,4,0,0,6,0,16],[80,100,0.8,0.77231,0.29203,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,6,0,0,4,0,15],[84,100,0.84,0.8616,0.20973,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,10,0,16],[88,100,0.88,0.84373,0.25346,0.85711,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,7,0,18],[92,100,0.92,0.79008,0.29031,0.82132,0.85714,1.0,0.14,1.0,0,15,0,0,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0,9,0,15],[96,100,0.96,0.80803,0.249,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,5,0,0,6,0,15],[100,100,1.0,0.76784,0.26183,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,5,0,0,14,0,8]]}]},{"i":"24ee9616c3d4e0de","q":"Petya and Vasya are given equal sets of  $ N$  weights, in which the masses of any two weights are in ratio at most  $ 1.25$ . Petya succeeded to divide his set into  $ 10$  groups of equal masses, while Vasya succeeded to divide his set into  $ 11$  groups of equal masses. Find the smallest possible  $ N$ 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0,1,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,12,0,0,2,0,1],[220,227,0.9692,0.55357,0.16269,0.42857,0.42857,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,17,0,0,2,0,0,9,0,0,3,0,0],[224,227,0.9868,0.54451,0.15321,0.42857,0.42859,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,15,0,0,3,0,0,11,0,0,1,0,0],[227,227,1.0,0.59375,0.16793,0.42857,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,13,0,0,2,0,1]]}]},{"i":"641cc418a3ba7b40","q":"Quadrangle  $ABCD$  is inscribed in a circle with diameter  $AC$ . Points  $K$  and  $M$  are projections of vertices  $A$  and  $C$ , respectively, onto line  $BD$ . A line parallel to  $BC$  is drawn through point  $K$  and intersecting  $AC$  at point  $P$ . Prove that angle  $KPM$  is a right angle.","t":[{"b":4,"e":0.0,"k":"flat","v":0.11607,"x":0.16965,"p":[[0,103,0.0,0.12947,0.08268,0.14286,0.14286,0.14287,0.0,0.42857,6,0,0,6,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,103,0.0388,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,103,0.0777,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,103,0.1165,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,103,0.1553,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,103,0.1942,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,103,0.233,0.13822,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,103,0.2718,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,103,0.3107,0.14277,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,103,0.3495,0.1517,0.07937,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,103,0.3883,0.13385,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,103,0.4272,0.13394,0.04971,0.14286,0.14286,0.1429,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,103,0.466,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,103,0.5049,0.12938,0.05484,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,103,0.5437,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,103,0.5825,0.1384,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,103,0.6214,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,103,0.6602,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,103,0.699,0.16491,0.15616,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,103,0.7379,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,103,0.7767,0.14286,0.16366,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[84,103,0.8155,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,103,0.8544,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,103,0.8932,0.13394,0.03458,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,103,0.932,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,103,0.9709,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[103,103,1.0,0.16965,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"flat","v":0.12054,"x":0.17848,"p":[[0,43,0.0,0.17848,0.14728,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[4,43,0.093,0.1696,0.08345,0.14286,0.14286,0.1429,0.0,0.43,2,0,0,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.15161,0.04975,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.17411,0.1504,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,43,0.4651,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.15161,0.07089,0.14286,0.14286,0.14287,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.16956,0.14034,0.14286,0.14286,0.14287,0.0,0.85714,3,0,0,3,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[32,43,0.7442,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.16964,0.06621,0.14286,0.14286,0.14287,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.15616,0.07458,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.16518,0.06297,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"61c3ccb899853c11","q":"Show that\n\\[\\left(a+2b+\\dfrac{2}{a+1}\\right)\\left(b+2a+\\dfrac{2}{b+1}\\right)\\geq 16\\]\nfor all positive real numbers  $a$  and  $b$  such  that  $ab\\geq 1$ 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,0,0,0,2],[8,78,0.1026,0.16516,0.18592,0.0,0.07143,0.28571,0.0,0.571,16,0,0,16,0,3,0,0,6,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[12,78,0.1538,0.24554,0.32778,0.0,0.14286,0.42858,0.0,1.0,15,3,0,15,0,6,0,0,2,0,0,3,0,0,0,0,0,3,0,0,0,0,3],[16,78,0.2051,0.18741,0.1692,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,11,0,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[20,78,0.2564,0.18304,0.17582,0.0,0.14286,0.42857,0.0,0.4286,12,0,0,12,0,8,0,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[24,78,0.3077,0.19643,0.24419,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,5,0,0,8,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[28,78,0.359,0.28572,0.34626,0.0,0.14286,0.42857,0.0,1.0,13,5,0,13,0,5,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,5],[32,78,0.4103,0.15179,0.2111,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,4,0,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[36,78,0.4615,0.16072,0.1948,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,5,0,0,4,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[40,78,0.5128,0.12947,0.21535,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,6,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[44,78,0.5641,0.10714,0.12877,0.0,0.0,0.17857,0.0,0.42857,17,0,0,17,0,7,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,78,0.6154,0.1875,0.29329,0.0,0.0,0.32143,0.0,1.0,19,2,0,19,0,3,0,0,2,0,0,5,0,0,0,0,0,0,0,0,1,0,2],[52,78,0.6667,0.22321,0.24468,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,6,0,0,3,0,0,6,0,0,0,0,0,4,0,0,0,0,0],[56,78,0.7179,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,78,0.7692,0.10714,0.19885,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,78,0.8205,0.08929,0.15872,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,3,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[68,78,0.8718,0.09822,0.17655,0.0,0.0,0.14287,0.0,0.71429,22,0,0,22,0,3,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[72,78,0.9231,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,78,0.9744,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[78,78,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"237ee485fee2a4ab","q":"Show that $r=2$ is the largest real number $r$ which satisfies the following condition:\n\nIf a sequence $a_{1}, a_{2}, \\ldots$ of positive integers fulfills the inequalities\n\n$$\na_{n} \\leq a_{n+2} \\leq \\sqrt{a_{n}^{2}+r a_{n+1}}\n$$\n\nfor every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2}=a_{n}$ for every $n \\geq M$.","t":[{"b":1,"e":0.571,"k":"flat","v":0.54459,"x":0.87946,"p":[[0,88,0.0,0.57142,0.2857,0.28571,0.4286,0.85714,0.143,1.0,0,6,0,0,0,1,0,0,10,0,0,6,0,0,2,0,0,3,0,0,4,0,6],[4,88,0.0455,0.80353,0.22521,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,2,0,16],[8,88,0.0909,0.79017,0.23955,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,8,0,0,2,0,15],[12,88,0.1364,0.875,0.20748,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,0,2,0,21],[16,88,0.1818,0.75,0.23958,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,10,0,0,6,0,9],[20,88,0.2273,0.79016,0.19882,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,11,0,0,3,0,12],[24,88,0.2727,0.80357,0.19804,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,6,0,12],[28,88,0.3182,0.78113,0.22327,0.71429,0.78571,1.0,0.14,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,9,0,0,4,0,12],[32,88,0.3636,0.82589,0.16263,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,4,0,13],[36,88,0.4091,0.77215,0.20789,0.67536,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,9,0,0,4,0,11],[40,88,0.4545,0.87945,0.14337,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,18],[44,88,0.5,0.87946,0.17536,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,4,0,19],[48,88,0.5455,0.68747,0.2354,0.571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,6,0,0,5,0,7],[52,88,0.5909,0.7991,0.18161,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,3,0,12],[56,88,0.6364,0.74329,0.20194,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,1,1,0,0,8,0,0,7,0,0,6,0,8],[60,88,0.6818,0.63392,0.2257,0.5354,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,4,0,0,6,0,0,12,0,0,2,0,4],[64,88,0.7273,0.70981,0.23552,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,1,0,0,5,0,0,11,0,0,3,0,8],[68,88,0.7727,0.77677,0.19213,0.71429,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,8,0,7],[72,88,0.8182,0.66069,0.23892,0.4286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,8,0,0,2,0,7],[76,88,0.8636,0.66963,0.14481,0.67857,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,21,0,0,2,0,1],[80,88,0.9091,0.65621,0.167,0.57132,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,17,0,0,0,0,3],[84,88,0.9545,0.59803,0.14904,0.5354,0.64071,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,5,0,0,8,0,0,15,0,0,1,0,0],[88,88,1.0,0.54459,0.18707,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,10,0,0,8,0,0,9,0,0,2,0,0]]},{"b":2,"e":0.571,"k":"rising","v":0.51344,"x":0.98214,"p":[[0,91,0.0,0.51344,0.29849,0.28571,0.4293,0.71429,0.0,1.0,3,4,2,3,0,1,0,0,8,0,0,5,0,0,2,0,0,7,0,0,2,0,4],[4,91,0.044,0.83482,0.22619,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,3,0,18],[8,91,0.0879,0.89731,0.13478,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,19],[12,91,0.1319,0.84375,0.21237,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,5,0,17],[16,91,0.1758,0.83034,0.20343,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,3,0,16],[20,91,0.2198,0.7857,0.19563,0.71429,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,14,0,0,1,0,12],[24,91,0.2637,0.79018,0.22299,0.71429,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,9,0,0,2,0,14],[28,91,0.3077,0.75893,0.24338,0.67857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,10,0,0,1,0,13],[32,91,0.3516,0.75892,0.23267,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,8,0,0,3,0,12],[36,91,0.3956,0.82587,0.21052,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,1,0,17],[40,91,0.4396,0.78125,0.17852,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,14,0,0,1,0,11],[44,91,0.4835,0.79463,0.15949,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,4,0,10],[48,91,0.5275,0.86161,0.13592,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,6,0,14],[52,91,0.5714,0.79909,0.20783,0.71429,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,9,0,0,2,0,14],[56,91,0.6154,0.7656,0.19255,0.71421,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,1,1,0,0,4,0,0,10,0,0,7,0,8],[60,91,0.6593,0.7857,0.22305,0.71429,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,10,0,0,1,0,14],[64,91,0.7033,0.77228,0.19844,0.71429,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,11,0,0,4,0,10],[68,91,0.7473,0.72311,0.22281,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,9,0,0,3,0,9],[72,91,0.7912,0.91071,0.15047,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,22],[76,91,0.8352,0.90401,0.18002,0.85714,1.0,1.0,0.357,1.0,0,23,0,0,0,0,0,0,0,0,1,1,0,0,2,0,0,2,0,0,3,0,23],[80,91,0.8791,0.93749,0.16345,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[84,91,0.9231,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[88,91,0.967,0.87051,0.19356,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,19],[91,91,1.0,0.91515,0.15516,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23]]}]},{"i":"3b1b385382df6de8","q":"Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.\r\n\r**Note by Darij:** A *cyclic quadrilateral*is a quadrilateral inscribed in a circle.","t":[{"b":4,"e":0.85714,"k":"flat","v":0.77231,"x":0.99554,"p":[[0,55,0.0,0.86159,0.28681,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[4,55,0.0727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,55,0.1455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,55,0.2182,0.92857,0.16752,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[16,55,0.2909,0.85267,0.25376,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,1,0,0,2,0,22],[20,55,0.3636,0.86161,0.25123,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,3,0,0,1,0,23],[24,55,0.4364,0.89728,0.18645,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,24],[28,55,0.5091,0.8482,0.22852,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,0,3,0,20],[32,55,0.5818,0.88393,0.25111,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[36,55,0.6545,0.92855,0.15975,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[40,55,0.7273,0.90625,0.16602,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,21],[44,55,0.8,0.81247,0.20028,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,4,0,0,7,0,13],[48,55,0.8727,0.84375,0.2,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,5,0,17],[52,55,0.9455,0.80802,0.23039,0.67857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,0,3,0,16],[55,55,1.0,0.77231,0.22265,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,6,0,0,6,0,11]]},{"b":5,"e":0.28571,"k":"falling","v":0.70982,"x":0.92856,"p":[[0,44,0.0,0.91071,0.24157,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[4,44,0.0909,0.91518,0.24964,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[8,44,0.1818,0.90177,0.18366,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,24],[12,44,0.2727,0.92856,0.15571,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[16,44,0.3636,0.8616,0.24868,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,20],[20,44,0.4545,0.83035,0.26592,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,3,0,0,0,0,0,2,0,0,2,0,0,6,0,18],[24,44,0.5455,0.90625,0.16982,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,7,0,21],[28,44,0.6364,0.86161,0.23003,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,0,3,0,21],[32,44,0.7273,0.85266,0.22726,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,0,5,0,19],[36,44,0.8182,0.74999,0.30094,0.5354,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,4,0,0,3,0,0,4,0,0,1,0,0,3,0,16],[40,44,0.9091,0.78584,0.26701,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,0,0,0,4,0,0,1,0,0,8,0,14],[44,44,1.0,0.70982,0.2461,0.57143,0.78571,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,4,0,0,9,0,7]]}]},{"i":"dcda11739a78dfae","q":"Show that there exist infinitely many positive integers $n$ such that\n\n$$\n\\frac{4^{n}+2^{n}+1}{n^{2}+n+1}\n$$\n\nis an integer.","t":[{"b":3,"e":0.0,"k":"falling","v":0.01339,"x":0.558,"p":[[0,62,0.0,0.43292,0.23288,0.2857,0.42857,0.57111,0.0,1.0,1,2,0,1,0,6,0,0,3,0,0,12,0,0,5,0,0,3,0,0,0,0,2],[4,62,0.0645,0.54018,0.18808,0.42857,0.42857,0.71429,0.1429,1.0,0,2,0,0,0,1,0,0,1,0,0,16,0,0,5,0,0,6,0,0,1,0,2],[8,62,0.129,0.39283,0.27431,0.14286,0.42857,0.57143,0.0,1.0,7,1,0,7,0,2,0,0,3,0,0,9,0,0,5,0,0,4,0,0,1,0,1],[12,62,0.1935,0.558,0.22968,0.42857,0.4998,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,1,0,0,13,0,0,5,0,0,6,0,0,2,0,3],[16,62,0.2581,0.49106,0.28333,0.42857,0.42857,0.60714,0.0,1.0,4,4,0,4,0,1,0,0,2,0,0,12,0,0,5,0,0,3,0,0,1,0,4],[20,62,0.3226,0.38393,0.23266,0.2857,0.42857,0.42858,0.0,1.0,5,1,0,5,0,2,0,0,2,0,0,19,0,0,0,0,0,2,0,0,1,0,1],[24,62,0.3871,0.37945,0.2687,0.14286,0.42857,0.4642,0.0,1.0,5,2,0,5,0,4,0,0,6,0,0,9,0,0,2,0,0,4,0,0,0,0,2],[28,62,0.4516,0.40624,0.23719,0.2857,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,2,0,0,3,0,0,12,0,0,4,0,0,5,0,0,1,0,0],[32,62,0.5161,0.40179,0.27302,0.14289,0.42857,0.46431,0.0,1.0,4,2,0,4,0,5,0,0,4,0,0,11,0,0,1,0,0,4,0,0,1,0,2],[36,62,0.5806,0.22768,0.21683,0.0,0.2143,0.42857,0.0,0.85714,12,0,0,12,0,4,0,0,4,0,0,11,0,0,0,0,0,0,0,0,1,0,0],[40,62,0.6452,0.45534,0.26349,0.42857,0.42857,0.57111,0.0,1.0,5,2,0,5,0,0,0,0,1,0,0,17,0,0,2,0,0,3,0,0,2,0,2],[44,62,0.7097,0.3705,0.26207,0.14286,0.42857,0.571,0.0,1.0,6,1,0,6,0,5,0,0,2,0,0,9,0,0,5,0,0,4,0,0,0,0,1],[48,62,0.7742,0.34819,0.30499,0.0,0.42857,0.571,0.0,1.0,11,2,0,11,0,1,0,0,2,0,0,9,0,0,3,0,0,4,0,0,0,0,2],[52,62,0.8387,0.13839,0.21275,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[56,62,0.9032,0.10267,0.18635,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,4,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[60,62,0.9677,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.0892,"x":0.558,"p":[[0,67,0.0,0.33918,0.20445,0.14286,0.42857,0.42857,0.0,0.71429,4,0,1,4,0,6,0,0,4,0,0,13,0,0,2,0,0,3,0,0,0,0,0],[4,67,0.0597,0.49558,0.21123,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,1,0,0,20,0,0,3,0,0,3,0,0,0,0,3],[8,67,0.1194,0.558,0.19019,0.42857,0.4286,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,15,0,0,5,0,0,7,0,0,0,0,3],[12,67,0.1791,0.51785,0.20437,0.42857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,17,0,0,2,0,0,7,0,0,2,0,1],[16,67,0.2388,0.48665,0.17074,0.42857,0.42857,0.46536,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,22,0,0,1,0,0,6,0,0,0,0,1],[20,67,0.2985,0.52229,0.21311,0.42857,0.42857,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,17,0,0,2,0,0,7,0,0,1,0,2],[24,67,0.3582,0.55354,0.2829,0.42857,0.42857,0.74996,0.0,1.0,2,6,0,2,0,1,0,0,2,0,0,13,0,0,4,0,0,2,0,0,2,0,6],[28,67,0.4179,0.45087,0.25779,0.42857,0.42857,0.4642,0.0,1.0,4,2,0,4,0,1,0,0,2,0,0,17,0,0,1,0,0,3,0,0,2,0,2],[32,67,0.4776,0.3259,0.21498,0.14286,0.42857,0.4286,0.0,0.71429,6,0,0,6,0,5,0,0,3,0,0,12,0,0,4,0,0,2,0,0,0,0,0],[36,67,0.5373,0.2857,0.28569,0.0,0.35714,0.42857,0.0,1.0,13,1,0,13,0,2,0,0,1,0,0,11,0,0,1,0,0,2,0,0,1,0,1],[40,67,0.597,0.29911,0.27283,0.0,0.28571,0.42857,0.0,1.0,10,2,0,10,0,3,0,0,4,0,0,11,0,0,1,0,0,1,0,0,0,0,2],[44,67,0.6567,0.29017,0.27311,0.0,0.35714,0.42857,0.0,1.0,11,1,0,11,0,4,0,0,1,0,0,11,0,0,2,0,0,1,0,0,1,0,1],[48,67,0.7164,0.26782,0.28511,0.0,0.14288,0.42857,0.0,1.0,13,1,0,13,0,4,0,0,2,0,0,6,0,0,4,0,0,1,0,0,1,0,1],[52,67,0.7761,0.20534,0.22568,0.0,0.14286,0.42857,0.0,0.85714,14,0,0,14,0,5,0,0,2,0,0,9,0,0,1,0,0,0,0,0,1,0,0],[56,67,0.8358,0.14286,0.17857,0.0,0.0,0.32143,0.0,0.42857,17,0,0,17,0,6,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.11607,0.17655,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,5,0,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[64,67,0.9552,0.0892,0.13713,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6fbf6b8c8a9916eb","q":"Prove that\n\n$$\n(1+a b c)\\left(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\right) \\geq 3+a+b+c\n$$\n\nfor any real numbers $a, b, c \\geq 1$.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.13821,"x":0.51785,"p":[[0,37,0.0,0.37946,0.37899,0.14286,0.14286,0.89286,0.14286,1.0,0,8,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[4,37,0.1081,0.51785,0.39729,0.14286,0.35714,1.0,0.0,1.0,1,11,0,1,0,14,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,11],[8,37,0.2162,0.41964,0.39438,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,10],[12,37,0.3243,0.375,0.36202,0.14286,0.14286,0.71429,0.0,1.0,1,6,0,1,0,21,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,6],[16,37,0.4324,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,37,0.6486,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.14277,"x":0.39286,"p":[[0,31,0.0,0.39286,0.3677,0.14286,0.14286,0.75,0.14286,1.0,0,7,0,0,0,21,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,7],[4,31,0.129,0.25437,0.25939,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,24,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[8,31,0.2581,0.33036,0.27534,0.14286,0.14286,0.46429,0.14286,1.0,0,1,0,0,0,19,0,0,3,0,0,2,0,0,3,0,0,0,0,0,4,0,1],[12,31,0.3871,0.22321,0.24984,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[16,31,0.5161,0.24543,0.24806,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[20,31,0.6452,0.24554,0.25313,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[24,31,0.7742,0.18741,0.17657,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[28,31,0.9032,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.17411,0.1504,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"f4615a2ec4974b2e","q":"Positive numbers  $\\alpha \u000b,\\beta \f, x_1, x_2,\\ldots, x_n$  ( $n \\geq 1$ ) satisfy  $x_1+x_2+\\cdots+x_n = 1$ . Prove that\n\\[\\sum_{i=1}^{n} \\frac{x_i^3}{\\alpha x_i+\\beta x_{i+1}} \\geq \\frac{1}{n(\\alpha+\\beta)}.\\]**Note.**  $x_{n+1}=x_1$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,32,0.0,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,32,0.125,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,54,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,54,0.0741,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[8,54,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,54,0.2222,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,54,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,54,0.3704,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,54,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,54,0.5185,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,54,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,54,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,54,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,54,0.8148,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,54,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,54,0.963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[54,54,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"05b72b18b041b048","q":"Solve  $2a^2+3a-44=3p^n$     in positive integers where  $p$  is a prime.","t":[{"b":0,"e":0.0,"k":"flat","v":0.43741,"x":0.87051,"p":[[0,61,0.0,0.86606,0.2313,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,21],[4,61,0.0656,0.43741,0.37283,0.105,0.35714,0.85714,0.0,1.0,8,6,0,8,0,2,0,0,6,0,0,6,0,0,0,0,0,0,0,0,4,0,6],[8,61,0.1311,0.7098,0.20974,0.5354,0.78571,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,4,0,0,12,0,4],[12,61,0.1967,0.53572,0.3481,0.28571,0.50001,0.85714,0.0,1.0,5,5,0,5,0,0,0,0,8,0,0,3,0,0,3,0,0,0,0,0,8,0,5],[16,61,0.2623,0.71875,0.24868,0.57143,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,2,0,0,4,0,0,4,0,0,13,0,5],[20,61,0.3279,0.76336,0.22479,0.67857,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,12,0,7],[24,61,0.3934,0.76784,0.2594,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,12,0,10],[28,61,0.459,0.83036,0.17655,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,17,0,9],[32,61,0.5246,0.76338,0.24384,0.67857,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,15,0,7],[36,61,0.5902,0.84821,0.18189,0.85714,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,18,0,9],[40,61,0.6557,0.8125,0.21558,0.85711,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,16,0,9],[44,61,0.7213,0.83481,0.16409,0.85711,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,17,0,9],[48,61,0.7869,0.85713,0.15974,0.85714,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,18,0,10],[52,61,0.8525,0.87051,0.1093,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,19,0,9],[56,61,0.918,0.80803,0.15406,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,16,0,6],[60,61,0.9836,0.79017,0.22864,0.82132,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,17,0,7],[61,61,1.0,0.79908,0.20783,0.857,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,20,0,5]]},{"b":7,"e":0.42857,"k":"falling","v":0.49551,"x":0.87053,"p":[[0,143,0.0,0.82588,0.32289,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,23],[4,143,0.028,0.83036,0.29111,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,4,0,21],[8,143,0.0559,0.74991,0.3519,0.53571,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,5,0,17],[12,143,0.0839,0.80804,0.30849,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,0,3,0,20],[16,143,0.1119,0.82589,0.33069,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[20,143,0.1399,0.87053,0.27284,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,23],[24,143,0.1678,0.68749,0.38373,0.28571,1.0,1.0,0.0,1.0,4,17,0,4,0,1,0,0,4,0,0,2,0,0,2,0,0,0,0,0,2,0,17],[28,143,0.1958,0.8125,0.32427,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,22],[32,143,0.2238,0.49551,0.35171,0.24999,0.49979,0.85714,0.0,1.0,6,4,0,6,0,2,0,0,6,0,0,2,0,0,3,0,0,3,0,0,6,0,4],[36,143,0.2517,0.53123,0.28623,0.28571,0.4998,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,8,0,0,5,0,0,3,0,0,2,0,0,11,0,0],[40,143,0.2797,0.57141,0.28793,0.39286,0.57143,0.85714,0.0,1.0,3,1,0,3,0,0,0,0,5,0,0,6,0,0,3,0,0,3,0,0,11,0,1],[44,143,0.3077,0.56695,0.26119,0.42857,0.57121,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,4,0,0,9,0,0,4,0,0,4,0,0,7,0,2],[48,143,0.3357,0.61607,0.3163,0.39286,0.71429,0.85714,0.0,1.0,4,3,0,4,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,0,11,0,3],[52,143,0.3636,0.64732,0.2696,0.53572,0.71429,0.85714,0.0,1.0,3,2,0,3,0,0,0,0,1,0,0,4,0,0,3,0,0,9,0,0,10,0,2],[56,143,0.3916,0.60714,0.26963,0.42857,0.71429,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,5,0,0,4,0,0,4,0,0,6,0,0,9,0,2],[60,143,0.4196,0.75446,0.18977,0.67857,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,0,17,0,3],[64,143,0.4476,0.72321,0.26711,0.67857,0.85714,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,2,0,0,1,0,0,4,0,0,15,0,5],[68,143,0.4755,0.79464,0.18189,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,8,0,0,13,0,7],[72,143,0.5035,0.70534,0.20184,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,7,0,0,13,0,2],[76,143,0.5315,0.75446,0.18638,0.67857,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,15,0,4],[80,143,0.5594,0.72321,0.19212,0.71429,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,12,0,0,15,0,0],[84,143,0.5874,0.69195,0.23448,0.67857,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,1,0,0,2,0,0,10,0,0,12,0,2],[88,143,0.6154,0.72767,0.24577,0.71429,0.857,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,7,0,0,14,0,4],[92,143,0.6434,0.64732,0.2448,0.42857,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,8,0,0,1,0,0,8,0,0,10,0,2],[96,143,0.6713,0.69196,0.22334,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,3,0,0,4,0,0,8,0,0,12,0,2],[100,143,0.6993,0.62052,0.29366,0.53539,0.71429,0.85714,0.0,1.0,4,1,0,4,0,0,0,0,3,0,0,1,0,0,3,0,0,9,0,0,11,0,1],[104,143,0.7273,0.61161,0.29066,0.42857,0.71429,0.85714,0.0,0.85714,4,0,0,4,0,0,0,0,2,0,0,4,0,0,2,0,0,7,0,0,13,0,0],[108,143,0.7552,0.62944,0.27861,0.42859,0.71429,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,3,0,0,3,0,0,4,0,0,4,0,0,15,0,0],[112,143,0.7832,0.66517,0.28707,0.57143,0.85712,0.85714,0.0,1.0,3,1,0,3,0,1,0,0,2,0,0,0,0,0,4,0,0,5,0,0,16,0,1],[116,143,0.8112,0.6741,0.26058,0.5354,0.71429,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,4,0,0,2,0,0,8,0,0,11,0,3],[120,143,0.8392,0.65177,0.23128,0.53539,0.71429,0.85704,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,4,0,0,4,0,0,11,0,0,6,0,3],[124,143,0.8671,0.6875,0.20958,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,11,0,0,10,0,2],[128,143,0.8951,0.62497,0.28958,0.28571,0.71429,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,7,0,0,2,0,0,2,0,0,4,0,0,13,0,2],[132,143,0.9231,0.64732,0.29011,0.42859,0.71429,0.85714,0.0,1.0,3,3,0,3,0,0,0,0,2,0,0,5,0,0,2,0,0,5,0,0,12,0,3],[136,143,0.951,0.67857,0.24999,0.42857,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,5,0,0,0,0,0,8,0,0,13,0,2],[140,143,0.979,0.63393,0.22286,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,12,0,0,4,0,3],[143,143,1.0,0.57594,0.21272,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,5,0,0,5,0,0,9,0,0,6,0,0]]}]},{"i":"b245799b9a1f1dd1","q":"Solve the following equation in  $\\mathbb{Z}$ :\n\n\\[3^{2a + 1}b^2 + 1 = 2^c\\]","t":[{"b":1,"e":0.42857,"k":"falling","v":0.32143,"x":0.70534,"p":[[0,81,0.0,0.68304,0.32288,0.39286,0.78571,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,6,0,0,5,0,0,1,0,0,2,0,0,2,0,14],[4,81,0.0494,0.63839,0.33117,0.42857,0.57143,1.0,0.0,1.0,1,12,0,1,0,3,0,0,2,0,0,10,0,0,0,0,0,2,0,0,2,0,12],[8,81,0.0988,0.58036,0.31326,0.42857,0.42857,1.0,0.0,1.0,1,10,0,1,0,2,0,0,3,0,0,14,0,0,1,0,0,0,0,0,1,0,10],[12,81,0.1481,0.70534,0.2878,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,10,0,0,2,0,0,3,0,0,2,0,13],[16,81,0.1975,0.60714,0.34256,0.42857,0.42859,1.0,0.0,1.0,2,12,0,2,0,2,0,0,3,0,0,11,0,0,0,0,0,1,0,0,1,0,12],[20,81,0.2469,0.625,0.30462,0.42857,0.4286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,4,0,0,11,0,0,1,0,0,2,0,0,1,0,11],[24,81,0.2963,0.66518,0.30222,0.42857,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,12,0,0,0,0,0,4,0,0,1,0,12],[28,81,0.3457,0.59821,0.29973,0.42857,0.42859,1.0,0.0,1.0,1,9,0,1,0,1,0,0,3,0,0,14,0,0,0,0,0,2,0,0,2,0,9],[32,81,0.3951,0.60719,0.31132,0.42857,0.42857,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,14,0,0,1,0,0,0,0,0,1,0,11],[36,81,0.4444,0.59376,0.28596,0.42857,0.4286,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,16,0,0,0,0,0,2,0,0,1,0,9],[40,81,0.4938,0.58034,0.24983,0.42857,0.42857,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,18,0,0,2,0,0,1,0,0,3,0,6],[44,81,0.5432,0.60714,0.31135,0.42857,0.42857,1.0,0.0,1.0,1,11,0,1,0,2,0,0,1,0,0,15,0,0,0,0,0,2,0,0,0,0,11],[48,81,0.5926,0.46875,0.21498,0.42857,0.42857,0.42857,0.0,1.0,1,3,0,1,0,0,0,0,5,0,0,21,0,0,0,0,0,1,0,0,1,0,3],[52,81,0.642,0.52232,0.21902,0.42857,0.42857,0.46431,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,20,0,0,1,0,0,0,0,0,4,0,3],[56,81,0.6914,0.46429,0.22304,0.42857,0.42857,0.42858,0.0,1.0,1,3,0,1,0,1,0,0,5,0,0,19,0,0,1,0,0,1,0,0,1,0,3],[60,81,0.7407,0.49554,0.2448,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,16,0,0,0,0,0,4,0,0,2,0,3],[64,81,0.7901,0.42857,0.19561,0.42857,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,4,0,0,3,0,0,22,0,0,0,0,0,0,0,0,1,0,2],[68,81,0.8395,0.37054,0.11769,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,5,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[72,81,0.8889,0.43304,0.25874,0.2857,0.42857,0.4286,0.14286,1.0,0,4,0,0,0,7,0,0,5,0,0,14,0,0,0,0,0,2,0,0,0,0,4],[76,81,0.9383,0.47321,0.29545,0.28571,0.42857,0.46429,0.0,1.0,1,6,0,1,0,5,0,0,5,0,0,13,0,0,1,0,0,0,0,0,1,0,6],[80,81,0.9877,0.33929,0.12752,0.14289,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,9,0,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.32143,0.12877,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,10,0,0,4,0,0,18,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"falling","v":0.34821,"x":0.92411,"p":[[0,41,0.0,0.81249,0.25365,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,2,0,0,5,0,17],[4,41,0.0976,0.92411,0.15965,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,25],[8,41,0.1951,0.8125,0.26592,0.64286,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,2,0,0,3,0,19],[12,41,0.2927,0.75892,0.27535,0.42859,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,2,0,0,2,0,16],[16,41,0.3902,0.70982,0.27545,0.42857,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,11,0,0,2,0,0,1,0,0,3,0,13],[20,41,0.4878,0.75892,0.27067,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,6,0,0,0,0,0,5,0,0,4,0,14],[24,41,0.5854,0.78124,0.2575,0.42857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,1,0,0,6,0,15],[28,41,0.6829,0.71429,0.26486,0.42857,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,12,0,0,1,0,0,2,0,0,4,0,12],[32,41,0.7805,0.55357,0.28959,0.42857,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,3,0,0,16,0,0,0,0,0,1,0,0,1,0,8],[36,41,0.878,0.44647,0.2714,0.28571,0.42857,0.4286,0.0,1.0,3,4,0,3,0,2,0,0,4,0,0,17,0,0,0,0,0,1,0,0,1,0,4],[40,41,0.9756,0.37946,0.09182,0.39286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,5,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.34821,0.1234,0.25,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,8,0,0,2,0,0,22,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d32c4f9a07e32608","q":"Prove that $\\frac{a^{2}-b c}{2 a^{2}+b c}+\\frac{b^{2}-c a}{2 b^{2}+c a}+\\frac{c^{2}-a b}{2 c^{2}+a b} \\leq 0$ for any real positive numbers $a, b, c$.","t":[{"b":1,"e":0.28571,"k":"falling","v":0.15178,"x":0.625,"p":[[0,34,0.0,0.625,0.39082,0.28571,0.85714,1.0,0.0,1.0,3,15,0,3,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0,2,0,15],[4,34,0.1176,0.57143,0.38631,0.28571,0.28571,1.0,0.0,1.0,3,14,0,3,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[8,34,0.2353,0.33482,0.26392,0.2857,0.28571,0.28571,0.0,1.0,5,3,0,5,0,0,0,0,22,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[12,34,0.3529,0.19196,0.20079,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,34,0.4706,0.19643,0.13243,0.0,0.28571,0.28571,0.0,0.28571,10,0,0,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.15178,0.13803,0.0,0.21428,0.28571,0.0,0.28571,14,0,0,14,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.23214,0.11152,0.2857,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.60268,"x":1.0,"p":[[0,21,0.0,0.60268,0.3792,0.28571,0.28571,1.0,0.0,1.0,2,15,0,2,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[4,21,0.1905,0.83482,0.29475,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[8,21,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,21,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,21,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,21,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[21,21,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"22efab1e6bdde186","q":"Suppose that  $A=1,2,$  or  $3$ . Let  $a$  and  $b$  be relatively prime integers such that  $a^{2}+Ab^2 =s^3$  for some integer  $s$ . Then, there are integers  $u$  and  $v$  such that  $s=u^2 +Av^2$ ,  $a =u^3 - 3Avu^2$ , and  $b=3u^{2}v -Av^3$ .","t":[{"b":1,"e":0.57143,"k":"flat","v":0.55354,"x":0.70536,"p":[[0,47,0.0,0.59819,0.1729,0.5354,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,17,0,0,2,0,0,2,0,3],[4,47,0.0851,0.65179,0.16342,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,2,0,0,3,0,4],[8,47,0.1702,0.65177,0.15948,0.57143,0.57143,0.60714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,1,0,0,3,0,4],[12,47,0.2553,0.66518,0.18423,0.57143,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,2,0,0,4,0,5],[16,47,0.3404,0.70536,0.19212,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,1,0,0,3,0,8],[20,47,0.4255,0.70089,0.18681,0.57143,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,1,0,0,8,0,5],[24,47,0.5106,0.61607,0.10972,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,4,0,0,2,0,1],[28,47,0.5957,0.58479,0.10326,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,26,0,0,1,0,0,1,0,1],[32,47,0.6809,0.57588,0.06666,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0,1,0,0],[36,47,0.766,0.55354,0.04724,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[40,47,0.8511,0.56247,0.07086,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0],[44,47,0.9362,0.5625,0.04971,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0],[47,47,1.0,0.55802,0.04163,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.56695,"x":0.61156,"p":[[0,20,0.0,0.61156,0.16458,0.571,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,15,0,0,7,0,0,2,0,2],[4,20,0.2,0.5982,0.12079,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,24,0,0,3,0,0,0,0,2],[8,20,0.4,0.56695,0.09771,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,26,0,0,0,0,0,2,0,0],[12,20,0.6,0.58034,0.08702,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0,0,0,1],[16,20,0.8,0.56695,0.0977,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,23,0,0,1,0,0,2,0,0],[20,20,1.0,0.58036,0.10062,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,1,0,1]]}]},{"i":"75bd716118d66eeb","q":"Suppose  $a,b,c\\in \\mathbb R^+$ . Prove that :\\[\\left(\\frac ab+\\frac bc+\\frac ca\\right)^2\\geq (a+b+c)\\left(\\frac1a+\\frac1b+\\frac1c\\right)\\]","t":[{"b":1,"e":0.85714,"k":"flat","v":0.61598,"x":0.82141,"p":[[0,106,0.0,0.77679,0.18189,0.71429,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,10,0,0,8,0,8],[4,106,0.0377,0.79018,0.23954,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,8,0,10],[8,106,0.0755,0.79464,0.26229,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,3,0,0,0,0,0,10,0,0,1,0,16],[12,106,0.1132,0.82141,0.16368,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,13,0,9],[16,106,0.1509,0.7366,0.26989,0.42857,0.85707,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,7,0,0,0,0,0,5,0,0,5,0,12],[20,106,0.1887,0.80357,0.25442,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,4,0,16],[24,106,0.2264,0.7857,0.22869,0.71429,0.78571,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,10,0,0,3,0,13],[28,106,0.2642,0.77232,0.22829,0.71429,0.78564,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,10,0,0,6,0,10],[32,106,0.3019,0.79464,0.18189,0.71429,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,12,0,0,6,0,10],[36,106,0.3396,0.79018,0.22299,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,7,0,0,5,0,13],[40,106,0.3774,0.74549,0.24155,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,7,0,0,4,0,11],[44,106,0.4151,0.78569,0.14729,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,5,0,8],[48,106,0.4528,0.76777,0.23101,0.67857,0.71429,1.0,0.14,1.0,0,13,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,10,0,0,1,0,13],[52,106,0.4906,0.79911,0.22829,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,11,0,0,4,0,13],[56,106,0.5283,0.76338,0.249,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,5,0,10],[60,106,0.566,0.73213,0.28292,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,1,0,0,2,0,0,10,0,0,5,0,10],[64,106,0.6038,0.74107,0.3102,0.67857,0.85714,1.0,0.0,1.0,3,12,0,3,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,12],[68,106,0.6415,0.81695,0.22655,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,7,0,0,6,0,14],[72,106,0.6792,0.79017,0.2082,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,8,0,0,10,0,9],[76,106,0.717,0.6964,0.28066,0.42857,0.78571,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,7,0,0,3,0,0,3,0,0,7,0,9],[80,106,0.7547,0.79465,0.19541,0.71429,0.78571,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,13,0,0,6,0,10],[84,106,0.7925,0.75,0.25754,0.71429,0.78571,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,9,0,0,6,0,10],[88,106,0.8302,0.61598,0.246,0.5354,0.71429,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,4,0,0,2,0,0,7,0,0,11,0,0,2,0,4],[92,106,0.8679,0.75893,0.19704,0.71429,0.78571,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,10,0,0,10,0,6],[96,106,0.9057,0.75893,0.24598,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,3,0,0,0,0,0,12,0,0,5,0,10],[100,106,0.9434,0.81696,0.1684,0.71429,0.85707,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,0,6,0,11],[104,106,0.9811,0.74552,0.25936,0.67857,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,1,0,0,3,0,0,9,0,0,4,0,11],[106,106,1.0,0.64726,0.22584,0.571,0.71429,0.74996,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,4,0,0,8,0,0,9,0,0,4,0,4]]},{"b":7,"e":0.71429,"k":"flat","v":0.55802,"x":0.86605,"p":[[0,75,0.0,0.7499,0.20544,0.71429,0.71429,0.89286,0.14,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,12,0,0,5,0,8],[4,75,0.0533,0.86605,0.16344,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,16],[8,75,0.1067,0.84821,0.12846,0.71429,0.85714,1.0,0.5714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,9,0,11],[12,75,0.16,0.8125,0.15335,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,11,0,8],[16,75,0.2133,0.74106,0.19047,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,12,0,0,10,0,4],[20,75,0.2667,0.63836,0.2575,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,4,0,0,5,0,0,2,0,0,8,0,0,7,0,4],[24,75,0.32,0.60714,0.33312,0.39286,0.71429,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,3,0,0,4,0,0,1,0,0,8,0,0,4,0,7],[28,75,0.3733,0.60264,0.32681,0.39286,0.71429,0.85714,0.0,1.0,3,6,0,3,0,3,0,0,2,0,0,3,0,0,4,0,0,5,0,0,6,0,6],[32,75,0.4267,0.62054,0.33429,0.42857,0.71429,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,1,0,0,3,0,0,3,0,0,7,0,0,5,0,7],[36,75,0.48,0.63839,0.3194,0.42857,0.71429,0.85714,0.0,1.0,3,6,0,3,0,3,0,0,1,0,0,2,0,0,0,0,0,12,0,0,5,0,6],[40,75,0.5333,0.64286,0.25505,0.42857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,3,0,0,4,0,0,4,0,0,9,0,0,6,0,4],[44,75,0.5867,0.55802,0.34692,0.25001,0.57121,0.85714,0.0,1.0,3,7,0,3,0,5,0,0,2,0,0,5,0,0,2,0,0,4,0,0,4,0,7],[48,75,0.64,0.61157,0.32387,0.42857,0.71429,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,2,0,0,3,0,0,4,0,0,8,0,0,3,0,7],[52,75,0.6933,0.66516,0.30433,0.5354,0.71429,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,0,0,0,4,0,0,1,0,0,12,0,0,4,0,7],[56,75,0.7467,0.70088,0.28874,0.53539,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,3,0,0,3,0,0,1,0,0,9,0,0,5,0,9],[60,75,0.8,0.75446,0.24804,0.57143,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,6,0,0,6,0,11],[64,75,0.8533,0.76339,0.28484,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,6,0,0,9,0,11],[68,75,0.9067,0.72766,0.25345,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,0,0,0,3,0,0,13,0,0,3,0,9],[72,75,0.96,0.71874,0.16936,0.71429,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,17,0,0,8,0,2],[75,75,1.0,0.70088,0.16889,0.57143,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,14,0,0,6,0,3]]}]},{"i":"d78fc7763f58f84e","q":"Suppose  $f: \\mathbb{R}^{+} \\mapsto \\mathbb{R}^{+}$  is a function such that  $\\frac{f(x)}{x}$  is increasing on  $\\mathbb{R}^{+}$ . For  $a,b,c>0$ , prove that   $$ 2\\left (\\frac{f(a)+f(b)}{a+b} + \\frac{f(b)+f(c)}{b+c}+ \\frac{f(c)+f(a)}{c+a} \\right) \\geq 3\\left(\\frac{f(a)+f(b)+f(c)}{a+b+c}\\right) + \\frac{f(a)}{a}+ \\frac{f(b)}{b}+ \\frac{f(c)}{c} $$","t":[{"b":2,"e":1.0,"k":"rising","v":0.73659,"x":1.0,"p":[[0,101,0.0,0.73659,0.26272,0.53539,0.78564,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,6,0,0,2,0,0,6,0,0,5,0,11],[4,101,0.0396,0.92411,0.20512,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[8,101,0.0792,0.91517,0.17447,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,25],[12,101,0.1188,0.87054,0.27516,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,25],[16,101,0.1584,0.88839,0.23347,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,24],[20,101,0.198,0.92411,0.17491,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[24,101,0.2376,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[28,101,0.2772,0.88839,0.23618,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,25],[32,101,0.3168,0.90625,0.23038,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,25],[36,101,0.3564,0.83036,0.27993,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,1,0,0,3,0,21],[40,101,0.396,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[44,101,0.4356,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[48,101,0.4752,0.89731,0.22086,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,24],[52,101,0.5149,0.94196,0.18161,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[56,101,0.5545,0.91072,0.23351,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,27],[60,101,0.5941,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[64,101,0.6337,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[68,101,0.6733,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,101,0.7129,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,101,0.7525,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[80,101,0.7921,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,101,0.8317,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,101,0.8713,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,101,0.9109,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,101,0.9505,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,101,0.9901,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[101,101,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":0.71429,"k":"flat","v":0.68748,"x":0.91518,"p":[[0,132,0.0,0.7589,0.22144,0.57143,0.78564,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,9,0,0,5,0,0,6,0,10],[4,132,0.0303,0.84375,0.28873,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,24],[8,132,0.0606,0.875,0.23077,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,22],[12,132,0.0909,0.91518,0.22548,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[16,132,0.1212,0.82589,0.27833,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,3,0,20],[20,132,0.1515,0.84822,0.22285,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,3,0,19],[24,132,0.1818,0.80803,0.28928,0.64286,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,5,0,0,0,0,0,2,0,0,2,0,20],[28,132,0.2121,0.875,0.23076,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,24],[32,132,0.2424,0.84375,0.2212,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,19],[36,132,0.2727,0.85268,0.23551,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,3,0,0,2,0,21],[40,132,0.303,0.81695,0.22655,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,8,0,0,1,0,17],[44,132,0.3333,0.86607,0.24468,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,4,0,21],[48,132,0.3636,0.83036,0.25614,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,3,0,0,2,0,20],[52,132,0.3939,0.88392,0.19379,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,2,0,22],[56,132,0.4242,0.79909,0.27401,0.67846,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,5,0,0,1,0,0,4,0,0,2,0,18],[60,132,0.4545,0.81696,0.2506,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,2,0,0,2,0,19],[64,132,0.4848,0.80357,0.30671,0.78571,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0,5,0,19],[68,132,0.5152,0.8482,0.22287,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,2,0,20],[72,132,0.5455,0.86607,0.19541,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,4,0,19],[76,132,0.5758,0.84375,0.20935,0.82143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,8,0,16],[80,132,0.6061,0.82143,0.26001,0.67857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,1,0,0,4,0,19],[84,132,0.6364,0.85712,0.22591,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,4,0,20],[88,132,0.6667,0.8214,0.24487,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,4,0,17],[92,132,0.697,0.77232,0.26692,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,1,0,0,1,0,0,7,0,0,4,0,14],[96,132,0.7273,0.83034,0.2156,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,4,0,17],[100,132,0.7576,0.72321,0.25238,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,6,0,0,1,0,12],[104,132,0.7879,0.76784,0.25192,0.53539,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,4,0,0,4,0,14],[108,132,0.8182,0.74553,0.30037,0.42857,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,5,0,0,1,0,0,4,0,0,3,0,15],[112,132,0.8485,0.7723,0.20474,0.71429,0.85707,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,8,0,0,8,0,9],[116,132,0.8788,0.80357,0.22798,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,8,0,0,3,0,15],[120,132,0.9091,0.73212,0.27375,0.5354,0.857,1.0,0.0,1.0,1,11,0,1,0,1,0,0,0,0,0,6,0,0,3,0,0,4,0,0,6,0,11],[124,132,0.9394,0.74998,0.2113,0.67857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,9,0,0,7,0,8],[128,132,0.9697,0.68748,0.22429,0.5354,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,7,0,0,4,0,7],[132,132,1.0,0.72767,0.23518,0.57132,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,4,0,10]]}]},{"i":"41edd94024a48e0c","q":"Prove that for each  $ n$ :\r\n\\[ \\sum_{k\\equal{}1}^n\\binom{n\\plus{}k\\minus{}1}{2k\\minus{}1}\\equal{}F_{2n}\\]","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.24107,"p":[[0,33,0.0,0.14732,0.34531,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[4,33,0.1212,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,33,0.2424,0.24107,0.4141,0.0,0.0,0.25,0.0,1.0,23,7,0,23,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7],[12,33,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.1875,"p":[[0,16,0.0,0.16071,0.36202,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,16,0.25,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,16,0.5,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,16,0.75,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,16,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1d7e3afb808a2c7c","q":"Suppose that the real numbers  $a_0,a_1,\\dots,a_n$  and  $x,$  with  $0<x<1,$  satisfy \\[\\frac{a_0}{1-x}+\\frac{a_1}{1-x^2}+\\cdots+\\frac{a_n}{1-x^{n+1}}=0.\\] Prove that there exists a real number  $y$  with  $0<y<1$  such that \\[a_0+a_1y+\\cdots+a_ny^n=0.\\]","t":[{"b":0,"e":0.0,"k":"flat","v":0.5625,"x":1.0,"p":[[0,66,0.0,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,66,0.0606,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,66,0.1212,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,66,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,66,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,66,0.303,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,66,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,66,0.4242,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,66,0.4848,0.86161,0.33021,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[36,66,0.5455,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[40,66,0.6061,0.74554,0.43114,0.64286,1.0,1.0,0.0,1.0,8,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[44,66,0.6667,0.74554,0.43114,0.64286,1.0,1.0,0.0,1.0,8,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[48,66,0.7273,0.83929,0.36202,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[52,66,0.7879,0.78125,0.4134,1.0,1.0,1.0,0.0,1.0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[56,66,0.8485,0.78125,0.4134,1.0,1.0,1.0,0.0,1.0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[60,66,0.9091,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[64,66,0.9697,0.5625,0.49608,0.0,1.0,1.0,0.0,1.0,14,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[66,66,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,16,0.0,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,16,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,16,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,16,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c43d1103618814b8","q":"Show that there are infinitely many pairs ( $\\mathrm{m}, \\mathfrak{n}$ ) of distinct strictly positive integers such that m ! n is a perfect square.","t":[{"b":5,"e":0.0,"k":"flat","v":0.07589,"x":0.20089,"p":[[0,27,0.0,0.20089,0.21387,0.0,0.0,0.42857,0.0,0.42857,17,0,1,17,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.13393,0.19865,0.0,0.0,0.42857,0.0,0.4286,22,0,0,22,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.14732,0.20355,0.0,0.0,0.42857,0.0,0.42857,21,0,0,21,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.07589,0.15966,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.11607,0.18708,0.0,0.0,0.32142,0.0,0.4286,23,0,0,23,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.09375,0.17717,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.09375,0.17717,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.14286,0.19885,0.0,0.0,0.42857,0.0,0.4286,21,0,0,21,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04018,"x":0.22321,"p":[[0,30,0.0,0.22321,0.2111,0.0,0.35714,0.42857,0.0,0.42857,15,0,0,15,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.16071,0.20748,0.0,0.0,0.42857,0.0,0.42857,20,0,0,20,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.06696,0.15561,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.10714,0.18558,0.0,0.0,0.10714,0.0,0.4286,24,0,0,24,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.14732,0.20355,0.0,0.0,0.42857,0.0,0.42857,21,0,0,21,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.08036,0.16728,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.10714,0.18558,0.0,0.0,0.10714,0.0,0.42857,24,0,0,24,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"18499e8c2a3a1c4a","q":"Solve in positive integers:  $\\frac{1}{x^2}+\\frac{y}{xz}+\\frac{1}{z^2}=\\frac{1}{2013}$  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chose a natural number  $X\\le100$ , and Sasha is trying to guess this number.  He can select two natural numbers  $M$  and  $N$  less than  $100$  and ask about  $\\gcd(X+M,N)$ .  Show that Sasha can determine Tanya's number with at most seven questions.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.2408,"x":0.85713,"p":[[0,72,0.0,0.38839,0.39967,0.0,0.28571,0.78571,0.0,1.0,11,8,10,11,0,4,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,8],[4,72,0.0556,0.84375,0.30169,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,23],[8,72,0.1111,0.80354,0.33266,0.57132,1.0,1.0,0.0,1.0,1,23,0,1,0,3,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,23],[12,72,0.1667,0.79464,0.3213,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,19],[16,72,0.2222,0.85713,0.29235,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,24],[20,72,0.2778,0.75891,0.34891,0.53539,1.0,1.0,0.0,1.0,1,20,0,1,0,4,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,20],[24,72,0.3333,0.59821,0.40159,0.14286,0.71429,1.0,0.0,1.0,2,14,0,2,0,8,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,14],[28,72,0.3889,0.76786,0.29397,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,3,0,0,7,0,0,0,0,0,1,0,18],[32,72,0.4444,0.71874,0.32437,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,1,0,0,2,0,0,5,0,0,1,0,0,3,0,15],[36,72,0.5,0.57141,0.37115,0.14286,0.57121,1.0,0.0,1.0,1,11,0,1,0,9,0,0,2,0,0,3,0,0,3,0,0,1,0,0,2,0,11],[40,72,0.5556,0.48661,0.36221,0.14286,0.28571,0.89286,0.0,1.0,2,8,0,2,0,9,0,0,6,0,0,0,0,0,5,0,0,0,0,0,2,0,8],[44,72,0.6111,0.34821,0.29653,0.14286,0.21428,0.57143,0.0,1.0,2,4,0,2,0,14,0,0,5,0,0,2,0,0,5,0,0,0,0,0,0,0,4],[48,72,0.6667,0.2408,0.19717,0.14286,0.14288,0.28571,0.0,1.0,3,1,0,3,0,14,0,0,12,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[52,72,0.7222,0.36607,0.19541,0.14286,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,8,0,0,9,0,0,1,0,0,12,0,0,1,0,0,0,0,0],[56,72,0.7778,0.4375,0.23941,0.28571,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,12,0,0,1,0,0,9,0,0,2,0,0,1,0,2],[60,72,0.8333,0.40613,0.28379,0.14286,0.28571,0.57143,0.0,1.0,1,3,0,1,0,12,0,0,4,0,0,0,0,0,10,0,0,2,0,0,0,0,3],[64,72,0.8889,0.47766,0.21902,0.28571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,10,0,0,1,0,0,13,0,0,3,0,0,0,0,2],[68,72,0.9444,0.34821,0.2111,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,15,0,0,3,0,0,0,0,0,13,0,0,1,0,0,0,0,0],[72,72,1.0,0.29463,0.20804,0.14286,0.14286,0.57143,0.0,0.71429,2,0,0,2,0,15,0,0,5,0,0,0,0,0,9,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.33027,"x":0.84821,"p":[[0,49,0.0,0.49107,0.44883,0.0,0.28571,1.0,0.0,1.0,9,13,8,9,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,13],[4,49,0.0816,0.84821,0.25238,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,1,0,0,3,0,21],[8,49,0.1633,0.79018,0.31132,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,5,0,0,0,0,0,2,0,0,1,0,0,2,0,20],[12,49,0.2449,0.74103,0.31833,0.571,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,2,0,0,7,0,0,0,0,0,0,0,18],[16,49,0.3265,0.71429,0.3312,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,3,0,0,3,0,0,3,0,0,1,0,0,2,0,16],[20,49,0.4082,0.75,0.34069,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,2,0,0,0,0,0,3,0,0,2,0,0,2,0,18],[24,49,0.4898,0.82589,0.28956,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,3,0,0,0,0,22],[28,49,0.5714,0.78124,0.27891,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,18],[32,49,0.6531,0.62946,0.31106,0.53571,0.57143,1.0,0.0,1.0,1,10,0,1,0,4,0,0,2,0,0,1,0,0,11,0,0,2,0,0,1,0,10],[36,49,0.7347,0.68301,0.33643,0.571,0.71429,1.0,0.0,1.0,2,14,0,2,0,3,0,0,2,0,0,0,0,0,7,0,0,4,0,0,0,0,14],[40,49,0.8163,0.58036,0.29437,0.28571,0.57143,0.78571,0.14286,1.0,0,8,0,0,0,5,0,0,4,0,0,1,0,0,12,0,0,2,0,0,0,0,8],[44,49,0.898,0.51783,0.28515,0.25001,0.57143,0.60714,0.14286,1.0,0,5,0,0,0,8,0,0,3,0,0,0,0,0,13,0,0,3,0,0,0,0,5],[48,49,0.9796,0.40625,0.24513,0.14286,0.49999,0.57143,0.0,1.0,1,1,0,1,0,10,0,0,4,0,0,1,0,0,13,0,0,1,0,0,1,0,1],[49,49,1.0,0.33027,0.20966,0.14286,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,14,0,0,4,0,0,1,0,0,11,0,0,1,0,0,0,0,0]]}]},{"i":"2d7e0aa2ec478d81","q":"The Wonder Island Intelligence Service has  $16$  spies in Tartu. Each of them watches on some of his colleagues. It is known that if spy  $A$  watches on spy  $B$ , then  $B$  does not watch on  $A$ . Moreover, any  $10$  spies can numbered in such a way that the first spy watches on the second, the second watches on the third and so on until the tenth watches on the first. Prove that any  $11$  spies can also be numbered is a similar manner.","t":[{"b":0,"e":0.2857,"k":"rising","v":0.04911,"x":0.37498,"p":[[0,55,0.0,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.18751,0.15336,0.0,0.21428,0.28571,0.0,0.57143,10,0,0,10,0,6,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,55,0.1455,0.1383,0.21274,0.0,0.0,0.17857,0.0,0.85714,19,0,0,19,0,5,0,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[12,55,0.2182,0.2008,0.21089,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,7,0,0,7,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[16,55,0.2909,0.27679,0.2257,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,7,0,0,7,0,0,7,0,0,1,0,0,2,0,0,1,0,0],[20,55,0.3636,0.29018,0.16164,0.24999,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,5,0,0,16,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[24,55,0.4364,0.21426,0.23686,0.0,0.14286,0.42858,0.0,0.71429,13,0,0,13,0,8,0,0,1,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[28,55,0.5091,0.37498,0.23889,0.25,0.28571,0.57111,0.0,1.0,3,1,0,3,0,5,0,0,9,0,0,6,0,0,5,0,0,2,0,0,1,0,1],[32,55,0.5818,0.34375,0.19516,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,10,0,0,4,0,0,7,0,0,2,0,0,0,0,0],[36,55,0.6545,0.32586,0.18635,0.24999,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,12,0,0,6,0,0,4,0,0,2,0,0,0,0,0],[40,55,0.7273,0.23205,0.12759,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,9,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.28125,0.05629,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.28571,0.09449,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,7,0,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.27682,0.1068,0.14289,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,9,0,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[55,55,1.0,0.25893,0.06622,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,7,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"rising","v":0.00893,"x":0.3125,"p":[[0,18,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.12492,0.18122,0.0,0.0,0.1429,0.0,0.71429,17,0,0,17,0,9,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,18,0.4444,0.26786,0.13243,0.14286,0.28571,0.42857,0.0,0.42857,2,0,0,2,0,10,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.27232,0.1488,0.14286,0.2857,0.32143,0.0,0.71429,3,0,0,3,0,7,0,0,14,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[16,18,0.8889,0.30795,0.11916,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,15,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[18,18,1.0,0.3125,0.13092,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,15,0,0,10,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"32ca4d78fddd0733","q":"Suppose that  $a, b, c$  are complex numbers, satisfying the system of equations   $$ a^2=b-c,  $$     $$ b^2=c-a,  $$     $$ c^2=a-b. $$   Compute all possible values of  $a+b+c$ .","t":[{"b":0,"e":0.85714,"k":"rising","v":0.37053,"x":0.67409,"p":[[0,232,0.0,0.37053,0.19186,0.24999,0.35714,0.57143,0.0,0.57143,2,0,1,2,0,6,0,0,8,0,0,3,0,0,13,0,0,0,0,0,0,0,0],[4,232,0.0172,0.49554,0.17122,0.42857,0.57143,0.57143,0.0,0.85714,1,0,1,1,0,1,0,0,5,0,0,4,0,0,18,0,0,2,0,0,1,0,0],[8,232,0.0345,0.53122,0.1525,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,4,0,0,19,0,0,2,0,0,2,0,0],[12,232,0.0517,0.54017,0.18116,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,3,0,0,16,0,0,4,0,0,3,0,0],[16,232,0.069,0.55356,0.19804,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,2,0,0,12,0,0,8,0,0,3,0,0],[20,232,0.0862,0.49553,0.19227,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,4,0,0,11,0,0,5,0,0,2,0,0],[24,232,0.1034,0.49554,0.17851,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,4,0,0,13,0,0,5,0,0,1,0,0],[28,232,0.1207,0.51784,0.17034,0.39288,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,1,0,0,19,0,0,3,0,0,0,0,1],[32,232,0.1379,0.50446,0.14279,0.42859,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,3,0,0,19,0,0,3,0,0,0,0,0],[36,232,0.1552,0.44632,0.1778,0.28593,0.57141,0.57143,0.0,0.57143,2,0,0,2,0,2,0,0,5,0,0,4,0,0,19,0,0,0,0,0,0,0,0],[40,232,0.1724,0.5089,0.17471,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,5,0,0,15,0,0,3,0,0,2,0,0],[44,232,0.1897,0.50445,0.15965,0.39286,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,3,0,0,17,0,0,3,0,0,1,0,0],[48,232,0.2069,0.49107,0.18189,0.28571,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,7,0,0,2,0,0,17,0,0,3,0,0,1,0,0],[52,232,0.2241,0.55803,0.16115,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,3,0,0,19,0,0,4,0,0,1,0,1],[56,232,0.2414,0.50444,0.1556,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,2,0,0,21,0,0,1,0,0,1,0,0],[60,232,0.2586,0.49553,0.1636,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,3,0,0,16,0,0,3,0,0,1,0,0],[64,232,0.2759,0.51339,0.18161,0.4286,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,3,0,0,18,0,0,4,0,0,1,0,0],[68,232,0.2931,0.54018,0.1461,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,5,0,0,15,0,0,6,0,0,1,0,0],[72,232,0.3103,0.53571,0.13832,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,2,0,0,20,0,0,5,0,0,0,0,0],[76,232,0.3276,0.51786,0.14174,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,3,0,0,19,0,0,4,0,0,0,0,0],[80,232,0.3448,0.50445,0.1749,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,3,0,0,15,0,0,3,0,0,2,0,0],[84,232,0.3621,0.50893,0.18877,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,3,0,0,13,0,0,5,0,0,2,0,0],[88,232,0.3793,0.52232,0.17717,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,0,0,0,17,0,0,4,0,0,2,0,0],[92,232,0.3966,0.50446,0.1636,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,3,0,0,19,0,0,2,0,0,1,0,0],[96,232,0.4138,0.52232,0.13176,0.53571,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,2,0,0,22,0,0,1,0,0,1,0,0],[100,232,0.431,0.45979,0.16261,0.28571,0.57141,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,3,0,0,18,0,0,1,0,0,0,0,0],[104,232,0.4483,0.45088,0.15197,0.28571,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,9,0,0,6,0,0,15,0,0,1,0,0,0,0,0],[108,232,0.4655,0.55357,0.18123,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,5,0,0,14,0,0,5,0,0,2,0,1],[112,232,0.4828,0.51783,0.15871,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,3,0,0,19,0,0,3,0,0,1,0,0],[116,232,0.5,0.44196,0.17261,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,12,0,0,3,0,0,11,0,0,4,0,0,0,0,0],[120,232,0.5172,0.46872,0.18291,0.28571,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,3,0,0,12,0,0,4,0,0,1,0,0],[124,232,0.5345,0.42857,0.19562,0.2857,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,10,0,0,3,0,0,12,0,0,2,0,0,1,0,0],[128,232,0.5517,0.45089,0.15612,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,6,0,0,8,0,0,13,0,0,2,0,0,0,0,0],[132,232,0.569,0.53125,0.1684,0.5357,0.57143,0.57143,0.14286,0.8571,0,0,0,0,0,3,0,0,2,0,0,3,0,0,18,0,0,5,0,0,1,0,0],[136,232,0.5862,0.55802,0.13054,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,1,0,0,23,0,0,2,0,0,2,0,0],[140,232,0.6034,0.54911,0.16013,0.53572,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,3,0,0,17,0,0,5,0,0,2,0,0],[144,232,0.6207,0.61607,0.10374,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,3,0,0,4,0,0],[148,232,0.6379,0.65177,0.14699,0.57143,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,6,0,0,4,0,2],[152,232,0.6552,0.58034,0.17104,0.57143,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,0,0,0,24,0,0,1,0,0,3,0,1],[156,232,0.6724,0.63838,0.1287,0.57143,0.57143,0.71429,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,6,0,0,2,0,2],[160,232,0.6897,0.62944,0.12808,0.57143,0.57143,0.60714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,4,0,0,2,0,2],[164,232,0.7069,0.59375,0.08827,0.57143,0.57143,0.57143,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,3,0,0,0,0,1],[168,232,0.7241,0.63837,0.1472,0.57143,0.57143,0.60714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,4,0,0,0,0,4],[172,232,0.7414,0.60714,0.07986,0.57143,0.57143,0.57143,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,4,0,0,2,0,0],[176,232,0.7586,0.64732,0.14279,0.57143,0.64286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,14,0,0,12,0,0,3,0,1],[180,232,0.7759,0.63391,0.11811,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,6,0,0,3,0,1],[184,232,0.7931,0.64732,0.10705,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,10,0,0,2,0,1],[188,232,0.8103,0.67409,0.11426,0.57143,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,12,0,0,4,0,1],[192,232,0.8276,0.62497,0.13244,0.57143,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,3,0,0,4,0,1],[196,232,0.8448,0.61607,0.0974,0.57143,0.57143,0.57143,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,5,0,0,1,0,1],[200,232,0.8621,0.64731,0.14717,0.57143,0.57143,0.71429,0.1429,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,9,0,0,4,0,1],[204,232,0.8793,0.63393,0.11259,0.57143,0.57143,0.71429,0.5714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,5,0,0,3,0,1],[208,232,0.8966,0.60714,0.11845,0.57143,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,6,0,0,1,0,1],[212,232,0.9138,0.65176,0.10679,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,8,0,0,5,0,0],[216,232,0.931,0.64732,0.16746,0.57143,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,5,0,0,3,0,3],[220,232,0.9483,0.625,0.09942,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,7,0,0,1,0,1],[224,232,0.9655,0.61607,0.13092,0.57143,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,4,0,0,3,0,1],[228,232,0.9828,0.62052,0.09852,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,3,0,0,4,0,0],[232,232,1.0,0.62053,0.10478,0.57143,0.57143,0.57143,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,4,0,0,2,0,1]]},{"b":4,"e":0.1429,"k":"falling","v":0.17857,"x":0.5625,"p":[[0,115,0.0,0.35259,0.21433,0.14286,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,9,0,0,3,0,0,5,0,0,11,0,0,1,0,0,0,0,0],[4,115,0.0348,0.56249,0.18188,0.57142,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,4,0,0,1,0,0,19,0,0,3,0,0,4,0,0],[8,115,0.0696,0.5625,0.13333,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,1,0,0,22,0,0,3,0,0,2,0,0],[12,115,0.1043,0.45533,0.16916,0.2857,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,7,0,0,6,0,0,14,0,0,1,0,0,1,0,0],[16,115,0.1391,0.5,0.13832,0.28571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,0,0,0,21,0,0,2,0,0,0,0,0],[20,115,0.1739,0.52232,0.16602,0.53571,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,2,0,0,21,0,0,2,0,0,0,0,1],[24,115,0.2087,0.51786,0.16269,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,3,0,0,18,0,0,2,0,0,2,0,0],[28,115,0.2435,0.48215,0.16268,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,3,0,0,16,0,0,2,0,0,1,0,0],[32,115,0.2783,0.45089,0.16016,0.28571,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,0,0,0,18,0,0,1,0,0,0,0,0],[36,115,0.313,0.49553,0.16746,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,1,0,0,19,0,0,2,0,0,1,0,0],[40,115,0.3478,0.54909,0.17536,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,4,0,0,17,0,0,3,0,0,2,0,1],[44,115,0.3826,0.47766,0.16213,0.28571,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,4,0,0,17,0,0,1,0,0,1,0,0],[48,115,0.4174,0.52679,0.10972,0.57143,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,3,0,0,1,0,0,27,0,0,0,0,0,0,0,0],[52,115,0.4522,0.47321,0.18363,0.28571,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,7,0,0,0,0,0,17,0,0,4,0,0,0,0,0],[56,115,0.487,0.51784,0.15046,0.57142,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,0,1,0,0,24,0,0,2,0,0,0,0,0],[60,115,0.5217,0.52677,0.15335,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,5,0,0,18,0,0,2,0,0,2,0,0],[64,115,0.5565,0.44643,0.23351,0.2857,0.57143,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,5,0,0,1,0,0,15,0,0,3,0,0,0,0,1],[68,115,0.5913,0.41518,0.18681,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,8,0,0,4,0,0,11,0,0,3,0,0,0,0,0],[72,115,0.6261,0.41964,0.18189,0.28571,0.50001,0.57143,0.0,0.57143,2,0,0,2,0,3,0,0,6,0,0,5,0,0,16,0,0,0,0,0,0,0,0],[76,115,0.6609,0.29016,0.1767,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,13,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[80,115,0.6957,0.33927,0.21353,0.14286,0.2857,0.57143,0.0,0.85714,1,0,0,1,0,11,0,0,9,0,0,0,0,0,9,0,0,1,0,0,1,0,0],[84,115,0.7304,0.26784,0.15867,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,11,0,0,13,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[88,115,0.7652,0.22312,0.14263,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,7,0,0,15,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[92,115,0.8,0.27678,0.14698,0.14286,0.2857,0.28571,0.14286,0.85714,0,0,0,0,0,11,0,0,16,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[96,115,0.8348,0.19634,0.14177,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,15,0,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[100,115,0.8696,0.23661,0.1411,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,12,0,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[104,115,0.9043,0.22766,0.12295,0.14286,0.21428,0.28571,0.0,0.571,2,0,0,2,0,14,0,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[108,115,0.9391,0.26339,0.13882,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,10,0,0,18,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[112,115,0.9739,0.17857,0.10714,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,15,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[115,115,1.0,0.20089,0.11214,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,0,13,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ad0b56cc21211938","q":"Prove that if $0<\\frac{a}{b}<b<2 a$ then\n\n$$\n\\frac{2 a b-a^{2}}{7 a b-3 b^{2}-2 a^{2}}+\\frac{2 a b-b^{2}}{7 a b-3 a^{2}-2 b^{2}} \\geq 1+\\frac{1}{4}\\left(\\frac{a}{b}-\\frac{b}{a}\\right)^{2}\n$$","t":[{"b":4,"e":0.57143,"k":"rising","v":0.43304,"x":0.61158,"p":[[0,127,0.0,0.43304,0.14501,0.28571,0.42857,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,11,0,0,13,0,0,5,0,0,2,0,0,1,0,0],[4,127,0.0315,0.51338,0.15509,0.42857,0.49979,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,12,0,0,12,0,0,2,0,0,1,0,1],[8,127,0.063,0.54015,0.12234,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,10,0,0,13,0,0,7,0,0,0,0,0],[12,127,0.0945,0.51339,0.12807,0.42857,0.42857,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,18,0,0,7,0,0,5,0,0,1,0,0],[16,127,0.126,0.5357,0.15566,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,11,0,0,12,0,0,4,0,0,1,0,1],[20,127,0.1575,0.49552,0.14278,0.42857,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,13,0,0,11,0,0,3,0,0,1,0,0],[24,127,0.189,0.50446,0.12869,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,16,0,0,11,0,0,1,0,0,2,0,0],[28,127,0.2205,0.5,0.12877,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,15,0,0,10,0,0,3,0,0,1,0,0],[32,127,0.252,0.46427,0.10713,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,18,0,0,8,0,0,2,0,0,0,0,0],[36,127,0.2835,0.46429,0.09449,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,19,0,0,9,0,0,1,0,0,0,0,0],[40,127,0.315,0.49552,0.12869,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,13,0,0,12,0,0,2,0,0,1,0,0],[44,127,0.3465,0.44196,0.13534,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,18,0,0,6,0,0,0,0,0,0,0,1],[48,127,0.378,0.46875,0.10248,0.42857,0.42857,0.46431,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,22,0,0,5,0,0,3,0,0,0,0,0],[52,127,0.4094,0.44197,0.10926,0.42857,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,21,0,0,5,0,0,0,0,0,1,0,0],[56,127,0.4409,0.46874,0.10852,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,17,0,0,9,0,0,2,0,0,0,0,0],[60,127,0.4724,0.46429,0.10714,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,18,0,0,8,0,0,2,0,0,0,0,0],[64,127,0.5039,0.46427,0.101,0.42857,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,20,0,0,7,0,0,2,0,0,0,0,0],[68,127,0.5354,0.48659,0.12298,0.42857,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,12,0,0,12,0,0,3,0,0,0,0,0],[72,127,0.5669,0.51786,0.13243,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,9,0,0,15,0,0,3,0,0,1,0,0],[76,127,0.5984,0.52677,0.14032,0.42857,0.4998,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,14,0,0,10,0,0,4,0,0,2,0,0],[80,127,0.6299,0.52229,0.12168,0.42857,0.5712,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,12,0,0,14,0,0,3,0,0,1,0,0],[84,127,0.6614,0.53121,0.10851,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,9,0,0,17,0,0,4,0,0,0,0,0],[88,127,0.6929,0.52677,0.12595,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,11,0,0,16,0,0,1,0,0,2,0,0],[92,127,0.7244,0.50445,0.12868,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,11,0,0,16,0,0,1,0,0,1,0,0],[96,127,0.7559,0.51339,0.15916,0.42857,0.50001,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,12,0,0,13,0,0,0,0,0,2,0,1],[100,127,0.7874,0.50445,0.12869,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,9,0,0,14,0,0,4,0,0,0,0,0],[104,127,0.8189,0.53568,0.14284,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,9,0,0,16,0,0,1,0,0,3,0,0],[108,127,0.8504,0.52676,0.11537,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,11,0,0,14,0,0,5,0,0,0,0,0],[112,127,0.8819,0.50446,0.14279,0.42857,0.57141,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,11,0,0,13,0,0,3,0,0,1,0,0],[116,127,0.9134,0.4732,0.11537,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,14,0,0,13,0,0,1,0,0,0,0,0],[120,127,0.9449,0.53122,0.13474,0.42857,0.571,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,14,0,0,12,0,0,4,0,0,0,0,1],[124,127,0.9764,0.57139,0.10101,0.571,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,22,0,0,2,0,0,2,0,0],[127,127,1.0,0.61158,0.11426,0.57143,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,0,4,0,0,4,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.41518,"x":0.56696,"p":[[0,78,0.0,0.42856,0.12876,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,20,0,0,3,0,0,1,0,0,1,0,0],[4,78,0.0513,0.47321,0.15746,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,13,0,0,7,0,0,4,0,0,1,0,0],[8,78,0.1026,0.49549,0.16743,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,9,0,0,13,0,0,2,0,0,2,0,0],[12,78,0.1538,0.49554,0.17491,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,13,0,0,5,0,0,4,0,0,3,0,0],[16,78,0.2051,0.46427,0.11293,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,13,0,0,12,0,0,1,0,0,0,0,0],[20,78,0.2564,0.49999,0.17496,0.42857,0.49979,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,12,0,0,11,0,0,3,0,0,2,0,0],[24,78,0.3077,0.51339,0.18509,0.42857,0.50001,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,12,0,0,8,0,0,6,0,0,2,0,0],[28,78,0.359,0.51339,0.17076,0.42857,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,17,0,0,4,0,0,5,0,0,3,0,0],[32,78,0.4103,0.50445,0.17122,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,11,0,0,8,0,0,5,0,0,2,0,0],[36,78,0.4615,0.51786,0.13716,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,10,0,0,13,0,0,4,0,0,1,0,0],[40,78,0.5128,0.50445,0.1556,0.42857,0.4286,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,16,0,0,7,0,0,4,0,0,2,0,0],[44,78,0.5641,0.41518,0.22406,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,6,0,0,9,0,0,5,0,0,2,0,0,3,0,0],[48,78,0.6154,0.45091,0.13879,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,13,0,0,10,0,0,2,0,0,0,0,0],[52,78,0.6667,0.52232,0.18423,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,10,0,0,11,0,0,4,0,0,3,0,0],[56,78,0.7179,0.54463,0.17655,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,9,0,0,9,0,0,5,0,0,4,0,0],[60,78,0.7692,0.52229,0.1268,0.42857,0.42859,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,1,0,0,16,0,0,9,0,0,5,0,0,1,0,0],[64,78,0.8205,0.51784,0.17405,0.42857,0.42857,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,11,0,0,7,0,0,5,0,0,3,0,0],[68,78,0.8718,0.50893,0.11259,0.42857,0.5,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,14,0,0,12,0,0,4,0,0,0,0,0],[72,78,0.9231,0.53569,0.16751,0.42857,0.57121,0.57143,0.1429,0.85714,0,0,0,0,0,1,0,0,2,0,0,11,0,0,12,0,0,2,0,0,4,0,0],[76,78,0.9744,0.56247,0.15126,0.42857,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,7,0,0,14,0,0,5,0,0,3,0,0],[78,78,1.0,0.56696,0.16164,0.53572,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,3,0,0,15,0,0,6,0,0,3,0,0]]}]},{"i":"ff045a0ee0eae801","q":"Prove that in triangle  $ABC$ , the foot of the altitude  $AH$ , the point of tangency of the inscribed circle with side  $BC$  and projections of point  $A$  on the bisectors  $\\angle B$  and  $\\angle C$  of the triangle lie on one circle.\n\n(Dmitry Prokopenko)","t":[{"b":2,"e":0.57143,"k":"flat","v":0.11152,"x":0.16509,"p":[[0,65,0.0,0.14259,0.08749,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.1517,0.08704,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,65,0.2462,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.14277,0.06187,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.14732,0.13592,0.14286,0.14286,0.14286,0.0,0.85714,4,0,0,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[28,65,0.4308,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.16062,0.12754,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,25,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,65,0.6154,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.16509,0.12933,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,28,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[56,65,0.8615,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.11589,0.06615,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.08482,"x":0.15616,"p":[[0,82,0.0,0.12929,0.07454,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,82,0.0488,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,82,0.0976,0.15616,0.09008,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,27,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,82,0.1463,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,82,0.1951,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,82,0.2439,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,82,0.2927,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,82,0.3415,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,82,0.3902,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,82,0.439,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,82,0.4878,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,82,0.5366,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,82,0.5854,0.12482,0.0778,0.14214,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,82,0.6341,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,82,0.6829,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,82,0.7317,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,82,0.7805,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,82,0.8293,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,82,0.878,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.143,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ee7102b15d1ef596","q":"Prove that if  $x>0$  and  $n\\in\\mathbb N,$  then we have \n\\[\\frac{x^n(x^{n+1}+1)}{x^n+1}\\leq\\left(\\frac {x+1}{2}\\right)^{2n+1}.\\]","t":[{"b":1,"e":0.0,"k":"flat","v":0.08482,"x":0.33482,"p":[[0,108,0.0,0.16071,0.20748,0.0,0.07143,0.2857,0.0,0.71429,16,0,0,16,0,7,0,0,2,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,108,0.037,0.26339,0.26752,0.0,0.21429,0.42857,0.0,0.85714,14,0,0,14,0,2,0,0,1,0,0,9,0,0,3,0,0,2,0,0,1,0,0],[8,108,0.0741,0.33036,0.25614,0.10714,0.35714,0.46429,0.0,1.0,8,1,0,8,0,3,0,0,5,0,0,8,0,0,5,0,0,2,0,0,0,0,1],[12,108,0.1111,0.1875,0.19377,0.0,0.14288,0.42857,0.0,0.57143,15,0,0,15,0,2,0,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[16,108,0.1481,0.21651,0.30172,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,4,0,0,1,0,1,4,0,0,2,0,0,0,0,0,1,0,2],[20,108,0.1852,0.20982,0.24481,0.0,0.07143,0.42857,0.0,0.71429,16,0,0,16,0,3,0,0,2,0,0,6,0,0,3,0,0,2,0,0,0,0,0],[24,108,0.2222,0.20536,0.23128,0.0,0.14286,0.42857,0.0,0.85714,14,0,0,14,0,6,0,0,1,0,0,8,0,0,2,0,0,0,0,0,1,0,0],[28,108,0.2593,0.23661,0.28033,0.0,0.0,0.42857,0.0,1.0,17,1,0,17,0,0,0,0,1,0,0,10,0,0,2,0,0,0,0,0,1,0,1],[32,108,0.2963,0.28572,0.25505,0.0,0.42857,0.46429,0.0,0.71429,12,0,0,12,0,3,0,0,0,0,0,9,0,0,6,0,0,2,0,0,0,0,0],[36,108,0.3333,0.30356,0.23621,0.0,0.42857,0.42857,0.0,0.71429,11,0,0,11,0,1,0,0,0,0,0,14,0,0,5,0,0,1,0,0,0,0,0],[40,108,0.3704,0.19196,0.24383,0.0,0.0,0.42857,0.0,0.71429,19,0,0,19,0,0,0,0,2,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[44,108,0.4074,0.16518,0.20858,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,3,0,0,1,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[48,108,0.4444,0.21429,0.26245,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,1,0,0,1,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[52,108,0.4815,0.21874,0.20509,0.0,0.14288,0.42857,0.0,0.57143,12,0,0,12,0,5,0,0,4,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[56,108,0.5185,0.21429,0.20516,0.0,0.21429,0.42857,0.0,0.57143,14,0,0,14,0,2,0,0,3,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[60,108,0.5556,0.14732,0.24086,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,6,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[64,108,0.5926,0.20982,0.23954,0.0,0.14286,0.42857,0.0,0.85714,15,0,0,15,0,4,0,0,2,0,0,7,0,0,3,0,0,0,0,0,1,0,0],[68,108,0.6296,0.28569,0.23955,0.0,0.42857,0.42857,0.0,0.71429,11,0,0,11,0,3,0,0,1,0,0,10,0,0,6,0,0,1,0,0,0,0,0],[72,108,0.6667,0.20536,0.24468,0.0,0.14286,0.42857,0.0,0.85714,15,0,0,15,0,4,0,0,4,0,0,6,0,0,0,0,0,2,0,0,1,0,0],[76,108,0.7037,0.20536,0.24727,0.0,0.0,0.42857,0.0,0.85714,17,0,0,17,0,2,0,0,1,0,0,8,0,0,3,0,0,0,0,0,1,0,0],[80,108,0.7407,0.24107,0.2683,0.0,0.07143,0.42857,0.0,0.85714,16,0,0,16,0,1,0,0,1,0,0,9,0,0,2,0,0,2,0,0,1,0,0],[84,108,0.7778,0.16518,0.22048,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,4,0,0,1,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[88,108,0.8148,0.16964,0.19704,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,7,0,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[92,108,0.8519,0.12499,0.19147,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,2,0,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[96,108,0.8889,0.18304,0.22654,0.0,0.07143,0.42857,0.0,0.71429,16,0,0,16,0,5,0,0,2,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[100,108,0.9259,0.33482,0.3126,0.0,0.35714,0.57143,0.0,0.85714,12,0,0,12,0,2,0,0,2,0,0,5,0,0,5,0,0,2,0,0,4,0,0],[104,108,0.963,0.22768,0.2392,0.0,0.1429,0.42857,0.0,0.71429,14,0,0,14,0,4,0,0,2,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[108,108,1.0,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.08036,"x":0.27231,"p":[[0,107,0.0,0.20534,0.22568,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,3,0,0,4,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[4,107,0.0374,0.27231,0.28427,0.0,0.28571,0.42857,0.0,1.0,14,1,0,14,0,2,0,0,0,0,0,11,0,0,2,0,0,1,0,0,1,0,1],[8,107,0.0748,0.23661,0.26633,0.0,0.0,0.42857,0.0,0.85714,17,0,0,17,0,0,0,0,0,0,0,10,0,0,3,0,0,1,0,0,1,0,0],[12,107,0.1121,0.21429,0.23419,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,5,0,0,2,0,0,7,0,0,2,0,0,2,0,0,0,0,0],[16,107,0.1495,0.25893,0.25364,0.0,0.28571,0.42857,0.0,1.0,13,1,0,13,0,2,0,0,2,0,0,11,0,0,3,0,0,0,0,0,0,0,1],[20,107,0.1869,0.20982,0.2082,0.0,0.14286,0.42857,0.0,0.57143,14,0,0,14,0,3,0,0,3,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[24,107,0.2243,0.21875,0.2172,0.0,0.14288,0.42857,0.0,0.57143,14,0,0,14,0,3,0,0,2,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[28,107,0.2617,0.22322,0.24984,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,4,0,0,0,0,0,9,0,0,1,0,0,3,0,0,0,0,0],[32,107,0.2991,0.125,0.1915,0.0,0.0,0.28571,0.0,0.57143,22,0,0,22,0,0,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[36,107,0.3364,0.18304,0.2237,0.0,0.0,0.42857,0.0,0.85714,17,0,0,17,0,2,0,0,3,0,0,9,0,0,0,0,0,0,0,0,1,0,0],[40,107,0.3738,0.15624,0.18333,0.0,0.07143,0.28571,0.0,0.571,16,0,0,16,0,5,0,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[44,107,0.4112,0.25,0.21429,0.0,0.28571,0.42857,0.0,0.57143,12,0,0,12,0,2,0,0,3,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[48,107,0.4486,0.1875,0.19377,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,2,0,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[52,107,0.486,0.08036,0.15126,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,107,0.5234,0.15179,0.18536,0.0,0.0,0.42857,0.0,0.4286,17,0,0,17,0,5,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[60,107,0.5607,0.15625,0.19352,0.0,0.0,0.32143,0.0,0.57143,18,0,0,18,0,2,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[64,107,0.5981,0.1875,0.20652,0.0,0.0,0.42857,0.0,0.4286,17,0,0,17,0,1,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[68,107,0.6355,0.20981,0.21422,0.0,0.21428,0.42857,0.0,0.57143,15,0,0,15,0,1,0,0,5,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[72,107,0.6729,0.15179,0.19212,0.0,0.0,0.42857,0.0,0.4286,19,0,0,19,0,1,0,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[76,107,0.7103,0.23214,0.26665,0.0,0.14286,0.42857,0.0,0.85714,15,0,0,15,0,3,0,0,3,0,0,5,0,0,4,0,0,0,0,0,2,0,0],[80,107,0.7477,0.17411,0.20119,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,4,0,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[84,107,0.785,0.20089,0.20782,0.0,0.14288,0.42857,0.0,0.71429,14,0,0,14,0,3,0,0,6,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[88,107,0.8224,0.125,0.23076,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[92,107,0.8598,0.19643,0.21053,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,3,0,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[96,107,0.8972,0.14732,0.21572,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,2,0,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[100,107,0.9346,0.16518,0.1992,0.0,0.07143,0.32143,0.0,0.71429,16,0,0,16,0,5,0,0,3,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[104,107,0.972,0.13393,0.18536,0.0,0.0,0.32143,0.0,0.4286,20,0,0,20,0,2,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[107,107,1.0,0.17857,0.18898,0.0,0.07143,0.42857,0.0,0.4286,16,0,0,16,0,1,0,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"005e5c13541bedc5","q":"Suppose  $X$  is a random variable that takes on only nonnegative integer values, with  $E[X]=1,$   $E[X^2]=2,$  and  $E[X^3]=5.$  (Here  $E[Y]$  denotes the expectation of the random variable  $Y.$ ) Determine the smallest possible value of the probability of the event  $X=0.$","t":[{"b":6,"e":1.0,"k":"flat","v":0.92857,"x":0.99554,"p":[[0,71,0.0,0.94643,0.15872,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[4,71,0.0563,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[8,71,0.1127,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[12,71,0.169,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[16,71,0.2254,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[20,71,0.2817,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[24,71,0.338,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[28,71,0.3944,0.92857,0.18211,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,26],[32,71,0.4507,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,71,0.507,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,71,0.5634,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[44,71,0.6197,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,71,0.6761,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,71,0.7324,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,71,0.7887,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[60,71,0.8451,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,71,0.9014,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,71,0.9577,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[71,71,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":7,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,438,0.0,0.85714,0.30514,1.0,1.0,1.0,0.0,1.0,3,25,3,3,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,25],[4,438,0.0091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,438,0.0183,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[12,438,0.0274,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,438,0.0365,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,438,0.0457,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,438,0.0548,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[28,438,0.0639,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,438,0.0731,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,438,0.0822,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[40,438,0.0913,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,438,0.1005,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[48,438,0.1096,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[52,438,0.1187,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,438,0.1279,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,438,0.137,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[64,438,0.1461,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[68,438,0.1553,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[72,438,0.1644,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[76,438,0.1735,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[80,438,0.1826,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[84,438,0.1918,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,438,0.2009,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[92,438,0.21,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[96,438,0.2192,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[100,438,0.2283,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[104,438,0.2374,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[108,438,0.2466,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[112,438,0.2557,0.92857,0.15567,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,25],[116,438,0.2648,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[120,438,0.274,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[124,438,0.2831,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[128,438,0.2922,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[132,438,0.3014,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[136,438,0.3105,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[140,438,0.3196,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[144,438,0.3288,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[148,438,0.3379,0.95535,0.11539,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[152,438,0.347,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[156,438,0.3562,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[160,438,0.3653,0.91072,0.22517,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,25],[164,438,0.3744,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[168,438,0.3836,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[172,438,0.3927,0.9375,0.18877,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[176,438,0.4018,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[180,438,0.411,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[184,438,0.4201,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[188,438,0.4292,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[192,438,0.4384,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[196,438,0.4475,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[200,438,0.4566,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[204,438,0.4658,0.92857,0.14725,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[208,438,0.4749,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[212,438,0.484,0.95536,0.16536,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[216,438,0.4932,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[220,438,0.5023,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[224,438,0.5114,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[228,438,0.5205,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,27],[232,438,0.5297,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[236,438,0.5388,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[240,438,0.5479,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[244,438,0.5571,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[248,438,0.5662,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[252,438,0.5753,0.94196,0.11214,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,22],[256,438,0.5845,0.92856,0.14289,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,6,0,23],[260,438,0.5936,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[264,438,0.6027,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[268,438,0.6119,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[272,438,0.621,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[276,438,0.6301,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[280,438,0.6393,0.93304,0.16746,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,24],[284,438,0.6484,0.95535,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[288,438,0.6575,0.93304,0.16746,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,24],[292,438,0.6667,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[296,438,0.6758,0.93304,0.15561,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,22],[300,438,0.6849,0.94196,0.1551,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[304,438,0.6941,0.91071,0.17768,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,5,0,23],[308,438,0.7032,0.92411,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,21],[312,438,0.7123,0.85714,0.25254,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0,6,0,20],[316,438,0.7215,0.91071,0.18814,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,20],[320,438,0.7306,0.94643,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[324,438,0.7397,0.93304,0.17852,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[328,438,0.7489,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[332,438,0.758,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[336,438,0.7671,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[340,438,0.7763,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[344,438,0.7854,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[348,438,0.7945,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[352,438,0.8037,0.96427,0.10719,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[356,438,0.8128,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[360,438,0.8219,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[364,438,0.8311,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[368,438,0.8402,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[372,438,0.8493,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[376,438,0.8584,0.95981,0.12496,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[380,438,0.8676,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[384,438,0.8767,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[388,438,0.8858,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[392,438,0.895,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[396,438,0.9041,0.91964,0.15947,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,20],[400,438,0.9132,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[404,438,0.9224,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[408,438,0.9315,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[412,438,0.9406,0.94196,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[416,438,0.9498,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,25],[420,438,0.9589,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[424,438,0.968,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[428,438,0.9772,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,25],[432,438,0.9863,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[436,438,0.9954,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[438,438,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ef88396f20f652fb","q":"Let $p$ and $q$ be two distinct prime numbers, such that $p<2q$ and $q<2p$. Prove that there exist two consecutive integers, one of which has $p$ as its largest prime factor and the other has $q$ as its largest prime factor.","t":[{"b":5,"e":0.28571,"k":"flat","v":0.38394,"x":0.78571,"p":[[0,46,0.0,0.46874,0.27718,0.28571,0.42857,0.60714,0.0,1.0,3,3,0,3,0,2,0,0,8,0,0,4,0,0,7,0,0,4,0,0,1,0,3],[4,46,0.087,0.64283,0.27433,0.53539,0.57143,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,2,0,0,4,0,0,9,0,0,6,0,0,1,0,8],[8,46,0.1739,0.78571,0.22016,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,11,0,0,2,0,13],[12,46,0.2609,0.76338,0.18075,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,12,0,0,5,0,8],[16,46,0.3478,0.76783,0.21056,0.67857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,10,0,0,3,0,11],[20,46,0.4348,0.72768,0.20935,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,12,0,0,3,0,8],[24,46,0.5217,0.66516,0.20706,0.57143,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,16,0,0,2,0,4],[28,46,0.6087,0.70982,0.19061,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,12,0,0,3,0,6],[32,46,0.6957,0.69196,0.21461,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,10,0,0,4,0,6],[36,46,0.7826,0.61607,0.22142,0.42857,0.57143,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,8,0,0,5,0,0,4,0,4],[40,46,0.8696,0.6205,0.23039,0.53539,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,4,0,0,10,0,0,7,0,0,3,0,4],[44,46,0.9565,0.46873,0.1931,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,6,0,0,7,0,0,4,0,0,1,0,1],[46,46,1.0,0.38394,0.14032,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,17,0,0,7,0,0,5,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.37497,"x":0.67857,"p":[[0,47,0.0,0.47319,0.22427,0.28571,0.571,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,6,0,0,5,0,0,10,0,0,6,0,0,0,0,1],[4,47,0.0851,0.67857,0.26726,0.57143,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,1,0,0,2,0,0,10,0,0,7,0,0,1,0,9],[8,47,0.1702,0.62945,0.21387,0.53539,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,7,0,0,10,0,0,9,0,0,0,0,5],[12,47,0.2553,0.60712,0.23146,0.57143,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,2,0,0,0,0,0,13,0,0,9,0,0,2,0,3],[16,47,0.3404,0.58927,0.22799,0.53539,0.57143,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,2,0,0,4,0,0,11,0,0,8,0,0,3,0,2],[20,47,0.4255,0.55802,0.25595,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,6,0,0,8,0,0,4,0,0,4,0,3],[24,47,0.5106,0.55804,0.24578,0.42857,0.57143,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,3,0,0,6,0,0,6,0,0,10,0,0,2,0,2],[28,47,0.5957,0.5625,0.29653,0.28571,0.71429,0.71429,0.0,1.0,4,3,0,4,0,0,0,0,6,0,0,0,0,0,5,0,0,11,0,0,3,0,3],[32,47,0.6809,0.59375,0.25281,0.39286,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,7,0,0,2,0,0,8,0,0,7,0,0,3,0,4],[36,47,0.766,0.58033,0.28333,0.42857,0.57143,0.74996,0.0,1.0,3,4,0,3,0,0,0,0,3,0,0,7,0,0,4,0,0,7,0,0,4,0,4],[40,47,0.8511,0.51784,0.2714,0.39286,0.57143,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,3,0,0,5,0,0,6,0,0,9,0,0,3,0,1],[44,47,0.9362,0.37497,0.26182,0.14286,0.42857,0.57143,0.0,1.0,4,2,0,4,0,7,0,0,4,0,0,6,0,0,8,0,0,1,0,0,0,0,2],[47,47,1.0,0.50886,0.15944,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,5,0,0,15,0,0,4,0,0,1,0,0]]}]},{"i":"a9b3f9c0e5933fcf","q":"Let $r$ be a positive integer, and let $a_{0}, a_{1}, \\ldots$ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \\in[m+1, m+r]$ such that  $$ a_{m}+a_{m+1}+\\cdots+a_{m+s}=a_{n}+a_{n+1}+\\cdots+a_{n+s} $$  Prove that the sequence is periodic, i. e. there exists some $p \\geqslant 1$ such that $a_{n+p}=a_{n}$ for all $n \\geqslant 0$.  (India)","t":[{"b":0,"e":0.57143,"k":"rising","v":0.10714,"x":0.46426,"p":[[0,52,0.0,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.30357,0.27374,0.14286,0.2143,0.42858,0.0,0.85714,6,0,0,6,0,10,0,0,6,0,0,3,0,0,2,0,0,1,0,0,4,0,0],[8,52,0.1538,0.24545,0.19642,0.14286,0.2143,0.28571,0.0,0.85714,5,0,0,5,0,11,0,0,10,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[12,52,0.2308,0.41068,0.28736,0.14286,0.42857,0.60714,0.0,1.0,4,1,0,4,0,8,0,0,2,0,0,4,0,0,6,0,0,5,0,0,2,0,1],[16,52,0.3077,0.34375,0.2854,0.14286,0.35714,0.57143,0.0,0.85714,7,0,0,7,0,8,0,0,1,0,0,6,0,0,4,0,0,3,0,0,3,0,0],[20,52,0.3846,0.44195,0.22405,0.28571,0.42857,0.60714,0.0,0.71429,2,0,0,2,0,4,0,0,6,0,0,5,0,0,7,0,0,8,0,0,0,0,0],[24,52,0.4615,0.46426,0.24999,0.28571,0.4286,0.60714,0.0,1.0,2,1,0,2,0,4,0,0,5,0,0,6,0,0,7,0,0,5,0,0,2,0,1],[28,52,0.5385,0.41961,0.29651,0.14286,0.35714,0.71429,0.0,0.85714,4,0,0,4,0,7,0,0,5,0,0,2,0,0,4,0,0,5,0,0,5,0,0],[32,52,0.6154,0.42408,0.22722,0.24999,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,4,0,0,6,0,0,7,0,0,7,0,0,0,0,0],[36,52,0.6923,0.33036,0.21558,0.14289,0.28571,0.57143,0.0,0.71429,6,0,0,6,0,3,0,0,9,0,0,4,0,0,9,0,0,1,0,0,0,0,0],[40,52,0.7692,0.42857,0.26,0.14289,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,5,0,0,3,0,0,5,0,0,6,0,0,8,0,0,1,0,0],[44,52,0.8462,0.35713,0.24742,0.14289,0.28571,0.57143,0.0,0.71429,5,0,0,5,0,6,0,0,6,0,0,4,0,0,5,0,0,6,0,0,0,0,0],[48,52,0.9231,0.27679,0.26229,0.14286,0.14286,0.42857,0.0,1.0,5,2,0,5,0,14,0,0,3,0,0,6,0,0,0,0,0,2,0,0,0,0,2],[52,52,1.0,0.32141,0.23957,0.14286,0.2857,0.57111,0.0,0.71429,5,0,0,5,0,8,0,0,7,0,0,3,0,0,4,0,0,5,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"rising","v":0.13822,"x":0.52678,"p":[[0,42,0.0,0.13822,0.0435,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.4509,0.25028,0.25001,0.4286,0.60714,0.0,0.85714,2,0,0,2,0,6,0,0,3,0,0,6,0,0,7,0,0,5,0,0,3,0,0],[8,42,0.1905,0.37054,0.22548,0.14286,0.42857,0.4643,0.0,0.85714,2,0,0,2,0,8,0,0,5,0,0,9,0,0,4,0,0,2,0,0,2,0,0],[12,42,0.2857,0.46875,0.21199,0.28571,0.4286,0.60714,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,5,0,0,7,0,0,6,0,0,2,0,0],[16,42,0.381,0.5,0.23958,0.28571,0.42859,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,5,0,0,8,0,0,6,0,0,3,0,0,6,0,0],[20,42,0.4762,0.48213,0.18471,0.39286,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,4,0,0,7,0,0,10,0,0,7,0,0,0,0,0],[24,42,0.5714,0.37054,0.21975,0.1429,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,8,0,0,6,0,0,3,0,0,6,0,0,0,0,0],[28,42,0.6667,0.42856,0.23689,0.2857,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,6,0,0,8,0,0,5,0,0,2,0,0,9,0,0,1,0,0],[32,42,0.7619,0.52678,0.25614,0.28571,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,7,0,0,4,0,0,5,0,0,8,0,0,4,0,1],[36,42,0.8571,0.40176,0.19701,0.2857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,8,0,0,6,0,0,6,0,0,5,0,0,0,0,0],[40,42,0.9524,0.375,0.26184,0.14286,0.35714,0.57143,0.0,0.85714,5,0,0,5,0,5,0,0,6,0,0,6,0,0,3,0,0,5,0,0,2,0,0],[42,42,1.0,0.42407,0.21863,0.28571,0.42857,0.60714,0.0,0.71429,2,0,0,2,0,3,0,0,9,0,0,6,0,0,4,0,0,8,0,0,0,0,0]]}]},{"i":"c8615d9c611712cc","q":"Solve in the integers the diophantine equation  $$ x^4-6x^2+1 = 7 \\cdot 2^y. $$","t":[{"b":2,"e":0.57143,"k":"falling","v":0.64731,"x":0.98661,"p":[[0,46,0.0,0.83036,0.21558,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,1,0,0,3,0,18],[4,46,0.087,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[8,46,0.1739,0.88392,0.19379,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,0,0,23],[12,46,0.2609,0.91071,0.16269,0.96425,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,1,0,24],[16,46,0.3478,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[20,46,0.4348,0.92411,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[24,46,0.5217,0.92409,0.16748,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,26],[28,46,0.6087,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,46,0.6957,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,46,0.7826,0.74997,0.22306,0.57143,0.57143,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,14],[40,46,0.8696,0.70982,0.20355,0.57143,0.57143,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,0,0,0,1,0,10],[44,46,0.9565,0.67409,0.17582,0.57143,0.57143,0.71429,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,2,0,0,0,0,7],[46,46,1.0,0.64731,0.17123,0.57143,0.57143,0.57143,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,0,0,0,6]]},{"b":4,"e":0.71429,"k":"flat","v":0.74554,"x":0.95536,"p":[[0,15,0.0,0.8125,0.26592,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,2,0,0,6,0,0,1,0,0,2,0,19],[4,15,0.2667,0.74554,0.22794,0.57143,0.64286,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,2,0,0,1,0,13],[8,15,0.5333,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,15,0.8,0.79463,0.21707,0.57143,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,7,0,0,1,0,15],[15,15,1.0,0.81694,0.18642,0.57143,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,15]]}]},{"i":"f1d8502df4d8d7e4","q":"Let $p$ be an odd prime, $h<p$ an integer, $e \\in\\{1,2\\}$.\nWe set $n=h \\cdot p^{e}+1$ and assume that:\n\n$$\n\\left\\{\\begin{array}{r}\nn \\mid 2^{n-1}-1 \\\\\nn \\nmid 2^{h}-1\n\\end{array}\\right.\n$$\n\nShow that $n$ is prime.","t":[{"b":2,"e":0.71429,"k":"rising","v":0.28576,"x":0.72766,"p":[[0,43,0.0,0.28576,0.15976,0.14289,0.28571,0.42857,0.0,0.57143,4,0,1,4,0,5,0,0,13,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[4,43,0.093,0.41964,0.15947,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,0,10,0,0,7,0,0,3,0,0,0,0,0],[8,43,0.186,0.44642,0.15464,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,8,0,0,11,0,0,9,0,0,3,0,0,0,0,0],[12,43,0.2791,0.44196,0.15714,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,13,0,0,7,0,0,2,0,0,1,0,0],[16,43,0.3721,0.47321,0.16535,0.42857,0.50001,0.57143,0.14286,0.857,0,0,0,0,0,3,0,0,4,0,0,9,0,0,13,0,0,2,0,0,1,0,0],[20,43,0.4651,0.49999,0.19561,0.42857,0.5712,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,8,0,0,11,0,0,3,0,0,3,0,0],[24,43,0.5581,0.64286,0.19233,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,9,0,0,9,0,0],[28,43,0.6512,0.63839,0.1636,0.57143,0.64286,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,2,0,0,12,0,0,10,0,0,6,0,0],[32,43,0.7442,0.63392,0.15947,0.57132,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,13,0,0,5,0,0],[36,43,0.8372,0.71427,0.14287,0.71429,0.71429,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,17,0,0,6,0,2],[40,43,0.9302,0.72766,0.13535,0.67857,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,11,0,0,13,0,0],[43,43,1.0,0.68301,0.12747,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,14,0,0,5,0,1]]},{"b":3,"e":0.0,"k":"falling","v":0.15616,"x":0.41516,"p":[[0,23,0.0,0.34822,0.1234,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,14,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[4,23,0.1739,0.41516,0.19017,0.28571,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,3,0,0,8,0,0,11,0,0,6,0,0,1,0,0,2,0,0],[8,23,0.3478,0.40624,0.16014,0.28571,0.42857,0.4642,0.0,0.71429,1,0,0,1,0,1,0,0,11,0,0,11,0,0,5,0,0,3,0,0,0,0,0],[12,23,0.5217,0.36607,0.15542,0.2857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,9,0,0,14,0,0,2,0,0,2,0,0,0,0,0],[16,23,0.6957,0.20536,0.13803,0.14286,0.14286,0.28571,0.0,0.4286,5,0,0,5,0,14,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.16062,0.12244,0.105,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,14,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.15616,0.13998,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,12,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"20093d7584af9435","q":"Let $x, y, z$ be positive integers such that $x \\neq y \\neq z \\neq x$. Prove that\n\n$$\n(x+y+z)(x y+y z+z x-2) \\geq 9 x y z\n$$\n\nWhen does the equality hold?","t":[{"b":1,"e":0.14286,"k":"falling","v":0.26338,"x":0.88839,"p":[[0,113,0.0,0.62051,0.30642,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,4,0,0,2,0,0,3,0,0,8,0,0,2,0,8],[4,113,0.0354,0.88839,0.17762,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,21],[8,113,0.0708,0.73212,0.29398,0.571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,6,0,0,0,0,0,5,0,0,4,0,0,1,0,15],[12,113,0.1062,0.79464,0.24984,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,8,0,0,3,0,15],[16,113,0.1416,0.87052,0.20002,0.82132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,5,0,19],[20,113,0.177,0.79462,0.24209,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,0,2,0,15],[24,113,0.2124,0.79016,0.26484,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,4,0,0,2,0,17],[28,113,0.2478,0.79464,0.24468,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,6,0,0,4,0,15],[32,113,0.2832,0.83929,0.21943,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,0,0,19],[36,113,0.3186,0.88838,0.1417,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,18],[40,113,0.354,0.79911,0.2621,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,9,0,0,2,0,16],[44,113,0.3894,0.82143,0.20516,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,2,0,16],[48,113,0.4248,0.79911,0.25966,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,6,0,0,2,0,17],[52,113,0.4602,0.77232,0.30275,0.53571,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,6,0,0,1,0,0,1,0,0,3,0,0,2,0,18],[56,113,0.4956,0.82142,0.23421,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,0,0,19],[60,113,0.531,0.88839,0.19475,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,4,0,21],[64,113,0.5664,0.79463,0.23942,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,4,0,0,2,0,16],[68,113,0.6018,0.76337,0.27576,0.57132,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,5,0,0,1,0,16],[72,113,0.6372,0.7857,0.25,0.71429,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,8,0,0,4,0,14],[76,113,0.6726,0.78125,0.2575,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,4,0,0,1,0,0,5,0,0,4,0,0,2,0,16],[80,113,0.708,0.7589,0.26593,0.57132,0.78564,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,6,0,0,1,0,15],[84,113,0.7434,0.76338,0.24901,0.57143,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,7,0,0,1,0,14],[88,113,0.7788,0.76339,0.25155,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,7,0,0,5,0,12],[92,113,0.8142,0.82143,0.25,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,6,0,0,4,0,17],[96,113,0.8496,0.72321,0.26949,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,7,0,0,2,0,12],[100,113,0.885,0.83035,0.25364,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,6,0,0,2,0,19],[104,113,0.9204,0.79017,0.27196,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,4,0,16],[108,113,0.9558,0.64732,0.3499,0.28571,0.71429,1.0,0.0,1.0,1,12,0,1,0,4,0,0,6,0,0,1,0,0,1,0,0,4,0,0,3,0,12],[112,113,0.9912,0.40625,0.26752,0.24999,0.28571,0.57143,0.0,1.0,1,3,0,1,0,7,0,0,11,0,0,2,0,0,5,0,0,3,0,0,0,0,3],[113,113,1.0,0.26338,0.16406,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,8,0,0,13,0,0,3,0,0,4,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.67854,"x":0.92411,"p":[[0,130,0.0,0.77674,0.25493,0.5713,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,6,0,0,2,0,15],[4,130,0.0308,0.85712,0.20518,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,0,5,0,18],[8,130,0.0615,0.76784,0.27375,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,4,0,0,0,0,0,5,0,0,5,0,0,1,0,16],[12,130,0.0923,0.79018,0.31539,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,5,0,0,0,0,0,3,0,0,0,0,0,1,0,21],[16,130,0.1231,0.83482,0.23175,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,0,6,0,17],[20,130,0.1538,0.67854,0.35356,0.28571,0.85714,1.0,0.0,1.0,1,16,0,1,0,3,0,0,6,0,0,1,0,0,3,0,0,2,0,0,0,0,16],[24,130,0.1846,0.78571,0.26486,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,3,0,0,4,0,16],[28,130,0.2154,0.83034,0.2354,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,2,0,19],[32,130,0.2462,0.92411,0.1636,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,25],[36,130,0.2769,0.74107,0.30396,0.57143,0.85707,1.0,0.0,1.0,1,14,0,1,0,2,0,0,3,0,0,0,0,0,3,0,0,6,0,0,3,0,14],[40,130,0.3077,0.81696,0.20589,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,7,0,0,5,0,14],[44,130,0.3385,0.74554,0.29393,0.53571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,2,0,0,3,0,0,4,0,0,1,0,16],[48,130,0.3692,0.90624,0.13654,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,20],[52,130,0.4,0.77232,0.23107,0.57143,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,7,0,0,1,0,14],[56,130,0.4308,0.82589,0.22513,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,3,0,0,2,0,18],[60,130,0.4615,0.79017,0.23415,0.67857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,6,0,0,4,0,14],[64,130,0.4923,0.8348,0.25283,0.82132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,5,0,19],[68,130,0.5231,0.80804,0.25153,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,4,0,0,3,0,17],[72,130,0.5538,0.73659,0.27226,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,6,0,0,0,0,0,7,0,0,2,0,0,4,0,13],[76,130,0.5846,0.8125,0.2683,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,0,2,0,19],[80,130,0.6154,0.86161,0.23003,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,1,0,22],[84,130,0.6462,0.89731,0.1864,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,21],[88,130,0.6769,0.85268,0.26119,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,0,2,0,22],[92,130,0.7077,0.85714,0.24485,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,0,0,23],[96,130,0.7385,0.875,0.1948,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,20],[100,130,0.7692,0.90625,0.17717,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,22],[104,130,0.8,0.8482,0.21708,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,5,0,18],[108,130,0.8308,0.85714,0.2342,0.85711,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,5,0,20],[112,130,0.8615,0.81249,0.27068,0.57143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,2,0,0,1,0,20],[116,130,0.8923,0.88838,0.20121,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,6,0,21],[120,130,0.9231,0.91963,0.14261,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[124,130,0.9538,0.91964,0.13803,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,22],[128,130,0.9846,0.89286,0.20825,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,1,0,24],[130,130,1.0,0.69194,0.27459,0.5713,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,6,0,0,0,0,0,7,0,0,3,0,0,6,0,9]]}]},{"i":"f7ed7dc8159a40f0","q":"Let sequences of real numbers  $(x_n)$  and  $(y_n)$  satisfy  $x_1 = y_1 = 1$  and  $x_{n+1} =\\frac{x_n + 2}{x_n + 1}$   and  $y_{n+1} = \\frac{y_n^2 + 2}{2y_n}$  for  $n = 1,2, ...$  Prove that  $y_{n+1} = x_{2^n}$  holds for  $n =0, 1,2, ... $","t":[{"b":1,"e":1.0,"k":"rising","v":0.75446,"x":0.99554,"p":[[0,70,0.0,0.75446,0.35934,0.28571,1.0,1.0,0.14286,1.0,0,20,0,0,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,20],[4,70,0.0571,0.94642,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[8,70,0.1143,0.87944,0.25031,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,24],[12,70,0.1714,0.8705,0.256,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,25],[16,70,0.2286,0.92411,0.21424,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[20,70,0.2857,0.9375,0.2111,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[24,70,0.3429,0.9241,0.20512,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[28,70,0.4,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[32,70,0.4571,0.9195,0.15563,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,25],[36,70,0.5143,0.94642,0.1372,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[40,70,0.5714,0.91071,0.1915,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,4,0,24],[44,70,0.6286,0.85266,0.25628,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,1,0,22],[48,70,0.6857,0.89285,0.25254,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,25],[52,70,0.7429,0.875,0.29613,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[56,70,0.8,0.83925,0.29181,0.82132,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,23],[60,70,0.8571,0.89732,0.20277,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,23],[64,70,0.9143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,70,0.9714,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[70,70,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.74105,"x":0.96875,"p":[[0,53,0.0,0.74105,0.33965,0.53539,1.0,1.0,0.14286,1.0,0,17,0,0,0,6,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,17],[4,53,0.0755,0.74107,0.3163,0.53569,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,2,0,0,2,0,0,2,0,0,4,0,0,2,0,16],[8,53,0.1509,0.80356,0.30041,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,19],[12,53,0.2264,0.91517,0.20784,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,26],[16,53,0.3019,0.88393,0.23807,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,24],[20,53,0.3774,0.875,0.26426,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,26],[24,53,0.4528,0.88839,0.2618,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[28,53,0.5283,0.89732,0.2402,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,26],[32,53,0.6038,0.90179,0.23538,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,26],[36,53,0.6792,0.90625,0.26392,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[40,53,0.7547,0.9375,0.18189,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,28],[44,53,0.8302,0.87052,0.27285,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,23],[48,53,0.9057,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[52,53,0.9811,0.94196,0.21086,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[53,53,1.0,0.82143,0.31339,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,22]]}]},{"i":"4c57b5cd341dcc0b","q":"Let the integer  $n \\ge 2$  , and  $x_1,x_2,\\cdots,x_n $  be real numbers such that  $\\sum_{k=1}^nx_k$  be integer  .  $d_k=\\underset{m\\in {Z}}{\\min}\\left|x_k-m\\right| $ ,  $1\\leq k\\leq n$  .Find the maximum  value of  $\\sum_{k=1}^nd_k$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,52,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,52,0.0769,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,52,0.1538,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,52,0.2308,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,52,0.3077,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,52,0.3846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,52,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,52,0.5385,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,52,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,52,0.6923,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.98213,"x":1.0,"p":[[0,26,0.0,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,26,0.1538,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,26,0.6154,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"7cb793659ba5afc8","q":"Suppose  $n\\ge 2$ . Consider the polynomial \\[Q_n(x) = 1-x^n - (1-x)^n .\\]\nShow that the equation  $Q_n(x) = 0$  has only two real roots, namely  $0$  and  $1$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.83927,"x":0.93303,"p":[[0,41,0.0,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[4,41,0.0976,0.92411,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[8,41,0.1951,0.88392,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,10],[12,41,0.2927,0.89286,0.09449,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,12],[16,41,0.3902,0.89741,0.08167,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,11],[20,41,0.4878,0.875,0.08564,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,20,0,8],[24,41,0.5854,0.85714,0.07143,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4],[28,41,0.6829,0.84375,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,2],[32,41,0.7805,0.84375,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,2],[36,41,0.878,0.85268,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,3],[40,41,0.9756,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[41,41,1.0,0.83927,0.05922,0.857,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,26,0,1]]},{"b":4,"e":0.85714,"k":"flat","v":0.83036,"x":0.9375,"p":[[0,11,0.0,0.92857,0.13364,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,0,20],[4,11,0.3636,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[8,11,0.7273,0.92857,0.14286,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[11,11,1.0,0.83036,0.05576,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0]]}]},{"i":"720de1e3d879a83c","q":"Let \\[ E_n=(a_1-a_2)(a_1-a_3)\\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\\ldots(a_2-a_n)+\\ldots+(a_n-a_1)(a_n-a_2)\\ldots(a_n-a_{n-1}). \\] Let  $S_n$  be the proposition that  $E_n\\ge0$  for all real  $a_i$ . Prove that  $S_n$  is true for  $n=3$  and  $5$ , but for no other  $n>2$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.31696,"x":0.81249,"p":[[0,32,0.0,0.31696,0.09933,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[4,32,0.125,0.80355,0.22233,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,3,0,0,6,0,14],[8,32,0.25,0.80354,0.22235,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,1,0,0,5,0,15],[12,32,0.375,0.81249,0.25113,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,2,0,0,1,0,19],[16,32,0.5,0.72763,0.24838,0.57142,0.71429,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,3,0,0,0,0,0,9,0,0,4,0,0,5,0,10],[20,32,0.625,0.7098,0.23003,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,7,0,0,4,0,8],[24,32,0.75,0.76334,0.24122,0.57143,0.78564,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,6,0,0,4,0,12],[28,32,0.875,0.71872,0.27777,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,0,0,0,9,0,0,3,0,0,3,0,12],[32,32,1.0,0.62494,0.29828,0.39286,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,4,0,0,2,0,0,8,0,0,2,0,0,4,0,8]]},{"b":5,"e":1.0,"k":"rising","v":0.3348,"x":0.83925,"p":[[0,30,0.0,0.3348,0.12166,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,26,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[4,30,0.1333,0.79905,0.20164,0.57143,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,0,4,0,14],[8,30,0.2667,0.74996,0.22018,0.57143,0.78564,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,8,0,0,5,0,0,7,0,9],[12,30,0.4,0.79007,0.2261,0.67857,0.85714,1.0,0.14,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,5,0,0,7,0,12],[16,30,0.5333,0.69192,0.25282,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,3,0,0,6,0,8],[20,30,0.6667,0.81248,0.18709,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,9,0,11],[24,30,0.8,0.70085,0.25346,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,1,0,0,9,0,0,4,0,0,6,0,8],[28,30,0.9333,0.83925,0.15876,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,7,0,13],[30,30,1.0,0.77228,0.21088,0.57143,0.857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,4,0,0,8,0,10]]}]},{"i":"a763451d2c72897d","q":"Let the circle  $ {\\omega}_{1}$  be internally tangent to another circle  $ {\\omega}_{2}$  at  $ N$ .Take a point  $ K$  on  $ {\\omega}_{1}$  and draw a tangent  $ AB$  which intersects  $ {\\omega}_{2}$  at  $ A$  and  $ B$ . Let  $M$  be the midpoint of the arc  $ AB$  which is on the opposite side of  $ N$ . Prove that, the circumradius of the  $ \\triangle KBM$  doesnt depend on the choice of  $ K$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.17857,"p":[[0,111,0.0,0.17411,0.18117,0.0,0.2143,0.28571,0.0,0.57143,15,0,3,15,0,1,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,111,0.036,0.17857,0.25754,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,1,0,0,9,0,0,0,0,0,2,0,0,0,0,0,1,0,1],[8,111,0.0721,0.07143,0.18558,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[12,111,0.1081,0.05804,0.16698,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,111,0.1441,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,111,0.1802,0.07589,0.17122,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[24,111,0.2162,0.05357,0.16656,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[28,111,0.2523,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,111,0.2883,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,111,0.3243,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,111,0.3604,0.04464,0.12079,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,111,0.3964,0.05804,0.14664,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,111,0.4324,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,111,0.4685,0.0625,0.15947,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,111,0.5045,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,111,0.5405,0.04911,0.18423,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,111,0.5766,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,111,0.6126,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,111,0.6486,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,111,0.6847,0.03125,0.09933,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,111,0.7207,0.03572,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,111,0.7568,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,111,0.8288,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,111,0.8649,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,111,0.9009,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,111,0.9369,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,111,0.973,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[111,111,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.25446,"p":[[0,101,0.0,0.25446,0.19475,0.14286,0.28571,0.28571,0.0,1.0,7,1,2,7,0,2,0,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[4,101,0.0396,0.12054,0.17169,0.0,0.0,0.2857,0.0,0.57143,20,0,0,20,0,1,0,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,101,0.0792,0.08482,0.17076,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,101,0.1188,0.07589,0.15146,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,101,0.1584,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,101,0.198,0.0625,0.13803,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,101,0.2376,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,101,0.2772,0.06696,0.17122,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[32,101,0.3168,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,101,0.3564,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,101,0.396,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,101,0.4356,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,101,0.4752,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,101,0.5149,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,101,0.5545,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,101,0.5941,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,101,0.6337,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,101,0.6733,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,101,0.7129,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,101,0.7525,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,101,0.7921,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,101,0.8317,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,101,0.8713,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,101,0.9109,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,101,0.9505,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,101,0.9901,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[101,101,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"058a04bf80014e33","q":"Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly  $10$  operations.","t":[{"b":1,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,74,0.0,0.94643,0.14174,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[4,74,0.0541,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,74,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,74,0.1622,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,74,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,74,0.2703,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[24,74,0.3243,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,74,0.3784,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[32,74,0.4324,0.97777,0.05166,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[36,74,0.4865,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[40,74,0.5405,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[44,74,0.5946,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,74,0.6486,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[52,74,0.7027,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,74,0.7568,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[60,74,0.8108,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[64,74,0.8649,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[68,74,0.9189,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[72,74,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[74,74,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.85714,"k":"falling","v":0.42847,"x":1.0,"p":[[0,53,0.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[4,53,0.0755,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,53,0.1509,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,53,0.2264,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,53,0.3019,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,53,0.3774,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,53,0.4528,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,53,0.5283,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[32,53,0.6038,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,53,0.6792,0.91964,0.20497,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,27],[40,53,0.7547,0.91071,0.21943,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[44,53,0.8302,0.91964,0.21706,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,28],[48,53,0.9057,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[52,53,0.9811,0.94643,0.17035,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[53,53,1.0,0.42847,0.27902,0.2857,0.28571,0.57143,0.14,1.0,0,5,0,0,0,4,0,0,18,0,0,0,0,0,4,0,0,1,0,0,0,0,5]]}]},{"i":"2c8cfa2012fb6935","q":"The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral  $q$ . Show that the sum of the lengths of each pair of opposite sides of  $q$  is equal.","t":[{"b":0,"e":0.0,"k":"falling","v":0.15615,"x":0.77679,"p":[[0,59,0.0,0.77679,0.28107,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,3,0,0,0,0,18],[4,59,0.0678,0.66515,0.4066,0.2857,0.92857,1.0,0.0,1.0,6,16,0,6,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,16],[8,59,0.1356,0.49088,0.41038,0.14214,0.42836,0.89286,0.0,1.0,6,8,0,6,0,9,0,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,8],[12,59,0.2034,0.5,0.44892,0.0,0.42857,1.0,0.0,1.0,10,11,0,10,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,11],[16,59,0.2712,0.49106,0.40553,0.14286,0.35714,1.0,0.0,1.0,4,11,0,4,0,10,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,11],[20,59,0.339,0.40178,0.37531,0.10714,0.28571,0.74996,0.0,1.0,8,5,0,8,0,6,0,0,6,0,0,0,0,0,1,0,0,3,0,0,3,0,5],[24,59,0.4068,0.33929,0.3458,0.0,0.21428,0.57143,0.0,1.0,10,4,0,10,0,6,0,0,3,0,0,4,0,0,2,0,0,2,0,0,1,0,4],[28,59,0.4746,0.29018,0.32631,0.0,0.14286,0.42857,0.0,1.0,11,3,0,11,0,6,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,3],[32,59,0.5424,0.38839,0.3867,0.0,0.2857,0.75,0.0,1.0,9,7,0,9,0,6,0,0,5,0,0,1,0,0,2,0,0,1,0,0,1,0,7],[36,59,0.6102,0.38393,0.39518,0.0,0.14286,0.85714,0.0,1.0,9,6,0,9,0,8,0,0,3,0,0,2,0,0,0,0,0,0,0,0,4,0,6],[40,59,0.678,0.3348,0.33617,0.0,0.21428,0.60682,0.0,1.0,9,2,0,9,0,7,0,0,5,0,0,2,0,0,1,0,0,2,0,0,4,0,2],[44,59,0.7458,0.24544,0.34115,0.0,0.14286,0.28579,0.0,1.0,14,4,0,14,0,8,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,4],[48,59,0.8136,0.24107,0.25111,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,6,0,0,3,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[52,59,0.8814,0.25897,0.28672,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0,2,0,2],[56,59,0.9492,0.24552,0.28844,0.0,0.14286,0.28571,0.0,1.0,11,2,0,11,0,8,0,0,6,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[59,59,1.0,0.15615,0.15711,0.0,0.14286,0.2857,0.0,0.571,12,0,0,12,0,10,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.41067,"x":0.90625,"p":[[0,55,0.0,0.90625,0.16213,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,23],[4,55,0.0727,0.46429,0.40406,0.14286,0.28571,0.89286,0.0,1.0,6,8,0,6,0,9,0,0,2,0,0,1,0,0,2,0,0,0,0,0,4,0,8],[8,55,0.1455,0.48206,0.38926,0.14286,0.28571,0.89286,0.0,1.0,5,8,0,5,0,7,0,0,5,0,0,1,0,0,1,0,0,2,0,0,3,0,8],[12,55,0.2182,0.61161,0.36287,0.2857,0.71429,1.0,0.0,1.0,2,12,0,2,0,4,0,0,6,0,0,1,0,0,2,0,0,4,0,0,1,0,12],[16,55,0.2909,0.41067,0.2961,0.14286,0.28571,0.60682,0.0,1.0,2,2,0,2,0,8,0,0,9,0,0,2,0,0,3,0,0,2,0,0,4,0,2],[20,55,0.3636,0.46866,0.31396,0.14289,0.42857,0.71429,0.0,1.0,2,5,0,2,0,7,0,0,5,0,0,5,0,0,2,0,0,6,0,0,0,0,5],[24,55,0.4364,0.70979,0.28902,0.53539,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,2,0,0,3,0,0,3,0,0,7,0,0,2,0,12],[28,55,0.5091,0.66516,0.32851,0.42857,0.71429,1.0,0.0,1.0,2,12,0,2,0,2,0,0,2,0,0,5,0,0,3,0,0,4,0,0,2,0,12],[32,55,0.5818,0.58033,0.30292,0.28571,0.57143,0.85714,0.0,1.0,1,6,0,1,0,4,0,0,4,0,0,4,0,0,4,0,0,6,0,0,3,0,6],[36,55,0.6545,0.68749,0.31225,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,4,0,0,5,0,0,3,0,0,3,0,0,2,0,13],[40,55,0.7273,0.66518,0.30849,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,4,0,0,1,0,0,3,0,0,3,0,0,7,0,0,4,0,9],[44,55,0.8,0.68738,0.26127,0.4286,0.71429,1.0,0.14,1.0,0,10,0,0,0,1,0,0,3,0,0,5,0,0,4,0,0,8,0,0,1,0,10],[48,55,0.8727,0.57143,0.27199,0.39286,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,5,0,0,6,0,0,6,0,0,5,0,0,1,0,6],[52,55,0.9455,0.57139,0.2945,0.28571,0.57141,0.75,0.0,1.0,1,5,0,1,0,4,0,0,4,0,0,4,0,0,4,0,0,7,0,0,3,0,5],[55,55,1.0,0.4866,0.24707,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,11,0,0,2,0,0,4,0,0,3,0,2]]}]},{"i":"3265266dfba475c3","q":"The Sequence  $\\{a_{n}\\}_{n \\geqslant 0}$   is defined by  $a_{0}=1, a_{1}=-4$  and  $a_{n+2}=-4a_{n+1}-7a_{n}$  , for  $n \\geqslant 0$ . Find the number of positive integer divisors of  $a^2_{50}-a_{49}a_{51}$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.86161,"x":0.91071,"p":[[0,38,0.0,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[4,38,0.1053,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[8,38,0.2105,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,38,0.3158,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[16,38,0.4211,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[20,38,0.5263,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[24,38,0.6316,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[28,38,0.7368,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[32,38,0.8421,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[36,38,0.9474,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[38,38,1.0,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5]]},{"b":4,"e":0.85714,"k":"flat","v":0.86607,"x":0.88839,"p":[[0,57,0.0,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[4,57,0.0702,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[8,57,0.1404,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[12,57,0.2105,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[16,57,0.2807,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[20,57,0.3509,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[24,57,0.4211,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[28,57,0.4912,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[32,57,0.5614,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[36,57,0.6316,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[40,57,0.7018,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[44,57,0.7719,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[48,57,0.8421,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[52,57,0.9123,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[56,57,0.9825,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[57,57,1.0,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2]]}]},{"i":"09261f11ec0aea76","q":"Prove that given three positive numbers, we can choose two of them, say  $x$  and  $y,$  with  $x >y$  such that   $$ \\frac{x-y}{1 +xy }<1. $$   Prove also  that if the number  $1$  that appears in the second member of the previous inequality is replaced by a lower number, even if very close to  $1$ , the previous proposition is false.","t":[{"b":4,"e":0.0,"k":"falling","v":0.14732,"x":0.45987,"p":[[0,21,0.0,0.45987,0.46666,0.0,0.42857,1.0,0.0,1.0,15,13,0,15,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13],[4,21,0.1905,0.30357,0.4385,0.0,0.0,0.89275,0.0,1.0,21,8,0,21,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,8],[8,21,0.381,0.14732,0.31841,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,2],[12,21,0.5714,0.22321,0.40396,0.0,0.0,0.07143,0.0,1.0,24,6,0,24,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[16,21,0.7619,0.14732,0.27545,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[20,21,0.9524,0.24552,0.39806,0.0,0.0,0.46418,0.0,1.0,22,5,0,22,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,5],[21,21,1.0,0.14732,0.28231,0.0,0.0,0.17857,0.0,1.0,23,2,0,23,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,2]]},{"b":5,"e":0.0,"k":"falling","v":0.03125,"x":0.58487,"p":[[0,34,0.0,0.58487,0.47695,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18],[4,34,0.1176,0.29911,0.44515,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[8,34,0.2353,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[12,34,0.3529,0.125,0.3004,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[16,34,0.4706,0.05362,0.14185,0.0,0.0,0.0,0.0,0.43,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.10714,0.25505,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[24,34,0.7059,0.14287,0.26728,0.0,0.0,0.1786,0.0,1.0,22,2,0,22,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[28,34,0.8235,0.12054,0.18595,0.0,0.0,0.32143,0.0,0.4286,22,0,0,22,0,1,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.12938,0.23517,0.0,0.0,0.21214,0.0,1.0,23,1,0,23,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[34,34,1.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fa355623c513bebd","q":"The sum of the angles $A$ and $C$ of a convex quadrilateral $A B C D$ is less than $180^{\\circ}$. Prove that\n\n$$\nA B \\cdot C D+A D \\cdot B C<A C(A B+A D)\n$$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,54,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,54,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,54,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,72,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,72,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,72,0.3889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,72,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,72,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3777179b20440e52","q":"Let $x, y, z$ be positive numbers such that\n\n$$\n\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=1\n$$\n\nShow that\n\n$$\n\\sqrt{x+y z}+\\sqrt{y+z x}+\\sqrt{z+x y} \\geq \\sqrt{x y z}+\\sqrt{x}+\\sqrt{y}+\\sqrt{z}\n$$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.87946,"x":0.91072,"p":[[0,7,0.0,0.91072,0.21943,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,25],[4,7,0.5714,0.87946,0.2372,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,4,0,23],[7,7,1.0,0.89284,0.06187,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]},{"b":3,"e":1.0,"k":"flat","v":0.89284,"x":0.99107,"p":[[0,4,0.0,0.89284,0.20827,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,24],[4,4,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"bb91a04ac0bea09f","q":"There are  $n$  boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of  $n$  can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than  $n$  dances?","t":[{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,35,0.0,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,26],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,35,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,17,0.0,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[4,17,0.2353,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,17,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,17,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,17,0.9412,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[17,17,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ccd27da909d09a6b","q":"The point  $C \\ne A, B$  belongs to a circle of diameter  $AB$ . Point  $D$  divides the shorter arc  $BC$  in  half. Lines  $AD$  and  $BC$  intersects at point  $E$ . Find the length of the segment  $AB$  if  $CE = 3$ ,  $BD = 2\\sqrt5$  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sphere  $ \\omega $  passes through the vertex  $S$  of the pyramid  $SABC$  and intersects with the edges  $SA,SB,SC$  at  $A_1,B_1,C_1$   other than  $S$ . The sphere  $ \\Omega $  is the circumsphere of the pyramid  $SABC$  and intersects with  $ \\omega $  circumferential,  lies on a plane which parallel to the plane  $(ABC)$ . \nPoints  $A_2,B_2,C_2$  are symmetry points of the points  $A_1,B_1,C_1$  respect to midpoints of the edges  $SA,SB,SC$  respectively. Prove that the points  $A$ ,  $B$ ,  $C$ ,  $A_2$ ,  $B_2$ , and   $C_2$  lie on a sphere.","t":[{"b":6,"e":0.0,"k":"flat","v":0.10705,"x":0.18299,"p":[[0,89,0.0,0.16955,0.11401,0.14214,0.14286,0.2857,0.0,0.42857,5,0,0,5,2,15,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,89,0.0449,0.16509,0.08075,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,89,0.0899,0.18299,0.09621,0.14286,0.14286,0.2857,0.0,0.43,2,0,0,2,0,21,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,89,0.1348,0.18071,0.07577,0.14286,0.14286,0.1786,0.07143,0.42857,0,0,0,0,1,23,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,89,0.1798,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,89,0.2247,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,89,0.2697,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,89,0.3146,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,89,0.3596,0.16072,0.07783,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,89,0.4045,0.16491,0.0808,0.14286,0.14286,0.1786,0.0,0.28571,3,0,0,3,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,89,0.4494,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,89,0.4944,0.165,0.08832,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,19,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,89,0.5393,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,89,0.5843,0.16482,0.08083,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,24,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,89,0.6292,0.15178,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,89,0.6742,0.15179,0.09408,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,21,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,89,0.7191,0.15625,0.09006,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,22,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.13375,0.07085,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.12482,0.06912,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.14715,0.05631,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.14268,0.07986,0.14286,0.14286,0.14287,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.11116,0.05883,0.14,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.15616,"x":0.22769,"p":[[0,39,0.0,0.20981,0.1128,0.14286,0.14288,0.2857,0.0,0.571,1,0,0,1,0,19,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,39,0.1026,0.16072,0.06916,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.17848,0.09453,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,25,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.2142,0.07994,0.14286,0.14293,0.28571,0.14,0.4286,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.22321,0.1234,0.14286,0.14286,0.28571,0.0,0.4286,1,0,0,1,0,19,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.20974,0.10098,0.14286,0.14288,0.28571,0.0,0.4286,1,0,0,1,0,18,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.22769,0.08645,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,15,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.21419,0.10716,0.14286,0.14286,0.28571,0.14,0.571,0,0,0,0,0,20,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[32,39,0.8205,0.20089,0.10012,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,22,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,39,0.9231,0.18304,0.07349,0.14286,0.14286,0.1786,0.14286,0.4286,0,0,0,0,0,24,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fa7322fc3f120277","q":"The set  $M$  consists of integers, the smallest of which is  $1$  and the greatest  $100$ . Each member of  $M$ , except  $1$ , is the sum of two (possibly identical) numbers in  $M$ . Of all such sets, find one with the smallest possible number of elements.","t":[{"b":0,"e":1.0,"k":"flat","v":0.67857,"x":0.97768,"p":[[0,156,0.0,0.69641,0.27607,0.42857,0.71429,1.0,0.0,1.0,1,11,1,1,0,0,0,0,1,0,0,9,0,0,3,0,0,4,0,0,3,0,11],[4,156,0.0256,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,156,0.0513,0.91964,0.15542,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,23],[12,156,0.0769,0.86606,0.24728,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,22],[16,156,0.1026,0.82587,0.24933,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,5,0,17],[20,156,0.1282,0.89732,0.21498,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,23],[24,156,0.1538,0.82594,0.26416,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,1,0,20],[28,156,0.1795,0.77677,0.31123,0.57143,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,2,0,18],[32,156,0.2051,0.82588,0.26424,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,3,0,19],[36,156,0.2308,0.86159,0.24088,0.82132,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,21],[40,156,0.2564,0.87052,0.22121,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,21],[44,156,0.2821,0.83927,0.26668,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,4,0,20],[48,156,0.3077,0.84374,0.27748,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,22],[52,156,0.3333,0.87498,0.24938,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[56,156,0.359,0.75445,0.3219,0.57132,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,3,0,16],[60,156,0.3846,0.84375,0.27283,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,22],[64,156,0.4103,0.82589,0.28735,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,4,0,20],[68,156,0.4359,0.79911,0.30485,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,0,5,0,18],[72,156,0.4615,0.77232,0.32705,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,18],[76,156,0.4872,0.67857,0.3481,0.42859,0.78571,1.0,0.0,1.0,4,12,0,4,0,0,0,0,3,0,0,3,0,0,1,0,0,5,0,0,4,0,12],[80,156,0.5128,0.82589,0.26901,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,4,0,0,1,0,0,3,0,0,2,0,20],[84,156,0.5385,0.83034,0.2096,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,2,0,17],[88,156,0.5641,0.80802,0.24121,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,4,0,0,1,0,18],[92,156,0.5897,0.85713,0.19887,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,5,0,18],[96,156,0.6154,0.84375,0.24053,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,5,0,0,2,0,20],[100,156,0.641,0.8125,0.25614,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,1,0,0,4,0,18],[104,156,0.6667,0.83927,0.21055,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,0,2,0,18],[108,156,0.6923,0.83481,0.21163,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,2,0,18],[112,156,0.7179,0.82588,0.19476,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,6,0,14],[116,156,0.7436,0.82588,0.2544,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,4,0,0,1,0,20],[120,156,0.7692,0.88392,0.17657,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,3,0,20],[124,156,0.7949,0.85259,0.24376,0.71429,1.0,1.0,0.14,1.0,0,21,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,4,0,0,2,0,21],[128,156,0.8205,0.80355,0.21056,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,2,0,15],[132,156,0.8462,0.83928,0.21354,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,3,0,18],[136,156,0.8718,0.89732,0.18292,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,4,0,22],[140,156,0.8974,0.90625,0.13651,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[144,156,0.9231,0.77679,0.24468,0.67857,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,8,0,0,1,0,15],[148,156,0.9487,0.83479,0.23179,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,6,0,0,1,0,19],[152,156,0.9744,0.78568,0.22871,0.57132,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,4,0,0,6,0,13],[156,156,1.0,0.72317,0.2473,0.571,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,5,0,0,1,0,12]]},{"b":2,"e":0.42857,"k":"falling","v":0.58036,"x":0.96429,"p":[[0,88,0.0,0.82588,0.21647,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,3,0,17],[4,88,0.0455,0.94643,0.17405,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[8,88,0.0909,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[12,88,0.1364,0.88837,0.19148,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,22],[16,88,0.1818,0.87946,0.17896,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,2,0,20],[20,88,0.2273,0.88392,0.16917,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,4,0,19],[24,88,0.2727,0.90625,0.16602,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,8,0,20],[28,88,0.3182,0.93304,0.16746,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[32,88,0.3636,0.94643,0.12752,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[36,88,0.4091,0.92857,0.16751,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[40,88,0.4545,0.91963,0.17837,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,26],[44,88,0.5,0.92857,0.15152,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[48,88,0.5455,0.91516,0.15919,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,23],[52,88,0.5909,0.93304,0.15146,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,25],[56,88,0.6364,0.91071,0.18123,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,24],[60,88,0.6818,0.90625,0.15407,0.82132,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,22],[64,88,0.7273,0.93304,0.15146,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[68,88,0.7727,0.88838,0.16653,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,20],[72,88,0.8182,0.88838,0.19477,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3,0,22],[76,88,0.8636,0.92856,0.13367,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,24],[80,88,0.9091,0.84821,0.21706,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,7,0,0,0,0,20],[84,88,0.9545,0.62944,0.28541,0.39286,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,8,0,0,5,0,0,6,0,0,2,0,0,1,0,10],[88,88,1.0,0.58036,0.26471,0.42857,0.4286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,7,0,0,11,0,0,3,0,0,1,0,0,4,0,6]]}]},{"i":"c28d1efb57772fed","q":"$5.$ Let x be a real number with  $0<x<1$  and let  $0.c_1c_2c_3...$  be the decimal expansion of x.Denote by  $B(x)$  the set of all subsequences of  $c_1c_2c_3$  that consist of 6 consecutive digits.\nFor instance ,  $B(\\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of  $B(x)$  as  $x$  varies among all irrational numbers with  $0<x<1$","t":[{"b":1,"e":0.85714,"k":"rising","v":0.70089,"x":0.90625,"p":[[0,17,0.0,0.70089,0.3524,0.67857,0.85714,0.85714,0.0,1.0,6,7,6,6,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,7],[4,17,0.2353,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[8,17,0.4706,0.90625,0.06785,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[12,17,0.7059,0.88392,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,9],[16,17,0.9412,0.88392,0.06622,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,7],[17,17,1.0,0.88838,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8]]},{"b":3,"e":0.85714,"k":"flat","v":0.75446,"x":0.91517,"p":[[0,32,0.0,0.75446,0.28175,0.85714,0.85714,0.85714,0.0,1.0,3,5,2,3,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,20,0,5],[4,32,0.125,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,32,0.25,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[12,32,0.375,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[16,32,0.5,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[20,32,0.625,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[24,32,0.75,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[28,32,0.875,0.87053,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[32,32,1.0,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6]]}]},{"i":"c55527cb6ddb8445","q":"$101$  wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.","t":[{"b":2,"e":1.0,"k":"flat","v":0.83927,"x":0.92857,"p":[[0,26,0.0,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[4,26,0.1538,0.87054,0.10926,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,19,0,9],[8,26,0.3077,0.85714,0.20825,0.85714,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,14,0,13],[12,26,0.4615,0.83927,0.17407,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,11,0,12],[16,26,0.6154,0.89286,0.14286,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,14],[20,26,0.7692,0.90623,0.15818,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,20],[24,26,0.9231,0.88393,0.20341,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,8,0,19],[26,26,1.0,0.85267,0.19061,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,4,0,17]]},{"b":3,"e":0.85714,"k":"falling","v":0.70532,"x":0.90625,"p":[[0,23,0.0,0.90625,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[4,23,0.1739,0.90625,0.14987,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,11,0,18],[8,23,0.3478,0.89286,0.15972,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,11,0,17],[12,23,0.5217,0.89731,0.1964,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,19],[16,23,0.6957,0.88392,0.12595,0.85714,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,14,0,13],[20,23,0.8696,0.80803,0.19103,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,11,0,10],[23,23,1.0,0.70532,0.21113,0.571,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,10,0,0,7,0,5]]}]},{"i":"17089e281448395b","q":"a, b, c are positive real numbers and a+b+c=k. Find the minimum value of  $ b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2 $","t":[{"b":0,"e":0.14286,"k":"falling","v":0.125,"x":0.80804,"p":[[0,90,0.0,0.80804,0.19759,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,3,0,15],[4,90,0.0444,0.72766,0.22121,0.57143,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,1,0,0,6,0,9],[8,90,0.0889,0.66518,0.20079,0.57143,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,0,0,0,19,0,0,0,0,0,7,0,4],[12,90,0.1333,0.69196,0.23176,0.57143,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,2,0,0,12,0,0,4,0,0,5,0,7],[16,90,0.1778,0.71875,0.21275,0.57143,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,15,0,0,1,0,0,6,0,8],[20,90,0.2222,0.70536,0.17474,0.57143,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,4,0,0,8,0,4],[24,90,0.2667,0.67411,0.17582,0.57143,0.57143,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,2,0,0,6,0,4],[28,90,0.3111,0.51338,0.274,0.35714,0.57143,0.57143,0.0,1.0,1,3,0,1,0,7,0,0,0,0,0,4,0,0,13,0,0,1,0,0,3,0,3],[32,90,0.3556,0.58036,0.26229,0.53571,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,6,0,0,0,0,0,2,0,0,14,0,0,1,0,0,6,0,3],[36,90,0.4,0.58929,0.27141,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,3,0,0,2,0,0,3,0,0,13,0,0,1,0,0,4,0,5],[40,90,0.4444,0.53571,0.29014,0.28571,0.57143,0.75,0.0,1.0,2,3,0,2,0,5,0,0,2,0,0,1,0,0,13,0,0,1,0,0,5,0,3],[44,90,0.4889,0.59375,0.26027,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,5,0,0,1,0,0,3,0,0,11,0,0,1,0,0,9,0,2],[48,90,0.5333,0.65179,0.2257,0.57143,0.57143,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,1,0,0,14,0,0,2,0,0,10,0,2],[52,90,0.5778,0.58929,0.28738,0.53571,0.57143,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,1,0,0,1,0,0,13,0,0,0,0,0,7,0,4],[56,90,0.6222,0.51339,0.31715,0.14286,0.57143,0.75,0.0,1.0,2,3,0,2,0,8,0,0,2,0,0,0,0,0,8,0,0,4,0,0,5,0,3],[60,90,0.6667,0.58482,0.27283,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,3,0,0,3,0,0,2,0,0,13,0,0,0,0,0,6,0,4],[64,90,0.7111,0.59374,0.27458,0.53539,0.57143,0.85714,0.0,1.0,1,3,0,1,0,4,0,0,2,0,0,1,0,0,11,0,0,3,0,0,7,0,3],[68,90,0.7556,0.59821,0.25111,0.57143,0.57143,0.75,0.0,1.0,1,4,0,1,0,2,0,0,3,0,0,0,0,0,16,0,0,2,0,0,4,0,4],[72,90,0.8,0.53122,0.29716,0.35714,0.57143,0.64286,0.0,1.0,2,4,0,2,0,6,0,0,0,0,0,3,0,0,13,0,0,0,0,0,4,0,4],[76,90,0.8444,0.60268,0.27371,0.57143,0.57143,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,2,0,0,0,0,0,15,0,0,1,0,0,6,0,4],[80,90,0.8889,0.44196,0.32996,0.14286,0.5,0.64286,0.0,1.0,2,3,0,2,0,13,0,0,0,0,0,1,0,0,8,0,0,0,0,0,5,0,3],[84,90,0.9333,0.48213,0.28959,0.14286,0.57143,0.57143,0.0,1.0,3,3,0,3,0,6,0,0,1,0,0,2,0,0,14,0,0,1,0,0,2,0,3],[88,90,0.9778,0.38393,0.33012,0.10714,0.42857,0.57143,0.0,1.0,8,2,0,8,0,7,0,0,0,0,0,3,0,0,7,0,0,2,0,0,3,0,2],[90,90,1.0,0.125,0.09942,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"falling","v":0.54018,"x":0.82589,"p":[[0,89,0.0,0.82589,0.18466,0.57143,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,7,0,14],[4,89,0.0449,0.71429,0.17857,0.57143,0.57143,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,7,0,6],[8,89,0.0899,0.70536,0.19212,0.57143,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,17,0,0,3,0,0,4,0,7],[12,89,0.1348,0.78125,0.17491,0.57143,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,10,0,8],[16,89,0.1798,0.62945,0.24448,0.57143,0.57143,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,3,0,0,14,0,0,2,0,0,6,0,4],[20,89,0.2247,0.66071,0.18472,0.57143,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,2,0,0,16,0,0,4,0,0,6,0,3],[24,89,0.2697,0.71429,0.28121,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,1,0,0,1,0,0,11,0,0,0,0,0,4,0,12],[28,89,0.3146,0.65179,0.23402,0.57143,0.57143,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,7,0,4],[32,89,0.3596,0.66964,0.21852,0.57143,0.57143,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,0,0,0,16,0,0,2,0,0,9,0,3],[36,89,0.4045,0.64732,0.23954,0.57143,0.57143,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,1,0,0,17,0,0,0,0,0,6,0,5],[40,89,0.4494,0.71873,0.16165,0.57143,0.57143,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,12,0,3],[44,89,0.4944,0.70089,0.20628,0.57143,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,16,0,0,1,0,0,7,0,6],[48,89,0.5393,0.72768,0.20628,0.57143,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,15,0,0,1,0,0,8,0,7],[52,89,0.5843,0.65625,0.15093,0.57143,0.57143,0.64286,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,5,0,3],[56,89,0.6292,0.62946,0.28983,0.57143,0.57143,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,0,0,0,2,0,0,12,0,0,3,0,0,2,0,8],[60,89,0.6742,0.72321,0.17105,0.57143,0.64286,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,1,0,0,11,0,4],[64,89,0.7191,0.63839,0.2696,0.57143,0.57143,0.85714,0.0,1.0,1,5,0,1,0,3,0,0,1,0,0,0,0,0,14,0,0,1,0,0,7,0,5],[68,89,0.764,0.58929,0.32291,0.25,0.64286,0.85714,0.0,1.0,2,3,0,2,0,6,0,0,1,0,0,1,0,0,6,0,0,2,0,0,11,0,3],[72,89,0.809,0.54018,0.34944,0.14286,0.57143,0.85714,0.0,1.0,4,5,0,4,0,5,0,0,2,0,0,2,0,0,6,0,0,1,0,0,7,0,5],[76,89,0.8539,0.64732,0.2448,0.57143,0.64286,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,2,0,0,9,0,0,1,0,0,14,0,1],[80,89,0.8989,0.62946,0.2547,0.57143,0.57143,0.85714,0.0,1.0,1,3,0,1,0,3,0,0,0,0,0,1,0,0,15,0,0,0,0,0,9,0,3],[84,89,0.9438,0.58036,0.20806,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,9,0,0,5,0,0,11,0,0,3,0,1],[88,89,0.9888,0.55357,0.23891,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,5,0,0,1,0,0,8,0,0,3,0,0,9,0,0,6,0,0],[89,89,1.0,0.58929,0.25939,0.42857,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,6,0,0,1,0,0,4,0,0,1,0,0,12,0,0,8,0,0]]}]},{"i":"3ef71db43ac88259","q":"There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.)\n\nEach person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a 1-best friend. More generally, if $n>1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.\n(a) Prove that every popular person is the best friend of a popular person.\n(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.\n\nOrigin. Romania (Dan Schwarz) (rephrasing by Geoff Smith).\nRemark. The original formulation of this problem was:\nGiven a function $f: X \\rightarrow X$, let us use the notations $f^{0}(X):=X, f^{n+1}(X):=f\\left(f^{n}(X)\\right)$ for $n \\geq 0$, and also $f^{\\omega}(X):=\\bigcap_{n \\geq 0} f^{n}(X)$. Let us now impose on $f$ that all its fibres $f^{-1}(y):=\\{x \\in X \\mid f(x)=y\\}$, for $y \\in f(X)$, are finite. Prove that $f\\left(f^{\\omega}(X)\\right)=f^{\\omega}(X)$.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.85268,"x":0.97777,"p":[[0,10,0.0,0.85268,0.32827,0.96429,1.0,1.0,0.0,1.0,4,24,4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,24],[4,10,0.4,0.91518,0.12299,0.82143,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[8,10,0.8,0.97777,0.06281,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[10,10,1.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]},{"b":6,"e":0.2857,"k":"flat","v":0.80354,"x":0.92857,"p":[[0,53,0.0,0.85268,0.29555,0.85714,1.0,1.0,0.0,1.0,3,22,3,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[4,53,0.0755,0.92857,0.15972,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[8,53,0.1509,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[12,53,0.2264,0.82816,0.21112,0.71429,0.96429,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,10,0,0,1,1,16],[16,53,0.3019,0.8079,0.21318,0.71321,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,9,0,0,0,0,16],[20,53,0.3774,0.84807,0.20194,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,0,0,19],[24,53,0.4528,0.86607,0.16728,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,2,0,18],[28,53,0.5283,0.8482,0.17837,0.71429,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,2,0,17],[32,53,0.6038,0.82143,0.24743,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,1,0,18],[36,53,0.6792,0.80355,0.20748,0.71421,0.85714,1.0,0.143,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,6,0,0,7,0,12],[40,53,0.7547,0.80354,0.2252,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,4,0,15],[44,53,0.8302,0.82593,0.20735,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,8,0,0,4,0,15],[48,53,0.9057,0.86147,0.16566,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,1,0,18],[52,53,0.9811,0.8079,0.21017,0.71321,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,7,0,0,4,0,14],[53,53,1.0,0.82589,0.18466,0.71429,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,12,0,0,1,0,15]]}]},{"i":"0b8aa1efa86ba5b3","q":"Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to  $2$  and the sum of its terms is  $200$ . How many progressions does Omar's list have?","t":[{"b":2,"e":0.57143,"k":"falling","v":0.5982,"x":0.9375,"p":[[0,56,0.0,0.9375,0.13803,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,26],[4,56,0.0714,0.77677,0.19213,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,3,0,12],[8,56,0.1429,0.6875,0.16146,0.57143,0.57143,0.75,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,5,0,0,3,0,5],[12,56,0.2143,0.65179,0.12846,0.57143,0.57143,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,0,0,0,3],[16,56,0.2857,0.67853,0.1557,0.57143,0.57143,0.71429,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,7,0,0,1,0,5],[20,56,0.3571,0.67407,0.19312,0.57143,0.57143,0.75,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,0,4,0,4],[24,56,0.4286,0.62495,0.17036,0.57143,0.57143,0.71429,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,0,0,0,22,0,0,4,0,0,3,0,2],[28,56,0.5,0.5982,0.06622,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,4,0,0,1,0,0],[32,56,0.5714,0.62944,0.09355,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,10,0,0,0,0,1],[36,56,0.6429,0.62052,0.09853,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,6,0,0,3,0,0],[40,56,0.7143,0.62944,0.10632,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,6,0,0,4,0,0],[44,56,0.7857,0.63835,0.09441,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,0,3,0,0],[48,56,0.8571,0.61159,0.08172,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,8,0,0,1,0,0],[52,56,0.9286,0.64286,0.10102,0.57143,0.57143,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,8,0,0,4,0,0],[56,56,1.0,0.60262,0.14168,0.57143,0.57143,0.60714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,23,0,0,5,0,0,3,0,0]]},{"b":4,"e":0.71429,"k":"falling","v":0.59372,"x":0.95089,"p":[[0,77,0.0,0.95089,0.13176,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[4,77,0.0519,0.86159,0.19395,0.57143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,21],[8,77,0.1039,0.7232,0.18878,0.57143,0.57143,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,0,2,0,9],[12,77,0.1558,0.79905,0.19192,0.57143,0.78571,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,5,0,0,2,0,14],[16,77,0.2078,0.62944,0.08647,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,9,0,0,2,0,0],[20,77,0.2597,0.62499,0.08565,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,8,0,0,2,0,0],[24,77,0.3117,0.61153,0.09609,0.57132,0.57143,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,7,0,0,2,0,0],[28,77,0.3636,0.61603,0.0833,0.57143,0.57143,0.60714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0],[32,77,0.4156,0.62049,0.0846,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,7,0,0,2,0,0],[36,77,0.4675,0.62051,0.11072,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,5,0,0,4,0,0],[40,77,0.5195,0.61606,0.07524,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0],[44,77,0.5714,0.59805,0.08311,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,3,0,0,2,0,0],[48,77,0.6234,0.60713,0.08749,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,2,0,0,3,0,0],[52,77,0.6753,0.61607,0.07523,0.57143,0.57143,0.71429,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0],[56,77,0.7273,0.61155,0.08174,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,5,0,0,2,0,0],[60,77,0.7792,0.64732,0.11285,0.57143,0.57143,0.71429,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,5,0,0,6,0,0],[64,77,0.8312,0.59372,0.06298,0.57143,0.57143,0.57143,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,6,0,0,0,0,0],[68,77,0.8831,0.61607,0.08328,0.57143,0.57143,0.60714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0],[72,77,0.9351,0.62051,0.07669,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,9,0,0,1,0,0],[76,77,0.987,0.62054,0.09852,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,6,0,0,3,0,0],[77,77,1.0,0.61607,0.08328,0.57143,0.57143,0.60714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0]]}]},{"i":"f063351114f95fb7","q":"$12$  distinct points are equally spaced around a circle. How many ways can Bryan choose  $3$  points (not in any order) out of these  $12$  points such that they form an acute triangle (Rotations of a set of points are considered distinct).\n\n*Proposed by Bryan Guo*","t":[{"b":1,"e":0.57143,"k":"falling","v":0.49106,"x":0.97768,"p":[[0,42,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,42,0.0952,0.88393,0.20025,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,4,0,21],[8,42,0.1905,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[12,42,0.2857,0.87052,0.21239,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,22],[16,42,0.381,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,26],[20,42,0.4762,0.86607,0.2141,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,21],[24,42,0.5714,0.88839,0.20434,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,0,0,24],[28,42,0.6667,0.91518,0.13767,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,21],[32,42,0.7619,0.88393,0.17655,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,1,0,21],[36,42,0.8571,0.875,0.20748,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,0,2,0,21],[40,42,0.9524,0.69195,0.23988,0.57132,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,4,0,0,3,0,9],[42,42,1.0,0.49106,0.27418,0.28571,0.49979,0.57143,0.0,1.0,2,5,0,2,0,0,0,0,12,0,0,2,0,0,10,0,0,1,0,0,0,0,5]]},{"b":3,"e":0.57143,"k":"falling","v":0.69196,"x":0.96875,"p":[[0,36,0.0,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,36,0.1111,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[8,36,0.2222,0.92409,0.14722,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,24],[12,36,0.3333,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,36,0.4444,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[20,36,0.5556,0.89732,0.18638,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,1,0,23],[24,36,0.6667,0.90624,0.17717,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,24],[28,36,0.7778,0.76786,0.26426,0.53572,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,5,0,0,1,0,16],[32,36,0.8889,0.83482,0.22899,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,2,0,19],[36,36,1.0,0.69196,0.16793,0.57143,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,11,0,0,0,0,6]]}]},{"i":"97464d9c93944aaf","q":"$ABC$  is an arbitrary triangle.  $A',B',C'$  are midpoints of arcs  $BC, AC, AB$ . Sides of triangle  $ABC$ , intersect sides of triangle  $A'B'C'$  at points  $P,Q,R,S,T,F$ . Prove that \\[\\frac{S_{PQRSTF}}{S_{ABC}}=1-\\frac{ab+ac+bc}{(a+b+c)^{2}}\\]","t":[{"b":2,"e":1.0,"k":"rising","v":0.625,"x":0.93303,"p":[[0,64,0.0,0.625,0.34022,0.42857,0.71429,0.89286,0.0,1.0,5,8,5,5,0,1,0,0,1,0,0,2,0,0,2,0,0,11,0,0,2,0,8],[4,64,0.0625,0.84375,0.24836,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,19],[8,64,0.125,0.80804,0.24383,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,8,0,0,1,0,17],[12,64,0.1875,0.81697,0.2321,0.71429,0.92857,1.0,0.1429,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,10,0,0,2,0,16],[16,64,0.25,0.86606,0.23404,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,0,3,0,21],[20,64,0.3125,0.74107,0.31428,0.53571,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,5,0,0,1,0,0,2,0,0,3,0,0,4,0,15],[24,64,0.375,0.64726,0.29665,0.5354,0.64286,1.0,0.0,1.0,1,10,0,1,0,2,0,0,4,0,0,1,0,0,8,0,0,6,0,0,0,0,10],[28,64,0.4375,0.75444,0.28625,0.57143,0.85714,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,5,0,0,0,0,16],[32,64,0.5,0.69642,0.27607,0.42859,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,4,0,0,4,0,0,1,0,0,10,0,0,2,0,10],[36,64,0.5625,0.62946,0.329,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,5,0,0,3,0,0,1,0,0,4,0,0,7,0,0,1,0,10],[40,64,0.625,0.67411,0.37327,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,4,0,0,4,0,0,1,0,0,2,0,0,2,0,0,1,0,16],[44,64,0.6875,0.68749,0.31429,0.53539,0.71429,1.0,0.0,1.0,1,12,0,1,0,3,0,0,3,0,0,1,0,0,3,0,0,8,0,0,1,0,12],[48,64,0.75,0.80356,0.27837,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,5,0,0,2,0,18],[52,64,0.8125,0.73661,0.31157,0.53571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,6,0,0,2,0,15],[56,64,0.875,0.79463,0.2695,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,4,0,0,1,0,18],[60,64,0.9375,0.79463,0.24985,0.71429,0.85707,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,10,0,0,2,0,15],[64,64,1.0,0.93303,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23]]},{"b":6,"e":0.71429,"k":"flat","v":0.49991,"x":0.88839,"p":[[0,71,0.0,0.65179,0.24984,0.57143,0.64286,0.75,0.0,1.0,1,7,1,1,0,1,0,0,1,0,0,4,0,0,9,0,0,8,0,0,1,0,7],[4,71,0.0563,0.88839,0.25688,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,25],[8,71,0.1127,0.84821,0.26229,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,22],[12,71,0.169,0.74107,0.30186,0.67857,0.71429,1.0,0.0,1.0,1,15,0,1,0,2,0,0,2,0,0,2,0,0,1,0,0,9,0,0,0,0,15],[16,71,0.2254,0.62052,0.33428,0.28571,0.71429,1.0,0.0,1.0,4,9,0,4,0,0,0,0,5,0,0,1,0,0,3,0,0,9,0,0,1,0,9],[20,71,0.2817,0.69197,0.30327,0.53572,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,3,0,0,2,0,0,1,0,0,10,0,0,3,0,10],[24,71,0.338,0.68759,0.30611,0.53572,0.71429,1.0,0.0,1.0,2,11,0,2,0,0,0,0,5,0,0,1,0,0,3,0,0,8,0,0,2,0,11],[28,71,0.3944,0.59812,0.31032,0.28571,0.57143,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,6,0,0,2,0,0,5,0,0,4,0,0,4,0,7],[32,71,0.4507,0.57141,0.31339,0.28571,0.57143,0.74996,0.0,1.0,1,7,0,1,0,5,0,0,4,0,0,3,0,0,4,0,0,7,0,0,1,0,7],[36,71,0.507,0.625,0.29613,0.39286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,5,0,0,5,0,0,1,0,0,9,0,0,0,0,9],[40,71,0.5634,0.62946,0.30275,0.39286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,4,0,0,2,0,0,4,0,0,8,0,0,2,0,8],[44,71,0.6197,0.56675,0.32655,0.28571,0.64286,0.75,0.0,1.0,2,7,0,2,0,4,0,0,6,0,0,0,0,0,4,0,0,8,0,0,1,0,7],[48,71,0.6761,0.49991,0.31954,0.28571,0.28571,0.71429,0.0,1.0,1,7,0,1,0,4,0,0,12,0,0,1,0,0,3,0,0,4,0,0,0,0,7],[52,71,0.7324,0.62945,0.29851,0.53539,0.71429,0.78571,0.0,1.0,2,8,0,2,0,2,0,0,3,0,0,1,0,0,6,0,0,10,0,0,0,0,8],[56,71,0.7887,0.64283,0.272,0.39286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,7,0,0,2,0,0,4,0,0,9,0,0,1,0,8],[60,71,0.8451,0.70536,0.20497,0.71429,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,18,0,0,1,0,6],[64,71,0.9014,0.68749,0.23266,0.57143,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,4,0,0,1,0,0,3,0,0,14,0,0,3,0,6],[68,71,0.9577,0.7723,0.17077,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,20,0,0,2,0,8],[71,71,1.0,0.69194,0.15201,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,20,0,0,3,0,2]]}]},{"i":"0db42031e6e1d0b3","q":"$P$  is an arbitrary point inside acute triangle  $ABC$ . Let  $A_1,B_1,C_1$  be the reflections of point  $P$  with respect to sides  $BC,CA,AB$ . Prove that the centroid of triangle  $A_1B_1C_1$  lies inside triangle  $ABC$ .","t":[{"b":0,"e":0.71429,"k":"rising","v":0.35714,"x":0.67854,"p":[[0,22,0.0,0.35714,0.26726,0.14286,0.28571,0.32143,0.14286,1.0,0,3,0,0,0,11,0,0,13,0,0,1,0,0,2,0,0,1,0,0,1,0,3],[4,22,0.1818,0.45982,0.3655,0.14286,0.28571,1.0,0.0,1.0,2,9,0,2,0,7,0,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,9],[8,22,0.3636,0.48659,0.30693,0.28571,0.28571,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,16,0,0,2,0,0,2,0,0,0,0,0,3,0,6],[12,22,0.5455,0.60267,0.34761,0.28571,0.64286,1.0,0.0,1.0,1,11,0,1,0,5,0,0,5,0,0,3,0,0,2,0,0,4,0,0,1,0,11],[16,22,0.7273,0.52232,0.30011,0.28571,0.28571,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,14,0,0,0,0,0,1,0,0,8,0,0,0,0,6],[20,22,0.9091,0.65622,0.19187,0.57143,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,10,0,0,13,0,0,1,0,4],[22,22,1.0,0.67854,0.19234,0.57143,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,13,0,0,1,0,5]]},{"b":6,"e":0.85714,"k":"rising","v":0.35265,"x":0.69641,"p":[[0,76,0.0,0.37945,0.25154,0.24999,0.28571,0.4642,0.14286,1.0,0,2,0,0,0,8,0,0,14,0,0,2,0,0,3,0,0,1,0,0,2,0,2],[4,76,0.0526,0.49106,0.33108,0.2857,0.28571,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,8],[8,76,0.1053,0.53571,0.35535,0.14286,0.35714,0.89286,0.14286,1.0,0,8,0,0,0,9,0,0,7,0,0,1,0,0,1,0,0,2,0,0,4,0,8],[12,76,0.1579,0.59818,0.35434,0.28571,0.64264,1.0,0.0,1.0,1,11,0,1,0,4,0,0,9,0,0,0,0,0,2,0,0,3,0,0,2,0,11],[16,76,0.2105,0.5357,0.34441,0.14286,0.57121,0.85714,0.0,1.0,2,6,0,2,0,7,0,0,4,0,0,2,0,0,2,0,0,5,0,0,4,0,6],[20,76,0.2632,0.35265,0.27888,0.14286,0.28571,0.4642,0.0,1.0,4,2,1,4,0,7,0,0,10,0,0,3,0,0,3,0,0,1,0,0,2,0,2],[24,76,0.3158,0.5179,0.34208,0.2857,0.35786,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,10,0,0,1,0,0,3,0,0,3,0,0,1,0,8],[28,76,0.3684,0.43301,0.30614,0.14286,0.28571,0.60714,0.0,1.0,1,5,0,1,0,8,0,0,10,0,0,1,0,0,4,0,0,3,0,0,0,0,5],[32,76,0.4211,0.49104,0.32524,0.24999,0.42857,0.74996,0.0,1.0,3,5,0,3,0,5,0,0,5,0,0,4,0,0,5,0,0,2,0,0,3,0,5],[36,76,0.4737,0.52229,0.34369,0.24999,0.571,0.85714,0.0,1.0,2,7,0,2,0,6,0,0,7,0,0,0,0,0,4,0,0,4,0,0,2,0,7],[40,76,0.5263,0.49552,0.33309,0.2857,0.35714,0.85714,0.0,1.0,2,5,0,2,0,5,0,0,9,0,0,3,0,0,1,0,0,2,0,0,5,0,5],[44,76,0.5789,0.45089,0.29475,0.2857,0.28571,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,12,0,0,3,0,0,2,0,0,3,0,0,2,0,4],[48,76,0.6316,0.45534,0.23807,0.28571,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,15,0,0,1,0,0,2,0,0,9,0,0,1,0,1],[52,76,0.6842,0.41507,0.24063,0.2857,0.28571,0.57143,0.0,1.0,1,2,0,1,0,3,0,0,15,0,0,3,0,0,3,0,0,5,0,0,0,0,2],[56,76,0.7368,0.51336,0.28315,0.28571,0.42836,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,13,0,0,0,0,0,5,0,0,4,0,0,3,0,4],[60,76,0.7895,0.61158,0.29284,0.42857,0.57143,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,4,0,0,4,0,0,9,0,0,0,0,0,5,0,7],[64,76,0.8421,0.69195,0.29039,0.57132,0.78569,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,4,0,0,0,0,0,3,0,0,6,0,0,9,0,7],[68,76,0.8947,0.69641,0.25939,0.57142,0.85714,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,1,0,0,6,0,0,1,0,0,16,0,3],[72,76,0.9474,0.66514,0.21314,0.5713,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,12,0,0,4,0,0,7,0,4],[76,76,1.0,0.65624,0.13768,0.57143,0.57143,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,8,0,1]]}]},{"i":"c31302a5b73c66f8","q":"**Q.** Determine all the functions  $f: \\mathbb{R} \\rightarrow \\mathbb{R}$  such that   $$ f(x) \\geqslant x+1, \\forall\\ x \\in \\mathbb{R}\\quad \\text{and}\\quad f(x+y) \\geqslant f(x) f(y), \\forall\\ x, y \\in \\mathbb{R} $$  *Proposed by TuZo*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.37045,"x":0.62947,"p":[[0,71,0.0,0.37945,0.25406,0.14286,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,13,0,0,3,0,0,2,0,0,7,0,0,4,0,0,2,0,0],[4,71,0.0563,0.60713,0.22304,0.42857,0.57143,0.75,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,7,0,0,10,0,0,4,0,0,5,0,3],[8,71,0.1127,0.62499,0.27607,0.42859,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,5,0,0,1,0,0,3,0,0,6,0,0,7,0,0,5,0,5],[12,71,0.169,0.54018,0.23347,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,3,0,0,6,0,0,8,0,0,7,0,0,2,0,2],[16,71,0.2254,0.62947,0.24448,0.42859,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,4,0,0,8,0,0,5,0,0,6,0,4],[20,71,0.2817,0.57143,0.27199,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,8,0,0,5,0,0,1,0,0,8,0,3],[24,71,0.338,0.54009,0.27383,0.39286,0.57143,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,3,0,0,6,0,0,7,0,0,5,0,0,2,0,4],[28,71,0.3944,0.47759,0.22205,0.39286,0.4286,0.57143,0.14,1.0,0,1,0,0,0,6,0,0,2,0,0,9,0,0,9,0,0,3,0,0,2,0,1],[32,71,0.4507,0.45971,0.22239,0.28571,0.42857,0.57143,0.14,1.0,0,1,0,0,0,6,0,0,4,0,0,8,0,0,9,0,0,2,0,0,2,0,1],[36,71,0.507,0.55356,0.23891,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,2,0,0,7,0,0,8,0,0,5,0,0,4,0,2],[40,71,0.5634,0.47321,0.22711,0.28571,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,7,0,0,3,0,0,5,0,0,9,0,0,7,0,0,0,0,1],[44,71,0.6197,0.56249,0.22286,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,8,0,0,9,0,0,5,0,0,2,0,3],[48,71,0.6761,0.53123,0.2237,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,4,0,0,3,0,0,5,0,0,12,0,0,3,0,0,4,0,1],[52,71,0.7324,0.54908,0.25532,0.42857,0.57121,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,7,0,0,8,0,0,4,0,0,2,0,4],[56,71,0.7887,0.52676,0.23807,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,7,0,0,9,0,0,3,0,0,5,0,1],[60,71,0.8451,0.39732,0.17399,0.28571,0.42857,0.46429,0.14286,0.85714,0,0,0,0,0,4,0,0,11,0,0,9,0,0,5,0,0,2,0,0,1,0,0],[64,71,0.9014,0.40177,0.19702,0.28571,0.42857,0.46418,0.14286,1.0,0,1,0,0,0,7,0,0,5,0,0,12,0,0,5,0,0,2,0,0,0,0,1],[68,71,0.9577,0.37045,0.24972,0.14286,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,12,0,0,4,0,0,5,0,0,6,0,0,0,0,0,4,0,0],[71,71,1.0,0.40624,0.17167,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,8,0,0,9,0,0,7,0,0,3,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"rising","v":0.32589,"x":0.64285,"p":[[0,51,0.0,0.32589,0.29931,0.14286,0.14286,0.57143,0.0,1.0,3,2,0,3,0,17,0,0,1,0,0,2,0,0,3,0,0,2,0,0,2,0,2],[4,51,0.0784,0.64285,0.22015,0.57143,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,4,0,0,12,0,0,3,0,0,6,0,4],[8,51,0.1569,0.63393,0.24468,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,5,0,0,9,0,0,4,0,0,5,0,5],[12,51,0.2353,0.59375,0.27458,0.42859,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,6,0,0,1,0,0,2,0,0,9,0,0,4,0,0,7,0,3],[16,51,0.3137,0.57134,0.23707,0.28571,0.57143,0.71429,0.14,1.0,0,3,0,0,0,1,0,0,8,0,0,3,0,0,6,0,0,9,0,0,2,0,3],[20,51,0.3922,0.55356,0.23076,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,8,0,0,7,0,0,4,0,0,6,0,1],[24,51,0.4706,0.52678,0.22711,0.42857,0.57143,0.60714,0.14286,1.0,0,2,0,0,0,3,0,0,4,0,0,8,0,0,9,0,0,3,0,0,3,0,2],[28,51,0.549,0.48661,0.21086,0.42857,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,10,0,0,8,0,0,5,0,0,1,0,1],[32,51,0.6275,0.47767,0.19433,0.42857,0.49979,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,2,0,0,9,0,0,11,0,0,3,0,0,2,0,0],[36,51,0.7059,0.49554,0.20198,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,14,0,0,6,0,0,3,0,0,1,0,2],[40,51,0.7843,0.38838,0.18976,0.25,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,8,0,0,6,0,0,8,0,0,8,0,0,1,0,0,1,0,0],[44,51,0.8627,0.49999,0.15971,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,8,0,0,17,0,0,1,0,0,0,0,1],[48,51,0.9412,0.62053,0.07668,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,12,0,0,0,0,0],[51,51,1.0,0.57581,0.07563,0.571,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,23,0,0,5,0,0,0,0,0]]}]},{"i":"0d5bb973fa214890","q":"(A.Akopyan, V.Dolnikov) Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.23215,"p":[[0,37,0.0,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.28571,14,0,2,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.18749,0.20956,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,3,0,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[8,37,0.2162,0.13384,0.3008,0.0,0.0,0.035,0.0,1.0,24,3,0,24,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[12,37,0.3243,0.13831,0.27078,0.0,0.0,0.14073,0.0,1.0,23,2,0,23,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[16,37,0.4324,0.125,0.22232,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[20,37,0.5405,0.23215,0.30252,0.0,0.14286,0.32143,0.0,1.0,13,3,0,13,0,8,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,3],[24,37,0.6486,0.1383,0.15355,0.0,0.14143,0.2857,0.0,0.4286,15,0,0,15,0,7,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.13393,0.14258,0.0,0.14286,0.2857,0.0,0.42857,15,0,0,15,0,6,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.16071,0.15465,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,9,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.07144,0.08749,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04912,"x":0.27231,"p":[[0,38,0.0,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.4286,23,0,6,23,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.27231,0.21235,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,8,0,0,3,0,0,11,0,0,2,0,0,0,0,0,1,0,0],[8,38,0.2105,0.20527,0.21413,0.0,0.14288,0.28571,0.0,1.0,11,1,0,11,0,6,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[12,38,0.3158,0.09812,0.17653,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,38,0.4211,0.0982,0.13565,0.0,0.0,0.14286,0.0,0.571,18,0,0,18,0,8,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,38,0.5263,0.08929,0.12242,0.0,0.0,0.1429,0.0,0.42857,19,0,0,19,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.1383,0.2461,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[28,38,0.7368,0.11152,0.19473,0.0,0.0,0.1429,0.0,0.71429,22,0,0,22,0,3,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[32,38,0.8421,0.07143,0.14286,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,38,0.9474,0.14722,0.17304,0.0,0.07,0.2857,0.0,0.571,16,0,0,16,0,5,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.04912,0.11073,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c723ff43c134de8c","q":"(1)  $D$  is an arbitary point in  $\\triangle{ABC}$ . Prove that:\r\n\\[ \\frac{BC}{\\min{AD,BD,CD}} \\geq \\{ \\begin{array}{c} \\displaystyle 2\\sin{A}, \\ \\angle{A}< 90^o   2, \\ \\angle{A} \\geq 90^o \\end{array}  \\]\r\n(2) $E$  is an arbitary point in convex quadrilateral  $ABCD$ . Denote  $k$  the ratio of the largest and least distances of any two points among  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $E$ . Prove that  $k \\geq 2\\sin{70^o}$ . Can equality be achieved?","t":[{"b":3,"e":0.2857,"k":"flat","v":0.10715,"x":0.46872,"p":[[0,65,0.0,0.10715,0.12877,0.0,0.07143,0.14287,0.0,0.42857,16,0,10,16,0,10,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.46872,0.23481,0.28571,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,15,0,0,1,0,0,4,0,0,7,0,0,2,0,1],[8,65,0.1231,0.4107,0.20438,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,10,0,0,6,0,0,6,0,0,4,0,0,1,0,0],[12,65,0.1846,0.34374,0.18849,0.2857,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,5,0,0,12,0,0,8,0,0,1,0,0,4,0,0,0,0,0],[16,65,0.2462,0.37053,0.23107,0.24999,0.28571,0.46431,0.0,0.85714,2,0,0,2,0,6,0,0,11,0,0,5,0,0,2,0,0,4,0,0,2,0,0],[20,65,0.3077,0.37497,0.20121,0.2857,0.28571,0.4642,0.0,0.85714,2,0,0,2,0,3,0,0,13,0,0,6,0,0,4,0,0,3,0,0,1,0,0],[24,65,0.3692,0.33479,0.23851,0.14286,0.28571,0.42858,0.0,1.0,4,1,0,4,0,5,0,0,13,0,0,3,0,0,2,0,0,4,0,0,0,0,1],[28,65,0.4308,0.44196,0.25343,0.2857,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,5,0,0,7,0,0,4,0,0,5,0,0,6,0,0,3,0,0],[32,65,0.4923,0.37942,0.2039,0.2857,0.28571,0.4286,0.0,1.0,1,1,0,1,0,4,0,0,13,0,0,7,0,0,3,0,0,3,0,0,0,0,1],[36,65,0.5538,0.33482,0.20079,0.25,0.28571,0.46431,0.0,0.71429,3,0,0,3,0,5,0,0,13,0,0,3,0,0,5,0,0,3,0,0,0,0,0],[40,65,0.6154,0.33928,0.14174,0.2857,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,2,0,0,22,0,0,5,0,0,1,0,0,1,0,0,1,0,0],[44,65,0.6769,0.29464,0.14698,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,5,0,0,18,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[48,65,0.7385,0.35714,0.2369,0.2857,0.28571,0.42857,0.0,1.0,2,1,0,2,0,5,0,0,16,0,0,2,0,0,1,0,0,4,0,0,1,0,1],[52,65,0.8,0.28125,0.16554,0.14286,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,9,0,0,14,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[56,65,0.8615,0.29007,0.16941,0.24999,0.2857,0.32143,0.0,0.71429,4,0,0,4,0,4,0,0,16,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[60,65,0.9231,0.30338,0.18827,0.14289,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,6,0,0,15,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[64,65,0.9846,0.22759,0.18855,0.14214,0.21435,0.28571,0.0,0.71429,7,0,0,7,0,9,0,0,11,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[65,65,1.0,0.20525,0.15947,0.105,0.2143,0.28571,0.0,0.5714,8,0,0,8,0,8,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"rising","v":0.0759,"x":0.43749,"p":[[0,71,0.0,0.0759,0.11285,0.0,0.0,0.14287,0.0,0.42857,20,0,14,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.37945,0.21608,0.2857,0.28571,0.57141,0.0,0.85714,2,0,0,2,0,4,0,0,13,0,0,3,0,0,5,0,0,4,0,0,1,0,0],[8,71,0.1127,0.40177,0.23537,0.2857,0.28571,0.60714,0.0,0.71429,4,0,0,4,0,1,0,0,12,0,0,3,0,0,4,0,0,8,0,0,0,0,0],[12,71,0.169,0.4107,0.2165,0.2857,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,15,0,0,3,0,0,4,0,0,6,0,0,1,0,0],[16,71,0.2254,0.299,0.20633,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,4,0,0,14,0,0,5,0,0,1,0,0,2,0,0,1,0,0],[20,71,0.2817,0.36605,0.23671,0.14289,0.28571,0.57111,0.0,0.85714,3,0,0,3,0,6,0,0,10,0,0,3,0,0,4,0,0,5,0,0,1,0,0],[24,71,0.338,0.32142,0.1675,0.24999,0.28571,0.4286,0.0,0.71429,2,0,0,2,0,6,0,0,12,0,0,7,0,0,4,0,0,1,0,0,0,0,0],[28,71,0.3944,0.33928,0.25191,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,3,0,0,11,0,0,5,0,0,1,0,0,4,0,0,2,0,0],[32,71,0.4507,0.34818,0.23124,0.14289,0.28571,0.571,0.0,0.71429,4,0,0,4,0,7,0,0,6,0,0,6,0,0,4,0,0,5,0,0,0,0,0],[36,71,0.507,0.33036,0.1729,0.2857,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,5,0,0,13,0,0,7,0,0,3,0,0,2,0,0,0,0,0],[40,71,0.5634,0.3347,0.20085,0.14289,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,7,0,0,12,0,0,4,0,0,3,0,0,4,0,0,0,0,0],[44,71,0.6197,0.43749,0.24205,0.2857,0.35714,0.71429,0.0,0.85714,1,0,0,1,0,5,0,0,10,0,0,3,0,0,3,0,0,8,0,0,2,0,0],[48,71,0.6761,0.30356,0.20746,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,9,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[52,71,0.7324,0.31691,0.22507,0.14286,0.28571,0.4642,0.0,0.85714,4,0,0,4,0,8,0,0,9,0,0,3,0,0,5,0,0,2,0,0,1,0,0],[56,71,0.7887,0.35709,0.23953,0.14286,0.28571,0.57111,0.0,0.71429,4,0,0,4,0,7,0,0,6,0,0,5,0,0,4,0,0,6,0,0,0,0,0],[60,71,0.8451,0.33022,0.21263,0.14289,0.28571,0.4642,0.0,0.857,4,0,0,4,0,5,0,0,11,0,0,4,0,0,6,0,0,1,0,0,1,0,0],[64,71,0.9014,0.33929,0.21352,0.14286,0.28571,0.46525,0.0,0.85714,3,0,0,3,0,6,0,0,11,0,0,4,0,0,5,0,0,2,0,0,1,0,0],[68,71,0.9577,0.32134,0.22874,0.14286,0.28571,0.4286,0.0,0.857,6,0,0,6,0,4,0,0,9,0,0,6,0,0,4,0,0,2,0,0,1,0,0],[71,71,1.0,0.23215,0.16268,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,10,0,0,12,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"765a0d95219011e2","q":"(Netherlands).\n\nLet $n \\geq 2$ be an integer, and let $a_{1}, a_{2}, \\ldots, a_{n}$ be positive integers. Show that there exist positive integers $b_{1}, b_{2}, \\ldots, b_{n}$ satisfying the following three conditions:\n\n1). $a_{i} \\leq b_{i}$ for $i=1,2, \\ldots, n$;\n2). the remainders of $b_{1}, b_{2}, \\ldots, b_{n}$ on division by $n$ are pairwise different; and\n3). $b_{1}+\\cdots+b_{n} \\leq n\\left(\\frac{n-1}{2}+\\left\\lfloor\\frac{a_{1}+\\cdots+a_{n}}{n}\\right\\rfloor\\right)$.\n(Here, $\\lfloor x\\rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)","t":[{"b":3,"e":0.42857,"k":"rising","v":0.41059,"x":0.77679,"p":[[0,119,0.0,0.41059,0.20756,0.25001,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,4,0,0,8,0,0,7,0,0,5,0,0,0,0,0],[4,119,0.0336,0.62723,0.22212,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,5,0,0,2,1,0,13,0,0,6,0,1],[8,119,0.0672,0.76339,0.1411,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,18,0,0,4,0,6],[12,119,0.1008,0.77679,0.13803,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,17,0,0,8,0,5],[16,119,0.1345,0.70089,0.22968,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,14,0,0,4,0,6],[20,119,0.1681,0.66964,0.23538,0.53571,0.71429,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,15,0,0,3,0,5],[24,119,0.2017,0.72767,0.25345,0.57143,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,0,0,0,2,0,0,5,0,0,6,0,0,8,0,8],[28,119,0.2353,0.7366,0.22899,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,13,0,0,5,0,8],[32,119,0.2689,0.69196,0.21756,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,14,0,0,4,0,5],[36,119,0.3025,0.75,0.19233,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,14,0,0,5,0,7],[40,119,0.3361,0.76339,0.20394,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,14,0,0,4,0,9],[44,119,0.3697,0.77232,0.17076,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,17,0,0,7,0,6],[48,119,0.4034,0.71428,0.19232,0.67857,0.71429,0.85704,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,15,0,0,4,0,5],[52,119,0.437,0.65625,0.23106,0.53572,0.71429,0.75,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,13,0,0,4,0,4],[56,119,0.4706,0.68749,0.20341,0.67857,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,17,0,0,3,0,4],[60,119,0.5042,0.68304,0.21048,0.67857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,15,0,0,6,0,3],[64,119,0.5378,0.74999,0.22589,0.67857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,10,0,0,4,0,10],[68,119,0.5714,0.6875,0.2126,0.57143,0.71429,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,15,0,0,3,0,5],[72,119,0.605,0.70087,0.17629,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,13,0,0,6,0,3],[76,119,0.6387,0.71203,0.26032,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,1,0,0,6,1,0,9,0,0,3,0,9],[80,119,0.6723,0.71875,0.21274,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,15,0,0,3,0,7],[84,119,0.7059,0.73884,0.24131,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,13,0,0,3,1,9],[88,119,0.7395,0.68076,0.21204,0.571,0.71429,0.85704,0.1429,1.0,0,4,0,0,0,1,0,0,2,0,0,3,0,1,3,0,0,13,0,0,5,0,4],[92,119,0.7731,0.64732,0.24996,0.42857,0.71429,0.75,0.14286,1.0,0,6,0,0,0,3,0,0,0,0,0,7,0,0,3,0,0,11,0,0,2,0,6],[96,119,0.8067,0.58929,0.24679,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,0,0,0,6,0,0,4,0,0,11,0,0,4,0,2],[100,119,0.8403,0.69197,0.16015,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,20,0,0,2,0,3],[104,119,0.8739,0.61829,0.23119,0.42857,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,7,0,1,3,0,0,13,0,0,1,0,4],[108,119,0.9076,0.65176,0.24207,0.42857,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,5,0,0,3,0,0,13,0,0,1,0,6],[112,119,0.9412,0.68525,0.21119,0.57143,0.71429,0.73214,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,14,1,0,2,0,5],[116,119,0.9748,0.62052,0.21609,0.42859,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,6,0,0,4,0,0,14,0,0,3,0,2],[119,119,1.0,0.64509,0.17266,0.57143,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,14,1,0,2,0,2]]},{"b":7,"e":0.71429,"k":"rising","v":0.45533,"x":0.76783,"p":[[0,48,0.0,0.45533,0.23806,0.2857,0.42859,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,4,0,0,6,0,0,8,0,0,5,0,0,1,0,1],[4,48,0.0833,0.75445,0.18294,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,16,0,0,4,0,7],[8,48,0.1667,0.72321,0.25985,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,2,0,0,1,0,0,11,0,0,6,0,8],[12,48,0.25,0.56687,0.27558,0.28571,0.71429,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,2,0,0,1,0,0,2,0,0,15,0,0,4,0,1],[16,48,0.3333,0.62054,0.26871,0.42857,0.71429,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,1,0,0,7,0,0,3,0,0,9,0,0,5,0,4],[20,48,0.4167,0.72766,0.18337,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,12,0,0,10,0,3],[24,48,0.5,0.62945,0.21086,0.53539,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,5,0,0,3,0,0,17,0,0,2,0,2],[28,48,0.5833,0.76783,0.1505,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,7,0,6],[32,48,0.6667,0.6607,0.20749,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,3,0,0,4,0,0,16,0,0,3,0,3],[36,48,0.75,0.76116,0.21548,0.71429,0.85714,0.89286,0.21429,1.0,0,8,0,0,0,0,0,1,1,0,0,3,0,0,2,0,0,8,0,0,9,0,8],[40,48,0.8333,0.75222,0.22016,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,1,2,0,0,11,0,0,5,0,9],[44,48,0.9167,0.73661,0.1992,0.67857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,13,0,0,3,0,8],[48,48,1.0,0.73437,0.24172,0.71429,0.82143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,3,0,0,0,0,0,8,1,0,9,0,7]]}]},{"i":"503b55203f59df54","q":"13. (IRE 3) Let $S$ be the set of all pairs $(m, n)$ of relatively prime positive integers $m, n$ with $n$ even and $m<n$. For $s=(m, n) \\in S$ write $n=2^{k} n_{0}$, where $k, n_{0}$ are positive integers with $n_{0}$ odd and define $f(s)=\\left(n_{0}, m+\\right.$ $n-n_{0}$ ). Prove that $f$ is a function from $S$ to $S$ and that for each $s=(m, n) \\in S$, there exists a positive integer $t \\leq \\frac{m+n+1}{4}$ such that $f^{t}(s)=s$, where  $$ f^{t}(s)=\\underbrace{(f \\circ f \\circ \\cdots \\circ f)}_{t \\text { times }}(s) . $$  If $m+n$ is a prime number that does not divide $2^{k}-1$ for $k=1,2, \\ldots, m+$ $n-2$, prove that the smallest value of $t$ that satisfies the above conditions is $\\left[\\frac{m+n+1}{4}\\right]$, where $[x]$ denotes the greatest integer less than or equal to $x$.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.20982,"x":0.62052,"p":[[0,35,0.0,0.25884,0.16923,0.14286,0.2143,0.28571,0.14,0.85714,0,0,0,0,0,16,0,0,12,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[4,35,0.1143,0.51339,0.25218,0.28571,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,8,0,0,6,0,0,5,0,0,5,0,0,2,0,3],[8,35,0.2286,0.61607,0.23808,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,4,0,0,3,0,0,9,0,0,7,0,2],[12,35,0.3429,0.5625,0.22286,0.39286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,3,0,0,8,0,0,8,0,0,4,0,1],[16,35,0.4571,0.59819,0.23266,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,7,0,0,8,0,0,3,0,0,6,0,3],[20,35,0.5714,0.62052,0.21312,0.42857,0.71429,0.75,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,7,0,0,3,0,0,10,0,0,7,0,1],[24,35,0.6857,0.58034,0.24984,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,3,0,0,6,0,0,9,0,0,6,0,1],[28,35,0.8,0.49107,0.26711,0.28571,0.42859,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,6,0,0,6,0,0,4,0,0,4,0,0,4,0,2],[32,35,0.9143,0.32125,0.18575,0.14286,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,11,0,0,12,0,0,2,0,0,4,0,0,3,0,0,0,0,0],[35,35,1.0,0.20982,0.07973,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,18,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.24107,"x":0.65624,"p":[[0,34,0.0,0.24107,0.10972,0.14286,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,15,0,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,34,0.1176,0.65624,0.26931,0.53539,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,1,0,0,9,0,0,3,0,0,4,0,8],[8,34,0.2353,0.60267,0.22228,0.5354,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,2,0,0,7,0,0,10,0,0,7,0,0],[12,34,0.3529,0.58033,0.2623,0.28571,0.57143,0.75,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,0,0,0,8,0,0,6,0,0,5,0,3],[16,34,0.4706,0.52675,0.21558,0.42857,0.571,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,7,0,0,9,0,0,5,0,0,3,0,1],[20,34,0.5882,0.55801,0.22119,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,5,0,0,6,0,0,9,0,0,5,0,0],[24,34,0.7059,0.49996,0.20515,0.28571,0.57121,0.71429,0.14286,0.71429,0,0,0,0,0,4,0,0,6,0,0,3,0,0,8,0,0,11,0,0,0,0,0],[28,34,0.8235,0.50444,0.20197,0.28571,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,3,0,0,12,0,0,5,0,0,1,0,1],[32,34,0.9412,0.45088,0.19268,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,11,0,0,3,0,0,8,0,0,7,0,0,0,0,0],[34,34,1.0,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a7049cdf8469e251","q":"11. (FIN $\\mathbf{2}^{\\prime}$ ) Let $f:[0,1] \\rightarrow \\mathbb{R}$ be continuous and satisfy:  $$ \\begin{aligned} b f(2 x) & =f(x), & & 0 \\leq x \\leq 1 / 2 \\\\ f(x) & =b+(1-b) f(2 x-1), & & 1 / 2 \\leq x \\leq 1 \\end{aligned} $$  where $b=\\frac{1+c}{2+c}, c>0$. Show that $0<f(x)-x<c$ for every $x, 0<x<1$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.8125,"x":0.96429,"p":[[0,70,0.0,0.91071,0.08565,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[4,70,0.0571,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[8,70,0.1143,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[12,70,0.1714,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,70,0.2286,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[20,70,0.2857,0.90625,0.10479,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15],[24,70,0.3429,0.90625,0.13651,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,19],[28,70,0.4,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[32,70,0.4571,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[36,70,0.5143,0.9241,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[40,70,0.5714,0.91963,0.10067,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[44,70,0.6286,0.92855,0.09455,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,18],[48,70,0.6857,0.91518,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[52,70,0.7429,0.90178,0.12079,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,14,0,15],[56,70,0.8,0.87945,0.18937,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,15],[60,70,0.8571,0.88391,0.12599,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,14],[64,70,0.9143,0.87946,0.11356,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,19,0,10],[68,70,0.9714,0.8125,0.2299,0.85714,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,16,0,9],[70,70,1.0,0.89731,0.11429,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,15,0,14]]},{"b":5,"e":0.57143,"k":"flat","v":0.87946,"x":0.95982,"p":[[0,54,0.0,0.88838,0.13236,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,14,0,14],[4,54,0.0741,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,54,0.1481,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[12,54,0.2222,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[16,54,0.2963,0.93748,0.09412,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[20,54,0.3704,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,54,0.4444,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[28,54,0.5185,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[32,54,0.5926,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16],[36,54,0.6667,0.90179,0.09062,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,19,0,12],[40,54,0.7407,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[44,54,0.8148,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[48,54,0.8889,0.88392,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,11],[52,54,0.963,0.89285,0.10102,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,12],[54,54,1.0,0.87946,0.12931,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,16,0,12]]}]},{"i":"579f55c5dc19c823","q":"(a) Prove that for every positive integer $n$, there exists a fraction $\\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \\leqslant \\sqrt{n}+1$ and $\\sqrt{n} \\leqslant \\frac{a}{b} \\leqslant \\sqrt{n+1}$. (b) Prove that there are infinitely many positive integers $n$ such that there is no fraction $\\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \\leqslant \\sqrt{n}$ and $\\sqrt{n} \\leqslant \\frac{a}{b} \\leqslant \\sqrt{n+1}$.","t":[{"b":3,"e":0.42857,"k":"falling","v":0.45536,"x":0.66514,"p":[[0,32,0.0,0.66514,0.21611,0.5713,0.64286,0.71429,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,4,0,0,11,0,0,9,0,0,1,0,6],[4,32,0.125,0.54018,0.21049,0.42857,0.5,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,13,0,0,10,0,0,0,0,0,4,0,2],[8,32,0.25,0.5625,0.2788,0.42857,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,3,0,0,12,0,0,5,0,0,0,0,0,2,0,7],[12,32,0.375,0.53124,0.2581,0.42857,0.42857,0.60714,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,13,0,0,4,0,0,1,0,0,2,0,5],[16,32,0.5,0.5491,0.29474,0.28571,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,11,0,0,8,0,0,1,0,0,1,0,0,3,0,7],[20,32,0.625,0.55356,0.20438,0.42857,0.4998,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,13,0,0,10,0,0,1,0,0,1,0,4],[24,32,0.75,0.4732,0.10373,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,15,0,0,12,0,0,1,0,0,0,0,0],[28,32,0.875,0.47766,0.20393,0.39286,0.42857,0.4642,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,16,0,0,4,0,0,0,0,0,1,0,3],[32,32,1.0,0.45536,0.15746,0.42857,0.42857,0.4286,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,22,0,0,3,0,0,0,0,0,0,0,2]]},{"b":7,"e":1.0,"k":"falling","v":0.24553,"x":0.77232,"p":[[0,39,0.0,0.77232,0.22829,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,2,0,0,5,0,13],[4,39,0.1026,0.49553,0.22863,0.42857,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,13,0,0,7,0,0,0,0,0,2,0,3],[8,39,0.2051,0.48212,0.2389,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,1,0,0,7,0,0,12,0,0,6,0,0,1,0,0,0,0,4],[12,39,0.3077,0.44641,0.23891,0.28571,0.42857,0.57111,0.0,1.0,1,3,0,1,0,1,0,0,12,0,0,9,0,0,4,0,0,1,0,0,1,0,3],[16,39,0.4103,0.54015,0.23887,0.39286,0.571,0.57143,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,7,0,0,10,0,0,2,0,0,0,0,5],[20,39,0.5128,0.45534,0.2822,0.28571,0.42857,0.46418,0.0,1.0,2,5,0,2,0,2,0,0,9,0,0,11,0,0,2,0,0,0,0,0,1,0,5],[24,39,0.6154,0.45089,0.29474,0.2857,0.42857,0.60714,0.0,1.0,3,3,0,3,0,3,0,0,8,0,0,8,0,0,2,0,0,1,0,0,4,0,3],[28,39,0.7179,0.50892,0.24726,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,12,0,0,6,0,0,1,0,0,3,0,3],[32,39,0.8205,0.39277,0.25263,0.2857,0.35714,0.42857,0.0,1.0,2,3,0,2,0,5,0,0,9,0,0,9,0,0,3,0,0,1,0,0,0,0,3],[36,39,0.9231,0.26339,0.22899,0.14286,0.21428,0.42857,0.0,0.85714,7,0,0,7,0,9,0,0,7,0,0,4,0,0,2,0,0,2,0,0,1,0,0],[39,39,1.0,0.24553,0.23481,0.14286,0.14288,0.28571,0.0,0.85714,7,0,0,7,0,12,0,0,6,0,0,2,0,0,1,0,0,3,0,0,1,0,0]]}]},{"i":"dcc69edc0ecdde26","q":"28. (USS 3) Prove that on the coordinate plane it is impossible to draw a closed broken line such that (i) the coordinates of each vertex are rational; (ii) the length each of its edges is 1 ; (iii) the line has an odd number of vertices.","t":[{"b":1,"e":1.0,"k":"rising","v":0.35713,"x":0.91964,"p":[[0,32,0.0,0.35713,0.39929,0.0,0.14286,0.71429,0.0,1.0,13,6,0,13,0,6,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,6],[4,32,0.125,0.74544,0.31707,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,3,0,0,1,0,0,1,0,0,6,0,0,1,0,0,3,0,16],[8,32,0.25,0.71872,0.34347,0.571,0.85714,1.0,0.0,1.0,3,14,0,3,0,2,0,0,1,0,0,1,0,0,4,0,0,2,0,0,5,0,14],[12,32,0.375,0.55357,0.39407,0.14286,0.71429,1.0,0.0,1.0,6,9,0,6,0,5,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,9],[16,32,0.5,0.7812,0.3093,0.571,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,1,0,0,1,0,0,4,0,0,3,0,0,2,0,18],[20,32,0.625,0.7098,0.33405,0.57143,0.85714,1.0,0.0,1.0,4,9,0,4,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,0,11,0,9],[24,32,0.75,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[28,32,0.875,0.84362,0.14009,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,10,0,11],[32,32,1.0,0.87942,0.13889,0.857,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,10,0,15]]},{"b":4,"e":1.0,"k":"rising","v":0.47319,"x":0.87945,"p":[[0,22,0.0,0.47319,0.37869,0.10714,0.571,0.75,0.0,1.0,8,7,0,8,0,4,0,0,1,0,0,2,0,0,6,0,0,3,0,0,1,0,7],[4,22,0.1818,0.62498,0.38589,0.25,0.78564,1.0,0.0,1.0,5,11,0,5,0,3,0,0,2,0,0,1,0,0,2,0,0,3,0,0,5,0,11],[8,22,0.3636,0.56695,0.36854,0.28571,0.57143,0.89286,0.0,1.0,5,8,0,5,0,2,0,0,5,0,0,1,0,0,4,0,0,2,0,0,5,0,8],[12,22,0.5455,0.79016,0.26483,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,0,9,0,13],[16,22,0.7273,0.80354,0.15051,0.71429,0.78564,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,7,0,9],[20,22,0.9091,0.84373,0.14448,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,8,0,12],[22,22,1.0,0.87945,0.14337,0.82143,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,8,0,16]]}]},{"i":"f535155a785a22fb","q":"23. (GBR 3) A finite set of (distinct) positive integers is called a \" $D S$-set\" if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some $D S$-set.","t":[{"b":0,"e":1.0,"k":"rising","v":0.23205,"x":0.89285,"p":[[0,52,0.0,0.23205,0.21057,0.14286,0.1429,0.28571,0.0,1.0,5,1,2,5,0,12,0,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[4,52,0.0769,0.53562,0.32938,0.28571,0.5,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,11,0,0,1,0,0,4,0,0,2,0,0,3,0,7],[8,52,0.1538,0.51785,0.39729,0.14286,0.28571,1.0,0.0,1.0,3,10,0,3,0,9,0,0,5,0,0,0,0,0,1,0,0,1,0,0,3,0,10],[12,52,0.2308,0.59821,0.31224,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,9,0,0,2,0,0,3,0,0,2,0,0,6,0,7],[16,52,0.3077,0.58929,0.33071,0.28571,0.57143,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,9,0,0,1,0,0,3,0,0,1,0,0,7,0,7],[20,52,0.3846,0.68302,0.29393,0.57132,0.78564,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,3,0,0,0,0,0,3,0,0,6,0,0,10,0,6],[24,52,0.4615,0.83482,0.26271,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,20],[28,52,0.5385,0.74106,0.31225,0.39288,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,6,0,0,1,0,0,1,0,0,1,0,0,7,0,14],[32,52,0.6154,0.80356,0.23076,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,2,0,0,9,0,13],[36,52,0.6923,0.88393,0.19704,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,5,0,21],[40,52,0.7692,0.87946,0.15198,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,11,0,15],[44,52,0.8462,0.89285,0.17496,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,18],[48,52,0.9231,0.83926,0.19484,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,7,0,15],[52,52,1.0,0.74997,0.20519,0.57143,0.57143,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,2,0,12]]},{"b":7,"e":0.71429,"k":"rising","v":0.21876,"x":0.70089,"p":[[0,45,0.0,0.21876,0.16746,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,45,0.0889,0.70089,0.32015,0.42857,0.85707,1.0,0.0,1.0,1,12,0,1,0,3,0,0,2,0,0,4,0,0,1,0,0,4,0,0,5,0,12],[8,45,0.1778,0.57143,0.31339,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,8,0,0,3,0,0,4,0,0,3,0,0,2,0,8],[12,45,0.2667,0.67855,0.29234,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,2,0,0,4,0,0,4,0,0,5,0,0,5,0,9],[16,45,0.3556,0.63386,0.30081,0.53539,0.57143,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,4,0,0,1,0,0,10,0,0,2,0,0,4,0,8],[20,45,0.4444,0.61151,0.33938,0.28571,0.64286,1.0,0.14,1.0,0,10,0,0,0,7,0,0,3,0,0,3,0,0,3,0,0,3,0,0,3,0,10],[24,45,0.5333,0.62499,0.30462,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,7,0,0,2,0,0,3,0,0,5,0,0,4,0,8],[28,45,0.6222,0.58479,0.30589,0.28571,0.571,0.85714,0.14286,1.0,0,6,0,0,0,5,0,0,4,0,0,6,0,0,3,0,0,2,0,0,6,0,6],[32,45,0.7111,0.58479,0.3038,0.28571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,4,0,0,3,0,0,7,0,0,1,0,0,7,0,5],[36,45,0.8,0.52231,0.28706,0.28571,0.4286,0.85704,0.0,1.0,1,3,0,1,0,5,0,0,3,0,0,8,0,0,5,0,0,1,0,0,6,0,3],[40,45,0.8889,0.55357,0.32488,0.28571,0.50001,0.85714,0.0,1.0,1,7,0,1,0,6,0,0,3,0,0,6,0,0,3,0,0,3,0,0,3,0,7],[44,45,0.9778,0.41517,0.27515,0.2857,0.28571,0.57111,0.0,1.0,1,3,0,1,0,5,0,0,14,0,0,3,0,0,2,0,0,2,0,0,2,0,3],[45,45,1.0,0.44195,0.24577,0.2857,0.28571,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,12,0,0,3,0,0,4,0,0,4,0,0,3,0,1]]}]},{"i":"c2c6f8e6fff034d1","q":"15. (GDR 1) Let $n$ be an integer greater than 1 . In the Cartesian coordinate system we consider all squares with integer vertices $(x, y)$ such that $1 \\leq$ $x, y \\leq n$. Denote by $p_{k}(k=0,1,2, \\ldots)$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\\sum_{k}(k-1) p_{k}=0$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,42,0.0,0.92857,0.15568,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[4,42,0.0952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,42,0.1905,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[12,42,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,42,0.381,0.95535,0.16538,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[20,42,0.4762,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,42,0.5714,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[28,42,0.6667,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,42,0.7619,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[36,42,0.8571,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[40,42,0.9524,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[42,42,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.85265,"x":1.0,"p":[[0,32,0.0,0.93304,0.14279,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[4,32,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,32,0.25,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,32,0.375,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[16,32,0.5,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[28,32,0.875,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,32,1.0,0.85265,0.20359,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,18]]}]},{"i":"f0034882a0e4c67b","q":"27. N4 (POL) Let $k$ be a fixed integer greater than 1 , and let $m=4 k^{2}-5$. Show that there exist positive integers $a$ and $b$ such that the sequence $\\left(x_{n}\\right)$ defined by  $$ x_{0}=a, \\quad x_{1}=b, \\quad x_{n+2}=x_{n+1}+x_{n} \\quad \\text { for } \\quad n=0,1,2, \\ldots $$  has all of its terms relatively prime to $m$.","t":[{"b":5,"e":0.71429,"k":"flat","v":0.48212,"x":0.83035,"p":[[0,37,0.0,0.65177,0.32131,0.42857,0.64286,1.0,0.0,1.0,2,10,0,2,0,3,0,0,0,0,0,5,0,0,6,0,0,2,0,0,4,0,10],[4,37,0.1081,0.63839,0.31539,0.42857,0.64286,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,0,0,0,5,0,0,5,0,0,5,0,0,0,0,11],[8,37,0.2162,0.48212,0.28064,0.14286,0.49979,0.60714,0.0,1.0,1,3,0,1,0,8,0,0,1,0,0,6,0,0,8,0,0,3,0,0,2,0,3],[12,37,0.3243,0.61606,0.27765,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,0,0,0,8,0,0,7,0,0,4,0,0,1,0,8],[16,37,0.4324,0.5,0.29233,0.28571,0.42857,0.60714,0.0,1.0,1,5,0,1,0,6,0,0,2,0,0,10,0,0,5,0,0,1,0,0,2,0,5],[20,37,0.5405,0.55804,0.23787,0.42857,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,4,0,0,0,0,0,8,0,0,13,0,0,1,0,0,2,0,4],[24,37,0.6486,0.82589,0.18118,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,3,0,14],[28,37,0.7568,0.83035,0.1729,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,4,0,14],[32,37,0.8649,0.80801,0.16606,0.71429,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,3,0,12],[36,37,0.973,0.7857,0.17859,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,5,0,10],[37,37,1.0,0.75,0.19562,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,11,0,0,3,0,9]]},{"b":6,"e":0.14286,"k":"falling","v":0.25,"x":0.64273,"p":[[0,34,0.0,0.63393,0.3443,0.42857,0.57143,1.0,0.0,1.0,3,12,0,3,0,2,0,0,2,0,0,4,0,0,6,0,0,2,0,0,1,0,12],[4,34,0.1176,0.64273,0.29447,0.4286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,1,0,0,6,0,0,6,0,0,4,0,0,1,0,10],[8,34,0.2353,0.54911,0.24251,0.42857,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,8,0,0,10,0,0,2,0,0,2,0,4],[12,34,0.3529,0.59821,0.32031,0.39286,0.57143,1.0,0.0,1.0,1,9,0,1,0,5,0,0,2,0,0,4,0,0,6,0,0,4,0,0,1,0,9],[16,34,0.4706,0.54464,0.2822,0.42857,0.42859,0.75,0.0,1.0,1,5,0,1,0,4,0,0,1,0,0,11,0,0,5,0,0,2,0,0,3,0,5],[20,34,0.5882,0.30804,0.15612,0.14286,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,2,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[24,34,0.7059,0.34366,0.20168,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,11,0,0,2,0,0,13,0,0,4,0,0,0,0,0,0,0,1],[28,34,0.8235,0.25,0.17496,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,18,0,0,2,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[32,34,0.9412,0.30355,0.19476,0.14286,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,13,0,0,4,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[34,34,1.0,0.32589,0.14827,0.14286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,4,0,0,16,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"f503b0b9e2ea77da","q":"27. C7 (IRE) Let $p>3$ be a prime number. For each nonempty subset $T$ of $\\{0,1,2,3, \\ldots, p-1\\}$ let $E(T)$ be the set of all $(p-1)$-tuples $\\left(x_{1}, \\ldots, x_{p-1}\\right)$, where each $x_{i} \\in T$ and $x_{1}+2 x_{2}+\\cdots+(p-1) x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that  $$ |E(\\{0,1,3\\})| \\geq|E(\\{0,1,2\\})|, $$  with equality if and only if $p=5$.","t":[{"b":1,"e":0.57143,"k":"rising","v":0.62497,"x":0.8705,"p":[[0,50,0.0,0.62497,0.11152,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,9,0,0,3,0,0],[4,50,0.08,0.82142,0.14728,0.71429,0.78571,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,5,0,11],[8,50,0.16,0.80357,0.24157,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,4,0,15],[12,50,0.24,0.81695,0.17217,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,8,0,11],[16,50,0.32,0.8705,0.15308,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,7,0,16],[20,50,0.4,0.77679,0.15947,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,10,0,0,12,0,5],[24,50,0.48,0.83033,0.16921,0.71429,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,6,0,13],[28,50,0.56,0.83928,0.14174,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,9,0,11],[32,50,0.64,0.84821,0.16342,0.71429,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,8,0,14],[36,50,0.72,0.79683,0.15482,0.67857,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,1,0,12,0,7],[40,50,0.8,0.84371,0.16512,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,7,0,14],[44,50,0.88,0.80798,0.15823,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,10,0,9],[48,50,0.96,0.77677,0.14698,0.71429,0.78564,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,11,0,5],[50,50,1.0,0.79463,0.11811,0.71429,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,16,0,3]]},{"b":6,"e":0.42857,"k":"flat","v":0.57139,"x":0.8482,"p":[[0,70,0.0,0.57139,0.18211,0.57132,0.57143,0.60714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,3,0,0,19,0,0,5,0,0,3,0,0],[4,70,0.0571,0.84371,0.1531,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,10,0,12],[8,70,0.1143,0.81694,0.16067,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,9],[12,70,0.1714,0.83482,0.16409,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,9,0,12],[16,70,0.2286,0.82142,0.15974,0.71429,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,8,0,11],[20,70,0.2857,0.81247,0.14915,0.71429,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,11,0,8],[24,70,0.3429,0.76335,0.19107,0.67857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,9,0,0,7,0,8],[28,70,0.4,0.80353,0.15878,0.71429,0.78571,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,6,0,10],[32,70,0.4571,0.8482,0.15949,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,9,0,13],[36,70,0.5143,0.84375,0.1488,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,6,0,13],[40,70,0.5714,0.82573,0.13252,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,9,0,9],[44,70,0.6286,0.79909,0.15512,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,6,0,9],[48,70,0.6857,0.80353,0.16272,0.71429,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,11,0,8],[52,70,0.7429,0.74996,0.20205,0.67857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,11,0,0,5,0,8],[56,70,0.8,0.73206,0.17775,0.571,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,9,0,0,6,0,6],[60,70,0.8571,0.70089,0.20316,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,8,0,0,5,0,6],[64,70,0.9143,0.79463,0.18879,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,10,0,9],[68,70,0.9714,0.67409,0.212,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,7,0,0,4,0,6],[70,70,1.0,0.69192,0.19922,0.571,0.71429,0.85704,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,10,0,0,3,0,6]]}]},{"i":"be56006b34dc6e45","q":"$P(x)$  is a polynomial in  $x$  with non-negative integer coefficients. If  $P(1)=5$  and  $P(P(1))=177$ , what is the sum of all possible values of  $P(10)$ ?","t":[{"b":3,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,82,0.0,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[4,82,0.0488,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[8,82,0.0976,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,82,0.1463,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,82,0.1951,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,82,0.2439,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,82,0.2927,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,82,0.3415,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[32,82,0.3902,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[36,82,0.439,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,82,0.4878,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,82,0.5366,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[48,82,0.5854,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,82,0.6341,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,82,0.6829,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[60,82,0.7317,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,82,0.7805,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,82,0.8293,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,82,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,82,0.9268,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[80,82,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[82,82,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":7,"e":0.85714,"k":"flat","v":0.86607,"x":0.96429,"p":[[0,70,0.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[4,70,0.0571,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[8,70,0.1143,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,70,0.1714,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[16,70,0.2286,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[20,70,0.2857,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[24,70,0.3429,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[28,70,0.4,0.89286,0.08748,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10],[32,70,0.4571,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[36,70,0.5143,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[40,70,0.5714,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[44,70,0.6286,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[48,70,0.6857,0.87946,0.08073,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,24,0,7],[52,70,0.7429,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[56,70,0.8,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[60,70,0.8571,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[64,70,0.9143,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[68,70,0.9714,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[70,70,1.0,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]}]},{"i":"47955e04eec03854","q":"A sequence $x_{1}, x_{2}, \\ldots$ is defined by $x_{1}=1$ and $x_{2 k}=-x_{k}, x_{2 k-1}=(-1)^{k+1} x_{k}$ for all $k \\geq 1$. Show that for all $n \\geq 1$ it holds: $x_{1}+x_{2}+\\ldots+x_{n} \\geq 0$.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.52678,"x":0.75888,"p":[[0,134,0.0,0.52678,0.26592,0.28571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,8,0,0,3,0,0,8,0,0,5,0,0,1,0,4],[4,134,0.0299,0.74541,0.22255,0.67857,0.85714,0.85714,0.14,1.0,0,5,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,6,0,0,13,0,5],[8,134,0.0597,0.6964,0.228,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,11,0,0,8,0,4],[12,134,0.0896,0.75888,0.20962,0.57143,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,3,0,0,10,0,8],[16,134,0.1194,0.74105,0.19704,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,9,0,0,8,0,6],[20,134,0.1493,0.68742,0.15752,0.57132,0.64286,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,7,0,0,6,0,3],[24,134,0.1791,0.6339,0.22851,0.57132,0.71429,0.85704,0.0,1.0,1,1,0,1,0,1,0,0,3,0,0,1,0,0,8,0,0,9,0,0,8,0,1],[28,134,0.209,0.54459,0.25862,0.42857,0.57143,0.71429,0.0,1.0,3,1,0,3,0,2,0,0,2,0,0,3,0,0,7,0,0,12,0,0,2,0,1],[32,134,0.2388,0.62051,0.23854,0.5354,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,3,0,0,7,0,0,10,0,0,4,0,3],[36,134,0.2687,0.64728,0.22584,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,0,0,0,8,0,0,13,0,0,5,0,2],[40,134,0.2985,0.68747,0.19047,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,9,0,0,4,0,5],[44,134,0.3284,0.6964,0.20125,0.67857,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,0,0,0,5,0,0,14,0,0,8,0,2],[48,134,0.3582,0.7053,0.17838,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,7,0,0,9,0,3],[52,134,0.3881,0.69193,0.18596,0.5713,0.71429,0.85711,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,9,0,0,8,0,3],[56,134,0.4179,0.70085,0.19354,0.57142,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,0,11,0,3],[60,134,0.4478,0.67852,0.15976,0.57143,0.71429,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,10,0,0,6,0,2],[64,134,0.4776,0.75,0.19561,0.71429,0.78571,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,9,0,0,11,0,5],[68,134,0.5075,0.71872,0.20357,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,6,0,0,9,0,5],[72,134,0.5373,0.63389,0.19865,0.42859,0.71429,0.74996,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,12,0,0,8,0,0],[76,134,0.5672,0.60708,0.19563,0.571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,2,0,0,12,0,0,9,0,0,4,0,1],[80,134,0.597,0.70536,0.13803,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,13,0,0,10,0,0],[84,134,0.6269,0.558,0.23786,0.42857,0.57141,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,4,0,0,6,0,0,8,0,0,6,0,0,5,0,1],[88,134,0.6567,0.56247,0.18536,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,7,0,0,5,0,0,11,0,0,3,0,0],[92,134,0.6866,0.63391,0.18536,0.57132,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,12,0,0,5,0,1],[96,134,0.7164,0.65617,0.17078,0.571,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,12,0,0,4,0,2],[100,134,0.7463,0.5848,0.22406,0.571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,1,0,0,13,0,0,7,0,0,3,0,2],[104,134,0.7761,0.60261,0.2012,0.571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,2,0,0,12,0,0,8,0,0,3,0,2],[108,134,0.806,0.64281,0.2143,0.571,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,5,0,0,7,0,0,9,0,0,8,0,1],[112,134,0.8358,0.63834,0.20199,0.571,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,5,0,0,9,0,0,6,0,0,10,0,0],[116,134,0.8657,0.61157,0.17941,0.5354,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,9,0,0,9,0,0,6,0,0],[120,134,0.8955,0.65621,0.17809,0.57143,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,1,0,0,11,0,0,10,0,0,5,0,2],[124,134,0.9254,0.62052,0.23038,0.5354,0.71429,0.75,0.0,1.0,1,1,0,1,0,1,0,0,3,0,0,3,0,0,6,0,0,10,0,0,7,0,1],[128,134,0.9552,0.64282,0.18559,0.571,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,4,0,0,6,0,0,14,0,0,6,0,0],[132,134,0.9851,0.57584,0.2004,0.5354,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,2,0,0,11,0,0,9,0,0,4,0,0],[134,134,1.0,0.54459,0.19703,0.42857,0.57141,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,1,0,0,8,0,0,12,0,0,6,0,0,3,0,0]]},{"b":3,"e":0.14,"k":"flat","v":0.5223,"x":0.60711,"p":[[0,10,0.0,0.53124,0.22083,0.39286,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,7,0,0,6,0,0,8,0,0,1,0,2],[4,10,0.4,0.60711,0.27664,0.53539,0.71429,0.74996,0.0,1.0,2,3,0,2,0,3,0,0,1,0,0,2,0,0,6,0,0,10,0,0,5,0,3],[8,10,0.8,0.5937,0.27689,0.571,0.71429,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,1,0,0,0,0,0,8,0,0,11,0,0,6,0,1],[10,10,1.0,0.5223,0.31865,0.24999,0.57143,0.71429,0.0,1.0,5,3,0,5,0,3,0,0,1,0,0,4,0,0,4,0,0,9,0,0,3,0,3]]}]},{"i":"39fd85220de6d561","q":"9. G3 (EST) ${ }^{\\mathrm{IMO} 1}$ A set $S$ of points in space will be called completely symmetric if it has at least three elements and satisfies the following condition: For every two distinct points $A, B$ from $S$ the perpendicular bisector of the segment $A B$ is an axis of symmetry for $S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, a regular tetrahedron, or a regular octahedron.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.51338,"x":0.76783,"p":[[0,30,0.0,0.51338,0.22548,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,8,0,0,7,0,0,7,0,0,2,0,1],[4,30,0.1333,0.70977,0.17674,0.71429,0.71429,0.857,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,16,0,0,7,0,2],[8,30,0.2667,0.73657,0.10782,0.71429,0.71429,0.85704,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,9,0,1],[12,30,0.4,0.72315,0.11263,0.71421,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,18,0,0,5,0,2],[16,30,0.5333,0.66955,0.14043,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,15,0,0,4,0,1],[20,30,0.6667,0.7187,0.16556,0.57143,0.71429,0.85704,0.1429,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,13,0,0,7,0,3],[24,30,0.8,0.73656,0.12931,0.71421,0.71429,0.857,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,0,0,9,0,2],[28,30,0.9333,0.76783,0.16659,0.57143,0.71429,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,8,0,7],[30,30,1.0,0.70085,0.14883,0.57143,0.71429,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,9,0,0,10,0,1]]},{"b":4,"e":0.571,"k":"flat","v":0.53122,"x":0.80342,"p":[[0,37,0.0,0.53122,0.16839,0.42857,0.571,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,10,0,0,9,0,0,5,0,0,3,0,0],[4,37,0.1081,0.80342,0.14627,0.71429,0.78564,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,8,0,8],[8,37,0.2162,0.7497,0.15162,0.67536,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,7,0,5],[12,37,0.3243,0.79687,0.14408,0.71429,0.78564,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,12,0,0,9,0,7],[16,37,0.4324,0.72545,0.1657,0.71429,0.71429,0.85714,0.1429,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,4,1,0,15,0,0,7,0,3],[20,37,0.5405,0.74774,0.17033,0.71429,0.71429,0.85714,0.1429,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,1,2,0,0,13,0,0,11,0,3],[24,37,0.6486,0.76769,0.16662,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,14,0,0,9,0,5],[28,37,0.7568,0.78879,0.10831,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,16,0,0,10,0,4],[32,37,0.8649,0.7476,0.17407,0.713,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,11,1,0,8,0,5],[36,37,0.973,0.68294,0.16272,0.57143,0.71429,0.857,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,11,0,0,8,0,1],[37,37,1.0,0.58036,0.16338,0.4286,0.57141,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,7,0,0,11,0,0,7,0,0,4,0,0]]}]},{"i":"728b55e8725b6857","q":"Let  $ a, b, c $  be positive real numbers less than or equal to  $ \\sqrt{2} $  such that  $ abc = 2 $ , prove that   $$  \\sqrt{2}\\displaystyle\\sum_{cyc}\\frac{ab + 3c}{3ab + c} \\ge a + b + c  $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,23,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,23,0.1739,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,23,0.3478,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,23,0.5217,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":0.571,"k":"flat","v":0.95088,"x":0.99554,"p":[[0,30,0.0,0.95088,0.16605,1.0,1.0,1.0,0.1429,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[4,30,0.1333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,30,0.2667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,30,0.4,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,30,0.5333,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,30,0.6667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,30,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,30,0.9333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[30,30,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"f500b6a17debbd9f","q":"1. (BUL 1) An integer sequence is defined by  $$ a_{n}=2 a_{n-1}+a_{n-2} \\quad(n>1), \\quad a_{0}=0, \\quad a_{1}=1 $$  Prove that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,52,0.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[4,52,0.0769,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[8,52,0.1538,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,52,0.2308,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[16,52,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,52,0.3846,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,52,0.4615,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,52,0.5385,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,52,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,52,0.6923,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,52,0.7692,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,61,0.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[4,61,0.0656,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,61,0.1311,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[12,61,0.1967,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,61,0.2623,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[20,61,0.3279,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,61,0.3934,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,61,0.459,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,61,0.5246,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,61,0.5902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,61,0.6557,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,61,0.7213,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,61,0.7869,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,61,0.8525,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,61,0.918,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,61,0.9836,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[61,61,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]}]},{"i":"9e93351c4382c168","q":"Consider a complete graph on  $4046$  nodes, whose edges are colored in some colors. Let's call this graph  $k$ -good if we can split all its nodes into  $2023$  pairs so that there are exactly  $k$  distinct colors among the colors of  $2023$  edges that connect the nodes from the same pairs.  Is it possible that the graph is  $999$ -good and  $1001$ -good but not  $1000$ -good?\n\n*Proposed by Anton Trygub*","t":[{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.21429,"p":[[0,70,0.0,0.21429,0.3677,0.0,0.0,0.28571,0.0,1.0,23,3,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,3],[4,70,0.0571,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,70,0.1143,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,70,0.1714,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,70,0.2286,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[20,70,0.2857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,70,0.4,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,70,0.4571,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,70,0.5143,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,70,0.5714,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[44,70,0.6286,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,70,0.6857,0.10714,0.27199,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,1],[52,70,0.7429,0.13393,0.32328,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[56,70,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,70,0.9143,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.30804,"p":[[0,26,0.0,0.30804,0.41819,0.0,0.0,0.71429,0.0,1.0,20,6,0,20,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,6],[4,26,0.1538,0.23661,0.40502,0.0,0.0,0.28571,0.0,1.0,23,6,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,6],[8,26,0.3077,0.125,0.31084,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[12,26,0.4615,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5b92ed22ef6e1deb","q":"Consider a group  $ G $  which has the property that any element of it, with the exception of the identity, has order  $ p\\ge 2. $  Prove that**a)**  $ p $  is prime.**b)**  $ G $  is commutative if any subset of  $ G $  having  $ p^2-1 $  elements contains at least  $ p $  elements that commute between themselves pairwise.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.21853,"x":0.79911,"p":[[0,24,0.0,0.21853,0.18568,0.14286,0.14286,0.1429,0.14,1.0,0,1,0,0,0,25,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[4,24,0.1667,0.35713,0.22867,0.14286,0.28571,0.4642,0.14286,1.0,0,1,0,0,0,11,0,0,9,0,0,4,0,0,4,0,0,2,0,0,1,0,1],[8,24,0.3333,0.79911,0.16311,0.71429,0.78571,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,7,0,9],[12,24,0.5,0.70978,0.23553,0.67846,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,12,0,0,6,0,6],[16,24,0.6667,0.71426,0.19235,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,11,0,0,6,0,5],[20,24,0.8333,0.78123,0.17853,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,0,6,0,9],[24,24,1.0,0.69637,0.17771,0.57132,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,11,0,0,5,0,4]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.39728,"p":[[0,69,0.0,0.18724,0.10983,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,27,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,69,0.058,0.39708,0.28524,0.14286,0.21428,0.60714,0.14,1.0,0,1,0,0,0,16,0,0,1,0,0,2,0,0,5,0,0,4,0,0,3,0,1],[8,69,0.1159,0.39728,0.25184,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,11,0,0,7,0,0,2,0,0,5,0,0,5,0,0,1,0,1],[12,69,0.1739,0.33034,0.23804,0.14286,0.2857,0.4642,0.14286,0.85714,0,0,0,0,0,15,0,0,8,0,0,1,0,0,2,0,0,4,0,0,2,0,0],[16,69,0.2319,0.38839,0.284,0.14286,0.28571,0.60714,0.14286,1.0,0,2,0,0,0,15,0,0,4,0,0,1,0,0,4,0,0,5,0,0,1,0,2],[20,69,0.2899,0.29004,0.1906,0.14286,0.14295,0.42857,0.14,0.71429,0,0,0,0,0,17,0,0,6,0,0,2,0,0,5,0,0,2,0,0,0,0,0],[24,69,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.2366,0.26869,0.0,0.14286,0.2857,0.0,0.85714,11,0,0,11,0,9,0,0,5,0,0,1,0,0,1,0,0,3,0,0,2,0,0]]}]},{"i":"e5ca3eba893ecc6d","q":"A circle $\\omega$ is inscribed in a quadrilateral $A B C D$. Let $I$ be the center of $\\omega$. Suppose that  $$ (A I+D I)^{2}+(B I+C I)^{2}=(A B+C D)^{2} $$  Prove that $A B C D$ is an isosceles trapezoid.","t":[{"b":2,"e":0.857,"k":"rising","v":0.39283,"x":0.75443,"p":[[0,169,0.0,0.39283,0.16749,0.28571,0.42857,0.571,0.0,0.71429,2,0,1,2,0,0,0,0,13,0,0,8,0,0,7,0,0,2,0,0,0,0,0],[4,169,0.0237,0.68746,0.20654,0.571,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,3,0,0,4,0,7],[8,169,0.0473,0.67851,0.1856,0.571,0.57143,0.857,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,6,0,0,4,0,5],[12,169,0.071,0.66964,0.21558,0.42857,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,9,0,0,5,0,0,6,0,0,6,0,5],[16,169,0.0947,0.6428,0.25507,0.53539,0.57143,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,5,0,0,13,0,0,1,0,0,2,0,8],[20,169,0.1183,0.64726,0.18894,0.571,0.57143,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,5,0,0,14,0,0,4,0,0,4,0,4],[24,169,0.142,0.65619,0.22264,0.4286,0.57143,0.857,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,9,0,0,8,0,0,5,0,0,2,0,7],[28,169,0.1657,0.66957,0.22714,0.4286,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,7,0,0,8,0,0,3,0,0,6,0,6],[32,169,0.1893,0.65623,0.23106,0.4286,0.64286,0.857,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,6,0,0,4,0,6],[36,169,0.213,0.68302,0.23619,0.42859,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,9,0,0,6,0,0,5,0,0,2,0,9],[40,169,0.2367,0.65622,0.17075,0.571,0.57143,0.85702,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,5,0,0,9,0,1],[44,169,0.2604,0.66512,0.18426,0.57132,0.64286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,9,0,0,3,0,4],[48,169,0.284,0.62049,0.2008,0.571,0.57143,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,2,0,0,16,0,0,2,0,0,5,0,3],[52,169,0.3077,0.62937,0.18163,0.571,0.57143,0.71429,0.1429,1.0,0,4,0,0,0,1,0,0,0,0,0,3,0,0,18,0,0,5,0,0,1,0,4],[56,169,0.3314,0.6741,0.22084,0.4286,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,4,0,0,2,0,8],[60,169,0.355,0.69194,0.20552,0.571,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,6,0,0,7,0,5],[64,169,0.3787,0.68747,0.18709,0.57132,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,6,0,0,7,0,4],[68,169,0.4024,0.63381,0.25256,0.42857,0.57143,0.85714,0.14,1.0,0,7,0,0,0,1,0,0,3,0,0,7,0,0,8,0,0,3,0,0,3,0,7],[72,169,0.426,0.64725,0.17493,0.571,0.57143,0.74996,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,17,0,0,3,0,0,5,0,3],[76,169,0.4497,0.64729,0.223,0.53539,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,11,0,0,4,0,0,3,0,6],[80,169,0.4734,0.63832,0.24997,0.53539,0.57143,0.85704,0.0,1.0,1,6,0,1,0,1,0,0,1,0,0,5,0,0,10,0,0,5,0,0,3,0,6],[84,169,0.497,0.66962,0.27535,0.4286,0.64286,1.0,0.0,1.0,1,9,0,1,0,0,0,0,3,0,0,6,0,0,6,0,0,3,0,0,4,0,9],[88,169,0.5207,0.65618,0.23655,0.571,0.57143,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,5,0,0,11,0,0,4,0,0,4,0,6],[92,169,0.5444,0.66514,0.17355,0.571,0.71429,0.85704,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,8,0,0,7,0,2],[96,169,0.568,0.70533,0.2447,0.571,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,3,0,0,4,0,0,8,0,0,4,0,0,3,0,10],[100,169,0.5917,0.65621,0.21088,0.42859,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,4,0,0,5,0,5],[104,169,0.6154,0.63838,0.20512,0.57132,0.57143,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,4,0,0,12,0,0,5,0,0,4,0,4],[108,169,0.6391,0.70087,0.20628,0.57132,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,6,0,0,8,0,5],[112,169,0.6627,0.67632,0.19065,0.57143,0.64286,0.85704,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,6,1,0,5,0,4],[116,169,0.6864,0.63393,0.19541,0.4286,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,8,0,0,10,0,0,6,0,0,3,0,4],[120,169,0.7101,0.72319,0.18536,0.57143,0.71429,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,10,0,0,5,0,6],[124,169,0.7337,0.72316,0.19545,0.57132,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,5,0,0,6,0,7],[128,169,0.7574,0.7231,0.18885,0.571,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,6,0,0,5,0,7],[132,169,0.7811,0.68749,0.23266,0.5354,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,3,0,0,8,0,6],[136,169,0.8047,0.66514,0.25658,0.42859,0.64286,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,5,0,0,5,0,0,6,0,0,4,0,0,4,0,8],[140,169,0.8284,0.66509,0.18769,0.571,0.57143,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,8,0,0,2,0,5],[144,169,0.8521,0.68744,0.21854,0.57132,0.71414,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,3,0,0,10,0,0,6,0,0,5,0,6],[148,169,0.8757,0.65177,0.18537,0.57142,0.57143,0.85711,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,4,0,0,6,0,3],[152,169,0.8994,0.69862,0.20343,0.5713,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,1,3,0,0,9,0,0,7,0,0,5,0,6],[156,169,0.9231,0.72764,0.20936,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,7,0,0,5,0,8],[160,169,0.9467,0.65167,0.22001,0.42859,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,6,0,0,8,0,0,4,0,0,7,0,4],[164,169,0.9704,0.75443,0.22372,0.57143,0.85712,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,3,0,0,9,0,9],[168,169,0.9941,0.71427,0.26487,0.53539,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,5,0,0,6,0,0,4,0,0,2,0,12],[169,169,1.0,0.74099,0.1971,0.57143,0.64286,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,3,0,0,3,0,10]]},{"b":3,"e":1.0,"k":"rising","v":0.38836,"x":0.76335,"p":[[0,115,0.0,0.38836,0.17938,0.28571,0.42857,0.571,0.0,0.71429,3,0,0,3,0,0,0,0,11,0,0,9,0,0,7,0,0,2,0,0,0,0,0],[4,115,0.0348,0.68298,0.24935,0.571,0.57143,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,5,0,0,13,0,0,2,0,0,1,0,10],[8,115,0.0696,0.59372,0.19595,0.5354,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,6,0,0,14,0,0,5,0,0,3,0,2],[12,115,0.1043,0.69637,0.20127,0.57132,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,4,0,0,2,0,8],[16,115,0.1391,0.67854,0.22017,0.5354,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,7,0,0,9,0,0,4,0,0,4,0,7],[20,115,0.1739,0.70982,0.20355,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,8,0,0,4,0,7],[24,115,0.2087,0.69637,0.20128,0.5713,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,8,0,0,4,0,6],[28,115,0.2435,0.68748,0.2299,0.5354,0.64286,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,3,0,0,6,0,7],[32,115,0.2783,0.6874,0.19382,0.571,0.57143,0.85704,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,2,0,0,3,0,7],[36,115,0.313,0.71872,0.19721,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,6,0,0,5,0,7],[40,115,0.3478,0.74994,0.22308,0.571,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,2,0,0,9,0,9],[44,115,0.3826,0.75,0.20203,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,5,0,0,6,0,9],[48,115,0.4174,0.65173,0.1673,0.571,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,8,0,0,5,0,2],[52,115,0.4522,0.67407,0.20898,0.571,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,6,0,0,10,0,0,5,0,0,4,0,6],[56,115,0.487,0.65623,0.17445,0.57143,0.57143,0.857,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,5,0,0,7,0,2],[60,115,0.5217,0.72761,0.20005,0.57143,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,4,0,0,3,0,9],[64,115,0.5565,0.71872,0.21276,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,12,0,0,4,0,0,5,0,8],[68,115,0.5913,0.65177,0.17474,0.57143,0.57143,0.85704,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,2,0,0,6,0,3],[72,115,0.6261,0.69192,0.20552,0.57143,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,3,0,0,4,0,7],[76,115,0.6609,0.67854,0.189,0.57143,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,4,0,0,5,0,5],[80,115,0.6957,0.66071,0.20124,0.57143,0.64286,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,3,0,0,12,0,0,9,0,0,3,0,4],[84,115,0.7304,0.6607,0.21944,0.57143,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,4,0,0,14,0,0,3,0,0,5,0,5],[88,115,0.7652,0.68749,0.22143,0.57143,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,10,0,0,0,0,8],[92,115,0.8,0.64283,0.25,0.53539,0.57143,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,6,0,0,9,0,0,4,0,0,7,0,4],[96,115,0.8348,0.683,0.20121,0.57143,0.71429,0.85714,0.1429,1.0,0,4,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,8,0,0,6,0,4],[100,115,0.8696,0.6339,0.17105,0.57132,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,3,0,0,4,0,3],[104,115,0.9043,0.67409,0.21499,0.57143,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,4,0,0,13,0,0,5,0,0,2,0,7],[108,115,0.9391,0.76335,0.18078,0.57143,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,4,0,0,8,0,8],[112,115,0.9739,0.67409,0.18293,0.57143,0.57143,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,4,0,0,2,0,6],[115,115,1.0,0.6562,0.14224,0.57143,0.57143,0.74996,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,4,0,0,7,0,1]]}]},{"i":"bc152b8accb62c41","q":"Consider a triangle  $ABC$  with  $\\angle ACB = 2 \\angle CAB $  and  $\\angle ABC> 90 ^ \\circ$ . Consider the perpendicular on  $AC$  that passes through  $A$  and intersects  $BC$  at  $D$ , prove that   $$ \\frac {1} {BC} - \\frac {2} {DC} = \\frac {1} {CA}  $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,56,0.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,56,0.0714,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,56,0.1429,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[12,56,0.2143,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,56,0.2857,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,56,0.3571,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,56,0.4286,0.9375,0.13804,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,23],[28,56,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,56,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,56,0.6429,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,56,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,56,0.7857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,56,0.8571,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.91517,"x":1.0,"p":[[0,106,0.0,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,106,0.0377,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,106,0.0755,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,106,0.1132,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,106,0.1509,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,106,0.1887,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,106,0.2264,0.94643,0.1915,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[28,106,0.2642,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,106,0.3019,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[36,106,0.3396,0.93303,0.19227,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[40,106,0.3774,0.9375,0.18189,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[44,106,0.4151,0.9375,0.15126,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[48,106,0.4528,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[52,106,0.4906,0.93304,0.13825,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,22],[56,106,0.5283,0.95089,0.1411,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[60,106,0.566,0.91518,0.17445,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,24],[64,106,0.6038,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[68,106,0.6415,0.9375,0.15541,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,23],[72,106,0.6792,0.91517,0.15093,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,9,0,20],[76,106,0.717,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[80,106,0.7547,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[84,106,0.7925,0.95089,0.13175,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[88,106,0.8302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,106,0.8679,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,106,0.9057,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,106,0.9434,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,106,0.9811,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[106,106,1.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"8f89e3b481e4fdb0","q":"Consider an acute triangle  $ABC$  and it's circumcircle  $\\omega$ . With center  $A$ , we construct a circle  $\\gamma$  that intersects arc  $AB$  of circle  $\\omega$  , that doesn't contain  $C$ , at point  $D$  and arc  $AC$  , that doesn't contain  $B$ , at point  $E$ . Suppose that the intersection point  $K$  of lines  $BE$  and  $CD$  lies on circle  $\\gamma$ . Prove that line  $AK$  is perpendicular on line  $BC$ 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equilateral triangle is divided into 25 congruent triangles enumerated with numbers from 1 to 25 . Prove that one can find two triangles having a common side and with the difference of the numbers assigned to them greater than 3 .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.04911,"x":0.4732,"p":[[0,51,0.0,0.04911,0.11633,0.0,0.0,0.0,0.0,0.4286,26,0,21,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,51,0.0784,0.4732,0.35792,0.14286,0.42857,0.85714,0.0,1.0,4,6,0,4,0,8,0,0,2,0,0,4,0,0,3,0,0,2,0,0,3,0,6],[8,51,0.1569,0.36598,0.31128,0.14286,0.28571,0.5,0.0,1.0,3,3,0,3,0,12,0,0,5,0,0,4,0,0,0,0,0,3,0,0,2,0,3],[12,51,0.2353,0.27679,0.29001,0.10714,0.14286,0.32143,0.0,1.0,8,2,0,8,0,10,0,0,6,0,0,2,0,0,1,0,0,2,0,0,1,0,2],[16,51,0.3137,0.42857,0.37286,0.14286,0.28571,0.85704,0.0,1.0,5,6,0,5,0,10,0,0,3,0,0,1,0,0,3,0,0,1,0,0,3,0,6],[20,51,0.3922,0.37505,0.35311,0.14286,0.14286,0.57143,0.0,1.0,6,5,0,6,0,11,0,0,1,0,0,4,0,0,3,0,0,0,0,0,2,0,5],[24,51,0.4706,0.3884,0.34669,0.14286,0.14286,0.71429,0.0,1.0,3,4,0,3,0,15,0,0,2,0,0,0,0,0,3,0,0,2,0,0,3,0,4],[28,51,0.549,0.41071,0.33264,0.14286,0.42857,0.60714,0.0,1.0,6,4,0,6,0,6,0,0,3,0,0,6,0,0,3,0,0,2,0,0,2,0,4],[32,51,0.6275,0.21875,0.26721,0.0,0.14286,0.28571,0.0,1.0,11,1,0,11,0,10,0,0,6,0,0,0,0,0,2,0,0,0,0,0,2,0,1],[36,51,0.7059,0.19643,0.28516,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,2],[40,51,0.7843,0.30356,0.31692,0.10714,0.14286,0.46431,0.0,1.0,8,2,0,8,0,11,0,0,3,0,0,2,0,0,1,0,0,3,0,0,2,0,2],[44,51,0.8627,0.14286,0.18558,0.0,0.14286,0.14287,0.0,1.0,12,1,0,12,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,51,0.9412,0.11589,0.10967,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.14287,0.12878,0.0,0.14286,0.14287,0.0,0.42857,10,0,0,10,0,15,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"rising","v":0.05793,"x":0.53571,"p":[[0,52,0.0,0.05793,0.11757,0.0,0.0,0.14071,0.0,0.571,23,0,20,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,52,0.0769,0.30801,0.25778,0.14286,0.21429,0.571,0.0,0.85714,6,0,0,6,0,10,0,0,4,0,0,3,0,0,4,0,0,4,0,0,1,0,0],[8,52,0.1538,0.37499,0.39729,0.0,0.14286,0.85704,0.0,1.0,9,6,0,9,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,4,0,6],[12,52,0.2308,0.48213,0.37585,0.14286,0.42857,0.85714,0.0,1.0,6,7,0,6,0,5,0,0,3,0,0,4,0,0,2,0,0,2,0,0,3,0,7],[16,52,0.3077,0.53571,0.33119,0.25,0.50001,0.85714,0.0,1.0,4,3,0,4,0,4,0,0,1,0,0,7,0,0,1,0,0,4,0,0,8,0,3],[20,52,0.3846,0.51339,0.33666,0.24999,0.50001,0.85714,0.0,1.0,3,4,0,3,0,5,0,0,6,0,0,2,0,0,2,0,0,4,0,0,6,0,4],[24,52,0.4615,0.49105,0.34429,0.14286,0.49979,0.85704,0.0,1.0,2,5,0,2,0,10,0,0,2,0,0,2,0,0,4,0,0,3,0,0,4,0,5],[28,52,0.5385,0.40616,0.30545,0.14286,0.28571,0.60714,0.0,1.0,3,2,0,3,0,9,0,0,5,0,0,4,0,0,3,0,0,2,0,0,4,0,2],[32,52,0.6154,0.36607,0.36585,0.14286,0.14286,0.60714,0.0,1.0,5,6,0,5,0,15,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,6],[36,52,0.6923,0.4933,0.36655,0.14286,0.49999,1.0,0.0,1.0,1,9,0,1,0,13,0,0,1,0,0,1,0,0,5,1,0,1,0,0,0,0,9],[40,52,0.7692,0.27223,0.23792,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,19,0,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[44,52,0.8462,0.26786,0.24936,0.14286,0.14286,0.28571,0.0,1.0,3,2,0,3,0,16,0,0,7,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[48,52,0.9231,0.26786,0.21943,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,17,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[52,52,1.0,0.24107,0.12595,0.14286,0.14288,0.32143,0.0,0.4286,1,0,0,1,0,16,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8374cf1433a4ed5b","q":"Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?","t":[{"b":5,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,48,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,48,0.0833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,48,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,48,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,48,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,48,0.4167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,48,0.5,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,48,0.5833,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,48,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,48,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,48,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,48,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,48,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,68,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,68,0.0588,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,68,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,68,0.1765,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,68,0.2353,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[20,68,0.2941,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[24,68,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,68,0.4118,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[32,68,0.4706,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,68,0.5294,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[40,68,0.5882,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[44,68,0.6471,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[48,68,0.7059,0.93304,0.22011,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[52,68,0.7647,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,68,0.8235,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[60,68,0.8824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,68,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,68,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e78ea63b5f2ebb3e","q":"Find all functions  $f\\colon \\mathbb{Z}^2 \\to [0, 1]$  such that for any integers  $x$  and  $y$ ,\n\\[f(x, y) = \\frac{f(x - 1, y) + f(x, y - 1)}{2}.\\]\n\n*Proposed by Yang Liu and Michael Kural*","t":[{"b":0,"e":0.0,"k":"flat","v":0.08036,"x":0.25436,"p":[[0,38,0.0,0.12277,0.16678,0.0,0.14286,0.14286,0.0,0.85714,14,0,0,14,1,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,38,0.1053,0.1875,0.20652,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,8,0,0,7,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[8,38,0.2105,0.22321,0.23941,0.0,0.14286,0.42857,0.0,1.0,12,1,0,12,0,6,0,0,5,0,0,5,0,0,3,0,0,0,0,0,0,0,1],[12,38,0.3158,0.23205,0.23626,0.105,0.14286,0.42857,0.0,1.0,8,1,0,8,0,13,0,0,2,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[16,38,0.4211,0.21429,0.20516,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,11,0,0,5,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[20,38,0.5263,0.2433,0.22222,0.14286,0.14286,0.375,0.0,0.71429,7,0,0,7,0,12,0,0,4,0,1,4,0,0,0,0,0,4,0,0,0,0,0],[24,38,0.6316,0.21429,0.19885,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,11,0,0,7,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[28,38,0.7368,0.25436,0.21649,0.14214,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,11,0,0,4,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[32,38,0.8421,0.24107,0.23808,0.0,0.14286,0.32143,0.0,1.0,9,1,0,9,0,9,0,0,6,0,0,2,0,0,5,0,0,0,0,0,0,0,1],[36,38,0.9474,0.08036,0.09407,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.10714,"x":0.28124,"p":[[0,28,0.0,0.10714,0.15152,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,10,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,28,0.1429,0.25444,0.23615,0.10714,0.14286,0.42857,0.0,1.0,8,1,0,8,0,10,0,0,3,0,0,6,0,0,4,0,0,0,0,0,0,0,1],[8,28,0.2857,0.24098,0.2486,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,9,0,0,3,0,0,5,0,0,3,0,0,1,0,0,0,0,1],[12,28,0.4286,0.28124,0.2244,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,9,0,0,6,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[16,28,0.5714,0.24107,0.26107,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,9,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[20,28,0.7143,0.21865,0.22012,0.14286,0.14286,0.17857,0.0,1.0,4,1,0,4,0,20,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[24,28,0.8571,0.22321,0.18877,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,18,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[28,28,1.0,0.20089,0.20782,0.10714,0.14286,0.17857,0.0,0.71429,8,0,0,8,0,16,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,0]]}]},{"i":"9e6f19599c6912e9","q":"Consider all finite sequences of positive real numbers each of whose terms is at most  $3$  and the sum of whose terms is more than  $100$ . For each such sequence, let  $S$  denote the sum of the subsequence whose sum is the closest to  $100$ , and define the *defect* of this sequence to be the value  $|S-100|$ . Find the maximum possible value of the defect.","t":[{"b":4,"e":0.0,"k":"falling","v":0.00893,"x":0.34374,"p":[[0,67,0.0,0.17402,0.14169,0.0,0.2857,0.28571,0.0,0.42857,12,0,8,12,0,2,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.33482,0.28707,0.21427,0.28571,0.42858,0.0,1.0,8,3,0,8,0,0,0,0,14,0,0,4,0,0,1,0,0,2,0,0,0,0,3],[8,67,0.1194,0.25893,0.15335,0.24999,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,2,0,0,18,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[12,67,0.1791,0.26785,0.12753,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,2,0,0,21,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,67,0.2388,0.34374,0.21085,0.2857,0.28571,0.42857,0.0,1.0,4,1,0,4,0,1,0,0,15,0,0,7,0,0,2,0,0,2,0,0,0,0,1],[20,67,0.2985,0.28571,0.21723,0.21427,0.28571,0.32143,0.0,0.857,8,0,0,8,0,0,0,0,16,0,0,4,0,0,1,0,0,2,0,0,1,0,0],[24,67,0.3582,0.21429,0.17496,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,0,0,0,17,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[28,67,0.4179,0.15177,0.19538,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,1,0,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[32,67,0.4776,0.1741,0.18117,0.0,0.21428,0.28571,0.0,0.57143,15,0,0,15,0,1,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[36,67,0.5373,0.11607,0.18363,0.0,0.0,0.28571,0.0,0.85714,20,0,0,20,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,67,0.597,0.13839,0.16164,0.0,0.0,0.28571,0.0,0.4286,18,0,0,18,0,0,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.14732,0.16935,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,2,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.12053,0.16409,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,1,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.16964,0.20025,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,0,0,0,10,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[56,67,0.8358,0.08036,0.13803,0.0,0.0,0.14287,0.0,0.42857,23,0,0,23,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.16964,0.19377,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,1,0,0,11,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[64,67,0.9552,0.12946,0.21829,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[67,67,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.24107,"p":[[0,62,0.0,0.17857,0.12877,0.0,0.2857,0.28571,0.0,0.28571,10,0,6,10,0,4,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.13839,0.21572,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,0,0,0,7,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[8,62,0.129,0.13839,0.16554,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,0,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,62,0.1935,0.19195,0.19757,0.0,0.28571,0.28571,0.0,0.85714,13,0,0,13,0,2,0,0,14,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[16,62,0.2581,0.12052,0.16012,0.0,0.0,0.28571,0.0,0.571,19,0,0,19,0,2,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,62,0.3226,0.15625,0.17985,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,2,0,0,11,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[24,62,0.3871,0.24107,0.14914,0.21427,0.28571,0.28571,0.0,0.4286,8,0,0,8,0,0,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.13839,0.22442,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,0,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[32,62,0.5161,0.10714,0.15152,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.12946,0.16115,0.0,0.0,0.28571,0.0,0.4286,19,0,0,19,0,0,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.12054,0.15198,0.0,0.0,0.28571,0.0,0.42857,19,0,0,19,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.10714,0.16366,0.0,0.0,0.2857,0.0,0.42857,22,0,0,22,0,0,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.07143,0.13832,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d90299d2528bc0a8","q":"Ana and Beto play against each other. Initially, Ana chooses a non-negative integer  $N$  and announces it to Beto. Next Beto writes a succession of  $2016$  numbers,  $1008$  of them equal to  $1$  and  $1008$  of them equal to  $-1$ . Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to  $N$ , Ana wins. If not, Beto wins. Determine all values of  $N$  for which Ana can ensure victory, no matter how Beto plays.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.65619,"x":0.84374,"p":[[0,23,0.0,0.82142,0.17857,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,13,0,10],[4,23,0.1739,0.76783,0.21653,0.57143,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,4,0,0,6,0,11],[8,23,0.3478,0.65619,0.24187,0.42857,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,4,0,0,7,0,0,5,0,0,5,0,6],[12,23,0.5217,0.74999,0.22017,0.71429,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,13,0,0,2,0,10],[16,23,0.6957,0.7857,0.16368,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,4,0,9],[20,23,0.8696,0.83928,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,3,0,15],[23,23,1.0,0.84374,0.13999,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,4,0,13]]},{"b":5,"e":0.71429,"k":"flat","v":0.68748,"x":0.88392,"p":[[0,89,0.0,0.75892,0.21852,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,4,0,0,12,0,7],[4,89,0.0449,0.88391,0.18366,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,19],[8,89,0.0899,0.88392,0.19704,0.85711,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,6,0,20],[12,89,0.1348,0.70981,0.33214,0.42857,1.0,1.0,0.0,1.0,1,17,1,1,0,1,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,17],[16,89,0.1798,0.78579,0.22306,0.71429,0.78564,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,9,0,0,3,0,13],[20,89,0.2247,0.77677,0.23675,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,6,0,0,6,0,12],[24,89,0.2697,0.74553,0.24675,0.57143,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,9,0,0,2,0,12],[28,89,0.3146,0.74554,0.26422,0.67857,0.78571,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,8,0,0,5,0,11],[32,89,0.3596,0.79016,0.21721,0.67857,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,7,0,0,4,0,13],[36,89,0.4045,0.72319,0.26713,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,4,0,0,0,0,0,6,0,0,6,0,0,5,0,10],[40,89,0.4494,0.73212,0.23353,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,7,0,0,8,0,0,3,0,10],[44,89,0.4944,0.74552,0.24416,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,0,0,0,5,0,0,9,0,0,3,0,11],[48,89,0.5393,0.74553,0.27137,0.57143,0.85712,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,5,0,0,0,0,0,3,0,0,6,0,0,5,0,12],[52,89,0.5843,0.72311,0.25759,0.57132,0.71429,1.0,0.14,1.0,0,10,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,6,0,0,5,0,10],[56,89,0.6292,0.70088,0.26813,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,0,0,0,5,0,0,8,0,0,4,0,9],[60,89,0.6742,0.68748,0.28221,0.53539,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,5,0,0,1,0,0,4,0,0,6,0,0,5,0,9],[64,89,0.7191,0.80357,0.25191,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,16],[68,89,0.764,0.75888,0.25114,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,4,0,0,7,0,11],[72,89,0.809,0.76786,0.18814,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,13,0,0,6,0,8],[76,89,0.8539,0.72762,0.18685,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,14,0,0,2,0,7],[80,89,0.8989,0.70982,0.20666,0.57143,0.71429,0.85704,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,13,0,0,2,0,7],[84,89,0.9438,0.74999,0.2113,0.71429,0.71429,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,13,0,0,5,0,8],[88,89,0.9888,0.77237,0.19178,0.67857,0.71429,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,11,0,0,2,0,11],[89,89,1.0,0.76338,0.17719,0.67857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,12,0,0,3,0,9]]}]},{"i":"b4517de7692713b5","q":"Find all pairs  $(a,\\, b)$  of positive integers such that  $2a-1$  and  $2b+1$  are coprime and  $a+b$  divides  $4ab+1.$","t":[{"b":1,"e":0.57143,"k":"falling","v":0.49107,"x":0.87054,"p":[[0,135,0.0,0.87054,0.24053,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,2,0,23],[4,135,0.0296,0.78571,0.29451,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,6,0,0,3,0,0,0,0,0,2,0,0,2,0,19],[8,135,0.0593,0.72321,0.333,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0,3,0,17],[12,135,0.0889,0.70089,0.31209,0.28571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,8,0,0,2,0,0,1,0,0,3,0,0,4,0,13],[16,135,0.1185,0.71427,0.31543,0.42857,0.85707,1.0,0.0,1.0,1,15,0,1,0,0,0,0,6,0,0,3,0,0,3,0,0,2,0,0,2,0,15],[20,135,0.1481,0.70089,0.32016,0.28571,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,3,0,0,0,0,16],[24,135,0.1778,0.74107,0.32031,0.28571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,2,0,0,1,0,18],[28,135,0.2074,0.73214,0.30878,0.39286,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,8,0,0,2,0,0,3,0,0,0,0,0,3,0,16],[32,135,0.237,0.66964,0.33776,0.28571,0.78571,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,16],[36,135,0.2667,0.73214,0.31084,0.42857,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,17],[40,135,0.2963,0.73214,0.32291,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,4,0,0,4,0,0,2,0,0,2,0,0,1,0,17],[44,135,0.3259,0.67411,0.31791,0.39286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,6,0,0,4,0,0,4,0,0,0,0,0,3,0,13],[48,135,0.3556,0.60268,0.30875,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,11,0,0,2,0,0,5,0,0,2,0,0,1,0,10],[52,135,0.3852,0.66069,0.30671,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,3,0,0,2,0,0,2,0,0,4,0,11],[56,135,0.4148,0.56696,0.31841,0.28571,0.42859,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,13,0,0,4,0,0,3,0,0,0,0,0,1,0,10],[60,135,0.4444,0.62946,0.32313,0.28571,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,10,0,0,4,0,0,3,0,0,0,0,0,2,0,12],[64,135,0.4741,0.68304,0.33643,0.28571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,8,0,0,3,0,0,1,0,0,1,0,0,2,0,15],[68,135,0.5037,0.64731,0.31132,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,7,0,0,3,0,0,5,0,0,1,0,0,3,0,11],[72,135,0.5333,0.52679,0.31428,0.28571,0.35714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,14,0,0,3,0,0,4,0,0,0,0,0,0,0,9],[76,135,0.563,0.65178,0.32525,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,10,0,0,3,0,0,2,0,0,0,0,0,4,0,12],[80,135,0.5926,0.56695,0.31841,0.28571,0.4998,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,12,0,0,2,0,0,5,0,0,1,0,0,0,0,10],[84,135,0.6222,0.61161,0.3219,0.28571,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,11,0,0,3,0,0,4,0,0,1,0,0,0,0,12],[88,135,0.6519,0.625,0.34395,0.28571,0.5,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,11,0,0,3,0,0,1,0,0,0,0,0,2,0,13],[92,135,0.6815,0.49553,0.30926,0.28571,0.28571,0.67857,0.14286,1.0,0,8,0,0,0,2,0,0,17,0,0,1,0,0,4,0,0,0,0,0,0,0,8],[96,135,0.7111,0.49107,0.28558,0.28571,0.28571,0.64286,0.2857,1.0,0,6,0,0,0,0,0,0,18,0,0,4,0,0,2,0,0,0,0,0,2,0,6],[100,135,0.7407,0.56249,0.33681,0.28571,0.42857,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,10,0,0,3,0,0,4,0,0,0,0,0,0,0,11],[104,135,0.7704,0.5,0.30305,0.28571,0.28571,0.67857,0.1429,1.0,0,8,0,0,0,1,0,0,17,0,0,3,0,0,3,0,0,0,0,0,0,0,8],[108,135,0.8,0.54464,0.31428,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,17,0,0,2,0,0,2,0,0,1,0,0,1,0,9],[112,135,0.8296,0.58928,0.35129,0.2857,0.28571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,15,0,0,0,0,0,1,0,0,0,0,0,2,0,12],[116,135,0.8593,0.61161,0.34113,0.28571,0.57143,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,12,0,0,1,0,0,3,0,0,1,0,0,0,0,13],[120,135,0.8889,0.6116,0.343,0.2857,0.5,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,14,0,0,1,0,0,2,0,0,0,0,0,1,0,13],[124,135,0.9185,0.58482,0.3542,0.28571,0.35714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,13,0,0,2,0,0,0,0,0,0,0,0,2,0,12],[128,135,0.9481,0.55357,0.35129,0.28571,0.28571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[132,135,0.9778,0.55802,0.33189,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,16,0,0,2,0,0,1,0,0,0,0,0,2,0,10],[135,135,1.0,0.66518,0.33996,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,1,0,0,3,0,14]]},{"b":2,"e":0.28571,"k":"falling","v":0.34375,"x":0.87055,"p":[[0,110,0.0,0.87055,0.26086,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0,2,0,24],[4,110,0.0364,0.76339,0.31665,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,9,0,0,0,0,0,2,0,0,0,0,0,2,0,19],[8,110,0.0727,0.7232,0.31732,0.39286,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,6,0,0,1,0,0,1,0,0,3,0,0,6,0,13],[12,110,0.1091,0.75444,0.28625,0.53539,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,2,0,0,6,0,14],[16,110,0.1455,0.6875,0.32427,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,11,0,0,2,0,0,0,0,0,2,0,0,3,0,14],[20,110,0.1818,0.78125,0.31941,0.28571,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,21],[24,110,0.2182,0.67857,0.33882,0.28571,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,1,0,0,2,0,0,0,0,16],[28,110,0.2545,0.66071,0.33072,0.28571,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,2,0,0,1,0,14],[32,110,0.2909,0.65178,0.3387,0.28571,0.78571,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,10,0,0,2,0,0,1,0,0,1,0,0,3,0,13],[36,110,0.3273,0.55803,0.31412,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,15,0,0,4,0,0,2,0,0,1,0,0,0,0,10],[40,110,0.3636,0.59821,0.31832,0.28571,0.42859,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,14,0,0,3,0,0,0,0,0,3,0,0,2,0,10],[44,110,0.4,0.41518,0.24317,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,23,0,0,3,0,0,0,0,0,1,0,0,2,0,3],[48,110,0.4364,0.34375,0.15916,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[52,110,0.4727,0.42411,0.23279,0.28571,0.28571,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,23,0,0,0,0,0,1,0,0,5,0,0,1,0,2],[56,110,0.5091,0.44196,0.26088,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,20,0,0,0,0,0,0,0,0,5,0,0,3,0,2],[60,110,0.5455,0.49106,0.28333,0.28571,0.28571,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,20,0,0,0,0,0,2,0,0,3,0,0,2,0,5],[64,110,0.5818,0.44643,0.23891,0.28571,0.28571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,21,0,0,1,0,0,1,0,0,4,0,0,4,0,1],[68,110,0.6182,0.4375,0.27185,0.28571,0.28571,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,23,0,0,2,0,0,0,0,0,1,0,0,1,0,5],[72,110,0.6545,0.53572,0.28793,0.28571,0.42857,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,14,0,0,6,0,0,1,0,0,3,0,0,1,0,7],[76,110,0.6909,0.49107,0.26471,0.28571,0.28571,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,18,0,0,2,0,0,2,0,0,3,0,0,4,0,3],[80,110,0.7273,0.40178,0.2299,0.28571,0.28571,0.32143,0.2857,1.0,0,3,0,0,0,0,0,0,24,0,0,2,0,0,0,0,0,3,0,0,0,0,3],[84,110,0.7636,0.46428,0.27199,0.28571,0.28571,0.60714,0.2857,1.0,0,5,0,0,0,0,0,0,20,0,0,3,0,0,1,0,0,2,0,0,1,0,5],[88,110,0.8,0.55343,0.28508,0.28571,0.42857,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,7,0,0,3,0,5],[92,110,0.8364,0.41516,0.24835,0.2857,0.28571,0.35704,0.14286,1.0,0,2,0,0,0,1,0,0,23,0,0,0,0,0,2,0,0,0,0,0,4,0,2],[96,110,0.8727,0.41963,0.23941,0.28571,0.28571,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,22,0,0,3,0,0,2,0,0,0,0,0,2,0,3],[100,110,0.9091,0.4509,0.23987,0.28571,0.28571,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,19,0,0,3,0,0,3,0,0,3,0,0,1,0,3],[104,110,0.9455,0.41964,0.21409,0.28571,0.28571,0.46429,0.14286,0.85714,0,0,0,0,0,1,0,0,19,0,0,4,0,0,1,0,0,3,0,0,4,0,0],[108,110,0.9818,0.38839,0.19638,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,5,0,0,2,0,0],[110,110,1.0,0.41964,0.22286,0.28571,0.28571,0.5,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,2,0,0,0,0,0,6,0,0,0,0,2]]}]},{"i":"7deaea633612f387","q":"Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are  $n\\geq 1$  marbles on the table, then the player whose turn it is removes  $k$  marbles, where  $k\\geq 1$  either is an even number with  $k\\leq \\frac{n}{2}$  or an odd number with  $\\frac{n}{2}\\leq k\\leq n$ . A player win the game if she removes the last marble from the table.\nDetermine the smallest number  $N\\geq 100000$  such that Berta can enforce a victory if there are exactly  $N$  marbles on the tale in the beginning.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.70534,"x":0.93304,"p":[[0,108,0.0,0.91964,0.13803,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,22],[4,108,0.037,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[8,108,0.0741,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[12,108,0.1111,0.93304,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[16,108,0.1481,0.89286,0.13832,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,8,0,17],[20,108,0.1852,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[24,108,0.2222,0.91518,0.11769,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,20],[28,108,0.2593,0.90612,0.13195,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[32,108,0.2963,0.90625,0.12169,0.82143,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,19],[36,108,0.3333,0.92409,0.11289,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[40,108,0.3704,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[44,108,0.4074,0.87054,0.12556,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,14],[48,108,0.4444,0.88839,0.11143,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,14],[52,108,0.4815,0.83929,0.12242,0.71429,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,10],[56,108,0.5185,0.84821,0.12846,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,12],[60,108,0.5556,0.88839,0.12234,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,16],[64,108,0.5926,0.89286,0.11294,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,15],[68,108,0.6296,0.87054,0.1488,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,15],[72,108,0.6667,0.86607,0.11811,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,10,0,12],[76,108,0.7037,0.91071,0.12753,0.85714,1.0,1.0,0.5714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[80,108,0.7407,0.84375,0.13533,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,12,0,10],[84,108,0.7778,0.83482,0.13415,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,6,0,11],[88,108,0.8148,0.78125,0.14718,0.71429,0.71429,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,18,0,0,3,0,8],[92,108,0.8519,0.73661,0.10779,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,0,3,0,3],[96,108,0.8889,0.77229,0.10635,0.71429,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,12,0,2],[100,108,0.9259,0.70534,0.10064,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,22,0,0,3,0,1],[104,108,0.963,0.70535,0.04971,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0],[108,108,1.0,0.71872,0.09099,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,0,4,0,1]]},{"b":5,"e":0.71429,"k":"falling","v":0.68749,"x":0.90179,"p":[[0,78,0.0,0.90179,0.1448,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,21],[4,78,0.0513,0.85268,0.13115,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,8,0,12],[8,78,0.1026,0.86161,0.12103,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,9,0,12],[12,78,0.1538,0.87946,0.12428,0.71429,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,7,0,15],[16,78,0.2051,0.8125,0.1357,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,6,0,9],[20,78,0.2564,0.84379,0.14432,0.71429,0.85714,1.0,0.43,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,7,0,12],[24,78,0.3077,0.81696,0.12492,0.71429,0.78571,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,8,0,8],[28,78,0.359,0.79911,0.14223,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,6,0,8],[32,78,0.4103,0.83482,0.11904,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,9,0,9],[36,78,0.4615,0.82143,0.12877,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,4,0,10],[40,78,0.5128,0.79018,0.12364,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,7,0,6],[44,78,0.5641,0.7366,0.10778,0.71429,0.71429,0.857,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,0,8,0,1],[48,78,0.6154,0.74997,0.11849,0.71429,0.71429,0.71429,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,23,0,0,1,0,5],[52,78,0.6667,0.75893,0.11538,0.71429,0.71429,0.85714,0.5714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,5,0,4],[56,78,0.7179,0.72768,0.07457,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,3,0,1],[60,78,0.7692,0.74107,0.09062,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,4,0,2],[64,78,0.8205,0.73214,0.11151,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,23,0,0,6,0,1],[68,78,0.8718,0.71429,0.11294,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,24,0,0,0,0,3],[72,78,0.9231,0.68749,0.1154,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,19,0,0,3,0,1],[76,78,0.9744,0.71428,0.07142,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,0,4,0,0],[78,78,1.0,0.69642,0.06918,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,1,0,0]]}]},{"i":"e10c516b731c824d","q":"Find all integers solutions for \r $ xy\\plus{}yz\\plus{}zx\\minus{}xyz\\equal{}2$","t":[{"b":5,"e":0.85714,"k":"flat","v":0.8436,"x":0.96429,"p":[[0,147,0.0,0.8436,0.17639,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,10,0,13],[4,147,0.0272,0.87052,0.15717,0.85711,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,10,0,15],[8,147,0.0544,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[12,147,0.0816,0.95545,0.07512,0.85929,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[16,147,0.1088,0.91517,0.12807,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,20],[20,147,0.1361,0.91071,0.12752,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,18],[24,147,0.1633,0.87933,0.12948,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,13,0,13],[28,147,0.1905,0.88837,0.16653,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,8,0,18],[32,147,0.2177,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[36,147,0.2449,0.9375,0.11811,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,22],[40,147,0.2721,0.9375,0.11259,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,21],[44,147,0.2993,0.95088,0.09857,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[48,147,0.3265,0.9375,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[52,147,0.3537,0.91962,0.09408,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[56,147,0.381,0.89732,0.13474,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,18],[60,147,0.4082,0.94642,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[64,147,0.4354,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[68,147,0.4626,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[72,147,0.4898,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[76,147,0.517,0.90179,0.13092,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,10,0,17],[80,147,0.5442,0.86157,0.20359,0.857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,15],[84,147,0.5714,0.91517,0.13767,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,18],[88,147,0.5986,0.92857,0.11294,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,19],[92,147,0.6259,0.88393,0.1729,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,8,0,18],[96,147,0.6531,0.92411,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20],[100,147,0.6803,0.95535,0.07525,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[104,147,0.7075,0.9375,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[108,147,0.7347,0.91517,0.12811,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,0,18],[112,147,0.7619,0.91526,0.12295,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,18],[116,147,0.7891,0.89732,0.14827,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,13,0,16],[120,147,0.8163,0.89284,0.14726,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,14,0,15],[124,147,0.8435,0.89731,0.16457,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,10,0,18],[128,147,0.8707,0.91518,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[132,147,0.898,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[136,147,0.9252,0.91517,0.12808,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,10,0,19],[140,147,0.9524,0.92857,0.11845,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,20],[144,147,0.9796,0.88839,0.18118,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,17],[147,147,1.0,0.85267,0.18381,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,9,0,15]]},{"b":6,"e":0.71429,"k":"rising","v":0.79013,"x":0.96875,"p":[[0,85,0.0,0.79013,0.19884,0.67857,0.85707,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,9,0,10],[4,85,0.0471,0.89286,0.15152,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,10,0,17],[8,85,0.0941,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[12,85,0.1412,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[16,85,0.1882,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,17],[20,85,0.2353,0.90623,0.1217,0.85711,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,13,0,16],[24,85,0.2824,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[28,85,0.3294,0.92854,0.1072,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[32,85,0.3765,0.95534,0.09746,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[36,85,0.4235,0.91964,0.13333,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,21],[40,85,0.4706,0.94643,0.09943,0.85714,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[44,85,0.5176,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[48,85,0.5647,0.89732,0.13474,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,17],[52,85,0.6118,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[56,85,0.6588,0.9107,0.12246,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,18],[60,85,0.7059,0.91516,0.1377,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,20],[64,85,0.7529,0.91071,0.19805,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,22],[68,85,0.8,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[72,85,0.8471,0.92411,0.13825,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,10,0,20],[76,85,0.8941,0.90178,0.1448,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,10,0,18],[80,85,0.9412,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[84,85,0.9882,0.90179,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,14],[85,85,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"6802a591c83ad47a","q":"For which integers  $N\\ge 3$  can we find  $N$  points on the plane such that no three are collinear, and for any triangle formed by three vertices of the points\u2019 convex hull, there is exactly one point within that triangle?","t":[{"b":0,"e":0.14286,"k":"flat","v":0.08929,"x":0.4464,"p":[[0,51,0.0,0.10705,0.17126,0.0,0.0,0.14286,0.0,0.57143,20,0,14,20,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,51,0.0784,0.43291,0.21875,0.2857,0.42859,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,4,0,0,9,0,0,9,0,0,3,0,0,0,0,1],[8,51,0.1569,0.39283,0.19559,0.28571,0.42857,0.4642,0.0,1.0,2,1,0,2,0,2,0,0,10,0,0,10,0,0,6,0,0,1,0,0,0,0,1],[12,51,0.2353,0.33026,0.18371,0.14286,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,10,0,0,7,0,0,8,0,0,4,0,0,2,0,0,0,0,0],[16,51,0.3137,0.4464,0.16267,0.28571,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,7,0,0,9,0,0,13,0,0,0,0,0,1,0,0],[20,51,0.3922,0.33481,0.17715,0.14289,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,7,0,0,8,0,0,9,0,0,5,0,0,1,0,0,0,0,0],[24,51,0.4706,0.433,0.20037,0.2857,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,3,0,0,9,0,0,10,0,0,4,0,0,0,0,0],[28,51,0.549,0.32587,0.2179,0.14286,0.2857,0.5711,0.0,0.71429,3,0,0,3,0,11,0,0,4,0,0,4,0,0,8,0,0,2,0,0,0,0,0],[32,51,0.6275,0.30801,0.21458,0.14286,0.28571,0.571,0.0,0.71429,5,0,0,5,0,8,0,0,6,0,0,4,0,0,8,0,0,1,0,0,0,0,0],[36,51,0.7059,0.3213,0.19236,0.1429,0.28571,0.42858,0.0,0.71429,3,0,0,3,0,6,0,0,12,0,0,4,0,0,5,0,0,2,0,0,0,0,0],[40,51,0.7843,0.14277,0.17857,0.0,0.07,0.2857,0.0,0.57143,16,0,0,16,0,7,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[44,51,0.8627,0.18303,0.14386,0.14286,0.14286,0.2857,0.0,0.571,6,0,0,6,0,17,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[48,51,0.9412,0.15598,0.18683,0.0,0.14286,0.14287,0.0,0.71429,11,0,0,11,0,15,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[51,51,1.0,0.08929,0.09943,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.12945,"x":0.48654,"p":[[0,103,0.0,0.12945,0.21234,0.0,0.0,0.1786,0.0,0.57143,22,0,13,22,0,2,0,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[4,103,0.0388,0.47318,0.16916,0.28571,0.49979,0.57143,0.14286,0.8571,0,0,0,0,0,2,0,0,7,0,0,7,0,0,12,0,0,3,0,0,1,0,0],[8,103,0.0777,0.48654,0.19513,0.28571,0.571,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,5,0,0,14,0,0,2,0,0,1,0,1],[12,103,0.1165,0.39729,0.21347,0.2857,0.42859,0.57143,0.0,0.71429,3,0,0,3,0,4,0,0,7,0,0,4,0,0,11,0,0,3,0,0,0,0,0],[16,103,0.1553,0.45515,0.22452,0.28571,0.571,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,4,0,0,6,0,0,13,0,0,3,0,0,0,0,1],[20,103,0.1942,0.45076,0.18261,0.39286,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,12,0,0,9,0,0,1,0,0,2,0,0],[24,103,0.233,0.41067,0.16265,0.2857,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,3,0,0,8,0,0,7,0,0,13,0,0,0,0,0,0,0,0],[28,103,0.2718,0.35712,0.202,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,8,0,0,7,0,0,3,0,0,11,0,0,1,0,0,0,0,0],[32,103,0.3107,0.33925,0.2012,0.14286,0.35714,0.4642,0.0,0.71429,2,0,0,2,0,10,0,0,4,0,0,8,0,0,6,0,0,2,0,0,0,0,0],[36,103,0.3495,0.29456,0.18544,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,11,0,0,5,0,0,7,0,0,6,0,0,0,0,0,0,0,0],[40,103,0.3883,0.33035,0.2065,0.14286,0.28571,0.42858,0.0,0.71429,4,0,0,4,0,6,0,0,7,0,0,9,0,0,3,0,0,3,0,0,0,0,0],[44,103,0.4272,0.29018,0.22441,0.14286,0.2857,0.42857,0.0,1.0,5,1,0,5,0,9,0,0,7,0,0,5,0,0,5,0,0,0,0,0,0,0,1],[48,103,0.466,0.33481,0.23583,0.14286,0.28571,0.57143,0.0,0.71429,6,0,0,6,0,5,0,0,7,0,0,3,0,0,8,0,0,3,0,0,0,0,0],[52,103,0.5049,0.34362,0.22552,0.14286,0.28571,0.571,0.0,1.0,2,1,0,2,0,11,0,0,4,0,0,5,0,0,9,0,0,0,0,0,0,0,1],[56,103,0.5437,0.34361,0.20785,0.14286,0.35714,0.571,0.0,0.71429,3,0,0,3,0,8,0,0,5,0,0,7,0,0,7,0,0,2,0,0,0,0,0],[60,103,0.5825,0.26322,0.22908,0.14286,0.14286,0.42858,0.0,1.0,5,1,0,5,0,14,0,0,3,0,0,4,0,0,5,0,0,0,0,0,0,0,1],[64,103,0.6214,0.27228,0.19994,0.14286,0.2143,0.42857,0.0,0.57143,5,0,0,5,0,11,0,0,5,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[68,103,0.6602,0.28123,0.21271,0.14286,0.2857,0.42858,0.0,0.71429,5,0,0,5,0,10,0,0,6,0,0,6,0,0,2,0,0,3,0,0,0,0,0],[72,103,0.699,0.24105,0.17285,0.14286,0.14295,0.28571,0.0,0.71429,4,0,0,4,0,13,0,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[76,103,0.7379,0.27231,0.19349,0.14286,0.21435,0.42857,0.0,0.57143,4,0,0,4,0,12,0,0,6,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[80,103,0.7767,0.20972,0.17852,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,18,0,0,3,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[84,103,0.8155,0.33927,0.21649,0.14286,0.28571,0.57111,0.0,0.71429,4,0,0,4,0,6,0,0,8,0,0,5,0,0,6,0,0,3,0,0,0,0,0],[88,103,0.8544,0.2142,0.175,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,19,0,0,2,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[92,103,0.8932,0.19626,0.11718,0.14286,0.14286,0.2857,0.0,0.4286,2,0,0,2,0,21,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[96,103,0.932,0.26322,0.17908,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,11,0,0,8,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[100,103,0.9709,0.25881,0.24075,0.0,0.2857,0.46418,0.0,0.71429,12,0,0,12,0,3,0,0,5,0,0,4,0,0,7,0,0,1,0,0,0,0,0],[103,103,1.0,0.28124,0.25374,0.0,0.2857,0.57143,0.0,0.57143,13,0,0,13,0,1,0,0,3,0,0,4,0,0,11,0,0,0,0,0,0,0,0]]}]},{"i":"5c2e27d850603f1d","q":"Given is a triangle  $ABC$  with incircle  $\\omega$ , tangent to  $BC, CA, AB$  at  $D, E, F$ . The perpendicular from  $B$  to  $BC$  meets  $EF$  at  $M$ , and the perpendicular from  $C$  to  $BC$  meets  $EF$  at  $N$ . Let  $DM$  and  $DN$  meet  $\\omega$  at  $P$  and  $Q$ . Prove that  $DP=DQ$ 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trapezoid  $ABCD$ , the sum of the lengths of the bases  $AB$  and  $CD$  is equal to the length of the diagonal  $BD$ . Let  $M$  denote the midpoint of  $BC$ , and let  $E$  denote the reflection of  $C$  about the line  $DM$ . Prove that  $\\angle AEB=\\angle ACD$ 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n}{n},\\binom{3 n}{n}, \\ldots,\\binom{n^{2}}{n}\\right\\}, \\quad \\text { for } n \\in \\mathbb{N}\n$$\n\na) Prove that there are infinitely many composite natural numbers $n$ such that $S_{n}$ is not a complete residue system modulo $n$.\n\nb) Prove that there are infinitely many composite natural numbers $n$ such that $S_{n}$ is a complete residue system modulo $n$.\n\n(Milos Milosevic)","t":[{"b":2,"e":0.571,"k":"flat","v":0.54912,"x":0.84817,"p":[[0,51,0.0,0.54912,0.24249,0.42857,0.57121,0.60714,0.0,1.0,1,4,1,1,0,0,0,0,6,0,0,7,0,0,10,0,0,2,0,0,2,0,4],[4,51,0.0784,0.84817,0.19547,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,2,0,18],[8,51,0.1569,0.80355,0.2044,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,5,0,13],[12,51,0.2353,0.78106,0.22874,0.67536,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,6,0,0,6,0,12],[16,51,0.3137,0.78107,0.21728,0.67536,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,9,0,0,2,0,13],[20,51,0.3922,0.78108,0.18901,0.713,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,13,0,0,0,0,12],[24,51,0.4706,0.79906,0.18514,0.71429,0.85707,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,8,0,0,6,0,11],[28,51,0.549,0.7946,0.19546,0.67857,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,9,0,0,2,0,13],[32,51,0.6275,0.69194,0.25282,0.57132,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,2,0,0,7,0,0,7,0,0,1,0,10],[36,51,0.7059,0.71425,0.23692,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,5,0,8],[40,51,0.7843,0.68301,0.26423,0.4286,0.71429,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,3,0,0,5,0,0,6,0,0,5,0,0,2,0,10],[44,51,0.8627,0.63389,0.25738,0.39286,0.71429,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,8,0,0,3,0,0,4,0,0,6,0,0,6,0,5],[48,51,0.9412,0.62494,0.28065,0.42857,0.57141,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,5,0,0,7,0,0,3,0,0,4,0,7],[51,51,1.0,0.61609,0.22987,0.4286,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,6,0,0,7,0,0,7,0,0,2,0,5]]},{"b":6,"e":1.0,"k":"rising","v":0.51312,"x":0.80348,"p":[[0,46,0.0,0.51312,0.21088,0.39286,0.571,0.71107,0.0,1.0,1,1,1,1,0,1,0,0,6,0,0,6,0,0,9,0,0,7,0,0,1,0,1],[4,46,0.087,0.74099,0.23544,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,0,0,0,8,0,0,8,0,0,2,0,11],[8,46,0.1739,0.75891,0.22429,0.57143,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,5,0,0,5,0,11],[12,46,0.2609,0.72312,0.22291,0.571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,5,0,0,4,0,9],[16,46,0.3478,0.68299,0.23349,0.571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,6,0,0,6,0,6],[20,46,0.4348,0.70976,0.18383,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,6,0,0,5,0,6],[24,46,0.5217,0.67406,0.23755,0.4286,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,6,0,0,6,0,6],[28,46,0.6087,0.80348,0.20133,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,6,0,13],[32,46,0.6957,0.78123,0.19881,0.71429,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,9,0,0,8,0,9],[36,46,0.7826,0.72302,0.20501,0.571,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,10,0,0,3,0,8],[40,46,0.8696,0.77227,0.23925,0.571,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,0,6,0,12],[44,46,0.9565,0.77676,0.22853,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,0,7,0,11],[46,46,1.0,0.70534,0.26472,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,7,0,0,2,0,11]]}]},{"i":"4a4c90f1cfa07048","q":"Given that  $g(n) = \\frac{1}{{2 + \\frac{1}{{3 + \\frac{1}{{... + \\frac{1}{{n - 1}}}}}}}}$  and  $k(n) = \\frac{1}{{2 + \\frac{1}{{3 + \\frac{1}{{... + \\frac{1}{{n - 1 + \\frac{1}{n}}}}}}}}}$ , for natural  $n$ . Prove that  $\\left| {g(n) - k(n)} \\right| \\le \\frac{1}{{(n - 1)!n!}}$ .","t":[{"b":1,"e":0.71429,"k":"falling","v":0.41071,"x":0.95982,"p":[[0,54,0.0,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,54,0.0741,0.41071,0.40837,0.0,0.42857,0.74996,0.0,1.0,15,4,0,15,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,4,0,4],[8,54,0.1481,0.64286,0.41955,0.10714,0.85714,1.0,0.0,1.0,8,14,0,8,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,14],[12,54,0.2222,0.58036,0.42399,0.0,0.71429,1.0,0.0,1.0,9,12,0,9,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,12],[16,54,0.2963,0.67856,0.383,0.42859,0.85714,1.0,0.0,1.0,6,14,0,6,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,4,0,14],[20,54,0.3704,0.81246,0.18369,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,7,0,12],[24,54,0.4444,0.83036,0.24598,0.82143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,15],[28,54,0.5185,0.69641,0.20125,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,7,0,0,10,0,3],[32,54,0.5926,0.74775,0.17221,0.67857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,11,0,0,7,1,5],[36,54,0.6667,0.64286,0.29881,0.53571,0.71429,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,0,10,0,4],[40,54,0.7407,0.83929,0.23891,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,17],[44,54,0.8148,0.69196,0.28596,0.57142,0.78571,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,11,0,5],[48,54,0.8889,0.79909,0.18163,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,10,0,9],[52,54,0.963,0.75891,0.18364,0.71429,0.78564,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,11,0,5],[54,54,1.0,0.76784,0.19151,0.67857,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,8,0,0,8,0,8]]},{"b":5,"e":1.0,"k":"rising","v":0.53572,"x":0.99107,"p":[[0,25,0.0,0.83482,0.34646,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[4,25,0.16,0.53572,0.46153,0.0,0.78571,1.0,0.0,1.0,12,13,0,12,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,13],[8,25,0.32,0.7857,0.33121,0.71429,1.0,1.0,0.0,1.0,4,17,0,4,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,17],[12,25,0.48,0.71875,0.39687,0.39286,1.0,1.0,0.0,1.0,6,18,0,6,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,18],[16,25,0.64,0.89286,0.26964,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,26],[20,25,0.8,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[24,25,0.96,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[25,25,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"2f8f585ffe0b05d8","q":"Given that there are  $24$  primes between  $3$  and  $100$ , inclusive, what is the number of ordered pairs  $(p, a)$  with  $p$  prime,  $3 \\le p < 100$ , and  $1 \\le a < p$  such that the sum\n\\[a+a^2+a^3+\\cdots+a^{(p-2)!} \\]is not divisible by  $p$ ?","t":[{"b":6,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,46,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,46,0.087,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,46,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,46,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,46,0.5217,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,46,0.6087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,46,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,46,0.7826,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,46,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,52,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,52,0.0769,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,52,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,52,0.2308,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,52,0.3077,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,52,0.3846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,52,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,52,0.5385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,52,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,52,0.6923,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,52,0.7692,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ee5cb1e77a6afce9","q":"In a triangle  $ABC$  a point  $D$  on segment  $BC$  is such that  $BD = 2DC$ . Let  $E$  be the foot of the altitude of  $ABC$  through  $ B$  and let  $\\ell$  be the line through  $C$  perpendicular on  $AC$ . Let  $O$  be the point on  $\\ell$  such that  $OB = OE$ . Prove that  $D$ ,  $E$ , and  $O$  are collinear.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.96875,"x":1.0,"p":[[0,47,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,47,0.0851,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,47,0.1702,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,47,0.2553,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,47,0.3404,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,47,0.4255,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,47,0.5106,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,47,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,47,0.6809,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,47,0.766,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[47,47,1.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,20,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,20,0.2,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,20,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,20,0.6,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"e0fbfa525203599e","q":"Find all functions  $f:\\mathbb{N} \\to \\mathbb{N}$  such that\\[f\\left(\\Big \\lceil \\frac{f(m)}{n} \\Big \\rceil\\right)=\\Big \\lceil \\frac{m}{f(n)} \\Big \\rceil\\]for all  $m,n \\in \\mathbb{N}$ .\n\n*Proposed by Md. Ashraful Islam Fahim*","t":[{"b":4,"e":0.71429,"k":"flat","v":0.55357,"x":0.66071,"p":[[0,75,0.0,0.64727,0.09441,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,14,0,0,0,0,1],[4,75,0.0533,0.66071,0.08564,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,16,0,0,2,0,0],[8,75,0.1067,0.61158,0.09607,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,10,0,0,1,0,0],[12,75,0.16,0.65177,0.10063,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,19,0,0,1,0,0],[16,75,0.2133,0.625,0.07784,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,13,0,0,0,0,0],[20,75,0.2667,0.59821,0.0974,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,11,0,0,0,0,0],[24,75,0.32,0.5625,0.10062,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,16,0,0,7,0,0,0,0,0],[28,75,0.3733,0.60265,0.08553,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,10,0,0,0,0,0],[32,75,0.4267,0.56691,0.06666,0.57132,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,3,0,0,0,0,0],[36,75,0.48,0.56695,0.09094,0.57132,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,0,6,0,0,0,0,0],[40,75,0.5333,0.62051,0.06787,0.57143,0.57143,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0],[44,75,0.5867,0.58479,0.08268,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,0,7,0,0,0,0,0],[48,75,0.64,0.59374,0.08828,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,9,0,0,0,0,0],[52,75,0.6933,0.62054,0.07668,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,12,0,0,0,0,0],[56,75,0.7467,0.57143,0.10102,0.53571,0.57143,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,16,0,0,8,0,0,0,0,0],[60,75,0.8,0.61603,0.06624,0.57143,0.57143,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0],[64,75,0.8533,0.58034,0.09407,0.57143,0.57143,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,0,8,0,0,0,0,0],[68,75,0.9067,0.60268,0.09932,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,12,0,0,0,0,0],[72,75,0.96,0.55357,0.09279,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,18,0,0,5,0,0,0,0,0],[75,75,1.0,0.58482,0.09689,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,0,9,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.56696,"x":0.64732,"p":[[0,16,0.0,0.62497,0.08566,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,14,0,0,0,0,0],[4,16,0.25,0.64732,0.07973,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,18,0,0,0,0,0],[8,16,0.5,0.60713,0.14726,0.57143,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,1,0,0,19,0,0,10,0,0,0,0,1],[12,16,0.75,0.57141,0.14286,0.57142,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,4,0,0,16,0,0,10,0,0,0,0,0],[16,16,1.0,0.56696,0.10999,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,0,0,0,0]]}]},{"i":"75139f9d726ae327","q":"Given the positive integer  $m \\geq 2$ ,  $n \\geq 3$ . Define the following set  $$ S = \\left\\{(a, b) | a \\in \\{1, 2, \\cdots, m\\}, b \\in \\{1, 2, \\cdots, n\\} \\right\\}. $$  Let  $A$  be a subset of  $S$ . If there does not exist positive integers  $x_1, x_2, y_1, y_2, y_3$  such that  $x_1 < x_2, y_1 < y_2 < y_3$  and  $$ (x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \\in A. $$  Determine the largest possible number of elements in  $A$ .","t":[{"b":5,"e":0.85714,"k":"rising","v":0.84375,"x":0.99554,"p":[[0,34,0.0,0.84375,0.27049,0.82143,1.0,1.0,0.0,1.0,2,20,2,2,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,20],[4,34,0.1176,0.88839,0.20743,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,22],[8,34,0.2353,0.93749,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[12,34,0.3529,0.97768,0.08073,1.0,1.0,1.0,0.5714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,34,0.4706,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,34,0.5882,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,34,0.7059,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[28,34,0.8235,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,34,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[34,34,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":0.42857,"k":"falling","v":0.6116,"x":0.8482,"p":[[0,30,0.0,0.8482,0.23943,0.71429,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,2,0,20],[4,30,0.1333,0.70089,0.20316,0.57143,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,15,0,0,0,0,7],[8,30,0.2667,0.6116,0.17215,0.42859,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,9,0,0,8,0,0,10,0,0,2,0,2],[12,30,0.4,0.62496,0.20439,0.571,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,5,0,0,10,0,0,11,0,0,0,0,4],[16,30,0.5333,0.67411,0.15251,0.67857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,21,0,0,1,0,2],[20,30,0.6667,0.69195,0.18596,0.57143,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,15,0,0,0,0,6],[24,30,0.8,0.64732,0.16745,0.53572,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,6,0,0,14,0,0,1,0,3],[28,30,0.9333,0.6116,0.1394,0.4286,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,15,0,0,0,0,1],[30,30,1.0,0.65176,0.14259,0.57132,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,18,0,0,0,0,2]]}]},{"i":"584aaf062c05b927","q":"Given a set of 9 points in the plane, no three collinear, show that for each point $\\mathrm{P}$ in the set, the number of triangles containing $\\mathrm{P}$ formed from the other 8 points in the set must be even.","t":[{"b":3,"e":0.71429,"k":"flat","v":0.51338,"x":0.80353,"p":[[0,29,0.0,0.54908,0.31157,0.42857,0.57143,0.75,0.0,1.0,6,2,6,6,0,0,0,0,1,0,0,4,0,0,6,0,0,7,0,0,6,0,2],[4,29,0.1379,0.76335,0.24646,0.57143,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,8,0,0,4,0,0,5,0,12],[8,29,0.2759,0.7366,0.30116,0.53571,0.85707,1.0,0.0,1.0,1,14,0,1,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,0,4,0,14],[12,29,0.4138,0.74552,0.29176,0.53539,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,4,0,0,4,0,0,2,0,0,3,0,15],[16,29,0.5517,0.77678,0.29437,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,2,0,0,5,0,16],[20,29,0.6897,0.80353,0.19154,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,9,0,11],[24,29,0.8276,0.6384,0.36243,0.42859,0.78571,1.0,0.0,1.0,5,11,0,5,0,0,0,0,2,0,0,5,0,0,3,0,0,1,0,0,5,0,11],[28,29,0.9655,0.51338,0.26692,0.42857,0.4998,0.71429,0.0,0.85714,4,0,0,4,0,1,0,0,1,0,0,10,0,0,4,0,0,6,0,0,6,0,0],[29,29,1.0,0.52677,0.24598,0.39286,0.4998,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,5,0,0,8,0,0,2,0,0,9,0,0,5,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.5089,"x":0.97321,"p":[[0,24,0.0,0.5089,0.3443,0.24999,0.57121,0.85714,0.0,1.0,6,4,6,6,0,2,0,0,4,0,0,2,0,0,5,0,0,4,0,0,5,0,4],[4,24,0.1667,0.70981,0.3164,0.57132,0.71429,1.0,0.0,1.0,1,14,0,1,0,3,0,0,2,0,0,1,0,0,5,0,0,5,0,0,1,0,14],[8,24,0.3333,0.81247,0.24077,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,3,0,0,0,0,19],[12,24,0.5,0.72756,0.29976,0.57132,0.71429,1.0,0.0,1.0,1,14,0,1,0,2,0,0,2,0,0,1,0,0,5,0,0,6,0,0,1,0,14],[16,24,0.6667,0.76783,0.24159,0.57143,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,7,0,0,3,0,13],[20,24,0.8333,0.79461,0.28109,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,0,3,0,17],[24,24,1.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30]]}]},{"i":"c2ad5e61b7b91740","q":"Given a real sequence  $\\left \\{ x_n \\right \\}_{n=1}^{\\infty}$  with  $x_1^2 = 1$ . Prove that for each integer  $n \\ge 2$ ,   $$ \\sum_{i|n}\\sum_{j|n}\\frac{x_ix_j}{\\textup{lcm} \\left ( i,j \\right )} \\ge \\prod_{\\mbox{\\tiny $\\begin{array}{c}\np \\: \\textup{is prime}  p|n \\end{array}$ } }\\left ( 1-\\frac{1}{p} \\right ).  $$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.64286,"x":0.84821,"p":[[0,45,0.0,0.69196,0.17536,0.67857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,18,0,0,3,0,3],[4,45,0.0889,0.80356,0.17034,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,12,0,8],[8,45,0.1778,0.69642,0.2896,0.57143,0.85714,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,2,0,0,1,0,0,3,0,0,6,0,0,12,0,5],[12,45,0.2667,0.64286,0.30305,0.53571,0.71429,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,1,0,0,3,0,0,7,0,0,5,0,0,5,0,7],[16,45,0.3556,0.80357,0.23891,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,6,0,0,6,0,14],[20,45,0.4444,0.79017,0.26483,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,4,0,0,5,0,15],[24,45,0.5333,0.84821,0.21706,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,8,0,15],[28,45,0.6222,0.80357,0.23351,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,6,0,0,4,0,15],[32,45,0.7111,0.78125,0.18893,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,0,3,0,11],[36,45,0.8,0.82143,0.21724,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,0,7,0,13],[40,45,0.8889,0.76772,0.24682,0.71429,0.85707,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,10,0,0,7,0,10],[44,45,0.9778,0.72768,0.19019,0.71429,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,17,0,0,3,0,6],[45,45,1.0,0.74985,0.2052,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,15,0,0,7,0,6]]},{"b":3,"e":0.71429,"k":"rising","v":0.72321,"x":0.93304,"p":[[0,56,0.0,0.72321,0.12846,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,21,0,0,1,0,4],[4,56,0.0714,0.77232,0.20158,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,8,0,0,4,0,11],[8,56,0.1429,0.79017,0.22864,0.67836,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,3,0,0,10,0,11],[12,56,0.2143,0.88392,0.2096,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,22],[16,56,0.2857,0.81696,0.27487,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,2,0,0,2,0,20],[20,56,0.3571,0.76786,0.26184,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,1,0,0,7,0,0,3,0,0,4,0,14],[24,56,0.4286,0.77232,0.23652,0.71429,0.85707,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,8,0,0,6,0,11],[28,56,0.5,0.77232,0.24186,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,4,0,0,6,0,12],[32,56,0.5714,0.75446,0.28624,0.53571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,3,0,0,7,0,13],[36,56,0.6429,0.81237,0.2408,0.71321,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,4,0,0,6,0,15],[40,56,0.7143,0.75893,0.25862,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,8,0,0,8,0,10],[44,56,0.7857,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[48,56,0.8571,0.89732,0.1394,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,10,0,17],[52,56,0.9286,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,13],[56,56,1.0,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23]]}]},{"i":"d405dc3515b8d2bc","q":"Let  $ ABC$  be a triangle with  $ AB \\equal{} AC$ . A ray  $ Ax$  is constructed in space such that the three planar angles of the trihedral angle  $ ABCx$  at its vertex  $ A$  are equal. If a point  $ S$  moves on  $ Ax$ , find the locus of the incenter of triangle  $ SBC$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.08928,"x":0.85713,"p":[[0,57,0.0,0.66738,0.29967,0.57132,0.67857,1.0,0.0,1.0,2,9,1,2,0,2,0,0,1,0,0,1,0,0,9,1,0,3,0,0,4,0,9],[4,57,0.0702,0.85713,0.20825,0.85711,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0,12,0,15],[8,57,0.1404,0.84818,0.18193,0.85711,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,13,0,13],[12,57,0.2105,0.71875,0.31438,0.57143,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,7,0,11],[16,57,0.2807,0.64282,0.28121,0.42857,0.64271,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,4,0,0,6,0,0,4,0,0,5,0,7],[20,57,0.3509,0.57142,0.34625,0.25002,0.57121,0.89286,0.0,1.0,1,8,0,1,0,7,0,0,3,0,0,4,0,0,3,0,0,1,0,0,5,0,8],[24,57,0.4211,0.4016,0.24355,0.14286,0.42857,0.4286,0.0,0.85714,2,0,0,2,0,7,0,0,3,0,0,13,0,0,1,0,0,2,0,0,4,0,0],[28,57,0.4912,0.2991,0.24316,0.14286,0.2857,0.42857,0.0,1.0,6,1,0,6,0,7,0,0,7,0,0,8,0,0,1,0,0,1,0,0,1,0,1],[32,57,0.5614,0.34803,0.24742,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,4,0,0,11,0,0,0,0,0,1,0,0,1,0,2],[36,57,0.6316,0.33027,0.23273,0.14286,0.2857,0.42858,0.0,0.85714,3,0,0,3,0,9,0,0,7,0,0,8,0,0,0,0,0,3,0,0,2,0,0],[40,57,0.7018,0.36807,0.22683,0.14289,0.42857,0.4286,0.0,0.85714,2,0,0,2,0,7,0,1,5,0,0,11,0,0,2,0,0,1,0,0,3,0,0],[44,57,0.7719,0.44643,0.23351,0.28571,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,5,0,0,5,0,0,15,0,0,1,0,0,3,0,0,0,0,3],[48,57,0.8421,0.33481,0.20703,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,9,0,0,6,0,0,10,0,0,3,0,0,0,0,0,2,0,0],[52,57,0.9123,0.14499,0.18771,0.0,0.0355,0.2857,0.0,0.71429,16,0,0,16,1,6,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[56,57,0.9825,0.08928,0.12242,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.11607,0.14032,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,9,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.35255,"x":0.74553,"p":[[0,50,0.0,0.55129,0.28651,0.41068,0.50071,0.75,0.0,1.0,2,4,1,2,0,2,0,0,3,0,1,8,0,0,3,0,0,5,0,0,4,0,4],[4,50,0.08,0.74553,0.27833,0.57143,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,7,0,11],[8,50,0.16,0.66071,0.3067,0.42857,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,2,0,0,7,0,0,1,0,0,4,0,0,7,0,8],[12,50,0.24,0.73661,0.28372,0.42857,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,7,0,0,1,0,0,2,0,0,7,0,12],[16,50,0.32,0.55355,0.28289,0.42857,0.57121,0.857,0.0,1.0,3,2,0,3,0,1,0,0,2,0,0,9,0,0,5,0,0,2,0,0,8,0,2],[20,50,0.4,0.65186,0.31925,0.42857,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,0,0,0,8,0,0,2,0,0,4,0,0,5,0,9],[24,50,0.48,0.71427,0.25754,0.53539,0.78571,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,7,0,0,9,0,7],[28,50,0.56,0.69195,0.31765,0.42857,0.857,1.0,0.0,1.0,3,9,0,3,0,0,0,0,2,0,0,5,0,0,1,0,0,3,0,0,9,0,9],[32,50,0.64,0.64283,0.30928,0.4286,0.71429,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,1,0,0,5,0,0,4,0,0,5,0,0,6,0,7],[36,50,0.72,0.61146,0.2884,0.42857,0.57143,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,0,0,0,10,0,0,3,0,0,5,0,0,3,0,7],[40,50,0.8,0.683,0.26663,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,6,0,0,3,0,0,4,0,0,8,0,7],[44,50,0.88,0.49551,0.32923,0.14286,0.4998,0.71429,0.0,1.0,4,6,0,4,0,5,0,0,1,0,0,6,0,0,7,0,0,2,0,0,1,0,6],[48,50,0.96,0.41067,0.27833,0.14286,0.42857,0.60607,0.0,1.0,1,2,0,1,0,11,0,0,2,0,0,9,0,0,1,0,0,4,0,0,2,0,2],[50,50,1.0,0.35255,0.20516,0.14286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,4,0,0,14,0,0,4,0,0,0,0,0,0,0,1]]}]},{"i":"ecd91d2fcc1a751f","q":"In triangle  $ABC$ , points  $M$  and  $N$  are on segments  $AB$  and  $AC$  respectively such that  $AM = MC$  and  $AN = NB$ . Let  $P$  be the point such that  $PB$  and  $PC$  are tangent to the circumcircle of  $ABC$ . Given that the perimeters of  $PMN$  and  $BCNM$  are  $21$  and  $29$  respectively, and that  $PB = 5$ , compute the length of  $BC$ 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431,0.0,0.71429,6,0,0,6,0,0,0,0,6,0,0,12,0,0,7,0,0,1,0,0,0,0,0],[348,394,0.8832,0.34375,0.18509,0.28571,0.42857,0.42857,0.0,0.71429,5,0,0,5,0,1,0,0,8,0,0,13,0,0,4,0,0,1,0,0,0,0,0],[352,394,0.8934,0.3125,0.23266,0.0,0.35714,0.57143,0.0,0.71429,9,0,0,9,0,2,0,0,5,0,0,7,0,0,8,0,0,1,0,0,0,0,0],[356,394,0.9036,0.38839,0.19308,0.28571,0.42857,0.57141,0.0,0.71429,3,0,0,3,0,3,0,0,6,0,0,10,0,0,8,0,0,2,0,0,0,0,0],[360,394,0.9137,0.33036,0.18707,0.28571,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,2,0,0,10,0,0,8,0,0,7,0,0,0,0,0,0,0,0],[364,394,0.9239,0.37947,0.17353,0.28571,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,1,0,0,8,0,0,14,0,0,4,0,0,2,0,0,0,0,0],[368,394,0.934,0.33036,0.18013,0.2857,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,2,0,0,13,0,0,7,0,0,5,0,0,1,0,0,0,0,0],[372,394,0.9442,0.35713,0.16365,0.28571,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,4,0,0,9,0,0,10,0,0,7,0,0,0,0,0,0,0,0],[376,394,0.9543,0.36606,0.17833,0.28571,0.42857,0.42858,0.0,0.71429,3,0,0,3,0,3,0,0,6,0,0,15,0,0,3,0,0,2,0,0,0,0,0],[380,394,0.9645,0.36607,0.20183,0.28571,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,4,0,0,7,0,0,12,0,0,3,0,0,2,0,0,1,0,0],[384,394,0.9746,0.31697,0.20743,0.14286,0.28571,0.4286,0.0,0.71429,5,0,0,5,0,6,0,0,6,0,0,9,0,0,4,0,0,2,0,0,0,0,0],[388,394,0.9848,0.38393,0.18708,0.28571,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,8,0,0,10,0,0,7,0,0,2,0,0,0,0,0],[392,394,0.9949,0.34821,0.2257,0.24999,0.42857,0.57143,0.0,0.71429,7,0,0,7,0,1,0,0,7,0,0,7,0,0,8,0,0,2,0,0,0,0,0],[394,394,1.0,0.37052,0.1816,0.28571,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,2,0,0,9,0,0,11,0,0,5,0,0,2,0,0,0,0,0]]}]},{"i":"4af457ab3b0e0b05","q":"Given an integer  $k\\geq 2$ , determine all functions  $f$  from the positive integers into themselves such that  $f(x_1)!+f(x_2)!+\\cdots f(x_k)!$  is divisibe by  $x_1!+x_2!+\\cdots x_k!$  for all positive integers  $x_1,x_2,\\cdots x_k$ . $Albania$","t":[{"b":6,"e":0.14286,"k":"flat","v":0.14732,"x":0.2232,"p":[[0,98,0.0,0.2232,0.18186,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,21,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[4,98,0.0408,0.17848,0.10105,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,20,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,98,0.0816,0.19643,0.11152,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,0,14,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,98,0.1224,0.20089,0.10012,0.14286,0.14286,0.2857,0.0,0.4286,1,0,0,1,0,20,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,98,0.1633,0.16955,0.06625,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,98,0.2041,0.19196,0.08459,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,20,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,98,0.2449,0.21411,0.11305,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,20,0,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,98,0.2857,0.20973,0.08744,0.14286,0.14286,0.28571,0.14,0.42857,0,0,0,0,0,19,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,98,0.3265,0.18304,0.08917,0.14286,0.14286,0.1786,0.0,0.42857,1,0,0,1,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,98,0.3673,0.19196,0.06785,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,98,0.4082,0.1875,0.06621,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,98,0.449,0.17848,0.06191,0.14286,0.14286,0.17857,0.14,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,98,0.4898,0.19196,0.07668,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,98,0.5306,0.16509,0.06301,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,98,0.5714,0.16955,0.06625,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,98,0.6122,0.18295,0.07354,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,98,0.6531,0.15178,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,98,0.6939,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,98,0.7347,0.17848,0.06191,0.14286,0.14286,0.17857,0.14,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,98,0.7755,0.17848,0.10105,0.14286,0.14286,0.17857,0.0,0.57143,2,0,0,2,0,22,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,98,0.8163,0.19197,0.09182,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,23,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[84,98,0.8571,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,98,0.898,0.16071,0.04725,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,98,0.9388,0.17411,0.05906,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,98,0.9796,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[98,98,1.0,0.17411,0.05906,0.14286,0.14286,0.14287,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.1875,"x":0.29018,"p":[[0,55,0.0,0.1875,0.10971,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,55,0.0727,0.21428,0.12372,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,18,0,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,55,0.1455,0.20981,0.12865,0.14286,0.14286,0.28571,0.0,0.571,3,0,0,3,0,16,0,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[12,55,0.2182,0.20089,0.08645,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.20518,0.10073,0.14286,0.14286,0.2857,0.14,0.57143,0,0,0,0,0,21,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,55,0.3636,0.2232,0.10057,0.14286,0.14286,0.28571,0.14286,0.571,0,0,0,0,0,17,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,55,0.4364,0.20536,0.07087,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.21429,0.10101,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,19,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,55,0.5818,0.2008,0.07024,0.14286,0.14286,0.28571,0.14,0.28571,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.23214,0.09279,0.14286,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,12,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.28571,0.14286,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,9,0,0,19,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[44,55,0.8,0.29018,0.09094,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,6,0,0,19,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.25893,0.06622,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,7,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.27232,0.06546,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,5,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9ed899357713ee5d","q":"Given an acute triangle  $ABC$  with  $AC>BC$  and the circumcenter of triangle  $ABC$  is  $O$ . The altitude of triangle  $ABC$  from  $C$  intersects  $AB$  and the circumcircle at  $D$  and  $E$ , respectively. A line which passed through  $O$  which is parallel to  $AB$  intersects  $AC$  at  $F$ . Show that the line  $CO$ , the line which passed through  $F$  and perpendicular to  $AC$ , and the line which passed through  $E$  and parallel with  $DO$  are concurrent.\n\n*Fajar Yuliawan, Bandung*","t":[{"b":0,"e":0.14,"k":"flat","v":0.13375,"x":0.17411,"p":[[0,116,0.0,0.14723,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,116,0.0345,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,116,0.069,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,116,0.1034,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,116,0.1379,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,116,0.1724,0.16071,0.05922,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,116,0.2069,0.15178,0.06121,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,116,0.2414,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,116,0.2759,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,116,0.3103,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,116,0.3448,0.15152,0.07939,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,22,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,116,0.3793,0.17411,0.05906,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,116,0.4138,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,116,0.4483,0.14732,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,116,0.4828,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,116,0.5172,0.17393,0.0691,0.14286,0.14286,0.17857,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,116,0.5517,0.13375,0.04968,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,116,0.5862,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,116,0.6207,0.16045,0.04735,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,116,0.6552,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,116,0.6897,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,116,0.7241,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,116,0.7586,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,116,0.7931,0.16955,0.06625,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,116,0.8276,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,116,0.8621,0.14705,0.05632,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,116,0.8966,0.15178,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,116,0.931,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,116,0.9655,0.16947,0.06629,0.14286,0.14286,0.14289,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,116,1.0,0.17411,0.05905,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.12045,"x":0.18741,"p":[[0,93,0.0,0.12045,0.08825,0.0,0.14286,0.14286,0.0,0.28571,9,0,1,9,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,93,0.043,0.16509,0.05191,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,93,0.086,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,93,0.129,0.16955,0.0558,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,93,0.172,0.16062,0.05925,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,93,0.2151,0.16045,0.05931,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,93,0.2581,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,93,0.3011,0.16518,0.0724,0.14286,0.14286,0.14287,0.0,0.28571,2,0,0,2,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,93,0.3441,0.18741,0.15748,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,93,0.3871,0.14268,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,93,0.4301,0.16054,0.05928,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,93,0.4731,0.17411,0.05905,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,93,0.5161,0.17411,0.06901,0.14286,0.14286,0.1786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the greatest positive integer  $n$  for which there exist  $n$  nonnegative integers  $x_1, x_2,\\ldots , x_n$ , not all zero, such that for any  $\\varepsilon_1, \\varepsilon_2, \\ldots, \\varepsilon_n$  from the set  $\\{-1, 0, 1\\}$ , not all zero,  $\\varepsilon_1 x_1 + \\varepsilon_2 x_2 + \\cdots + \\varepsilon_n x_n$  is not divisible by  $n^3$ .","t":[{"b":1,"e":0.0,"k":"falling","v":0.01786,"x":0.98661,"p":[[0,48,0.0,0.8079,0.26876,0.67536,1.0,1.0,0.0,1.0,2,17,1,2,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,3,0,17],[4,48,0.0833,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,48,0.1667,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,48,0.25,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,48,0.3333,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[20,48,0.4167,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[24,48,0.5,0.85045,0.33233,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,25],[28,48,0.5833,0.44196,0.4783,0.0,0.07143,1.0,0.0,1.0,16,12,0,16,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,12],[32,48,0.6667,0.62054,0.46099,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,18],[36,48,0.75,0.50893,0.4935,0.0,0.64286,1.0,0.0,1.0,15,16,0,15,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[40,48,0.8333,0.37054,0.47897,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,11],[44,48,0.9167,0.15179,0.34244,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[48,48,1.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.79016,"x":1.0,"p":[[0,40,0.0,0.79016,0.19229,0.57143,0.78571,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,3,0,13],[4,40,0.1,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,40,0.2,0.96205,0.17676,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,30],[12,40,0.3,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,40,0.4,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,40,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,40,0.6,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,40,0.7,0.98884,0.04414,1.0,1.0,1.0,0.78571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,30],[32,40,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6037fb48e671756d","q":"For a positive integer  $n$  denote  $F_n(x_1,x_2,\\ldots,x_n) = 1 + x_1 + x_1x_2 + \\cdots +x_1x_2\\ldots x_n$ . For any real numbers  $x_1\\geq x_2 \\geq \\ldots \\geq x_k \\geq 0$  prove that\n\\[ \\prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\\ldots,x_k) \\geq \\prod_{i=1}^k F_i(x_i,x_i,\\ldots,x_i)\\]","t":[{"b":1,"e":0.0,"k":"flat","v":0.05348,"x":0.14286,"p":[[0,36,0.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.09822,0.12078,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,36,0.5556,0.10045,0.10543,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,11,0,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.06688,0.10699,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.12054,0.09522,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.14286,0.10101,0.10714,0.14286,0.1786,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.12482,0.11149,0.0,0.14286,0.17857,0.0,0.28571,12,0,0,12,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.04018,"x":0.1874,"p":[[0,72,0.0,0.07125,0.07126,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.10705,0.12369,0.0,0.07,0.14287,0.0,0.42857,16,0,0,16,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.06695,0.11279,0.0,0.0,0.14286,0.0,0.571,20,0,1,20,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,72,0.3333,0.0982,0.14028,0.0,0.0,0.14286,0.0,0.571,18,0,0,18,0,9,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,72,0.3889,0.08036,0.10677,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.08482,0.10012,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.09822,0.11538,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.12043,0.13877,0.0,0.14286,0.14286,0.0,0.571,14,0,0,14,0,12,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,72,0.6667,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.15179,0.15542,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,11,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[56,72,0.7778,0.09822,0.13092,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.10706,0.1383,0.0,0.07,0.14287,0.0,0.57143,16,0,0,16,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,72,0.8889,0.12947,0.12037,0.0,0.14286,0.1786,0.0,0.4286,12,0,0,12,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.1874,0.17289,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,12,0,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[72,72,1.0,0.16963,0.14476,0.10714,0.14286,0.2857,0.0,0.571,8,0,0,8,0,15,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"84b3416ec5387e8e","q":"In a building there are  $119$  inhabitants who live in  $120$  apartments (several inhabitants can live in the same apartment). We call an apartment *overcrowded* if  $15$  or more people live in it. Every day in some overcrowded apartment (if there is one) its inhabitants have a fight and yes they all go to live in a different apartment (which may or may not be already inhabited). Should you always terminate this process?","t":[{"b":2,"e":0.0,"k":"volatile","v":0.27223,"x":0.77679,"p":[[0,40,0.0,0.27223,0.4321,0.0,0.0,0.67857,0.0,1.0,22,8,0,22,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8],[4,40,0.1,0.77679,0.41178,0.96429,1.0,1.0,0.0,1.0,7,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[8,40,0.2,0.64285,0.46702,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,18],[12,40,0.3,0.58036,0.48173,0.0,0.92857,1.0,0.0,1.0,13,16,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,16],[16,40,0.4,0.60714,0.46566,0.0,0.92857,1.0,0.0,1.0,11,16,0,11,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,16],[20,40,0.5,0.47321,0.48107,0.0,0.21428,1.0,0.0,1.0,15,13,0,15,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,13],[24,40,0.6,0.45982,0.49065,0.0,0.0,1.0,0.0,1.0,17,13,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,13],[28,40,0.7,0.5491,0.4793,0.0,0.9285,1.0,0.0,1.0,13,16,0,13,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,16],[32,40,0.8,0.375,0.46941,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,10],[36,40,0.9,0.66071,0.45422,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,19],[40,40,1.0,0.3125,0.46351,0.0,0.0,1.0,0.0,1.0,22,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10]]},{"b":6,"e":0.0,"k":"volatile","v":0.0,"x":0.69196,"p":[[0,27,0.0,0.24107,0.40789,0.0,0.0,0.28571,0.0,1.0,22,7,0,22,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,27,0.1481,0.69196,0.45049,0.0,1.0,1.0,0.0,1.0,9,21,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,21],[8,27,0.2963,0.52232,0.48394,0.0,0.85714,1.0,0.0,1.0,14,14,0,14,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,14],[12,27,0.4444,0.36607,0.45728,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,9],[16,27,0.5926,0.26339,0.42425,0.0,0.0,0.75,0.0,1.0,23,6,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,6],[20,27,0.7407,0.28571,0.43448,0.0,0.0,0.85714,0.0,1.0,22,7,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,7],[24,27,0.8889,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[27,27,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"457531c02573860b","q":"Find the greatest integer  $A$  for which in any permutation of the numbers  $1, 2, \\ldots , 100$  there exist ten consecutive numbers whose sum is at least  $A$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.09821,"x":0.8839,"p":[[0,92,0.0,0.51786,0.37754,0.35714,0.42857,1.0,0.0,1.0,7,10,2,7,0,1,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,10],[4,92,0.0435,0.79909,0.32901,0.53539,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,22],[8,92,0.087,0.74554,0.34761,0.4286,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,3,0,0,5,0,0,0,0,0,0,0,0,2,0,19],[12,92,0.1304,0.83928,0.27606,0.82132,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,0,2,0,22],[16,92,0.1739,0.77232,0.33476,0.57143,1.0,1.0,0.0,1.0,2,20,0,2,0,2,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,20],[20,92,0.2174,0.75,0.32341,0.42857,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,3,0,16],[24,92,0.2609,0.8839,0.19708,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,3,0,22],[28,92,0.3043,0.73659,0.36962,0.5354,1.0,1.0,0.0,1.0,4,18,1,4,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,18],[32,92,0.3478,0.68302,0.35126,0.42859,0.78564,1.0,0.0,1.0,4,13,0,4,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,0,3,0,13],[36,92,0.3913,0.59375,0.40265,0.28571,0.64286,1.0,0.0,1.0,7,13,0,7,0,0,0,0,3,0,0,5,0,0,1,0,0,1,0,0,2,0,13],[40,92,0.4348,0.67411,0.42142,0.25,1.0,1.0,0.0,1.0,7,18,0,7,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,18],[44,92,0.4783,0.67856,0.40563,0.39286,0.92857,1.0,0.0,1.0,7,16,0,7,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,16],[48,92,0.5217,0.50446,0.44318,0.0,0.57143,1.0,0.0,1.0,12,11,0,12,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,11],[52,92,0.5652,0.41965,0.39759,0.0,0.28586,1.0,0.0,1.0,10,9,0,10,0,2,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,9],[56,92,0.6087,0.36161,0.40404,0.0,0.28571,0.75,0.0,1.0,15,7,0,15,0,0,0,0,3,0,0,5,0,0,0,0,0,1,0,0,1,0,7],[60,92,0.6522,0.30803,0.38649,0.0,0.07143,0.71429,0.0,1.0,16,5,0,16,0,2,0,0,4,0,0,1,0,0,0,0,0,3,0,0,1,0,5],[64,92,0.6957,0.24551,0.31788,0.0,0.07143,0.42857,0.0,1.0,16,3,0,16,0,2,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,3],[68,92,0.7391,0.33027,0.34342,0.0,0.28571,0.46431,0.0,1.0,12,3,0,12,0,1,0,0,8,0,0,3,0,0,1,0,0,1,0,0,3,0,3],[72,92,0.7826,0.1964,0.26663,0.0,0.0,0.32143,0.0,1.0,17,1,0,17,0,3,0,0,4,0,0,4,0,0,2,0,0,0,0,0,1,0,1],[76,92,0.8261,0.14286,0.16751,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[80,92,0.8696,0.125,0.17035,0.0,0.0,0.2857,0.0,0.57143,19,0,0,19,0,3,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[84,92,0.913,0.18303,0.25059,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,1,0,0,10,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[88,92,0.9565,0.09821,0.15746,0.0,0.0,0.17857,0.0,0.4286,22,0,0,22,0,2,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[92,92,1.0,0.16963,0.19374,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,2,0,0,9,0,0,2,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"falling","v":0.10714,"x":0.93304,"p":[[0,90,0.0,0.57589,0.3302,0.42857,0.42857,0.85714,0.0,1.0,4,7,1,4,0,1,0,0,0,0,0,13,0,0,0,0,0,2,0,0,5,0,7],[4,90,0.0444,0.83929,0.27837,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,22],[8,90,0.0889,0.56695,0.36331,0.28571,0.42857,1.0,0.0,1.0,4,10,0,4,0,3,0,0,2,0,0,8,0,0,1,0,0,2,0,0,2,0,10],[12,90,0.1333,0.43749,0.28107,0.28571,0.42857,0.57143,0.0,1.0,3,3,0,3,0,3,0,0,8,0,0,8,0,0,3,0,0,2,0,0,2,0,3],[16,90,0.1778,0.47768,0.32656,0.28571,0.42857,0.71429,0.0,1.0,5,6,0,5,0,2,0,0,4,0,0,9,0,0,2,0,0,4,0,0,0,0,6],[20,90,0.2222,0.70088,0.29094,0.42857,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,5,0,0,2,0,0,6,0,0,2,0,12],[24,90,0.2667,0.74998,0.34257,0.42857,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,5,0,0,3,0,0,1,0,0,0,0,0,2,0,19],[28,90,0.3111,0.73213,0.31289,0.53539,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,0,2,0,15],[32,90,0.3556,0.8125,0.33586,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,22],[36,90,0.4,0.91955,0.25266,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[40,90,0.4444,0.89284,0.26965,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,27],[44,90,0.4889,0.83929,0.31693,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,25],[48,90,0.5333,0.91518,0.21683,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,26],[52,90,0.5778,0.93304,0.19556,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[56,90,0.6222,0.75,0.4165,0.71429,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,22],[60,90,0.6667,0.88393,0.25614,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,25],[64,90,0.7111,0.85714,0.30514,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[68,90,0.7556,0.77679,0.37105,0.64286,1.0,1.0,0.0,1.0,4,22,0,4,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,22],[72,90,0.8,0.76339,0.37561,0.64286,1.0,1.0,0.0,1.0,5,20,0,5,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,20],[76,90,0.8444,0.61161,0.43775,0.10714,0.92857,1.0,0.0,1.0,8,16,0,8,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,16],[80,90,0.8889,0.53571,0.43886,0.0,0.71429,1.0,0.0,1.0,10,12,0,10,0,1,0,0,4,0,0,0,0,0,0,0,0,3,0,0,2,0,12],[84,90,0.9333,0.625,0.43412,0.21429,1.0,1.0,0.0,1.0,8,17,0,8,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,17],[88,90,0.9778,0.30803,0.36089,0.0,0.2857,0.42857,0.0,1.0,14,5,0,14,0,1,0,0,7,0,0,3,0,0,1,0,0,0,0,0,1,0,5],[90,90,1.0,0.10714,0.12877,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b0db830573b215e9","q":"In a directed graph with  $2013$  vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach  $k$  vertices using at most two edges. Find the maximum value of  $k$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.17857,"p":[[0,48,0.0,0.17857,0.37457,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[4,48,0.0833,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,48,0.1667,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,48,0.25,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,48,0.3333,0.0625,0.2257,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[20,48,0.4167,0.08482,0.26693,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[24,48,0.5,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,48,0.5833,0.0625,0.2257,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[32,48,0.6667,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[36,48,0.75,0.04911,0.19759,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,48,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,48,1.0,0.0625,0.2257,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.11607,"p":[[0,33,0.0,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,33,0.1212,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.11607,0.30813,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[20,33,0.6061,0.11607,0.3102,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[24,33,0.7273,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,33,0.8485,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,33,0.9697,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[33,33,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"ffc3351474e66821","q":"If the set  $S = \\{1,2,3,\u2026,16\\}$  is partitioned into  $n$  subsets, there must be a subset in which elements  $a, b, c$  (can be the same) exist, satisfying  $a+ b=c$ . Find the maximum value of  $n$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.30802,"x":0.47313,"p":[[0,78,0.0,0.31696,0.07771,0.28571,0.28571,0.28571,0.1429,0.57143,0,0,0,0,0,1,0,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,78,0.0513,0.4285,0.18893,0.28571,0.42857,0.5711,0.0,1.0,2,1,2,2,0,0,0,0,10,0,0,7,0,0,12,0,0,0,0,0,0,0,1],[8,78,0.1026,0.44637,0.21648,0.28571,0.571,0.57143,0.0,1.0,3,1,3,3,0,0,0,0,9,0,0,3,0,0,14,0,0,2,0,0,0,0,1],[12,78,0.1538,0.43745,0.21407,0.28571,0.42857,0.57143,0.0,1.0,2,2,2,2,0,0,0,0,11,0,0,6,0,0,11,0,0,0,0,0,0,0,2],[16,78,0.2051,0.4286,0.16365,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,5,0,0,11,0,0,0,0,0,0,0,1],[20,78,0.2564,0.39727,0.14604,0.28571,0.35714,0.571,0.0,0.57143,1,0,1,1,0,0,0,0,15,0,0,5,0,0,11,0,0,0,0,0,0,0,0],[24,78,0.3077,0.42407,0.18026,0.28571,0.42857,0.57143,0.0,0.71429,3,0,3,3,0,0,0,0,7,0,0,8,0,0,13,0,0,1,0,0,0,0,0],[28,78,0.359,0.38389,0.14475,0.28571,0.42857,0.4286,0.0,0.57143,2,0,2,2,0,0,0,0,11,0,0,12,0,0,7,0,0,0,0,0,0,0,0],[32,78,0.4103,0.42405,0.19389,0.28571,0.4286,0.57143,0.0,0.71429,3,0,3,3,0,1,0,0,7,0,0,6,0,0,13,0,0,2,0,0,0,0,0],[36,78,0.4615,0.43743,0.17468,0.39286,0.4998,0.57143,0.0,0.57143,3,0,3,3,0,0,0,0,5,0,0,8,0,0,16,0,0,0,0,0,0,0,0],[40,78,0.5128,0.40622,0.13412,0.28571,0.35714,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,6,0,0,9,0,0,1,0,0,0,0,0],[44,78,0.5641,0.47313,0.17286,0.39286,0.4998,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,7,0,0,8,0,0,14,0,0,1,0,0,0,0,1],[48,78,0.6154,0.42853,0.17125,0.28571,0.42857,0.5711,0.0,1.0,1,1,1,1,0,0,0,0,11,0,0,9,0,0,10,0,0,0,0,0,0,0,1],[52,78,0.6667,0.45977,0.12229,0.28571,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[56,78,0.7179,0.42406,0.14048,0.28571,0.42859,0.5711,0.0,0.57143,1,0,1,1,0,0,0,0,10,0,0,9,0,0,12,0,0,0,0,0,0,0,0],[60,78,0.7692,0.44633,0.13708,0.28571,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,5,0,0,14,0,0,1,0,0,0,0,0],[64,78,0.8205,0.34372,0.15912,0.2857,0.28571,0.4642,0.0,0.57143,2,0,1,2,0,2,0,0,17,0,0,3,0,0,8,0,0,0,0,0,0,0,0],[68,78,0.8718,0.37944,0.20077,0.2857,0.42859,0.57143,0.0,0.57143,4,0,0,4,0,3,0,0,6,0,0,6,0,0,13,0,0,0,0,0,0,0,0],[72,78,0.9231,0.38378,0.20031,0.2857,0.42857,0.57143,0.0,0.71429,4,0,3,4,0,1,0,0,10,0,0,4,0,0,12,0,0,1,0,0,0,0,0],[76,78,0.9744,0.33481,0.22476,0.2857,0.28571,0.42858,0.0,1.0,6,1,0,6,0,0,0,0,14,0,0,5,0,0,5,0,0,1,0,0,0,0,1],[78,78,1.0,0.30802,0.21755,0.10714,0.28571,0.4642,0.0,0.71429,8,0,0,8,0,1,0,0,10,0,0,5,0,0,7,0,0,1,0,0,0,0,0]]},{"b":2,"e":0.571,"k":"flat","v":0.29464,"x":0.48211,"p":[[0,56,0.0,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.44637,0.15868,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,7,0,0,12,0,0,0,0,0,0,0,1],[8,56,0.1429,0.48211,0.26904,0.2857,0.42857,0.57143,0.0,1.0,2,5,1,2,0,0,0,0,11,0,0,5,0,0,9,0,0,0,0,0,0,0,5],[12,56,0.2143,0.41063,0.15862,0.28571,0.42857,0.571,0.0,0.57143,2,0,2,2,0,0,0,0,10,0,0,8,0,0,12,0,0,0,0,0,0,0,0],[16,56,0.2857,0.40617,0.19591,0.28571,0.42857,0.57111,0.0,0.85714,3,0,2,3,0,0,0,0,12,0,0,3,0,0,13,0,0,0,0,0,1,0,0],[20,56,0.3571,0.42408,0.14051,0.28571,0.4286,0.57143,0.0,0.57143,1,0,1,1,0,0,0,0,10,0,0,9,0,0,12,0,0,0,0,0,0,0,0],[24,56,0.4286,0.45978,0.12231,0.28571,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[28,56,0.5,0.46867,0.1606,0.28571,0.571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,4,0,0,16,0,0,0,0,0,0,0,1],[32,56,0.5714,0.47765,0.13173,0.42857,0.4998,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,9,0,0,15,0,0,0,0,0,1,0,0],[36,56,0.6429,0.4508,0.11346,0.39286,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,11,0,0,13,0,0,0,0,0,0,0,0],[40,56,0.7143,0.43301,0.16933,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,5,0,0,10,0,0,1,0,0,0,0,1],[44,56,0.7857,0.41961,0.13331,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,7,0,0,10,0,0,1,0,0,0,0,0],[48,56,0.8571,0.38837,0.11421,0.28571,0.35714,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[52,56,0.9286,0.42408,0.14051,0.28571,0.42857,0.5714,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,7,0,0,9,0,0,2,0,0,0,0,0],[56,56,1.0,0.39283,0.15565,0.28571,0.28571,0.42858,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,7,0,0,6,0,0,0,0,0,0,0,1]]}]},{"i":"8c71997df8f903d7","q":"If  $x$  is a real number, let  $\\lfloor x \\rfloor$  be the greatest integer that is less than or equal to  $x$ .  If  $n$  is a positive integer, let  $S(n)$  be defined by\n\\[\n  S(n) \n    = \\left\\lfloor \\frac{n}{10^{\\lfloor \\log n \\rfloor}} \\right\\rfloor\n        + 10 \\left( n - 10^{\\lfloor \\log n \\rfloor} \n               \\cdot \\left\\lfloor \\frac{n}{10^{\\lfloor \\log n \\rfloor}} \\right\\rfloor\n             \\right) \\, .\n\\]\n(All the logarithms are base 10.)  How many integers  $n$  from 1 to 2011 (inclusive) satisfy  $S(S(n)) = n$ ?","t":[{"b":2,"e":0.85714,"k":"falling","v":0.83482,"x":1.0,"p":[[0,30,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,30,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,30,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[30,30,1.0,0.83482,0.17536,0.85714,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,7]]},{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,115,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,115,0.0348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,115,0.0696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,115,0.1043,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,115,0.1391,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,115,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,115,0.2087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,115,0.2435,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,115,0.2783,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,115,0.313,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,115,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,115,0.3826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,115,0.4174,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,115,0.4522,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,115,0.487,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,115,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,115,0.5565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,115,0.5913,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,115,0.6261,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,115,0.6609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,115,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,115,0.7304,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,115,0.7652,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,115,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,115,0.8348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,115,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,115,0.9043,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,115,0.9391,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,115,0.9739,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[115,115,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"033f05289f125c1d","q":"In a square  $ABCD$ ,  $E$  is a point on diagonal  $BD$ .  $P$  and  $Q$  are the circumcentres of  $\\triangle ABE$  and  $\\triangle ADE$  respectively. Prove that  $APEQ$  is a square.","t":[{"b":2,"e":1.0,"k":"rising","v":0.78125,"x":0.95982,"p":[[0,34,0.0,0.78125,0.29877,0.64286,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,3,0,0,0,0,0,5,0,0,0,0,19],[4,34,0.1176,0.94643,0.15047,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,28],[8,34,0.2353,0.88839,0.17029,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,1,0,21],[12,34,0.3529,0.9241,0.18553,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[16,34,0.4706,0.89286,0.2369,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,25],[20,34,0.5882,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[24,34,0.7059,0.95982,0.10249,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[28,34,0.8235,0.87053,0.2,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,6,0,0,1,0,21],[32,34,0.9412,0.92856,0.14729,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,25],[34,34,1.0,0.95088,0.12169,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.79911,"x":0.91964,"p":[[0,45,0.0,0.79911,0.28984,0.64286,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,3,0,0,1,0,20],[4,45,0.0889,0.90624,0.21013,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,26],[8,45,0.1778,0.85714,0.22016,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,6,0,0,2,0,20],[12,45,0.2667,0.83482,0.23987,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,1,0,20],[16,45,0.3556,0.87946,0.20858,0.82132,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,2,0,22],[20,45,0.4444,0.80357,0.28291,0.64286,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,5,0,0,3,0,0,0,0,0,3,0,0,1,0,20],[24,45,0.5333,0.90625,0.18073,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,24],[28,45,0.6222,0.91964,0.16342,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,0,0,25],[32,45,0.7111,0.84822,0.2141,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,7,0,0,0,0,20],[36,45,0.8,0.82576,0.24159,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,6,0,0,1,0,19],[40,45,0.8889,0.87053,0.20935,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,5,0,0,2,0,21],[44,45,0.9778,0.81696,0.2506,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0,5,0,0,1,0,19],[45,45,1.0,0.85268,0.26603,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,24]]}]},{"i":"23e2b7a519bfb202","q":"In acute triangle  $ ABC$ , show that:\r\n\r $ \\sin^3{A}\\cos^2{(B \\minus{} C)} \\plus{} \\sin^3{B}\\cos^2{(C \\minus{} A)} \\plus{} \\sin^3{C}\\cos^2{(A \\minus{} B)} \\leq 3\\sin{A} \\sin{B} \\sin{C}$ \r\n\r\nand find out when the equality holds.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.10259,"x":0.37054,"p":[[0,69,0.0,0.10259,0.13937,0.0,0.14143,0.14286,0.0,0.71429,15,0,0,15,0,14,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,69,0.058,0.24107,0.22427,0.14286,0.14286,0.2857,0.0,0.85714,3,0,0,3,0,19,0,0,4,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[8,69,0.1159,0.24545,0.20901,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,19,0,0,5,0,0,3,0,0,0,0,0,1,0,0,2,0,0],[12,69,0.1739,0.25,0.15152,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,20,0,0,2,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[16,69,0.2319,0.25447,0.26422,0.14286,0.14286,0.42857,0.0,0.85714,7,0,0,7,0,14,0,0,2,0,0,5,0,0,0,0,0,0,0,0,4,0,0],[20,69,0.2899,0.27223,0.26336,0.14286,0.14286,0.32142,0.0,0.85714,4,0,0,4,0,17,0,0,3,0,0,3,0,0,0,0,0,1,0,0,4,0,0],[24,69,0.3478,0.22768,0.17076,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,17,0,0,5,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[28,69,0.4058,0.37054,0.25218,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,13,0,0,4,0,0,9,0,0,0,0,0,1,0,0,5,0,0],[32,69,0.4638,0.22321,0.18536,0.14286,0.14286,0.32143,0.0,0.85714,4,0,0,4,0,18,0,0,2,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[36,69,0.5217,0.21875,0.1988,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,18,0,0,5,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[40,69,0.5797,0.23206,0.20128,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,19,0,0,3,0,0,5,0,0,0,0,0,0,0,0,2,0,0],[44,69,0.6377,0.22768,0.20159,0.14286,0.14286,0.2857,0.0,0.85714,3,0,0,3,0,20,0,0,2,0,0,5,0,0,0,0,0,0,0,0,2,0,0],[48,69,0.6957,0.28126,0.22441,0.14286,0.14286,0.42857,0.0,0.85714,3,0,0,3,0,16,0,0,0,0,0,10,0,0,0,0,0,1,0,0,2,0,0],[52,69,0.7536,0.25446,0.19144,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,18,0,0,6,0,0,5,0,0,0,0,0,0,0,0,2,0,0],[56,69,0.8116,0.29018,0.21572,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,18,0,0,5,0,0,5,0,0,0,0,0,2,0,0,2,0,0],[60,69,0.8696,0.31255,0.21854,0.14286,0.2857,0.42857,0.0,0.85714,1,0,0,1,0,14,0,0,5,0,0,8,0,0,0,0,0,2,0,0,2,0,0],[64,69,0.9275,0.27232,0.23244,0.14286,0.14286,0.42857,0.0,0.85714,4,0,0,4,0,14,0,0,5,0,0,5,0,0,0,0,0,2,0,0,2,0,0],[68,69,0.9855,0.20982,0.1636,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,18,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[69,69,1.0,0.1875,0.13092,0.14286,0.14286,0.2857,0.0,0.71429,3,0,0,3,0,20,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.06696,"x":0.23652,"p":[[0,12,0.0,0.06696,0.09438,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.23652,0.2248,0.14286,0.14286,0.32143,0.0,0.85714,5,0,0,5,0,17,0,0,2,0,0,5,0,0,0,0,0,1,0,0,2,0,0],[8,12,0.6667,0.23215,0.22517,0.14286,0.14286,0.32143,0.0,0.85714,5,0,0,5,0,18,0,0,1,0,0,5,0,0,0,0,0,1,0,0,2,0,0],[12,12,1.0,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ea80bb6d888ec143","q":"Find all triples  $(a, b, c)$  of positive integers for which  $\\frac{32a + 3b + 48c}{4abc}$  is also an integer.","t":[{"b":3,"e":0.4286,"k":"flat","v":0.41964,"x":0.78125,"p":[[0,445,0.0,0.54911,0.26027,0.42857,0.42857,0.75,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,16,0,0,3,0,0,1,0,0,3,0,5],[4,445,0.009,0.78125,0.32337,0.64286,1.0,1.0,0.0,1.0,3,18,3,3,0,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,4,0,18],[8,445,0.018,0.74554,0.33831,0.42857,1.0,1.0,0.0,1.0,3,18,3,3,0,0,0,0,0,0,0,8,0,0,0,0,0,1,0,0,2,0,18],[12,445,0.027,0.76339,0.30432,0.57143,0.85714,1.0,0.0,1.0,3,14,3,3,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,6,0,14],[16,445,0.036,0.66964,0.36498,0.42857,0.78571,1.0,0.0,1.0,5,14,5,5,0,0,0,0,0,0,0,7,0,0,1,0,0,3,0,0,2,0,14],[20,445,0.0449,0.66071,0.37585,0.42857,0.85714,1.0,0.0,1.0,5,13,5,5,0,1,0,0,1,0,0,4,0,0,3,0,0,0,0,0,5,0,13],[24,445,0.0539,0.64732,0.29877,0.42857,0.71429,0.89286,0.0,1.0,3,8,3,3,0,0,0,0,0,0,0,8,0,0,4,0,0,5,0,0,4,0,8],[28,445,0.0629,0.60272,0.3741,0.42857,0.71429,0.89286,0.0,1.0,7,8,7,7,0,0,0,0,0,0,0,6,0,0,1,0,0,3,0,0,7,0,8],[32,445,0.0719,0.77232,0.28089,0.67857,0.85714,1.0,0.0,1.0,2,12,2,2,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,10,0,12],[36,445,0.0809,0.73661,0.24251,0.42859,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,2,0,0,4,0,12],[40,445,0.0899,0.71875,0.25376,0.42857,0.85714,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,9,0,0,1,0,0,3,0,0,11,0,7],[44,445,0.0989,0.68304,0.33261,0.42857,0.85714,1.0,0.0,1.0,4,10,4,4,0,0,0,0,0,0,0,6,0,0,3,0,0,1,0,0,8,0,10],[48,445,0.1079,0.67857,0.32927,0.42857,0.85714,1.0,0.0,1.0,3,12,3,3,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,5,0,12],[52,445,0.1169,0.53124,0.31589,0.42857,0.42857,0.85714,0.0,1.0,5,3,5,5,0,1,0,0,1,0,0,10,0,0,1,0,0,5,0,0,6,0,3],[56,445,0.1258,0.65179,0.31529,0.42857,0.71429,1.0,0.0,1.0,3,9,3,3,0,0,0,0,0,0,0,11,0,0,1,0,0,2,0,0,6,0,9],[60,445,0.1348,0.70981,0.2612,0.42857,0.78571,1.0,0.0,1.0,1,9,1,1,0,0,0,0,0,0,0,9,0,0,3,0,0,3,0,0,7,0,9],[64,445,0.1438,0.63393,0.35163,0.42857,0.71429,1.0,0.0,1.0,5,11,5,5,0,0,0,0,0,0,0,8,0,0,1,0,0,5,0,0,2,0,11],[68,445,0.1528,0.70982,0.31234,0.4286,0.85714,1.0,0.0,1.0,3,11,3,3,0,0,0,0,0,0,0,6,0,0,3,0,0,2,0,0,7,0,11],[72,445,0.1618,0.65165,0.36757,0.42857,0.78571,1.0,0.0,1.0,5,13,5,5,0,0,0,0,1,0,0,7,0,0,1,0,0,2,0,0,3,0,13],[76,445,0.1708,0.66964,0.28221,0.42857,0.71429,1.0,0.0,1.0,1,9,1,1,0,0,0,0,1,0,0,13,0,0,0,0,0,2,0,0,6,0,9],[80,445,0.1798,0.63397,0.30077,0.42857,0.71429,0.85714,0.0,1.0,3,6,3,3,0,0,0,0,0,0,0,11,0,0,1,0,0,3,0,0,8,0,6],[84,445,0.1888,0.6563,0.28759,0.42857,0.71429,0.85714,0.0,1.0,2,7,2,2,0,0,0,0,0,0,0,12,0,0,1,0,0,2,0,0,8,0,7],[88,445,0.1978,0.66071,0.28065,0.42857,0.71429,0.85714,0.0,1.0,2,7,2,2,0,0,0,0,0,0,0,10,0,0,3,0,0,3,0,0,7,0,7],[92,445,0.2067,0.70535,0.24469,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,4,0,0,4,0,10],[96,445,0.2157,0.66517,0.26632,0.42857,0.71429,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,1,0,0,11,0,0,2,0,0,3,0,0,7,0,7],[100,445,0.2247,0.74554,0.29609,0.42859,0.85714,1.0,0.0,1.0,2,14,2,2,0,0,0,0,0,0,0,7,0,0,1,0,0,4,0,0,4,0,14],[104,445,0.2337,0.625,0.29179,0.42857,0.50001,0.89286,0.0,1.0,2,8,2,2,0,0,0,0,0,0,0,14,0,0,3,0,0,0,0,0,5,0,8],[108,445,0.2427,0.64732,0.25997,0.42857,0.57143,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,13,0,0,3,0,0,4,0,0,0,0,10],[112,445,0.2517,0.73661,0.28146,0.42857,0.85714,1.0,0.0,1.0,1,13,1,1,0,0,0,0,0,0,0,10,0,0,2,0,0,0,0,0,6,0,13],[116,445,0.2607,0.62054,0.28928,0.42857,0.57143,0.85714,0.0,1.0,2,7,2,2,0,0,0,0,2,0,0,10,0,0,4,0,0,2,0,0,5,0,7],[120,445,0.2697,0.63839,0.29877,0.42857,0.5,1.0,0.0,1.0,2,9,2,2,0,0,0,0,0,0,0,14,0,0,2,0,0,0,0,0,5,0,9],[124,445,0.2787,0.63392,0.29219,0.42857,0.57143,0.85714,0.0,1.0,2,7,2,2,0,0,0,0,1,0,0,12,0,0,2,0,0,1,0,0,7,0,7],[128,445,0.2876,0.67857,0.23419,0.42857,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,4,0,0,6,0,7],[132,445,0.2966,0.59821,0.27765,0.42857,0.5,0.85714,0.0,1.0,1,5,1,1,0,2,0,0,1,0,0,12,0,0,2,0,0,3,0,0,6,0,5],[136,445,0.3056,0.67857,0.25254,0.42857,0.64286,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,2,0,0,5,0,9],[140,445,0.3146,0.56696,0.2461,0.42857,0.42857,0.85714,0.0,1.0,2,2,2,2,0,0,0,0,0,0,0,15,0,0,4,0,0,2,0,0,7,0,2],[144,445,0.3236,0.67411,0.27254,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,10,0,0,0,0,0,5,0,0,3,0,10],[148,445,0.3326,0.69195,0.27919,0.42857,0.71429,1.0,0.0,1.0,1,9,1,1,0,1,0,0,1,0,0,7,0,0,3,0,0,4,0,0,6,0,9],[152,445,0.3416,0.58482,0.26088,0.42857,0.42857,0.85714,0.0,1.0,1,4,1,1,0,1,0,0,1,0,0,15,0,0,1,0,0,3,0,0,6,0,4],[156,445,0.3506,0.6875,0.22142,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,11,0,0,4,0,0,2,0,0,10,0,5],[160,445,0.3596,0.63393,0.23402,0.42857,0.5,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,15,0,0,2,0,0,1,0,0,9,0,4],[164,445,0.3685,0.66518,0.22759,0.42857,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,1,0,0,7,0,6],[168,445,0.3775,0.62053,0.28259,0.42857,0.42857,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,2,0,0,15,0,0,0,0,0,2,0,0,4,0,8],[172,445,0.3865,0.58927,0.22232,0.42857,0.42857,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,19,0,0,3,0,0,2,0,0,3,0,5],[176,445,0.3955,0.67411,0.25313,0.42857,0.57143,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,14,0,0,4,0,0,1,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an acute triangle  $ABC$  with  $AC > AB$ , let  $D$  be the projection of  $A$  on  $BC$ , and let  $E$  and  $F$  be the projections of  $D$  on  $AB$  and  $AC$ , respectively. Let  $G$  be the intersection point of the lines  $AD$  and  $EF$ . Let  $H$  be the second intersection point of the line  $AD$  and the circumcircle of triangle  $ABC$ . Prove that \\[AG \\cdot AH=AD^2\\]","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,64,0.125,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,64,0.1875,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,64,0.25,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,64,0.3125,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,1,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,64,0.375,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,64,0.4375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,64,0.5625,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,64,0.8125,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[56,64,0.875,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,64,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,128,0.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,128,0.0312,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,128,0.0625,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,128,0.0938,0.0625,0.2257,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[16,128,0.125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,128,0.1562,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,128,0.1875,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,128,0.2188,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,128,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,128,0.2812,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,128,0.3125,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,128,0.3438,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,128,0.375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,128,0.4062,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,128,0.4375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,128,0.4688,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,128,0.5,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[68,128,0.5312,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[72,128,0.5625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,128,0.5938,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[80,128,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,128,0.6562,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[88,128,0.6875,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[92,128,0.7188,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,128,0.75,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[100,128,0.7812,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,128,0.8125,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[108,128,0.8438,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,128,0.875,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,128,0.9062,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[120,128,0.9375,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[124,128,0.9688,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,128,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78b2da5a81306f8c","q":"Let  $\\mathcal{F}$  be a family of (distinct) subsets of the set  $\\{1,2,\\dots,n\\}$  such that for all  $A$ ,  $B\\in \\mathcal{F}$ ,we have that  $A^C\\cup B\\in \\mathcal{F}$ , where  $A^C$  is the set of all members of  ${1,2,\\dots,n}$  that are not in  $A$ . \n\nProve that every  $k\\in {1,2,\\dots,n}$  appears in at least half of the sets in  $\\mathcal{F}$ . \n\n*Stijn Cambie, Mohammad Javad Moghaddas Mehr*","t":[{"b":2,"e":0.4286,"k":"rising","v":0.29463,"x":0.56696,"p":[[0,21,0.0,0.29909,0.24052,0.14286,0.28571,0.42857,0.0,1.0,7,1,3,7,0,4,0,0,11,0,0,4,0,0,3,0,0,2,0,0,0,0,1],[4,21,0.1905,0.38391,0.2511,0.2857,0.28571,0.57143,0.0,1.0,5,1,0,5,0,1,0,0,11,0,0,5,0,0,5,0,0,3,0,0,1,0,1],[8,21,0.381,0.29463,0.24467,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,2,0,0,11,0,0,3,0,0,2,0,0,5,0,0,0,0,0],[12,21,0.5714,0.42853,0.17854,0.28571,0.42857,0.57111,0.0,0.85714,2,0,0,2,0,1,0,0,6,0,0,12,0,0,9,0,0,1,0,0,1,0,0],[16,21,0.7619,0.44191,0.18332,0.28571,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,6,0,0,8,0,0,10,0,0,4,0,0,0,0,0],[20,21,0.9524,0.56696,0.13592,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,9,0,0,9,0,0,12,0,0,0,0,0],[21,21,1.0,0.54909,0.14772,0.42857,0.57141,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,11,0,0,6,0,0,12,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.25889,"x":0.42411,"p":[[0,52,0.0,0.25889,0.18701,0.10714,0.28571,0.28571,0.0,0.71429,8,0,5,8,0,1,0,0,17,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[4,52,0.0769,0.32588,0.26542,0.0,0.28571,0.57143,0.0,0.85714,10,0,0,10,0,1,0,0,6,0,0,5,0,0,6,0,0,3,0,0,1,0,0],[8,52,0.1538,0.42411,0.33213,0.0,0.42857,0.60714,0.0,1.0,9,3,0,9,0,1,0,0,3,0,0,4,0,0,7,0,0,3,0,0,2,0,3],[12,52,0.2308,0.37053,0.2942,0.10714,0.28571,0.57143,0.0,1.0,8,2,0,8,0,1,0,0,9,0,0,3,0,0,4,0,0,5,0,0,0,0,2],[16,52,0.3077,0.36159,0.24738,0.24999,0.35714,0.57111,0.0,1.0,6,1,0,6,0,2,0,0,8,0,0,7,0,0,5,0,0,3,0,0,0,0,1],[20,52,0.3846,0.4107,0.24155,0.28571,0.42857,0.57143,0.0,0.857,5,0,0,5,0,1,0,0,6,0,0,9,0,0,4,0,0,6,0,0,1,0,0],[24,52,0.4615,0.37054,0.27632,0.14289,0.28571,0.57143,0.0,1.0,6,2,0,6,0,3,0,0,8,0,0,6,0,0,3,0,0,4,0,0,0,0,2],[28,52,0.5385,0.37049,0.21971,0.2857,0.28571,0.571,0.0,0.85714,3,0,0,3,0,4,0,0,11,0,0,4,0,0,6,0,0,3,0,0,1,0,0],[32,52,0.6154,0.32585,0.26778,0.14286,0.28571,0.4642,0.0,1.0,7,1,0,7,0,6,0,0,5,0,0,6,0,0,4,0,0,2,0,0,1,0,1],[36,52,0.6923,0.29018,0.21275,0.0,0.28571,0.4286,0.0,0.71429,9,0,0,9,0,0,0,0,11,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[40,52,0.7692,0.37941,0.21605,0.2857,0.42857,0.571,0.0,0.71429,5,0,0,5,0,1,0,0,8,0,0,7,0,0,8,0,0,3,0,0,0,0,0],[44,52,0.8462,0.30355,0.23073,0.10714,0.28571,0.42858,0.0,0.71429,8,0,0,8,0,3,0,0,8,0,0,6,0,0,4,0,0,3,0,0,0,0,0],[48,52,0.9231,0.37052,0.17444,0.28571,0.35714,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,12,0,0,10,0,0,3,0,0,3,0,0,0,0,0],[52,52,1.0,0.39286,0.12372,0.28571,0.42857,0.46431,0.14286,0.57143,0,0,0,0,0,1,0,0,14,0,0,9,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"94006ce33cab0a13","q":"Given are positive integers $r$ and $k$ and an infinite sequence of positive integers $a_{1} \\leq a_{2} \\leq \\ldots$ such that $\\frac{r}{a_{r}}=k+1$. Prove that there is a $t$ such that $\\frac{t}{a_{t}}=k$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.83034,"x":0.95981,"p":[[0,53,0.0,0.83034,0.27995,0.85711,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,17],[4,53,0.0755,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,53,0.1509,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[12,53,0.2264,0.88839,0.1504,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,11,0,16],[16,53,0.3019,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[20,53,0.3774,0.94642,0.07785,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[24,53,0.4528,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[28,53,0.5283,0.89732,0.1439,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,12,0,16],[32,53,0.6038,0.9241,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,21],[36,53,0.6792,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[40,53,0.7547,0.91518,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16],[44,53,0.8302,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[48,53,0.9057,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[52,53,0.9811,0.92856,0.07144,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[53,53,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":6,"e":1.0,"k":"flat","v":0.83482,"x":0.97321,"p":[[0,27,0.0,0.83482,0.27919,0.82143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,5,0,19],[4,27,0.1481,0.93302,0.11841,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[8,27,0.2963,0.93302,0.11839,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23],[12,27,0.4444,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[16,27,0.5926,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[20,27,0.7407,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[24,27,0.8889,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[27,27,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"c1bad0fbb9340743","q":"Let  $\\mathbb{N}$  denote the set of positive integers. Find all functions  $f:\\mathbb{N}\\longrightarrow\\mathbb{N}$  such that\n\\[n+f(m)\\mid f(n)+nf(m)\\]\nfor all  $m,n\\in \\mathbb{N}$ *Proposed by Dorlir Ahmeti, Albania*","t":[{"b":3,"e":0.0,"k":"flat","v":0.70981,"x":0.8482,"p":[[0,138,0.0,0.73664,0.35731,0.42965,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,18],[4,138,0.029,0.76337,0.16603,0.71429,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,14,0,0,9,0,5],[8,138,0.058,0.76338,0.15408,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,16,0,2],[12,138,0.087,0.78122,0.22584,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,0,12,0,8],[16,138,0.1159,0.79017,0.21124,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,10,0,0,14,0,6],[20,138,0.1449,0.79449,0.14268,0.71429,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,0,12,0,5],[24,138,0.1739,0.81249,0.14032,0.82132,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,21,0,3],[28,138,0.2029,0.72768,0.25595,0.71429,0.85714,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,1,0,0,3,0,0,0,0,0,9,0,0,12,0,5],[32,138,0.2319,0.70981,0.22725,0.71429,0.71429,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,4,0,0,1,0,0,11,0,0,11,0,3],[36,138,0.2609,0.74996,0.23958,0.71429,0.85707,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,15,0,4],[40,138,0.2899,0.79907,0.14663,0.71429,0.857,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,16,0,4],[44,138,0.3188,0.70983,0.21268,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,4,0,0,1,0,0,11,0,0,12,0,2],[48,138,0.3478,0.76338,0.18765,0.71429,0.78564,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,14,0,0,13,0,3],[52,138,0.3768,0.76785,0.25442,0.71429,0.85714,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,18,0,5],[56,138,0.4058,0.82142,0.15567,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,15,0,7],[60,138,0.4348,0.82588,0.13709,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,17,0,6],[64,138,0.4638,0.78125,0.19227,0.71429,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,15,0,4],[68,138,0.4928,0.80355,0.09941,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,20,0,1],[72,138,0.5217,0.80354,0.11715,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,15,0,4],[76,138,0.5507,0.82587,0.17401,0.85714,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,5],[80,138,0.5797,0.79017,0.19227,0.71429,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,17,0,4],[84,138,0.6087,0.77231,0.20158,0.71429,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,8,0,0,16,0,4],[88,138,0.6377,0.7945,0.15133,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,16,0,4],[92,138,0.6667,0.80802,0.06786,0.71429,0.85714,0.85714,0.714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,21,0,0],[96,138,0.6957,0.77232,0.17807,0.71429,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,11,0,0,12,0,5],[100,138,0.7246,0.81694,0.16067,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,15,0,7],[104,138,0.7536,0.72766,0.20935,0.71429,0.85707,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,9,0,0,16,0,1],[108,138,0.7826,0.83927,0.09942,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,6],[112,138,0.8116,0.76338,0.18073,0.71429,0.85707,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,17,0,1],[116,138,0.8406,0.75892,0.25364,0.82132,0.85714,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,20,0,4],[120,138,0.8696,0.76784,0.20747,0.71429,0.85707,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,9,0,0,17,0,3],[124,138,0.8986,0.81249,0.22142,0.85708,0.85714,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,5],[128,138,0.9275,0.81695,0.12492,0.85714,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,28,0,0],[132,138,0.9565,0.8482,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,1],[136,138,0.9855,0.81694,0.16457,0.857,0.85714,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,2],[138,138,1.0,0.80354,0.18814,0.857,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,22,0,3]]},{"b":6,"e":0.0,"k":"falling","v":0.40164,"x":0.84371,"p":[[0,81,0.0,0.72766,0.32997,0.42857,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,1,0,0,4,0,0,1,0,0,4,0,0,3,0,15],[4,81,0.0494,0.74999,0.22589,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,10,0,0,12,0,5],[8,81,0.0988,0.79016,0.19879,0.71429,0.857,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,11,0,0,10,0,8],[12,81,0.1481,0.84371,0.12561,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,13,0,9],[16,81,0.1975,0.83927,0.14619,0.82143,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,16,0,8],[20,81,0.2469,0.8125,0.21558,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,14,0,9],[24,81,0.2963,0.7991,0.23381,0.71429,0.85714,1.0,0.0,1.0,2,9,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,11,0,9],[28,81,0.3457,0.82586,0.1546,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,15,0,8],[32,81,0.3951,0.74551,0.27137,0.71429,0.85714,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,0,13,0,7],[36,81,0.4444,0.71204,0.26992,0.64286,0.85707,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,1,0,1,4,0,0,0,0,0,6,0,0,13,0,5],[40,81,0.4938,0.76784,0.24157,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,0,12,0,8],[44,81,0.5432,0.80803,0.15815,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,13,0,7],[48,81,0.5926,0.7098,0.28232,0.67846,0.85714,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,0,13,0,5],[52,81,0.642,0.74998,0.19234,0.67857,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,7,0,0,12,0,5],[56,81,0.6914,0.82587,0.19801,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,13],[60,81,0.7407,0.73214,0.2714,0.57142,0.85714,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,11,0,8],[64,81,0.7901,0.73213,0.2165,0.67857,0.857,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,6,0,0,14,0,4],[68,81,0.8395,0.75892,0.2126,0.71429,0.78564,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,10,0,0,8,0,8],[72,81,0.8889,0.61606,0.3415,0.42857,0.71429,0.85714,0.0,1.0,5,4,0,5,0,2,0,0,0,0,0,3,0,0,2,0,0,5,0,0,11,0,4],[76,81,0.9383,0.66517,0.33618,0.42857,0.85707,1.0,0.0,1.0,3,9,0,3,0,2,0,0,1,0,0,5,0,0,1,0,0,3,0,0,8,0,9],[80,81,0.9877,0.40164,0.31617,0.0,0.42857,0.71107,0.0,1.0,9,1,0,9,0,1,0,0,3,0,0,8,0,0,2,0,0,4,0,0,4,0,1],[81,81,1.0,0.40179,0.32031,0.0,0.42857,0.50002,0.0,1.0,9,2,0,9,0,1,0,0,1,0,0,13,0,0,0,0,0,2,0,0,4,0,2]]}]},{"i":"68cdc47d1e403b3d","q":"Find prime numbers  $p$ ,  $q$ ,  $r$  and  $s$ , pairwise distinct, such that their sum is prime number and numbers  $p^2+qr$  and  $p^2+qs$  are perfect squares","t":[{"b":1,"e":1.0,"k":"flat","v":0.90625,"x":0.96875,"p":[[0,10,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,10,0.4,0.92857,0.13833,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,21],[8,10,0.8,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[10,10,1.0,0.90625,0.06785,0.85714,0.85714,1.0,0.8571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11]]},{"b":4,"e":1.0,"k":"flat","v":0.94196,"x":0.97768,"p":[[0,17,0.0,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[4,17,0.2353,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[8,17,0.4706,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,17,0.7059,0.94643,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[16,17,0.9412,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[17,17,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"78a2a4def627770f","q":"Find the complex numbers  $ z$  for which the series \r\n\\[ 1 \\plus{} \\frac {z}{2!} \\plus{} \\frac {z(z \\plus{} 1)}{3!} \\plus{} \\frac {z(z \\plus{} 1)(z \\plus{} 2)}{4!} \\plus{} \\cdots \\plus{} \\frac {z(z \\plus{} 1)\\cdots(z \\plus{} n)}{(n \\plus{} 2)!} \\plus{} \\cdots\\]\r\nconverges and find its sum.","t":[{"b":5,"e":1.0,"k":"flat","v":0.88839,"x":0.95982,"p":[[0,7,0.0,0.88839,0.198,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,22],[4,7,0.5714,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[7,7,1.0,0.90177,0.18366,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,23]]},{"b":6,"e":0.71429,"k":"flat","v":0.89732,"x":0.96873,"p":[[0,14,0.0,0.90179,0.1729,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,21],[4,14,0.2857,0.89732,0.17582,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,22],[8,14,0.5714,0.94196,0.13767,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[12,14,0.8571,0.95982,0.11426,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[14,14,1.0,0.96873,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28]]}]},{"i":"311b24b29c125a75","q":"Let  $N$  be a positive integer. Suppose given any real  $x\\in (0,1)$  with decimal representation  $0.a_1a_2a_3a_4\\cdots$ , one can color the digits  $a_1,a_2,\\cdots$  with  $N$  colors so that the following hold:\n    1. each color is used at least once;\n    2. for any color, if we delete all the digits in  $x$  except those of this color, the resulting decimal number is rational.\nFind the least possible value of  $N$ .\n\n*~Sutanay Bhattacharya*","t":[{"b":1,"e":0.0,"k":"falling","v":0.04018,"x":0.43302,"p":[[0,50,0.0,0.40625,0.20238,0.28571,0.42857,0.57143,0.0,0.71429,4,0,2,4,0,1,0,0,5,0,0,11,0,0,8,0,0,3,0,0,0,0,0],[4,50,0.08,0.43302,0.358,0.0,0.42859,0.71429,0.0,1.0,9,5,0,9,0,2,0,0,2,0,0,6,0,0,4,0,0,2,0,0,2,0,5],[8,50,0.16,0.39732,0.36897,0.0,0.35714,0.71429,0.0,1.0,12,2,0,12,0,2,0,0,2,0,0,2,0,0,1,0,0,7,0,0,4,0,2],[12,50,0.24,0.23661,0.35104,0.0,0.0,0.2857,0.0,1.0,17,4,0,17,0,4,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[16,50,0.32,0.20982,0.32924,0.0,0.0,0.32143,0.0,1.0,20,2,0,20,0,2,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,2],[20,50,0.4,0.14732,0.27545,0.0,0.0,0.10714,0.0,1.0,24,1,0,24,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,1],[24,50,0.48,0.09375,0.23854,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[28,50,0.56,0.08036,0.22851,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[32,50,0.64,0.14284,0.31133,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[36,50,0.72,0.14285,0.25252,0.0,0.0,0.1786,0.0,0.71429,23,0,0,23,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,0,0,0,0],[40,50,0.8,0.06696,0.14279,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,50,0.88,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.45982,"p":[[0,41,0.0,0.40624,0.18594,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,11,0,0,5,0,0,10,0,0,1,0,0,1,0,0],[4,41,0.0976,0.45982,0.40521,0.0,0.42857,0.85704,0.0,1.0,11,7,0,11,0,2,0,0,1,0,0,3,0,0,1,0,0,5,0,0,2,0,7],[8,41,0.1951,0.30804,0.39787,0.0,0.0,0.60714,0.0,1.0,17,5,0,17,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,5],[12,41,0.2927,0.17857,0.30929,0.0,0.0,0.1786,0.0,1.0,21,2,0,21,0,3,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,2],[16,41,0.3902,0.20079,0.33476,0.0,0.0,0.21429,0.0,1.0,21,2,0,21,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,2],[20,41,0.4878,0.31696,0.34944,0.0,0.28571,0.71429,0.0,1.0,14,1,0,14,0,1,0,0,7,0,0,0,0,0,1,0,0,3,0,0,5,0,1],[24,41,0.5854,0.14731,0.26601,0.0,0.0,0.17857,0.0,0.85714,23,0,0,23,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0],[28,41,0.6829,0.17411,0.30667,0.0,0.0,0.21429,0.0,0.85714,23,0,0,23,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0],[32,41,0.7805,0.21429,0.30723,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,3,0,0,4,0,0,0,0,0,1,0,0,4,0,0,1,0,1],[36,41,0.878,0.07143,0.16366,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,41,0.9756,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8e6fc1f4dc2f7cdf","q":"Let  $P$  be a point in the interior of an acute triangle  $ABC$ , and let  $Q$  be its isogonal conjugate.  Denote by  $\\omega_P$  and  $\\omega_Q$  the circumcircles of triangles  $BPC$  and  $BQC$ , respectively.  Suppose the circle with diameter  $\\overline{AP}$  intersects  $\\omega_P$  again at  $M$ , and line  $AM$  intersects  $\\omega_P$  again at  $X$ .  Similarly, suppose the circle with diameter  $\\overline{AQ}$  intersects  $\\omega_Q$  again at  $N$ , and line  $AN$  intersects  $\\omega_Q$  again at  $Y$ .\n\nProve that lines  $MN$  and  $XY$  are parallel.\n\n(Here, the points  $P$  and  $Q$  are *isogonal conjugates* with respect to  $\\triangle ABC$  if the internal angle bisectors of  $\\angle BAC$ ,  $\\angle CBA$ , and  $\\angle ACB$  also bisect the angles  $\\angle PAQ$ ,  $\\angle PBQ$ , and  $\\angle PCQ$ , respectively.  For example, the orthocenter is the isogonal conjugate of the circumcenter.)\n\n*Proposed by Sammy Luo*","t":[{"b":4,"e":0.14286,"k":"rising","v":0.15626,"x":0.31696,"p":[[0,47,0.0,0.15626,0.15303,0.0,0.14286,0.1786,0.0,0.4286,11,0,0,11,0,13,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.29452,0.17838,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,11,0,0,13,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[8,47,0.1702,0.30803,0.17169,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,8,0,0,11,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[12,47,0.2553,0.28572,0.21429,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,11,0,0,7,0,0,5,0,0,1,0,0,4,0,0,0,0,0],[16,47,0.3404,0.28104,0.20673,0.14286,0.2857,0.42857,0.0,0.857,4,0,0,4,0,10,0,0,9,0,0,4,0,0,3,0,0,1,0,0,1,0,0],[20,47,0.4255,0.30786,0.18951,0.14286,0.2857,0.42858,0.0,0.71429,1,0,0,1,0,11,0,0,11,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[24,47,0.5106,0.31696,0.18117,0.25,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,12,0,0,9,0,0,0,0,0,3,0,0,0,0,0],[28,47,0.5957,0.28561,0.16372,0.14286,0.2857,0.32143,0.0,0.71429,2,0,0,2,0,9,0,0,13,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[32,47,0.6809,0.22759,0.17811,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,13,0,0,9,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[36,47,0.766,0.24106,0.19043,0.14286,0.2857,0.28571,0.0,0.71429,7,0,0,7,0,7,0,0,12,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[40,47,0.8511,0.25893,0.14913,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,7,0,0,19,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[44,47,0.9362,0.24552,0.18975,0.14286,0.2857,0.28571,0.0,0.85714,5,0,0,5,0,9,0,0,14,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[47,47,1.0,0.3125,0.18707,0.14286,0.28571,0.32143,0.0,0.71429,2,0,0,2,0,7,0,0,15,0,0,2,0,0,3,0,0,3,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.17402,"x":0.40175,"p":[[0,84,0.0,0.17402,0.13711,0.14214,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,16,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,84,0.0476,0.30354,0.17031,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,6,0,0,12,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[8,84,0.0952,0.35258,0.24225,0.14289,0.28571,0.57111,0.0,1.0,4,1,0,4,0,6,0,0,8,0,0,5,0,0,5,0,0,3,0,0,0,0,1],[12,84,0.1429,0.35712,0.20513,0.25,0.28571,0.4642,0.0,0.85714,1,0,0,1,0,7,0,0,12,0,0,4,0,0,4,0,0,3,0,0,1,0,0],[16,84,0.1905,0.33463,0.14576,0.2857,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,6,0,0,15,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[20,84,0.2381,0.26784,0.17765,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,7,0,0,16,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[24,84,0.2857,0.29469,0.16731,0.14289,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,9,0,0,10,0,0,9,0,0,0,0,0,2,0,0,0,0,0],[28,84,0.3333,0.31695,0.17027,0.2857,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,2,0,0,15,0,0,5,0,0,6,0,0,0,0,0,0,0,0],[32,84,0.381,0.27677,0.18187,0.14286,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,4,0,0,17,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[36,84,0.4286,0.25,0.13363,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,11,0,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[40,84,0.4762,0.29016,0.17305,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,13,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[44,84,0.5238,0.32143,0.20203,0.14289,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,5,0,0,11,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[48,84,0.5714,0.25893,0.19045,0.14286,0.2143,0.32143,0.0,0.71429,4,0,0,4,0,12,0,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[52,84,0.619,0.32589,0.20589,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,10,0,0,6,0,0,2,0,0,4,0,0,0,0,0],[56,84,0.6667,0.35714,0.24222,0.14286,0.28571,0.57143,0.0,1.0,1,1,0,1,0,11,0,0,8,0,0,3,0,0,3,0,0,5,0,0,0,0,1],[60,84,0.7143,0.29454,0.13811,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,14,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[64,84,0.7619,0.299,0.17267,0.14286,0.2857,0.32143,0.0,0.71429,1,0,0,1,0,10,0,0,13,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[68,84,0.8095,0.3214,0.15147,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,16,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[72,84,0.8571,0.30804,0.18249,0.1429,0.28571,0.32143,0.0,0.71429,2,0,0,2,0,7,0,0,15,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[76,84,0.9048,0.30354,0.18119,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,6,0,0,14,0,0,4,0,0,3,0,0,2,0,0,0,0,0],[80,84,0.9524,0.25428,0.15052,0.14286,0.2857,0.28571,0.0,0.857,1,0,0,1,0,13,0,0,13,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[84,84,1.0,0.40175,0.21849,0.25,0.42857,0.57111,0.0,0.71429,2,0,0,2,0,6,0,0,5,0,0,8,0,0,5,0,0,6,0,0,0,0,0]]}]},{"i":"96f6166ba1ac57a7","q":"If  $a$  and  $b$  are positive integers such that\n\\[\n  \\sqrt{8 + \\sqrt{32 + \\sqrt{768}}} = a \\cos \\frac{\\pi}{b} \\, ,\n\\]\ncompute the ordered pair  $(a, b)$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.83927,"x":1.0,"p":[[0,28,0.0,0.83927,0.14176,0.85711,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,17,0,8],[4,28,0.1429,0.84374,0.13054,0.85711,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,19,0,7],[8,28,0.2857,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[12,28,0.4286,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[16,28,0.5714,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[20,28,0.7143,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":0.85714,"k":"flat","v":0.68304,"x":0.84821,"p":[[0,39,0.0,0.83034,0.13095,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,21,0,5],[4,39,0.1026,0.77231,0.12301,0.71429,0.85714,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,17,0,1],[8,39,0.2051,0.72768,0.15714,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,11,0,0,7,0,4],[12,39,0.3077,0.68304,0.17029,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,12,0,0,7,0,2],[16,39,0.4103,0.70088,0.17986,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,11,0,0,6,0,4],[20,39,0.5128,0.73213,0.19151,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,7,0,0,10,0,5],[24,39,0.6154,0.82142,0.13363,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,18,0,5],[28,39,0.7179,0.79909,0.15512,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,12,0,7],[32,39,0.8205,0.80356,0.13243,0.71429,0.857,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,11,0,6],[36,39,0.9231,0.84821,0.13333,0.82132,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,15,0,9],[39,39,1.0,0.7857,0.14288,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,15,0,4]]}]},{"i":"f1622a0356736ea7","q":"If $a, b, c$ are positive real numbers such that $a b c=1$, prove that\n\n$$\na^{b+c} b^{c+a} c^{a+b} \\leq 1\n$$","t":[{"b":2,"e":0.71429,"k":"flat","v":0.6339,"x":0.70089,"p":[[0,24,0.0,0.65179,0.11258,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,21,0,0,1,0,0],[4,24,0.1667,0.6339,0.12847,0.571,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,22,0,0,0,0,0],[8,24,0.3333,0.67411,0.10249,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,27,0,0,0,0,0],[12,24,0.5,0.66517,0.11071,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,26,0,0,0,0,0],[16,24,0.6667,0.70089,0.08268,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,27,0,0,2,0,0],[20,24,0.8333,0.67857,0.09449,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,28,0,0,0,0,0],[24,24,1.0,0.67408,0.10251,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,27,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.63393,"x":0.69642,"p":[[0,31,0.0,0.63393,0.13333,0.42857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,21,0,0,1,0,0],[4,31,0.129,0.66518,0.16214,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,22,0,0,2,0,1],[8,31,0.2581,0.63839,0.12364,0.53571,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,23,0,0,0,0,0],[12,31,0.3871,0.69196,0.08073,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,27,0,0,1,0,0],[16,31,0.5161,0.63393,0.12846,0.57142,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,22,0,0,0,0,0],[20,31,0.6452,0.6875,0.0974,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,25,0,0,2,0,0],[24,31,0.7742,0.66504,0.13646,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,26,0,0,1,0,0],[28,31,0.9032,0.69642,0.05925,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0],[31,31,1.0,0.66964,0.09062,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,25,0,0,0,0,0]]}]},{"i":"1004a096744a11cb","q":"Let  $P_1 \\cdots P_{27}$  be a regular  $27$ -gon. How many triples of positive integers  $(a, b, c)$  with  $1 \\le a < b < c \\le 27$  are there such that  $P_a P_b P_c$  is acute? $\\textit{Remark:}$   $P_a P_b P_c$  refers to a triangle, not an angle.","t":[{"b":0,"e":0.4286,"k":"flat","v":0.46428,"x":0.71875,"p":[[0,34,0.0,0.47767,0.41436,0.14286,0.28571,1.0,0.0,1.0,7,10,0,7,0,6,0,0,5,0,0,0,0,0,0,0,0,3,0,0,1,0,10],[4,34,0.1176,0.71875,0.34346,0.28571,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,8,0,0,2,0,0,0,0,0,1,0,0,1,0,18],[8,34,0.2353,0.66964,0.35792,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,7,0,0,3,0,0,0,0,0,1,0,0,1,0,16],[12,34,0.3529,0.5,0.32341,0.28571,0.28571,0.75,0.0,1.0,1,7,0,1,0,4,0,0,12,0,0,2,0,0,2,0,0,3,0,0,1,0,7],[16,34,0.4706,0.46428,0.3312,0.14286,0.28571,0.71429,0.0,1.0,1,6,0,1,0,8,0,0,10,0,0,0,0,0,2,0,0,4,0,0,1,0,6],[20,34,0.5882,0.5625,0.32131,0.28571,0.5,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,9,0,0,3,0,0,4,0,0,2,0,0,1,0,9],[24,34,0.7059,0.62946,0.34415,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,6,0,0,3,0,0,1,0,0,3,0,0,2,0,12],[28,34,0.8235,0.68304,0.30459,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,5,0,0,7,0,0,3,0,0,0,0,0,3,0,13],[32,34,0.9412,0.48214,0.29613,0.2857,0.42857,0.71429,0.0,1.0,1,5,0,1,0,4,0,0,10,0,0,5,0,0,2,0,0,4,0,0,1,0,5],[34,34,1.0,0.47768,0.28259,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,6,0,0,7,0,0,8,0,0,3,0,0,1,0,0,3,0,4]]},{"b":6,"e":0.42857,"k":"falling","v":0.31696,"x":0.63393,"p":[[0,73,0.0,0.63393,0.39113,0.28571,0.85714,1.0,0.0,1.0,4,13,0,4,0,3,0,0,5,0,0,0,0,0,1,0,0,2,0,0,4,0,13],[4,73,0.0548,0.60713,0.35893,0.28571,0.57121,1.0,0.0,1.0,1,12,0,1,0,3,0,0,10,0,0,1,0,0,2,0,0,0,0,0,3,0,12],[8,73,0.1096,0.58929,0.3549,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,6,0,0,2,0,0,2,0,0,2,0,0,2,0,11],[12,73,0.1644,0.62946,0.33666,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,11,0,0,2,0,0,0,0,0,3,0,0,2,0,12],[16,73,0.2192,0.51339,0.36221,0.14286,0.35714,0.85714,0.0,1.0,3,6,0,3,0,6,0,0,7,0,0,1,0,0,1,0,0,2,0,0,6,0,6],[20,73,0.274,0.53125,0.36111,0.2857,0.35714,1.0,0.0,1.0,2,9,0,2,0,5,0,0,9,0,0,2,0,0,0,0,0,3,0,0,2,0,9],[24,73,0.3288,0.57143,0.36596,0.28571,0.50001,1.0,0.0,1.0,2,11,0,2,0,4,0,0,8,0,0,2,0,0,2,0,0,1,0,0,2,0,11],[28,73,0.3836,0.58036,0.37617,0.2857,0.50001,1.0,0.0,1.0,2,13,0,2,0,5,0,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,13],[32,73,0.4384,0.52231,0.33808,0.14289,0.42857,0.85714,0.0,1.0,1,5,0,1,0,8,0,0,5,0,0,3,0,0,1,0,0,3,0,0,6,0,5],[36,73,0.4932,0.59375,0.37306,0.28571,0.64286,1.0,0.0,1.0,3,11,0,3,0,2,0,0,9,0,0,1,0,0,1,0,0,1,0,0,4,0,11],[40,73,0.5479,0.6249,0.32892,0.28571,0.64286,1.0,0.14,1.0,0,10,0,0,0,4,0,0,7,0,0,2,0,0,3,0,0,2,0,0,4,0,10],[44,73,0.6027,0.4732,0.33775,0.25,0.28571,0.85714,0.0,1.0,2,5,0,2,0,6,0,0,10,0,0,2,0,0,1,0,0,1,0,0,5,0,5],[48,73,0.6575,0.53125,0.343,0.28571,0.28571,1.0,0.0,1.0,1,9,0,1,0,4,0,0,12,0,0,0,0,0,3,0,0,2,0,0,1,0,9],[52,73,0.7123,0.58035,0.29653,0.28571,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,7,0,0,3,0,0,2,0,0,6,0,0,5,0,5],[56,73,0.7671,0.5491,0.29689,0.28571,0.57121,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,9,0,0,2,0,0,4,0,0,4,0,0,4,0,5],[60,73,0.8219,0.60705,0.35904,0.2857,0.64286,1.0,0.0,1.0,1,12,0,1,0,6,0,0,4,0,0,3,0,0,2,0,0,3,0,0,1,0,12],[64,73,0.8767,0.48661,0.31916,0.2857,0.28571,0.85714,0.14286,1.0,0,5,0,0,0,6,0,0,13,0,0,1,0,0,0,0,0,3,0,0,4,0,5],[68,73,0.9315,0.39284,0.26725,0.24999,0.28571,0.42857,0.0,1.0,2,3,0,2,0,6,0,0,9,0,0,8,0,0,2,0,0,1,0,0,1,0,3],[72,73,0.9863,0.31696,0.23619,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,10,0,0,11,0,0,5,0,0,0,0,0,1,0,0,2,0,1],[73,73,1.0,0.37945,0.23584,0.24999,0.28571,0.46418,0.0,0.85714,1,0,0,1,0,7,0,0,11,0,0,5,0,0,3,0,0,1,0,0,4,0,0]]}]},{"i":"cbbd9d4e24b09e3e","q":"Let  $S$  be a nonempty finite set, and  $\\mathcal {F}$  be a collection of subsets of  $S$  such that the following conditions are met:\n(i)  $\\mathcal {F}$   $\\setminus$  { $S$ }  $\\neq$   $\\emptyset$  ;\n(ii) if  $F_1, F_2 \\in \\mathcal {F}$ , then  $F_1 \\cap F_2 \\in \\mathcal {F}$  and  $F_1 \\cup F_2 \\in \\mathcal {F}$ .\n\nProve that there exists  $a \\in S$  which belongs to at most half of the elements of  $\\mathcal {F}$ .","t":[{"b":6,"e":0.71429,"k":"rising","v":0.04018,"x":0.66517,"p":[[0,63,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,4,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.34821,0.3173,0.0,0.28571,0.71429,0.0,0.85714,11,0,0,11,0,3,0,0,3,0,0,3,0,0,3,0,0,6,0,0,3,0,0],[8,63,0.127,0.41072,0.37415,0.0,0.35714,0.75,0.0,1.0,9,5,1,9,0,5,0,0,2,0,0,4,0,0,2,0,0,2,0,0,3,0,5],[12,63,0.1905,0.44639,0.34021,0.14286,0.49979,0.60714,0.0,1.0,6,5,0,6,0,5,0,0,3,0,0,2,0,0,8,0,0,2,0,0,1,0,5],[16,63,0.254,0.45534,0.30605,0.14286,0.42857,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,3,0,0,5,0,0,5,0,0,5,0,0,3,0,2],[20,63,0.3175,0.41518,0.30589,0.14286,0.42857,0.71429,0.0,1.0,6,2,0,6,0,5,0,0,3,0,0,4,0,0,5,0,0,6,0,0,1,0,2],[24,63,0.381,0.46426,0.36596,0.0,0.57121,0.71429,0.0,1.0,9,4,0,9,0,2,0,0,3,0,0,0,0,0,5,0,0,6,0,0,3,0,4],[28,63,0.4444,0.49554,0.28344,0.24999,0.50001,0.71429,0.0,1.0,2,3,0,2,0,6,0,0,1,0,0,7,0,0,5,0,0,7,0,0,1,0,3],[32,63,0.5079,0.50442,0.34066,0.14286,0.57121,0.71429,0.0,1.0,4,6,0,4,0,7,0,0,0,0,0,2,0,0,7,0,0,6,0,0,0,0,6],[36,63,0.5714,0.45534,0.3719,0.14286,0.42857,0.85714,0.0,1.0,6,6,0,6,0,7,0,0,2,0,0,3,0,0,3,0,0,2,0,0,3,0,6],[40,63,0.6349,0.3482,0.32524,0.14286,0.21428,0.57143,0.0,1.0,6,3,0,6,0,10,0,0,5,0,0,1,0,0,3,0,0,2,0,0,2,0,3],[44,63,0.6984,0.37054,0.34967,0.0,0.28571,0.60714,0.0,1.0,10,3,0,10,0,4,0,0,4,0,0,2,0,0,4,0,0,2,0,0,3,0,3],[48,63,0.7619,0.43293,0.35985,0.14214,0.42857,0.71429,0.0,1.0,7,5,0,7,0,7,0,0,0,0,0,4,0,0,3,0,0,5,0,0,1,0,5],[52,63,0.8254,0.30801,0.31963,0.0,0.14286,0.57111,0.0,1.0,11,2,0,11,0,6,0,0,3,0,0,2,0,0,4,0,0,3,0,0,1,0,2],[56,63,0.8889,0.52232,0.30849,0.28571,0.57143,0.71429,0.0,1.0,4,4,0,4,0,2,0,0,5,0,0,2,0,0,5,0,0,9,0,0,1,0,4],[60,63,0.9524,0.66517,0.21903,0.57143,0.71429,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,1,0,0,2,0,0,4,0,0,18,0,0,1,0,4],[63,63,1.0,0.66515,0.09854,0.67857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,24,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.44639,"p":[[0,92,0.0,0.12946,0.20316,0.0,0.0,0.14286,0.0,0.85714,17,0,6,17,0,10,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[4,92,0.0435,0.42857,0.34069,0.10714,0.57143,0.71429,0.0,0.85714,8,0,0,8,0,5,0,0,2,0,0,0,0,0,5,0,0,5,0,0,7,0,0],[8,92,0.087,0.35712,0.3388,0.10714,0.21428,0.71429,0.0,1.0,8,2,0,8,0,8,0,0,4,0,0,1,0,0,1,0,0,5,0,0,3,0,2],[12,92,0.1304,0.32589,0.30142,0.0,0.28571,0.57143,0.0,0.85714,11,0,0,11,0,4,0,0,2,0,0,3,0,0,7,0,0,2,0,0,3,0,0],[16,92,0.1739,0.31249,0.36671,0.0,0.14286,0.57143,0.0,1.0,15,4,0,15,0,3,0,0,1,0,0,3,0,0,3,0,0,2,0,0,1,0,4],[20,92,0.2174,0.31694,0.26177,0.0,0.28571,0.571,0.0,0.85714,9,0,0,9,0,4,0,0,4,0,0,6,0,0,5,0,0,3,0,0,1,0,0],[24,92,0.2609,0.44639,0.27373,0.14286,0.571,0.71429,0.0,0.85714,4,0,0,4,0,5,0,0,4,0,0,1,0,0,8,0,0,8,0,0,2,0,0],[28,92,0.3043,0.20089,0.22548,0.0,0.14286,0.32143,0.0,0.85714,13,0,0,13,0,7,0,0,4,0,0,4,0,0,3,0,0,0,0,0,1,0,0],[32,92,0.3478,0.07588,0.15557,0.0,0.0,0.03571,0.0,0.571,24,0,0,24,0,4,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,92,0.3913,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,92,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,92,0.4783,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,92,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,92,0.5652,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,92,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,92,0.6522,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,92,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,92,0.7391,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,92,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,92,0.8261,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,92,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,92,0.913,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,92,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,92,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ae91a805b9d4dc3d","q":"Let  $a_0,b_0$  be positive integers, and define  $a_{i+1}=a_i+\\lfloor\\sqrt{b_i}\\rfloor$  and  $b_{i+1}=b_i+\\lfloor\\sqrt{a_i}\\rfloor$  for all  $i\\ge0$ . Show that there exists a positive integer  $n$  such that  $a_n=b_n$ .\n\n*David Yang.*","t":[{"b":2,"e":1.0,"k":"rising","v":0.61158,"x":0.87946,"p":[[0,43,0.0,0.61158,0.31387,0.571,0.71429,0.85714,0.0,1.0,5,3,4,5,0,1,0,0,0,0,0,1,0,0,6,0,0,8,0,0,8,0,3],[4,43,0.093,0.87946,0.23176,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,6,0,21],[8,43,0.186,0.808,0.20081,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,0,14,0,9],[12,43,0.2791,0.76338,0.23314,0.67857,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,4,0,11],[16,43,0.3721,0.74998,0.2113,0.57143,0.857,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,5,0,0,10,0,7],[20,43,0.4651,0.76338,0.24901,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,5,0,12],[24,43,0.5581,0.73658,0.21164,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,6,0,0,8,0,7],[28,43,0.6512,0.75443,0.21795,0.67857,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,0,13,0,6],[32,43,0.7442,0.73659,0.1927,0.57143,0.85707,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,6,0,0,14,0,3],[36,43,0.8372,0.71428,0.17857,0.71429,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,15,0,0,9,0,2],[40,43,0.9302,0.79908,0.13772,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,14,0,5],[43,43,1.0,0.77232,0.07873,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,1]]},{"b":4,"e":0.85714,"k":"flat","v":0.63836,"x":0.80802,"p":[[0,30,0.0,0.63836,0.3369,0.571,0.71429,0.85714,0.0,1.0,6,5,5,6,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,9,0,5],[4,30,0.1333,0.79018,0.14719,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,14,0,5],[8,30,0.2667,0.80802,0.16984,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,9,0,0,10,0,9],[12,30,0.4,0.79908,0.15514,0.71429,0.78571,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,7,0,9],[16,30,0.5333,0.79018,0.17122,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,14,0,6],[20,30,0.6667,0.76784,0.1627,0.71429,0.85707,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,13,0,4],[24,30,0.8,0.72768,0.16506,0.71429,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,13,0,0,10,0,2],[28,30,0.9333,0.63839,0.22864,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,1,0,0,3,0,0,3,0,0,15,0,0,6,0,1],[30,30,1.0,0.78125,0.15561,0.71429,0.78564,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,0,10,0,6]]}]},{"i":"cb94b8d8c0335e93","q":"Let  $\\Omega$   be the inscribed circle of a triangle  $\\vartriangle ABC$ . Let  $D, E$  and  $F$  be the tangency points of  $\\Omega$   and the sides  $BC, CA$  and  $AB$ , respectively, and let  $AD, BE$  and  $CF$  intersect  $\\Omega$  at  $K, L$  and  $M$ , respectively, such that  $D, E, F, K, L$  and  $M$  are all distinct. The tangent line of  $\\Omega$   at  $K$  intersects  $EF$  at  $X$ , the tangent line of  $\\Omega$   at  $L$  intersects  $DE$  at  $Y$  , and the tangent line of  $\\Omega$   at M intersects  $DF$  at  $Z$ . Prove that  $X,Y$  and  $Z$  are collinear.","t":[{"b":0,"e":1.0,"k":"rising","v":0.66962,"x":1.0,"p":[[0,51,0.0,0.66962,0.24599,0.42857,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,10,0,0,8,0,0,2,0,0,1,0,10],[4,51,0.0784,0.77231,0.27167,0.67846,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0,6,0,0,3,0,15],[8,51,0.1569,0.77676,0.33871,0.571,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,1,0,0,2,0,0,3,0,0,1,0,0,0,0,21],[12,51,0.2353,0.80802,0.27805,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,0,3,0,19],[16,51,0.3137,0.84821,0.26229,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,20],[20,51,0.3922,0.82143,0.26487,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,4,0,0,1,0,20],[24,51,0.4706,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,51,0.549,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[32,51,0.6275,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,51,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,51,0.9412,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[51,51,1.0,0.96879,0.11122,1.0,1.0,1.0,0.43,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29]]},{"b":7,"e":1.0,"k":"rising","v":0.59375,"x":0.99554,"p":[[0,43,0.0,0.59819,0.20959,0.42857,0.57143,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,9,0,0,14,0,0,0,0,0,2,0,5],[4,43,0.093,0.90179,0.22142,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[8,43,0.186,0.8125,0.29545,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,21],[12,43,0.2791,0.67411,0.38504,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,8,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,17],[16,43,0.3721,0.70089,0.37519,0.28571,1.0,1.0,0.0,1.0,2,18,0,2,0,4,0,0,3,0,0,2,0,0,0,0,0,3,0,0,0,0,18],[20,43,0.4651,0.60705,0.38642,0.2857,0.57144,1.0,0.0,1.0,3,14,0,3,0,4,0,0,4,0,0,5,0,0,0,0,0,1,0,0,1,0,14],[24,43,0.5581,0.59375,0.39303,0.24999,0.64284,1.0,0.0,1.0,5,13,0,5,0,3,0,0,2,0,0,5,0,0,1,0,0,2,0,0,1,0,13],[28,43,0.6512,0.62045,0.41906,0.14286,0.92857,1.0,0.0,1.0,4,16,0,4,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,16],[32,43,0.7442,0.71875,0.38711,0.5,1.0,1.0,0.0,1.0,4,19,0,4,0,3,0,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,19],[36,43,0.8372,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,43,0.9302,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[43,43,1.0,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30]]}]},{"i":"67185e254a668476","q":"Let  $a_0, b_0, c_0$   be complex numbers, and define \\begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n  b_{n+1} &= b_n^2 + 2c_na_n  c_{n+1} &= c_n^2 + 2a_nb_n\\end{align*}for all nonnegative integers  $n.$  \n\nSuppose that  $\\max{\\{|a_n|, |b_n|, |c_n|\\}} \\leq 2022$  for all  $n.$  Prove that   $$ |a_0|^2 + |b_0|^2 + |c_0|^2 \\leq 1. $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.26786,"x":0.94643,"p":[[0,41,0.0,0.26786,0.31693,0.0,0.14286,0.32143,0.0,1.0,12,3,0,12,0,6,0,0,6,0,0,2,0,0,0,0,0,3,0,0,0,0,3],[4,41,0.0976,0.93304,0.21124,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[8,41,0.1951,0.90625,0.23854,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,26],[12,41,0.2927,0.89731,0.22373,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,26],[16,41,0.3902,0.83928,0.28065,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,21],[20,41,0.4878,0.83929,0.30671,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,23],[24,41,0.5854,0.92411,0.21124,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[28,41,0.6829,0.86607,0.28107,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,24],[32,41,0.7805,0.87945,0.26754,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,25],[36,41,0.878,0.94643,0.17767,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[40,41,0.9756,0.84374,0.26813,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,4,0,21],[41,41,1.0,0.83027,0.33414,1.0,1.0,1.0,0.14,1.0,0,25,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,25]]},{"b":7,"e":1.0,"k":"rising","v":0.24107,"x":0.98214,"p":[[0,79,0.0,0.24107,0.35072,0.0,0.0,0.32143,0.0,1.0,17,4,0,17,0,5,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[4,79,0.0506,0.94642,0.13717,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[8,79,0.1013,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[12,79,0.1519,0.86161,0.28901,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,23],[16,79,0.2025,0.83929,0.27141,0.82143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0,5,0,19],[20,79,0.2532,0.90179,0.21558,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,23],[24,79,0.3038,0.86607,0.21706,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,21],[28,79,0.3544,0.78125,0.335,0.42859,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,3,0,20],[32,79,0.4051,0.79911,0.309,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,1,0,0,2,0,0,0,0,0,5,0,0,2,0,19],[36,79,0.4557,0.82589,0.28287,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,3,0,20],[40,79,0.5063,0.84374,0.26332,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,21],[44,79,0.557,0.79464,0.3213,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,4,0,19],[48,79,0.6076,0.82143,0.28347,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,20],[52,79,0.6582,0.75892,0.30606,0.5354,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,3,0,0,2,0,0,2,0,0,4,0,16],[56,79,0.7089,0.74552,0.33832,0.5354,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,18],[60,79,0.7595,0.85714,0.31944,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[64,79,0.8101,0.77679,0.29001,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,17],[68,79,0.8608,0.95089,0.16602,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[72,79,0.9114,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[76,79,0.962,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[79,79,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"1d09f94e11cbef4d","q":"Let  $\\mathcal{S}$  be the set of all positive integers which are both a multiple of  $3$  and have at least one digit that is a  $1$ . For example,  $123$  is in  $\\mathcal{S}$  and  $450$  is not. The probability that a randomly chosen  $3$ -digit positive integer is in  $\\mathcal{S}$  can be written as  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .\n\n*Proposed by GammaZero*","t":[{"b":0,"e":0.28571,"k":"flat","v":0.29464,"x":0.45536,"p":[[0,111,0.0,0.38839,0.16458,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,21,0,0,2,0,0,8,0,0,0,0,0,0,0,1],[4,111,0.036,0.45536,0.26351,0.28571,0.28571,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,21,0,0,0,0,0,5,0,0,1,0,0,0,0,5],[8,111,0.0721,0.40178,0.19704,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,0,0,0,8,0,0,0,0,0,0,0,2],[12,111,0.1081,0.38393,0.21558,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,25,0,0,1,0,0,3,0,0,0,0,0,0,0,3],[16,111,0.1441,0.40625,0.19269,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,20,0,0,3,0,0,7,0,0,0,0,0,0,0,2],[20,111,0.1802,0.45088,0.26027,0.28571,0.28571,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,21,0,0,0,0,0,6,0,0,0,0,0,0,0,5],[24,111,0.2162,0.34821,0.11811,0.28571,0.28571,0.28571,0.28571,0.57143,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[28,111,0.2523,0.39286,0.21724,0.28571,0.28571,0.32143,0.2857,1.0,0,3,0,0,0,0,0,0,24,0,0,1,0,0,4,0,0,0,0,0,0,0,3],[32,111,0.2883,0.29911,0.05486,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,111,0.3243,0.38393,0.19377,0.28571,0.28571,0.35714,0.2857,1.0,0,2,0,0,0,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,2],[40,111,0.3604,0.33482,0.14555,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[44,111,0.3964,0.31251,0.08328,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[48,111,0.4324,0.33928,0.11152,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[52,111,0.4685,0.33036,0.10374,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[56,111,0.5045,0.3125,0.08328,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[60,111,0.5405,0.36159,0.15964,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,0,0,0,6,0,0,0,0,0,0,0,1],[64,111,0.5766,0.29464,0.04971,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,111,0.6126,0.35268,0.15561,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,26,0,0,0,0,0,5,0,0,0,0,0,0,0,1],[72,111,0.6486,0.33928,0.11152,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[76,111,0.6847,0.36161,0.12364,0.28571,0.28571,0.46429,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,1,0,0,8,0,0,0,0,0,0,0,0],[80,111,0.7207,0.33929,0.10564,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[84,111,0.7568,0.33482,0.10479,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[88,111,0.7928,0.36607,0.11259,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,20,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[92,111,0.8288,0.31251,0.07523,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[96,111,0.8649,0.34821,0.11811,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[100,111,0.9009,0.36607,0.17105,0.28571,0.28571,0.46429,0.14286,1.0,0,1,0,0,0,2,0,0,21,0,0,1,0,0,7,0,0,0,0,0,0,0,1],[104,111,0.9369,0.37054,0.12299,0.28571,0.28571,0.46431,0.2857,0.57143,0,0,0,0,0,0,0,0,21,0,0,3,0,0,8,0,0,0,0,0,0,0,0],[108,111,0.973,0.37946,0.13651,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,21,0,0,2,0,0,8,0,0,1,0,0,0,0,0],[111,111,1.0,0.33928,0.11152,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.33036,"x":0.47321,"p":[[0,101,0.0,0.47321,0.27534,0.28571,0.28571,0.57143,0.2857,1.0,0,6,0,0,0,0,0,0,20,0,0,0,0,0,6,0,0,0,0,0,0,0,6],[4,101,0.0396,0.40178,0.21852,0.28571,0.28571,0.46429,0.2857,1.0,0,3,0,0,0,0,0,0,23,0,0,1,0,0,5,0,0,0,0,0,0,0,3],[8,101,0.0792,0.4107,0.19804,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,21,0,0,0,0,0,9,0,0,0,0,0,0,0,2],[12,101,0.1188,0.45982,0.2618,0.28571,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,1,0,0,18,0,0,1,0,0,7,0,0,0,0,0,0,0,5],[16,101,0.1584,0.39285,0.19233,0.28571,0.28571,0.46428,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,2,0,0,6,0,0,0,0,0,0,0,2],[20,101,0.198,0.42411,0.22156,0.28571,0.28571,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,21,0,0,0,0,0,8,0,0,0,0,0,0,0,3],[24,101,0.2376,0.35713,0.15566,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[28,101,0.2772,0.42411,0.20355,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,20,0,0,0,0,0,9,0,0,1,0,0,0,0,2],[32,101,0.3168,0.4375,0.21998,0.28571,0.28571,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,19,0,0,1,0,0,9,0,0,0,0,0,0,0,3],[36,101,0.3564,0.37499,0.19148,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,2],[40,101,0.396,0.36607,0.12846,0.28571,0.28571,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[44,101,0.4356,0.3616,0.19228,0.28571,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,1,0,0,25,0,0,0,0,0,4,0,0,0,0,0,0,0,2],[48,101,0.4752,0.37946,0.16602,0.28571,0.28571,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,8,0,0,0,0,0,0,0,1],[52,101,0.5149,0.375,0.1915,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,2],[56,101,0.5545,0.38839,0.16841,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,22,0,0,0,0,0,9,0,0,0,0,0,0,0,1],[60,101,0.5941,0.37947,0.16602,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,8,0,0,0,0,0,0,0,1],[64,101,0.6337,0.33482,0.10479,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[68,101,0.6733,0.33036,0.10374,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[72,101,0.7129,0.33482,0.10479,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[76,101,0.7525,0.37052,0.1631,0.28571,0.28571,0.35704,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,0,0,1],[80,101,0.7921,0.34375,0.15093,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[84,101,0.8317,0.36607,0.12846,0.28571,0.28571,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[88,101,0.8713,0.39286,0.13832,0.28571,0.28571,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,20,0,0,0,0,0,12,0,0,0,0,0,0,0,0],[92,101,0.9109,0.3482,0.11809,0.28571,0.28571,0.28571,0.28571,0.57143,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[96,101,0.9505,0.35714,0.12372,0.28571,0.28571,0.35714,0.2857,0.57143,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[100,101,0.9901,0.37053,0.16312,0.28571,0.28571,0.35714,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0,0,0,1],[101,101,1.0,0.33928,0.11152,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"2f188a690c9715fd","q":"Let  $k$  be a positive integer. Show that there are infinitely many positive integer solutions  $(m, n)$  to $(m - n)^2 = kmn + m + n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.94642,"x":1.0,"p":[[0,53,0.0,0.96428,0.12878,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,53,0.0755,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,53,0.1509,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[12,53,0.2264,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,53,0.3019,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[20,53,0.3774,0.94642,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[24,53,0.4528,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[28,53,0.5283,0.94643,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[32,53,0.6038,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,53,0.6792,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,53,0.7547,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,53,0.8302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,53,0.9057,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,53,0.9811,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[53,53,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":0.85714,"k":"flat","v":0.80354,"x":0.99554,"p":[[0,43,0.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[4,43,0.093,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,43,0.186,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,43,0.2791,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,43,0.3721,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[20,43,0.4651,0.9375,0.18877,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[24,43,0.5581,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,43,0.6512,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[32,43,0.7442,0.92409,0.11841,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,20],[36,43,0.8372,0.86158,0.10407,0.85714,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,21,0,7],[40,43,0.9302,0.80354,0.13718,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,19,0,4],[43,43,1.0,0.8571,0.14733,0.85711,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,13,0,12]]}]},{"i":"a21758c016bf5089","q":"Let  $f : \\mathbb{Z} \\to \\mathbb{Z}$  be a function satisfying  $f(0) \\ne 0$ ,  $f(1) = 0$  and  $(i) f(xy) + f(x)f(y) = f(x) + f(y)$  $(ii)\\left(f(x-y) - f(0)\\right ) f(x)f(y) = 0 $ for all  $x,y \\in \\mathbb{Z}$ , simultaneously. $(a)$  Find the set of all possible values of the function  $f$ . $(b)$  If  $f(10) \\ne 0$  and  $f(2) = 0$ , find the set of all integers  $n$  such that  $f(n) \\ne 0$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.87053,"x":0.9241,"p":[[0,64,0.0,0.87053,0.14445,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,15,0,12],[4,64,0.0625,0.88392,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,13],[8,64,0.125,0.89732,0.10249,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,13],[12,64,0.1875,0.88393,0.14033,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,15,0,13],[16,64,0.25,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[20,64,0.3125,0.88393,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,10],[24,64,0.375,0.88839,0.12745,0.82143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16],[28,64,0.4375,0.90178,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[32,64,0.5,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[36,64,0.5625,0.90179,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,14],[40,64,0.625,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[44,64,0.6875,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[48,64,0.75,0.88392,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,11],[52,64,0.8125,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[56,64,0.875,0.88839,0.15865,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,14],[60,64,0.9375,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[64,64,1.0,0.88392,0.12078,0.85708,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,11,0,14]]},{"b":3,"e":1.0,"k":"flat","v":0.7991,"x":0.91071,"p":[[0,70,0.0,0.7991,0.15511,0.71429,0.85714,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,11,0,7],[4,70,0.0571,0.875,0.12242,0.82132,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,13],[8,70,0.1143,0.90625,0.09852,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,14],[12,70,0.1714,0.88839,0.11701,0.85714,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,14],[16,70,0.2286,0.89732,0.10853,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,15],[20,70,0.2857,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,12],[24,70,0.3429,0.85267,0.14934,0.82132,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,13,0,11],[28,70,0.4,0.85714,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,11,0,12],[32,70,0.4571,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[36,70,0.5143,0.83927,0.10564,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,17,0,6],[40,70,0.5714,0.83479,0.12933,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,16,0,7],[44,70,0.6286,0.88839,0.09932,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,12],[48,70,0.6857,0.87946,0.1017,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,15,0,11],[52,70,0.7429,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,18,0,11],[56,70,0.8,0.85268,0.09771,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,20,0,6],[60,70,0.8571,0.84375,0.12037,0.82132,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,17,0,7],[64,70,0.9143,0.85254,0.11016,0.71429,0.85714,1.0,0.71,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,13,0,9],[68,70,0.9714,0.875,0.11152,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,11],[70,70,1.0,0.88839,0.11701,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,13]]}]},{"i":"cd1f4e0a69d7a31e","q":"Let $A, B, C, D$ and $E$ be five points located in this order on a circle $\\Omega$, such that $(CD)$ is parallel to $(BE)$ and $(AB)$ is parallel to $(DE)$. Let $X, Y$ and $Z$ be the midpoints of the segments $[BD]$, $[CE]$ and $[AE]$ respectively.\nProve that the line $(AE)$ is tangent to the circumcircle of $XYZ$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.87946,"p":[[0,117,0.0,0.87946,0.24513,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,22],[4,117,0.0342,0.38839,0.42444,0.0,0.0,0.71429,0.0,1.0,17,6,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,6],[8,117,0.0684,0.28125,0.36681,0.0,0.0,0.71429,0.0,1.0,17,3,0,17,0,4,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,3],[12,117,0.1026,0.39286,0.41802,0.0,0.14288,0.71429,0.0,1.0,14,7,0,14,0,3,0,0,1,0,0,0,0,0,1,0,0,6,0,0,0,0,7],[16,117,0.1368,0.23205,0.33647,0.0,0.0,0.5,0.0,1.0,19,1,0,19,0,3,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,1],[20,117,0.1709,0.16071,0.30252,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,3,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,2],[24,117,0.2051,0.23661,0.33808,0.0,0.0,0.71429,0.0,1.0,19,1,0,19,0,3,0,0,1,0,0,0,0,0,0,0,0,7,0,0,1,0,1],[28,117,0.2393,0.33036,0.38372,0.0,0.0,0.71429,0.0,1.0,17,3,0,17,0,1,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,3],[32,117,0.2735,0.37054,0.41321,0.0,0.07143,0.71429,0.0,1.0,16,6,0,16,0,1,0,0,1,0,0,1,0,0,0,0,0,7,0,0,0,0,6],[36,117,0.3077,0.20089,0.38773,0.0,0.0,0.03571,0.0,1.0,24,6,0,24,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[40,117,0.3419,0.25,0.34442,0.0,0.0,0.71429,0.0,1.0,18,2,0,18,0,3,0,0,2,0,0,0,0,0,0,0,0,7,0,0,0,0,2],[44,117,0.3761,0.32134,0.38964,0.0,0.14143,0.71429,0.0,1.0,15,5,0,15,0,5,0,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,5],[48,117,0.4103,0.09375,0.23854,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[52,117,0.4444,0.20089,0.33093,0.0,0.0,0.17857,0.0,1.0,19,3,0,19,0,5,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3],[56,117,0.4786,0.19196,0.32065,0.0,0.0,0.21429,0.0,1.0,21,2,0,21,0,3,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,2],[60,117,0.5128,0.17411,0.33452,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,2],[64,117,0.547,0.11598,0.2635,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[68,117,0.5812,0.24554,0.37667,0.0,0.0,0.32143,0.0,1.0,20,5,0,20,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,5],[72,117,0.6154,0.16071,0.31288,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,2],[76,117,0.6496,0.10714,0.26245,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[80,117,0.6838,0.09375,0.23854,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[84,117,0.7179,0.27679,0.37617,0.0,0.0,0.71429,0.0,1.0,17,4,0,17,0,5,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,4],[88,117,0.7521,0.17411,0.33261,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[92,117,0.7863,0.03563,0.17491,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[96,117,0.8205,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,117,0.8547,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[104,117,0.8889,0.06696,0.21424,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[108,117,0.9231,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[112,117,0.9573,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,117,0.9915,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[117,117,1.0,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.00893,"x":0.77679,"p":[[0,69,0.0,0.77679,0.333,0.71429,1.0,1.0,0.0,1.0,3,19,3,3,0,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,1,0,19],[4,69,0.058,0.38393,0.44812,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,9],[8,69,0.1159,0.32589,0.40757,0.0,0.07143,0.71429,0.0,1.0,16,6,0,16,0,4,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,6],[12,69,0.1739,0.38393,0.43366,0.0,0.07143,0.78571,0.0,1.0,16,8,0,16,0,1,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,8],[16,69,0.2319,0.42411,0.4622,0.0,0.14286,1.0,0.0,1.0,15,11,0,15,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,11],[20,69,0.2899,0.20089,0.35867,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[24,69,0.3478,0.25447,0.33069,0.0,0.07143,0.5,0.0,1.0,16,2,0,16,0,4,0,0,3,0,0,1,0,0,0,0,0,6,0,0,0,0,2],[28,69,0.4058,0.21875,0.36067,0.0,0.0,0.17857,0.0,1.0,20,4,0,20,0,4,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,4],[32,69,0.4638,0.17411,0.29824,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,6,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,2],[36,69,0.5217,0.22321,0.35703,0.0,0.0,0.39285,0.0,1.0,21,3,0,21,0,2,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,3],[40,69,0.5797,0.18303,0.31991,0.0,0.0,0.17857,0.0,1.0,20,3,0,20,0,4,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[44,69,0.6377,0.09821,0.21852,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[48,69,0.6957,0.06696,0.17852,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[52,69,0.7536,0.12054,0.21756,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,5,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[56,69,0.8116,0.15179,0.2922,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[60,69,0.8696,0.09375,0.2412,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[64,69,0.9275,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c5b59b59ac2ae0db","q":"Let  $k$  and  $m$ , with  $k > m$ , be positive integers such that the number  $km(k^2 - m^2)$  is divisible by  $k^3 - m^3$ . Prove that  $(k - m)^3 > 3km$ 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 $a_1,a_2,\\ldots ,a_9$  be any non-negative numbers such that  $a_1=a_9=0$  and at least one of the numbers is non-zero. Prove that for some  $i$ ,  $2\\le i\\le 8$ , the inequality  $a_{i-1}+a_{i+1}<2a_i$  holds. Will the statement remain true if we change the number  $2$  in the last inequality to  $1.9$ ?","t":[{"b":3,"e":0.57143,"k":"falling","v":0.64732,"x":0.90624,"p":[[0,106,0.0,0.86161,0.23003,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,0,0,23],[4,106,0.0377,0.87946,0.16409,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,13,0,15],[8,106,0.0755,0.75893,0.30813,0.4286,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,8,0,0,1,0,0,0,0,0,5,0,16],[12,106,0.1132,0.78125,0.2448,0.4286,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,7,0,14],[16,106,0.1509,0.64732,0.33117,0.42857,0.71429,1.0,0.0,1.0,3,10,0,3,0,1,0,0,0,0,0,10,0,0,2,0,0,0,0,0,6,0,10],[20,106,0.1887,0.90624,0.15407,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,10,0,19],[24,106,0.2264,0.84821,0.20183,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,0,9,0,16],[28,106,0.2642,0.88393,0.16145,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,9,0,17],[32,106,0.3019,0.86161,0.1838,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,10,0,15],[36,106,0.3396,0.86161,0.18723,0.8214,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,7,0,17],[40,106,0.3774,0.78125,0.23954,0.53571,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,3,0,0,7,0,13],[44,106,0.4151,0.85268,0.22442,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,4,0,20],[48,106,0.4528,0.83482,0.21756,0.82143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,8,0,16],[52,106,0.4906,0.76339,0.24382,0.42857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,1,0,0,6,0,13],[56,106,0.5283,0.88393,0.18363,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,5,0,20],[60,106,0.566,0.86607,0.19865,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,5,0,19],[64,106,0.6038,0.78123,0.21721,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,2,0,0,9,0,11],[68,106,0.6415,0.79911,0.22548,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,1,0,0,9,0,13],[72,106,0.6792,0.85267,0.18724,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,9,0,15],[76,106,0.717,0.78572,0.23958,0.4286,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,3,0,0,6,0,14],[80,106,0.7547,0.83929,0.22232,0.67857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,3,0,19],[84,106,0.7925,0.74105,0.23266,0.42859,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,3,0,0,7,0,10],[88,106,0.8302,0.73659,0.25282,0.42857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,2,0,0,5,0,12],[92,106,0.8679,0.89285,0.14725,0.85714,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,10,0,17],[96,106,0.9057,0.89732,0.16065,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,20],[100,106,0.9434,0.6875,0.24598,0.42857,0.57143,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,7,0,0,2,0,0,1,0,11],[104,106,0.9811,0.66076,0.26422,0.42857,0.4293,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,2,0,0,1,0,11],[106,106,1.0,0.6607,0.23623,0.42857,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,6,0,0,0,0,9]]},{"b":6,"e":1.0,"k":"flat","v":0.79911,"x":0.97321,"p":[[0,81,0.0,0.79911,0.27166,0.42859,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,7,0,0,1,0,0,2,0,0,0,0,20],[4,81,0.0494,0.88392,0.16538,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,7,0,18],[8,81,0.0988,0.90179,0.16146,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,7,0,20],[12,81,0.1481,0.81696,0.20589,0.82143,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,13,0,11],[16,81,0.1975,0.87054,0.20628,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,5,0,20],[20,81,0.2469,0.81249,0.21261,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,5,0,15],[24,81,0.2963,0.84375,0.19352,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,8,0,15],[28,81,0.3457,0.85268,0.20666,0.82143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,18],[32,81,0.3951,0.83929,0.22232,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,5,0,18],[36,81,0.4444,0.87946,0.20238,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,7,0,20],[40,81,0.4938,0.90625,0.16213,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,22],[44,81,0.5432,0.88839,0.18118,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,6,0,20],[48,81,0.5926,0.84822,0.2141,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,0,8,0,17],[52,81,0.642,0.80804,0.21902,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,3,0,0,9,0,13],[56,81,0.6914,0.80804,0.23585,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,0,10,0,13],[60,81,0.7407,0.84821,0.2141,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,1,0,0,6,0,18],[64,81,0.7901,0.85714,0.19562,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,5,0,18],[68,81,0.8395,0.89286,0.16366,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,7,0,19],[72,81,0.8889,0.85714,0.21429,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,3,0,20],[76,81,0.9383,0.94643,0.13716,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[80,81,0.9877,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[81,81,1.0,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24]]}]},{"i":"6a87284ad84ff263","q":"Let  $k$  be a positive integer. A sequence of integers  $a_1, a_2, \\cdots$  is called  $k$ -pop if the following holds: for every  $n \\in \\mathbb{N}$ ,  $a_n$  is equal to the number of distinct elements in the set  $\\{a_1, \\cdots , a_{n+k} \\}$ . Determine, as a function of  $k$ , how many  $k$ -pop sequences there are.\n\n*Proposed by Sutanay Bhattacharya*","t":[{"b":1,"e":0.14286,"k":"falling","v":0.2008,"x":0.75,"p":[[0,85,0.0,0.72768,0.42762,0.14286,1.0,1.0,0.0,1.0,6,22,2,6,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,22],[4,85,0.0471,0.70973,0.3986,0.14286,1.0,1.0,0.0,1.0,1,20,0,1,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,20],[8,85,0.0941,0.72768,0.38525,0.39286,1.0,1.0,0.0,1.0,3,20,0,3,0,4,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,20],[12,85,0.1412,0.75,0.38132,0.57143,1.0,1.0,0.0,1.0,2,21,0,2,0,6,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,21],[16,85,0.1882,0.70982,0.3838,0.35714,1.0,1.0,0.0,1.0,2,19,0,2,0,6,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,19],[20,85,0.2353,0.69643,0.40049,0.14286,1.0,1.0,0.0,1.0,3,19,0,3,0,6,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,19],[24,85,0.2824,0.67857,0.41496,0.14286,1.0,1.0,0.14286,1.0,0,20,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[28,85,0.3294,0.63393,0.40079,0.14286,0.85714,1.0,0.0,1.0,1,16,0,1,0,10,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,16],[32,85,0.3765,0.48661,0.42387,0.14286,0.14286,1.0,0.0,1.0,3,12,0,3,0,15,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,12],[36,85,0.4235,0.70536,0.39599,0.14286,1.0,1.0,0.0,1.0,1,19,0,1,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,19],[40,85,0.4706,0.55357,0.43704,0.14286,0.57143,1.0,0.0,1.0,4,15,0,4,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,15],[44,85,0.5176,0.625,0.38919,0.14286,0.78571,1.0,0.14286,1.0,0,16,0,0,0,10,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,16],[48,85,0.5647,0.70536,0.38289,0.14286,1.0,1.0,0.0,1.0,1,18,0,1,0,8,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,18],[52,85,0.6118,0.71875,0.3668,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,6,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,19],[56,85,0.6588,0.63839,0.40404,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,10,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,17],[60,85,0.7059,0.5892,0.43859,0.14286,1.0,1.0,0.0,1.0,2,17,0,2,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[64,85,0.7529,0.56249,0.37447,0.14286,0.42857,1.0,0.0,1.0,1,12,0,1,0,9,0,0,2,0,0,5,0,0,1,0,0,2,0,0,0,0,12],[68,85,0.8,0.60714,0.40721,0.14286,0.78571,1.0,0.0,1.0,1,16,0,1,0,11,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,16],[72,85,0.8471,0.60705,0.3978,0.14286,0.71429,1.0,0.14,1.0,0,14,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,14],[76,85,0.8941,0.29464,0.29437,0.14286,0.14286,0.21429,0.14286,1.0,0,4,0,0,0,24,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,4],[80,85,0.9412,0.2008,0.15513,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,27,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[84,85,0.9882,0.23214,0.19805,0.14286,0.14286,0.17857,0.0,1.0,1,1,0,1,0,23,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[85,85,1.0,0.2008,0.14227,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,27,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.46875,"x":1.0,"p":[[0,42,0.0,0.6875,0.42173,0.25,1.0,1.0,0.0,1.0,5,20,1,5,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,20],[4,42,0.0952,0.60268,0.42966,0.14286,1.0,1.0,0.0,1.0,3,17,0,3,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,17],[8,42,0.1905,0.67857,0.41803,0.14286,1.0,1.0,0.0,1.0,4,19,0,4,0,6,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,19],[12,42,0.2857,0.57143,0.43448,0.14286,0.78571,1.0,0.0,1.0,3,15,0,3,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,15],[16,42,0.381,0.60268,0.40364,0.14286,0.71429,1.0,0.0,1.0,2,14,0,2,0,10,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,14],[20,42,0.4762,0.46875,0.38998,0.14286,0.2143,1.0,0.0,1.0,3,9,0,3,0,13,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,9],[24,42,0.5714,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,42,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,42,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,42,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,42,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"036410d6ebab50f4","q":"Let $ABCD$ be a parallelogram. The internal bisector of angle $\\widehat{BAC}$ intersects the segment $[BC]$ at $E$, while its external bisector intersects the line $(CD)$ at $F$. Let $M$ be the midpoint of the segment $[AE]$.\nProve that the lines $(EF)$ and $(BM)$ are parallel.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.12054,"x":0.35704,"p":[[0,43,0.0,0.32589,0.15663,0.14289,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,9,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[4,43,0.093,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,16,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.17411,0.11143,0.14286,0.14286,0.28571,0.0,0.4286,6,0,0,6,0,14,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.20537,0.1234,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,14,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.20088,0.13763,0.14286,0.14286,0.28571,0.0,0.571,4,0,0,4,0,17,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,43,0.4651,0.21848,0.15574,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,15,0,0,8,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[24,43,0.5581,0.19196,0.08458,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,20,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.24991,0.16757,0.14286,0.14286,0.42857,0.0,0.57143,3,0,0,3,0,15,0,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[32,43,0.7442,0.35704,0.11859,0.2857,0.42857,0.42857,0.14,0.57143,0,0,0,0,0,4,0,0,11,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[36,43,0.8372,0.31241,0.14043,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[40,43,0.9302,0.32143,0.15972,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,9,0,0,8,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[43,43,1.0,0.28571,0.13363,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.06697,"x":0.32588,"p":[[0,135,0.0,0.32588,0.15248,0.1429,0.28571,0.42858,0.0,0.57143,1,0,0,1,0,8,0,0,8,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[4,135,0.0296,0.19625,0.09952,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,18,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,135,0.0593,0.18741,0.09066,0.14286,0.14286,0.2857,0.0,0.4286,1,0,0,1,0,22,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,135,0.0889,0.21429,0.12877,0.14286,0.14286,0.28571,0.0,0.4286,4,0,0,4,0,13,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,135,0.1185,0.16518,0.09522,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,135,0.1481,0.16071,0.09942,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,135,0.1778,0.16518,0.09522,0.14286,0.14286,0.1786,0.0,0.42857,4,0,0,4,0,20,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,135,0.2074,0.12946,0.10326,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,135,0.237,0.20081,0.13299,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,16,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,135,0.2667,0.15179,0.08702,0.14286,0.14286,0.1429,0.0,0.28571,5,0,0,5,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,135,0.2963,0.16062,0.08566,0.14286,0.14286,0.1786,0.0,0.28571,4,0,0,4,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,135,0.3259,0.14286,0.11294,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,135,0.3556,0.16072,0.10564,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,17,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,135,0.3852,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,135,0.4148,0.12045,0.10779,0.0,0.14286,0.14286,0.0,0.286,12,0,0,12,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,135,0.4444,0.12492,0.12751,0.0,0.14286,0.1429,0.0,0.57143,12,0,0,12,0,14,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,135,0.4741,0.125,0.11152,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,135,0.5037,0.08929,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,135,0.5333,0.10268,0.12492,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,135,0.563,0.11598,0.10372,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,135,0.5926,0.10259,0.08914,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,135,0.6222,0.10714,0.10101,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,135,0.6519,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,135,0.6815,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,135,0.7111,0.11607,0.0974,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,135,0.7407,0.12491,0.10563,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,17,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,135,0.7704,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,135,0.8,0.13393,0.13803,0.0,0.14286,0.2857,0.0,0.4286,14,0,0,14,0,8,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,135,0.8296,0.10714,0.10101,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,135,0.8593,0.06697,0.07974,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,135,0.8889,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,135,0.9185,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,135,0.9481,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,135,0.9778,0.07589,0.09438,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[135,135,1.0,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cab0d13fcfc0c229","q":"Let $ABC$ be a triangle such that $AB < AC$, $H$ its orthocenter, $\\Gamma$ its circumcircle, and $d$ the tangent to $\\Gamma$ at $A$. Consider the circle centered at $B$ passing through $A$. It intersects $d$ at $D$ and $(AC)$ at $E$.\nShow that $D$, $E$, and $H$ are collinear.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.04464,"x":0.58482,"p":[[0,126,0.0,0.19643,0.30462,0.0,0.07143,0.17857,0.0,1.0,16,2,3,16,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,2],[4,126,0.0317,0.0625,0.14698,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,126,0.0635,0.05802,0.14218,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,126,0.0952,0.10705,0.1956,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[16,126,0.127,0.10714,0.19885,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,126,0.1587,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,126,0.1905,0.04464,0.10971,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,126,0.2222,0.05358,0.08566,0.0,0.0,0.14286,0.0,0.286,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,126,0.254,0.13839,0.20355,0.0,0.0,0.28571,0.0,0.85714,18,0,0,18,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[36,126,0.2857,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,126,0.3175,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,126,0.3492,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,126,0.381,0.13839,0.14054,0.0,0.14286,0.17857,0.0,0.57143,12,0,0,12,0,12,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,126,0.4127,0.14722,0.13588,0.0,0.14286,0.1786,0.0,0.571,10,0,0,10,0,14,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,126,0.4444,0.14286,0.17857,0.0,0.14286,0.1786,0.0,0.57143,15,0,0,15,0,9,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[60,126,0.4762,0.15178,0.17473,0.0,0.14286,0.2857,0.0,0.57143,14,0,0,14,0,8,0,0,7,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[64,126,0.5079,0.15179,0.15947,0.0,0.14286,0.2857,0.0,0.5714,13,0,0,13,0,9,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[68,126,0.5397,0.13384,0.12846,0.0,0.14286,0.17857,0.0,0.4286,12,0,0,12,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,126,0.5714,0.14732,0.14933,0.0,0.14286,0.1786,0.0,0.57143,11,0,0,11,0,13,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[76,126,0.6032,0.12045,0.16792,0.0,0.0,0.2857,0.0,0.57143,19,0,0,19,0,3,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[80,126,0.6349,0.17409,0.16647,0.0,0.14286,0.28571,0.0,0.57143,11,0,0,11,0,9,0,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[84,126,0.6667,0.14732,0.15765,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,7,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[88,126,0.6984,0.16964,0.20341,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,4,0,0,6,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[92,126,0.7302,0.34817,0.23937,0.14286,0.42857,0.57143,0.0,0.71429,7,0,0,7,0,4,0,0,3,0,0,6,0,0,10,0,0,2,0,0,0,0,0],[96,126,0.7619,0.38835,0.19306,0.28571,0.42857,0.57143,0.0,0.57143,3,0,0,3,0,4,0,0,5,0,0,7,0,0,13,0,0,0,0,0,0,0,0],[100,126,0.7937,0.41514,0.19348,0.28571,0.42857,0.5711,0.0,0.71429,3,0,0,3,0,2,0,0,4,0,0,12,0,0,8,0,0,3,0,0,0,0,0],[104,126,0.8254,0.49551,0.13823,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,7,0,0,16,0,0,3,0,0,0,0,0],[108,126,0.8571,0.47768,0.19103,0.42857,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,1,0,0,1,0,0,7,0,0,17,0,0,3,0,0,0,0,0],[112,126,0.8889,0.55356,0.12242,0.57132,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,5,0,0,19,0,0,6,0,0,0,0,0],[116,126,0.9206,0.51337,0.16311,0.42859,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,1,0,0,5,0,0,19,0,0,4,0,0,0,0,0],[120,126,0.9524,0.58482,0.09689,0.57143,0.57143,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,0,8,0,0,0,0,0],[124,126,0.9841,0.57589,0.12619,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,2,0,0,22,0,0,7,0,0,0,0,0],[126,126,1.0,0.57136,0.07143,0.571,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,0,4,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.02232,"x":0.14277,"p":[[0,116,0.0,0.14277,0.25754,0.0,0.0,0.14286,0.0,1.0,17,1,2,17,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[4,116,0.0345,0.07589,0.17122,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,116,0.069,0.05804,0.18509,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,116,0.1034,0.12054,0.25532,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[16,116,0.1379,0.11161,0.21349,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,8,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[20,116,0.1724,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,116,0.2069,0.05804,0.11769,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,116,0.2414,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,116,0.2759,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,116,0.3103,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,116,0.3448,0.06687,0.12864,0.0,0.0,0.14071,0.0,0.57143,23,0,0,23,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,116,0.3793,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,116,0.4138,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,116,0.4483,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,116,0.4828,0.0625,0.13803,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[60,116,0.5172,0.05804,0.12807,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,116,0.5517,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,116,0.5862,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,116,0.6207,0.05804,0.18509,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,116,0.6552,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,116,0.6897,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,116,0.7241,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,116,0.7586,0.08027,0.14254,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[92,116,0.7931,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,116,0.8276,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,116,0.8621,0.04,0.06395,0.0,0.0,0.14,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,116,0.8966,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,116,0.931,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,116,0.9655,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,116,1.0,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"98a3b03977ec8af6","q":"Let $A B C$ be an acute triangle with the property $\\angle B A C=45^{\\circ}$. Let $D$ be the foot of the perpendicular from $C$ to $A B$. Let $P$ be an interior point of the line segment $C D$. Prove that the lines $A P$ and $B C$ are perpendicular to each other if and only if $|A P|=|B C|$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.85714,"x":0.99554,"p":[[0,28,0.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[4,28,0.1429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,28,0.2857,0.91071,0.21651,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,25],[12,28,0.4286,0.9642,0.15201,1.0,1.0,1.0,0.14,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[16,28,0.5714,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[20,28,0.7143,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[24,28,0.8571,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[28,28,1.0,0.85714,0.26245,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,21]]},{"b":6,"e":1.0,"k":"flat","v":0.90624,"x":0.98661,"p":[[0,15,0.0,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,15,0.2667,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,15,0.5333,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,15,0.8,0.94196,0.20782,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[15,15,1.0,0.90624,0.18425,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,23]]}]},{"i":"ebf41379b1553e92","q":"Let $ABCD$ be a trapezoid such that $(AB)$ and $(CD)$ are parallel. Suppose that $AD < CD$ and that $ABCD$ is inscribed in a circle $\\Gamma$. Let $P$ be a point on $\\Gamma$ such that $(DP)$ is parallel to $(AC)$. The tangent to $\\Gamma$ at $D$ intersects $(AB)$ at $E$, and the chords $[BP]$ and $[CD]$ intersect at $Q$. Show that $EQ = AC$.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.08027,"x":0.24999,"p":[[0,145,0.0,0.24999,0.22866,0.14286,0.2143,0.28571,0.0,1.0,6,1,0,6,0,10,0,0,11,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[4,145,0.0276,0.1383,0.11564,0.0,0.14286,0.2857,0.0,0.28571,11,0,0,11,0,11,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,145,0.0552,0.17848,0.16753,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,7,0,0,11,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,145,0.0828,0.12054,0.10779,0.0,0.14286,0.14287,0.0,0.28571,12,0,0,12,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,145,0.1103,0.13839,0.11564,0.0,0.14286,0.17857,0.0,0.42857,10,0,0,10,0,14,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,145,0.1379,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,145,0.1655,0.10259,0.09602,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,145,0.1931,0.11607,0.10374,0.0,0.14286,0.14287,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,145,0.2207,0.11161,0.11701,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,145,0.2483,0.10714,0.12371,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,145,0.2759,0.09821,0.09061,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,145,0.3034,0.17848,0.20205,0.0,0.14286,0.2857,0.0,0.85714,12,0,0,12,0,8,0,0,9,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[48,145,0.331,0.10266,0.12487,0.0,0.14286,0.14286,0.0,0.571,15,0,0,15,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,145,0.3586,0.12491,0.09941,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,145,0.3862,0.11589,0.13089,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,15,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[60,145,0.4138,0.09813,0.08323,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,145,0.4414,0.09812,0.1037,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,145,0.469,0.12052,0.1135,0.0,0.14286,0.14286,0.0,0.571,10,0,0,10,0,19,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,145,0.4966,0.10268,0.11425,0.0,0.07143,0.14287,0.0,0.28571,16,0,0,16,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,145,0.5241,0.13393,0.14258,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,15,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[80,145,0.5517,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,145,0.5793,0.12036,0.09519,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,145,0.6069,0.20536,0.19212,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,14,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[92,145,0.6345,0.17839,0.12882,0.14214,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,15,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[96,145,0.6621,0.18303,0.10853,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,14,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,145,0.6897,0.125,0.15465,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[104,145,0.7172,0.15625,0.11495,0.10714,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,14,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,145,0.7448,0.12929,0.16113,0.0,0.14286,0.14286,0.0,0.85714,12,0,0,12,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[112,145,0.7724,0.15178,0.12846,0.0,0.14286,0.2857,0.0,0.42857,11,0,0,11,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[116,145,0.8,0.19188,0.15818,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,16,0,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[120,145,0.8276,0.18295,0.12995,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,17,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[124,145,0.8552,0.1875,0.11539,0.14286,0.14286,0.28571,0.0,0.4286,3,0,0,3,0,20,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[128,145,0.8828,0.17848,0.07147,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,145,0.9103,0.20088,0.11209,0.14286,0.14286,0.2857,0.0,0.571,2,0,0,2,0,18,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[136,145,0.9379,0.18304,0.09606,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[140,145,0.9655,0.22321,0.08702,0.14286,0.21428,0.28571,0.14286,0.42857,0,0,0,0,0,16,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[144,145,0.9931,0.17839,0.07994,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[145,145,1.0,0.20536,0.10062,0.14286,0.1429,0.2857,0.0,0.42857,2,0,0,2,0,16,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.08036,"x":0.18304,"p":[[0,108,0.0,0.17857,0.18898,0.0,0.14286,0.2857,0.0,0.85714,10,0,4,10,0,12,0,0,6,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[4,108,0.037,0.14286,0.12372,0.0,0.14286,0.17857,0.0,0.42857,10,0,0,10,0,14,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,108,0.0741,0.15179,0.15542,0.0,0.14286,0.1786,0.0,0.57143,11,0,0,11,0,13,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,108,0.1111,0.16071,0.13716,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,14,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,108,0.1481,0.13393,0.14258,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,9,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,108,0.1852,0.14286,0.13832,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,17,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,108,0.2222,0.16955,0.13094,0.105,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,13,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,108,0.2593,0.13839,0.13592,0.0,0.14286,0.2857,0.0,0.4286,14,0,0,14,0,6,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,108,0.2963,0.125,0.09279,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,108,0.3333,0.13393,0.13333,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,10,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,108,0.3704,0.08473,0.07866,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,108,0.4074,0.125,0.14174,0.0,0.14286,0.1786,0.0,0.42857,15,0,0,15,0,9,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,108,0.4444,0.09822,0.10972,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,108,0.4815,0.11607,0.11538,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,13,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,108,0.5185,0.11161,0.1461,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,12,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,108,0.5556,0.11143,0.09262,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,108,0.5926,0.10259,0.10245,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,108,0.6296,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,108,0.6667,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,108,0.7037,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,108,0.7407,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,108,0.7778,0.125,0.09279,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,108,0.8148,0.14723,0.11564,0.105,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,17,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,108,0.8519,0.15178,0.11258,0.10714,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,108,0.8889,0.18304,0.11425,0.14286,0.1429,0.2857,0.0,0.42857,6,0,0,6,0,12,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,108,0.9259,0.15178,0.11811,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,13,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,108,0.963,0.15178,0.11258,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,108,1.0,0.15183,0.12349,0.0,0.14286,0.2857,0.0,0.43,9,0,0,9,0,14,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ca41da68daaa1528","q":"Let $C_{k}=\\frac{1}{k+1}\\binom{2 k}{k}$ denote the $k^{\\text {th }}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set\n\n$$\n\\left\\{\\sum_{k=1}^{p-1} C_{k} n^{k} \\mid n \\in\\{1,2, \\ldots, p-1\\}\\right\\}\n$$\n\nare divisible by $p$.","t":[{"b":0,"e":0.28571,"k":"rising","v":0.19197,"x":0.70089,"p":[[0,53,0.0,0.19197,0.19103,0.0,0.14286,0.32143,0.0,0.57143,13,0,0,13,0,5,0,0,6,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[4,53,0.0755,0.62945,0.22548,0.57132,0.57143,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,4,0,0,2,0,0,11,0,0,6,0,0,4,0,4],[8,53,0.1509,0.69643,0.25191,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,4,0,0,8,0,0,5,0,0,2,0,10],[12,53,0.2264,0.60266,0.2618,0.42857,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,3,0,0,5,0,0,7,0,0,7,0,0,1,0,6],[16,53,0.3019,0.60714,0.25504,0.42857,0.57143,0.74996,0.143,1.0,0,6,0,0,0,1,0,0,5,0,0,7,0,0,5,0,0,6,0,0,2,0,6],[20,53,0.3774,0.62054,0.28259,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,6,0,0,4,0,0,5,0,0,2,0,8],[24,53,0.4528,0.60268,0.26422,0.42857,0.57143,0.75,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,7,0,0,7,0,0,5,0,0,2,0,6],[28,53,0.5283,0.65625,0.2854,0.42859,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,4,0,0,2,0,0,5,0,0,6,0,0,4,0,8],[32,53,0.6038,0.66071,0.24157,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,7,0,0,5,0,0,6,0,0,5,0,6],[36,53,0.6792,0.62945,0.26211,0.53539,0.71429,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,1,0,0,4,0,0,11,0,0,6,0,3],[40,53,0.7547,0.70089,0.27049,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,3,0,0,5,0,0,5,0,0,7,0,8],[44,53,0.8302,0.52232,0.26633,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,9,0,0,6,0,0,5,0,0,4,0,0,2,0,4],[48,53,0.9057,0.558,0.22119,0.42857,0.571,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,4,0,0,10,0,0,7,0,0,6,0,0,0,0,4],[52,53,0.9811,0.58034,0.23673,0.42857,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,9,0,0,3,0,0,7,0,0,4,0,3],[53,53,1.0,0.46426,0.24221,0.28571,0.42857,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,9,0,0,7,0,0,6,0,0,4,0,0,1,0,2]]},{"b":5,"e":0.42857,"k":"rising","v":0.23214,"x":0.74103,"p":[[0,46,0.0,0.23214,0.20124,0.10714,0.14286,0.42857,0.0,0.71429,8,0,0,8,0,11,0,0,2,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[4,46,0.087,0.59375,0.21461,0.42857,0.57143,0.71429,0.1429,1.0,0,4,0,0,0,1,0,0,3,0,0,7,0,0,8,0,0,9,0,0,0,0,4],[8,46,0.1739,0.62505,0.25688,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,1,0,0,10,0,0,5,0,0,5,0,0,2,0,7],[12,46,0.2609,0.74103,0.20344,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,8,0,0,5,0,8],[16,46,0.3478,0.61607,0.24598,0.42857,0.57143,0.75,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,10,0,0,6,0,0,6,0,0,3,0,5],[20,46,0.4348,0.57589,0.2435,0.42857,0.5,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,12,0,0,4,0,0,6,0,0,1,0,5],[24,46,0.5217,0.68749,0.24599,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,7,0,0,5,0,0,7,0,0,2,0,9],[28,46,0.6087,0.61606,0.23266,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,8,0,0,6,0,0,9,0,0,1,0,5],[32,46,0.6957,0.63393,0.22851,0.42857,0.57143,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,11,0,0,4,0,0,7,0,0,2,0,6],[36,46,0.7826,0.56695,0.15764,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,11,0,0,9,0,0,10,0,0,0,0,1],[40,46,0.8696,0.61158,0.15251,0.57142,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,4,0,0,15,0,0,9,0,0,2,0,1],[44,46,0.9565,0.54018,0.14167,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,11,0,0,10,0,0,9,0,0,0,0,0],[46,46,1.0,0.58927,0.15872,0.53539,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,0,14,0,0,8,0,0,0,0,2]]}]},{"i":"a482a8a993597b8e","q":"Let $A B C$ be a triangle, and $M$ an interior point such that $\\angle M A B=10^{\\circ}, \\angle M B A=$ $20^{\\circ}, \\angle M A C=40^{\\circ}$ and $\\angle M C A=30^{\\circ}$. Prove that the triangle is isosceles.","t":[{"b":2,"e":1.0,"k":"rising","v":0.78572,"x":0.99553,"p":[[0,56,0.0,0.78572,0.35714,0.64286,1.0,1.0,0.14286,1.0,0,23,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,23],[4,56,0.0714,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[8,56,0.1429,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,56,0.2143,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,56,0.2857,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,56,0.3571,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,56,0.4286,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,56,0.5,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,56,0.5714,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,56,0.6429,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,56,0.7143,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,56,0.7857,0.97319,0.08336,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[48,56,0.8571,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[52,56,0.9286,0.98661,0.04164,1.0,1.0,1.0,0.8571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[56,56,1.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":3,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,71,0.0,0.85714,0.28122,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,24],[4,71,0.0563,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,71,0.1127,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,71,0.169,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,71,0.2254,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,71,0.2817,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[24,71,0.338,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[28,71,0.3944,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,71,0.4507,0.95535,0.13571,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[36,71,0.507,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,71,0.5634,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,71,0.6197,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,71,0.6761,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[52,71,0.7324,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,71,0.7887,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,71,0.8451,0.99107,0.03459,1.0,1.0,1.0,0.8571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,71,0.9014,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,71,0.9577,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[71,71,1.0,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"5e211ed6c06a5717","q":"Let  $n$  be a positive integer and let  $k$  be an odd positive integer. Moreover, let  $a,b$  and  $c$  be integers (not necessarily positive) satisfying the equations\n\\[a^n+kb=b^n+kc=c^n+ka \\]\nProve that  $a=b=c$ .","t":[{"b":5,"e":0.0,"k":"falling","v":0.05357,"x":0.70982,"p":[[0,86,0.0,0.46428,0.31339,0.2857,0.28571,0.64286,0.0,1.0,1,7,0,1,0,2,0,0,17,0,0,3,0,0,1,0,0,0,0,0,1,0,7],[4,86,0.0465,0.70982,0.33784,0.42857,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,4,0,0,6,0,0,1,0,0,2,0,0,0,0,17],[8,86,0.093,0.52676,0.35613,0.28571,0.4286,0.89286,0.0,1.0,6,8,0,6,0,0,0,0,4,0,0,8,0,0,2,0,0,2,0,0,2,0,8],[12,86,0.1395,0.53569,0.32537,0.28571,0.49979,0.78571,0.0,1.0,3,8,0,3,0,1,0,0,8,0,0,4,0,0,5,0,0,3,0,0,0,0,8],[16,86,0.186,0.5089,0.34244,0.28571,0.42857,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,7,0,0,6,0,0,5,0,0,0,0,0,1,0,8],[20,86,0.2326,0.40625,0.36615,0.0,0.35714,0.60714,0.0,1.0,9,7,0,9,0,2,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,7],[24,86,0.2791,0.33036,0.38538,0.0,0.21429,0.5,0.0,1.0,14,6,0,14,0,2,0,0,5,0,0,3,0,0,0,0,0,1,0,0,1,0,6],[28,86,0.3256,0.41069,0.37071,0.0,0.28571,0.64254,0.0,1.0,9,7,0,9,0,1,0,0,8,0,0,4,0,0,2,0,0,0,0,0,1,0,7],[32,86,0.3721,0.30356,0.36201,0.0,0.14293,0.57111,0.0,1.0,14,5,0,14,0,3,0,0,5,0,0,1,0,0,3,0,0,1,0,0,0,0,5],[36,86,0.4186,0.21429,0.29014,0.0,0.14286,0.28571,0.0,1.0,15,3,0,15,0,3,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[40,86,0.4651,0.27231,0.31002,0.0,0.21428,0.42857,0.0,1.0,13,3,0,13,0,3,0,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,3],[44,86,0.5116,0.37946,0.36001,0.0,0.28571,0.42858,0.0,1.0,9,7,0,9,0,1,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,7],[48,86,0.5581,0.29018,0.34346,0.0,0.28571,0.42858,0.0,1.0,15,4,0,15,0,0,0,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,4],[52,86,0.6047,0.37054,0.40855,0.0,0.2857,0.89286,0.0,1.0,13,8,0,13,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,8],[56,86,0.6512,0.35267,0.33499,0.10714,0.28571,0.42858,0.0,1.0,8,5,0,8,0,3,0,0,11,0,0,3,0,0,1,0,0,0,0,0,1,0,5],[60,86,0.6977,0.38838,0.34852,0.0,0.28571,0.57111,0.0,1.0,9,6,0,9,0,0,0,0,10,0,0,4,0,0,2,0,0,1,0,0,0,0,6],[64,86,0.7442,0.29464,0.29219,0.0,0.28571,0.42857,0.0,1.0,10,2,0,10,0,4,0,0,6,0,0,8,0,0,0,0,0,0,0,0,2,0,2],[68,86,0.7907,0.30804,0.36089,0.0,0.2143,0.42858,0.0,1.0,12,6,0,12,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[72,86,0.8372,0.3125,0.36846,0.0,0.28571,0.42857,0.0,1.0,14,6,0,14,0,1,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,6],[76,86,0.8837,0.28116,0.34535,0.0,0.14286,0.32143,0.0,1.0,13,5,0,13,0,4,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,5],[80,86,0.9302,0.14731,0.22439,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,2,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[84,86,0.9767,0.18304,0.343,0.0,0.0,0.2857,0.0,1.0,23,4,0,23,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[86,86,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"rising","v":0.38375,"x":0.65624,"p":[[0,44,0.0,0.38375,0.26367,0.2857,0.28571,0.42857,0.0,1.0,1,4,0,1,0,4,0,0,18,0,0,3,0,0,1,0,0,1,0,0,0,0,4],[4,44,0.0909,0.61607,0.38038,0.28571,0.71429,1.0,0.0,1.0,4,13,0,4,0,1,0,0,7,0,0,2,0,0,1,0,0,2,0,0,2,0,13],[8,44,0.1818,0.62057,0.36175,0.28571,0.57121,1.0,0.0,1.0,3,14,0,3,0,1,0,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,14],[12,44,0.2727,0.64285,0.37457,0.28571,0.78564,1.0,0.0,1.0,4,14,0,4,0,1,0,0,4,0,0,4,0,0,2,0,0,1,0,0,2,0,14],[16,44,0.3636,0.62052,0.35645,0.28571,0.64264,1.0,0.0,1.0,2,12,0,2,0,2,0,0,7,0,0,4,0,0,1,0,0,1,0,0,3,0,12],[20,44,0.4545,0.5357,0.3896,0.24999,0.57121,1.0,0.0,1.0,7,11,0,7,0,1,0,0,4,0,0,3,0,0,5,0,0,1,0,0,0,0,11],[24,44,0.5455,0.64286,0.34626,0.42857,0.57143,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,10,0,0,0,0,0,1,0,0,2,0,13],[28,44,0.6364,0.65624,0.32706,0.42857,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,3,0,0,7,0,0,2,0,0,3,0,0,2,0,12],[32,44,0.7273,0.59374,0.31361,0.42857,0.42859,1.0,0.0,1.0,1,9,0,1,0,2,0,0,4,0,0,11,0,0,1,0,0,1,0,0,3,0,9],[36,44,0.8182,0.62054,0.34183,0.28571,0.57143,1.0,0.0,1.0,2,11,0,2,0,2,0,0,5,0,0,5,0,0,3,0,0,1,0,0,3,0,11],[40,44,0.9091,0.63844,0.34987,0.39286,0.71429,1.0,0.0,1.0,1,13,0,1,0,5,0,0,2,0,0,6,0,0,1,0,0,3,0,0,1,0,13],[44,44,1.0,0.5625,0.31731,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,11,0,0,6,0,0,1,0,0,1,0,0,2,0,9]]}]},{"i":"7fab0eefdaa73f54","q":"Let  $n$  be a positive integer. Alice and Bob play the following game. First, Alice picks  $n + 1$  subsets  $A_1,...,A_{n+1}$  of  $\\{1,... ,2^n\\}$  each of size  $2^{n-1}$ . Second, Bob picks  $n + 1$  arbitrary integers  $a_1,...,a_{n+1}$ . Finally, Alice picks an integer  $t$ . Bob wins if there exists an integer  $1 \\le i \\le n + 1$  and  $s \\in A_i$  such that  $s + a_i \\equiv t$  (mod  $2^n$ ). Otherwise, Alice wins.\nFind all values of  $n$  where Alice has a winning strategy.","t":[{"b":3,"e":0.14286,"k":"volatile","v":0.34822,"x":0.94643,"p":[[0,24,0.0,0.34822,0.34981,0.14286,0.14286,0.35714,0.0,1.0,3,6,3,3,0,16,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,6],[4,24,0.1667,0.94643,0.20748,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,24,0.3333,0.92402,0.22043,1.0,1.0,1.0,0.14,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[12,24,0.5,0.92857,0.22588,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[16,24,0.6667,0.91071,0.23077,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,27],[20,24,0.8333,0.94196,0.19516,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[24,24,1.0,0.43723,0.3831,0.14286,0.21428,1.0,0.14,1.0,0,10,0,0,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,10]]},{"b":5,"e":0.28571,"k":"flat","v":0.2008,"x":0.95536,"p":[[0,79,0.0,0.34366,0.38447,0.14286,0.14286,0.46429,0.0,1.0,5,8,5,5,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,79,0.0506,0.95536,0.14914,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[8,79,0.1013,0.86607,0.3008,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[12,79,0.1519,0.47768,0.42948,0.14286,0.21428,1.0,0.0,1.0,5,12,0,5,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,12],[16,79,0.2025,0.50446,0.42104,0.14286,0.28571,1.0,0.0,1.0,3,13,0,3,0,12,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,13],[20,79,0.2532,0.57589,0.44818,0.14286,0.9285,1.0,0.0,1.0,6,16,0,6,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,16],[24,79,0.3038,0.55804,0.45226,0.14286,0.78571,1.0,0.0,1.0,6,16,0,6,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16],[28,79,0.3544,0.45973,0.45144,0.105,0.14286,1.0,0.0,1.0,8,13,0,8,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[32,79,0.4051,0.76786,0.37415,0.28571,1.0,1.0,0.0,1.0,1,23,0,1,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[36,79,0.4557,0.8258,0.33087,1.0,1.0,1.0,0.14,1.0,0,25,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[40,79,0.5063,0.84821,0.31122,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,25],[44,79,0.557,0.875,0.29179,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[48,79,0.6076,0.80804,0.33429,0.82143,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[52,79,0.6582,0.87054,0.28428,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[56,79,0.7089,0.85268,0.30406,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[60,79,0.7595,0.85714,0.29881,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[64,79,0.8101,0.80802,0.32461,0.67846,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,23],[68,79,0.8608,0.62938,0.40078,0.14286,0.92857,1.0,0.14,1.0,0,16,0,0,0,11,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,16],[72,79,0.9114,0.42848,0.38139,0.14286,0.2857,1.0,0.0,1.0,3,9,0,3,0,12,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,9],[76,79,0.962,0.29902,0.28878,0.14286,0.14286,0.28571,0.0,1.0,1,4,0,1,0,18,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[79,79,1.0,0.2008,0.08651,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f56f2d46918bc759","q":"Let  $n$  be a positive integer. There are  $2018n+1$  cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city  $C$  and integer  $1\\le i\\le 2018,$  there are exactly  $n$  cities that are a distance  $i$  away from  $C.$  (The *distance* between two cities is the least number of roads on any path between the two cities.)\n    \nFor which  $n$  is it possible for Mark to achieve this?\n\n*Proposed by Michael Ren*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.17393,"x":0.42411,"p":[[0,35,0.0,0.42411,0.28231,0.28571,0.28571,0.32143,0.0,1.0,1,5,1,1,0,0,0,0,23,0,0,1,0,0,0,0,0,1,0,0,1,0,5],[4,35,0.1143,0.29018,0.12619,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,23,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[8,35,0.2286,0.34821,0.19212,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,22,0,0,0,0,0,2,0,0,3,0,0,0,0,1],[12,35,0.3429,0.27668,0.13339,0.14286,0.28571,0.28571,0.14,0.71429,0,0,0,0,0,10,0,0,18,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[16,35,0.4571,0.29018,0.14054,0.24999,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,8,0,0,19,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[20,35,0.5714,0.26339,0.1017,0.24999,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,8,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,35,0.6857,0.27232,0.13533,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,10,0,0,19,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[28,35,0.8,0.21866,0.09446,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,17,0,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,35,0.9143,0.17393,0.09938,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,26,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[35,35,1.0,0.2008,0.15919,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,22,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"falling","v":0.19196,"x":0.57588,"p":[[0,57,0.0,0.57588,0.32436,0.28571,0.49979,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,14,0,0,1,0,0,5,0,0,0,0,0,0,0,11],[4,57,0.0702,0.37946,0.24383,0.28571,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,2,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,4],[8,57,0.1404,0.3616,0.23952,0.2857,0.28571,0.28571,0.0,1.0,1,3,1,1,0,3,0,0,22,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[12,57,0.2105,0.2991,0.16115,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,3,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,57,0.2807,0.27678,0.14698,0.24999,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,8,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,57,0.3509,0.2633,0.17175,0.14286,0.2857,0.28571,0.14,1.0,0,1,0,0,0,13,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[24,57,0.4211,0.29902,0.17269,0.24999,0.28571,0.28571,0.14,1.0,0,1,0,0,0,8,0,0,20,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[28,57,0.4912,0.29018,0.21275,0.14286,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,13,0,0,15,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[32,57,0.5614,0.25884,0.13578,0.14286,0.28571,0.28571,0.14,0.71429,0,0,0,0,0,13,0,0,16,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[36,57,0.6316,0.24999,0.14284,0.14286,0.2857,0.28571,0.14286,0.857,0,0,0,0,0,14,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[40,57,0.7018,0.24552,0.18975,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,17,0,0,11,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[44,57,0.7719,0.27231,0.18334,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,15,0,0,12,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[48,57,0.8421,0.25892,0.15333,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,15,0,0,13,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[52,57,0.9123,0.22768,0.17445,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,16,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[56,57,0.9825,0.22768,0.10012,0.14286,0.21428,0.28571,0.14286,0.57143,0,0,0,0,0,16,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[57,57,1.0,0.19196,0.09852,0.14286,0.14286,0.17857,0.14286,0.57143,0,0,0,0,0,24,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b073384942e022e6","q":"Let $S$ be the set of all triangles $A B C$ for which  $$ 5\\left(\\frac{1}{A P}+\\frac{1}{B Q}+\\frac{1}{C R}\\right)-\\frac{3}{\\min \\{A P, B Q, C R\\}}=\\frac{6}{r} $$  where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $A B, B C, C A$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.85713,"x":0.99107,"p":[[0,87,0.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,87,0.046,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,87,0.092,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,87,0.1379,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,87,0.1839,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,87,0.2299,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[24,87,0.2759,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,87,0.3218,0.95535,0.15336,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[32,87,0.3678,0.9598,0.09612,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[36,87,0.4138,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,87,0.4598,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[44,87,0.5057,0.94643,0.13243,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[48,87,0.5517,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,87,0.5977,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,87,0.6437,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[60,87,0.6897,0.95088,0.11073,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[64,87,0.7356,0.91071,0.15465,0.85711,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,5,0,21],[68,87,0.7816,0.92856,0.11298,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,20],[72,87,0.8276,0.90624,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[76,87,0.8736,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[80,87,0.9195,0.93303,0.14279,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[84,87,0.9655,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[87,87,1.0,0.85713,0.16369,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,8,0,14]]},{"b":4,"e":1.0,"k":"falling","v":0.79013,"x":1.0,"p":[[0,90,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,90,0.0444,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,90,0.0889,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,90,0.1333,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,90,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,90,0.2222,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,90,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,90,0.3111,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[32,90,0.3556,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,90,0.4,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,90,0.4444,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[44,90,0.4889,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[48,90,0.5333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,90,0.5778,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[56,90,0.6222,0.96428,0.10715,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[60,90,0.6667,0.94196,0.19186,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[64,90,0.7111,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[68,90,0.7556,0.91964,0.15947,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[72,90,0.8,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[76,90,0.8444,0.95089,0.13651,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[80,90,0.8889,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,90,0.9333,0.93749,0.15129,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[88,90,0.9778,0.88839,0.25187,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,24],[90,90,1.0,0.79013,0.23417,0.71429,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,4,0,0,10,0,11]]}]},{"i":"4951fe3d6bc0149d","q":"Let $\\Gamma$ be a circle and $P$ a point outside $\\Gamma$. The tangents to $\\Gamma$ from $P$ touch $\\Gamma$ at $A$ and $B$. Let $K$ be a point distinct from $A$ and $B$ on the segment $[A B]$. The circumcircle of triangle $P B K$ intersects the circle $\\Gamma$ at point $T$. Let $P'$ be the symmetric point of $P$ with respect to point $A$. Show that $\\widehat{\\mathrm{PBT}}=\\widehat{\\mathrm{P}^{\\prime} \\mathrm{KA}}$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.03571,"x":0.13393,"p":[[0,77,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,77,0.1039,0.07563,0.07951,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,77,0.1558,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,77,0.2078,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,77,0.3117,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,77,0.3636,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,77,0.4156,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,77,0.4675,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,77,0.5195,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,77,0.5714,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,77,0.6234,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.09812,0.06616,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.04464,"x":0.13375,"p":[[0,65,0.0,0.06687,0.08729,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.08455,0.06994,0.0,0.14143,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.13375,0.17834,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,65,0.6154,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.1025,0.0734,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ec12fce72929fef0","q":"Let $S$ be the set of all ordered pairs $(a, b)$ of integers with $0<2 a<2 b<2017$ such that $a^{2}+b^{2}$ is a multiple of 2017. Prove that\n\n$$\n\\sum_{(a, b) \\in S} a=\\frac{1}{2} \\sum_{(a, b) \\in S} b\n$$","t":[{"b":3,"e":0.2857,"k":"flat","v":0.19635,"x":0.44641,"p":[[0,86,0.0,0.26777,0.25695,0.14286,0.14286,0.32143,0.0,1.0,3,2,0,3,0,18,0,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[4,86,0.0465,0.42411,0.28901,0.14286,0.28571,0.71429,0.0,1.0,2,1,0,2,0,7,0,0,9,0,0,2,0,0,3,0,0,3,0,0,5,0,1],[8,86,0.093,0.44641,0.28063,0.14289,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,7,0,0,6,0,0,3,0,0,3,0,0,6,0,0,5,0,0],[12,86,0.1395,0.28571,0.29233,0.14286,0.14288,0.28571,0.0,1.0,7,2,0,7,0,10,0,0,8,0,0,1,0,0,1,0,0,1,0,0,2,0,2],[16,86,0.186,0.44641,0.32092,0.14289,0.35714,0.74996,0.0,1.0,3,3,0,3,0,7,0,0,6,0,0,3,0,0,4,0,0,1,0,0,5,0,3],[20,86,0.2326,0.40177,0.23807,0.2857,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,4,0,0,11,0,0,6,0,0,2,0,0,4,0,0,3,0,0],[24,86,0.2791,0.4241,0.31436,0.14286,0.42857,0.71429,0.0,1.0,5,2,0,5,0,5,0,0,5,0,0,6,0,0,1,0,0,4,0,0,4,0,2],[28,86,0.3256,0.37499,0.32878,0.14286,0.28571,0.60714,0.0,1.0,4,3,0,4,0,10,0,0,8,0,0,0,0,0,2,0,0,1,0,0,4,0,3],[32,86,0.3721,0.42853,0.35173,0.10714,0.42859,0.71429,0.0,1.0,8,4,0,8,0,4,0,0,3,0,0,2,0,0,5,0,0,4,0,0,2,0,4],[36,86,0.4186,0.42399,0.29563,0.24999,0.35714,0.60714,0.0,1.0,5,2,0,5,0,3,0,0,8,0,0,2,0,0,6,0,0,4,0,0,2,0,2],[40,86,0.4651,0.26775,0.24682,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,5,0,0,4,0,0,1,0,0,3,0,0,0,0,1],[44,86,0.5116,0.28122,0.28897,0.14286,0.14286,0.32143,0.0,1.0,7,2,0,7,0,11,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,2],[48,86,0.5581,0.29463,0.26228,0.14286,0.21428,0.46418,0.0,1.0,6,1,0,6,0,10,0,0,7,0,0,1,0,0,3,0,0,4,0,0,0,0,1],[52,86,0.6047,0.24552,0.20273,0.14286,0.2857,0.28571,0.0,0.857,7,0,0,7,0,8,0,0,10,0,0,3,0,0,3,0,0,0,0,0,1,0,0],[56,86,0.6512,0.36161,0.2879,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,9,0,0,8,0,0,0,0,0,4,0,0,2,0,0,5,0,0],[60,86,0.6977,0.27901,0.28812,0.05357,0.14288,0.4286,0.0,1.0,8,1,0,8,1,9,0,0,5,0,0,2,0,0,3,0,0,0,0,0,3,0,1],[64,86,0.7442,0.25445,0.25436,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,11,0,0,8,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[68,86,0.7907,0.23651,0.19436,0.14286,0.14288,0.28571,0.0,0.85714,3,0,0,3,0,17,0,0,7,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[72,86,0.8372,0.20536,0.22286,0.0,0.14286,0.2857,0.0,0.85714,9,0,0,9,0,13,0,0,5,0,0,1,0,0,1,0,0,2,0,0,1,0,0],[76,86,0.8837,0.22321,0.22569,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,15,0,0,6,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[80,86,0.9302,0.19641,0.15459,0.14286,0.14286,0.2857,0.0,0.714,4,0,0,4,0,19,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[84,86,0.9767,0.21652,0.1651,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,1,16,0,0,7,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[86,86,1.0,0.19635,0.14621,0.14286,0.14286,0.1429,0.0,0.71429,2,0,0,2,0,23,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.05357,"x":0.44196,"p":[[0,53,0.0,0.20527,0.16729,0.14286,0.14286,0.1429,0.0,0.857,2,0,0,2,0,23,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[4,53,0.0755,0.42857,0.3234,0.14286,0.35714,0.71429,0.0,1.0,5,2,0,5,0,6,0,0,5,0,0,3,0,0,2,0,0,5,0,0,4,0,2],[8,53,0.1509,0.3482,0.30079,0.14286,0.28571,0.57111,0.0,1.0,6,1,0,6,0,7,0,0,8,0,0,2,0,0,2,0,0,2,0,0,4,0,1],[12,53,0.2264,0.44196,0.28428,0.14286,0.35714,0.71429,0.0,1.0,1,2,0,1,0,8,0,0,7,0,0,3,0,0,4,0,0,4,0,0,3,0,2],[16,53,0.3019,0.27232,0.31003,0.0,0.14286,0.28571,0.0,1.0,10,3,0,10,0,7,0,0,9,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[20,53,0.3774,0.28125,0.25874,0.14286,0.14286,0.28571,0.0,0.85714,5,0,0,5,0,13,0,0,7,0,0,0,0,0,1,0,0,4,0,0,2,0,0],[24,53,0.4528,0.39283,0.32339,0.14286,0.28571,0.60714,0.0,1.0,5,3,0,5,0,7,0,0,7,0,0,2,0,0,3,0,0,2,0,0,3,0,3],[28,53,0.5283,0.26338,0.20236,0.14286,0.2857,0.28571,0.0,1.0,3,1,0,3,0,12,0,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[32,53,0.6038,0.29907,0.27044,0.14286,0.28571,0.46418,0.0,1.0,7,1,0,7,0,8,0,0,7,0,0,2,0,0,5,0,0,0,0,0,2,0,1],[36,53,0.6792,0.27679,0.31326,0.0,0.14286,0.42857,0.0,1.0,10,2,0,10,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0,3,0,2],[40,53,0.7547,0.25893,0.31831,0.0,0.14286,0.42857,0.0,1.0,13,3,0,13,0,6,0,0,4,0,0,3,0,0,2,0,0,0,0,0,1,0,3],[44,53,0.8302,0.2991,0.2681,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,7,0,0,9,0,0,2,0,0,2,0,0,3,0,0,1,0,1],[48,53,0.9057,0.15179,0.15947,0.0,0.14286,0.1786,0.0,0.71429,11,0,0,11,0,13,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[52,53,0.9811,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0c210e1412e03141","q":"Let  $p_1, p_2, \\ldots$  be a sequence of primes such that  $p_1 =2$  and for  $n\\geq 1, p_{n+1}$  is the largest prime factor of  $p_1 p_2 \\ldots p_n +1$  . Prove that  $p_n \\not= 5$  for any  $n$ .","t":[{"b":5,"e":0.0,"k":"volatile","v":0.01339,"x":0.7767,"p":[[0,23,0.0,0.7767,0.32542,0.42859,1.0,1.0,0.0,1.0,2,20,1,2,0,1,0,0,0,0,0,6,0,0,1,0,0,1,0,0,1,0,20],[4,23,0.1739,0.59821,0.36846,0.35714,0.64286,1.0,0.0,1.0,4,10,0,4,0,4,0,0,0,0,0,6,0,0,2,0,0,2,0,0,4,0,10],[8,23,0.3478,0.68302,0.35127,0.42857,0.78571,1.0,0.0,1.0,4,13,0,4,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,0,3,0,13],[12,23,0.5217,0.63838,0.39928,0.14286,0.85714,1.0,0.0,1.0,4,14,0,4,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,0,4,0,14],[16,23,0.6957,0.66071,0.36553,0.42857,0.78571,1.0,0.0,1.0,4,13,0,4,0,2,0,0,1,0,0,4,0,0,2,0,0,3,0,0,3,0,13],[20,23,0.8696,0.625,0.40049,0.35714,0.78571,1.0,0.0,1.0,6,14,0,6,0,2,0,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,14],[23,23,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.70987,"x":0.98661,"p":[[0,46,0.0,0.70987,0.34156,0.42857,0.92857,1.0,0.0,1.0,3,16,2,3,0,0,0,0,2,0,0,6,0,0,1,0,0,3,0,0,1,0,16],[4,46,0.087,0.91518,0.1551,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23],[8,46,0.1739,0.91964,0.21997,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[12,46,0.2609,0.82143,0.31944,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,22],[16,46,0.3478,0.91964,0.15542,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,23],[20,46,0.4348,0.87945,0.25283,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,24],[24,46,0.5217,0.85713,0.21726,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,0,6,0,19],[28,46,0.6087,0.88839,0.23072,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,23],[32,46,0.6957,0.79469,0.29648,0.42857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,2,0,20],[36,46,0.7826,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,46,0.8696,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,46,0.9565,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[46,46,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"29c8e4789fa8d0ce","q":"Let  $x,y$  be integer number with  $x,y\\neq-1$  so that  $\\frac{x^{4}-1}{y+1}+\\frac{y^{4}-1}{x+1}\\in\\mathbb{Z}$ . Prove that  $x^{4}y^{44}-1$  is divisble by  $x+1$","t":[{"b":1,"e":0.42857,"k":"flat","v":0.17411,"x":0.22759,"p":[[0,22,0.0,0.20081,0.16701,0.14214,0.14288,0.28571,0.0,0.71429,7,0,0,7,0,13,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.22759,0.12306,0.14286,0.14286,0.32143,0.14,0.4286,0,0,0,0,0,21,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.18741,0.13573,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,18,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.22322,0.18189,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,12,0,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[16,22,0.7273,0.21867,0.14723,0.14286,0.14286,0.42857,0.0,0.42857,4,0,0,4,0,16,0,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.17411,0.09269,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,26,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.17411,0.08552,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.11607,"x":0.25447,"p":[[0,34,0.0,0.20088,0.19837,0.0,0.14286,0.32143,0.0,0.71429,10,0,0,10,0,11,0,0,3,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[4,34,0.1176,0.20536,0.15542,0.14286,0.14286,0.1786,0.0,0.71429,3,0,0,3,0,21,0,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[8,34,0.2353,0.1875,0.16536,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,16,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[12,34,0.3529,0.25447,0.18118,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,15,0,0,4,0,0,8,0,0,0,0,0,2,0,0,0,0,0],[16,34,0.4706,0.17849,0.10105,0.14286,0.14286,0.14292,0.0,0.42857,2,0,0,2,0,23,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.17411,0.12745,0.14286,0.14286,0.14287,0.0,0.4286,5,0,0,5,0,20,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.19197,0.11071,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.17411,0.11143,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,23,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.11607,0.10972,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,20,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[34,34,1.0,0.13839,0.10403,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,21,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78e9e16becefa304","q":"Let  $n \\geq 3$  be an integer. A labelling of the  $n$  vertices, the  $n$  sides and the interior of a regular  $n$ -gon by  $2n + 1$  distinct integers is called *memorable* if the following conditions hold:\n(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.\n(b) The interior of the  $n$ -gon has a label that is the arithmetic mean of the labels of all the vertices.\nDetermine all integers  $n \\geq 3$  for which there exists a memorable labelling of a regular  $n$ -gon consisting of  $2n + 1$  consecutive integers.","t":[{"b":5,"e":0.2857,"k":"falling","v":0.27232,"x":0.77231,"p":[[0,74,0.0,0.59821,0.16145,0.57143,0.57143,0.71429,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,4,0,0,16,0,0,9,0,0,1,0,1],[4,74,0.0541,0.77231,0.21976,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,4,0,0,8,0,10],[8,74,0.1081,0.74105,0.27302,0.67857,0.85707,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,0,0,0,3,0,0,7,0,0,7,0,10],[12,74,0.1622,0.77229,0.23383,0.57143,0.85707,1.0,0.1429,1.0,0,12,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,0,5,0,12],[16,74,0.2162,0.70534,0.22286,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,10,0,0,4,0,7],[20,74,0.2703,0.71874,0.23821,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,7,0,0,6,0,8],[24,74,0.3243,0.6741,0.26301,0.4286,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,5,0,0,3,0,0,5,0,0,3,0,0,9,0,6],[28,74,0.3784,0.64284,0.29666,0.28571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,7,0,0,3,0,0,2,0,0,5,0,0,5,0,8],[32,74,0.4324,0.69642,0.32094,0.42857,0.78571,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,3,0,0,2,0,0,5,0,0,2,0,0,2,0,14],[36,74,0.4865,0.58033,0.23129,0.39286,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,8,0,0,3,0,0,10,0,0,5,0,0,2,0,4],[40,74,0.5405,0.54015,0.30458,0.2857,0.57121,0.74996,0.14286,1.0,0,5,0,0,0,7,0,0,5,0,0,2,0,0,5,0,0,5,0,0,3,0,5],[44,74,0.5946,0.52677,0.29545,0.28571,0.42859,0.85714,0.14286,1.0,0,5,0,0,0,5,0,0,7,0,0,5,0,0,5,0,0,1,0,0,4,0,5],[48,74,0.6486,0.43747,0.26948,0.2857,0.28571,0.60714,0.14286,1.0,0,2,0,0,0,7,0,0,11,0,0,2,0,0,4,0,0,3,0,0,3,0,2],[52,74,0.7027,0.5625,0.28558,0.28571,0.57144,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,11,0,0,3,0,0,0,0,0,7,0,0,5,0,4],[56,74,0.7568,0.45982,0.24415,0.28571,0.28571,0.60714,0.14286,1.0,0,2,0,0,0,2,0,0,16,0,0,1,0,0,5,0,0,4,0,0,2,0,2],[60,74,0.8108,0.35712,0.18895,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,16,0,0,6,0,0,3,0,0,0,0,0,1,0,1],[64,74,0.8649,0.33478,0.16976,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,5,0,0,19,0,0,3,0,0,4,0,0,0,0,0,0,0,1],[68,74,0.9189,0.44641,0.24677,0.2857,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,15,0,0,2,0,0,6,0,0,1,0,0,3,0,2],[72,74,0.973,0.30357,0.10565,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,25,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[74,74,1.0,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.33927,"x":0.74553,"p":[[0,116,0.0,0.58475,0.14446,0.57143,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,2,0,0,18,0,0,9,0,0,1,0,0],[4,116,0.0345,0.6339,0.30501,0.42859,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,7,0,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,6],[8,116,0.069,0.72766,0.22689,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,7,0,0,4,0,9],[12,116,0.1034,0.72317,0.23131,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,6,0,0,6,0,8],[16,116,0.1379,0.74553,0.24153,0.67836,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,8,0,0,8,0,8],[20,116,0.1724,0.66067,0.29829,0.53539,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,2,0,0,1,0,0,5,0,0,6,0,0,5,0,8],[24,116,0.2069,0.67409,0.30772,0.57132,0.71429,1.0,0.0,1.0,1,10,0,1,0,4,0,0,1,0,0,1,0,0,6,0,0,6,0,0,3,0,10],[28,116,0.2414,0.57592,0.29336,0.39286,0.57143,0.85704,0.14286,1.0,0,4,0,0,0,7,0,0,1,0,0,4,0,0,6,0,0,4,0,0,6,0,4],[32,116,0.2759,0.56247,0.30916,0.25,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,8,0,0,2,0,0,2,0,0,7,0,0,3,0,0,5,0,5],[36,116,0.3103,0.63836,0.30302,0.39288,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,3,0,0,1,0,0,6,0,0,5,0,0,4,0,8],[40,116,0.3448,0.63839,0.31538,0.39288,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,2,0,0,2,0,0,5,0,0,3,0,0,6,0,8],[44,116,0.3793,0.71427,0.29881,0.57132,0.71429,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,2,0,0,1,0,0,3,0,0,7,0,0,3,0,12],[48,116,0.4138,0.49554,0.34253,0.14286,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,13,0,0,2,0,0,0,0,0,6,0,0,2,0,0,3,0,6],[52,116,0.4483,0.63837,0.32337,0.35714,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,8,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,8],[56,116,0.4828,0.55357,0.31288,0.24999,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,8,0,0,3,0,0,3,0,0,4,0,0,3,0,0,7,0,4],[60,116,0.5172,0.60714,0.29014,0.39286,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,4,0,0,4,0,0,5,0,0,4,0,0,5,0,6],[64,116,0.5517,0.61593,0.32423,0.28571,0.71214,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,6,0,0,1,0,0,3,0,0,5,0,0,3,0,9],[68,116,0.5862,0.47319,0.27993,0.14286,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,4,0,0,4,0,0,6,0,0,3,0,0,4,0,2],[72,116,0.6207,0.40616,0.31165,0.14286,0.28571,0.57143,0.0,1.0,1,4,0,1,0,13,0,0,5,0,0,0,0,0,6,0,0,2,0,0,1,0,4],[76,116,0.6552,0.34821,0.25238,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,14,0,0,8,0,0,1,0,0,4,0,0,2,0,0,2,0,1],[80,116,0.6897,0.33927,0.25938,0.14286,0.21428,0.5711,0.0,0.85714,1,0,0,1,0,15,0,0,5,0,0,2,0,0,4,0,0,1,0,0,4,0,0],[84,116,0.7241,0.37499,0.29826,0.14286,0.2857,0.57111,0.14286,1.0,0,4,0,0,0,14,0,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,4],[88,116,0.7586,0.56248,0.27649,0.28571,0.57143,0.85704,0.14286,1.0,0,4,0,0,0,2,0,0,10,0,0,2,0,0,5,0,0,4,0,0,5,0,4],[92,116,0.7931,0.55802,0.24053,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,4,0,0,8,0,0,6,0,0,4,0,2],[96,116,0.8276,0.44196,0.24836,0.28571,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,14,0,0,6,0,0,2,0,0,3,0,0,1,0,3],[100,116,0.8621,0.41516,0.2509,0.28571,0.28571,0.57111,0.14286,1.0,0,2,0,0,0,5,0,0,15,0,0,3,0,0,2,0,0,3,0,0,2,0,2],[104,116,0.8966,0.40174,0.24595,0.28571,0.28571,0.57111,0.0,1.0,2,1,0,2,0,5,0,0,10,0,0,4,0,0,6,0,0,2,0,0,2,0,1],[108,116,0.931,0.61607,0.30605,0.42859,0.64286,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,2,0,0,3,0,0,6,0,0,3,0,0,8,0,5],[112,116,0.9655,0.5,0.24744,0.28571,0.50001,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,12,0,0,2,0,0,7,0,0,5,0,0,1,0,3],[116,116,1.0,0.47307,0.21543,0.28571,0.42857,0.60607,0.14286,1.0,0,1,0,0,0,2,0,0,10,0,0,8,0,0,4,0,0,5,0,0,2,0,1]]}]},{"i":"b09583aa1e021350","q":"Let $a, b, c$ be positive real numbers such that $a^{3}+b^{3}=c^{3}$. Prove that\n\n$$\na^{2}+b^{2}-c^{2}>6(c-a)(c-b)\n$$","t":[{"b":0,"e":0.571,"k":"flat","v":0.62496,"x":0.71875,"p":[[0,39,0.0,0.66956,0.18709,0.57132,0.57143,0.857,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,14,0,0,4,0,0,9,0,2],[4,39,0.1026,0.68749,0.16146,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,7,0,0,8,0,2],[8,39,0.2051,0.71427,0.20824,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,6,0,0,8,0,0,10,0,4],[12,39,0.3077,0.69642,0.15048,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,4,0,0,13,0,0],[16,39,0.4103,0.71429,0.16366,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,7,0,0,9,0,3],[20,39,0.5128,0.66071,0.22798,0.57143,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,2,0,0,12,0,0,5,0,0,5,0,5],[24,39,0.6154,0.70085,0.16118,0.57143,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,2,0,0,15,0,0],[28,39,0.7179,0.64732,0.13825,0.57143,0.57143,0.75,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,3,0,0,8,0,0],[32,39,0.8205,0.71875,0.145,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,7,0,0,14,0,0],[36,39,0.9231,0.71427,0.17497,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,4,0,0,11,0,3],[39,39,1.0,0.62496,0.14618,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,1,0,0,18,0,0,5,0,0,6,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.59372,"x":0.75,"p":[[0,128,0.0,0.70088,0.19018,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,6,0,0,10,0,3],[4,128,0.0312,0.7232,0.15948,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,9,0,0,9,0,3],[8,128,0.0625,0.67856,0.18558,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,7,0,0,7,0,3],[12,128,0.0938,0.66513,0.20081,0.57132,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,1,0,0,11,0,2],[16,128,0.125,0.71875,0.2004,0.57143,0.78571,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,4,0,0,13,0,3],[20,128,0.1562,0.6607,0.15047,0.57143,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,3,0,0,8,0,1],[24,128,0.1875,0.65624,0.15511,0.57143,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,18,0,0,3,0,0,8,0,1],[28,128,0.2188,0.71872,0.16556,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,6,0,0,10,0,3],[32,128,0.25,0.67855,0.19885,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,0,0,0,13,0,0,6,0,0,9,0,2],[36,128,0.2812,0.68749,0.20959,0.57143,0.64286,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,12,0,0,4,0,0,7,0,5],[40,128,0.3125,0.7098,0.1838,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,9,0,0,9,0,3],[44,128,0.3438,0.75,0.16366,0.57143,0.71429,0.85714,0.5714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,8,0,6],[48,128,0.375,0.64729,0.18554,0.57143,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,4,0,0,9,0,1],[52,128,0.4062,0.71874,0.14055,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,12,0,0,10,0,1],[56,128,0.4375,0.7232,0.18878,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,7,0,0,8,0,5],[60,128,0.4688,0.625,0.1948,0.57142,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,2,0,0,9,0,0,9,0,0,6,0,1],[64,128,0.5,0.65625,0.19186,0.57143,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,4,0,0,10,0,1],[68,128,0.5312,0.71872,0.13595,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,9,0,0,11,0,1],[72,128,0.5625,0.65176,0.16729,0.57143,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,15,0,0,7,0,0,5,0,2],[76,128,0.5938,0.69196,0.18595,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,6,0,0,12,0,1],[80,128,0.625,0.70533,0.19214,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,2,0,0,11,0,0,4,0,0,11,0,3],[84,128,0.6562,0.6741,0.18638,0.57143,0.64286,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,13,0,0,5,0,0,9,0,2],[88,128,0.6875,0.71425,0.13834,0.57143,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,0,12,0,1],[92,128,0.7188,0.69196,0.18249,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,10,0,0,6,0,0,12,0,1],[96,128,0.75,0.65176,0.19866,0.57132,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,3,0,0,12,0,0,3,0,0,9,0,2],[100,128,0.7812,0.66068,0.1915,0.57143,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,1,0,0,14,0,0,3,0,0,9,0,2],[104,128,0.8125,0.6696,0.15748,0.57143,0.64286,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,7,0,0,8,0,1],[108,128,0.8438,0.62498,0.1948,0.57143,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,0,0,0,14,0,0,6,0,0,5,0,2],[112,128,0.875,0.625,0.20124,0.42857,0.64286,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,4,0,0,5,0,0,7,0,0,9,0,0,5,0,2],[116,128,0.9062,0.60267,0.21938,0.42857,0.57143,0.85704,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,2,0,0,8,0,0,6,0,0,9,0,0],[120,128,0.9375,0.65625,0.18851,0.57143,0.64286,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,4,0,0,12,0,0],[124,128,0.9688,0.66964,0.18707,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,8,0,0,10,0,1],[128,128,1.0,0.59372,0.22899,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,9,0,0,6,0,1]]}]},{"i":"5dcb320397672fc2","q":"Let $a, b, c$ be positive real numbers with $a b c=1$. Prove that\n\n$$\n\\frac{a}{a^{2}+2}+\\frac{b}{b^{2}+2}+\\frac{c}{c^{2}+2} \\leq 1\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.01786,"x":0.1875,"p":[[0,91,0.0,0.14286,0.31135,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[4,91,0.044,0.09821,0.25862,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[8,91,0.0879,0.14732,0.3164,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[12,91,0.1319,0.08482,0.18162,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[16,91,0.1758,0.11607,0.2683,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[20,91,0.2198,0.1875,0.36498,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[24,91,0.2637,0.10268,0.22654,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[28,91,0.3077,0.0625,0.19865,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,91,0.3516,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,91,0.3956,0.125,0.26904,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[40,91,0.4396,0.08482,0.21683,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[44,91,0.4835,0.11607,0.2299,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[48,91,0.5275,0.06696,0.20198,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[52,91,0.5714,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,91,0.6154,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,91,0.6593,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[64,91,0.7033,0.04464,0.12595,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[68,91,0.7473,0.07589,0.2082,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[72,91,0.7912,0.05357,0.18814,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[76,91,0.8352,0.04464,0.12595,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[80,91,0.8791,0.13393,0.3008,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[84,91,0.9231,0.08929,0.25692,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[88,91,0.967,0.13392,0.23672,0.0,0.0,0.21429,0.0,0.857,23,0,0,23,0,1,0,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[91,91,1.0,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.16518,"p":[[0,73,0.0,0.16518,0.34646,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[4,73,0.0548,0.05804,0.1984,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[8,73,0.1096,0.06696,0.20198,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,73,0.1644,0.07143,0.20825,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[16,73,0.2192,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.09375,0.26392,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[24,73,0.3288,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,73,0.3836,0.07142,0.16747,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,73,0.4384,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.06697,0.1988,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,73,0.5479,0.0625,0.2111,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[44,73,0.6027,0.09821,0.27302,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[48,73,0.6575,0.12054,0.28372,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[52,73,0.7123,0.08036,0.20806,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[56,73,0.7671,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,73,0.8219,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[68,73,0.9315,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"80257e8458cd45bc","q":"Let $a, b, c$ be positive real numbers. Prove that\n\n$$\n\\frac{1}{a b(b+1)(c+1)}+\\frac{1}{b c(c+1)(a+1)}+\\frac{1}{c a(a+1)(b+1)} \\geq \\frac{3}{(1+a b c)^{2}}\n$$","t":[{"b":1,"e":0.2857,"k":"falling","v":0.33481,"x":0.90622,"p":[[0,119,0.0,0.86161,0.26841,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,24],[4,119,0.0336,0.90622,0.22763,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,25],[8,119,0.0672,0.77231,0.27863,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,1,0,0,5,0,0,4,0,0,2,0,16],[12,119,0.1008,0.88393,0.25614,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,24],[16,119,0.1345,0.89731,0.15666,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,20],[20,119,0.1681,0.77232,0.3533,0.67857,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,2,0,0,2,0,20],[24,119,0.2017,0.82589,0.30037,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,22],[28,119,0.2353,0.80357,0.30671,0.53572,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,0,2,0,21],[32,119,0.2689,0.85714,0.24744,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3,0,21],[36,119,0.3025,0.76784,0.31895,0.42857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,20],[40,119,0.3361,0.89732,0.24804,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,25],[44,119,0.3697,0.87499,0.18474,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,2,0,20],[48,119,0.4034,0.75891,0.3102,0.67846,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,3,0,0,2,0,0,1,0,0,7,0,0,0,0,17],[52,119,0.437,0.79017,0.27891,0.57132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,3,0,0,2,0,18],[56,119,0.4706,0.83482,0.28146,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,2,0,0,3,0,21],[60,119,0.5042,0.76783,0.3288,0.571,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,3,0,0,1,0,0,2,0,0,2,0,0,3,0,18],[64,119,0.5378,0.81692,0.25816,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,1,0,19],[68,119,0.5714,0.84821,0.2878,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,2,0,22],[72,119,0.605,0.70089,0.33189,0.42857,0.78571,1.0,0.0,1.0,2,15,0,2,0,0,0,0,5,0,0,4,0,0,1,0,0,4,0,0,1,0,15],[76,119,0.6387,0.82589,0.27603,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,4,0,0,0,0,0,2,0,0,3,0,0,1,0,21],[80,119,0.6723,0.73661,0.32362,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,5,0,0,2,0,0,2,0,0,3,0,0,1,0,17],[84,119,0.7059,0.69196,0.38484,0.28571,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,3,0,0,0,0,0,0,0,0,6,0,0,1,0,16],[88,119,0.7395,0.83479,0.28374,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,22],[92,119,0.7731,0.77679,0.24984,0.67857,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,8,0,0,1,0,15],[96,119,0.8067,0.79454,0.26972,0.57143,1.0,1.0,0.14,1.0,0,18,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,4,0,0,1,0,18],[100,119,0.8403,0.80802,0.27806,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,0,1,0,20],[104,119,0.8739,0.45981,0.31488,0.14286,0.4286,0.71429,0.0,1.0,4,4,0,4,0,5,0,0,5,0,0,3,0,0,4,0,0,7,0,0,0,0,4],[108,119,0.9076,0.433,0.29553,0.2857,0.35714,0.60707,0.0,1.0,4,4,0,4,0,2,0,0,10,0,0,5,0,0,3,0,0,4,0,0,0,0,4],[112,119,0.9412,0.34372,0.30061,0.14286,0.28571,0.571,0.0,1.0,7,2,0,7,0,6,0,0,7,0,0,3,0,0,2,0,0,4,0,0,1,0,2],[116,119,0.9748,0.33481,0.23036,0.14289,0.28571,0.42857,0.0,1.0,3,1,0,3,0,7,0,0,11,0,0,4,0,0,3,0,0,3,0,0,0,0,1],[119,119,1.0,0.36156,0.22294,0.14286,0.28571,0.571,0.0,0.857,2,0,0,2,0,8,0,0,8,0,0,4,0,0,6,0,0,3,0,0,1,0,0]]},{"b":4,"e":1.0,"k":"flat","v":0.84375,"x":1.0,"p":[[0,178,0.0,0.91518,0.21682,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,27],[4,178,0.0225,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[8,178,0.0449,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[12,178,0.0674,0.94196,0.17807,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[16,178,0.0899,0.87499,0.24938,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[20,178,0.1124,0.8616,0.27314,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,24],[24,178,0.1348,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[28,178,0.1573,0.95982,0.1439,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[32,178,0.1798,0.90625,0.19434,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,25],[36,178,0.2022,0.875,0.27606,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,25],[40,178,0.2247,0.84375,0.27049,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,4,0,0,1,0,22],[44,178,0.2472,0.93749,0.19544,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[48,178,0.2697,0.875,0.25442,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[52,178,0.2921,0.86607,0.27418,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,24],[56,178,0.3146,0.90625,0.22759,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,26],[60,178,0.3371,0.90179,0.25364,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[64,178,0.3596,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,178,0.382,0.9375,0.2141,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[72,178,0.4045,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,178,0.427,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[80,178,0.4494,0.92857,0.17857,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[84,178,0.4719,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[88,178,0.4944,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[92,178,0.5169,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[96,178,0.5393,0.95534,0.13095,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[100,178,0.5618,0.89284,0.19887,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,22],[104,178,0.5843,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[108,178,0.6067,0.91963,0.17107,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[112,178,0.6292,0.95536,0.12595,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[116,178,0.6517,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[120,178,0.6742,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[124,178,0.6966,0.92409,0.13829,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,23],[128,178,0.7191,0.93749,0.1285,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[132,178,0.7416,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[136,178,0.764,0.93302,0.19559,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[140,178,0.7865,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[144,178,0.809,0.95534,0.14917,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[148,178,0.8315,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[152,178,0.8539,0.94642,0.1505,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[156,178,0.8764,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[160,178,0.8989,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[164,178,0.9213,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[168,178,0.9438,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[172,178,0.9663,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[176,178,0.9888,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[178,178,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"536c536683af1c7a","q":"Let  ${ABC}$  be a triangle with  $\\angle BAC={{60}^{{}^\\circ }}$ . Let  $D$  and  $E$  be the feet of the perpendiculars from  ${A}$  to the external angle bisectors of  $\\angle ABC$  and  $\\angle ACB$ , respectively. Let  ${O}$  be the circumcenter of the triangle  ${ABC}$ . Prove that the circumcircles of the triangles  ${ADE}$ and  ${BOC}$  are tangent to each other.","t":[{"b":3,"e":0.1429,"k":"flat","v":0.09821,"x":0.76339,"p":[[0,105,0.0,0.33929,0.3004,0.0,0.28586,0.71429,0.0,0.71429,12,0,4,12,0,0,0,0,5,0,0,4,0,0,1,0,0,10,0,0,0,0,0],[4,105,0.0381,0.62052,0.26633,0.67846,0.71429,0.71429,0.0,1.0,4,3,0,4,0,0,0,0,0,0,0,3,0,0,1,0,0,21,0,0,0,0,3],[8,105,0.0762,0.27231,0.30379,0.0,0.2857,0.46418,0.0,1.0,15,1,0,15,0,0,0,0,7,0,0,2,0,0,1,0,0,6,0,0,0,0,1],[12,105,0.1143,0.27231,0.28427,0.0,0.28571,0.46418,0.0,0.71429,15,0,0,15,0,0,0,0,4,0,0,5,0,0,2,0,0,6,0,0,0,0,0],[16,105,0.1524,0.32143,0.30514,0.0,0.28571,0.71429,0.0,1.0,12,1,0,12,0,0,0,0,8,0,0,3,0,0,0,0,0,8,0,0,0,0,1],[20,105,0.1905,0.21428,0.29014,0.0,0.0,0.32143,0.0,0.71429,19,0,0,19,0,0,0,0,5,0,0,1,0,0,0,0,0,7,0,0,0,0,0],[24,105,0.2286,0.09821,0.21558,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[28,105,0.2667,0.24107,0.33012,0.0,0.0,0.60714,0.0,1.0,20,1,0,20,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,0,0,0,1],[32,105,0.3048,0.66518,0.29366,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,1,0,0,1,0,0,9,0,0,0,0,11],[36,105,0.3429,0.56697,0.29769,0.28571,0.64286,0.74996,0.0,1.0,1,6,0,1,0,0,0,0,13,0,0,1,0,0,1,0,0,8,0,0,2,0,6],[40,105,0.381,0.53571,0.32537,0.28571,0.28571,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,17,0,0,1,0,0,0,0,0,3,0,0,2,0,8],[44,105,0.419,0.76339,0.29366,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,5,0,0,1,0,0,0,0,0,8,0,0,1,0,16],[48,105,0.4571,0.74104,0.26108,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,4,0,0,0,0,0,2,0,0,10,0,0,5,0,10],[52,105,0.4952,0.73214,0.24936,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,13,0,0,1,0,11],[56,105,0.5333,0.75446,0.1996,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,16,0,0,2,0,9],[60,105,0.5714,0.72321,0.22285,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,16,0,0,2,0,8],[64,105,0.6095,0.65625,0.25719,0.39286,0.71429,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0,14,0,0,1,0,7],[68,105,0.6476,0.62495,0.19805,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,16,0,0,2,0,2],[72,105,0.6857,0.56696,0.26603,0.28571,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,2,0,0,0,0,0,12,0,0,0,0,5],[76,105,0.7238,0.61607,0.22711,0.39286,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,2,0,0,2,0,0,15,0,0,2,0,3],[80,105,0.7619,0.6116,0.27718,0.28571,0.71429,0.85704,0.14286,1.0,0,6,0,0,0,1,0,0,9,0,0,3,0,0,1,0,0,9,0,0,3,0,6],[84,105,0.8,0.53571,0.24484,0.28571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,12,0,0,3,0,0,0,0,0,11,0,0,4,0,1],[88,105,0.8381,0.40177,0.18705,0.2857,0.28571,0.57111,0.1429,0.857,0,0,0,0,0,1,0,0,20,0,0,2,0,0,3,0,0,5,0,0,1,0,0],[92,105,0.8762,0.4686,0.22356,0.28571,0.28571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,1,0,0,2,0,0,9,0,0,1,0,1],[96,105,0.9143,0.31696,0.14607,0.2857,0.28571,0.28579,0.0,0.71429,2,0,0,2,0,1,0,0,22,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[100,105,0.9524,0.42855,0.23419,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,21,0,0,0,0,0,1,0,0,7,0,0,0,0,2],[104,105,0.9905,0.50891,0.26948,0.28571,0.28571,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,18,0,0,0,0,0,1,0,0,8,0,0,1,0,4],[105,105,1.0,0.35714,0.24222,0.2857,0.28571,0.4286,0.0,0.85714,4,0,0,4,0,2,0,0,16,0,0,3,0,0,1,0,0,3,0,0,3,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.11161,"x":0.61593,"p":[[0,60,0.0,0.31695,0.31689,0.0,0.28571,0.60714,0.0,1.0,14,1,2,14,0,0,0,0,4,0,0,3,0,0,3,0,0,7,0,0,0,0,1],[4,60,0.0667,0.49551,0.28343,0.28571,0.57121,0.71429,0.0,1.0,5,1,0,5,0,1,0,0,4,0,0,4,0,0,3,0,0,13,0,0,1,0,1],[8,60,0.1333,0.33036,0.31428,0.0,0.28571,0.71429,0.0,0.71429,13,0,0,13,0,1,0,0,4,0,0,2,0,0,1,0,0,11,0,0,0,0,0],[12,60,0.2,0.23661,0.26871,0.0,0.14285,0.42858,0.0,0.71429,16,0,0,16,0,0,0,0,6,0,0,4,0,0,1,0,0,5,0,0,0,0,0],[16,60,0.2667,0.22768,0.32705,0.0,0.0,0.60714,0.0,1.0,20,1,0,20,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,0,0,0,1],[20,60,0.3333,0.11161,0.18808,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,1,0,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[24,60,0.4,0.17857,0.24744,0.0,0.0,0.28571,0.0,0.71429,19,0,0,19,0,0,0,0,7,0,0,2,0,0,0,0,0,4,0,0,0,0,0],[28,60,0.4667,0.21874,0.29009,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,0,0,0,6,0,0,2,0,0,1,0,0,4,0,0,0,0,1],[32,60,0.5333,0.22767,0.2786,0.0,0.0,0.32144,0.0,0.71429,17,0,0,17,0,0,0,0,7,0,0,1,0,0,1,0,0,6,0,0,0,0,0],[36,60,0.6,0.61593,0.13561,0.42857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,18,0,0,1,0,0],[40,60,0.6667,0.41067,0.25689,0.14286,0.42857,0.60714,0.0,1.0,1,1,0,1,0,11,0,0,2,0,0,5,0,0,5,0,0,7,0,0,0,0,1],[44,60,0.7333,0.57125,0.1514,0.42857,0.57141,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,9,0,0,5,0,0,15,0,0,0,0,0],[48,60,0.8,0.58033,0.15542,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,5,0,0,13,0,0,0,0,1],[52,60,0.8667,0.58481,0.13533,0.42859,0.64286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,3,0,0,16,0,0,0,0,0],[56,60,0.9333,0.58928,0.17034,0.42857,0.64286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,14,0,0,1,0,1],[60,60,1.0,0.59818,0.12596,0.4286,0.64286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,16,0,0,0,0,0]]}]},{"i":"e981c2aef888a4f4","q":"Let  $n>6$  be a perfect number, and let  $n=p_1^{e_1}\\cdot\\cdot\\cdot p_k^{e_k}$  be its prime factorisation with  $1<p_1<\\dots <p_k$ . Prove that  $e_1$  is an even number.\nA number  $n$  is *perfect* if  $s(n)=2n$ , where  $s(n)$  is the sum of the divisors of  $n$ .\n\n(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)","t":[{"b":3,"e":0.2857,"k":"flat","v":0.33929,"x":0.5357,"p":[[0,23,0.0,0.5357,0.21724,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,20,0,0,4,0,0,1,0,0,3,0,3],[4,23,0.1739,0.44641,0.30251,0.24999,0.42857,0.71429,0.0,1.0,7,1,0,7,0,1,0,0,2,0,0,9,0,0,3,0,0,5,0,0,4,0,1],[8,23,0.3478,0.52231,0.25904,0.42857,0.42859,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,1,0,0,12,0,0,4,0,0,6,0,0,3,0,2],[12,23,0.5217,0.48658,0.30274,0.28571,0.42857,0.71429,0.0,1.0,4,3,0,4,0,3,0,0,2,0,0,10,0,0,3,0,0,3,0,0,4,0,3],[16,23,0.6957,0.41963,0.28779,0.25,0.42857,0.57143,0.0,1.0,5,3,0,5,0,3,0,0,4,0,0,10,0,0,4,0,0,2,0,0,1,0,3],[20,23,0.8696,0.33929,0.14173,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,5,0,0,5,0,0,19,0,0,1,0,0,0,0,0,0,0,0],[23,23,1.0,0.38839,0.12492,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,9,0,0,21,0,0,0,0,0,0,0,0,1,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.41955,"x":0.5134,"p":[[0,28,0.0,0.5134,0.18851,0.42857,0.42857,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,20,0,0,5,0,0,3,0,0,1,0,2],[4,28,0.1429,0.43304,0.23003,0.28571,0.42857,0.46431,0.0,1.0,2,2,0,2,0,3,0,0,4,0,0,15,0,0,4,0,0,1,0,0,1,0,2],[8,28,0.2857,0.5,0.26486,0.42857,0.42857,0.60714,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,15,0,0,2,0,0,2,0,0,2,0,4],[12,28,0.4286,0.46875,0.26302,0.28571,0.42857,0.4286,0.0,1.0,1,5,0,1,0,3,0,0,5,0,0,16,0,0,1,0,0,1,0,0,0,0,5],[16,28,0.5714,0.41955,0.1889,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,2,0,0,6,0,0,19,0,0,0,0,0,2,0,0,1,0,1],[20,28,0.7143,0.41964,0.17835,0.42857,0.42857,0.42857,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,23,0,0,0,0,0,0,0,0,0,0,2],[24,28,0.8571,0.4375,0.11258,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[28,28,1.0,0.43304,0.02486,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"78f486aaf04c7005","q":"Let  $n_1, n_2, \\cdots , n_{51}$  be distinct natural numbers each of which has exactly  $2023$  positive integer factors. For instance,  $2^{2022}$  has exactly  $2023$  positive integer factors  $1,2, 2^{2}, 2^{3}, \\cdots 2^{2021}, 2^{2022}$ . Assume that no prime larger than  $11$  divides any of the  $n_{i}$ 's. Show that there must be some perfect cube among the  $n_{i}$ 's.","t":[{"b":0,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,35,0.0,0.92411,0.20512,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[4,35,0.1143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,35,0.2286,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,35,0.3429,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,35,0.4571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,35,0.5714,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[24,35,0.6857,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,24,0.0,0.92411,0.19228,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,27],[4,24,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,24,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"dbc7215e2e9798f1","q":"Let  $n>1$  be a positive integer. Each cell of an  $n\\times n$  table contains an integer. Suppose that the following conditions are satisfied:\n[list=1]\n[*] Each number in the table is congruent to  $1$  modulo  $n$ .\n[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to  $n$  modulo  $n^2$ .\n[/list]\nLet  $R_i$  be the product of the numbers in the  $i^{\\text{th}}$  row, and  $C_j$  be the product of the number in the  $j^{\\text{th}}$  column. Prove that the sums  $R_1+\\hdots R_n$  and  $C_1+\\hdots C_n$  are congruent modulo  $n^4$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.10714,"x":0.22756,"p":[[0,55,0.0,0.14277,0.13832,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,17,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,55,0.0727,0.125,0.15465,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,13,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,55,0.1455,0.16965,0.1448,0.10714,0.14286,0.1786,0.0,0.4286,8,0,0,8,0,16,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.19643,0.15047,0.14286,0.14286,0.1786,0.0,0.57143,4,0,0,4,0,20,0,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[16,55,0.2909,0.14723,0.12619,0.14214,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,21,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,55,0.3636,0.14284,0.13828,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,18,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,55,0.4364,0.14731,0.14497,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,18,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,55,0.5091,0.1607,0.1417,0.14286,0.14286,0.14286,0.0,0.571,7,0,0,7,0,20,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,55,0.5818,0.13839,0.13592,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,17,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,55,0.6545,0.10714,0.12877,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,13,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.21874,0.17849,0.14286,0.14286,0.42857,0.0,0.57143,6,0,0,6,0,15,0,0,2,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[44,55,0.8,0.14286,0.13363,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,16,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.20089,0.16698,0.14286,0.14286,0.32143,0.0,0.57143,6,0,0,6,0,17,0,0,1,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[52,55,0.9455,0.22756,0.17444,0.14286,0.14286,0.42857,0.0,0.57143,5,0,0,5,0,15,0,0,3,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[55,55,1.0,0.15177,0.15538,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,16,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.14732,"x":0.21429,"p":[[0,62,0.0,0.1607,0.12748,0.14286,0.14286,0.1786,0.0,0.571,7,0,0,7,0,17,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,62,0.0645,0.19197,0.18423,0.14286,0.14286,0.1429,0.0,0.71429,6,0,0,6,0,19,0,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[8,62,0.129,0.19643,0.15872,0.14286,0.14286,0.17857,0.0,0.57143,5,0,0,5,0,19,0,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[12,62,0.1935,0.15625,0.13533,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,20,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,62,0.2581,0.14732,0.13592,0.10714,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,20,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,62,0.3226,0.17411,0.15458,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,18,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,62,0.3871,0.21429,0.21429,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,15,0,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[28,62,0.4516,0.16964,0.1448,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,20,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[32,62,0.5161,0.20973,0.20514,0.14286,0.14286,0.1786,0.0,1.0,5,1,0,5,0,19,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[36,62,0.5806,0.18304,0.17941,0.10714,0.14286,0.17857,0.0,0.71429,8,0,0,8,0,16,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[40,62,0.6452,0.15167,0.1425,0.105,0.14286,0.14286,0.0,0.571,8,0,0,8,0,19,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[44,62,0.7097,0.18303,0.12992,0.14286,0.14286,0.17857,0.0,0.57143,4,0,0,4,0,20,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[48,62,0.7742,0.20536,0.14258,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,20,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[52,62,0.8387,0.16518,0.12428,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,21,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[56,62,0.9032,0.15179,0.12846,0.10714,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,18,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.15625,0.12556,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.19633,0.16269,0.14286,0.14286,0.17857,0.0,0.57143,5,0,0,5,0,19,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"55ba6890cd1c369e","q":"Let  $p(x)$  and  $q(x)$  non constant real polynomials of degree at most  $n$  ( $n > 1$ ). Show that there exists a non zero polynomial  $F(x,y)$  in two variables with real coefficients of degree at most  $2n-2,$  such that  $F(p(t),q(t)) = 0$  for every  $t\\in \\mathbb{R}$ .","t":[{"b":3,"e":0.57143,"k":"flat","v":0.85265,"x":1.0,"p":[[0,24,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,24,0.1667,0.91518,0.22829,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,27],[8,24,0.3333,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,24,0.5,0.88392,0.22992,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,24],[16,24,0.6667,0.88393,0.18013,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,21],[20,24,0.8333,0.85265,0.17312,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,3,0,17],[24,24,1.0,0.875,0.16269,0.85711,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,8,0,17]]},{"b":5,"e":1.0,"k":"flat","v":0.86159,"x":0.98214,"p":[[0,24,0.0,0.91516,0.18512,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,0,0,26],[4,24,0.1667,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,24,0.3333,0.90178,0.25862,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,27],[12,24,0.5,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,24,0.6667,0.93749,0.14702,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[20,24,0.8333,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[24,24,1.0,0.86159,0.15357,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,7,0,15]]}]},{"i":"12a7eb8942ab832b","q":"Let  $p$  be a prime number. \nProve that it is possible to choose a permutation  $a_1, a_2,...,a_p$  of  $1,2,...,p$  such that the numbers  $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$  all have different remainder upon division by  $p$ .","t":[{"b":3,"e":0.57143,"k":"falling","v":0.43297,"x":0.67406,"p":[[0,28,0.0,0.62945,0.41475,0.14286,0.92857,1.0,0.0,1.0,5,16,0,5,0,5,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,16],[4,28,0.1429,0.67406,0.29286,0.571,0.57143,1.0,0.0,1.0,1,11,0,1,0,3,0,0,0,0,0,1,0,0,14,0,0,0,0,0,2,0,11],[8,28,0.2857,0.54014,0.28287,0.571,0.57143,0.57143,0.0,1.0,4,5,0,4,0,2,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,5],[12,28,0.4286,0.59813,0.23267,0.571,0.57143,0.57143,0.0,1.0,1,5,0,1,0,2,0,0,0,0,0,2,0,0,20,0,0,1,0,0,1,0,5],[16,28,0.5714,0.6651,0.23316,0.57132,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,9],[20,28,0.7143,0.55353,0.23891,0.5354,0.57143,0.57143,0.0,1.0,1,4,0,1,0,3,0,0,1,0,0,3,0,0,19,0,0,0,0,0,1,0,4],[24,28,0.8571,0.43297,0.24346,0.24999,0.571,0.57143,0.0,1.0,5,1,0,5,0,3,0,0,1,0,0,3,0,0,19,0,0,0,0,0,0,0,1],[28,28,1.0,0.45533,0.30395,0.25,0.57121,0.57143,0.0,1.0,7,4,0,7,0,1,0,0,1,0,0,5,0,0,14,0,0,0,0,0,0,0,4]]},{"b":4,"e":0.571,"k":"flat","v":0.56246,"x":0.68298,"p":[[0,12,0.0,0.64732,0.43593,0.14286,1.0,1.0,0.0,1.0,6,18,0,6,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,18],[4,12,0.3333,0.68298,0.24935,0.5713,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,1,0,0,17,0,0,0,0,0,0,0,11],[8,12,0.6667,0.68297,0.24678,0.571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,4,0,0,14,0,0,0,0,0,2,0,10],[12,12,1.0,0.56246,0.24206,0.5354,0.57143,0.57143,0.0,1.0,1,5,0,1,0,3,0,0,0,0,0,4,0,0,19,0,0,0,0,0,0,0,5]]}]},{"i":"d8c87f4af202ed55","q":"Let $a \\in] 0 ; 1\\left[\\right.$ and $n>0$ be an integer. We denote $f_{n}$ the function defined on $\\mathbb{R}$ by $f_{n}(x)=x+\\frac{x^{2}}{n}$, for all real $x$. Prove that\n\n$$\n\\frac{a(1-a) n^{2}+2 a^{2} n+a^{3}}{(1-a)^{2} n^{2}+a(2-a) n+a^{2}}<\\underbrace{\\left(f_{n} \\circ f_{n} \\circ \\cdots \\circ f_{n}\\right)}_{n}(a)<\\frac{a n+a^{2}}{(1-a) n+a} .\n$$","t":[{"b":5,"e":0.0,"k":"falling","v":0.15621,"x":0.91517,"p":[[0,70,0.0,0.8482,0.1673,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,2,0,16],[4,70,0.0571,0.70982,0.37027,0.39286,0.92857,1.0,0.0,1.0,4,16,0,4,0,1,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,16],[8,70,0.1143,0.91517,0.16701,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,23],[12,70,0.1714,0.78568,0.31745,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,18],[16,70,0.2286,0.85268,0.29121,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,23],[20,70,0.2857,0.79909,0.30901,0.67857,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,2,0,19],[24,70,0.3429,0.66071,0.40524,0.2857,0.85714,1.0,0.0,1.0,6,15,0,6,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,4,0,15],[28,70,0.4,0.62942,0.40226,0.21429,0.78564,1.0,0.0,1.0,8,12,0,8,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,4,0,12],[32,70,0.4571,0.52676,0.42474,0.0,0.571,1.0,0.0,1.0,9,11,0,9,0,2,0,0,3,0,0,1,0,0,3,0,0,0,0,0,3,0,11],[36,70,0.5143,0.74553,0.31891,0.57143,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,3,0,0,1,0,0,2,0,0,3,0,0,6,0,14],[40,70,0.5714,0.49555,0.36242,0.2857,0.42859,1.0,0.0,1.0,5,9,0,5,0,2,0,0,8,0,0,3,0,0,4,0,0,1,0,0,0,0,9],[44,70,0.6286,0.50444,0.35171,0.25,0.571,0.85714,0.0,1.0,6,6,0,6,0,2,0,0,4,0,0,3,0,0,6,0,0,2,0,0,3,0,6],[48,70,0.6857,0.39279,0.32337,0.0,0.571,0.57143,0.0,1.0,10,3,0,10,0,1,0,0,4,0,0,0,0,0,12,0,0,2,0,0,0,0,3],[52,70,0.7429,0.366,0.36055,0.0,0.42857,0.57143,0.0,1.0,14,4,0,14,0,0,0,0,1,0,0,2,0,0,10,0,0,0,0,0,1,0,4],[56,70,0.8,0.38835,0.39159,0.0,0.571,0.57143,0.0,1.0,15,5,0,15,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,2,0,5],[60,70,0.8571,0.33482,0.33237,0.0,0.2857,0.57143,0.0,1.0,12,2,0,12,0,1,0,0,7,0,0,1,0,0,5,0,0,1,0,0,3,0,2],[64,70,0.9143,0.38835,0.32385,0.0,0.571,0.57143,0.0,1.0,11,1,0,11,0,0,0,0,4,0,0,0,0,0,12,0,0,0,0,0,4,0,1],[68,70,0.9714,0.29007,0.27301,0.0,0.28571,0.571,0.0,0.57143,14,0,0,14,0,1,0,0,2,0,0,0,0,0,15,0,0,0,0,0,0,0,0],[70,70,1.0,0.15621,0.25085,0.0,0.0,0.4642,0.0,0.57143,23,0,0,23,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"falling","v":0.45969,"x":0.93749,"p":[[0,46,0.0,0.77232,0.24964,0.71429,0.71429,1.0,0.0,1.0,1,12,1,1,0,1,0,0,1,0,0,0,0,0,2,0,0,12,0,0,3,0,12],[4,46,0.087,0.90625,0.20705,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,23],[8,46,0.1739,0.93749,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[12,46,0.2609,0.84375,0.26812,0.82143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,5,0,19],[16,46,0.3478,0.89284,0.16754,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,20],[20,46,0.4348,0.85266,0.19395,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,7,0,16],[24,46,0.5217,0.7857,0.3093,0.71429,1.0,1.0,0.0,1.0,3,17,1,3,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,3,0,17],[28,46,0.6087,0.82588,0.27604,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,5,0,18],[32,46,0.6957,0.81696,0.25059,0.71429,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,7,0,0,3,0,17],[36,46,0.7826,0.7232,0.34431,0.49968,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,5,0,0,0,0,0,2,0,0,3,0,0,4,0,15],[40,46,0.8696,0.66515,0.33619,0.39288,0.71429,1.0,0.0,1.0,2,13,0,2,0,2,0,0,4,0,0,1,0,0,6,0,0,3,0,0,1,0,13],[44,46,0.9565,0.77232,0.25218,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,3,0,0,3,0,0,7,0,0,4,0,13],[46,46,1.0,0.45969,0.22523,0.28571,0.4286,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,4,0,0,11,0,0,0,0,0,3,0,1]]}]},{"i":"74851c49c6f20680","q":"Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m, n)$ for which $m^{2}+a n+b$ and $n^{2}+a m+b$ are both perfect squares. Prove that $a$ divides $2 b$.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.29463,"x":0.95089,"p":[[0,80,0.0,0.70982,0.37709,0.42857,0.85714,1.0,0.0,1.0,5,15,1,5,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,6,0,15],[4,80,0.05,0.69196,0.39787,0.28571,1.0,1.0,0.0,1.0,4,18,0,4,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,18],[8,80,0.1,0.64732,0.38463,0.28571,0.85714,1.0,0.0,1.0,3,12,0,3,0,4,0,0,5,0,0,0,0,0,0,0,0,1,0,0,7,0,12],[12,80,0.15,0.90179,0.25111,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,25],[16,80,0.2,0.87053,0.27049,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,22],[20,80,0.25,0.90179,0.26351,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,26],[24,80,0.3,0.95089,0.17717,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[28,80,0.35,0.89732,0.21199,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,22],[32,80,0.4,0.84375,0.28428,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,22],[36,80,0.45,0.82143,0.28794,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,8,0,18],[40,80,0.5,0.68749,0.38867,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,17],[44,80,0.55,0.82589,0.30037,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,21],[48,80,0.6,0.7991,0.32313,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,4,0,20],[52,80,0.65,0.85714,0.24485,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,8,0,19],[56,80,0.7,0.94643,0.15872,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[60,80,0.75,0.86161,0.28679,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[64,80,0.8,0.85268,0.29121,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,24],[68,80,0.85,0.77679,0.31326,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,4,0,0,0,0,0,1,0,0,3,0,0,3,0,18],[72,80,0.9,0.53571,0.36422,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,10,0,0,4,0,0,5,0,0,0,0,0,1,0,0,2,0,10],[76,80,0.95,0.39285,0.29014,0.14286,0.28571,0.57111,0.14286,1.0,0,4,0,0,0,13,0,0,5,0,0,5,0,0,4,0,0,0,0,0,1,0,4],[80,80,1.0,0.29463,0.14696,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,13,0,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.04455,"x":0.9107,"p":[[0,67,0.0,0.9107,0.23625,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[4,67,0.0597,0.75,0.37965,0.39288,1.0,1.0,0.0,1.0,3,20,0,3,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,20],[8,67,0.1194,0.60714,0.41955,0.24999,0.85714,1.0,0.0,1.0,7,14,0,7,0,1,0,0,4,0,0,2,0,0,0,0,0,1,0,0,3,0,14],[12,67,0.1791,0.69629,0.38754,0.39285,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,16],[16,67,0.2388,0.75445,0.37327,0.67846,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,18],[20,67,0.2985,0.47321,0.43218,0.0,0.28571,1.0,0.0,1.0,9,11,0,9,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0,2,0,11],[24,67,0.3582,0.53125,0.44783,0.14286,0.57135,1.0,0.0,1.0,7,13,0,7,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,13],[28,67,0.4179,0.45982,0.44854,0.0,0.28571,1.0,0.0,1.0,12,11,0,12,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,11],[32,67,0.4776,0.54018,0.46116,0.0,0.85714,1.0,0.0,1.0,11,13,0,11,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,13],[36,67,0.5373,0.58482,0.44084,0.0,0.85714,1.0,0.0,1.0,9,13,0,9,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,13],[40,67,0.597,0.68304,0.42218,0.28571,1.0,1.0,0.0,1.0,7,18,0,7,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,3,0,18],[44,67,0.6567,0.52679,0.44812,0.0,0.5,1.0,0.0,1.0,9,13,0,9,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,13],[48,67,0.7164,0.375,0.4222,0.0,0.14286,0.85714,0.0,1.0,14,7,0,14,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,7],[52,67,0.7761,0.4866,0.44011,0.0,0.28571,1.0,0.0,1.0,10,11,0,10,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,3,0,11],[56,67,0.8358,0.47768,0.44408,0.0,0.35714,1.0,0.0,1.0,12,11,0,12,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,0,2,0,11],[60,67,0.8955,0.16518,0.25532,0.0,0.14286,0.14286,0.0,1.0,15,2,0,15,0,10,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[64,67,0.9552,0.20536,0.35344,0.0,0.0,0.1786,0.0,1.0,19,5,0,19,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[67,67,1.0,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f32b431180d2f764","q":"Let $a_{0}, b_{0}, c_{0}, a, b, c$ be integers such that $\\operatorname{gcd}\\left(a_{0}, b_{0}, c_{0}\\right)=\\operatorname{gcd}(a, b, c)=1$. Prove that there exists a positive integer $n$ and integers $a_{1}, a_{2}, \\ldots, a_{n}=a, b_{1}, b_{2}, \\ldots, b_{n}=b, c_{1}, c_{2}, \\ldots, c_{n}=c$ such that for all $1 \\leq i \\leq n, a_{i-1} a_{i}+b_{i-1} b_{i}+c_{i-1} c_{i}=1$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.33919,"x":0.47768,"p":[[0,38,0.0,0.33919,0.32296,0.0,0.2857,0.57143,0.0,1.0,9,3,0,9,0,4,0,0,8,0,0,2,0,0,2,0,0,3,0,0,1,0,3],[4,38,0.1053,0.4686,0.3198,0.14289,0.42857,0.71107,0.0,1.0,2,6,0,2,0,7,0,0,5,0,0,5,0,0,4,0,0,3,0,0,0,0,6],[8,38,0.2105,0.47768,0.27574,0.2857,0.42857,0.71429,0.0,1.0,1,3,0,1,0,5,0,0,7,0,0,6,0,0,2,0,0,7,0,0,1,0,3],[12,38,0.3158,0.46426,0.2857,0.28571,0.42857,0.57143,0.0,1.0,3,3,0,3,0,4,0,0,5,0,0,5,0,0,8,0,0,2,0,0,2,0,3],[16,38,0.4211,0.41515,0.24834,0.14286,0.42857,0.57143,0.0,1.0,2,2,0,2,0,7,0,0,3,0,0,8,0,0,8,0,0,2,0,0,0,0,2],[20,38,0.5263,0.42411,0.25874,0.2857,0.42857,0.57143,0.0,1.0,3,2,0,3,0,4,0,0,6,0,0,8,0,0,4,0,0,5,0,0,0,0,2],[24,38,0.6316,0.42401,0.32834,0.14286,0.28571,0.71429,0.0,1.0,4,4,0,4,0,8,0,0,5,0,0,2,0,0,3,0,0,5,0,0,1,0,4],[28,38,0.7368,0.40168,0.28228,0.14286,0.42857,0.4642,0.0,1.0,3,3,0,3,0,7,0,0,4,0,0,10,0,0,2,0,0,2,0,0,1,0,3],[32,38,0.8421,0.42408,0.30405,0.14286,0.42857,0.57143,0.0,1.0,5,3,0,5,0,4,0,0,5,0,0,6,0,0,5,0,0,2,0,0,2,0,3],[36,38,0.9474,0.40625,0.22047,0.28571,0.28571,0.42858,0.0,1.0,1,2,0,1,0,1,0,0,16,0,0,7,0,0,3,0,0,1,0,0,1,0,2],[38,38,1.0,0.39285,0.25999,0.25002,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,8,0,0,13,0,0,1,0,0,6,0,0,0,0,0,1,0,3]]},{"b":7,"e":0.57143,"k":"rising","v":0.29463,"x":0.58482,"p":[[0,39,0.0,0.29463,0.3173,0.0,0.14286,0.57111,0.0,1.0,10,3,0,10,0,8,0,0,4,0,0,1,0,0,4,0,0,2,0,0,0,0,3],[4,39,0.1026,0.55802,0.31003,0.28571,0.57143,0.71429,0.0,1.0,3,6,0,3,0,2,0,0,5,0,0,2,0,0,5,0,0,9,0,0,0,0,6],[8,39,0.2051,0.57585,0.29984,0.42857,0.571,0.85704,0.0,1.0,3,7,0,3,0,0,0,0,3,0,0,8,0,0,7,0,0,2,0,0,2,0,7],[12,39,0.3077,0.50893,0.28333,0.28571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,6,0,0,3,0,0,5,0,0,8,0,0,0,0,4],[16,39,0.4103,0.58482,0.29312,0.39286,0.57143,0.71429,0.0,1.0,1,7,0,1,0,3,0,0,4,0,0,5,0,0,4,0,0,8,0,0,0,0,7],[20,39,0.5128,0.55801,0.30169,0.39286,0.57143,0.71429,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,4,0,0,8,0,0,5,0,0,1,0,6],[24,39,0.6154,0.50436,0.29674,0.25001,0.57143,0.71429,0.0,1.0,4,3,0,4,0,4,0,0,1,0,0,3,0,0,9,0,0,7,0,0,1,0,3],[28,39,0.7179,0.45981,0.29175,0.14286,0.57143,0.71429,0.0,1.0,5,1,0,5,0,5,0,0,1,0,0,3,0,0,5,0,0,12,0,0,0,0,1],[32,39,0.8205,0.5177,0.1982,0.42857,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,3,0,0,4,0,0,13,0,0,9,0,0,0,0,0],[36,39,0.9231,0.57572,0.18019,0.571,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,0,0,0,3,0,0,12,0,0,14,0,0,0,0,0],[39,39,1.0,0.52677,0.2034,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,4,0,0,0,0,0,5,0,0,11,0,0,11,0,0,0,0,0]]}]},{"i":"1f534c358e6cd5e2","q":"Let $P$ be a (non-self-intersecting) polygon in the plane. Let $C_{1}, \\ldots, C_{n}$ be circles in the plane whose interiors cover the interior of $P$. For $1 \\leq i \\leq n$, let $r_{i}$ be the radius of $C_{i}$. Prove that there is a single circle of radius $r_{1}+\\cdots+r_{n}$ whose interior covers the interior of $P$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13839,"x":0.25,"p":[[0,10,0.0,0.13839,0.2004,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,9,0,0,4,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[4,10,0.4,0.25,0.25254,0.14286,0.14286,0.2857,0.0,0.85714,4,0,0,4,0,19,0,0,2,0,0,3,0,0,0,0,0,0,0,0,4,0,0],[8,10,0.8,0.20089,0.20782,0.14286,0.14286,0.14286,0.0,0.85714,4,0,0,4,0,22,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[10,10,1.0,0.18286,0.15255,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.57143,"k":"rising","v":0.29902,"x":0.76785,"p":[[0,13,0.0,0.29902,0.29962,0.0,0.21429,0.42857,0.0,1.0,9,1,0,9,0,7,0,0,5,0,0,5,0,0,1,0,0,0,0,0,4,0,1],[4,13,0.3077,0.49553,0.39927,0.10714,0.64286,0.85714,0.0,1.0,8,3,0,8,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,0,12,0,3],[8,13,0.6154,0.76785,0.21354,0.57143,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,0,0,0,17,0,6],[12,13,0.9231,0.74997,0.21726,0.57132,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,0,0,0,13,0,7],[13,13,1.0,0.73658,0.23988,0.571,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,6,0,0,6,0,0,1,0,0,9,0,9]]}]},{"i":"ca43c1fc845409b0","q":"Let  $x, y, z$  be non-negative real numbers such that  $x + y + z \\leq 1$ . Prove the inequality\n\\[\n6xyz \\leq x(1 - x) + y(1 - y) + z(1 - z),\n\\]\nand determine when equality holds.","t":[{"b":0,"e":1.0,"k":"flat","v":0.74106,"x":0.92857,"p":[[0,42,0.0,0.90179,0.19045,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,2,0,24],[4,42,0.0952,0.76786,0.27141,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,3,0,0,2,0,16],[8,42,0.1905,0.8616,0.21866,0.71421,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,2,0,0,2,0,21],[12,42,0.2857,0.87053,0.17985,0.85713,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,8,0,17],[16,42,0.381,0.86607,0.20497,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,7,0,18],[20,42,0.4762,0.74106,0.2822,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,0,0,0,5,0,0,1,0,0,6,0,13],[24,42,0.5714,0.85713,0.19887,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,3,0,0,3,0,19],[28,42,0.6667,0.87497,0.18128,0.78561,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,4,0,20],[32,42,0.7619,0.88839,0.18466,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,3,0,22],[36,42,0.8571,0.89284,0.1713,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,2,0,22],[40,42,0.9524,0.92857,0.14726,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,25],[42,42,1.0,0.92857,0.14726,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,25]]},{"b":7,"e":1.0,"k":"flat","v":0.79463,"x":0.99107,"p":[[0,15,0.0,0.85712,0.2287,0.57143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,0,0,0,1,0,22],[4,15,0.2667,0.79463,0.25491,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,6,0,0,3,0,0,2,0,17],[8,15,0.5333,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[12,15,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[15,15,1.0,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28]]}]},{"i":"6fb8e56c244fb42d","q":"Let $f$ be a non-constant polynomial with integer coefficients. Prove that there is an integer $n$ such that $f(n)$ has at least 2004 distinct prime factors.","t":[{"b":4,"e":1.0,"k":"flat","v":0.89286,"x":0.99554,"p":[[0,41,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,41,0.0976,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,41,0.1951,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[12,41,0.2927,0.94643,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[16,41,0.3902,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[20,41,0.4878,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[24,41,0.5854,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[28,41,0.6829,0.93304,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[32,41,0.7805,0.91964,0.12846,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,20],[36,41,0.878,0.89286,0.12877,0.82143,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,17],[40,41,0.9756,0.92857,0.10715,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[41,41,1.0,0.89286,0.12372,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,9,0,16]]},{"b":5,"e":1.0,"k":"flat","v":0.94196,"x":0.97321,"p":[[0,17,0.0,0.95536,0.1357,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,17,0.2353,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,17,0.4706,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[12,17,0.7059,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[16,17,0.9412,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[17,17,1.0,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21]]}]},{"i":"f1156cd4f36dfbc6","q":"Let $a_{2}, \\ldots, a_{n}$ be $n-1$ positive real numbers, where $n \\geq 3$, such that $a_{2} a_{3} \\cdots a_{n}=1$. Prove that  $$ \\left(1+a_{2}\\right)^{2}\\left(1+a_{3}\\right)^{3} \\cdots\\left(1+a_{n}\\right)^{n}>n^{n} . $$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,31,0.0,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,31,0.129,0.12054,0.29257,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,31,0.2581,0.08036,0.19212,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,31,0.3871,0.06697,0.20198,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,31,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.15179,"p":[[0,55,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.15179,0.333,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[8,55,0.1455,0.09822,0.25111,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,55,0.2182,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.08482,0.24707,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,55,0.3636,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.04455,0.09052,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.05358,0.10566,0.0,0.0,0.0,0.0,0.286,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.04912,0.10481,0.0,0.0,0.0,0.0,0.286,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7ee121c5781d8660","q":"Let $a, b, c$ be positive real numbers such that $a b c(a+b+c)=3$. Prove the inequality\n\n$$\n(a+b)(b+c)(c+a) \\geq 8\n$$\n\nand determine all cases when equality holds.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.24536,"x":0.375,"p":[[0,17,0.0,0.24536,0.15675,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,16,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.32581,0.13487,0.14286,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,11,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.375,0.1171,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.37054,0.15093,0.35715,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,8,0,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0],[16,17,0.9412,0.34375,0.17807,0.14286,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,12,0,0,0,0,0,18,0,0,0,0,0,1,0,0,1,0,0],[17,17,1.0,0.33036,0.13571,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,11,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.30356,"x":0.36161,"p":[[0,34,0.0,0.32589,0.18293,0.14286,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,1,0,0,17,0,0,1,0,0,0,0,0,1,0,0],[4,34,0.1176,0.34822,0.12846,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,9,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.36161,0.15561,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,1,0,0,20,0,0,0,0,0,2,0,0,0,0,0],[12,34,0.3529,0.35268,0.18205,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,11,0,0,0,0,0,19,0,0,1,0,0,0,0,0,0,0,1],[16,34,0.4706,0.30356,0.14614,0.14286,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,14,0,0,1,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[20,34,0.5882,0.33929,0.13243,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,10,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.34366,0.17817,0.14286,0.42857,0.42857,0.14,0.85714,0,0,0,0,0,12,0,0,0,0,0,18,0,0,0,0,0,1,0,0,1,0,0],[28,34,0.8235,0.30357,0.13243,0.14286,0.35714,0.42857,0.14286,0.4286,0,0,0,0,0,12,0,0,4,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.34375,0.18851,0.14286,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,12,0,0,2,0,0,15,0,0,0,0,0,2,0,0,1,0,0],[34,34,1.0,0.33036,0.14914,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,2,0,0,18,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"24549d1c57b7ee78","q":"Let $m$ and $n$ be integers such that $0 \\leqslant m \\leqslant 2 n$. Prove that the integer\n\n$$\n2^{2 n+2}+2^{m+2}+1\n$$\n\nis a perfect square if, and only if, $m=n$.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.62498,"x":0.8482,"p":[[0,66,0.0,0.62498,0.18471,0.57132,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,19,0,0,3,0,0],[4,66,0.0606,0.8125,0.19377,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,5,0,13],[8,66,0.1212,0.74107,0.16917,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,13,0,0,7,0,5],[12,66,0.1818,0.6964,0.23624,0.571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,9,0,0,1,0,9],[16,66,0.2424,0.799,0.21711,0.71429,0.78571,1.0,0.14,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,10,0,0,2,0,14],[20,66,0.303,0.79464,0.18877,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,11,0,0,3,0,12],[24,66,0.3636,0.73214,0.16656,0.71429,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,17,0,0,4,0,5],[28,66,0.4242,0.79017,0.1988,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,12,0,0,4,0,11],[32,66,0.4848,0.78125,0.17122,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,0,5,0,9],[36,66,0.5455,0.8482,0.17476,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,4,0,16],[40,66,0.6061,0.75892,0.16537,0.67857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,13,0,0,3,0,8],[44,66,0.6667,0.77241,0.19843,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,13,0,0,2,0,11],[48,66,0.7273,0.75,0.24743,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,6,0,0,3,0,12],[52,66,0.7879,0.68302,0.22228,0.57132,0.71429,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,12,0,0,1,0,7],[56,66,0.8485,0.75446,0.24545,0.67857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,1,0,0,4,0,0,9,0,0,4,0,11],[60,66,0.9091,0.73658,0.20553,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,8,0,0,5,0,8],[64,66,0.9697,0.83928,0.14617,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,6,0,12],[66,66,1.0,0.75893,0.14914,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,0,2,0,7]]},{"b":2,"e":0.71429,"k":"flat","v":0.63388,"x":0.86159,"p":[[0,62,0.0,0.63388,0.24728,0.42857,0.71429,0.74996,0.0,1.0,1,4,1,1,0,1,0,0,2,0,0,6,0,0,3,0,0,11,0,0,4,0,4],[4,62,0.0645,0.8214,0.15976,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,2,0,13],[8,62,0.129,0.7946,0.19217,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,9,0,0,7,0,10],[12,62,0.1935,0.79018,0.1636,0.71429,0.71429,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,3,0,10],[16,62,0.2581,0.86159,0.15767,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,5,0,16],[20,62,0.3226,0.75446,0.1684,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,17,0,0,3,0,7],[24,62,0.3871,0.82588,0.2105,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,0,4,0,16],[28,62,0.4516,0.83034,0.17657,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,14],[32,62,0.5161,0.78125,0.19227,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,12,0,0,6,0,9],[36,62,0.5806,0.76338,0.21313,0.71429,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,13,0,0,3,0,10],[40,62,0.6452,0.78568,0.13837,0.71429,0.71429,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,6,0,7],[44,62,0.7097,0.82589,0.15865,0.71429,0.78571,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,3,0,13],[48,62,0.7742,0.82589,0.16263,0.71429,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,3,0,13],[52,62,0.8387,0.73661,0.2372,0.67857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,10,0,0,5,0,9],[56,62,0.9032,0.81249,0.16919,0.71429,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,1,0,13],[60,62,0.9677,0.74552,0.20744,0.67857,0.71429,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,11,0,0,5,0,8],[62,62,1.0,0.76338,0.16604,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,14,0,0,5,0,7]]}]},{"i":"87d3edbc13227c23","q":"Let  $n,s,t$  be three positive integers, and let  $A_1,\\ldots, A_s, B_1,\\ldots, B_t$  be non-necessarily distinct subsets of  $\\{1,2,\\ldots,n\\}$ . For any subset  $S$  of  $\\{1,\\ldots,n\\}$ , define  $f(S)$  to be the number of  $i\\in\\{1,\\ldots,s\\}$  with  $S\\subseteq A_i$  and  $g(S)$  to be the number of  $j\\in\\{1,\\ldots,t\\}$  with  $S\\subseteq B_j$ . Assume that for any  $1\\leq x<y\\leq n$ , we have  $f(\\{x,y\\})=g(\\{x,y\\})$ . Show that if  $t<n$ , then there exists some  $1\\leq x\\leq n$  so that  $f(\\{x\\})\\geq g(\\{x\\})$ .\n\n*Proposed by usjl*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.19634,"x":0.875,"p":[[0,44,0.0,0.84375,0.3038,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,25],[4,44,0.0909,0.8125,0.33586,0.82143,1.0,1.0,0.0,1.0,1,23,1,1,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[8,44,0.1818,0.875,0.29397,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,27],[12,44,0.2727,0.19634,0.15875,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,26,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,44,0.3636,0.25445,0.17397,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,17,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[20,44,0.4545,0.24552,0.13471,0.14286,0.14288,0.28571,0.14286,0.57143,0,0,0,0,0,17,0,0,10,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[24,44,0.5455,0.28125,0.2461,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,18,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[28,44,0.6364,0.23652,0.20084,0.14286,0.14286,0.28571,0.14,1.0,0,1,0,0,0,22,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[32,44,0.7273,0.25,0.11294,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,14,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[36,44,0.8182,0.20536,0.11259,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,23,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,44,0.9091,0.23652,0.16987,0.14286,0.14286,0.28571,0.14,1.0,0,1,0,0,0,19,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[44,44,1.0,0.21875,0.10092,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,36,0.0,0.89286,0.28571,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,36,0.1111,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,36,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,36,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,36,0.5556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,36,0.6667,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3ac64c7e928a5477","q":"Let $n$ and $M$ be positive integers such that $M>n^{n-1}$. Prove that there are $n$ distinct primes $p_{1}, p_{2}, p_{3}, \\ldots, p_{n}$ such that $p_{j}$ divides $M+j$ for $1 \\leq j \\leq n$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.10268,"x":0.26786,"p":[[0,57,0.0,0.26786,0.18814,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,6,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[4,57,0.0702,0.26337,0.24248,0.0,0.21428,0.4286,0.0,0.71429,12,0,0,12,0,4,0,0,1,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[8,57,0.1404,0.22319,0.25735,0.0,0.14286,0.46418,0.0,0.85714,15,0,0,15,0,4,0,0,3,0,0,2,0,0,7,0,0,0,0,0,1,0,0],[12,57,0.2105,0.18304,0.1996,0.0,0.14286,0.42857,0.0,0.57143,13,0,0,13,0,9,0,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[16,57,0.2807,0.21871,0.26717,0.0,0.0,0.4642,0.0,0.85714,17,0,0,17,0,3,0,0,0,0,0,4,0,0,7,0,0,0,0,0,1,0,0],[20,57,0.3509,0.19642,0.2223,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,3,0,0,2,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[24,57,0.4211,0.1607,0.21051,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,5,0,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[28,57,0.4912,0.125,0.18472,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,6,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[32,57,0.5614,0.26338,0.26988,0.0,0.14288,0.42858,0.0,0.85714,12,0,0,12,0,5,0,0,3,0,0,5,0,0,4,0,0,1,0,0,2,0,0],[36,57,0.6316,0.20981,0.23413,0.0,0.14286,0.4286,0.0,0.57143,15,0,0,15,0,5,0,0,0,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[40,57,0.7018,0.23658,0.22187,0.0,0.2143,0.42857,0.0,0.57143,12,0,0,12,0,4,0,0,5,0,0,5,0,0,6,0,0,0,0,0,0,0,0],[44,57,0.7719,0.20981,0.25499,0.0,0.07143,0.4286,0.0,0.85714,16,0,0,16,0,4,0,0,2,0,0,3,0,0,6,0,0,0,0,0,1,0,0],[48,57,0.8421,0.16516,0.22616,0.0,0.0,0.32142,0.0,0.57143,18,0,0,18,0,5,0,0,1,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[52,57,0.9123,0.11607,0.16146,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,7,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[56,57,0.9825,0.10268,0.16457,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[57,57,1.0,0.19195,0.2131,0.0,0.14286,0.42857,0.0,0.57143,14,0,0,14,0,7,0,0,1,0,0,6,0,0,4,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.05357,"x":0.27231,"p":[[0,47,0.0,0.26339,0.20238,0.0,0.42857,0.42857,0.0,0.57143,11,0,0,11,0,1,0,0,3,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[4,47,0.0851,0.24104,0.26588,0.0,0.14286,0.57141,0.0,0.71429,15,0,0,15,0,4,0,0,1,0,0,1,0,0,10,0,0,1,0,0,0,0,0],[8,47,0.1702,0.26786,0.24155,0.0,0.2857,0.571,0.0,0.57143,12,0,0,12,0,3,0,0,3,0,0,5,0,0,9,0,0,0,0,0,0,0,0],[12,47,0.2553,0.23215,0.22232,0.0,0.1429,0.42858,0.0,0.57143,12,0,0,12,0,5,0,0,4,0,0,5,0,0,6,0,0,0,0,0,0,0,0],[16,47,0.3404,0.27231,0.25841,0.0,0.21428,0.57111,0.0,0.85714,11,0,0,11,0,5,0,0,4,0,0,3,0,0,7,0,0,1,0,0,1,0,0],[20,47,0.4255,0.17857,0.19233,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,8,0,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[24,47,0.5106,0.13837,0.18717,0.0,0.07143,0.14287,0.0,0.57143,16,0,0,16,0,9,0,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[28,47,0.5957,0.20978,0.24734,0.0,0.14286,0.46418,0.0,0.71429,15,0,0,15,0,6,0,0,1,0,0,2,0,0,7,0,0,1,0,0,0,0,0],[32,47,0.6809,0.1741,0.21644,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[36,47,0.766,0.12054,0.17169,0.0,0.0,0.14287,0.0,0.57143,18,0,0,18,0,7,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[40,47,0.8511,0.10268,0.1931,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[44,47,0.9362,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.16515,0.19263,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,6,0,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"baf8bb62e1e05f1a","q":"Let $n$ and $k$ be positive integers. There are $n k$ objects (of the same size) and $k$ boxes, each of which can hold $n$ objects. Each object is coloured in one of $k$ different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.","t":[{"b":4,"e":0.57143,"k":"rising","v":0.13393,"x":0.44643,"p":[[0,50,0.0,0.13393,0.21998,0.0,0.0,0.17857,0.0,0.71429,20,0,10,20,0,4,0,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[4,50,0.08,0.41964,0.40238,0.0,0.28571,0.85714,0.0,1.0,13,5,0,13,0,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,4,0,5],[8,50,0.16,0.41964,0.40554,0.0,0.35714,0.85714,0.0,1.0,13,6,0,13,0,0,0,0,3,0,0,3,0,0,2,0,0,1,0,0,4,0,6],[12,50,0.24,0.39732,0.33832,0.0,0.42857,0.71429,0.0,1.0,11,1,0,11,0,1,0,0,2,0,0,5,0,0,1,0,0,8,0,0,3,0,1],[16,50,0.32,0.36161,0.34439,0.0,0.28571,0.71429,0.0,1.0,10,2,0,10,0,4,0,0,6,0,0,0,0,0,2,0,0,5,0,0,3,0,2],[20,50,0.4,0.30803,0.36615,0.0,0.07143,0.71429,0.0,1.0,16,2,0,16,0,2,0,0,2,0,0,2,0,0,0,0,0,5,0,0,3,0,2],[24,50,0.48,0.37045,0.39105,0.0,0.14288,0.71429,0.0,1.0,13,4,0,13,0,4,0,0,2,0,0,0,0,0,0,0,0,7,0,0,2,0,4],[28,50,0.56,0.38391,0.34889,0.0,0.28571,0.71429,0.0,1.0,11,2,0,11,0,2,0,0,5,0,0,0,0,0,1,0,0,10,0,0,1,0,2],[32,50,0.64,0.4017,0.3263,0.14214,0.35714,0.71429,0.0,1.0,7,1,0,7,0,6,0,0,3,0,0,3,0,0,2,0,0,6,0,0,4,0,1],[36,50,0.72,0.44643,0.32093,0.10714,0.42857,0.71429,0.0,1.0,8,1,0,8,0,1,0,0,4,0,0,4,0,0,0,0,0,12,0,0,2,0,1],[40,50,0.8,0.38839,0.36287,0.0,0.28571,0.71429,0.0,1.0,10,3,0,10,0,5,0,0,2,0,0,2,0,0,2,0,0,5,0,0,3,0,3],[44,50,0.88,0.27232,0.29529,0.0,0.21428,0.42857,0.0,1.0,13,1,0,13,0,3,0,0,4,0,0,7,0,0,0,0,0,2,0,0,2,0,1],[48,50,0.96,0.35714,0.22016,0.24999,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,5,0,0,9,0,0,9,0,0,2,0,0,2,0,0,2,0,0],[50,50,1.0,0.30793,0.21762,0.14286,0.2857,0.57111,0.0,0.71429,5,0,0,5,0,9,0,0,4,0,0,5,0,0,8,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.12045,"x":0.62946,"p":[[0,68,0.0,0.12045,0.21161,0.0,0.0,0.14286,0.0,0.85714,20,0,9,20,0,7,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[4,68,0.0588,0.52679,0.43071,0.0,0.71429,0.89286,0.0,1.0,12,8,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,5,0,8],[8,68,0.1176,0.38393,0.41563,0.0,0.21429,0.85714,0.0,1.0,15,5,0,15,0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,5],[12,68,0.1765,0.52229,0.33996,0.28571,0.571,0.85714,0.0,1.0,6,4,0,6,0,0,0,0,6,0,0,3,0,0,2,0,0,6,0,0,5,0,4],[16,68,0.2353,0.51339,0.38109,0.0,0.71429,0.85714,0.0,1.0,9,6,0,9,0,0,0,0,3,0,0,3,0,0,0,0,0,8,0,0,3,0,6],[20,68,0.2941,0.625,0.33072,0.28571,0.71429,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,5,0,0,1,0,0,2,0,0,8,0,0,5,0,7],[24,68,0.3529,0.62946,0.2854,0.42857,0.71429,0.85714,0.0,1.0,3,3,0,3,0,0,0,0,4,0,0,2,0,0,1,0,0,12,0,0,7,0,3],[28,68,0.4118,0.61607,0.31225,0.39286,0.71429,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,3,0,0,2,0,0,2,0,0,8,0,0,8,0,4],[32,68,0.4706,0.47321,0.28669,0.2857,0.5,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,10,0,0,1,0,0,3,0,0,9,0,0,3,0,1],[36,68,0.5294,0.4732,0.2911,0.28571,0.49979,0.71429,0.0,1.0,4,2,0,4,0,1,0,0,10,0,0,1,0,0,4,0,0,8,0,0,2,0,2],[40,68,0.5882,0.5982,0.22428,0.42857,0.71429,0.71429,0.1429,1.0,0,2,0,0,0,2,0,0,4,0,0,5,0,0,2,0,0,15,0,0,2,0,2],[44,68,0.6471,0.50893,0.34799,0.28571,0.57143,0.85714,0.0,1.0,6,3,0,6,0,1,0,0,7,0,0,2,0,0,0,0,0,6,0,0,7,0,3],[48,68,0.7059,0.51339,0.27166,0.28571,0.50001,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,3,0,0,7,0,0,1,0,0,12,0,0,1,0,2],[52,68,0.7647,0.49552,0.2789,0.2857,0.57121,0.71429,0.0,1.0,2,1,0,2,0,4,0,0,7,0,0,2,0,0,4,0,0,8,0,0,4,0,1],[56,68,0.8235,0.46427,0.28121,0.2857,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,1,0,0,9,0,0,4,0,0,3,0,0,5,0,0,6,0,0],[60,68,0.8824,0.58929,0.28738,0.39286,0.71429,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,5,0,0,5,0,0,1,0,0,11,0,0,2,0,5],[64,68,0.9412,0.57589,0.26603,0.28571,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,7,0,0,1,0,0,1,0,0,13,0,0,7,0,0],[68,68,1.0,0.52679,0.20958,0.39286,0.50001,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,8,0,0,3,0,0,10,0,0,3,0,0]]}]},{"i":"24e5d53fadbed200","q":"Players  $A$  and  $B$  play the following game:  $A$  tosses a coin  $n$  times, and  $B$  does  $n+1$  times. The player who obtains more \u201dheads\u201d wins; or in the case of equal balances,  $A$  is assigned victory. Find the values of  $n$  for which this game is fair (i.e. both players have equal chances for victory).","t":[{"b":1,"e":1.0,"k":"rising","v":0.53125,"x":0.98661,"p":[[0,11,0.0,0.53125,0.37836,0.14286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,9],[4,11,0.3636,0.91071,0.18123,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[8,11,0.7273,0.91071,0.13243,0.82132,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,21],[11,11,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":3,"e":0.71429,"k":"rising","v":0.58035,"x":0.92857,"p":[[0,13,0.0,0.58035,0.35881,0.14286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,12,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,8],[4,13,0.3077,0.91518,0.22548,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,26],[8,13,0.6154,0.92856,0.12376,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,23],[12,13,0.9231,0.90625,0.16214,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,3,0,22],[13,13,1.0,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24]]}]},{"i":"67375a5f6720d4b3","q":"Let $n$ be a fixed integer with $n \\geqslant 2$. We say that two polynomials $P$ and $Q$ with real coefficients are block-similar if for each $i \\in\\{1,2, \\ldots, n\\}$ the sequences  $$ \\begin{aligned} & P(2015 i), P(2015 i-1), \\ldots, P(2015 i-2014) \\quad \\text { and } \\\\ & Q(2015 i), Q(2015 i-1), \\ldots, Q(2015 i-2014) \\end{aligned} $$  are permutations of each other. (a) Prove that there exist distinct block-similar polynomials of degree $n+1$. (b) Prove that there do not exist distinct block-similar polynomials of degree $n$.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.15179,"x":0.45982,"p":[[0,36,0.0,0.15179,0.14698,0.0,0.14286,0.14287,0.0,0.71429,9,0,0,9,0,16,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,36,0.1111,0.38391,0.21556,0.2857,0.35729,0.4642,0.0,1.0,1,1,0,1,0,6,0,0,9,0,0,8,0,0,5,0,0,1,0,0,1,0,1],[8,36,0.2222,0.45982,0.24153,0.2857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,11,0,0,6,0,0,0,0,0,8,0,0,2,0,1],[12,36,0.3333,0.37945,0.21311,0.2857,0.28571,0.4642,0.0,0.85714,1,0,0,1,0,6,0,0,11,0,0,6,0,0,2,0,0,5,0,0,1,0,0],[16,36,0.4444,0.45981,0.30037,0.2857,0.35714,0.71429,0.0,1.0,2,3,0,2,0,5,0,0,9,0,0,4,0,0,1,0,0,5,0,0,3,0,3],[20,36,0.5556,0.41072,0.28959,0.1429,0.28571,0.71429,0.0,1.0,2,3,0,2,0,7,0,0,9,0,0,5,0,0,0,0,0,5,0,0,1,0,3],[24,36,0.6667,0.44195,0.23785,0.2857,0.35714,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,12,0,0,5,0,0,4,0,0,4,0,0,1,0,2],[28,36,0.7778,0.41968,0.21998,0.2857,0.28571,0.50107,0.14286,1.0,0,1,0,0,0,3,0,0,15,0,0,6,0,0,0,0,0,6,0,0,1,0,1],[32,36,0.8889,0.39731,0.23886,0.2857,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,4,0,0,12,0,0,5,0,0,2,0,0,4,0,0,3,0,0],[36,36,1.0,0.41071,0.26665,0.2857,0.28571,0.46431,0.14286,1.0,0,2,0,0,0,6,0,0,15,0,0,3,0,0,1,0,0,1,0,0,4,0,2]]},{"b":5,"e":0.42857,"k":"rising","v":0.16063,"x":0.4464,"p":[[0,58,0.0,0.16063,0.17036,0.0,0.14286,0.1429,0.0,0.71429,10,0,1,10,0,15,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,58,0.069,0.4464,0.26665,0.2857,0.35714,0.60714,0.0,1.0,2,2,0,2,0,3,0,0,11,0,0,3,0,0,5,0,0,4,0,0,2,0,2],[8,58,0.1379,0.43304,0.27078,0.2857,0.28571,0.57143,0.0,1.0,1,3,0,1,0,4,0,0,13,0,0,4,0,0,3,0,0,2,0,0,2,0,3],[12,58,0.2069,0.37945,0.21608,0.14286,0.28571,0.4642,0.14286,0.85714,0,0,0,0,0,9,0,0,8,0,0,7,0,0,3,0,0,3,0,0,2,0,0],[16,58,0.2759,0.34375,0.17807,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,14,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[20,58,0.3448,0.34822,0.15946,0.2857,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,5,0,0,16,0,0,6,0,0,2,0,0,3,0,0,0,0,0],[24,58,0.4138,0.33034,0.2065,0.14289,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,7,0,0,12,0,0,6,0,0,2,0,0,1,0,0,2,0,0],[28,58,0.4828,0.31696,0.18115,0.14297,0.28571,0.42858,0.0,0.85714,1,0,0,1,0,8,0,0,14,0,0,4,0,0,3,0,0,1,0,0,1,0,0],[32,58,0.5517,0.41072,0.20124,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,11,0,0,8,0,0,3,0,0,5,0,0,1,0,0],[36,58,0.6207,0.30804,0.18249,0.14289,0.28571,0.32143,0.0,0.85714,1,0,0,1,0,8,0,0,15,0,0,5,0,0,1,0,0,0,0,0,2,0,0],[40,58,0.6897,0.36158,0.21717,0.14289,0.28571,0.4286,0.14286,1.0,0,1,0,0,0,9,0,0,10,0,0,7,0,0,2,0,0,2,0,0,1,0,1],[44,58,0.7586,0.39728,0.1947,0.2857,0.28571,0.4642,0.14286,0.85714,0,0,0,0,0,3,0,0,16,0,0,5,0,0,3,0,0,3,0,0,2,0,0],[48,58,0.8276,0.39733,0.15458,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,2,0,0,12,0,0,13,0,0,2,0,0,2,0,0,1,0,0],[52,58,0.8966,0.37497,0.17765,0.2857,0.35714,0.42858,0.14286,0.85714,0,0,0,0,0,6,0,0,10,0,0,10,0,0,3,0,0,2,0,0,1,0,0],[56,58,0.9655,0.33474,0.19111,0.1429,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,9,0,0,12,0,0,7,0,0,1,0,0,1,0,0,2,0,0],[58,58,1.0,0.36143,0.16384,0.2857,0.35714,0.4286,0.14,0.71429,0,0,0,0,0,7,0,0,9,0,0,10,0,0,4,0,0,2,0,0,0,0,0]]}]},{"i":"bf86e6009243f5c9","q":"PER   A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the numbers $P(1), P(2)$, ..., $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.48214,"x":0.83033,"p":[[0,71,0.0,0.68302,0.24153,0.67857,0.71429,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,0,0,0,2,0,0,3,0,0,17,0,0,1,0,6],[4,71,0.0563,0.82143,0.26486,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,19],[8,71,0.1127,0.76338,0.27343,0.71429,0.71429,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,10,0,0,0,0,15],[12,71,0.169,0.83033,0.25616,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,20],[16,71,0.2254,0.77232,0.274,0.67857,0.9285,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0,7,0,0,1,0,16],[20,71,0.2817,0.66963,0.30397,0.67846,0.71429,1.0,0.0,1.0,2,9,2,2,0,3,0,0,1,0,0,1,0,0,1,0,0,15,0,0,0,0,9],[24,71,0.338,0.7232,0.27185,0.67857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,12,0,0,1,0,11],[28,71,0.3944,0.75892,0.27993,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,11,0,0,1,0,14],[32,71,0.4507,0.66964,0.27301,0.4286,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,0,0,0,6,0,0,3,0,0,11,0,0,0,0,9],[36,71,0.507,0.74552,0.23888,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,13,0,0,1,0,11],[40,71,0.5634,0.70089,0.28203,0.71429,0.71429,1.0,0.0,1.0,1,10,1,1,0,3,0,0,0,0,0,3,0,0,0,0,0,15,0,0,0,0,10],[44,71,0.6197,0.65177,0.2788,0.53539,0.71429,0.75,0.14286,1.0,0,7,0,0,0,5,0,0,1,0,0,2,0,0,2,0,0,14,0,0,1,0,7],[48,71,0.6761,0.60267,0.24675,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,5,0,0,2,0,0,16,0,0,1,0,3],[52,71,0.7324,0.58036,0.23402,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,0,0,0,9,0,0,2,0,0,14,0,0,0,0,3],[56,71,0.7887,0.57142,0.23958,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,2,0,0,6,0,0,4,0,0,13,0,0,0,0,3],[60,71,0.8451,0.59372,0.23987,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,2,0,0,4,0,0,3,0,0,16,0,0,0,0,3],[64,71,0.9014,0.60264,0.21939,0.571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,0,0,0,2,0,0,3,0,0,21,0,0,0,0,1],[68,71,0.9577,0.60709,0.18899,0.571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,4,0,0,6,0,0,18,0,0,0,0,1],[71,71,1.0,0.48214,0.24936,0.14286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,9,0,0,1,0,0,5,0,0,5,0,0,11,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"rising","v":0.66068,"x":1.0,"p":[[0,32,0.0,0.66068,0.23352,0.571,0.71429,0.71429,0.0,1.0,1,5,1,1,0,1,0,0,1,0,0,4,0,0,3,0,0,16,0,0,1,0,5],[4,32,0.125,0.92857,0.14286,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,25],[8,32,0.25,0.88392,0.2096,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,0,0,0,23],[12,32,0.375,0.87054,0.22968,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,0,0,0,23],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,32,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"793c93adc0fbcd88","q":"Positive real numbers $x, y, z$ satisfy $x y z+x y+y z+z x=x+y+z+1$. Prove that  $$ \\frac{1}{3}\\left(\\sqrt{\\frac{1+x^{2}}{1+x}}+\\sqrt{\\frac{1+y^{2}}{1+y}}+\\sqrt{\\frac{1+z^{2}}{1+z}}\\right) \\leq\\left(\\frac{x+y+z}{3}\\right)^{5 / 8} . $$","t":[{"b":0,"e":0.14286,"k":"flat","v":0.16062,"x":0.37054,"p":[[0,106,0.0,0.16062,0.11711,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,106,0.0377,0.27231,0.23515,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,19,0,0,0,0,0,3,0,0,2,0,0,5,0,0,0,0,0],[8,106,0.0755,0.25893,0.21261,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,18,0,0,1,0,0,6,0,0,0,0,0,4,0,0,0,0,0],[12,106,0.1132,0.20536,0.18189,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,25,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[16,106,0.1509,0.21875,0.18552,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,23,0,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[20,106,0.1887,0.21875,0.19556,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[24,106,0.2264,0.24554,0.21498,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,24,0,0,0,0,0,2,0,0,0,0,0,5,0,0,0,0,0],[28,106,0.2642,0.21875,0.19556,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[32,106,0.3019,0.17857,0.14725,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,23,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[36,106,0.3396,0.25,0.19885,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,20,0,0,1,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[40,106,0.3774,0.25,0.19233,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,22,0,0,0,0,0,4,0,0,3,0,0,2,0,0,0,0,0],[44,106,0.4151,0.25446,0.22513,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,20,0,0,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0],[48,106,0.4528,0.37054,0.24707,0.14286,0.42857,0.60714,0.0,0.71429,2,0,0,2,0,12,0,0,1,0,0,7,0,0,2,0,0,8,0,0,0,0,0],[52,106,0.4906,0.29018,0.20355,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,17,0,0,2,0,0,8,0,0,0,0,0,4,0,0,0,0,0],[56,106,0.5283,0.29464,0.23941,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,16,0,0,2,0,0,4,0,0,1,0,0,6,0,0,0,0,0],[60,106,0.566,0.26786,0.21943,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,18,0,0,0,0,0,6,0,0,1,0,0,4,0,0,0,0,0],[64,106,0.6038,0.25893,0.16536,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,18,0,0,2,0,0,9,0,0,1,0,0,1,0,0,0,0,0],[68,106,0.6415,0.27679,0.2141,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,18,0,0,1,0,0,6,0,0,1,0,0,4,0,0,0,0,0],[72,106,0.6792,0.33482,0.22192,0.14286,0.2143,0.42858,0.14286,0.71429,0,0,0,0,0,16,0,0,2,0,0,7,0,0,1,0,0,6,0,0,0,0,0],[76,106,0.717,0.26339,0.21461,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,20,0,0,0,0,0,5,0,0,1,0,0,4,0,0,0,0,0],[80,106,0.7547,0.25893,0.20341,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,15,0,0,2,0,0,8,0,0,0,0,0,3,0,0,0,0,0],[84,106,0.7925,0.23214,0.17768,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,17,0,0,1,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[88,106,0.8302,0.27679,0.20497,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,19,0,0,1,0,0,7,0,0,0,0,0,4,0,0,0,0,0],[92,106,0.8679,0.30357,0.22517,0.14286,0.14286,0.42858,0.0,0.71429,1,0,0,1,0,17,0,0,3,0,0,5,0,0,0,0,0,6,0,0,0,0,0],[96,106,0.9057,0.28125,0.21275,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,17,0,0,2,0,0,6,0,0,1,0,0,4,0,0,0,0,0],[100,106,0.9434,0.20536,0.16728,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,24,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[104,106,0.9811,0.25893,0.19704,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,21,0,0,0,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[106,106,1.0,0.28125,0.17672,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,17,0,0,5,0,0,6,0,0,2,0,0,2,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.13839,"x":0.25,"p":[[0,58,0.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.23214,0.21651,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,24,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0],[8,58,0.1379,0.19195,0.15403,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,25,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[12,58,0.2069,0.25,0.26245,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,24,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,2],[16,58,0.2759,0.2142,0.17132,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,23,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[20,58,0.3448,0.24107,0.16917,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,23,0,0,0,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[24,58,0.4138,0.20982,0.18205,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,24,0,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[28,58,0.4828,0.23214,0.18123,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,25,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0],[32,58,0.5517,0.20536,0.18877,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,21,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[36,58,0.6207,0.18304,0.1439,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,24,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[40,58,0.6897,0.20534,0.19209,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,20,0,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[44,58,0.7586,0.19196,0.14987,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,25,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[48,58,0.8276,0.19643,0.17768,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[52,58,0.8966,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b751aa6e61b51b2","q":"Let $\\mathrm{k}^{3}=2$ and let $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ be any rational numbers such that $\\mathrm{x}+\\mathrm{y} \\mathrm{k}+\\mathrm{z} \\mathrm{k}^{2}$ is non-zero. Show that there are rational numbers $\\mathrm{u}, \\mathrm{v}, \\mathrm{w}$ such that $\\left(\\mathrm{x}+\\mathrm{yk}+\\mathrm{z} \\mathrm{k}^{2}\\right)\\left(\\mathrm{u}+\\mathrm{v} \\mathrm{k}+\\mathrm{w}^{2}\\right)=1$.","t":[{"b":3,"e":0.71429,"k":"falling","v":0.72767,"x":0.98661,"p":[[0,41,0.0,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,41,0.0976,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[8,41,0.1951,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,41,0.2927,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[16,41,0.3902,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[20,41,0.4878,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,41,0.5854,0.92411,0.16746,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[28,41,0.6829,0.82589,0.13236,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,11],[32,41,0.7805,0.77665,0.11818,0.71429,0.71429,0.85704,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,3,0,6],[36,41,0.878,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[40,41,0.9756,0.74554,0.08552,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,3],[41,41,1.0,0.72767,0.06549,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,1]]},{"b":7,"e":0.71429,"k":"falling","v":0.67857,"x":0.98213,"p":[[0,46,0.0,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,46,0.087,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[8,46,0.1739,0.81696,0.13474,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,1,0,11],[12,46,0.2609,0.71429,0.14725,0.71429,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,2],[16,46,0.3478,0.67857,0.18211,0.71429,0.71429,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,1],[20,46,0.4348,0.72768,0.05486,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,1],[24,46,0.5217,0.72754,0.0549,0.71429,0.71429,0.71429,0.71,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,1],[28,46,0.6087,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[32,46,0.6957,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,46,0.7826,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[40,46,0.8696,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[44,46,0.9565,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[46,46,1.0,0.71875,0.02486,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0]]}]},{"i":"c11da27d3bdd3458","q":"Let $a, b$ be two distinct real numbers and let $c$ be a positive real number such that\n\n$$\na^{4}-2019 a=b^{4}-2019 b=c .\n$$\n\nProve that $-\\sqrt{c}<a b<0$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,13,0.0,0.93304,0.19556,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[4,13,0.3077,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,13,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":0.57143,"k":"falling","v":0.48214,"x":0.96429,"p":[[0,46,0.0,0.9375,0.16728,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,27],[4,46,0.087,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,46,0.1739,0.70982,0.2977,0.42857,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,8,0,0,3,0,0,1,0,0,1,0,15],[12,46,0.2609,0.78124,0.26723,0.42857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,0,0,19],[16,46,0.3478,0.60715,0.27664,0.42857,0.4286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,13,0,0,3,0,0,0,0,0,2,0,9],[20,46,0.4348,0.5625,0.23128,0.42857,0.42857,0.57143,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,18,0,0,5,0,0,0,0,0,1,0,6],[24,46,0.5217,0.66072,0.2714,0.42857,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,12,0,0,6,0,0,0,0,0,0,0,12],[28,46,0.6087,0.55357,0.23623,0.42857,0.42857,0.60714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,16,0,0,4,0,0,2,0,0,0,0,6],[32,46,0.6957,0.51341,0.15912,0.42857,0.5005,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,14,0,0,14,0,0,0,0,0,0,0,2],[36,46,0.7826,0.54911,0.26513,0.42857,0.42859,0.57143,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,10,0,0,8,0,0,0,0,0,0,0,7],[40,46,0.8696,0.48214,0.17768,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,14,0,0,11,0,0,0,0,0,0,0,2],[44,46,0.9565,0.68749,0.25365,0.5354,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,6,0,0,12,0,0,0,0,0,0,0,12],[46,46,1.0,0.58033,0.25238,0.42857,0.57143,0.57143,0.14286,1.0,0,7,0,0,0,1,0,0,6,0,0,4,0,0,14,0,0,0,0,0,0,0,7]]}]},{"i":"147f3e2fae2fcc1e","q":"Prove that for all positive integer $n$, there is a positive integer $m$, that $7^{n} \\mid 3^{m}+5^{m}-1$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.29018,"x":1.0,"p":[[0,33,0.0,0.42856,0.38795,0.14286,0.14288,0.89286,0.0,1.0,4,8,1,4,0,13,0,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,8],[4,33,0.1212,0.29018,0.31234,0.14286,0.14286,0.28571,0.0,1.0,2,5,0,2,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,33,0.2424,0.50893,0.41023,0.14286,0.2857,1.0,0.0,1.0,1,13,0,1,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[12,33,0.3636,0.53571,0.39769,0.14286,0.28571,1.0,0.14286,1.0,0,12,0,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0,2,0,12],[16,33,0.4848,0.75893,0.37191,0.28571,1.0,1.0,0.0,1.0,1,22,0,1,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,22],[20,33,0.6061,0.85268,0.31234,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.42857,"k":"flat","v":0.16518,"x":0.37937,"p":[[0,21,0.0,0.37937,0.3939,0.14286,0.14286,1.0,0.0,1.0,4,9,0,4,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[4,21,0.1905,0.16518,0.17536,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,21,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,21,0.381,0.23214,0.22517,0.14286,0.14286,0.28571,0.0,1.0,4,2,0,4,0,16,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[12,21,0.5714,0.25446,0.30249,0.14286,0.14286,0.17857,0.0,1.0,7,2,0,7,0,17,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,2],[16,21,0.7619,0.24104,0.3081,0.0,0.14286,0.2857,0.0,1.0,10,3,0,10,0,13,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,3],[20,21,0.9524,0.21427,0.22301,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,10,0,0,2,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[21,21,1.0,0.2567,0.25869,0.0,0.14286,0.44643,0.0,0.71429,12,0,0,12,0,5,0,0,4,0,0,3,0,1,3,0,0,4,0,0,0,0,0]]}]},{"i":"7b6ad37a8341cf56","q":"Prove: there are polynomials  $S_1, S_2, \\ldots$  in the variables  $x_1, x_2, \\ldots,y_1,  y_2,\\ldots$  with integer coefficients satisfying, for every integer  $n \\ge 1$ ,   $$ \\sum_{d \\mid n} d \\cdot S_d ^{n/d}=\\sum_{d \\mid n} d \\cdot (x_d ^{n/d}+y_d ^{n/d}) \\quad (*) $$   \nHere, the sums run through the positive divisors  $d$  of  $n$ .\nFor example, the first two polynomials are  $S_1 = x_1 + y_1$  and  $S_2 = x_2 + y_2 - x_1y_1$ , which verify identity $(*)$  for  $n = 2$ :  $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \\cdot(x_2 + y_2)$ .","t":[{"b":1,"e":0.2857,"k":"flat","v":0.14278,"x":0.33927,"p":[[0,27,0.0,0.14278,0.10716,0.105,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.26783,0.28736,0.0,0.28571,0.571,0.0,1.0,15,1,0,15,0,0,0,0,5,0,0,3,0,0,6,0,0,2,0,0,0,0,1],[8,27,0.2963,0.21652,0.23384,0.0,0.1429,0.28571,0.0,0.85714,12,0,0,12,0,5,0,0,9,0,0,3,0,0,0,0,0,1,1,0,1,0,0],[12,27,0.4444,0.25446,0.24932,0.10714,0.2857,0.28571,0.0,1.0,8,2,0,8,0,6,0,0,13,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[16,27,0.5926,0.33927,0.24678,0.24999,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,1,0,0,10,0,0,8,0,0,1,0,0,3,0,0,2,0,0],[20,27,0.7407,0.20973,0.17855,0.105,0.21428,0.28571,0.0,0.85714,8,0,0,8,0,8,0,0,12,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[24,27,0.8889,0.22313,0.25741,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,10,0,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,2],[27,27,1.0,0.25446,0.24151,0.14286,0.2857,0.28571,0.0,1.0,7,1,0,7,0,7,0,0,14,0,0,1,0,0,0,0,0,0,0,0,2,0,1]]},{"b":2,"e":0.28571,"k":"flat","v":0.06696,"x":0.28121,"p":[[0,21,0.0,0.16955,0.1208,0.105,0.14286,0.2857,0.0,0.4286,8,0,0,8,0,11,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.28121,0.27771,0.0,0.28571,0.46418,0.0,1.0,12,1,0,12,0,2,0,0,7,0,0,3,0,0,4,0,0,3,0,0,0,0,1],[8,21,0.381,0.21875,0.29011,0.0,0.0,0.46429,0.0,0.85714,18,0,0,18,0,2,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,0],[12,21,0.5714,0.16519,0.22619,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,5,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[16,21,0.7619,0.17857,0.17857,0.0,0.21428,0.28571,0.0,0.57143,14,0,0,14,0,2,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[20,21,0.9524,0.125,0.25191,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[21,21,1.0,0.06696,0.15966,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"345ffd30615ac593","q":"Prove the inequality \\[ \\sin^n (2x) + \\left( \\sin^n x - \\cos^n x \\right)^2 \\le 1. \\]","t":[{"b":0,"e":0.14286,"k":"falling","v":0.08929,"x":0.58035,"p":[[0,56,0.0,0.46411,0.34643,0.14286,0.28571,0.89286,0.14,1.0,0,8,0,0,0,11,0,0,7,0,0,3,0,0,2,0,0,0,0,0,1,0,8],[4,56,0.0714,0.53571,0.44176,0.10714,0.57143,1.0,0.0,1.0,8,13,0,8,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,13],[8,56,0.1429,0.50891,0.40238,0.14286,0.49979,1.0,0.0,1.0,6,11,0,6,0,6,0,0,2,0,0,2,0,0,4,0,0,1,0,0,0,0,11],[12,56,0.2143,0.56695,0.38046,0.14286,0.57121,1.0,0.0,1.0,2,10,0,2,0,9,0,0,1,0,0,3,0,0,2,0,0,1,0,0,4,0,10],[16,56,0.2857,0.56697,0.4439,0.0,0.78571,1.0,0.0,1.0,9,14,0,9,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,14],[20,56,0.3571,0.49107,0.44741,0.0,0.42857,1.0,0.0,1.0,10,11,0,10,0,6,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,11],[24,56,0.4286,0.58035,0.44167,0.14286,0.85707,1.0,0.0,1.0,6,15,0,6,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,15],[28,56,0.5,0.38393,0.39357,0.0,0.14286,0.85714,0.0,1.0,9,6,0,9,0,9,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,6],[32,56,0.5714,0.47322,0.39194,0.14286,0.42857,1.0,0.0,1.0,5,9,0,5,0,9,0,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,9],[36,56,0.6429,0.42857,0.41033,0.10714,0.21428,1.0,0.0,1.0,8,9,0,8,0,8,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,9],[40,56,0.7143,0.33036,0.39679,0.0,0.14286,0.60714,0.0,1.0,12,7,0,12,0,9,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,7],[44,56,0.7857,0.29464,0.33108,0.14286,0.14286,0.32143,0.0,1.0,7,4,0,7,0,14,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,4],[48,56,0.8571,0.43741,0.42405,0.0,0.14288,1.0,0.0,1.0,9,9,0,9,0,8,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,9],[52,56,0.9286,0.40625,0.40423,0.14286,0.14286,0.89286,0.0,1.0,7,8,0,7,0,12,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,8],[56,56,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.09375,"x":0.85258,"p":[[0,49,0.0,0.45981,0.38587,0.14286,0.28571,1.0,0.0,1.0,3,10,0,3,0,10,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,10],[4,49,0.0816,0.85258,0.259,0.71429,1.0,1.0,0.14,1.0,0,23,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,23],[8,49,0.1633,0.77679,0.31731,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3,0,18],[12,49,0.2449,0.73214,0.38755,0.35714,1.0,1.0,0.0,1.0,2,20,0,2,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,20],[16,49,0.3265,0.7499,0.36786,0.53539,1.0,1.0,0.0,1.0,2,19,0,2,0,5,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,19],[20,49,0.4082,0.64732,0.38793,0.14286,0.78571,1.0,0.0,1.0,3,15,0,3,0,6,0,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,15],[24,49,0.4898,0.60267,0.41455,0.14286,0.71421,1.0,0.0,1.0,5,15,0,5,0,5,0,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,15],[28,49,0.5714,0.5223,0.38566,0.14286,0.42857,1.0,0.0,1.0,5,9,0,5,0,6,0,0,1,0,0,5,0,0,2,0,0,1,0,0,3,0,9],[32,49,0.6531,0.59375,0.36089,0.39286,0.57143,1.0,0.0,1.0,4,11,0,4,0,3,0,0,1,0,0,5,0,0,5,0,0,2,0,0,1,0,11],[36,49,0.7347,0.65625,0.40226,0.14286,0.92857,1.0,0.0,1.0,3,16,0,3,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,16],[40,49,0.8163,0.48214,0.41611,0.14286,0.14286,1.0,0.0,1.0,4,11,0,4,0,13,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,11],[44,49,0.898,0.44644,0.41303,0.14286,0.2143,1.0,0.0,1.0,7,9,0,7,0,9,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,9],[48,49,0.9796,0.19643,0.30252,0.0,0.14286,0.14286,0.0,1.0,15,3,0,15,0,10,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[49,49,1.0,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6863b897294bfb32","q":"Prove the inequality  $$ \\sum_{i<j} \\frac{a_{i} a_{j}}{a_{i}+a_{j}} \\leq \\frac{n}{2\\left(a_{1}+a_{2}+\\cdots+a_{n}\\right)} \\sum_{i<j} a_{i} a_{j} $$  for positive real numbers $a_{1}, a_{2}, \\ldots, a_{n}$.  (Serbia)","t":[{"b":5,"e":1.0,"k":"rising","v":0.37053,"x":0.96875,"p":[[0,69,0.0,0.44196,0.45367,0.0,0.14286,1.0,0.0,1.0,13,9,0,13,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,9],[4,69,0.058,0.73661,0.42724,0.64286,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,21],[8,69,0.1159,0.6875,0.41717,0.25,0.92857,1.0,0.0,1.0,7,16,0,7,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,16],[12,69,0.1739,0.52232,0.45402,0.0,0.57143,1.0,0.0,1.0,10,13,0,10,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,13],[16,69,0.2319,0.54018,0.43702,0.0,0.78571,1.0,0.0,1.0,11,9,0,11,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,7,0,9],[20,69,0.2899,0.40179,0.46213,0.0,0.07143,1.0,0.0,1.0,16,9,0,16,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,9],[24,69,0.3478,0.50446,0.43739,0.0,0.64286,1.0,0.0,1.0,11,9,0,11,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,9],[28,69,0.4058,0.37053,0.44444,0.0,0.0,0.85714,0.0,1.0,18,7,0,18,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,7],[32,69,0.4638,0.37946,0.44837,0.0,0.07143,1.0,0.0,1.0,16,9,0,16,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,9],[36,69,0.5217,0.38393,0.4467,0.0,0.14286,1.0,0.0,1.0,14,10,0,14,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,10],[40,69,0.5797,0.45089,0.45891,0.0,0.14286,1.0,0.0,1.0,14,9,0,14,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,9],[44,69,0.6377,0.87054,0.26812,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,21],[48,69,0.6957,0.78571,0.32927,0.82143,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,16],[52,69,0.7536,0.83036,0.26592,0.85714,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,13,0,14],[56,69,0.8116,0.69196,0.38814,0.57143,0.85714,1.0,0.0,1.0,6,11,0,6,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,11],[60,69,0.8696,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[64,69,0.9275,0.9283,0.10769,0.85714,1.0,1.0,0.71,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[68,69,0.9855,0.89732,0.15663,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,7,0,19],[69,69,1.0,0.83036,0.16917,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,8,0,12]]},{"b":6,"e":0.0,"k":"falling","v":0.16072,"x":0.82143,"p":[[0,162,0.0,0.71429,0.40721,0.35714,1.0,1.0,0.0,1.0,6,17,0,6,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,17],[4,162,0.0247,0.82143,0.34626,0.85714,1.0,1.0,0.0,1.0,4,21,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,21],[8,162,0.0494,0.73214,0.40838,0.35714,1.0,1.0,0.0,1.0,5,21,0,5,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,21],[12,162,0.0741,0.64285,0.45034,0.0,0.92857,1.0,0.0,1.0,9,16,0,9,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,16],[16,162,0.0988,0.73661,0.39787,0.71429,1.0,1.0,0.0,1.0,7,17,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,17],[20,162,0.1235,0.73661,0.41971,0.57143,1.0,1.0,0.0,1.0,7,21,0,7,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,21],[24,162,0.1481,0.72321,0.39599,0.57143,1.0,1.0,0.0,1.0,5,17,0,5,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,17],[28,162,0.1728,0.75446,0.39324,0.85714,1.0,1.0,0.0,1.0,6,18,0,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,18],[32,162,0.1975,0.79017,0.37284,0.85714,1.0,1.0,0.0,1.0,5,20,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,20],[36,162,0.2222,0.7098,0.40325,0.46396,1.0,1.0,0.0,1.0,6,17,0,6,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,17],[40,162,0.2469,0.60714,0.45457,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,15],[44,162,0.2716,0.59821,0.46075,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,15],[48,162,0.2963,0.74554,0.37752,0.67857,1.0,1.0,0.0,1.0,4,18,0,4,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,18],[52,162,0.321,0.54464,0.47305,0.0,0.85714,1.0,0.0,1.0,11,16,0,11,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16],[56,162,0.3457,0.68303,0.41147,0.14286,0.92857,1.0,0.0,1.0,6,16,0,6,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,16],[60,162,0.3704,0.64732,0.43151,0.10714,0.85714,1.0,0.0,1.0,8,15,0,8,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,15],[64,162,0.3951,0.46429,0.46153,0.0,0.35714,1.0,0.0,1.0,15,10,0,15,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,10],[68,162,0.4198,0.47321,0.46351,0.0,0.14286,1.0,0.0,1.0,12,11,0,12,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,11],[72,162,0.4444,0.43304,0.45943,0.0,0.14286,1.0,0.0,1.0,14,9,0,14,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,9],[76,162,0.4691,0.47768,0.44694,0.0,0.57143,1.0,0.0,1.0,13,9,0,13,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,9],[80,162,0.4938,0.54018,0.45978,0.0,0.78571,1.0,0.0,1.0,11,13,0,11,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,13],[84,162,0.5185,0.58482,0.45647,0.0,0.85714,1.0,0.0,1.0,11,13,0,11,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,13],[88,162,0.5432,0.46429,0.44464,0.0,0.28571,1.0,0.0,1.0,12,9,0,12,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,9],[92,162,0.5679,0.54017,0.44854,0.0,0.78564,1.0,0.0,1.0,10,12,0,10,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,12],[96,162,0.5926,0.5,0.45876,0.0,0.35714,1.0,0.0,1.0,11,14,0,11,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,14],[100,162,0.6173,0.35267,0.42854,0.0,0.07143,0.85714,0.0,1.0,16,7,0,16,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,7],[104,162,0.642,0.36607,0.45728,0.0,0.0,0.89286,0.0,1.0,18,8,0,18,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,8],[108,162,0.6667,0.44196,0.43354,0.0,0.14286,0.89286,0.0,1.0,11,8,0,11,0,6,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,8],[112,162,0.6914,0.45089,0.4793,0.0,0.14286,1.0,0.0,1.0,15,13,0,15,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,13],[116,162,0.716,0.25893,0.38372,0.0,0.0,0.32143,0.0,1.0,17,5,0,17,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,5],[120,162,0.7407,0.22759,0.37265,0.0,0.0,0.14286,0.0,1.0,19,4,0,19,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,4],[124,162,0.7654,0.28125,0.41263,0.0,0.0,0.42857,0.0,1.0,18,7,0,18,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,7],[128,162,0.7901,0.26786,0.37923,0.0,0.0,0.46429,0.0,1.0,17,4,0,17,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,4],[132,162,0.8148,0.51786,0.45281,0.0,0.64286,1.0,0.0,1.0,12,11,0,12,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,11],[136,162,0.8395,0.34375,0.44587,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,9],[140,162,0.8642,0.36598,0.45312,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,9],[144,162,0.8889,0.27678,0.39437,0.0,0.0,0.64282,0.0,1.0,18,4,0,18,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,4],[148,162,0.9136,0.38393,0.43218,0.0,0.14286,1.0,0.0,1.0,13,9,0,13,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,9],[152,162,0.9383,0.42411,0.42929,0.0,0.14288,0.89286,0.0,1.0,11,8,0,11,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,8],[156,162,0.963,0.19643,0.29179,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,6,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,1],[160,162,0.9877,0.16072,0.27141,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,8,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[162,162,1.0,0.2767,0.17112,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,10,0,0,9,0,0,7,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"28d595cf0833f83d","q":"Let $f$ be a function from the set of positive integers to itself such that, for every $n$, the number of positive integer divisors of $n$ is equal to $f(f(n))$. For example, $f(f(6))=4$ and $f(f(25))=3$. Prove that if $p$ is prime then $f(p)$ is also prime.","t":[{"b":1,"e":0.0,"k":"falling","v":0.16518,"x":0.76786,"p":[[0,22,0.0,0.75445,0.26781,0.857,0.85714,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0,20,0,5],[4,22,0.1818,0.76786,0.25939,0.85714,0.85714,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,22,0,4],[8,22,0.3636,0.70535,0.27418,0.53571,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,6,0,0,1,0,0,2,0,0,1,0,0,17,0,4],[12,22,0.5455,0.55354,0.29825,0.28571,0.57143,0.85704,0.0,1.0,2,1,0,2,0,3,0,0,6,0,0,2,0,0,5,0,0,2,0,0,11,0,1],[16,22,0.7273,0.22767,0.27165,0.0,0.14288,0.32143,0.0,0.85714,15,0,0,15,0,2,0,0,7,0,0,3,0,0,2,0,0,0,0,0,3,0,0],[20,22,0.9091,0.26337,0.22331,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,2,0,0,10,0,0,5,0,0,4,0,0,0,0,0,1,0,0],[22,22,1.0,0.16518,0.16793,0.0,0.2143,0.2857,0.0,0.57143,15,0,0,15,0,1,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.85714,"k":"flat","v":0.71428,"x":0.87946,"p":[[0,43,0.0,0.75446,0.25311,0.71429,0.85714,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,19,0,4],[4,43,0.093,0.77231,0.2392,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,0,15,0,7],[8,43,0.186,0.71428,0.29233,0.71429,0.85714,0.85714,0.0,1.0,3,2,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,6,0,0,19,0,2],[12,43,0.2791,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[16,43,0.3721,0.85714,2e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[20,43,0.4651,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[24,43,0.5581,0.85714,2e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[28,43,0.6512,0.85714,0.0,0.85714,0.85714,0.85714,0.85714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[32,43,0.7442,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[36,43,0.8372,0.85713,3e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[40,43,0.9302,0.85713,3e-05,0.85714,0.85714,0.85714,0.857,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[43,43,1.0,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1]]}]},{"i":"839ccf311a655f4b","q":"Show that for any integer $n \\geq 1$, there exists $m$ a multiple of $n$ such that the sum of the digits of $m$ is $n$.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.30344,"x":0.74554,"p":[[0,41,0.0,0.66518,0.32657,0.57143,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,2,0,0,0,0,0,9,0,0,3,0,0,4,0,10],[4,41,0.0976,0.74554,0.19475,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,4,0,0,3,0,10],[8,41,0.1951,0.69195,0.15613,0.57143,0.57143,0.71429,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,8,0,0,2,0,5],[12,41,0.2927,0.63392,0.24468,0.57143,0.57143,0.75,0.0,1.0,1,7,1,1,0,1,0,0,1,0,0,2,0,0,17,0,0,2,0,0,1,0,7],[16,41,0.3902,0.6183,0.1953,0.57143,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,4,0,0,17,0,0,5,1,0,0,0,4],[20,41,0.4878,0.59374,0.17169,0.57143,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,20,0,0,4,0,0,0,0,3],[24,41,0.5854,0.5714,0.13832,0.57132,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,3,0,0,19,0,0,6,0,0,0,0,1],[28,41,0.6829,0.625,0.18472,0.57143,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,4,0,0,17,0,0,5,0,0,1,0,4],[32,41,0.7805,0.54018,0.18118,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,6,0,0,17,0,0,2,0,0,2,0,1],[36,41,0.878,0.38838,0.19309,0.28571,0.35714,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,10,0,0,6,0,0,6,0,0,4,0,0,0,0,0],[40,41,0.9756,0.30344,0.19806,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,11,0,0,5,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[41,41,1.0,0.37043,0.19849,0.2857,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,5,0,0,10,0,0,5,0,0,7,0,0,3,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.34822,"x":0.70536,"p":[[0,36,0.0,0.70536,0.3272,0.57143,0.78571,1.0,0.0,1.0,4,13,0,4,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,3,0,13],[4,36,0.1111,0.64284,0.19562,0.57143,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,1,0,0,18,0,0,5,0,0,1,0,5],[8,36,0.2222,0.63836,0.17491,0.57143,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,2,0,0,18,0,0,5,0,0,3,0,3],[12,36,0.3333,0.45982,0.31081,0.14286,0.50001,0.57143,0.0,1.0,5,3,0,5,0,5,0,0,0,0,0,6,0,0,9,0,0,1,0,0,3,0,3],[16,36,0.4444,0.49554,0.2624,0.28571,0.50001,0.57143,0.0,1.0,1,4,0,1,0,4,0,0,5,0,0,6,0,0,9,0,0,3,0,0,0,0,4],[20,36,0.5556,0.49107,0.3071,0.39285,0.50001,0.60714,0.0,1.0,5,5,0,5,0,2,0,0,1,0,0,8,0,0,8,0,0,3,0,0,0,0,5],[24,36,0.6667,0.46427,0.26726,0.24999,0.57121,0.60714,0.0,1.0,3,2,0,3,0,5,0,0,2,0,0,5,0,0,9,0,0,6,0,0,0,0,2],[28,36,0.7778,0.40179,0.24335,0.2857,0.42929,0.57143,0.0,0.71429,6,0,0,6,0,1,0,0,6,0,0,4,0,0,10,0,0,5,0,0,0,0,0],[32,36,0.8889,0.50446,0.19228,0.42857,0.57143,0.60714,0.0,0.71429,2,0,0,2,0,1,0,0,2,0,0,8,0,0,11,0,0,8,0,0,0,0,0],[36,36,1.0,0.34822,0.18536,0.2857,0.42857,0.42857,0.0,0.71429,5,0,0,5,0,1,0,0,7,0,0,14,0,0,4,0,0,1,0,0,0,0,0]]}]},{"i":"5b670f4f8286ab5d","q":"Sixty seven points lie inside a regular hexagon with side length  $2$  centimetres. Prove that a circular coin of radius  $1$  centimetre can be placed to cover at least twelve of the points.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.63837,"x":0.90625,"p":[[0,46,0.0,0.68304,0.40991,0.28571,0.92857,1.0,0.0,1.0,7,16,5,7,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,4,0,16],[4,46,0.087,0.90625,0.23853,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[8,46,0.1739,0.77679,0.37954,0.71429,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,22],[12,46,0.2609,0.74092,0.35074,0.57143,1.0,1.0,0.0,1.0,4,17,0,4,0,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,3,0,17],[16,46,0.3478,0.83926,0.28962,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,21],[20,46,0.4348,0.87054,0.27976,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,25],[24,46,0.5217,0.75,0.31542,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,1,0,0,0,0,0,4,0,0,5,0,0,3,0,15],[28,46,0.6087,0.83033,0.22714,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,3,0,17],[32,46,0.6957,0.8616,0.2004,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,19],[36,46,0.7826,0.77675,0.28109,0.57143,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,3,0,16],[40,46,0.8696,0.69638,0.29614,0.57132,0.64286,1.0,0.0,1.0,2,11,0,2,0,1,0,0,1,0,0,1,0,0,11,0,0,1,0,0,4,0,11],[44,46,0.9565,0.67852,0.31341,0.571,0.71429,1.0,0.0,1.0,3,10,0,3,0,1,0,0,1,0,0,1,0,0,8,0,0,4,0,0,4,0,10],[46,46,1.0,0.63837,0.2474,0.57143,0.57143,0.71429,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,0,0,0,18,0,0,4,0,0,0,0,7]]},{"b":7,"e":0.57143,"k":"flat","v":0.7857,"x":0.89731,"p":[[0,34,0.0,0.85714,0.29667,0.96429,1.0,1.0,0.0,1.0,2,24,1,2,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,24],[4,34,0.1176,0.89731,0.22655,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,24],[8,34,0.2353,0.88838,0.15461,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,2,0,20],[12,34,0.3529,0.81249,0.25863,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,0,0,0,4,0,0,6,0,0,1,0,18],[16,34,0.4706,0.83481,0.20239,0.67857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,4,0,0,3,0,17],[20,34,0.5882,0.7857,0.23958,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,4,0,14],[24,34,0.7059,0.875,0.16269,0.71429,1.0,1.0,0.5714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,5,0,18],[28,34,0.8235,0.86606,0.20499,0.67857,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,2,0,21],[32,34,0.9412,0.808,0.18079,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,4,0,13],[34,34,1.0,0.8125,0.24074,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,5,0,0,4,0,16]]}]},{"i":"c20d10b33633afc7","q":"Show that the equation  $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$  has infintely many solutions in integers  $x,y,z$ .","t":[{"b":3,"e":0.0,"k":"falling","v":0.05357,"x":0.93747,"p":[[0,67,0.0,0.87054,0.32996,1.0,1.0,1.0,0.0,1.0,4,27,3,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[4,67,0.0597,0.90179,0.27765,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[8,67,0.1194,0.93747,0.15548,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,27],[12,67,0.1791,0.79911,0.38608,0.85714,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,23],[16,67,0.2388,0.45088,0.46167,0.0,0.21428,1.0,0.0,1.0,15,10,0,15,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,10],[20,67,0.2985,0.59375,0.45752,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,15],[24,67,0.3582,0.58481,0.45226,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,15],[28,67,0.4179,0.69643,0.40994,0.39286,1.0,1.0,0.0,1.0,7,17,0,7,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,17],[32,67,0.4776,0.71427,0.40721,0.5354,1.0,1.0,0.0,1.0,7,19,0,7,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,19],[36,67,0.5373,0.56695,0.4758,0.0,0.85714,1.0,0.0,1.0,13,15,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,15],[40,67,0.597,0.65625,0.45437,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,18],[44,67,0.6567,0.23214,0.37585,0.0,0.0,0.42857,0.0,1.0,22,4,0,22,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,4],[48,67,0.7164,0.16963,0.33585,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,3],[52,67,0.7761,0.12945,0.31207,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[56,67,0.8358,0.21428,0.35534,0.0,0.0,0.46429,0.0,1.0,23,2,0,23,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,2],[60,67,0.8955,0.05357,0.18123,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[64,67,0.9552,0.19642,0.31893,0.0,0.0,0.42857,0.0,1.0,22,1,0,22,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,3,0,1],[67,67,1.0,0.18302,0.34298,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,3]]},{"b":7,"e":0.0,"k":"falling","v":0.07142,"x":0.9375,"p":[[0,85,0.0,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,85,0.0471,0.59821,0.47439,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,17],[8,85,0.0941,0.67855,0.41496,0.32143,0.92857,1.0,0.0,1.0,8,16,0,8,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,16],[12,85,0.1412,0.625,0.45562,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,17],[16,85,0.1882,0.55803,0.4783,0.0,0.85714,1.0,0.0,1.0,13,15,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,15],[20,85,0.2353,0.49107,0.46694,0.0,0.57143,1.0,0.0,1.0,14,12,0,14,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,12],[24,85,0.2824,0.69643,0.43995,0.0,1.0,1.0,0.0,1.0,9,19,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,19],[28,85,0.3294,0.35268,0.47108,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[32,85,0.3765,0.57143,0.47782,0.0,0.92857,1.0,0.0,1.0,13,16,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,16],[36,85,0.4235,0.45981,0.46392,0.0,0.35693,1.0,0.0,1.0,15,11,0,15,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,11],[40,85,0.4706,0.7232,0.40711,0.53539,1.0,1.0,0.0,1.0,7,19,0,7,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,19],[44,85,0.5176,0.60268,0.46939,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,17],[48,85,0.5647,0.54018,0.45138,0.0,0.57141,1.0,0.0,1.0,11,14,0,11,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,14],[52,85,0.6118,0.52679,0.48239,0.0,0.78571,1.0,0.0,1.0,14,15,0,14,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,15],[56,85,0.6588,0.48661,0.4763,0.0,0.5,1.0,0.0,1.0,15,13,0,15,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,13],[60,85,0.7059,0.59821,0.46075,0.0,0.9285,1.0,0.0,1.0,11,16,0,11,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,16],[64,85,0.7529,0.12946,0.29312,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[68,85,0.8,0.24554,0.36984,0.0,0.0,0.57143,0.0,1.0,21,4,0,21,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,1,0,4],[72,85,0.8471,0.125,0.28065,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[76,85,0.8941,0.1875,0.35072,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,3],[80,85,0.9412,0.15177,0.31929,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,2],[84,85,0.9882,0.15177,0.33106,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[85,85,1.0,0.07142,0.22865,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1]]}]},{"i":"2db223c2589b07a2","q":"Prove that there exist nonconstant polynomials $f$ and $g$ with integer coefficients such that, for infinitely many primes $p$, there are no integers $x$ and $y$ with $p \\mid f(x)-g(y)$.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.14286,"x":0.85266,"p":[[0,44,0.0,0.14286,0.31135,0.0,0.0,0.0,0.0,1.0,26,2,26,26,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[4,44,0.0909,0.85266,0.15767,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,12,0,12],[8,44,0.1818,0.81249,0.19703,0.82132,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,17,0,7],[12,44,0.2727,0.84375,0.23787,0.85714,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,13,0,14],[16,44,0.3636,0.76784,0.21651,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,8,0,0,14,0,5],[20,44,0.4545,0.76335,0.22479,0.57143,0.85707,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,0,11,0,8],[24,44,0.5455,0.81688,0.19318,0.67846,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,10,0,12],[28,44,0.6364,0.81694,0.16842,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,13,0,9],[32,44,0.7273,0.76784,0.18473,0.67857,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,10,0,7],[36,44,0.8182,0.66512,0.19436,0.42857,0.64286,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,5,0,0,8,0,3],[40,44,0.9091,0.76784,0.16656,0.71421,0.85707,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,8,0,0,12,0,5],[44,44,1.0,0.77677,0.15125,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,19,0,2]]},{"b":6,"e":1.0,"k":"rising","v":0.16963,"x":0.96429,"p":[[0,56,0.0,0.16963,0.33203,0.0,0.0,0.0,0.0,1.0,25,2,24,25,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,2],[4,56,0.0714,0.80353,0.18474,0.82132,0.85714,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,0,18,0,6],[8,56,0.1429,0.83478,0.1389,0.82132,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,16,0,8],[12,56,0.2143,0.73653,0.19275,0.5713,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,5,0,0,9,0,6],[16,56,0.2857,0.7812,0.22015,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,3,0,0,8,0,11],[20,56,0.3571,0.83479,0.17539,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,7,0,13],[24,56,0.4286,0.8482,0.15947,0.82132,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,13,0,11],[28,56,0.5,0.76782,0.23626,0.57143,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,1,0,0,7,0,0,2,0,0,11,0,9],[32,56,0.5714,0.83927,0.15046,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,13,0,10],[36,56,0.6429,0.75444,0.22373,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,5,0,0,6,0,10],[40,56,0.7143,0.73657,0.20552,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,4,0,0,7,0,8],[44,56,0.7857,0.83033,0.20028,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,4,0,16],[48,56,0.8571,0.7767,0.16737,0.57143,0.85707,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,11,0,7],[52,56,0.9286,0.81693,0.215,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,11,0,11],[56,56,1.0,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28]]}]},{"i":"e8396887cd279fe7","q":"The integer number  $n > 1$  is given and a set  $S \\subset \\{0, 1, 2, \\ldots, n-1\\}$  with  $|S| > \\frac{3}{4} n$ . Prove that there exist integer numbers  $a, b, c$  such that the remainders after the division by  $n$  of the numbers:\n\\[a, b, c, a+b, b+c, c+a, a+b+c\\]\nbelong to  $S$ .","t":[{"b":5,"e":1.0,"k":"rising","v":0.60268,"x":0.85713,"p":[[0,25,0.0,0.60268,0.35307,0.28571,0.71429,0.85714,0.0,1.0,6,5,0,6,0,1,0,0,2,0,0,1,0,0,0,0,0,10,0,0,7,0,5],[4,25,0.16,0.76339,0.26392,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,4,0,0,0,0,0,1,0,0,8,0,0,7,0,11],[8,25,0.32,0.77006,0.25553,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,0,0,0,7,0,0,5,0,0,3,1,13],[12,25,0.48,0.75446,0.26782,0.71429,0.78564,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,5,0,11],[16,25,0.64,0.85713,0.14288,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,11,0,12],[20,25,0.8,0.83036,0.16917,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,10,0,0,7,0,12],[24,25,0.96,0.79462,0.17475,0.71429,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,8,0,9],[25,25,1.0,0.77007,0.15231,0.71429,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,7,1,5]]},{"b":6,"e":0.571,"k":"rising","v":0.54463,"x":0.91964,"p":[[0,29,0.0,0.54463,0.36975,0.1429,0.71429,0.85714,0.0,1.0,7,4,0,7,0,2,0,0,1,0,0,4,0,0,1,0,0,5,0,0,6,2,4],[4,29,0.1379,0.85491,0.21613,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,0,3,1,18],[8,29,0.2759,0.75446,0.29931,0.71429,0.85714,1.0,0.0,1.0,3,12,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,6,0,12],[12,29,0.4138,0.86606,0.13803,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,8,0,14],[16,29,0.5517,0.81026,0.23118,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,5,1,13],[20,29,0.6897,0.86827,0.14304,0.82132,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,1,0,4,0,0,10,0,14],[24,29,0.8276,0.87051,0.14,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,7,0,15],[28,29,0.9655,0.91295,0.16243,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,1,19],[29,29,1.0,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17]]}]},{"i":"05f4e8d85f5f71c2","q":"Prove that, for every integer  $n \\geq 3$ , there exist  $n$  positive composite integers that form an arithmetic progression and are pairwise coprime.**Note**: A positive integer is called **composite** if it can be expressed as the product of two integers greater than 1.","t":[{"b":0,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,49,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,49,0.0816,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,49,0.1633,0.92857,0.17128,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,25],[12,49,0.2449,0.91518,0.25218,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[16,49,0.3265,0.91071,0.17768,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,1,0,24],[20,49,0.4082,0.91072,0.17767,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,24],[24,49,0.4898,0.90179,0.1729,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,23],[28,49,0.5714,0.91071,0.18472,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,25],[32,49,0.6531,0.93304,0.18552,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,28],[36,49,0.7347,0.90179,0.16145,0.82143,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,2,0,22],[40,49,0.8163,0.91518,0.15093,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,23],[44,49,0.898,0.88393,0.16146,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,20],[48,49,0.9796,0.875,0.17035,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,19],[49,49,1.0,0.87945,0.15615,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,2,0,19]]},{"b":2,"e":0.57143,"k":"falling","v":0.61161,"x":1.0,"p":[[0,32,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,32,0.125,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,32,0.25,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,32,0.375,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[16,32,0.5,0.92857,0.24484,1.0,1.0,1.0,0.0,1.0,2,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[20,32,0.625,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,26],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,0.89284,0.20205,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,24],[32,32,1.0,0.61161,0.27718,0.42857,0.5,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,12,0,0,6,0,0,0,0,0,0,0,10]]}]},{"i":"0bee4921b4a1e46a","q":"Prove that, for all real numbers $x, y, z$ :\n\n$$\n\\frac{x^{2}-y^{2}}{2 x^{2}+1}+\\frac{y^{2}-z^{2}}{2 y^{2}+1}+\\frac{z^{2}-x^{2}}{2 z^{2}+1} \\leq(x+y+z)^{2}\n$$\n\nWhen the equality holds?","t":[{"b":2,"e":0.28571,"k":"falling","v":0.44641,"x":0.89732,"p":[[0,107,0.0,0.80803,0.28707,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,19],[4,107,0.0374,0.80804,0.26633,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,7,0,0,0,0,0,1,0,0,3,0,19],[8,107,0.0748,0.69643,0.30671,0.42857,0.64286,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,8,0,0,4,0,0,1,0,0,0,0,15],[12,107,0.1121,0.89732,0.20589,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,24],[16,107,0.1495,0.73214,0.27607,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,6,0,0,2,0,0,4,0,0,2,0,14],[20,107,0.1869,0.79464,0.28557,0.64286,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,19],[24,107,0.2243,0.78125,0.26722,0.53572,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,5,0,0,1,0,17],[28,107,0.2617,0.61161,0.32189,0.28571,0.50001,1.0,0.0,1.0,1,11,0,1,0,1,0,0,7,0,0,7,0,0,2,0,0,2,0,0,1,0,11],[32,107,0.2991,0.74106,0.28446,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,2,0,0,1,0,16],[36,107,0.3364,0.70089,0.28428,0.42857,0.64286,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,10,0,0,3,0,0,1,0,0,1,0,14],[40,107,0.3738,0.57143,0.23419,0.42857,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,15,0,0,1,0,0,6,0,0,1,0,5],[44,107,0.4112,0.70536,0.29437,0.42857,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,5,0,0,8,0,0,1,0,0,2,0,0,2,0,14],[48,107,0.4486,0.61161,0.27947,0.42857,0.50001,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,6,0,0,9,0,0,1,0,0,4,0,0,4,0,7],[52,107,0.486,0.67411,0.27947,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,10,0,0,1,0,0,5,0,0,0,0,12],[56,107,0.5234,0.70089,0.28651,0.42857,0.78564,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,9,0,0,2,0,0,1,0,0,3,0,13],[60,107,0.5607,0.62054,0.3001,0.39286,0.64286,1.0,0.0,1.0,1,9,0,1,0,0,0,0,7,0,0,7,0,0,1,0,0,5,0,0,2,0,9],[64,107,0.5981,0.64286,0.28122,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,7,0,0,7,0,0,1,0,0,7,0,0,0,0,10],[68,107,0.6355,0.56696,0.26119,0.28571,0.42857,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,8,0,0,3,0,0,5,0,0,1,0,6],[72,107,0.6729,0.65179,0.25238,0.42857,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,12,0,0,3,0,0,5,0,0,1,0,9],[76,107,0.7103,0.62499,0.28516,0.42857,0.4998,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,5,0,0,10,0,0,1,0,0,4,0,0,2,0,9],[80,107,0.7477,0.62052,0.28707,0.39286,0.57121,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,8,0,0,7,0,0,3,0,0,4,0,0,0,0,10],[84,107,0.785,0.62498,0.27374,0.42857,0.64286,0.89275,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,9,0,0,2,0,0,7,0,0,1,0,8],[88,107,0.8224,0.51786,0.26905,0.28571,0.42857,0.60714,0.14286,1.0,0,6,0,0,0,2,0,0,7,0,0,13,0,0,2,0,0,1,0,0,1,0,6],[92,107,0.8598,0.57141,0.27664,0.39286,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,2,0,0,6,0,0,9,0,0,2,0,0,6,0,0,0,0,7],[96,107,0.8972,0.59375,0.27225,0.42857,0.42857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,7,0,0,11,0,0,1,0,0,3,0,0,3,0,7],[100,107,0.9346,0.5625,0.2765,0.39286,0.42857,0.78571,0.28571,1.0,0,8,0,0,0,0,0,0,8,0,0,13,0,0,0,0,0,3,0,0,0,0,8],[104,107,0.972,0.44641,0.17767,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,8,0,0,17,0,0,1,0,0,3,0,0,1,0,1],[107,107,1.0,0.52232,0.22759,0.42857,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,1,0,0,4,0,0,17,0,0,2,0,0,3,0,0,1,0,4]]},{"b":5,"e":0.42857,"k":"falling","v":0.53571,"x":0.92411,"p":[[0,117,0.0,0.92411,0.18205,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,27],[4,117,0.0342,0.85714,0.23958,0.78571,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0,23],[8,117,0.0684,0.78125,0.28568,0.53571,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,2,0,0,2,0,18],[12,117,0.1026,0.87946,0.25028,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,25],[16,117,0.1368,0.91964,0.19865,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[20,117,0.1709,0.65179,0.34615,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,5,0,0,3,0,0,1,0,0,4,0,0,0,0,14],[24,117,0.2051,0.70982,0.34346,0.39286,0.92857,1.0,0.0,1.0,1,16,0,1,0,3,0,0,4,0,0,2,0,0,2,0,0,2,0,0,2,0,16],[28,117,0.2393,0.70089,0.33381,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,17],[32,117,0.2735,0.69643,0.33645,0.39286,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,17],[36,117,0.3077,0.60267,0.33261,0.39286,0.57143,1.0,0.0,1.0,1,10,0,1,0,5,0,0,2,0,0,6,0,0,4,0,0,2,0,0,2,0,10],[40,117,0.3419,0.64732,0.33309,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,2,0,0,6,0,0,4,0,0,2,0,0,4,0,0,0,0,13],[44,117,0.3761,0.68293,0.325,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,2,0,0,3,0,0,6,0,0,2,0,0,3,0,0,1,0,14],[48,117,0.4103,0.6116,0.31589,0.28571,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,9,0,0,5,0,0,0,0,0,3,0,0,4,0,9],[52,117,0.4444,0.54464,0.32031,0.28571,0.42857,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,8,0,0,7,0,0,1,0,0,1,0,0,4,0,7],[56,117,0.4786,0.66964,0.32427,0.39286,0.71429,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,6,0,0,6,0,0,1,0,0,2,0,0,1,0,14],[60,117,0.5128,0.59375,0.30745,0.28571,0.42859,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,12,0,0,6,0,0,0,0,0,2,0,0,3,0,9],[64,117,0.547,0.65625,0.31513,0.42857,0.64286,1.0,0.0,1.0,1,13,0,1,0,1,0,0,3,0,0,10,0,0,1,0,0,3,0,0,0,0,13],[68,117,0.5812,0.62054,0.32657,0.28571,0.5,1.0,0.0,1.0,1,12,0,1,0,0,0,0,9,0,0,6,0,0,2,0,0,1,0,0,1,0,12],[72,117,0.6154,0.62946,0.2942,0.42857,0.42859,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,10,0,0,0,0,0,4,0,0,0,0,11],[76,117,0.6496,0.75893,0.29974,0.42857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,4,0,0,8,0,0,0,0,0,1,0,0,0,0,19],[80,117,0.6838,0.57143,0.31744,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,8,0,0,8,0,0,0,0,0,2,0,0,2,0,9],[84,117,0.7179,0.70982,0.2977,0.42857,0.78571,1.0,0.0,1.0,1,14,0,1,0,0,0,0,2,0,0,9,0,0,2,0,0,2,0,0,2,0,14],[88,117,0.7521,0.68304,0.32875,0.39286,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,7,0,0,7,0,0,0,0,0,1,0,0,0,0,16],[92,117,0.7863,0.59375,0.29257,0.42857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,5,0,0,14,0,0,0,0,0,2,0,0,0,0,10],[96,117,0.8205,0.73214,0.28515,0.42857,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,5,0,0,6,0,0,0,0,0,4,0,0,3,0,14],[100,117,0.8547,0.66964,0.31428,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,6,0,0,8,0,0,0,0,0,3,0,0,0,0,14],[104,117,0.8889,0.70089,0.29312,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,10,0,0,1,0,0,1,0,0,2,0,14],[108,117,0.9231,0.66518,0.27804,0.42857,0.64286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,8,0,0,3,0,0,4,0,0,1,0,11],[112,117,0.9573,0.62946,0.30064,0.42857,0.42857,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,14,0,0,0,0,0,2,0,0,1,0,11],[116,117,0.9915,0.53571,0.26486,0.39286,0.42857,0.64286,0.14286,1.0,0,6,0,0,0,1,0,0,7,0,0,13,0,0,3,0,0,0,0,0,2,0,6],[117,117,1.0,0.66071,0.25939,0.42857,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,5,0,0,2,0,9]]}]},{"i":"8208a9afce826548","q":"There exist positive integers  $N, M$  such that  $N$ 's remainders modulo the four integers  $6, 36,$   $216,$  and  $M$  form an increasing nonzero geometric sequence in that order. Find the smallest possible value of  $M$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,135,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,135,0.0296,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,135,0.0593,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,135,0.0889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,135,0.1185,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,135,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,135,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,135,0.2074,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,135,0.237,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,135,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,135,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,135,0.3259,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,135,0.3556,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,135,0.3852,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,135,0.4148,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,135,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,135,0.4741,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,135,0.5037,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,135,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,135,0.563,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,135,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,135,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,135,0.6519,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,135,0.6815,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,135,0.7111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,135,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,135,0.7704,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,135,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,135,0.8296,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,135,0.8593,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,135,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,135,0.9185,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[128,135,0.9481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,135,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[135,135,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,139,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,139,0.0288,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,139,0.0576,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,139,0.0863,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,139,0.1151,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,139,0.1439,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,139,0.1727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,139,0.2014,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,139,0.2302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,139,0.259,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,139,0.2878,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,139,0.3165,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,139,0.3453,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,139,0.3741,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,139,0.4029,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,139,0.4317,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,139,0.4604,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,139,0.4892,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,139,0.518,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,139,0.5468,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,139,0.5755,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,139,0.6043,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,139,0.6331,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,139,0.6619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,139,0.6906,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,139,0.7194,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,139,0.7482,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,139,0.777,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,139,0.8058,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,139,0.8345,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,139,0.8633,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,139,0.8921,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,139,0.9209,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,139,0.9496,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[136,139,0.9784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[139,139,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"beb1d24eb299e5a9","q":"Prove that  $$ \\lim_{n\\to\\infty}n^2\\left(\\int^1_0\\sqrt[n]{1+x^n}\\text dx-1\\right)=\\frac{\\pi^2}{12}. $$","t":[{"b":2,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,14,0.0,0.91071,0.21943,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,27],[4,14,0.2857,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,14,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,14,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[14,14,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,6,0.0,0.92411,0.17491,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,26],[4,6,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[6,6,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"fd819a5207d9ee50","q":"There is a safe that can be opened by entering a secret code consisting of  $n$  digits, each of them is  $0$  or  $1$ . Initially,  $n$  zeros were entered, and the safe is closed (so, all zeros is not the secret code).\n\nIn one attempt, you can enter an arbitrary sequence of  $n$  digits, each of them is  $0$  or  $1$ . If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click.\n\nFind the smallest number of attempts that is sufficient to open the safe in all cases.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.51114,"x":0.79909,"p":[[0,24,0.0,0.51114,0.38011,0.10714,0.57121,0.85714,0.0,1.0,8,7,7,8,0,2,0,0,1,0,1,3,0,0,2,0,0,6,0,0,2,0,7],[4,24,0.1667,0.78112,0.22892,0.57143,0.85707,1.0,0.14,1.0,0,13,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,6,0,0,4,0,13],[8,24,0.3333,0.67407,0.27255,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,3,0,0,8,0,7],[12,24,0.5,0.79909,0.22829,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,9,0,11],[16,24,0.6667,0.74107,0.22142,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,6,0,0,10,0,7],[20,24,0.8333,0.69192,0.2882,0.5354,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,3,0,0,6,0,0,3,0,0,5,0,10],[24,24,1.0,0.79909,0.25219,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,1,0,0,5,0,16]]},{"b":7,"e":0.71429,"k":"flat","v":0.42861,"x":0.82143,"p":[[0,52,0.0,0.48442,0.35657,0.14289,0.42929,0.73214,0.0,1.0,7,6,6,7,0,2,0,0,3,0,0,5,0,0,3,0,0,4,1,0,1,0,6],[4,52,0.0769,0.82143,0.27199,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,0,5,0,18],[8,52,0.1538,0.79911,0.29203,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,3,0,0,2,0,19],[12,52,0.2308,0.69641,0.30671,0.57143,0.71429,1.0,0.0,1.0,3,10,0,3,0,0,0,0,2,0,0,1,0,0,6,0,0,5,0,0,5,0,10],[16,52,0.3077,0.78572,0.25253,0.53572,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,2,0,0,7,0,14],[20,52,0.3846,0.66503,0.26411,0.42857,0.57143,0.89286,0.14,1.0,0,8,0,0,0,2,0,0,1,0,0,7,0,0,7,0,0,2,0,0,5,0,8],[24,52,0.4615,0.72322,0.26949,0.57143,0.78571,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,3,0,0,6,0,0,4,0,0,6,0,10],[28,52,0.5385,0.70981,0.27545,0.42857,0.71429,1.0,0.1429,1.0,0,11,0,0,0,2,0,0,1,0,0,7,0,0,2,0,0,5,0,0,4,0,11],[32,52,0.6154,0.58023,0.33692,0.2857,0.57121,0.85714,0.0,1.0,1,7,0,1,0,6,0,0,4,0,0,3,0,0,3,0,0,2,0,0,6,0,7],[36,52,0.6923,0.45535,0.34706,0.14286,0.42859,0.71429,0.0,1.0,7,5,0,7,0,3,0,0,3,0,0,5,0,0,3,0,0,5,0,0,1,0,5],[40,52,0.7692,0.54015,0.26422,0.39286,0.57121,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,4,0,0,6,0,0,8,0,0,3,0,0,4,0,3],[44,52,0.8462,0.42861,0.22303,0.25,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,7,0,0,2,0,0,11,0,0,5,0,0,4,0,0,2,0,0],[48,52,0.9231,0.54902,0.3034,0.28571,0.57141,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,6,0,0,4,0,0,4,0,0,4,0,0,4,0,5],[52,52,1.0,0.57126,0.23974,0.42857,0.571,0.71429,0.14,1.0,0,4,0,0,0,3,0,0,3,0,0,5,0,0,10,0,0,6,0,0,1,0,4]]}]},{"i":"7aae81ef6775469a","q":"Through a point $P$ exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at $A$ and $B$, and the tangent touches the circle at $C$ on the same side of the diameter thorugh $P$ as $A$ and $B$. The projection of $C$ on the diameter is $Q$. Prove that $Q C$ bisects $\\angle A Q B$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.03125,"x":0.16072,"p":[[0,67,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.13831,0.0977,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,67,0.1194,0.14688,0.09097,0.14,0.14286,0.14286,0.0,0.4286,5,0,0,5,0,22,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.16072,0.09942,0.14286,0.14286,0.14287,0.0,0.57143,3,0,0,3,0,24,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,67,0.2985,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.08911,0.07772,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.09813,0.07518,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.12036,0.08068,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.09795,0.07508,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.10714,"x":0.1875,"p":[[0,62,0.0,0.1875,0.1729,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,23,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[4,62,0.0645,0.1517,0.10063,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,27,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,62,0.129,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,62,0.1935,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,62,0.2581,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,62,0.3226,0.12947,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.11116,0.07754,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.14268,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.12027,0.07232,0.14,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.14733,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.14268,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7df65868a8741b30","q":"On a natural number  $n$  you are allowed to operations :  $(1)$  multiply  $n$  by  $2$  or  $(2)$  subtract  $3$  from  $n$ . For example starting with  $8$  you can reach  $13$  as follows :  $8 \\longrightarrow 16 \\longrightarrow 13$ . You need two steps and you cannot do in less than two steps. Starting from  $11$ , what is the least number of steps required to reach  $121$ ?","t":[{"b":1,"e":0.57143,"k":"flat","v":0.875,"x":0.98437,"p":[[0,58,0.0,0.875,0.18814,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,14,0,15],[4,58,0.069,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,58,0.1379,0.94195,0.11774,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[12,58,0.2069,0.96428,0.12878,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[16,58,0.2759,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,58,0.3448,0.95088,0.11073,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[24,58,0.4138,0.92857,0.16366,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[28,58,0.4828,0.91964,0.20806,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,25],[32,58,0.5517,0.9598,0.09612,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[36,58,0.6207,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[40,58,0.6897,0.95535,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[44,58,0.7586,0.98437,0.04277,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,28],[48,58,0.8276,0.97544,0.06826,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,2,0,28],[52,58,0.8966,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[56,58,0.9655,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[58,58,1.0,0.9464,0.09949,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.85937,"x":1.0,"p":[[0,101,0.0,0.875,0.16656,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,11,0,15],[4,101,0.0396,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,101,0.0792,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,101,0.1188,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,101,0.1584,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,101,0.198,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,101,0.2376,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,101,0.2772,0.93303,0.19557,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[32,101,0.3168,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,101,0.3564,0.9241,0.21424,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[40,101,0.396,0.9732,0.08335,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[44,101,0.4356,0.91964,0.2141,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[48,101,0.4752,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[52,101,0.5149,0.85937,0.27692,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,1,22],[56,101,0.5545,0.91964,0.17105,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,23],[60,101,0.5941,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,101,0.6337,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[68,101,0.6733,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,101,0.7129,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,101,0.7525,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[80,101,0.7921,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,101,0.8317,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,101,0.8713,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,101,0.9109,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,101,0.9505,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,101,0.9901,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[101,101,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"bb8cca8654e7ae62","q":"We are given sufficiently many stones of the forms of a rectangle  $2\\times 1$  and square  $1\\times 1$ . Let  $n > 3$  be a natural number. In how many ways can one tile a rectangle  $3 \\times n$  using these stones, so that no two  $2 \\times 1$  rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?","t":[{"b":0,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,15,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,15,0.2667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[15,15,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":3,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,23,0.0,0.91518,0.21683,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,25],[4,23,0.1739,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,23,0.3478,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,23,0.8696,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[23,23,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"0814e248ca9e06e3","q":"Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, El\u00edas is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?\n\nClarification: If  $A$  is a friend of  $B$  then  $B$  is a friend of  $A$ .","t":[{"b":3,"e":0.28571,"k":"falling","v":0.20981,"x":0.99107,"p":[[0,47,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,47,0.0851,0.82589,0.36897,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[8,47,0.1702,0.86161,0.32827,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,27],[12,47,0.2553,0.91071,0.24157,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[16,47,0.3404,0.77232,0.33855,0.53571,1.0,1.0,0.0,1.0,2,20,0,2,0,2,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,20],[20,47,0.4255,0.83481,0.26514,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,1,0,21],[24,47,0.5106,0.71874,0.36855,0.39286,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,19],[28,47,0.5957,0.76776,0.376,0.71429,1.0,1.0,0.0,1.0,4,21,0,4,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,21],[32,47,0.6809,0.58479,0.40146,0.14286,0.64286,1.0,0.0,1.0,6,13,0,6,0,3,0,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,13],[36,47,0.766,0.59819,0.39032,0.2857,0.71429,1.0,0.0,1.0,5,13,0,5,0,2,0,0,5,0,0,1,0,0,2,0,0,4,0,0,0,0,13],[40,47,0.8511,0.4775,0.37064,0.14286,0.28571,1.0,0.0,1.0,2,9,0,2,0,10,0,0,5,0,0,3,0,0,0,0,0,3,0,0,0,0,9],[44,47,0.9362,0.41061,0.35676,0.14286,0.28571,0.60714,0.0,1.0,5,7,0,5,0,8,0,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,7],[47,47,1.0,0.20981,0.20196,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,11,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.27679,"x":0.98661,"p":[[0,55,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,55,0.0727,0.79464,0.33681,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,1,0,21],[8,55,0.1455,0.83036,0.32427,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,24],[12,55,0.2182,0.82142,0.31542,0.82132,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,22],[16,55,0.2909,0.75446,0.39161,0.71429,1.0,1.0,0.0,1.0,5,20,0,5,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,20],[20,55,0.3636,0.79911,0.35149,0.75,1.0,1.0,0.0,1.0,3,23,0,3,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,23],[24,55,0.4364,0.51339,0.39425,0.24999,0.42857,1.0,0.0,1.0,6,11,0,6,0,2,0,0,7,0,0,4,0,0,1,0,0,0,0,0,1,0,11],[28,55,0.5091,0.58034,0.41793,0.14286,0.64286,1.0,0.0,1.0,7,14,0,7,0,3,0,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,14],[32,55,0.5818,0.39286,0.43154,0.0,0.21428,1.0,0.0,1.0,13,10,0,13,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,10],[36,55,0.6545,0.375,0.38424,0.0,0.2143,0.85714,0.0,1.0,9,6,0,9,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,0,3,0,6],[40,55,0.7273,0.4107,0.36552,0.0,0.42857,0.71429,0.0,1.0,9,6,0,9,0,4,0,0,2,0,0,5,0,0,3,0,0,3,0,0,0,0,6],[44,55,0.8,0.33473,0.4019,0.0,0.14145,0.71429,0.0,1.0,15,7,0,15,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,0,0,0,7],[48,55,0.8727,0.51763,0.39261,0.105,0.571,1.0,0.0,1.0,8,10,0,8,0,2,0,0,1,0,0,4,0,0,5,0,0,2,0,0,0,0,10],[52,55,0.9455,0.48661,0.45014,0.0,0.28571,1.0,0.0,1.0,11,13,0,11,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,13],[55,55,1.0,0.27679,0.25985,0.0,0.2857,0.42857,0.0,0.85714,10,0,0,10,0,5,0,0,6,0,0,5,0,0,1,0,0,4,0,0,1,0,0]]}]},{"i":"4b3dcaa0ac6259e3","q":"We are given an infinite set of points in the plane such that any two of them have a distance of at most one. Prove that all the axes of symmetry of this set are concurrent, provided that there are at least two of them. \n\n*Proposed by David Anghel*","t":[{"b":5,"e":0.2857,"k":"flat","v":0.11161,"x":0.33036,"p":[[0,16,0.0,0.19196,0.27574,0.0,0.0,0.2857,0.0,1.0,17,2,0,17,0,3,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[4,16,0.25,0.33036,0.36846,0.0,0.2857,0.39286,0.0,1.0,12,5,0,12,0,2,0,0,10,0,0,0,0,0,0,0,0,1,0,0,2,0,5],[8,16,0.5,0.21428,0.32341,0.0,0.0,0.28571,0.0,1.0,18,4,0,18,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,16,0.75,0.11161,0.13709,0.0,0.0,0.28571,0.0,0.28571,19,0,0,19,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.31249,0.15335,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,1]]},{"b":6,"e":0.857,"k":"rising","v":0.3125,"x":0.92857,"p":[[0,37,0.0,0.37946,0.38234,0.0,0.28571,0.71429,0.0,1.0,12,6,0,12,0,0,0,0,9,0,0,0,0,0,0,0,0,5,0,0,0,0,6],[4,37,0.1081,0.3125,0.24338,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,2,0,0,13,0,0,4,0,0,2,0,0,3,0,0,0,0,1],[8,37,0.2162,0.37491,0.35496,0.105,0.28571,0.5,0.0,1.0,8,6,0,8,0,2,0,0,13,0,0,1,0,0,0,0,0,1,0,0,1,0,6],[12,37,0.3243,0.4375,0.35523,0.2857,0.28571,0.74996,0.0,1.0,7,6,0,7,0,0,0,0,12,0,0,2,0,0,1,0,0,2,0,0,2,0,6],[16,37,0.4324,0.42857,0.4072,0.0,0.28571,0.89286,0.0,1.0,11,8,0,11,0,1,0,0,6,0,0,2,0,0,1,0,0,1,0,0,2,0,8],[20,37,0.5405,0.33929,0.34947,0.0,0.28571,0.5,0.0,1.0,12,4,0,12,0,1,0,0,7,0,0,4,0,0,0,0,0,3,0,0,1,0,4],[24,37,0.6486,0.79018,0.33879,0.28571,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[28,37,0.7568,0.84822,0.31731,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,24],[32,37,0.8649,0.83482,0.34462,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,25],[36,37,0.973,0.92857,0.20203,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[37,37,1.0,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28]]}]},{"i":"210d01aff014d5a1","q":"Say that a polynomial with real coefficients in two variable,  $ x,y,$  is *balanced* if the average value of the polynomial on each circle centered at the origin is  $ 0.$  The balanced polynomials of degree at most  $ 2009$  form a vector space  $ V$  over  $ \\mathbb{R}.$  Find the dimension of  $ V.$","t":[{"b":4,"e":1.0,"k":"flat","v":0.90179,"x":0.99107,"p":[[0,16,0.0,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[4,16,0.25,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,16,0.5,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[12,16,0.75,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[16,16,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":5,"e":0.85714,"k":"flat","v":0.86607,"x":0.9375,"p":[[0,56,0.0,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[4,56,0.0714,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,56,0.1429,0.91518,0.09354,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,15],[12,56,0.2143,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,56,0.2857,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[20,56,0.3571,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[24,56,0.4286,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[28,56,0.5,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[32,56,0.5714,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[36,56,0.6429,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[40,56,0.7143,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[44,56,0.7857,0.88839,0.05906,0.85714,0.85714,0.85714,0.8571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[48,56,0.8571,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[52,56,0.9286,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[56,56,1.0,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2]]}]},{"i":"d37c0163f426074b","q":"Two equal circles  $S_1$  and  $S_2$  meet at two different points. The line  $\\ell$  intersects  $S_1$  at points  $A,C$  and  $S_2$  at points  $B,D$  respectively  (the order on  $\\ell$ :  $A,B,C,D$ ) . Define circles  $\\Gamma_1$  and  $\\Gamma_2$  as follows: both  $\\Gamma_1$  and  $\\Gamma_2$  touch  $S_1$  internally and  $S_2$  externally, both  $\\Gamma_1$  and  $\\Gamma_2$  line  $\\ell$ ,  $\\Gamma_1$  and  $\\Gamma_2$  lie in the different halfplanes relatively to line  $\\ell$ . Suppose that  $\\Gamma_1$  and  $\\Gamma_2$  touch each other. Prove that  $AB=CD$ .\n\nI. Voronovich","t":[{"b":3,"e":0.14286,"k":"flat","v":0.06241,"x":0.1317,"p":[[0,47,0.0,0.06241,0.09399,0.0,0.0,0.14286,0.0,0.28571,21,0,3,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.1317,0.11024,0.0,0.14286,0.14287,0.0,0.4286,10,0,0,10,0,15,0,1,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.08259,0.0996,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,1,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.09366,0.11067,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.08027,0.09401,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.10268,0.08171,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.07813,0.09341,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,1,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.10715,0.18898,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,47,0.8511,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.07804,0.08246,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,14,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14,"k":"flat","v":0.03116,"x":0.18304,"p":[[0,75,0.0,0.06697,0.10092,0.0,0.0,0.14286,0.0,0.28571,21,0,2,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.09822,0.10972,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,75,0.1067,0.07143,0.09942,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,75,0.16,0.09822,0.09062,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,75,0.2133,0.14723,0.13592,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,9,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,75,0.2667,0.13152,0.10117,0.0,0.14286,0.14287,0.0,0.28571,9,0,0,9,1,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,75,0.32,0.11598,0.11536,0.0,0.14286,0.14287,0.0,0.4286,13,0,0,13,0,13,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,75,0.3733,0.10714,0.11845,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,75,0.4267,0.18304,0.15251,0.0,0.14288,0.28571,0.0,0.57143,10,0,0,10,0,7,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,75,0.48,0.125,0.13243,0.0,0.14286,0.1786,0.0,0.42857,14,0,0,14,0,10,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.13839,0.15765,0.0,0.07143,0.28571,0.0,0.4286,16,0,0,16,0,5,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.13839,0.16164,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,6,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[52,75,0.6933,0.1384,0.13592,0.0,0.14286,0.2857,0.0,0.4286,13,0,0,13,0,9,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,75,0.7467,0.08928,0.12242,0.0,0.0,0.17857,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.1049,0.11292,0.0,0.10693,0.1429,0.0,0.28571,15,0,0,15,1,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.09375,0.11633,0.0,0.0,0.1429,0.0,0.28571,18,0,0,18,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c968e60555b6aab1","q":"We call a divisor  $d$  of a positive integer  $n$  *special* if  $d + 1$  is also a divisor of  $n$ . Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.","t":[{"b":0,"e":0.57143,"k":"falling","v":0.57585,"x":0.94196,"p":[[0,60,0.0,0.92411,0.19228,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[4,60,0.0667,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[8,60,0.1333,0.90624,0.12686,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,18],[12,60,0.2,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[16,60,0.2667,0.875,0.15872,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,13,0,14],[20,60,0.3333,0.8125,0.17655,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,9,0,10],[24,60,0.4,0.78122,0.20514,0.71429,0.78571,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,9,0,0,5,0,11],[28,60,0.4667,0.75891,0.19705,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,9,0,0,5,0,9],[32,60,0.5333,0.69195,0.23176,0.5354,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,7,0,0,6,0,0,7,0,0,3,0,8],[36,60,0.6,0.57585,0.23003,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,6,0,0,9,0,0,7,0,0,2,0,3],[40,60,0.6667,0.5848,0.23517,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,8,0,0,8,0,0,5,0,0,4,0,3],[44,60,0.7333,0.65622,0.22264,0.571,0.71429,0.85714,0.1429,1.0,0,4,0,0,0,1,0,0,3,0,0,3,0,0,7,0,0,9,0,0,5,0,4],[48,60,0.8,0.58926,0.21053,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,12,0,0,4,0,0,7,0,0,6,0,1],[52,60,0.8667,0.58036,0.2141,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,1,0,0,5,0,0,8,0,0,12,0,0,2,0,1],[56,60,0.9333,0.67408,0.1864,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,13,0,0,4,0,3],[60,60,1.0,0.62499,0.13717,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,10,0,0,4,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.90178,"x":0.99554,"p":[[0,30,0.0,0.90178,0.21852,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,24],[4,30,0.1333,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[8,30,0.2667,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,30,0.4,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,30,0.5333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,30,0.6667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,30,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,30,0.9333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[30,30,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"019c34bc3faf0773","q":"What is the least number of moves it takes a knight to get from one corner of an  $n\\times n$  chessboard, where  $n\\ge 4$ , to the diagonally opposite corner?","t":[{"b":0,"e":1.0,"k":"rising","v":0.71872,"x":0.98213,"p":[[0,76,0.0,0.71872,0.14055,0.57143,0.71429,0.857,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,13,0,0,6,0,3],[4,76,0.0526,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[8,76,0.1053,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[12,76,0.1579,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,76,0.2105,0.93749,0.10678,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[20,76,0.2632,0.91516,0.15919,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24],[24,76,0.3158,0.95076,0.09884,1.0,1.0,1.0,0.71,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[28,76,0.3684,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[32,76,0.4211,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[36,76,0.4737,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,76,0.5263,0.91963,0.14261,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[44,76,0.5789,0.90622,0.14993,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,21],[48,76,0.6316,0.91517,0.12299,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,20],[52,76,0.6842,0.91517,0.1377,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[56,76,0.7368,0.92409,0.12368,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[60,76,0.7895,0.85265,0.16169,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,9,0,14],[64,76,0.8421,0.94194,0.11219,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[68,76,0.8947,0.92409,0.13828,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,23],[72,76,0.9474,0.91959,0.14268,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,6,0,22],[76,76,1.0,0.94629,0.12268,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26]]},{"b":4,"e":1.0,"k":"flat","v":0.70981,"x":0.94196,"p":[[0,48,0.0,0.70981,0.14502,0.57143,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,7,0,3],[4,48,0.0833,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[8,48,0.1667,0.91963,0.15129,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24],[12,48,0.25,0.87945,0.14338,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,17],[16,48,0.3333,0.92854,0.13839,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[20,48,0.4167,0.87498,0.19483,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,20],[24,48,0.5,0.81248,0.17292,0.71429,0.85707,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,5,0,12],[28,48,0.5833,0.8705,0.14451,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,5,0,16],[32,48,0.6667,0.8571,0.13369,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,10,0,12],[36,48,0.75,0.8839,0.14485,0.82143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,7,0,17],[40,48,0.8333,0.89283,0.13836,0.82143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,18],[44,48,0.9167,0.85489,0.15822,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,6,0,0,6,0,15],[48,48,1.0,0.82809,0.17118,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,3,1,0,4,0,0,11,0,11]]}]},{"i":"8aff3a822f31d9bf","q":"Three prime numbers  $p,q,r$  and a positive integer  $n$  are given such that the numbers \n\\[ \\frac{p+n}{qr}, \\frac{q+n}{rp}, \\frac{r+n}{pq} \\] \nare integers. Prove that  $p=q=r $ .\n\n*Nazar Agakhanov*","t":[{"b":1,"e":0.857,"k":"flat","v":0.75445,"x":0.90624,"p":[[0,13,0.0,0.75445,0.23209,0.71429,0.85707,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,0,15,0,5],[4,13,0.3077,0.75893,0.26351,0.53572,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,3,0,0,4,0,14],[8,13,0.6154,0.8348,0.19599,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,9,0,14],[12,13,0.9231,0.90624,0.15815,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,6,0,21],[13,13,1.0,0.86158,0.16556,0.857,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,10,0,15]]},{"b":4,"e":1.0,"k":"flat","v":0.73213,"x":0.90176,"p":[[0,34,0.0,0.81696,0.21199,0.85714,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,16,0,9],[4,34,0.1176,0.82142,0.19561,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,12,0,11],[8,34,0.2353,0.85266,0.15357,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,15,0,11],[12,34,0.3529,0.74998,0.18557,0.57143,0.857,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,6,0,0,12,0,5],[16,34,0.4706,0.76786,0.2165,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,5,0,0,8,0,10],[20,34,0.5882,0.75445,0.16458,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,8,0,0,16,0,2],[24,34,0.7059,0.73213,0.18813,0.67857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,9,0,0,11,0,4],[28,34,0.8235,0.79464,0.19212,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,0,11,0,9],[32,34,0.9412,0.81249,0.18363,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,12,0,10],[34,34,1.0,0.90176,0.14036,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,12,0,17]]}]},{"i":"e371bbd30dd8132a","q":"Solve the following equation for real values of  $x$ :\n\n    \\[\n    2 \\left( 5^x + 6^x - 3^x \\right) = 7^x + 9^x.\n    \\]","t":[{"b":0,"e":0.71429,"k":"falling","v":0.34372,"x":0.84372,"p":[[0,65,0.0,0.57141,0.2988,0.28571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,6,0,0,3,0,0,5,0,0,4,0,0,4,0,0,5,0,5],[4,65,0.0615,0.70982,0.26841,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,5,0,0,3,0,11],[8,65,0.1231,0.83034,0.22143,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,3,0,0,6,0,16],[12,65,0.1846,0.84372,0.19023,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,3,0,17],[16,65,0.2462,0.79016,0.22864,0.71429,0.857,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,6,0,12],[20,65,0.3077,0.73661,0.29474,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,5,0,0,4,0,13],[24,65,0.3692,0.75893,0.25614,0.67857,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,8,0,0,4,0,12],[28,65,0.4308,0.83035,0.20652,0.71421,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,4,0,0,6,0,15],[32,65,0.4923,0.79464,0.25238,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,3,0,0,7,0,14],[36,65,0.5538,0.78569,0.27201,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,5,0,0,3,0,16],[40,65,0.6154,0.77902,0.23101,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,8,1,0,6,0,11],[44,65,0.6769,0.66518,0.28259,0.42859,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,6,0,0,3,0,0,5,0,0,5,0,0,2,0,10],[48,65,0.7385,0.79911,0.23652,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,5,0,0,9,0,12],[52,65,0.8,0.58036,0.31932,0.28571,0.64286,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,9,0,0,0,0,0,3,0,0,6,0,0,3,0,7],[56,65,0.8615,0.70536,0.26229,0.53571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,6,0,0,7,0,8],[60,65,0.9231,0.37944,0.3022,0.14286,0.28571,0.57143,0.0,1.0,4,4,0,4,0,8,0,0,7,0,0,1,0,0,8,0,0,0,0,0,0,0,4],[64,65,0.9846,0.40613,0.23183,0.24999,0.3573,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,8,0,0,3,0,0,8,0,0,4,0,0,0,0,1],[65,65,1.0,0.34372,0.20156,0.14286,0.2857,0.571,0.14286,0.71429,0,0,0,0,0,11,0,0,10,0,0,2,0,0,5,0,0,4,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.50437,"x":0.91964,"p":[[0,51,0.0,0.50437,0.30522,0.2857,0.42857,0.75,0.0,1.0,2,5,0,2,0,3,0,0,8,0,0,4,0,0,6,0,0,1,0,0,3,0,5],[4,51,0.0784,0.8125,0.20958,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,10,0,0,5,0,13],[8,51,0.1569,0.72766,0.22407,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,9,0,0,5,0,8],[12,51,0.2353,0.87053,0.18336,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,3,0,19],[16,51,0.3137,0.81696,0.20896,0.71429,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,15,0,0,0,0,15],[20,51,0.3922,0.86606,0.15544,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,3,0,17],[24,51,0.4706,0.78124,0.22865,0.57143,0.85714,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,6,0,0,4,0,13],[28,51,0.549,0.79909,0.24187,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,2,0,0,4,0,16],[32,51,0.6275,0.86607,0.19865,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,4,0,19],[36,51,0.7059,0.80357,0.26666,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,2,0,0,3,0,18],[40,51,0.7843,0.83929,0.21053,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,5,0,0,4,0,17],[44,51,0.8627,0.91964,0.14698,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,8,0,21],[48,51,0.9412,0.89286,0.15567,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,19],[51,51,1.0,0.83929,0.20124,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,11,0,14]]}]},{"i":"0d89a3ea3507ee2b","q":"We say that a  $2023$ -tuple of nonnegative integers  $(a_1,\\hdots,a_{2023})$  is *sweet* if the following conditions hold:\n\n\n\n- $a_1+\\hdots+a_{2023}=2023$\n- $\\frac{a_1}{2}+\\frac{a_2}{2^2}+\\hdots+\\frac{a_{2023}}{2^{2023}}\\le 1$\n\n\nDetermine the greatest positive integer  $L$  so that \\[a_1+2a_2+\\hdots+2023a_{2023}\\ge L\\] holds for every sweet  $2023$ -tuple  $(a_1,\\hdots,a_{2023})$ *Ivan Novak*","t":[{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.79464,"p":[[0,109,0.0,0.42857,0.35355,0.0,0.42857,0.71429,0.0,1.0,9,3,6,9,0,3,0,0,2,0,0,3,0,0,5,0,0,3,0,0,4,0,3],[4,109,0.0367,0.53125,0.35756,0.28571,0.35714,1.0,0.0,1.0,1,10,0,1,0,6,0,0,9,0,0,2,0,0,2,0,0,1,0,0,1,0,10],[8,109,0.0734,0.53124,0.29501,0.28571,0.42857,0.75,0.14286,1.0,0,6,0,0,0,3,0,0,10,0,0,6,0,0,1,0,0,4,0,0,2,0,6],[12,109,0.1101,0.65179,0.31122,0.28571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,6,0,0,2,0,0,4,0,0,3,0,0,4,0,10],[16,109,0.1468,0.58929,0.34209,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,7,0,0,4,0,0,4,0,0,2,0,0,3,0,0,2,0,10],[20,109,0.1835,0.74107,0.3163,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,4,0,0,1,0,0,4,0,0,2,0,0,4,0,15],[24,109,0.2202,0.65179,0.35524,0.28571,0.71429,1.0,0.0,1.0,2,13,0,2,0,4,0,0,3,0,0,1,0,0,5,0,0,2,0,0,2,0,13],[28,109,0.2569,0.58036,0.31931,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,4,0,0,5,0,0,3,0,0,2,0,0,5,0,7],[32,109,0.2936,0.65625,0.31916,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,4,0,0,1,0,0,3,0,0,4,0,0,6,0,9],[36,109,0.3303,0.61161,0.36637,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,7,0,0,6,0,0,2,0,0,1,0,0,0,0,0,4,0,12],[40,109,0.367,0.65179,0.36411,0.28571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,7,0,0,3,0,0,3,0,0,2,0,0,1,0,0,1,0,15],[44,109,0.4037,0.66518,0.36001,0.28571,0.78571,1.0,0.0,1.0,1,15,0,1,0,5,0,0,3,0,0,3,0,0,2,0,0,2,0,0,1,0,15],[48,109,0.4404,0.60713,0.34069,0.28571,0.64286,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,6,0,0,1,0,0,3,0,0,3,0,0,3,0,10],[52,109,0.4771,0.63839,0.33117,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,4,0,0,1,0,0,3,0,0,3,0,0,6,0,9],[56,109,0.5138,0.61607,0.34523,0.25,0.71429,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,3,0,0,0,0,0,4,0,0,4,0,0,3,0,10],[60,109,0.5505,0.70536,0.34981,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,4,0,0,1,0,0,3,0,0,1,0,0,1,0,17],[64,109,0.5872,0.73661,0.28146,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,2,0,0,0,0,0,6,0,0,4,0,0,5,0,12],[68,109,0.6239,0.66964,0.29761,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,5,0,0,5,0,0,1,0,0,5,0,0,4,0,10],[72,109,0.6606,0.79464,0.27879,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,0,0,0,4,0,0,3,0,0,5,0,16],[76,109,0.6972,0.66071,0.34947,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,6,0,0,4,0,0,1,0,0,3,0,0,3,0,0,1,0,14],[80,109,0.7339,0.69196,0.35912,0.39286,0.85714,1.0,0.0,1.0,2,15,0,2,0,4,0,0,2,0,0,1,0,0,4,0,0,1,0,0,3,0,15],[84,109,0.7706,0.53125,0.41686,0.10714,0.57143,1.0,0.0,1.0,8,11,0,8,0,2,0,0,5,0,0,0,0,0,2,0,0,2,0,0,2,0,11],[88,109,0.8073,0.57589,0.41108,0.24999,0.64286,1.0,0.0,1.0,7,13,0,7,0,1,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,13],[92,109,0.844,0.58929,0.43117,0.14286,0.85714,1.0,0.0,1.0,7,14,0,7,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,14],[96,109,0.8807,0.22768,0.35867,0.0,0.0,0.35714,0.0,1.0,21,3,0,21,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,1,0,3],[100,109,0.9174,0.30804,0.40265,0.0,0.0,0.60714,0.0,1.0,18,6,0,18,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,6],[104,109,0.9541,0.42857,0.43741,0.0,0.21429,1.0,0.0,1.0,13,9,0,13,0,3,0,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,9],[108,109,0.9908,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[109,109,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.37045,"x":0.57589,"p":[[0,67,0.0,0.55357,0.34209,0.25,0.57143,0.85714,0.0,1.0,4,5,2,4,0,4,0,0,2,0,0,3,0,0,4,0,0,4,0,0,6,0,5],[4,67,0.0597,0.50445,0.34253,0.14286,0.35714,0.85714,0.0,1.0,1,7,0,1,0,8,0,0,7,0,0,1,0,0,3,0,0,3,0,0,2,0,7],[8,67,0.1194,0.45982,0.30249,0.14286,0.28571,0.75,0.14286,1.0,0,3,0,0,0,9,0,0,8,0,0,2,0,0,4,0,0,1,0,0,5,0,3],[12,67,0.1791,0.55357,0.31894,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,7,0,0,2,0,0,3,0,0,5,0,0,2,0,7],[16,67,0.2388,0.55804,0.32214,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,6,0,0,3,0,0,5,0,0,2,0,0,2,0,8],[20,67,0.2985,0.57589,0.33404,0.25001,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,3,0,0,4,0,0,1,0,0,4,0,0,5,0,7],[24,67,0.3582,0.48661,0.33476,0.14286,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,10,0,0,5,0,0,5,0,0,2,0,0,1,0,0,2,0,7],[28,67,0.4179,0.48661,0.34416,0.14286,0.42857,0.85714,0.0,1.0,1,7,0,1,0,10,0,0,3,0,0,6,0,0,1,0,0,2,0,0,2,0,7],[32,67,0.4776,0.5,0.32143,0.14286,0.42857,0.75,0.14286,1.0,0,6,0,0,0,9,0,0,5,0,0,4,0,0,3,0,0,3,0,0,2,0,6],[36,67,0.5373,0.45089,0.29474,0.14286,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,10,0,0,5,0,0,5,0,0,3,0,0,4,0,0,1,0,4],[40,67,0.597,0.5,0.33503,0.24999,0.28571,0.85714,0.14286,1.0,0,6,0,0,0,8,0,0,9,0,0,2,0,0,2,0,0,0,0,0,5,0,6],[44,67,0.6567,0.48214,0.32488,0.28571,0.28571,0.85714,0.0,1.0,1,5,0,1,0,6,0,0,10,0,0,3,0,0,2,0,0,0,0,0,5,0,5],[48,67,0.7164,0.4375,0.31931,0.14286,0.28571,0.71429,0.14286,1.0,0,5,0,0,0,13,0,0,4,0,0,3,0,0,3,0,0,3,0,0,1,0,5],[52,67,0.7761,0.37945,0.27573,0.14286,0.28571,0.57111,0.14286,1.0,0,2,0,0,0,14,0,0,4,0,0,5,0,0,4,0,0,0,0,0,3,0,2],[56,67,0.8358,0.46875,0.32583,0.14286,0.42857,0.75,0.14286,1.0,0,5,0,0,0,12,0,0,3,0,0,4,0,0,3,0,0,2,0,0,3,0,5],[60,67,0.8955,0.37045,0.2877,0.14286,0.28571,0.46429,0.0,1.0,1,4,0,1,0,12,0,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,4],[64,67,0.9552,0.53125,0.29931,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,5,0,0,5,0,0,4,0,0,5,0,0,5,0,0,2,0,5],[67,67,1.0,0.57589,0.29339,0.39286,0.57143,0.75,0.0,1.0,2,5,0,2,0,2,0,0,4,0,0,5,0,0,4,0,0,7,0,0,3,0,5]]}]},{"i":"e0500e048cc25a99","q":"Prove that there doesn\u2019t exist any prime  $p$  such that every power of  $p$  is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).","t":[{"b":6,"e":0.28571,"k":"falling","v":0.2008,"x":0.77678,"p":[[0,35,0.0,0.77678,0.35163,0.28571,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,22],[4,35,0.1143,0.74554,0.38255,0.28571,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,22],[8,35,0.2286,0.55803,0.35778,0.2857,0.28571,1.0,0.0,1.0,1,12,0,1,0,2,0,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,12],[12,35,0.3429,0.25,0.21129,0.14286,0.21428,0.28571,0.0,1.0,2,2,0,2,0,14,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,35,0.4571,0.23214,0.07784,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,13,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.24554,0.06423,0.14289,0.2857,0.28571,0.14286,0.286,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.2008,0.07879,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,17,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.2366,0.06785,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.20982,0.09438,0.14286,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,11,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.23884,"x":0.83036,"p":[[0,19,0.0,0.83036,0.33775,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,25],[4,19,0.2105,0.76786,0.33835,0.28571,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,20],[8,19,0.4211,0.62045,0.40982,0.24999,1.0,1.0,0.0,1.0,2,17,0,2,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[12,19,0.6316,0.23884,0.07907,0.14286,0.28571,0.28571,0.0,0.35714,1,0,0,1,0,9,0,0,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.27233,0.07457,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,6,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.24107,0.06622,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"00b9154211027037","q":"Suppose line  $\\ell$  and four points  $A,B,C,D$  lies on  $\\ell$ . Suppose that circles  $\\omega_1 , \\omega_2$  passes through  $A,B$  and circles  $\\omega'_1 , \\omega'_2$  passes through  $C,D$ . If  $\\omega_1 \\perp \\omega'_1$  and  $\\omega_2 \\perp \\omega'_2$  then prove that lines  $O_1O'_2 , O_2O'_1 , \\ell $  are concurrent where  $O_1,O_2,O'_1,O'_2$  are center of  $\\omega_1 , \\omega_2 , \\omega'_1 , \\omega'_2$ .","t":[{"b":6,"e":0.571,"k":"flat","v":0.36603,"x":0.45981,"p":[[0,56,0.0,0.36603,0.1284,0.2857,0.28571,0.42858,0.14286,0.57143,0,0,0,0,0,2,0,0,17,0,0,6,0,0,7,0,0,0,0,0,0,0,0],[4,56,0.0714,0.38834,0.12985,0.28571,0.28571,0.571,0.14286,0.57143,0,0,0,0,0,1,0,0,16,0,0,6,0,0,9,0,0,0,0,0,0,0,0],[8,56,0.1429,0.43744,0.14692,0.28571,0.42857,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,13,0,0,6,0,0,12,0,0,0,0,0,1,0,0],[12,56,0.2143,0.45981,0.13235,0.28571,0.4286,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,9,0,0,12,0,0,2,0,0,0,0,0],[16,56,0.2857,0.4464,0.1275,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,9,0,0,12,0,0,1,0,0,0,0,0],[20,56,0.3571,0.44639,0.11147,0.39286,0.4286,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,12,0,0,12,0,0,0,0,0,0,0,0],[24,56,0.4286,0.43748,0.12835,0.28571,0.42929,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,6,0,0,14,0,0,0,0,0,0,0,0],[28,56,0.5,0.43746,0.12842,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,6,0,0,14,0,0,0,0,0,0,0,0],[32,56,0.5714,0.41512,0.15709,0.28571,0.42857,0.571,0.0,0.71429,1,0,0,1,0,0,0,0,13,0,0,7,0,0,9,0,0,2,0,0,0,0,0],[36,56,0.6429,0.41068,0.14614,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,4,0,0,9,0,0,2,0,0,0,0,0],[40,56,0.7143,0.42854,0.17126,0.28571,0.42857,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,7,0,0,7,0,0,2,0,0,0,0,1],[44,56,0.7857,0.43299,0.14497,0.28571,0.42857,0.57141,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,7,0,0,11,0,0,0,0,0,1,0,0],[48,56,0.8571,0.41511,0.12548,0.28571,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,7,0,0,11,0,0,0,0,0,0,0,0],[52,56,0.9286,0.45084,0.1433,0.28571,0.4998,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,8,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[56,56,1.0,0.40619,0.11895,0.28571,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,9,0,0,9,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.39728,"x":0.47316,"p":[[0,33,0.0,0.39728,0.12229,0.28571,0.35714,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,7,0,0,9,0,0,0,0,0,0,0,0],[4,33,0.1212,0.42409,0.15354,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,15,0,0,6,0,0,9,0,0,1,0,0,1,0,0],[8,33,0.2424,0.42409,0.14052,0.28571,0.42857,0.4642,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,12,0,0,6,0,0,1,0,0,1,0,0],[12,33,0.3636,0.45083,0.13876,0.28571,0.49979,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,4,0,0,15,0,0,1,0,0,0,0,0],[16,33,0.4848,0.40172,0.12587,0.28571,0.35714,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,6,0,0,10,0,0,0,0,0,0,0,0],[20,33,0.6061,0.4285,0.1287,0.28571,0.42857,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,6,0,0,13,0,0,0,0,0,0,0,0],[24,33,0.7273,0.41959,0.171,0.28571,0.28571,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,17,0,0,4,0,0,9,0,0,1,0,0,0,0,1],[28,33,0.8485,0.4196,0.13329,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,12,0,0,7,0,0,12,0,0,0,0,0,0,0,0],[32,33,0.9697,0.44635,0.11143,0.39286,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,12,0,0,12,0,0,0,0,0,0,0,0],[33,33,1.0,0.47316,0.13567,0.28571,0.571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,6,0,0,15,0,0,2,0,0,0,0,0]]}]},{"i":"4dcc2907c311fb06","q":"Starting with any  $n$ -tuple  $R_0$ ,  $n\\ge 1$ , of symbols from  $A,B,C$ , we define a sequence  $R_0, R_1, R_2,\\ldots,$  according to the following rule: If  $R_j= (x_1,x_2,\\ldots,x_n)$ , then  $R_{j+1}= (y_1,y_2,\\ldots,y_n)$ , where  $y_i=x_i$  if  $x_i=x_{i+1}$  (taking  $x_{n+1}=x_1$ ) and  $y_i$  is the symbol other than  $x_i, x_{i+1}$  if  $x_i\\neq x_{i+1}$ . Find all positive integers  $n>1$  for which there exists some integer  $m>0$  such that  $R_m=R_0$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.91962,"x":1.0,"p":[[0,35,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,35,0.1143,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,35,0.2286,0.91962,0.22574,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,28],[12,35,0.3429,0.95089,0.16213,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[32,35,0.9143,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[35,35,1.0,0.91963,0.10678,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19]]},{"b":7,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,13,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,13,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,13,0.6154,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1d2225bd71a945e9","q":"Suppose the integers  $1,2,\\ldots 10$  are split into two disjoint collections  $a_1,a_2, \\ldots a_5$  and  $b_1 , \\ldots b_5$  such that  $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$  \r\n(i) Show that the larger number in any pair  $\\{ a_j, b_j \\}$  ,  $1 \\leq j \\leq 5$  is at least  $6$ .\r\n(ii) Show that  $\\sum_{i=1} ^{5} | a_i - b_i|$  = 25 for every such partition.","t":[{"b":3,"e":0.71429,"k":"rising","v":0.30356,"x":0.81249,"p":[[0,17,0.0,0.30356,0.28063,0.0,0.2857,0.57143,0.0,0.71429,11,0,0,11,0,4,0,0,4,0,0,2,0,0,5,0,0,6,0,0,0,0,0],[4,17,0.2353,0.57588,0.39846,0.14286,0.78564,0.85714,0.0,1.0,7,7,0,7,0,3,0,0,2,0,0,0,0,0,1,0,0,3,0,0,9,0,7],[8,17,0.4706,0.6116,0.37836,0.2857,0.71429,1.0,0.0,1.0,4,11,0,4,0,3,0,0,4,0,0,2,0,0,1,0,0,3,0,0,4,0,11],[12,17,0.7059,0.63391,0.39437,0.24999,0.857,1.0,0.0,1.0,7,10,0,7,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,0,8,0,10],[16,17,0.9412,0.75446,0.27487,0.71429,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,9,0,0,9,0,9],[17,17,1.0,0.81249,0.24337,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,4,0,0,10,0,13]]},{"b":6,"e":0.85714,"k":"rising","v":0.35268,"x":0.72318,"p":[[0,17,0.0,0.35268,0.31336,0.0,0.28571,0.57143,0.0,1.0,10,2,0,10,0,1,0,0,8,0,0,2,0,0,4,0,0,4,0,0,1,0,2],[4,17,0.2353,0.68747,0.34706,0.42857,0.85707,1.0,0.0,1.0,3,11,0,3,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,0,8,0,11],[8,17,0.4706,0.72318,0.23675,0.57143,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,2,0,0,17,0,3],[12,17,0.7059,0.59818,0.33775,0.39286,0.64286,0.85714,0.0,1.0,5,4,0,5,0,1,0,0,2,0,0,2,0,0,6,0,0,1,0,0,11,0,4],[16,17,0.9412,0.64734,0.27426,0.53575,0.71429,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,3,0,0,2,0,0,7,0,0,3,0,0,10,0,4],[17,17,1.0,0.6875,0.32818,0.67857,0.85714,0.85714,0.0,1.0,5,3,0,5,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,19,0,3]]}]},{"i":"6acc94d23416d047","q":"Let $n$ be an odd integer, and let $S$ be a set of $n$ points in the plane with integer coordinates. Consider a permutation $\\mathrm{f}: \\mathrm{S} \\rightarrow \\mathrm{S}$ that satisfies the following property:\nFor any pair of points $A$ and $B$ belonging to $S$, the distance between $f(A)$ and $f(B)$ is greater than or equal to the distance between $A$ and $B$.\nShow that there exists a point $X$, belonging to $S$, such that $f(X)=X$.","t":[{"b":1,"e":0.1429,"k":"flat","v":0.0759,"x":0.18295,"p":[[0,25,0.0,0.18295,0.17217,0.14286,0.14286,0.2857,0.0,1.0,5,1,0,5,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,25,0.16,0.15179,0.14698,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,0,13,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,25,0.32,0.10715,0.14286,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,13,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,25,0.48,0.11143,0.08546,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.12054,0.13415,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,18,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,25,0.8,0.12927,0.12549,0.0,0.14286,0.14287,0.0,0.571,11,0,0,11,0,15,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,25,0.96,0.08911,0.07772,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.143,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.09375,"x":0.19179,"p":[[0,12,0.0,0.19179,0.17358,0.14286,0.14286,0.2857,0.0,1.0,5,1,0,5,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,12,0.3333,0.11599,0.13569,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,17,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,12,0.6667,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce402d4042282196","q":"Prove that for every prime number  $p\\ge5$ ,\n\n(a)  $p^3$  divides  $\\binom{2p}p-2$ ;\n(b)  $p^3$  divides  $\\binom{kp}p-k$  for every natural number  $k$ .","t":[{"b":0,"e":0.42857,"k":"falling","v":0.57589,"x":0.9375,"p":[[0,62,0.0,0.83929,0.22517,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,3,0,0,0,0,20],[4,62,0.0645,0.87054,0.21535,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,21],[8,62,0.129,0.85713,0.27201,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,24],[12,62,0.1935,0.84822,0.2765,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,23],[16,62,0.2581,0.9375,0.15947,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,27],[20,62,0.3226,0.89286,0.21129,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,24],[24,62,0.3871,0.8125,0.27994,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,20],[28,62,0.4516,0.86159,0.23005,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,1,0,0,3,0,21],[32,62,0.5161,0.88838,0.21351,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,23],[36,62,0.5806,0.90622,0.19438,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,25],[40,62,0.6452,0.88839,0.23072,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,24],[44,62,0.7097,0.85714,0.19233,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,4,0,18],[48,62,0.7742,0.83482,0.25028,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,0,0,0,1,0,21],[52,62,0.8387,0.69195,0.33524,0.39286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,3,0,0,1,0,0,5,0,0,2,0,0,1,0,15],[56,62,0.9032,0.77232,0.274,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,2,0,0,5,0,15],[60,62,0.9677,0.59819,0.29761,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,6,0,0,0,0,0,9,0,0,4,0,0,1,0,8],[62,62,1.0,0.57589,0.31234,0.28571,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,5,0,0,7,0,0,2,0,0,2,0,0,7,0,0,2,0,7]]},{"b":1,"e":1.0,"k":"flat","v":0.6875,"x":0.87946,"p":[[0,31,0.0,0.77232,0.2809,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,2,0,0,0,0,18],[4,31,0.129,0.80357,0.26904,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,1,0,0,0,0,20],[8,31,0.2581,0.87946,0.24772,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,24],[12,31,0.3871,0.76786,0.28959,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,2,0,0,2,0,17],[16,31,0.5161,0.73213,0.26185,0.57132,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,3,0,0,4,0,12],[20,31,0.6452,0.82142,0.25505,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,6,0,0,2,0,0,2,0,19],[24,31,0.7742,0.6875,0.3102,0.57143,0.71429,1.0,0.0,1.0,1,13,0,1,0,3,0,0,2,0,0,0,0,0,9,0,0,4,0,0,0,0,13],[28,31,0.9032,0.74554,0.30037,0.53571,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,4,0,0,3,0,0,4,0,0,1,0,16],[31,31,1.0,0.77232,0.28762,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,2,0,0,3,0,0,3,0,0,5,0,15]]}]},{"i":"3f7630e906aa7b76","q":"Suppose that the sequence  $\\{a_n\\}$  of positive integers satisfies the following conditions: \n\n\n- For an integer  $i \\geq 2022$ , define  $a_i$  as the smallest positive integer  $x$  such that  $x+\\sum_{k=i-2021}^{i-1}a_k$  is a perfect square.\n- There exists infinitely many positive integers  $n$  such that  $a_n=4\\times 2022-3$ .\n\nProve that there exists a positive integer  $N$  such that  $\\sum_{k=n}^{n+2021}a_k$  is constant for every integer  $n \\geq N$ . \nAnd determine the value of  $\\sum_{k=N}^{N+2021}a_k$ .","t":[{"b":3,"e":0.71429,"k":"flat","v":0.55804,"x":0.69642,"p":[[0,70,0.0,0.55804,0.17627,0.57143,0.57143,0.60714,0.0,0.85714,2,0,2,2,0,0,0,0,1,0,0,2,0,0,19,0,0,7,0,0,1,0,0],[4,70,0.0571,0.65177,0.16343,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,20,0,0,0,0,2],[8,70,0.1143,0.66963,0.16146,0.57143,0.71429,0.71429,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,1,0,0,7,0,0,21,0,0,0,0,2],[12,70,0.1714,0.60713,0.15568,0.5713,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,6,0,0,7,0,0,18,0,0,0,0,0],[16,70,0.2286,0.64282,0.11296,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,21,0,0,0,0,0],[20,70,0.2857,0.61607,0.18707,0.4286,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,6,0,0,4,0,0,17,0,0,0,0,2],[24,70,0.3429,0.61159,0.15251,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,19,0,0,0,0,0],[28,70,0.4,0.62054,0.1411,0.57143,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,11,0,0,15,0,0,0,0,1],[32,70,0.4571,0.65624,0.07874,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,20,0,0,0,0,0],[36,70,0.5143,0.67409,0.1025,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,22,0,0,0,0,1],[40,70,0.5714,0.62946,0.11214,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,18,0,0,0,0,0],[44,70,0.6286,0.62945,0.14223,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,17,0,0,0,0,1],[48,70,0.6857,0.64283,0.10103,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,20,0,0,0,0,0],[52,70,0.7429,0.64729,0.11839,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,22,0,0,0,0,0],[56,70,0.8,0.63391,0.11812,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,20,0,0,0,0,0],[60,70,0.8571,0.65625,0.08645,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,21,0,0,0,0,0],[64,70,0.9143,0.63387,0.10065,0.57132,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,18,0,0,0,0,0],[68,70,0.9714,0.64284,0.07987,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,17,0,0,0,0,0],[70,70,1.0,0.69642,0.04728,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.50445,"x":0.8125,"p":[[0,81,0.0,0.50445,0.25996,0.49968,0.57143,0.57143,0.0,1.0,4,2,2,4,0,2,0,0,2,0,0,0,0,0,17,0,0,5,0,0,0,0,2],[4,81,0.0494,0.58482,0.21237,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,1,0,0,5,0,0,6,0,0,15,0,0,1,0,1],[8,81,0.0988,0.58482,0.21829,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,1,0,0,2,0,0,7,0,0,17,0,0,0,0,1],[12,81,0.1481,0.61606,0.21852,0.57132,0.71429,0.71429,0.0,1.0,2,2,1,2,0,0,0,0,1,0,0,4,0,0,6,0,0,16,0,0,1,0,2],[16,81,0.1975,0.71875,0.20973,0.71429,0.71429,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,19,0,0,0,0,7],[20,81,0.2469,0.78125,0.1636,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,16,0,0,1,0,10],[24,81,0.2963,0.65179,0.27418,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,6,0,0,1,0,0,11,0,0,1,0,8],[28,81,0.3457,0.69197,0.22899,0.57143,0.71429,0.75,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,3,0,0,4,0,0,15,0,0,2,0,6],[32,81,0.3951,0.77232,0.22829,0.71429,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,4,0,11],[36,81,0.4444,0.78125,0.18205,0.71429,0.71429,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,15,0,0,1,0,11],[40,81,0.4938,0.72322,0.20806,0.67857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,15,0,0,1,0,8],[44,81,0.5432,0.8125,0.20025,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,11,0,0,4,0,13],[48,81,0.5926,0.77679,0.15947,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,19,0,0,1,0,9],[52,81,0.642,0.79911,0.15093,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,0,0,11],[56,81,0.6914,0.78125,0.15146,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,19,0,0,1,0,9],[60,81,0.7407,0.75893,0.12595,0.71429,0.71429,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,22,0,0,3,0,5],[64,81,0.7901,0.8125,0.14032,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,1,0,11],[68,81,0.8395,0.78571,0.13363,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,2,0,8],[72,81,0.8889,0.78571,0.15152,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,0,0,0,10],[76,81,0.9383,0.77232,0.11769,0.71429,0.71429,0.75,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,2,0,6],[80,81,0.9877,0.80356,0.15049,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,4,0,10],[81,81,1.0,0.75446,0.12993,0.71429,0.71429,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,0,1,0,6]]}]},{"i":"ed749d66969e4533","q":"Let $p$ be a prime. A complete residue class modulo $p$ is a set containing at least one element equivalent to $k(\\bmod p)$ for all $k$.\n(a) (20) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.\n(b) (25) Show that there exists an $n \\leq p^{2}$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.","t":[{"b":3,"e":0.85714,"k":"falling","v":0.7857,"x":1.0,"p":[[0,37,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,37,0.1081,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[8,37,0.2162,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[12,37,0.3243,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[16,37,0.4324,0.89286,0.19885,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,20],[20,37,0.5405,0.9241,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[24,37,0.6486,0.91964,0.14258,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,9,0,20],[28,37,0.7568,0.91071,0.13717,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[32,37,0.8649,0.81247,0.20653,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,0,13,0,10],[36,37,0.973,0.7857,0.22587,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,4,0,0,14,0,8],[37,37,1.0,0.79908,0.18163,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,0,18,0,5]]},{"b":6,"e":1.0,"k":"flat","v":0.79908,"x":1.0,"p":[[0,48,0.0,0.94197,0.20781,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[4,48,0.0833,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,48,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,48,0.25,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,48,0.3333,0.94195,0.19189,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[20,48,0.4167,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,48,0.5,0.96874,0.06903,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[28,48,0.5833,0.91962,0.14702,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,22],[32,48,0.6667,0.93748,0.10068,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[36,48,0.75,0.86159,0.20668,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,19],[40,48,0.8333,0.89283,0.17499,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,18],[44,48,0.9167,0.89283,0.11296,0.85711,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,14],[48,48,1.0,0.79908,0.19515,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,0,16,0,7]]}]},{"i":"4ef6a9c322cf8b9f","q":"Let $r$ be the radius of the inscribed circle of triangle $A B C$. Take a point $D$ on side $B C$, and let $r_{1}$ and $r_{2}$ be the inradii of triangles $A B D$ and $A C D$. Prove that $r, r_{1}$, and $r_{2}$ can always be the side lengths of a 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triangle  $ ABC$  acutangle, with  $ m \\angle ACB\\leq\\ m \\angle ABC$ .  $ M$  the midpoint of side  $ BC$  and  $ P$  a point over the side  $ MC$ . Let  $ C_{1}$  the circunference with center  $ C$ . Let  $ C_{2}$  the circunference with center  $ B$ .  $ P$  is a point of  $ C_{1}$  and  $ C_{2}$ . Let  $ X$  a point on the opposite semiplane than  $ B$  respecting with the straight line  $ AP$ ; Let  $ Y$  the intersection of side  $ XB$  with  $ C_{2}$  and  $ Z$  the intersection of side  $ XC$  with  $ C_{1}$ . Let  $ m\\angle PAX \\equal{} \\alpha$  and  $ m\\angle ABC \\equal{} \\beta$ . Find the geometric place of  $ X$  if it satisfies the following conditions:\r $ (a) \\frac {XY}{XZ} \\equal{} \\frac {XC \\plus{} CP}{XB \\plus{} BP}$ \r $ (b) \\cos(\\alpha) \\equal{} AB\\cdot \\frac {\\sin(\\beta )}{AP}$","t":[{"b":2,"e":0.42857,"k":"flat","v":0.62498,"x":0.79017,"p":[[0,41,0.0,0.76785,0.09942,0.71429,0.78564,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,16,0,0],[4,41,0.0976,0.75003,0.14275,0.71429,0.857,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,18,0,0],[8,41,0.1951,0.74998,0.17496,0.71429,0.78564,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,14,0,0,16,0,0],[12,41,0.2927,0.72311,0.18565,0.71429,0.78564,0.85714,0.14,0.85714,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,9,0,0,16,0,0],[16,41,0.3902,0.76337,0.15816,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,0,20,0,0],[20,41,0.4878,0.75891,0.14915,0.71429,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,17,0,1],[24,41,0.5854,0.76784,0.16655,0.71429,0.85707,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,19,0,0],[28,41,0.6829,0.78571,0.15972,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,19,0,1],[32,41,0.7805,0.79017,0.11284,0.71429,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,20,0,0],[36,41,0.878,0.77677,0.13335,0.71429,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,20,0,0],[40,41,0.9756,0.65174,0.17106,0.5354,0.71429,0.85704,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,9,0,0,9,0,0],[41,41,1.0,0.62498,0.16267,0.4286,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,8,0,0,7,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.76338,"x":0.99554,"p":[[0,41,0.0,0.76338,0.12681,0.71429,0.857,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,17,0,0],[4,41,0.0976,0.76338,0.12681,0.71429,0.857,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,17,0,0],[8,41,0.1951,0.78568,0.0875,0.71429,0.857,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,18,0,0],[12,41,0.2927,0.7991,0.10012,0.71429,0.85714,0.85714,0.5714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,17,0,2],[16,41,0.3902,0.79015,0.08739,0.71429,0.85707,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,19,0,0],[20,41,0.4878,0.79017,0.08736,0.71429,0.85707,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,19,0,0],[24,41,0.5854,0.76772,0.1666,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,17,0,1],[28,41,0.6829,0.83034,0.13095,0.71429,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,13,0,8],[32,41,0.7805,0.89732,0.11971,0.82143,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,17],[36,41,0.878,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[40,41,0.9756,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[41,41,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"7b7f449046264367","q":"Let \\( \\mathbb{R} \\) denote the set of real numbers. Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that\n\n\\[\nf(x^2) - y f(y) = f(x+y)(f(x) - y)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\).","t":[{"b":6,"e":0.42857,"k":"flat","v":0.58482,"x":0.80356,"p":[[0,127,0.0,0.64732,0.2575,0.42857,0.49999,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,15,0,0,3,0,0,1,0,0,3,0,9],[4,127,0.0315,0.70536,0.25738,0.42857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,3,0,0,4,0,11],[8,127,0.063,0.73661,0.26752,0.42857,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,8,0,0,1,0,0,5,0,0,3,0,13],[12,127,0.0945,0.76786,0.23623,0.42857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,3,0,0,7,0,12],[16,127,0.126,0.76786,0.22517,0.57143,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,8,0,0,2,0,13],[20,127,0.1575,0.74107,0.24074,0.42857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,3,0,0,6,0,11],[24,127,0.189,0.69196,0.24512,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,3,0,0,5,0,9],[28,127,0.2205,0.75447,0.2237,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,8,0,0,2,0,12],[32,127,0.252,0.78125,0.22299,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,4,0,0,7,0,12],[36,127,0.2835,0.78571,0.22588,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,4,0,0,6,0,13],[40,127,0.315,0.80356,0.21652,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,3,0,15],[44,127,0.3465,0.72768,0.23787,0.42857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,6,0,0,3,0,11],[48,127,0.378,0.75897,0.23259,0.53607,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,7,0,0,2,0,13],[52,127,0.4094,0.61607,0.23808,0.42857,0.4286,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,17,0,0,4,0,0,3,0,0,0,0,8],[56,127,0.4409,0.75,0.2369,0.42857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,5,0,0,6,0,11],[60,127,0.4724,0.68749,0.25112,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,4,0,0,3,0,10],[64,127,0.5039,0.62946,0.22263,0.42857,0.5,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,6,0,0,4,0,5],[68,127,0.5354,0.69197,0.22618,0.42857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,11,0,0,2,0,0,8,0,0,3,0,8],[72,127,0.5669,0.64286,0.22303,0.42857,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,6,0,0,3,0,6],[76,127,0.5984,0.6875,0.24338,0.42857,0.64286,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,4,0,0,2,0,10],[80,127,0.6299,0.69195,0.25782,0.42857,0.57143,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,2,0,0,1,0,12],[84,127,0.6614,0.58482,0.21237,0.42857,0.42857,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,7,0,0,0,0,5],[88,127,0.6929,0.70074,0.24317,0.42857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,5,0,0,3,0,10],[92,127,0.7244,0.66517,0.23038,0.42857,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,11,0,0,3,0,0,7,0,0,3,0,7],[96,127,0.7559,0.60268,0.22513,0.42857,0.42859,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,16,0,0,3,0,0,4,0,0,3,0,5],[100,127,0.7874,0.63839,0.21424,0.42857,0.64286,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,8,0,0,3,0,5],[104,127,0.8189,0.61606,0.22428,0.42857,0.4286,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,17,0,0,2,0,0,4,0,0,4,0,5],[108,127,0.8504,0.64285,0.23419,0.42857,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,15,0,0,3,0,0,4,0,0,3,0,7],[112,127,0.8819,0.65179,0.23128,0.42857,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,5,0,0,3,0,7],[116,127,0.9134,0.64732,0.22864,0.42857,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,6,0,0,2,0,7],[120,127,0.9449,0.69196,0.22335,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,4,0,0,8,0,6],[124,127,0.9764,0.71426,0.19563,0.571,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,5,0,0,11,0,4],[127,127,1.0,0.63393,0.18536,0.42857,0.57143,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,6,0,0,7,0,0,6,0,2]]},{"b":7,"e":0.85714,"k":"flat","v":0.55808,"x":0.80803,"p":[[0,105,0.0,0.73214,0.27606,0.42857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,12,0,0,1,0,0,1,0,0,2,0,15],[4,105,0.0381,0.76786,0.24157,0.42859,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,2,0,0,6,0,13],[8,105,0.0762,0.64732,0.24739,0.42857,0.50001,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,16,0,0,2,0,0,4,0,0,1,0,9],[12,105,0.1143,0.80357,0.23077,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,2,0,0,6,0,15],[16,105,0.1524,0.69197,0.23987,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,6,0,0,3,0,9],[20,105,0.1905,0.74107,0.25862,0.42857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,3,0,0,4,0,13],[24,105,0.2286,0.68304,0.2618,0.42857,0.57143,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,1,0,0,3,0,11],[28,105,0.2667,0.64732,0.23141,0.42857,0.64286,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,7,0,0,2,0,7],[32,105,0.3048,0.55808,0.22117,0.42857,0.42857,0.60714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,23,0,0,1,0,0,1,0,0,2,0,5],[36,105,0.3429,0.70982,0.21274,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,6,0,0,10,0,5],[40,105,0.381,0.71428,0.20202,0.42859,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,10,0,0,8,0,5],[44,105,0.419,0.7366,0.17536,0.57143,0.78564,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,7,0,0,13,0,3],[48,105,0.4571,0.75,0.17128,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,11,0,0,11,0,4],[52,105,0.4952,0.80357,0.14617,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,13,0,6],[56,105,0.5333,0.78125,0.13355,0.71429,0.85707,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,0,14,0,3],[60,105,0.5714,0.76339,0.16602,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,9,0,0,13,0,4],[64,105,0.6095,0.74553,0.1665,0.71429,0.85707,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,8,0,0,15,0,2],[68,105,0.6476,0.74554,0.1665,0.67857,0.78571,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,13,0,3],[72,105,0.6857,0.80803,0.13651,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,19,0,4],[76,105,0.7238,0.77231,0.12807,0.71429,0.85707,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,17,0,1],[80,105,0.7619,0.75,0.19233,0.57143,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,5,0,0,13,0,5],[84,105,0.8,0.78125,0.13355,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,0,14,0,3],[88,105,0.8381,0.77676,0.15127,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,0,16,0,3],[92,105,0.8762,0.72321,0.15542,0.67857,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,0,15,0,0],[96,105,0.9143,0.76339,0.16213,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,18,0,2],[100,105,0.9524,0.78125,0.16745,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,12,0,6],[104,105,0.9905,0.76772,0.12248,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,16,0,1],[105,105,1.0,0.78125,0.15146,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,17,0,3]]}]},{"i":"c236b57fc20baa99","q":"Let the real numbers $a, b, c, d$ satisfy the relations $a+b+c+d=6$ and $a^{2}+b^{2}+c^{2}+d^{2}=12$. Prove that  $$ 36 \\leq 4\\left(a^{3}+b^{3}+c^{3}+d^{3}\\right)-\\left(a^{4}+b^{4}+c^{4}+d^{4}\\right) \\leq 48 $$  (Ukraine)","t":[{"b":2,"e":0.0,"k":"flat","v":0.17411,"x":0.39286,"p":[[0,26,0.0,0.33036,0.411,0.0,0.14286,0.78571,0.0,1.0,13,8,0,13,0,7,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[4,26,0.1538,0.39286,0.45737,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,11],[8,26,0.3077,0.17411,0.36375,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[12,26,0.4615,0.33482,0.4159,0.0,0.0,0.57145,0.0,1.0,17,8,0,17,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,8],[16,26,0.6154,0.21429,0.39123,0.0,0.0,0.10714,0.0,1.0,24,6,0,24,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,6],[20,26,0.7692,0.17857,0.31135,0.0,0.0,0.42857,0.0,1.0,22,3,0,22,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[24,26,0.9231,0.18304,0.36637,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[26,26,1.0,0.19643,0.36727,0.0,0.0,0.10714,0.0,1.0,24,5,0,24,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,5]]},{"b":7,"e":0.0,"k":"falling","v":0.05357,"x":0.35268,"p":[[0,82,0.0,0.3125,0.40632,0.0,0.14286,0.57143,0.0,1.0,12,8,0,12,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[4,82,0.0488,0.35268,0.42855,0.0,0.0,0.85714,0.0,1.0,17,7,0,17,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,7],[8,82,0.0976,0.28572,0.40721,0.0,0.0,0.42858,0.0,1.0,20,7,0,20,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,7],[12,82,0.1463,0.25446,0.43114,0.0,0.0,0.35714,0.0,1.0,23,8,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[16,82,0.1951,0.16518,0.33902,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[20,82,0.2439,0.24107,0.38538,0.0,0.0,0.42857,0.0,1.0,22,5,0,22,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,5],[24,82,0.2927,0.20536,0.36585,0.0,0.0,0.32142,0.0,1.0,23,5,0,23,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[28,82,0.3415,0.125,0.25938,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,1],[32,82,0.3902,0.20982,0.36767,0.0,0.0,0.42857,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,5],[36,82,0.439,0.14286,0.33312,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[40,82,0.4878,0.33482,0.4159,0.0,0.0,0.57143,0.0,1.0,17,8,0,17,0,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,8],[44,82,0.5366,0.11161,0.25439,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,1],[48,82,0.5854,0.26339,0.38318,0.0,0.0,0.42857,0.0,1.0,19,6,0,19,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,6],[52,82,0.6341,0.20982,0.34623,0.0,0.0,0.42857,0.0,1.0,22,4,0,22,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,4],[56,82,0.6829,0.09821,0.22428,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[60,82,0.7317,0.19196,0.33045,0.0,0.0,0.42857,0.0,1.0,22,3,0,22,0,1,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,3],[64,82,0.7805,0.24107,0.36323,0.0,0.0,0.42857,0.0,1.0,20,5,0,20,0,0,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,5],[68,82,0.8293,0.07143,0.16751,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[72,82,0.878,0.11607,0.26108,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,1],[76,82,0.9268,0.12946,0.27283,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[80,82,0.9756,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.0625,0.14698,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"86db56a5a955ee23","q":"Let the sequence  $(a_n)$  be constructed in the following way:  $$ a_1=1,\\mbox{ }a_2=1,\\mbox{ }a_{n+2}=a_{n+1}+\\frac{1}{a_n},\\mbox{ }n=1,2,\\ldots. $$  Prove that  $a_{180}>19$ .\n\n*(Folklore)*","t":[{"b":3,"e":1.0,"k":"flat","v":0.88838,"x":0.99107,"p":[[0,108,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,108,0.037,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,108,0.0741,0.95088,0.13654,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[12,108,0.1111,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,108,0.1481,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,108,0.1852,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[24,108,0.2222,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[28,108,0.2593,0.95089,0.16213,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[32,108,0.2963,0.95088,0.1048,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[36,108,0.3333,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,108,0.3704,0.97321,0.07526,1.0,1.0,1.0,0.714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,108,0.4074,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[48,108,0.4444,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[52,108,0.4815,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[56,108,0.5185,0.91518,0.19186,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,25],[60,108,0.5556,0.9464,0.1225,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[64,108,0.5926,0.90179,0.14914,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,20],[68,108,0.6296,0.91963,0.15129,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[72,108,0.6667,0.90178,0.16917,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,21],[76,108,0.7037,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[80,108,0.7407,0.93749,0.15947,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[84,108,0.7778,0.91963,0.17476,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,23],[88,108,0.8148,0.91071,0.1948,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,22],[92,108,0.8519,0.94643,0.13243,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[96,108,0.8889,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[100,108,0.9259,0.88838,0.17765,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,21],[104,108,0.963,0.93303,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[108,108,1.0,0.89731,0.12996,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,17]]},{"b":5,"e":0.71429,"k":"falling","v":0.54463,"x":0.99554,"p":[[0,83,0.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,83,0.0482,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,83,0.0964,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[12,83,0.1446,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[16,83,0.1928,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,83,0.241,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,83,0.2892,0.91964,0.18536,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,25],[28,83,0.3373,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[32,83,0.3855,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,83,0.4337,0.93749,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[40,83,0.4819,0.89732,0.24545,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,26],[44,83,0.5301,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23],[48,83,0.5783,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,83,0.6265,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[56,83,0.6747,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[60,83,0.7229,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[64,83,0.7711,0.9241,0.15146,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,23],[68,83,0.8193,0.91071,0.12753,0.85711,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[72,83,0.8675,0.72322,0.3213,0.53572,0.85714,1.0,0.0,1.0,2,13,0,2,0,2,0,0,1,0,0,3,0,0,2,0,0,4,0,0,5,0,13],[76,83,0.9157,0.80357,0.28959,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,20],[80,83,0.9639,0.62498,0.3549,0.28571,0.71429,1.0,0.0,1.0,4,9,0,4,0,2,0,0,3,0,0,2,0,0,2,0,0,5,0,0,5,0,9],[83,83,1.0,0.54463,0.38205,0.14289,0.49979,1.0,0.0,1.0,5,9,0,5,0,4,0,0,3,0,0,4,0,0,2,0,0,1,0,0,4,0,9]]}]},{"i":"513bc011fc0f4d80","q":"The diagonals  $AC$  and  $BD$  of a cyclic quadrilateral  $ABCD$  intersect at  $P$ . Let  $O$  be the circumcenter of triangle  $APB$  and  $H$  be the orthocenter of triangle  $CPD$ . Show that the points  $H,P,O$  are collinear.","t":[{"b":0,"e":0.0,"k":"falling","v":0.18304,"x":0.55804,"p":[[0,98,0.0,0.50893,0.37786,0.14286,0.42857,0.89286,0.0,1.0,4,8,1,4,0,7,0,0,3,0,0,4,0,0,0,0,0,3,0,0,3,0,8],[4,98,0.0408,0.52223,0.37908,0.14286,0.42859,0.89286,0.0,1.0,5,8,0,5,0,5,0,0,3,0,0,4,0,0,0,0,0,4,0,0,3,0,8],[8,98,0.0816,0.53124,0.41068,0.14286,0.42857,1.0,0.0,1.0,5,11,0,5,0,8,0,0,0,0,0,4,0,0,1,0,0,0,0,0,3,0,11],[12,98,0.1224,0.39277,0.38138,0.105,0.21428,0.75,0.0,1.0,8,7,0,8,0,8,0,0,1,0,0,6,0,0,0,0,0,1,0,0,1,0,7],[16,98,0.1633,0.54465,0.36672,0.25,0.42859,1.0,0.0,1.0,3,10,0,3,0,5,0,0,3,0,0,8,0,0,0,0,0,1,0,0,2,0,10],[20,98,0.2041,0.47322,0.33775,0.14289,0.42857,0.71429,0.0,1.0,4,7,0,4,0,5,0,0,2,0,0,11,0,0,0,0,0,3,0,0,0,0,7],[24,98,0.2449,0.55804,0.32214,0.42857,0.42859,1.0,0.0,1.0,2,9,0,2,0,3,0,0,2,0,0,12,0,0,1,0,0,3,0,0,0,0,9],[28,98,0.2857,0.5,0.33693,0.14286,0.42857,0.78571,0.0,1.0,2,8,0,2,0,7,0,0,2,0,0,9,0,0,2,0,0,2,0,0,0,0,8],[32,98,0.3265,0.55357,0.3531,0.14286,0.42857,1.0,0.0,1.0,2,9,0,2,0,7,0,0,0,0,0,9,0,0,0,0,0,3,0,0,2,0,9],[36,98,0.3673,0.4375,0.38949,0.14286,0.42857,1.0,0.0,1.0,6,9,0,6,0,9,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,9],[40,98,0.4082,0.54018,0.36199,0.14286,0.42857,1.0,0.0,1.0,3,10,0,3,0,6,0,0,1,0,0,8,0,0,1,0,0,3,0,0,0,0,10],[44,98,0.449,0.45536,0.38703,0.14286,0.28571,1.0,0.0,1.0,4,9,0,4,0,10,0,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,9],[48,98,0.4898,0.41954,0.38128,0.14214,0.42857,0.78571,0.0,1.0,7,8,0,7,0,8,0,0,0,0,0,7,0,0,1,0,0,1,0,0,0,0,8],[52,98,0.5306,0.54464,0.38205,0.14286,0.42859,1.0,0.0,1.0,4,11,0,4,0,6,0,0,1,0,0,6,0,0,1,0,0,3,0,0,0,0,11],[56,98,0.5714,0.45089,0.3914,0.14286,0.42857,0.85714,0.0,1.0,7,7,0,7,0,8,0,0,0,0,0,4,0,0,0,0,0,4,0,0,2,0,7],[60,98,0.6122,0.54465,0.38703,0.14286,0.42857,1.0,0.0,1.0,2,12,0,2,0,9,0,0,1,0,0,7,0,0,0,0,0,0,0,0,1,0,12],[64,98,0.6531,0.50893,0.37447,0.14286,0.42857,0.89286,0.0,1.0,3,8,0,3,0,8,0,0,3,0,0,5,0,0,0,0,0,1,0,0,4,0,8],[68,98,0.6939,0.42411,0.3754,0.14286,0.28571,0.78571,0.0,1.0,5,8,0,5,0,9,0,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,8],[72,98,0.7347,0.40161,0.34536,0.14286,0.21428,0.71429,0.0,1.0,3,5,0,3,0,13,0,0,2,0,0,4,0,0,1,0,0,2,0,0,2,0,5],[76,98,0.7755,0.37947,0.32065,0.14286,0.2857,0.4286,0.0,1.0,4,5,0,4,0,9,0,0,5,0,0,7,0,0,0,0,0,2,0,0,0,0,5],[80,98,0.8163,0.32143,0.3312,0.14286,0.14288,0.42857,0.0,1.0,6,5,0,6,0,12,0,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,5],[84,98,0.8571,0.19635,0.24159,0.105,0.14286,0.14287,0.0,1.0,8,2,0,8,0,18,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[88,98,0.898,0.2633,0.26994,0.14286,0.14286,0.32143,0.0,1.0,5,3,0,5,0,15,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[92,98,0.9388,0.33027,0.34712,0.0,0.14286,0.4286,0.0,1.0,9,5,0,9,0,8,0,0,3,0,0,5,0,0,0,0,0,2,0,0,0,0,5],[96,98,0.9796,0.24106,0.2586,0.10714,0.14286,0.42857,0.0,1.0,8,2,0,8,0,13,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,2],[98,98,1.0,0.18304,0.16457,0.14286,0.14286,0.1786,0.0,0.71429,7,0,0,7,0,17,0,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"falling","v":0.11161,"x":0.49107,"p":[[0,55,0.0,0.35259,0.37967,0.14286,0.14286,0.53571,0.0,1.0,6,7,0,6,0,15,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,7],[4,55,0.0727,0.49107,0.38122,0.14286,0.42857,1.0,0.0,1.0,3,9,0,3,0,10,0,0,1,0,0,6,0,0,0,0,0,1,0,0,2,0,9],[8,55,0.1455,0.33036,0.36672,0.10714,0.14286,0.42857,0.0,1.0,8,6,0,8,0,12,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,6],[12,55,0.2182,0.31242,0.31836,0.14286,0.14286,0.42857,0.0,1.0,3,5,0,3,0,18,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,5],[16,55,0.2909,0.30348,0.31496,0.14214,0.14286,0.42857,0.0,1.0,7,4,0,7,0,12,0,0,1,0,0,7,0,0,0,0,0,1,0,0,0,0,4],[20,55,0.3636,0.24108,0.28446,0.0,0.14286,0.42857,0.0,1.0,9,3,0,9,0,13,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[24,55,0.4364,0.47322,0.36846,0.14286,0.42857,1.0,0.0,1.0,2,9,0,2,0,11,0,0,2,0,0,6,0,0,0,0,0,2,0,0,0,0,9],[28,55,0.5091,0.3392,0.34028,0.14214,0.14286,0.42857,0.0,1.0,7,5,0,7,0,10,0,0,3,0,0,5,0,0,0,0,0,2,0,0,0,0,5],[32,55,0.5818,0.29019,0.31234,0.14286,0.14286,0.42858,0.0,1.0,6,3,0,6,0,16,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,3],[36,55,0.6545,0.23661,0.25657,0.14286,0.14286,0.2857,0.0,1.0,5,2,0,5,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[40,55,0.7273,0.21429,0.21129,0.10714,0.14286,0.42857,0.0,1.0,8,1,0,8,0,13,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[44,55,0.8,0.21429,0.26964,0.0,0.14286,0.2857,0.0,1.0,10,2,0,10,0,13,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[48,55,0.8727,0.18304,0.20277,0.0,0.14286,0.1786,0.0,1.0,9,1,0,9,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[52,55,0.9455,0.18295,0.20279,0.0,0.14286,0.17857,0.0,1.0,9,1,0,9,0,15,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[55,55,1.0,0.11161,0.18118,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,8,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"c72cffad8004f45a","q":"Suppose $n \\geq 3$ is a positive integer. Let $a_{1}<a_{2}<\\cdots<a_{n}$ be an increasing sequence of positive real numbers, and let $a_{n+1}=a_{1}$. Prove that\n\n$$\n\\sum_{k=1}^{n} \\frac{a_{k}}{a_{k+1}}>\\sum_{k=1}^{n} \\frac{a_{k+1}}{a_{k}} .\n$$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.05356,"p":[[0,25,0.0,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,25,0.16,0.03572,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,25,0.32,0.05356,0.18119,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[12,25,0.48,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"rising","v":0.00893,"x":0.44187,"p":[[0,29,0.0,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,29,0.1379,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.10713,0.202,0.0,0.0,0.07143,0.0,0.71429,24,0,0,24,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[12,29,0.4138,0.41068,0.38753,0.0,0.35714,0.74996,0.0,1.0,11,6,0,11,0,3,0,0,2,0,0,2,0,0,5,0,0,1,0,0,2,0,6],[16,29,0.5517,0.3973,0.3724,0.0,0.42857,0.71429,0.0,1.0,11,6,0,11,0,2,0,0,2,0,0,6,0,0,2,0,0,3,0,0,0,0,6],[20,29,0.6897,0.26338,0.29473,0.0,0.14286,0.46418,0.0,1.0,12,1,0,12,0,7,0,0,3,0,0,2,0,0,3,0,0,3,0,0,1,0,1],[24,29,0.8276,0.44187,0.36316,0.14214,0.35714,0.71429,0.0,1.0,7,6,0,7,0,4,0,0,5,0,0,3,0,0,2,0,0,4,0,0,1,0,6],[28,29,0.9655,0.3214,0.33879,0.0,0.2857,0.571,0.0,1.0,12,3,0,12,0,3,0,0,5,0,0,3,0,0,3,0,0,1,0,0,2,0,3],[29,29,1.0,0.30799,0.28592,0.0,0.28571,0.57143,0.0,1.0,12,1,0,12,0,1,0,0,6,0,0,1,0,0,9,0,0,2,0,0,0,0,1]]}]},{"i":"c1ac1f68a391a691","q":"The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $A B \\| E C$ and $A C \\| E D$. The line tangent to the circle $c$ at $E$ meets the line $A B$ at $P$. The lines $B D$ and $E C$ meet at $Q$. Prove that $|A C|=|P Q|$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08036,"x":0.2767,"p":[[0,70,0.0,0.2767,0.2599,0.14286,0.14286,0.42857,0.0,1.0,3,2,3,3,0,19,0,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,2],[4,70,0.0571,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,70,0.1143,0.14277,0.18211,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,70,0.1714,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,70,0.2286,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,19,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.10715,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.1383,0.07563,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.13393,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.12946,0.05486,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.14277,0.07143,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.15626,0.10326,0.14286,0.14286,0.14286,0.0,0.4286,4,0,0,4,0,24,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.13831,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04465,"x":0.19634,"p":[[0,71,0.0,0.19634,0.23079,0.14286,0.14286,0.14286,0.0,1.0,6,1,3,6,0,21,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[4,71,0.0563,0.12492,0.15045,0.0,0.14286,0.14287,0.0,0.57143,14,0,0,14,0,12,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,71,0.1127,0.11143,0.11697,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,15,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,71,0.169,0.11152,0.17027,0.0,0.07,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,71,0.2254,0.10268,0.12492,0.0,0.07143,0.14286,0.0,0.4286,16,0,1,16,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,71,0.2817,0.13393,0.13333,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,71,0.338,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.15179,0.13803,0.0,0.14286,0.14287,0.0,0.57143,9,0,0,9,0,16,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[32,71,0.4507,0.10705,0.07982,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.125,0.14174,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,14,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,71,0.5634,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.09362,0.11638,0.0,0.07,0.14286,0.0,0.43,16,0,0,16,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.13393,0.13803,0.0,0.14286,0.14286,0.0,0.5714,11,0,0,11,0,16,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[56,71,0.7887,0.08464,0.11205,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.07571,0.07112,0.0,0.14,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.11598,0.06618,0.14214,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.13813,0.02482,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"375c0eaa5b440e60","q":"The incentre of the triangle  $ABC$  is  $I.$  The lines  $AI, BI$  and  $CI$  meet the circumcircle of the triangle  $ABC$  also at points  $D, E$  and  $F,$  respectively. \nProve that  $AD$  and  $EF$  are perpendicular.","t":[{"b":0,"e":0.85714,"k":"falling","v":0.53568,"x":0.97321,"p":[[0,91,0.0,0.94643,0.17035,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[4,91,0.044,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,91,0.0879,0.85714,0.31744,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,26],[12,91,0.1319,0.93304,0.21124,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[16,91,0.1758,0.80803,0.33429,0.78561,1.0,1.0,0.0,1.0,1,23,0,1,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,23],[20,91,0.2198,0.83927,0.29614,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,23],[24,91,0.2637,0.76339,0.36702,0.5,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,22],[28,91,0.3077,0.9241,0.22012,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,28],[32,91,0.3516,0.91517,0.24187,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,28],[36,91,0.3956,0.85267,0.28457,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,24],[40,91,0.4396,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,91,0.4835,0.87945,0.26991,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,26],[48,91,0.5275,0.95089,0.16213,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[52,91,0.5714,0.93304,0.22864,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[56,91,0.6154,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[60,91,0.6593,0.92857,0.19885,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[64,91,0.7033,0.94643,0.17768,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[68,91,0.7473,0.92857,0.17857,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,26],[72,91,0.7912,0.86606,0.25987,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,24],[76,91,0.8352,0.88839,0.26663,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,26],[80,91,0.8791,0.75445,0.25813,0.53539,0.78564,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,6,0,0,2,0,14],[84,91,0.9231,0.53568,0.29666,0.28571,0.4998,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,10,0,0,3,0,0,4,0,0,5,0,0,1,0,6],[88,91,0.967,0.70088,0.28652,0.42857,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,3,0,0,0,0,14],[91,91,1.0,0.54909,0.31766,0.2857,0.42857,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,10,0,0,4,0,0,4,0,0,1,0,0,2,0,8]]},{"b":3,"e":1.0,"k":"falling","v":0.52665,"x":1.0,"p":[[0,119,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,119,0.0336,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,119,0.0672,0.92411,0.24218,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[12,119,0.1008,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[16,119,0.1345,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[20,119,0.1681,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[24,119,0.2017,0.9375,0.20806,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[28,119,0.2353,0.93304,0.2172,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[32,119,0.2689,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,119,0.3025,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[40,119,0.3361,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[44,119,0.3697,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[48,119,0.4034,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,119,0.437,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[56,119,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,119,0.5042,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,119,0.5378,0.96427,0.12376,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[68,119,0.5714,0.96429,0.12371,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[72,119,0.605,0.9107,0.2075,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,26],[76,119,0.6387,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[80,119,0.6723,0.79464,0.34798,0.64286,1.0,1.0,0.0,1.0,2,23,0,2,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,23],[84,119,0.7059,0.83928,0.28959,0.82132,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,0,1,0,23],[88,119,0.7395,0.86161,0.32436,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[92,119,0.7731,0.89732,0.26058,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,26],[96,119,0.8067,0.88379,0.23817,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,25],[100,119,0.8403,0.79911,0.34967,0.71429,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,23],[104,119,0.8739,0.71875,0.3668,0.42857,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,0,1,0,18],[108,119,0.9076,0.73659,0.33142,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,4,0,0,2,0,0,2,0,0,2,0,0,2,0,17],[112,119,0.9412,0.8482,0.23128,0.71429,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,6,0,0,3,0,19],[116,119,0.9748,0.71871,0.30408,0.571,0.71429,1.0,0.0,1.0,1,13,0,1,0,2,0,0,3,0,0,1,0,0,3,0,0,7,0,0,2,0,13],[119,119,1.0,0.52665,0.33021,0.25001,0.4998,0.75,0.0,1.0,2,7,0,2,0,6,0,0,3,0,0,5,0,0,4,0,0,4,0,0,1,0,7]]}]},{"i":"8db1ce46fd2d0a4c","q":"Let $p$ be a prime. Given a sequence of positive integers $b_{1}$ through $b_{n}$, exactly one of which is divisible by $p$, show that when\n\n$$\n\\frac{1}{b_{1}}+\\frac{1}{b_{2}}+\\ldots+\\frac{1}{b_{n}}\n$$\n\nis written as a fraction in lowest terms, then its denominator is divisible by $p$. Use this fact to explain why no prime $p$ is ever juicy.","t":[{"b":2,"e":0.4286,"k":"flat","v":0.47325,"x":0.60267,"p":[[0,26,0.0,0.47325,0.14477,0.42857,0.4293,0.57143,0.0,0.71429,2,0,2,2,0,0,0,0,0,0,0,15,0,0,14,0,0,1,0,0,0,0,0],[4,26,0.1538,0.5625,0.1181,0.4286,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,0,4,0,0,2,0,0],[8,26,0.3077,0.56698,0.12098,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,9,0,0,11,0,0,0,0,0],[12,26,0.4615,0.60265,0.12234,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,0,10,0,0,2,0,0],[16,26,0.6154,0.54462,0.12595,0.42857,0.571,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,15,0,0,9,0,0,7,0,0,1,0,0],[20,26,0.7692,0.60267,0.11143,0.57132,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,14,0,0,0,0,0],[24,26,0.9231,0.58033,0.10677,0.571,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,17,0,0,7,0,0,1,0,0],[26,26,1.0,0.58912,0.13231,0.42859,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,13,0,0,7,0,0,3,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.46429,"x":0.51339,"p":[[0,30,0.0,0.51339,0.12299,0.42857,0.42857,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,20,0,0,6,0,0,5,0,0,1,0,0],[4,30,0.1333,0.49999,0.09448,0.42857,0.42859,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,10,0,0,3,0,0,0,0,0],[8,30,0.2667,0.4866,0.08644,0.42857,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,18,0,0,12,0,0,1,0,0,0,0,0],[12,30,0.4,0.49554,0.09432,0.42857,0.42859,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,19,0,0,12,0,0,0,0,0,1,0,0],[16,30,0.5333,0.48213,0.09277,0.42857,0.4286,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,17,0,0,12,0,0,1,0,0,0,0,0],[20,30,0.6667,0.47767,0.0918,0.42857,0.42857,0.57111,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,0,0,0,1,0,0],[24,30,0.8,0.48661,0.11214,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,18,0,0,10,0,0,1,0,0,1,0,0],[28,30,0.9333,0.491,0.07889,0.42857,0.4286,0.57143,0.42857,0.71,0,0,0,0,0,0,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0],[30,30,1.0,0.46429,0.07143,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0]]}]},{"i":"2f50320bdb8782f3","q":"There are  $n^2$  segments in the plane (read walls), no two of which are parallel or intersecting. \n\nProve that there are at least  $n$  points in the plane such that no two of them see each other (meaning there is a wall separating them).","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.05347,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.05347,0.12741,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,26,0.6154,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"rising","v":0.00446,"x":0.25427,"p":[[0,41,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.09821,0.14032,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.07142,0.14281,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,41,0.2927,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.09375,0.16982,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,41,0.4878,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.25427,0.31086,0.0,0.14,0.42857,0.0,1.0,15,1,0,15,0,3,0,0,5,0,0,2,0,0,1,0,0,3,0,0,2,0,1],[28,41,0.6829,0.12054,0.17536,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,4,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[32,41,0.7805,0.1517,0.18537,0.0,0.07,0.28571,0.0,0.71429,16,0,0,16,0,5,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[36,41,0.878,0.16509,0.1927,0.0,0.14286,0.2857,0.0,0.71429,14,0,0,14,0,8,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[40,41,0.9756,0.13839,0.16935,0.0,0.14286,0.17857,0.0,0.57143,15,0,0,15,0,9,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[41,41,1.0,0.165,0.2024,0.0,0.07,0.2857,0.0,0.71429,16,0,0,16,0,5,0,0,4,0,0,5,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"94b45d3fb9fa25c1","q":"The tangents to the circumcircle of  $\\triangle ABC$  at  $B$  and  $C$  meet at  $D$ . The circumcircle of  $\\triangle BCD$  meets sides  $AC$  and  $AB$  again at  $E$  and  $F$  respectively. Let  $O$  be the circumcentre of  $\\triangle ABC$ . Show that  $AO$  is perpendicular to  $EF$ .","t":[{"b":3,"e":0.71429,"k":"falling","v":0.49552,"x":0.89732,"p":[[0,70,0.0,0.89732,0.20896,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,24],[4,70,0.0571,0.7008,0.31224,0.4286,0.78564,1.0,0.0,1.0,1,13,0,1,0,3,0,0,0,0,0,5,0,0,5,0,0,2,0,0,3,0,13],[8,70,0.1143,0.76785,0.23891,0.57143,0.78564,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,11,0,0,3,0,0,2,0,14],[12,70,0.1714,0.70534,0.31123,0.5354,0.71429,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,1,0,0,3,0,0,7,0,0,2,0,0,0,0,15],[16,70,0.2286,0.65167,0.34629,0.42857,0.64286,1.0,0.0,1.0,2,13,0,2,0,4,0,0,1,0,0,3,0,0,6,0,0,2,0,0,1,0,13],[20,70,0.2857,0.68293,0.33657,0.42859,0.78571,1.0,0.0,1.0,1,14,1,1,0,4,0,0,2,0,0,2,0,0,6,0,0,1,0,0,2,0,14],[24,70,0.3429,0.66964,0.2976,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,1,0,0,2,0,0,12,0,0,0,0,0,1,0,12],[28,70,0.4,0.70979,0.2612,0.5713,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,3,0,0,11,0,0,1,0,0,2,0,12],[32,70,0.4571,0.74998,0.29234,0.5713,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,4,0,0,1,0,16],[36,70,0.5143,0.75,0.29233,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,1,0,0,4,0,15],[40,70,0.5714,0.70537,0.3498,0.50007,0.85714,1.0,0.0,1.0,3,15,0,3,0,1,0,0,4,0,0,0,0,0,3,0,0,4,0,0,2,0,15],[44,70,0.6286,0.73659,0.32755,0.5713,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,0,1,0,17],[48,70,0.6857,0.79908,0.25472,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,4,0,0,2,0,17],[52,70,0.7429,0.85258,0.25652,0.71429,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,23],[56,70,0.8,0.88393,0.18708,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,20],[60,70,0.8571,0.62033,0.29393,0.28593,0.57143,0.89286,0.14,1.0,0,8,0,0,0,2,0,0,7,0,0,3,0,0,6,0,0,2,0,0,4,0,8],[64,70,0.9143,0.51337,0.29636,0.28571,0.42859,0.71429,0.0,1.0,2,6,0,2,0,1,0,0,10,0,0,4,0,0,5,0,0,4,0,0,0,0,6],[68,70,0.9714,0.62945,0.31106,0.28571,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,7,0,0,5,0,0,3,0,0,3,0,0,1,0,11],[70,70,1.0,0.49552,0.21124,0.28571,0.4286,0.71429,0.14286,0.71429,0,0,0,0,0,3,0,0,8,0,0,6,0,0,1,0,0,14,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.75,"x":1.0,"p":[[0,86,0.0,0.92411,0.1988,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[4,86,0.0465,0.83026,0.27787,0.57143,1.0,1.0,0.14,1.0,0,22,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,0,0,0,1,0,22],[8,86,0.093,0.79911,0.2942,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,19],[12,86,0.1395,0.90179,0.16917,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,23],[16,86,0.186,0.80356,0.31085,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,20],[20,86,0.2326,0.75,0.31135,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,3,0,0,6,0,0,0,0,0,2,0,17],[24,86,0.2791,0.82589,0.25438,0.57143,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,0,0,0,0,0,21],[28,86,0.3256,0.82588,0.2544,0.57143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,1,0,0,0,0,21],[32,86,0.3721,0.76339,0.35285,0.57143,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,1,0,20],[36,86,0.4186,0.81249,0.29546,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,21],[40,86,0.4651,0.87499,0.23353,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,0,0,0,1,0,24],[44,86,0.5116,0.90177,0.20027,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,24],[48,86,0.5581,0.88837,0.2165,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,24],[52,86,0.6047,0.90625,0.17717,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,24],[56,86,0.6512,0.92855,0.14729,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[60,86,0.6977,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,86,0.7442,0.94642,0.17038,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[68,86,0.7907,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[72,86,0.8372,0.92409,0.18208,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,27],[76,86,0.8837,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[80,86,0.9302,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[84,86,0.9767,0.91518,0.21683,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,25],[86,86,1.0,0.87944,0.22339,0.8214,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,22]]}]},{"i":"c18f207da8aa4dae","q":"There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that  $A$  gives  $B$  a coin and  $B$  gives  $A$  a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer  $n$ , there exists a person who will give away coins during the  $n$ th minute. What is the smallest number of coins that could be at the party?\n\n*Proposed by Ray Li*","t":[{"b":6,"e":0.0,"k":"falling","v":0.02232,"x":0.29911,"p":[[0,58,0.0,0.29911,0.16115,0.28571,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,0,0,0,17,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[4,58,0.069,0.24107,0.22428,0.0,0.21428,0.42857,0.0,0.71429,12,0,0,12,0,4,0,0,2,0,0,12,0,0,0,0,0,2,0,0,0,0,0],[8,58,0.1379,0.20089,0.18509,0.0,0.2857,0.28571,0.0,0.71429,12,0,0,12,0,3,0,0,11,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[12,58,0.2069,0.16518,0.17169,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,5,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,58,0.2759,0.23214,0.29179,0.0,0.14286,0.28571,0.0,1.0,14,3,0,14,0,3,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[20,58,0.3448,0.26786,0.29179,0.0,0.28571,0.42857,0.0,1.0,12,3,0,12,0,3,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,3],[24,58,0.4138,0.125,0.29397,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[28,58,0.4828,0.06696,0.14279,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.08036,0.13803,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.05804,0.12299,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.10714,0.19233,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,4,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[52,58,0.8966,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.05804,0.07017,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.06241,"x":0.32143,"p":[[0,82,0.0,0.25438,0.14174,0.24999,0.28571,0.28579,0.0,0.42857,6,0,1,6,0,2,0,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,82,0.0488,0.32143,0.17857,0.28571,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,1,0,0,11,0,0,13,0,0,0,0,0,2,0,0,0,0,0],[8,82,0.0976,0.32143,0.21724,0.24999,0.28571,0.42857,0.0,1.0,5,1,0,5,0,3,0,0,11,0,0,10,0,0,0,0,0,2,0,0,0,0,1],[12,82,0.1463,0.24553,0.22654,0.0,0.28571,0.42857,0.0,1.0,12,1,0,12,0,0,0,0,9,0,0,10,0,0,0,0,0,0,0,0,0,0,1],[16,82,0.1951,0.23214,0.17035,0.0,0.28571,0.42857,0.0,0.42857,9,0,0,9,0,4,0,0,9,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[20,82,0.2439,0.21875,0.18205,0.0,0.28571,0.42857,0.0,0.57143,11,0,0,11,0,3,0,0,9,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[24,82,0.2927,0.12938,0.15302,0.0,0.07,0.2857,0.0,0.42857,16,0,0,16,0,7,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,82,0.3415,0.21428,0.17128,0.0,0.28571,0.28571,0.0,0.57143,11,0,0,11,0,1,0,0,14,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[32,82,0.3902,0.14286,0.23145,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,1,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[36,82,0.439,0.17411,0.22513,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,1,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[40,82,0.4878,0.10268,0.16841,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[44,82,0.5366,0.20973,0.24221,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,5,0,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,1],[48,82,0.5854,0.21875,0.2172,0.0,0.28571,0.32143,0.0,0.71429,13,0,0,13,0,2,0,0,9,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[52,82,0.6341,0.22321,0.24206,0.0,0.2857,0.42857,0.0,1.0,15,1,0,15,0,0,0,0,6,0,0,9,0,0,1,0,0,0,0,0,0,0,1],[56,82,0.6829,0.19634,0.19152,0.0,0.14286,0.32143,0.0,0.71429,12,0,0,12,0,6,0,0,6,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[60,82,0.7317,0.19196,0.20079,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,3,0,0,8,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[64,82,0.7805,0.13393,0.17105,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,3,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[68,82,0.8293,0.11161,0.1665,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,5,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[72,82,0.878,0.06241,0.11252,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.11589,0.20022,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,9,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[80,82,0.9756,0.0625,0.12846,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.12054,0.20858,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,7,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"d5b5b97133264247","q":"A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged.\nFind the maximum possible contingents such that the above condition still holds?\n\n*Raymond Christopher Sitorus, Singapore*","t":[{"b":1,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,43,0.0,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,43,0.093,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,43,0.186,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,43,0.2791,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,43,0.3721,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,43,0.4651,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,43,0.5581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,43,0.6512,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,43,0.7442,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,43,0.8372,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,43,0.9302,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[43,43,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.79018,"x":1.0,"p":[[0,52,0.0,0.79018,0.39927,1.0,1.0,1.0,0.0,1.0,6,25,3,6,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[4,52,0.0769,0.94634,0.20783,1.0,1.0,1.0,0.14,1.0,0,30,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,52,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,52,0.2308,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,52,0.3077,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,52,0.3846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,52,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,52,0.5385,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,52,0.6154,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,52,0.6923,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,52,0.7692,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,52,0.8462,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[48,52,0.9231,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[52,52,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]}]},{"i":"7d765955cf6e0dc6","q":"ALB\n\na) Show that the product of all differences of possible couples of six given positive integers is divisible by 960 (original from Albania).\n\nb) Show that the product of all differences of possible couples of six given positive integers. is divisible by 34560 (modified by problem selecting committee).","t":[{"b":2,"e":0.71429,"k":"flat","v":0.94643,"x":0.99554,"p":[[0,35,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,35,0.1143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,35,0.2286,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,35,0.3429,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,35,0.4571,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,35,0.5714,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[24,35,0.6857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,35,0.8,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[32,35,0.9143,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[35,35,1.0,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28]]},{"b":5,"e":0.57143,"k":"falling","v":0.69195,"x":0.98214,"p":[[0,32,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,32,0.125,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[8,32,0.25,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,32,0.375,0.94197,0.20158,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[16,32,0.5,0.75893,0.25862,0.57143,0.78571,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0,0,1,0,15],[20,32,0.625,0.79909,0.24449,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,0,0,0,1,0,18],[24,32,0.75,0.85713,0.1856,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,5,0,18],[28,32,0.875,0.79911,0.20472,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,3,0,15],[32,32,1.0,0.69195,0.1927,0.57143,0.57143,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,9]]}]},{"i":"4cd970cf17d0e978","q":"There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school  can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.","t":[{"b":4,"e":0.0,"k":"volatile","v":0.0,"x":0.8482,"p":[[0,32,0.0,0.06697,0.17491,0.0,0.0,0.0,0.0,0.71429,27,0,13,27,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,32,0.125,0.8482,0.29654,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,23],[8,32,0.25,0.59375,0.39947,0.28571,0.64286,1.0,0.0,1.0,7,12,0,7,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,3,0,12],[12,32,0.375,0.32143,0.37457,0.0,0.14286,0.60714,0.0,1.0,15,4,0,15,0,2,0,0,3,0,0,2,0,0,2,0,0,2,0,0,2,0,4],[16,32,0.5,0.16062,0.30463,0.0,0.0,0.17857,0.0,1.0,22,3,0,22,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[20,32,0.625,0.12054,0.28146,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[24,32,0.75,0.11607,0.2299,0.0,0.0,0.07143,0.0,1.0,24,1,0,24,0,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[28,32,0.875,0.07143,0.19233,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,32,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.03125,"x":0.96429,"p":[[0,93,0.0,0.08482,0.21975,0.0,0.0,0.0,0.0,1.0,26,1,9,26,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[4,93,0.043,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[8,93,0.086,0.70089,0.38525,0.39286,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,0,2,0,18],[12,93,0.129,0.91071,0.24936,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[16,93,0.172,0.66071,0.3973,0.39286,0.92857,1.0,0.0,1.0,6,16,0,6,0,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,16],[20,93,0.2151,0.53571,0.41033,0.21427,0.42857,1.0,0.0,1.0,8,11,0,8,0,0,0,0,6,0,0,3,0,0,1,0,0,0,0,0,3,0,11],[24,93,0.2581,0.64732,0.38297,0.42857,0.78571,1.0,0.0,1.0,6,14,0,6,0,0,0,0,1,0,0,4,0,0,4,0,0,1,0,0,2,0,14],[28,93,0.3011,0.57589,0.43077,0.0,0.71429,1.0,0.0,1.0,9,13,0,9,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,13],[32,93,0.3441,0.57589,0.40007,0.24999,0.57143,1.0,0.0,1.0,6,13,0,6,0,2,0,0,4,0,0,2,0,0,3,0,0,2,0,0,0,0,13],[36,93,0.3871,0.51784,0.37923,0.10714,0.57121,1.0,0.0,1.0,8,9,0,8,0,1,0,0,1,0,0,5,0,0,5,0,0,3,0,0,0,0,9],[40,93,0.4301,0.37946,0.35284,0.0,0.28571,0.60714,0.0,1.0,11,4,0,11,0,1,0,0,6,0,0,1,0,0,5,0,0,3,0,0,1,0,4],[44,93,0.4731,0.5,0.39448,0.0,0.57143,1.0,0.0,1.0,10,9,0,10,0,0,0,0,2,0,0,1,0,0,8,0,0,2,0,0,0,0,9],[48,93,0.5161,0.3482,0.32129,0.0,0.28571,0.57143,0.0,1.0,11,3,0,11,0,1,0,0,5,0,0,4,0,0,7,0,0,0,0,0,1,0,3],[52,93,0.5591,0.35268,0.3589,0.0,0.2857,0.57143,0.0,1.0,11,5,0,11,0,4,0,0,3,0,0,4,0,0,4,0,0,0,0,0,1,0,5],[56,93,0.6022,0.3482,0.31121,0.0,0.28571,0.57143,0.0,1.0,9,3,0,9,0,3,0,0,6,0,0,4,0,0,6,0,0,0,0,0,1,0,3],[60,93,0.6452,0.29911,0.33381,0.0,0.21428,0.46429,0.0,1.0,13,4,0,13,0,3,0,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,4],[64,93,0.6882,0.32143,0.35714,0.0,0.28571,0.46429,0.0,1.0,14,5,0,14,0,0,0,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,5],[68,93,0.7312,0.38393,0.36846,0.0,0.28571,0.57143,0.0,1.0,9,7,0,9,0,4,0,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,7],[72,93,0.7742,0.35268,0.3589,0.0,0.28571,0.57143,0.0,1.0,11,5,0,11,0,3,0,0,6,0,0,1,0,0,5,0,0,0,0,0,1,0,5],[76,93,0.8172,0.44196,0.41551,0.0,0.28571,1.0,0.0,1.0,11,9,0,11,0,2,0,0,4,0,0,1,0,0,3,0,0,1,0,0,1,0,9],[80,93,0.8602,0.26339,0.32165,0.0,0.14286,0.42857,0.0,1.0,15,3,0,15,0,2,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,3],[84,93,0.9032,0.17411,0.2618,0.0,0.0,0.28571,0.0,1.0,17,2,0,17,0,4,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[88,93,0.9462,0.15625,0.23787,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,5,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[92,93,0.9892,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[93,93,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2773a65980d9f571","q":"The real numbers  $a_0, a_1, \\dots, a_{2013}$  and  $b_0, b_1, \\dots, b_{2013}$  satisfy  $a_{n} = \\frac{1}{63} \\sqrt{2n+2} + a_{n-1}$  and  $b_{n} = \\frac{1}{96} \\sqrt{2n+2} - b_{n-1}$  for every integer  $n = 1, 2, \\dots, 2013$ .  If  $a_0 = b_{2013}$  and  $b_0 = a_{2013}$ , compute \\[ \\sum_{k=1}^{2013} \\left( a_kb_{k-1} - a_{k-1}b_k \\right). \\]*Proposed by Evan Chen*","t":[{"b":6,"e":1.0,"k":"flat","v":0.83482,"x":1.0,"p":[[0,69,0.0,0.84821,0.31529,1.0,1.0,1.0,0.0,1.0,3,25,2,3,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,25],[4,69,0.058,0.90625,0.26392,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[8,69,0.1159,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[12,69,0.1739,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[16,69,0.2319,0.91071,0.22798,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,27],[20,69,0.2899,0.92411,0.20511,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[24,69,0.3478,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[28,69,0.4058,0.83927,0.30253,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,23],[32,69,0.4638,0.83482,0.30116,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,23],[36,69,0.5217,0.85268,0.32827,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[40,69,0.5797,0.94643,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[44,69,0.6377,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,69,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,69,0.7536,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,69,0.8116,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,69,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,69,0.9275,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,69,0.9855,0.94642,0.11156,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[69,69,1.0,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28]]},{"b":7,"e":0.42857,"k":"flat","v":0.74107,"x":0.98661,"p":[[0,180,0.0,0.83482,0.32754,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,25],[4,180,0.0222,0.89286,0.25754,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,26],[8,180,0.0444,0.90179,0.26592,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,26],[12,180,0.0667,0.96874,0.12238,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[16,180,0.0889,0.93304,0.21424,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[20,180,0.1111,0.88393,0.24598,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,25],[24,180,0.1333,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,180,0.1556,0.88839,0.25935,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,26],[32,180,0.1778,0.90625,0.25657,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[36,180,0.2,0.9375,0.2141,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[40,180,0.2222,0.875,0.25692,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,25],[44,180,0.2444,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[48,180,0.2667,0.91518,0.26693,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[52,180,0.2889,0.86161,0.28901,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,25],[56,180,0.3111,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[60,180,0.3333,0.90625,0.23313,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,26],[64,180,0.3556,0.87499,0.25444,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,24],[68,180,0.3778,0.85268,0.30615,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,25],[72,180,0.4,0.90179,0.22428,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,25],[76,180,0.4222,0.86161,0.26841,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,23],[80,180,0.4444,0.86607,0.26229,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,24],[84,180,0.4667,0.85714,0.24484,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,21],[88,180,0.4889,0.9241,0.20512,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[92,180,0.5111,0.84822,0.2878,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,24],[96,180,0.5333,0.85714,0.29451,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,24],[100,180,0.5556,0.91518,0.19186,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,25],[104,180,0.5778,0.79018,0.3813,0.92857,1.0,1.0,0.0,1.0,4,24,0,4,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[108,180,0.6,0.77679,0.35524,0.53571,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,22],[112,180,0.6222,0.82589,0.33832,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,23],[116,180,0.6444,0.79464,0.37105,0.82143,1.0,1.0,0.0,1.0,4,23,0,4,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[120,180,0.6667,0.83929,0.26183,0.71429,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,0,0,0,0,22],[124,180,0.6889,0.77678,0.37617,0.67857,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,22],[128,180,0.7111,0.74107,0.39518,0.42857,1.0,1.0,0.0,1.0,6,21,0,6,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,21],[132,180,0.7333,0.76786,0.33264,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,2,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,19],[136,180,0.7556,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[140,180,0.7778,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[144,180,0.8,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[148,180,0.8222,0.95536,0.1448,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[152,180,0.8444,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[156,180,0.8667,0.91518,0.17445,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,25],[160,180,0.8889,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[164,180,0.9111,0.91518,0.19186,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,26],[168,180,0.9333,0.89286,0.18558,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,22],[172,180,0.9556,0.85268,0.21275,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,1,0,20],[176,180,0.9778,0.89284,0.19235,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,0,0,24],[180,180,1.0,0.82588,0.17401,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,5,0,13]]}]},{"i":"c1f7c4817481768c","q":"A triangle  $ABC$  is given. A circle  $\\Gamma$ , passing through  $A$ , is tangent to side  $BC$  at point  $P$  and intersects sides  $AB$  and  $AC$  at  $M$  and  $N$  respectively. Prove that the smaller arcs  $MP$  and  $NP$  of  $\\Gamma$  are equal iff  $\\Gamma$  is tangent to the circumcircle of  $\\Delta ABC$  at  $A$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.18294,"x":0.39731,"p":[[0,63,0.0,0.35249,0.21736,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,11,0,0,3,0,0,11,0,0,3,0,0,2,0,0,0,0,1],[4,63,0.0635,0.39731,0.14165,0.2857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,4,0,0,7,0,0,14,0,0,6,0,0,1,0,0,0,0,0],[8,63,0.127,0.38377,0.21535,0.2857,0.42857,0.4642,0.0,0.857,2,0,0,2,0,5,0,0,8,0,0,9,0,0,4,0,0,2,0,0,2,0,0],[12,63,0.1905,0.35713,0.21128,0.14286,0.35714,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,6,0,0,10,0,0,3,0,0,2,0,0,0,0,1],[16,63,0.254,0.30793,0.17903,0.14286,0.28571,0.42858,0.0,0.57143,3,0,0,3,0,8,0,0,8,0,0,7,0,0,6,0,0,0,0,0,0,0,0],[20,63,0.3175,0.33473,0.2191,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,8,0,0,6,0,0,9,0,0,4,0,0,1,0,0,0,0,1],[24,63,0.381,0.35709,0.21733,0.14286,0.42857,0.42895,0.0,0.85714,2,0,0,2,0,9,0,0,4,0,0,10,0,0,3,0,0,3,0,0,1,0,0],[28,63,0.4444,0.36159,0.21421,0.25,0.42857,0.4286,0.0,1.0,3,1,0,3,0,5,0,0,7,0,0,10,0,0,5,0,0,1,0,0,0,0,1],[32,63,0.5079,0.29456,0.18196,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,10,0,0,6,0,0,9,0,0,3,0,0,1,0,0,0,0,0],[36,63,0.5714,0.33919,0.22805,0.14286,0.28571,0.42858,0.0,1.0,3,1,0,3,0,7,0,0,8,0,0,9,0,0,2,0,0,1,0,0,1,0,1],[40,63,0.6349,0.30357,0.1948,0.14286,0.2857,0.42857,0.0,0.85714,2,0,0,2,0,11,0,0,8,0,0,5,0,0,5,0,0,0,0,0,1,0,0],[44,63,0.6984,0.33036,0.18707,0.14286,0.35714,0.42857,0.0,0.85714,2,0,0,2,0,8,0,0,6,0,0,13,0,0,1,0,0,1,0,0,1,0,0],[48,63,0.7619,0.21847,0.1637,0.14214,0.14286,0.32143,0.0,0.57143,5,0,0,5,0,15,0,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[52,63,0.8254,0.26329,0.16017,0.14286,0.2857,0.32164,0.0,0.57143,2,0,0,2,0,13,0,0,9,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[56,63,0.8889,0.29461,0.171,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,12,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[60,63,0.9524,0.18294,0.14826,0.14214,0.14286,0.2857,0.0,0.571,7,0,0,7,0,15,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[63,63,1.0,0.21878,0.17494,0.10714,0.14288,0.32143,0.0,0.57143,8,0,0,8,0,9,0,0,7,0,0,6,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.2679,"x":0.45536,"p":[[0,65,0.0,0.39286,0.21724,0.14286,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,9,0,0,4,0,0,13,0,0,0,0,0,5,0,0,0,0,1],[4,65,0.0615,0.42405,0.23,0.24999,0.42857,0.57141,0.0,1.0,1,1,0,1,0,7,0,0,4,0,0,7,0,0,9,0,0,2,0,0,1,0,1],[8,65,0.1231,0.4196,0.23126,0.24999,0.42857,0.571,0.14286,1.0,0,1,0,0,0,8,0,0,5,0,0,9,0,0,5,0,0,2,0,0,2,0,1],[12,65,0.1846,0.4107,0.24418,0.14286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,6,0,0,6,0,0,4,0,0,4,0,0,3,0,0],[16,65,0.2462,0.36573,0.1853,0.28571,0.42857,0.4642,0.0,0.71,3,0,0,3,0,4,0,0,6,0,0,11,0,0,7,0,0,1,0,0,0,0,0],[20,65,0.3077,0.4464,0.2335,0.2857,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,5,0,0,10,0,0,5,0,0,4,0,0,4,0,0,4,0,0],[24,65,0.3692,0.37043,0.18533,0.2857,0.42857,0.4286,0.14,1.0,0,1,0,0,0,7,0,0,8,0,0,12,0,0,3,0,0,1,0,0,0,0,1],[28,65,0.4308,0.38831,0.1865,0.2857,0.42857,0.4286,0.14,1.0,0,1,0,0,0,6,0,0,7,0,0,14,0,0,2,0,0,2,0,0,0,0,1],[32,65,0.4923,0.41503,0.2325,0.2857,0.42857,0.571,0.0,1.0,1,2,0,1,0,6,0,0,6,0,0,9,0,0,6,0,0,2,0,0,0,0,2],[36,65,0.5538,0.39728,0.17758,0.2857,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,6,0,0,7,0,0,10,0,0,7,0,0,1,0,0,1,0,0],[40,65,0.6154,0.45536,0.16915,0.39293,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,4,0,0,10,0,0,10,0,0,4,0,0,0,0,0],[44,65,0.6769,0.36598,0.21118,0.14296,0.42857,0.42858,0.0,1.0,2,1,0,2,0,7,0,0,3,0,0,16,0,0,2,0,0,0,0,0,1,0,1],[48,65,0.7385,0.42843,0.2141,0.2857,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,5,0,0,8,0,0,10,0,0,3,0,0,4,0,0,1,0,1],[52,65,0.8,0.42404,0.14493,0.28571,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,11,0,0,10,0,0,1,0,0,0,0,0],[56,65,0.8615,0.32567,0.15286,0.24999,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,10,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[60,65,0.9231,0.32589,0.17579,0.1429,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,10,0,0,10,0,0,8,0,0,2,0,0,1,0,0,1,0,0],[64,65,0.9846,0.27672,0.15133,0.14286,0.2143,0.42857,0.14,0.57143,0,0,0,0,0,16,0,0,5,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[65,65,1.0,0.2679,0.15051,0.14286,0.2143,0.42857,0.14286,0.71429,0,0,0,0,0,16,0,0,7,0,0,7,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"360aaf534646e150","q":"A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities  $$ P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) . $$  Prove that there exists a polynomial $F(t)$ in one variable such that  $$ P(x, y, z)=F\\left(x^{2}+y^{2}+z^{2}-x y z\\right) . $$","t":[{"b":0,"e":0.14286,"k":"flat","v":0.16518,"x":0.32125,"p":[[0,27,0.0,0.29017,0.14498,0.14286,0.28571,0.42857,0.0,0.571,1,0,0,1,0,12,0,0,5,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[4,27,0.1481,0.30804,0.16793,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,3,0,0,13,0,0,0,0,0,2,0,0,0,0,0],[8,27,0.2963,0.32125,0.16771,0.14286,0.35714,0.42857,0.14,0.71429,0,0,0,0,0,13,0,0,3,0,0,12,0,0,3,0,0,1,0,0,0,0,0],[12,27,0.4444,0.20973,0.14283,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,25,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[16,27,0.5926,0.27213,0.14893,0.14286,0.14286,0.42857,0.14,0.57143,0,0,0,0,0,17,0,0,3,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[20,27,0.7407,0.20983,0.12869,0.14286,0.14286,0.1429,0.14286,0.57143,0,0,0,0,0,25,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[24,27,0.8889,0.16518,0.0724,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.1429,"k":"flat","v":0.19643,"x":0.37938,"p":[[0,55,0.0,0.32134,0.14297,0.14286,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,7,0,0,4,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.37938,0.13666,0.42857,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,7,0,0,0,0,0,23,0,0,1,0,0,1,0,0,0,0,0],[8,55,0.1455,0.3392,0.17045,0.14286,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,11,0,0,4,0,0,13,0,0,2,0,0,2,0,0,0,0,0],[12,55,0.2182,0.33929,0.15465,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,2,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[16,55,0.2909,0.30357,0.14617,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,1,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[20,55,0.3636,0.33481,0.17715,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,13,0,0,1,0,0,14,0,0,2,0,0,2,0,0,0,0,0],[24,55,0.4364,0.29902,0.15312,0.14286,0.35714,0.42857,0.14,0.57143,0,0,0,0,0,15,0,0,1,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[28,55,0.5091,0.35258,0.16755,0.14286,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,10,0,0,3,0,0,15,0,0,2,0,0,2,0,0,0,0,0],[32,55,0.5818,0.36607,0.16342,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,0,0,0,17,0,0,4,0,0,1,0,0,0,0,0],[36,55,0.6545,0.33929,0.15047,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,3,0,0,17,0,0,1,0,0,1,0,0,0,0,0],[40,55,0.7273,0.34821,0.13803,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,2,0,0,19,0,0,2,0,0,0,0,0,0,0,0],[44,55,0.8,0.35714,0.18898,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,12,0,0,1,0,0,13,0,0,3,0,0,3,0,0,0,0,0],[48,55,0.8727,0.30357,0.13716,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,13,0,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.25438,0.13716,0.14286,0.14286,0.42857,0.14,0.42857,0,0,0,0,0,19,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.19643,0.10565,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,25,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8ebd6b8db4f96378","q":"A quadrilateral  $ABCD$  without parallel sides is circumscribed around a circle with centre  $O$ . Prove that  $O$  is a point of intersection of middle lines of quadrilateral  $ABCD$  (i.e. barycentre of points  $A,\\,B,\\,C,\\,D$ ) iff  $OA\\cdot OC=OB\\cdot OD$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.16505,"x":0.39284,"p":[[0,36,0.0,0.18741,0.13573,0.14214,0.14288,0.28571,0.0,0.57143,7,0,0,7,0,11,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,36,0.1111,0.36158,0.27193,0.14286,0.28571,0.57143,0.0,1.0,2,2,0,2,0,12,0,0,5,0,0,2,0,0,7,0,0,1,0,0,1,0,2],[8,36,0.2222,0.35259,0.29669,0.14286,0.28571,0.57143,0.0,1.0,6,2,0,6,0,7,0,0,6,0,0,4,0,0,2,0,0,4,0,0,1,0,2],[12,36,0.3333,0.39284,0.25999,0.25,0.35714,0.57111,0.0,1.0,3,2,0,3,0,5,0,0,8,0,0,7,0,0,4,0,0,2,0,0,1,0,2],[16,36,0.4444,0.30356,0.29395,0.14286,0.1429,0.4642,0.0,1.0,7,3,0,7,0,10,0,0,4,0,0,3,0,0,5,0,0,0,0,0,0,0,3],[20,36,0.5556,0.29454,0.25493,0.14214,0.2857,0.4286,0.0,1.0,7,1,0,7,0,8,0,0,5,0,0,5,0,0,5,0,0,0,0,0,1,0,1],[24,36,0.6667,0.37499,0.21942,0.2857,0.28571,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,13,0,0,6,0,0,4,0,0,0,0,0,2,0,1],[28,36,0.7778,0.28571,0.31134,0.0,0.14286,0.57143,0.0,1.0,11,2,0,11,0,7,0,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,2],[32,36,0.8889,0.16505,0.21453,0.0,0.14286,0.1786,0.0,0.857,14,0,0,14,0,10,0,0,3,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[36,36,1.0,0.19187,0.23854,0.0,0.14286,0.2857,0.0,0.85714,13,0,0,13,0,9,0,0,4,0,0,2,0,0,2,0,0,0,0,0,2,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.19643,"x":0.45087,"p":[[0,75,0.0,0.19643,0.09279,0.14286,0.14288,0.28571,0.0,0.28571,3,0,0,3,0,14,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.33481,0.28482,0.14286,0.2857,0.4642,0.0,1.0,5,2,0,5,0,10,0,0,4,0,0,5,0,0,3,0,0,2,0,0,1,0,2],[8,75,0.1067,0.34374,0.28314,0.14286,0.28571,0.42857,0.0,1.0,4,3,0,4,0,9,0,0,6,0,0,7,0,0,2,0,0,0,0,0,1,0,3],[12,75,0.16,0.45087,0.3136,0.14286,0.42859,0.60714,0.0,1.0,2,5,0,2,0,9,0,0,3,0,0,4,0,0,6,0,0,3,0,0,0,0,5],[16,75,0.2133,0.37944,0.23583,0.25,0.42857,0.4642,0.0,1.0,3,1,0,3,0,5,0,0,7,0,0,9,0,0,4,0,0,2,0,0,1,0,1],[20,75,0.2667,0.32589,0.26056,0.14286,0.28571,0.4286,0.0,1.0,4,2,0,4,0,11,0,0,3,0,0,7,0,0,4,0,0,1,0,0,0,0,2],[24,75,0.32,0.41516,0.33188,0.14286,0.28571,0.57143,0.0,1.0,5,5,0,5,0,6,0,0,6,0,0,3,0,0,5,0,0,1,0,0,1,0,5],[28,75,0.3733,0.37041,0.27171,0.14286,0.28571,0.571,0.0,1.0,3,3,0,3,0,8,0,0,7,0,0,4,0,0,7,0,0,0,0,0,0,0,3],[32,75,0.4267,0.32587,0.27945,0.14286,0.2857,0.42858,0.0,1.0,5,2,0,5,0,9,0,0,7,0,0,4,0,0,2,0,0,2,0,0,1,0,2],[36,75,0.48,0.32579,0.23489,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,7,0,0,8,0,0,8,0,0,2,0,0,1,0,0,1,0,1],[40,75,0.5333,0.31249,0.24855,0.14286,0.28571,0.46418,0.0,1.0,6,1,0,6,0,7,0,0,7,0,0,4,0,0,5,0,0,2,0,0,0,0,1],[44,75,0.5867,0.35256,0.189,0.24999,0.28571,0.46418,0.0,0.71429,1,0,0,1,0,7,0,0,11,0,0,5,0,0,5,0,0,3,0,0,0,0,0],[48,75,0.64,0.31694,0.24672,0.14286,0.2857,0.42858,0.0,1.0,3,1,0,3,0,11,0,0,8,0,0,3,0,0,3,0,0,2,0,0,1,0,1],[52,75,0.6933,0.35712,0.202,0.2857,0.28571,0.4286,0.0,1.0,2,1,0,2,0,5,0,0,10,0,0,9,0,0,4,0,0,1,0,0,0,0,1],[56,75,0.7467,0.40625,0.20858,0.28571,0.42857,0.46431,0.0,1.0,1,1,0,1,0,3,0,0,11,0,0,9,0,0,4,0,0,2,0,0,1,0,1],[60,75,0.8,0.41957,0.1887,0.28571,0.42857,0.57025,0.0,0.85714,1,0,0,1,0,3,0,0,7,0,0,12,0,0,6,0,0,1,0,0,2,0,0],[64,75,0.8533,0.29911,0.11495,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,6,0,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.28572,0.13363,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[72,75,0.96,0.39732,0.17029,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,10,0,0,15,0,0,1,0,0,2,0,0,0,0,1],[75,75,1.0,0.33929,0.13717,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,3,0,0,10,0,0,15,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"da154871d351ce38","q":"Numbers  $1, 2,\\dots , 101$  are written in the cells of a  $101\\times 101$  square board so that each number is repeated  $101$  times. Prove that there exists either a column or a row containing at least  $11$  different numbers.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.66504,"x":0.94196,"p":[[0,20,0.0,0.875,0.18123,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,0,0,21],[4,20,0.2,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[8,20,0.4,0.81249,0.21854,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,1,0,17],[12,20,0.6,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[16,20,0.8,0.94195,0.13771,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,27],[20,20,1.0,0.66504,0.08451,0.57143,0.71429,0.71429,0.42857,0.8571,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,1,0,0]]},{"b":4,"e":1.0,"k":"flat","v":0.86159,"x":1.0,"p":[[0,28,0.0,0.86159,0.19063,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,0,0,20],[4,28,0.1429,0.87946,0.17896,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,1,0,21],[8,28,0.2857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,28,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,28,0.5714,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,28,0.7143,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"0a0b9b323fb8d887","q":"An  $\\textrm{alien}$  script has  $n$  letters  $b_1,b_2,\\dots,b_n$ . For some  $k<n/2$  assume that all words formed by any of the  $k$  letters (written left to right) are meaningful. These words are called  $k$ -words. Such a  $k$ -word is considered  $\\textbf{sacred}$  if:\n\ni. no letter appears twice and,\nii. if a letter  $b_i$  appears in the word then the letters  $b_{i-1}$  and  $b_{i+1}$  do not appear. (Here  $b_{n+1} = b_1$  and  $b_0 = b_n$ ).\n\nFor example, if  $n = 7$  and  $k = 3$  then  $b_1b_3b_6, b_3b_1b_6, b_2b_4b_6$  are sacred  $3$ -words. On the other hand  $b_1b_7b_4, b_2b_2b_6$  are not sacred. \nWhat is the total number of sacred  $k$ -words?\nUse your formula to find the answer for  $n = 10$  and  $k = 4$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.91964,"x":0.98661,"p":[[0,24,0.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,24,0.1667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,24,0.3333,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[12,24,0.5,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[16,24,0.6667,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[20,24,0.8333,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[24,24,1.0,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15]]},{"b":6,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,15,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,15,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4f137944606e5784","q":"An *arithmetic sequence* is an infinite sequence of the form  $a_n=a_0+n\\cdot d$  with  $d\\neq 0$ .\n\nA *geometric sequence* is an infinite sequence of the form  $b_n=b_0 \\cdot q^n$  where  $q\\neq 1,0,-1$ .\n\n[list=a]\n[*] Does every arithmetic sequence of **integers** have an infinite subsequence which is geometric?\n[*] Does every arithmetic sequence of **real numbers** have an infinite subsequence which is geometric?\n[/list]","t":[{"b":6,"e":0.57143,"k":"falling","v":0.63393,"x":0.89732,"p":[[0,27,0.0,0.89732,0.16065,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,20],[4,27,0.1481,0.84372,0.17987,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,7,0,15],[8,27,0.2963,0.81696,0.21793,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,2,0,0,6,0,15],[12,27,0.4444,0.85714,0.18558,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,6,0,17],[16,27,0.5926,0.72767,0.21238,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,5,0,0,2,0,10],[20,27,0.7407,0.75446,0.21199,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,4,0,0,4,0,11],[24,27,0.8889,0.63393,0.17474,0.57143,0.57143,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,15,0,0,3,0,0,6,0,2],[27,27,1.0,0.65177,0.16343,0.57142,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,8,0,0,5,0,2]]},{"b":7,"e":0.85714,"k":"flat","v":0.74103,"x":0.90177,"p":[[0,11,0.0,0.90177,0.15338,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,20],[4,11,0.3636,0.79459,0.21713,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,0,6,0,13],[8,11,0.7273,0.74103,0.19705,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,3,0,0,7,0,8],[11,11,1.0,0.77231,0.20474,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,3,0,12]]}]},{"i":"a72f25da9aefd25a","q":"4. (CZS 2) Assume that the set of all positive integers is decomposed into $r$ (disjoint) subsets $A_{1} \\cup A_{2} \\cup \\cdots A_{r}=\\mathbb{N}$. Prove that one of them, say $A_{i}$, has the following property: There exists a positive $m$ such that for any $k$ one can find numbers $a_{1}, a_{2}, \\ldots, a_{k}$ in $A_{i}$ with $0<a_{j+1}-a_{j} \\leq m$ $(1 \\leq j \\leq k-1)$.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.15178,"x":0.31679,"p":[[0,44,0.0,0.19643,0.17405,0.0,0.2857,0.28571,0.0,0.85714,10,0,8,10,0,4,0,0,17,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,44,0.0909,0.22321,0.26949,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,5,0,0,6,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[8,44,0.1818,0.18304,0.20896,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,6,0,0,5,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[12,44,0.2727,0.19192,0.23578,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,0,0,0,5,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[16,44,0.3636,0.24106,0.26106,0.0,0.14288,0.32143,0.0,0.71429,13,0,0,13,0,4,0,0,7,0,0,1,0,0,2,0,0,5,0,0,0,0,0],[20,44,0.4545,0.22101,0.26809,0.0,0.0,0.42857,0.0,0.85714,17,0,0,17,0,1,0,0,3,0,0,6,0,0,1,1,0,2,0,0,1,0,0],[24,44,0.5455,0.22321,0.21998,0.0,0.14286,0.42857,0.0,0.85714,11,0,0,11,0,6,0,0,6,0,0,7,0,0,0,0,0,1,0,0,1,0,0],[28,44,0.6364,0.24107,0.30813,0.0,0.07143,0.42857,0.0,0.85714,16,0,0,16,0,4,0,0,2,0,0,3,0,0,2,0,0,1,0,0,4,0,0],[32,44,0.7273,0.183,0.25307,0.0,0.0,0.32143,0.0,0.85714,17,0,0,17,0,6,0,0,1,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[36,44,0.8182,0.15178,0.23401,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[40,44,0.9091,0.26325,0.25779,0.0,0.14286,0.42858,0.0,1.0,9,1,0,9,0,8,0,0,5,0,0,4,0,0,4,0,0,0,0,0,1,0,1],[44,44,1.0,0.31679,0.31901,0.0,0.2857,0.60714,0.0,1.0,10,2,0,10,0,5,0,0,7,0,0,1,0,0,1,0,0,5,0,0,1,0,2]]},{"b":6,"e":0.14286,"k":"flat","v":0.14731,"x":0.30804,"p":[[0,42,0.0,0.16963,0.1491,0.0,0.21428,0.28571,0.0,0.571,12,0,10,12,0,4,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,42,0.0952,0.16518,0.21756,0.0,0.07143,0.2857,0.0,0.71429,16,0,0,16,0,6,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[8,42,0.1905,0.14731,0.23276,0.0,0.0,0.2857,0.0,0.85714,20,0,0,20,0,3,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[12,42,0.2857,0.30804,0.31965,0.0,0.28571,0.42858,0.0,1.0,12,1,0,12,0,2,0,0,7,0,0,4,0,0,0,0,0,2,0,0,4,0,1],[16,42,0.381,0.20088,0.24962,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,6,0,0,6,0,0,2,0,0,1,0,0,1,0,0,2,0,0],[20,42,0.4762,0.2545,0.24676,0.0,0.14286,0.42895,0.0,0.71429,9,0,0,9,0,10,0,0,3,0,0,3,0,0,3,0,0,4,0,0,0,0,0],[24,42,0.5714,0.24105,0.29108,0.0,0.14286,0.32143,0.0,0.85714,15,0,0,15,0,2,0,0,7,0,0,2,0,0,2,0,0,0,0,0,4,0,0],[28,42,0.6667,0.17411,0.24932,0.0,0.0,0.32142,0.0,0.71429,19,0,0,19,0,3,0,0,2,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[32,42,0.7619,0.14732,0.22156,0.0,0.0,0.17857,0.0,0.85714,18,0,0,18,0,6,0,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[36,42,0.8571,0.21427,0.28792,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,2,0,0,6,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[40,42,0.9524,0.2455,0.24799,0.10714,0.14286,0.32144,0.0,0.85714,8,0,0,8,0,12,0,0,4,0,0,2,0,0,3,0,0,1,0,0,2,0,0],[42,42,1.0,0.27676,0.21106,0.14286,0.28571,0.32143,0.0,0.85714,4,0,0,4,0,10,0,0,10,0,0,4,0,0,2,0,0,0,0,0,2,0,0]]}]},{"i":"e9ef72acb57cbfba","q":"An \\textit{island} is a contiguous set of at least two equal digits. Let  $b(n)$  be the number of islands in the binary representation of  $n$ . For example,  $2020_{10} = 11111100100_2$ , so  $b(2020) = 3$ . Compute\n\\[b(1) + b(2) + \\cdots + b(2^{2020}).\\]","t":[{"b":0,"e":1.0,"k":"rising","v":0.83929,"x":1.0,"p":[[0,42,0.0,0.83929,0.24419,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,4,0,0,5,0,18],[4,42,0.0952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,42,0.1905,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[12,42,0.2857,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,42,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,42,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,42,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.70982,"x":1.0,"p":[[0,71,0.0,0.875,0.17768,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,8,0,17],[4,71,0.0563,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[8,71,0.1127,0.92857,0.19233,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,28],[12,71,0.169,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[16,71,0.2254,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[20,71,0.2817,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,71,0.338,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[28,71,0.3944,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,71,0.4507,0.70982,0.32436,0.39286,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,8,0,0,6,0,0,0,0,0,0,0,0,1,0,17],[36,71,0.507,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,71,0.5634,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,71,0.6197,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,71,0.6761,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,71,0.7324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,71,0.7887,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,71,0.8451,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,71,0.9014,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,71,0.9577,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[71,71,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"f54d49973c833d6c","q":"16. (NET 1) ${ }^{\\text {IMO6 }}$ Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides are a permutation of the numbers $1^{2}, 2^{2}, \\ldots, 1989^{2}, 1990^{2}$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.1384,"x":0.16072,"p":[[0,32,0.0,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.16072,0.06916,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.14723,0.05631,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.1383,"x":0.17857,"p":[[0,49,0.0,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.15179,0.04972,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.16955,0.07527,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.17402,0.08556,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.17411,0.08552,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.16072,0.05923,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.17857,0.08748,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,27,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"89516a8a7a235e45","q":"6. (POL) We are given $n \\geq 3$ points in the plane. Let $d$ be the maximal distance between two of the given points. Prove that the number of pairs of points whose distance is equal to $d$ is less than or equal to $n$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.21875,"p":[[0,13,0.0,0.21875,0.19227,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,13,0,0,9,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[4,13,0.3077,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.16945,"p":[[0,25,0.0,0.16945,0.13092,0.14286,0.14286,0.14286,0.0,0.571,5,0,0,5,0,21,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.14277,0.18898,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,25,0.32,0.08482,0.17807,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,25,0.48,0.11607,0.18707,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,0,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,25,0.64,0.04911,0.16213,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,25,0.8,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cf707faef8f19336","q":"7. C1 (BRA) ${ }^{\\mathrm{IMO} 1}$ Let $A$ be a 101-element subset of the set $S=\\{1,2, \\ldots$, $1000000\\}$. Prove that there exist numbers $t_{1}, t_{2}, \\ldots, t_{100}$ in $S$ such that the sets  $$ A_{j}=\\left\\{x+t_{j} \\mid x \\in A\\right\\}, \\quad j=1,2, \\ldots, 100 $$  are pairwise disjoint.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.41518,"x":0.85713,"p":[[0,38,0.0,0.50892,0.28557,0.28571,0.49979,0.71429,0.0,1.0,1,5,1,1,0,3,0,0,9,0,0,3,0,0,7,0,0,3,0,0,1,0,5],[4,38,0.1053,0.82143,0.22016,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,15,0,10],[8,38,0.2105,0.6607,0.2896,0.42857,0.85707,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,4,0,0,5,0,0,4,0,0,0,0,0,11,0,6],[12,38,0.3158,0.70982,0.32632,0.42857,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,4,0,0,5,0,0,1,0,0,2,0,0,4,0,14],[16,38,0.4211,0.67857,0.30093,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,3,0,0,2,0,0,3,0,0,5,0,0,6,0,9],[20,38,0.5263,0.85713,0.22589,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,0,9,0,17],[24,38,0.6316,0.68302,0.30038,0.42857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,2,0,0,3,0,0,5,0,0,1,0,0,8,0,9],[28,38,0.7368,0.61158,0.32584,0.39286,0.57143,1.0,0.0,1.0,1,10,1,1,0,4,0,0,3,0,0,5,0,0,5,0,0,2,0,0,2,0,10],[32,38,0.8421,0.57588,0.28901,0.28571,0.49979,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,6,0,0,7,0,0,3,0,0,3,0,0,4,0,6],[36,38,0.9474,0.49106,0.1954,0.42857,0.4286,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,1,0,0,13,0,0,9,0,0,1,0,0,4,0,0],[38,38,1.0,0.41518,0.24578,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,8,0,0,8,0,0,3,0,0,2,0,0,3,0,1]]},{"b":4,"e":0.85714,"k":"rising","v":0.48661,"x":0.86607,"p":[[0,44,0.0,0.48661,0.30901,0.28571,0.28571,0.75,0.0,1.0,1,5,1,1,0,4,0,0,12,0,0,2,0,0,3,0,0,2,0,0,3,0,5],[4,44,0.0909,0.83928,0.15047,0.85714,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,23,0,5],[8,44,0.1818,0.75892,0.19377,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,23,0,0],[12,44,0.2727,0.75893,0.21852,0.82143,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,0,22,0,2],[16,44,0.3636,0.75892,0.23538,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,6,0,0,16,0,5],[20,44,0.4545,0.61607,0.27765,0.39286,0.71429,0.85714,0.0,1.0,1,1,0,1,0,2,0,0,5,0,0,3,0,0,4,0,0,2,0,0,14,0,1],[24,44,0.5455,0.78571,0.14286,0.71429,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,21,0,1],[28,44,0.6364,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[32,44,0.7273,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[36,44,0.8182,0.85714,0.0,0.85714,0.85714,0.85714,0.85714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0],[40,44,0.9091,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[44,44,1.0,0.85714,0.07143,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,27,0,3]]}]},{"i":"9b117929ef7472fd","q":"5. (CZS 3) Let $M$ be an interior point of the tetrahedron $A B C D$. Prove that  $$ \\begin{aligned} & \\overrightarrow{M A} \\operatorname{vol}(M B C D)+\\overrightarrow{M B} \\operatorname{vol}(M A C D) \\\\ & \\quad+\\overrightarrow{M C} \\operatorname{vol}(M A B D)+\\overrightarrow{M D} \\operatorname{vol}(M A B C)=0 \\end{aligned} $$  ( $\\operatorname{vol}(P Q R S)$ denotes the volume of the tetrahedron $P Q R S)$.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.75891,"x":0.96429,"p":[[0,14,0.0,0.88839,0.21349,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,23],[4,14,0.2857,0.75891,0.33012,0.57132,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,4,0,0,0,0,0,4,0,0,1,0,0,1,0,19],[8,14,0.5714,0.86161,0.2977,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,24],[12,14,0.8571,0.93304,0.16746,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[14,14,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.71874,"x":0.94196,"p":[[0,18,0.0,0.91964,0.20183,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,25],[4,18,0.2222,0.875,0.28291,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,24],[8,18,0.4444,0.71874,0.36332,0.5354,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,2,0,0,1,0,0,5,0,0,0,0,0,1,0,18],[12,18,0.6667,0.89286,0.20516,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,3,0,23],[16,18,0.8889,0.91518,0.18851,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,25],[18,18,1.0,0.94196,0.14664,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26]]}]},{"i":"f751dab0f4987946","q":"There are eight different symbols designed on  $n\\geq 2$  different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any  $k$  symbols  $(1\\leq k\\leq 7)$  the number of shirts containing at least one of the  $k$  symbols is even. Determine the value of  $n$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,31,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,31,0.129,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,31,0.2581,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,31,0.3871,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,31,0.5161,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,31,0.6452,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,31,0.7742,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.96874,"x":1.0,"p":[[0,31,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,31,0.129,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,31,0.2581,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[12,31,0.3871,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,31,0.5161,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,31,0.6452,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,31,0.7742,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,31,0.9032,0.98661,0.07457,1.0,1.0,1.0,0.5714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e598a9b6e4c72a60","q":"19. (USS 2) Let $\\alpha, \\beta, \\gamma$ be positive real numbers such that $\\alpha+\\beta+\\gamma<\\pi$, $\\alpha+\\beta>\\gamma, \\beta+\\gamma>\\alpha, \\gamma+\\alpha>\\beta$. Prove that with the segments of lengths $\\sin \\alpha, \\sin \\beta, \\sin \\gamma$ we can construct a triangle and that its area is not greater than  $$ \\frac{1}{8}(\\sin 2 \\alpha+\\sin 2 \\beta+\\sin 2 \\gamma) $$","t":[{"b":3,"e":0.42857,"k":"rising","v":0.13839,"x":0.35714,"p":[[0,44,0.0,0.16071,0.09781,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,2,24,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.24554,0.2321,0.14286,0.14286,0.17857,0.0,1.0,1,1,0,1,0,23,0,0,2,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[8,44,0.1818,0.29464,0.3071,0.14286,0.14286,0.17857,0.0,1.0,1,4,0,1,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,4],[12,44,0.2727,0.20535,0.20182,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[16,44,0.3636,0.20089,0.15093,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,27,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[20,44,0.4545,0.19643,0.19805,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[24,44,0.5455,0.25,0.24221,0.14286,0.14286,0.14287,0.14286,1.0,0,2,0,0,0,25,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[28,44,0.6364,0.22768,0.21387,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,23,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,1],[32,44,0.7273,0.17857,0.13363,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[36,44,0.8182,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.33036,0.25614,0.14286,0.21428,0.42857,0.0,1.0,1,3,0,1,0,15,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,3],[44,44,1.0,0.35714,0.25,0.14286,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,12,0,0,6,0,0,10,0,0,0,0,0,1,0,0,0,0,3]]},{"b":7,"e":0.28571,"k":"flat","v":0.13393,"x":0.31696,"p":[[0,54,0.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.24107,0.18707,0.14286,0.14286,0.42857,0.0,1.0,1,1,0,1,0,21,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[8,54,0.1481,0.20981,0.18203,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[12,54,0.2222,0.24554,0.22084,0.14286,0.14286,0.17857,0.14286,1.0,0,2,0,0,0,24,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[16,54,0.2963,0.24554,0.2402,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[20,54,0.3704,0.31696,0.28512,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,19,0,0,1,0,0,6,0,0,0,0,0,1,0,0,1,0,3],[24,54,0.4444,0.29464,0.25237,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,21,0,0,1,0,0,6,0,0,0,0,0,0,0,0,3,0,1],[28,54,0.5185,0.23661,0.25407,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,25,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[32,54,0.5926,0.26786,0.24936,0.14286,0.14286,0.21432,0.14286,1.0,0,2,0,0,0,24,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[36,54,0.6667,0.25,0.22304,0.14286,0.14286,0.21429,0.14286,1.0,0,2,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[40,54,0.7407,0.21875,0.17491,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,25,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[44,54,0.8148,0.21875,0.22011,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,26,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[48,54,0.8889,0.22312,0.18881,0.14286,0.14286,0.21429,0.0,1.0,2,1,0,2,0,22,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[52,54,0.963,0.21429,0.11845,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,23,0,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.19643,0.10565,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,25,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c6c1a191120165c1","q":"5. (FRG 1) ${ }^{\\mathrm{IMO1}}$ The set $S=\\{2,5,13\\}$ has the property that for every $a, b \\in S, a \\neq b$, the number $a b-1$ is a perfect square. Show that for every positive integer $d$ not in $S$, the set $S \\cup\\{d\\}$ does not have the above property.","t":[{"b":4,"e":0.42857,"k":"flat","v":0.35714,"x":0.54911,"p":[[0,58,0.0,0.39286,0.22868,0.14286,0.42857,0.57143,0.0,0.85714,2,0,2,2,0,7,0,0,6,0,0,5,0,0,7,0,0,4,0,0,1,0,0],[4,58,0.069,0.49106,0.26711,0.28571,0.42857,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,5,0,0,10,0,0,2,0,0,5,0,0,2,0,3],[8,58,0.1379,0.35714,0.19233,0.28571,0.35714,0.42857,0.0,0.85714,2,0,0,2,0,5,0,0,9,0,0,11,0,0,2,0,0,2,0,0,1,0,0],[12,58,0.2069,0.41964,0.21706,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,9,0,0,7,0,0,7,0,0,2,0,0,1,0,1],[16,58,0.2759,0.48214,0.27141,0.25,0.42857,0.71429,0.0,1.0,1,2,0,1,0,7,0,0,2,0,0,8,0,0,3,0,0,7,0,0,2,0,2],[20,58,0.3448,0.45089,0.28371,0.25001,0.42857,0.71429,0.0,1.0,1,3,0,1,0,7,0,0,7,0,0,5,0,0,2,0,0,6,0,0,1,0,3],[24,58,0.4138,0.40625,0.19597,0.28571,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,11,0,0,6,0,0,5,0,0,4,0,0,1,0,0],[28,58,0.4828,0.39737,0.24416,0.14286,0.42857,0.46536,0.0,0.85714,2,0,0,2,0,7,0,0,5,0,0,10,0,0,1,0,0,4,0,0,3,0,0],[32,58,0.5517,0.38839,0.22654,0.28571,0.42857,0.42857,0.0,1.0,3,1,0,3,0,4,0,0,5,0,0,14,0,0,2,0,0,2,0,0,1,0,1],[36,58,0.6207,0.43303,0.26362,0.2857,0.42857,0.46429,0.14286,1.0,0,4,0,0,0,7,0,0,7,0,0,10,0,0,2,0,0,2,0,0,0,0,4],[40,58,0.6897,0.375,0.21354,0.25,0.42857,0.42857,0.0,1.0,1,1,1,1,0,7,0,0,6,0,0,14,0,0,0,0,0,2,0,0,1,0,1],[44,58,0.7586,0.54911,0.25532,0.42857,0.5,0.75,0.14286,1.0,0,2,0,0,0,4,0,0,3,0,0,9,0,0,4,0,0,4,0,0,6,0,2],[48,58,0.8276,0.375,0.24679,0.14289,0.28571,0.42858,0.0,1.0,2,1,0,2,0,7,0,0,9,0,0,7,0,0,0,0,0,5,0,0,1,0,1],[52,58,0.8966,0.40625,0.2969,0.14286,0.28571,0.46429,0.0,1.0,2,4,0,2,0,8,0,0,7,0,0,7,0,0,1,0,0,2,0,0,1,0,4],[56,58,0.9655,0.46875,0.24546,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,12,0,0,9,0,0,0,0,0,5,0,0,1,0,3],[58,58,1.0,0.41965,0.18188,0.28571,0.28571,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,4,0,0,1,0,0,8,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.20982,"x":0.59821,"p":[[0,51,0.0,0.40622,0.22045,0.24999,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,8,0,0,7,0,0,6,0,0,6,0,0,4,0,0,0,0,1],[4,51,0.0784,0.51338,0.29636,0.28571,0.42857,0.71429,0.0,1.0,1,5,0,1,0,5,0,0,5,0,0,7,0,0,2,0,0,6,0,0,1,0,5],[8,51,0.1569,0.40624,0.18934,0.28571,0.42857,0.57111,0.0,0.71429,1,0,0,1,0,6,0,0,3,0,0,13,0,0,5,0,0,4,0,0,0,0,0],[12,51,0.2353,0.59821,0.27764,0.39286,0.64286,0.74996,0.14286,1.0,0,6,0,0,0,3,0,0,5,0,0,5,0,0,3,0,0,8,0,0,2,0,6],[16,51,0.3137,0.41955,0.24216,0.24999,0.42857,0.57143,0.14,1.0,0,1,0,0,0,8,0,0,7,0,0,7,0,0,3,0,0,4,0,0,2,0,1],[20,51,0.3922,0.39281,0.2501,0.25,0.35714,0.46536,0.0,1.0,2,2,0,2,0,6,0,0,8,0,0,8,0,0,2,0,0,4,0,0,0,0,2],[24,51,0.4706,0.43308,0.24868,0.25,0.42857,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,3,0,0,11,0,0,4,0,0,3,0,0,1,0,2],[28,51,0.549,0.46429,0.24223,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,7,0,0,9,0,0,2,0,0,6,0,0,1,0,2],[32,51,0.6275,0.52677,0.24598,0.39286,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,9,0,0,2,0,0,10,0,0,0,0,3],[36,51,0.7059,0.42857,0.23146,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,8,0,0,9,0,0,2,0,0,4,0,0,2,0,1],[40,51,0.7843,0.375,0.24157,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,10,0,0,4,0,0,5,0,0,2,0,0,0,0,2],[44,51,0.8627,0.34375,0.20159,0.14286,0.28571,0.42858,0.14286,1.0,0,1,0,0,0,9,0,0,12,0,0,5,0,0,3,0,0,2,0,0,0,0,1],[48,51,0.9412,0.30804,0.21756,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,14,0,0,11,0,0,1,0,0,2,0,0,3,0,0,0,0,1],[51,51,1.0,0.20982,0.07973,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,18,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b0c88e5ef6be0452","q":"Determine all integers  $n>1$  such that every power of  $n$  has an odd number of digits.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.89286,"x":1.0,"p":[[0,30,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,30,0.1333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,30,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,30,0.4,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,30,0.5333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[28,30,0.9333,0.89286,0.13363,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,19],[30,30,1.0,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20]]},{"b":7,"e":1.0,"k":"flat","v":0.95089,"x":0.98214,"p":[[0,17,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,17,0.2353,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,17,0.4706,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[12,17,0.7059,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[16,17,0.9412,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[17,17,1.0,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26]]}]},{"i":"005899ae0404aa14","q":"Anne-Marie has a deck of  $16$  cards, each with a distinct positive factor of  $2002$  written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops?\n\n*Proposed by Michael Ren.*","t":[{"b":3,"e":0.57143,"k":"falling","v":0.60268,"x":0.98214,"p":[[0,76,0.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[4,76,0.0526,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,76,0.1053,0.91071,0.16269,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,0,0,24],[12,76,0.1579,0.86607,0.18189,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,1,0,20],[16,76,0.2105,0.70089,0.17627,0.57143,0.57143,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,4,0,0,2,0,7],[20,76,0.2632,0.66964,0.1729,0.57143,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,4,0,5],[24,76,0.3158,0.72321,0.19212,0.57143,0.57143,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,3,0,9],[28,76,0.3684,0.72768,0.19352,0.57143,0.57143,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,4,0,9],[32,76,0.4211,0.70089,0.17985,0.57143,0.57143,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,2,0,0,3,0,7],[36,76,0.4737,0.64732,0.15561,0.57143,0.57143,0.60714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,2,0,0,2,0,4],[40,76,0.5263,0.67857,0.15972,0.57143,0.57143,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,2,0,0,5,0,4],[44,76,0.5789,0.64732,0.13356,0.57143,0.57143,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,3,0,0,4,0,2],[48,76,0.6316,0.70536,0.18189,0.57143,0.57143,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,1,0,0,4,0,7],[52,76,0.6842,0.67411,0.17941,0.57143,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,3,0,6],[56,76,0.7368,0.64286,0.14286,0.57143,0.57143,0.57143,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,1,0,0,3,0,3],[60,76,0.7895,0.66964,0.15335,0.57143,0.57143,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,0,6,0,3],[64,76,0.8421,0.69643,0.17768,0.57143,0.57143,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,5,0,6],[68,76,0.8947,0.63393,0.14258,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,2,0,0,2,0,3],[72,76,0.9474,0.61161,0.10853,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,3,0,0,2,0,1],[76,76,1.0,0.60268,0.10555,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,1,0,0,2,0,1]]},{"b":5,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,43,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,43,0.093,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[8,43,0.186,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,43,0.2791,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,43,0.3721,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,43,0.4651,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,43,0.5581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,43,0.6512,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,43,0.7442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,43,0.8372,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[40,43,0.9302,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[43,43,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"5d7e215e8ed4c26b","q":"Denote  $g(k)$  as the greatest odd divisor of  $k$ . Put  $f(k) = \\dfrac{k}{2} + \\dfrac{k}{g(k)}$  for  $k$  even, and  $2^{(k+1)/2}$  for  $k$  odd. Define the sequence  $x_1, x_2, x_3, ...$  by  $x_1 = 1$ ,  $x_{n+1} = f(x_n)$ . Find  $n$  such that  $x_n = 800$ 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8571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,11,0,0,4,0,1],[48,101,0.4752,0.60268,0.17762,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,11,0,0,3,0,0,14,0,0,2,0,1],[52,101,0.5149,0.5759,0.19061,0.4286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,7,0,0,8,0,0,11,0,0,0,0,2],[56,101,0.5545,0.63392,0.14257,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,18,0,0,2,0,0],[60,101,0.5941,0.5759,0.14053,0.42859,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,14,0,0,0,0,0],[64,101,0.6337,0.56249,0.14258,0.42857,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,5,0,0,12,0,0,11,0,0,0,0,0],[68,101,0.6733,0.58929,0.13243,0.4286,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,7,0,0,15,0,0,0,0,0],[72,101,0.7129,0.64732,0.15966,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,13,0,0,6,0,0],[76,101,0.7525,0.61605,0.14914,0.571,0.64286,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,13,0,0,3,0,0],[80,101,0.7921,0.64284,0.12373,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,15,0,0,3,0,0],[84,101,0.8317,0.62054,0.11071,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,17,0,0,0,0,0],[88,101,0.8713,0.6473,0.11286,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,17,0,0,2,0,0],[92,101,0.9109,0.64285,0.12876,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,16,0,0,3,0,0],[96,101,0.9505,0.65179,0.1234,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,17,0,0,3,0,0],[100,101,0.9901,0.62052,0.12169,0.57132,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,16,0,0,1,0,0],[101,101,1.0,0.625,0.1171,0.57142,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,16,0,0,1,0,0]]}]},{"i":"b35811c2fcd925f8","q":"$a,b$  and  $c$  are positive real numbers so that  $\\sum_{\\text{cyc}} (a+b)^2=2\\sum_{\\text{cyc}} a +6abc$ . Prove that  $$ \\sum_{\\text{cyc}} (a-b)^2\\leq\\left|2\\sum_{\\text{cyc}} a -6abc\\right|. $$","t":[{"b":2,"e":0.71429,"k":"flat","v":0.25893,"x":0.48661,"p":[[0,84,0.0,0.45982,0.17399,0.28571,0.42857,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,14,0,0,1,0,0,5,0,0,2,0,0],[4,84,0.0476,0.31696,0.13709,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[8,84,0.0952,0.32143,0.16752,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,21,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[12,84,0.1429,0.38839,0.23209,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,17,0,0,3,0,0,0,0,0,4,0,0,2,0,1],[16,84,0.1905,0.4374,0.28116,0.28571,0.28571,0.4642,0.14,1.0,0,5,0,0,0,3,0,0,18,0,0,3,0,0,1,0,0,1,0,0,1,0,5],[20,84,0.2381,0.35267,0.23685,0.2857,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,6,0,0,20,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[24,84,0.2857,0.33482,0.18074,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,20,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[28,84,0.3333,0.30802,0.15196,0.28571,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,5,0,0,20,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[32,84,0.381,0.35713,0.1383,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,16,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[36,84,0.4286,0.34821,0.24468,0.2857,0.28571,0.28571,0.0,1.0,1,3,0,1,0,6,0,0,18,0,0,2,0,0,1,0,0,1,0,0,0,0,3],[40,84,0.4762,0.37053,0.19186,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,3,0,0,19,0,0,5,0,0,1,0,0,1,0,0,3,0,0],[44,84,0.5238,0.39732,0.25688,0.2857,0.28571,0.42858,0.14286,1.0,0,3,0,0,0,6,0,0,15,0,0,4,0,0,1,0,0,2,0,0,1,0,3],[48,84,0.5714,0.38839,0.21498,0.28571,0.28571,0.42857,0.0,1.0,1,1,0,1,0,1,0,0,20,0,0,3,0,0,1,0,0,4,0,0,1,0,1],[52,84,0.619,0.39285,0.21128,0.28571,0.28571,0.46429,0.14286,1.0,0,1,0,0,0,4,0,0,16,0,0,4,0,0,3,0,0,3,0,0,1,0,1],[56,84,0.6667,0.45088,0.23449,0.28571,0.28571,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,15,0,0,3,0,0,2,0,0,8,0,0,1,0,1],[60,84,0.7143,0.39286,0.23419,0.28571,0.28571,0.46431,0.0,1.0,1,2,0,1,0,2,0,0,19,0,0,2,0,0,3,0,0,2,0,0,1,0,2],[64,84,0.7619,0.25893,0.09061,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,6,0,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,84,0.8095,0.35713,0.21427,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,6,0,0,18,0,0,2,0,0,1,0,0,3,0,0,1,0,1],[72,84,0.8571,0.37945,0.24119,0.28571,0.28571,0.46418,0.0,1.0,2,2,0,2,0,4,0,0,14,0,0,4,0,0,3,0,0,3,0,0,0,0,2],[76,84,0.9048,0.45534,0.26107,0.28571,0.35714,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,9,0,0,2,0,0,3,0,0,9,0,0,0,0,2],[80,84,0.9524,0.48661,0.21387,0.28571,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,7,0,0,8,0,0,1,0,0,13,0,0,0,0,0],[84,84,1.0,0.39284,0.16365,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,4,0,0,10,0,0,12,0,0,2,0,0,4,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"falling","v":0.27678,"x":0.5357,"p":[[0,223,0.0,0.5357,0.27433,0.28571,0.42857,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,13,0,0,6,0,0,2,0,0,4,0,0,1,0,6],[4,223,0.0179,0.33928,0.26666,0.2857,0.28571,0.28571,0.0,1.0,3,4,2,3,0,3,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[8,223,0.0359,0.28571,0.05051,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,223,0.0538,0.33482,0.23038,0.2857,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,7,0,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[16,223,0.0717,0.31695,0.20432,0.28571,0.28571,0.28571,0.0,1.0,2,2,0,2,0,3,0,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[20,223,0.0897,0.35714,0.18898,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,23,0,0,3,0,0,0,0,0,2,0,0,1,0,1],[24,223,0.1076,0.39285,0.27894,0.2857,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,5,0,0,20,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[28,223,0.1256,0.37053,0.26453,0.2857,0.28571,0.28571,0.0,1.0,2,3,1,2,0,3,0,0,20,0,0,1,0,0,0,0,0,2,0,0,1,0,3],[32,223,0.1435,0.32143,0.21724,0.2857,0.28571,0.28571,0.0,1.0,2,2,0,2,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[36,223,0.1614,0.38391,0.22427,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,24,0,0,1,0,0,2,0,0,1,0,0,0,0,3],[40,223,0.1794,0.46427,0.29014,0.28571,0.28571,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,19,0,0,1,0,0,1,0,0,2,0,0,2,0,5],[44,223,0.1973,0.43303,0.25626,0.28571,0.28571,0.46429,0.14286,1.0,0,4,0,0,0,1,0,0,20,0,0,3,0,0,2,0,0,1,0,0,1,0,4],[48,223,0.2152,0.44196,0.28203,0.28571,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,3,0,0,18,0,0,2,0,0,2,0,0,1,0,0,1,0,5],[52,223,0.2332,0.38839,0.21793,0.28571,0.28571,0.32143,0.14286,1.0,0,2,0,0,0,1,0,0,23,0,0,2,0,0,1,0,0,2,0,0,1,0,2],[56,223,0.2511,0.33036,0.15746,0.2857,0.28571,0.32164,0.14286,1.0,0,1,0,0,0,4,0,0,20,0,0,5,0,0,2,0,0,0,0,0,0,0,1],[60,223,0.2691,0.36161,0.2525,0.28571,0.28571,0.28571,0.0,1.0,1,4,0,1,0,3,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[64,223,0.287,0.39732,0.24415,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,3,0,0,20,0,0,2,0,0,2,0,0,1,0,0,1,0,3],[68,223,0.3049,0.39731,0.26662,0.28571,0.28571,0.42858,0.0,1.0,2,4,0,2,0,2,0,0,17,0,0,4,0,0,2,0,0,1,0,0,0,0,4],[72,223,0.3229,0.42857,0.28347,0.28571,0.28571,0.57143,0.0,1.0,1,5,0,1,0,2,0,0,19,0,0,1,0,0,2,0,0,2,0,0,0,0,5],[76,223,0.3408,0.4375,0.25739,0.28571,0.28571,0.46431,0.2857,1.0,0,4,0,0,0,0,0,0,22,0,0,2,0,0,1,0,0,2,0,0,1,0,4],[80,223,0.3587,0.35714,0.24744,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,7,0,0,18,0,0,2,0,0,1,0,0,0,0,0,1,0,3],[84,223,0.3767,0.4241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Tu has a large rectangular table. On it, there are \ufb01nitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in \ufb01nitely many steps?","t":[{"b":4,"e":0.4286,"k":"volatile","v":0.20527,"x":0.79461,"p":[[0,9,0.0,0.20527,0.14261,0.14286,0.14286,0.1429,0.0,0.57143,1,0,1,1,0,24,0,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[4,9,0.4444,0.76338,0.23855,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,10,0,0,4,0,11],[8,9,0.8889,0.79461,0.23944,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,4,0,0,5,0,14],[9,9,1.0,0.73212,0.30672,0.5354,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,3,0,0,5,0,0,2,0,0,1,0,16]]},{"b":7,"e":0.14286,"k":"volatile","v":0.25444,"x":0.74996,"p":[[0,13,0.0,0.25444,0.17024,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,21,0,0,3,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[4,13,0.3077,0.74996,0.26246,0.67857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,9,0,0,3,0,12],[8,13,0.6154,0.52676,0.33203,0.14286,0.57121,0.85704,0.14286,1.0,0,6,0,0,0,10,0,0,3,0,0,2,0,0,5,0,0,2,0,0,4,0,6],[12,13,0.9231,0.5357,0.31338,0.14286,0.64286,0.74996,0.14286,1.0,0,3,0,0,0,11,0,0,0,0,0,2,0,0,3,0,0,8,0,0,5,0,3],[13,13,1.0,0.58917,0.30265,0.39286,0.57143,0.85704,0.0,1.0,1,6,0,1,0,5,0,0,2,0,0,3,0,0,6,0,0,6,0,0,3,0,6]]}]},{"i":"f5c5a34ce7f3d9a7","q":"Determine all positive integers  $k$  such that \\[\\frac{d(n^{2})}{d(n)}= k\\] for some  $n \\in \\mathbb{N}$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.30803,"x":0.74554,"p":[[0,93,0.0,0.38839,0.12492,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,13,0,0,14,0,0,2,0,0,2,0,0,0,0,0],[4,93,0.043,0.68746,0.20343,0.571,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,11,0,0,6,0,4],[8,93,0.086,0.67408,0.22935,0.5354,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,6,0,0,7,0,5],[12,93,0.129,0.66515,0.23586,0.42859,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,8,0,0,6,0,5],[16,93,0.172,0.70533,0.1819,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,8,0,0,11,0,2],[20,93,0.2151,0.70531,0.19545,0.571,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,8,0,0,5,0,6],[24,93,0.2581,0.74554,0.18808,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,11,0,0,9,0,5],[28,93,0.3011,0.68301,0.22229,0.57132,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,9,0,0,4,0,6],[32,93,0.3441,0.67854,0.20517,0.5354,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,8,0,0,5,0,5],[36,93,0.3871,0.68749,0.16145,0.57143,0.71429,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,13,0,0,6,0,2],[40,93,0.4301,0.71872,0.20041,0.57132,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,7,0,0,9,0,5],[44,93,0.4731,0.7098,0.16936,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,11,0,0,6,0,4],[48,93,0.5161,0.68747,0.20653,0.571,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,8,0,0,4,0,6],[52,93,0.5591,0.65395,0.2166,0.53539,0.67857,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,4,0,0,7,1,0,5,0,0,8,0,3],[56,93,0.6022,0.6741,0.21793,0.42859,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,9,0,0,5,0,0,5,0,0,7,0,5],[60,93,0.6452,0.64727,0.21425,0.571,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,6,0,0,10,0,1],[64,93,0.6882,0.67856,0.22304,0.57132,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,11,0,0,3,0,6],[68,93,0.7312,0.62939,0.17808,0.571,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,10,0,0,5,0,1],[72,93,0.7742,0.62946,0.2854,0.28571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,9,0,0,2,0,0,2,0,0,7,0,0,4,0,7],[76,93,0.8172,0.4866,0.22547,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,11,0,0,8,0,0,4,0,0,4,0,0,2,0,2],[80,93,0.8602,0.54909,0.27918,0.28571,0.4286,0.85704,0.14286,1.0,0,4,0,0,0,2,0,0,10,0,0,5,0,0,2,0,0,4,0,0,5,0,4],[84,93,0.9032,0.65161,0.22849,0.42857,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,11,0,0,3,0,5],[88,93,0.9462,0.41517,0.23243,0.28571,0.28571,0.57111,0.14286,0.85714,0,0,0,0,0,3,0,0,18,0,0,2,0,0,2,0,0,2,0,0,5,0,0],[92,93,0.9892,0.33035,0.13092,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,1,0,0,25,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[93,93,1.0,0.30803,0.06298,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.41964,"x":0.74996,"p":[[0,97,0.0,0.41964,0.12846,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,15,0,0,3,0,0,3,0,0,0,0,0],[4,97,0.0412,0.70087,0.22121,0.5713,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,6,0,0,7,0,6],[8,97,0.0825,0.7232,0.23403,0.57132,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,9,0,0,4,0,9],[12,97,0.1237,0.74548,0.18813,0.57132,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,8,0,0,7,0,7],[16,97,0.1649,0.71425,0.1429,0.57143,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,12,0,0,6,0,3],[20,97,0.2062,0.70088,0.18337,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,12,0,0,4,0,5],[24,97,0.2474,0.74996,0.19236,0.57143,0.78571,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,9,0,7],[28,97,0.2887,0.61604,0.25365,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,7,0,0,5,0,0,6,0,0,5,0,0,3,0,6],[32,97,0.3299,0.65625,0.23107,0.42857,0.71429,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,11,0,0,2,0,6],[36,97,0.3711,0.61154,0.22655,0.42857,0.5712,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,8,0,0,10,0,0,2,0,0,5,0,4],[40,97,0.4124,0.66963,0.21853,0.57132,0.64286,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,5,0,0,6,0,5],[44,97,0.4536,0.61158,0.22654,0.42859,0.57143,0.75,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,5,0,0,8,0,0,6,0,0,5,0,3],[48,97,0.4948,0.64282,0.19563,0.53539,0.71429,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,9,0,0,6,0,2],[52,97,0.5361,0.65175,0.19213,0.571,0.64286,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,7,0,0,7,0,2],[56,97,0.5773,0.62497,0.17405,0.42859,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,7,0,0,6,0,0,12,0,0,4,0,1],[60,97,0.6186,0.64731,0.23686,0.42859,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,2,0,0,3,0,0,11,0,0,5,0,4],[64,97,0.6598,0.71875,0.21274,0.67857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,11,0,0,8,0,5],[68,97,0.701,0.66963,0.20341,0.57143,0.64286,0.85704,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,4,0,0,10,0,0,7,0,0,4,0,5],[72,97,0.7423,0.6473,0.223,0.4286,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,4,0,0,5,0,0,8,0,0,7,0,3],[76,97,0.7835,0.66962,0.18708,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,8,0,0,8,0,2],[80,97,0.8247,0.6607,0.1915,0.5354,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,11,0,0,3,0,4],[84,97,0.866,0.61158,0.19638,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,9,0,0,4,0,2],[88,97,0.9072,0.65616,0.1708,0.571,0.57143,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,5,0,0,9,0,1],[92,97,0.9485,0.66515,0.21012,0.571,0.71429,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,10,0,0,3,0,5],[96,97,0.9897,0.52676,0.18363,0.42857,0.4286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,14,0,0,2,0,0,9,0,0,1,0,1],[97,97,1.0,0.44191,0.17981,0.28571,0.42857,0.5711,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,7,0,0,7,0,0,4,0,0,1,0,0]]}]},{"i":"34a43541d511ba35","q":"Are there four distinct positive integers such that adding the product of any two of them to 2006 yields a perfect square?","t":[{"b":0,"e":0.85714,"k":"flat","v":0.83482,"x":0.96875,"p":[[0,18,0.0,0.83482,0.36089,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[4,18,0.2222,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[8,18,0.4444,0.92857,0.20825,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[12,18,0.6667,0.94196,0.16311,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[16,18,0.8889,0.92856,0.15568,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,24],[18,18,1.0,0.83927,0.22234,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,0,14,0,13]]},{"b":2,"e":0.42857,"k":"flat","v":0.82588,"x":0.97767,"p":[[0,13,0.0,0.87947,0.27225,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,25],[4,13,0.3077,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,13,0.6154,0.86607,0.28557,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,23],[12,13,0.9231,0.82588,0.2362,0.85714,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,10],[13,13,1.0,0.84373,0.19353,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,15,0,12]]}]},{"i":"150870fcc6baabcd","q":"Archimedes planned to count all of the prime numbers between  $2$  and  $1000$  using the Sieve of Eratosthenes as follows:\n(a) List the integers from  $2$  to  $1000$ .\n(b) Circle the smallest number in the list and call this  $p$ .\n(c) Cross out all multiples of  $p$  in the list except for  $p$  itself.\n(d) Let  $p$  be the smallest number remaining that is neither circled nor crossed out. Circle  $p$ .\n(e) Repeat steps  $(c)$  and  $(d)$  until each number is either circled or crossed out.\n\nAt the end of this process, the circled numbers are prime and the crossed out numbers are composite.\n\nUnfortunately, while crossing out\u000b the multiples of  $2$ , Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides  $2$ ). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes \ffinished equals the number of primes from  $2$  to  $1000$  (including  $2$ ), then what is the largest possible prime that Archimedes accidentally crossed out?","t":[{"b":4,"e":0.71429,"k":"flat","v":0.54463,"x":0.79909,"p":[[0,45,0.0,0.62052,0.34184,0.28571,0.49979,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,14],[4,45,0.0889,0.79909,0.28764,0.57132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,2,0,0,1,0,20],[8,45,0.1778,0.62943,0.26931,0.42857,0.57121,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,6,0,0,7,0,0,6,0,0,3,0,0,1,0,9],[12,45,0.2667,0.70088,0.28652,0.4286,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,2,0,0,5,0,0,4,0,0,1,0,13],[16,45,0.3556,0.65625,0.28984,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,7,0,0,1,0,0,5,0,0,1,0,11],[20,45,0.4444,0.66071,0.27141,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,6,0,0,2,0,0,8,0,0,0,0,10],[24,45,0.5333,0.60717,0.28569,0.28571,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,10,0,0,4,0,0,4,0,0,5,0,0,0,0,9],[28,45,0.6222,0.6428,0.20827,0.5354,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,3,0,0,4,0,0,14,0,0,3,0,3],[32,45,0.7111,0.6205,0.22192,0.42859,0.64286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,3,0,0,7,0,0,10,0,0,2,0,4],[36,45,0.8,0.54463,0.26592,0.28571,0.42859,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,4,0,0,2,0,0,7,0,0,1,0,5],[40,45,0.8889,0.69643,0.24157,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,2,0,0,3,0,0,13,0,0,0,0,9],[44,45,0.9778,0.69642,0.11153,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,26,0,0,1,0,1],[45,45,1.0,0.66964,0.11538,0.67857,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,22,0,0,2,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.51339,"x":0.98214,"p":[[0,66,0.0,0.51339,0.30275,0.28571,0.28571,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,18,0,0,3,0,0,1,0,0,2,0,0,0,0,8],[4,66,0.0606,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,66,0.1212,0.78121,0.27199,0.57143,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,3,0,0,0,0,0,5,0,0,5,0,0,2,0,16],[12,66,0.1818,0.83481,0.25283,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,2,0,20],[16,66,0.2424,0.86161,0.24087,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,1,0,22],[20,66,0.303,0.77679,0.28558,0.67857,1.0,1.0,0.0,1.0,1,17,1,1,0,0,0,0,3,0,0,3,0,0,1,0,0,6,0,0,1,0,17],[24,66,0.3636,0.87054,0.25092,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,25],[28,66,0.4242,0.80804,0.25904,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,3,0,0,3,0,18],[32,66,0.4848,0.70534,0.31932,0.49968,0.71429,1.0,0.0,1.0,2,13,2,2,0,0,0,0,6,0,0,0,0,0,2,0,0,7,0,0,2,0,13],[36,66,0.5455,0.77232,0.2547,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,6,0,0,2,0,15],[40,66,0.6061,0.83928,0.25939,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,3,0,0,2,0,21],[44,66,0.6667,0.80356,0.2442,0.71429,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,1,0,16],[48,66,0.7273,0.83034,0.20027,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,12,0,0,1,0,16],[52,66,0.7879,0.72319,0.27186,0.571,0.71429,1.0,0.0,1.0,1,12,1,1,0,0,0,0,3,0,0,3,0,0,3,0,0,9,0,0,1,0,12],[56,66,0.8485,0.70533,0.24729,0.57132,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,11,0,0,2,0,9],[60,66,0.9091,0.65173,0.19543,0.571,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,9,0,0,13,0,0,1,0,4],[64,66,0.9697,0.70979,0.12106,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,23,0,0,2,0,2],[66,66,1.0,0.6964,0.10568,0.57143,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,20,0,0,1,0,2]]}]},{"i":"4b273bf9f76ecefe","q":"Consider three fixed spheres  $S_1, S_2, S_3$  with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each  $S_i$  along a great circle of  $S_i$ . \n\n*Stere Ianu\u0219*","t":[{"b":3,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,35,0.0,0.86607,0.15947,0.82132,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,10,0,14],[4,35,0.1143,0.92857,0.08748,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[8,35,0.2286,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[12,35,0.3429,0.92397,0.1131,0.85714,1.0,1.0,0.71,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[16,35,0.4571,0.91517,0.10013,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[20,35,0.5714,0.9107,0.0928,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":0.85714,"k":"flat","v":0.83033,"x":0.95089,"p":[[0,20,0.0,0.88392,0.18363,0.85714,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,15],[4,20,0.2,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[8,20,0.4,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[12,20,0.6,0.90178,0.10376,0.85714,0.85714,1.0,0.714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15],[16,20,0.8,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[20,20,1.0,0.83033,0.0662,0.857,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,24,0,1]]}]},{"i":"720a1c750e5e80ef","q":"Determine the least integer  $k$  for which the following story could hold true:\nIn a chess tournament with  $24$  players, every pair of players plays at least  $2$  and at most  $k$  games against each other. At the end of the tournament, it turns out that every player has played a different number of games.","t":[{"b":0,"e":0.14286,"k":"volatile","v":0.35712,"x":0.71429,"p":[[0,10,0.0,0.35712,0.18555,0.28571,0.42857,0.42857,0.0,0.57143,5,0,0,5,0,2,0,0,3,0,0,16,0,0,6,0,0,0,0,0,0,0,0],[4,10,0.4,0.71429,0.30305,0.71429,0.85714,0.85714,0.0,1.0,4,5,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,14,0,5],[8,10,0.8,0.52232,0.36528,0.10714,0.64286,0.85714,0.0,1.0,8,1,0,8,0,2,0,0,0,0,0,4,0,0,2,0,0,2,0,0,13,0,1],[10,10,1.0,0.44643,0.3973,0.0,0.57143,0.85714,0.0,1.0,11,2,0,11,0,4,0,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,2]]},{"b":3,"e":1.0,"k":"rising","v":0.34375,"x":0.78125,"p":[[0,66,0.0,0.34375,0.17807,0.25,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,4,0,0,4,0,0,15,0,0,5,0,0,0,0,0,0,0,0],[4,66,0.0606,0.72768,0.26573,0.71429,0.85714,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,1,0,0,1,0,0,0,0,0,10,0,0,12,0,5],[8,66,0.1212,0.78125,0.20511,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,16,0,5],[12,66,0.1818,0.66964,0.27301,0.64286,0.71429,0.85714,0.0,1.0,3,2,0,3,0,0,0,0,1,0,0,4,0,0,0,0,0,10,0,0,12,0,2],[16,66,0.2424,0.67409,0.24804,0.5354,0.71429,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,0,0,0,6,0,0,4,0,0,5,0,0,13,0,2],[20,66,0.303,0.62052,0.31259,0.5354,0.71429,0.85714,0.0,1.0,5,1,0,5,0,0,0,0,2,0,0,1,0,0,3,0,0,7,0,0,13,0,1],[24,66,0.3636,0.63393,0.28557,0.4286,0.71429,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,1,0,0,5,0,0,3,0,0,7,0,0,8,0,4],[28,66,0.4242,0.62945,0.28984,0.42857,0.71429,0.85714,0.0,1.0,2,3,0,2,0,3,0,0,0,0,0,5,0,0,3,0,0,6,0,0,10,0,3],[32,66,0.4848,0.72765,0.20002,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,0,13,0,2],[36,66,0.5455,0.67855,0.24486,0.571,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,4,0,0,4,0,0,10,0,0,6,0,5],[40,66,0.6061,0.70087,0.29529,0.53539,0.85707,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,1,0,0,3,0,0,2,0,0,5,0,0,9,0,8],[44,66,0.6667,0.71871,0.21869,0.57143,0.85714,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,1,0,0,5,0,0,6,0,0,15,0,2],[48,66,0.7273,0.69641,0.21354,0.57132,0.857,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,5,0,0,3,0,0,5,0,0,17,0,0],[52,66,0.7879,0.66957,0.26832,0.571,0.85707,0.85714,0.0,1.0,2,2,0,2,0,1,0,0,1,0,0,3,0,0,6,0,0,2,0,0,15,0,2],[56,66,0.8485,0.70536,0.19865,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,5,0,0,8,0,5],[60,66,0.9091,0.73214,0.23351,0.67857,0.85714,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,6,0,0,14,0,4],[64,66,0.9697,0.70982,0.22724,0.57143,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,6,0,0,15,0,2],[66,66,1.0,0.65179,0.22851,0.57143,0.64286,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,5,0,0,9,0,0,4,0,0,10,0,2]]}]},{"i":"b293f342cdbee383","q":"Every natural number is coloured in one of the  $ k$  colors. Prove that there exist four distinct natural numbers  $ a, b, c, d$ , all coloured in the same colour, such that  $ ad \\equal{} bc$ ,  $ \\displaystyle \\frac b a$  is power of 2 and  $ \\displaystyle \\frac c a$  is power of 3.","t":[{"b":0,"e":0.571,"k":"flat","v":0.35714,"x":0.53125,"p":[[0,49,0.0,0.35714,0.15568,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[4,49,0.0816,0.45536,0.27302,0.28571,0.28571,0.46429,0.2857,1.0,0,6,0,0,0,0,0,0,20,0,0,4,0,0,2,0,0,0,0,0,0,0,6],[8,49,0.1633,0.4375,0.25739,0.28571,0.28571,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,20,0,0,6,0,0,0,0,0,1,0,0,0,0,5],[12,49,0.2449,0.4732,0.2635,0.28571,0.42857,0.4286,0.2857,1.0,0,6,0,0,0,0,0,0,15,0,0,10,0,0,1,0,0,0,0,0,0,0,6],[16,49,0.3265,0.45088,0.23175,0.28571,0.42857,0.42858,0.2857,1.0,0,4,0,0,0,0,0,0,15,0,0,10,0,0,2,0,0,1,0,0,0,0,4],[20,49,0.4082,0.45089,0.23176,0.28571,0.35714,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,16,0,0,9,0,0,0,0,0,3,0,0,1,0,3],[24,49,0.4898,0.47768,0.22759,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,12,0,0,11,0,0,3,0,0,1,0,0,2,0,3],[28,49,0.5714,0.41964,0.17474,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,13,0,0,14,0,0,3,0,0,0,0,0,0,0,2],[32,49,0.6531,0.46428,0.25,0.28571,0.42857,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,15,0,0,10,0,0,1,0,0,1,0,0,0,0,5],[36,49,0.7347,0.48661,0.24707,0.28571,0.42857,0.46431,0.2857,1.0,0,5,0,0,0,0,0,0,12,0,0,12,0,0,1,0,0,2,0,0,0,0,5],[40,49,0.8163,0.51786,0.26905,0.28571,0.42857,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,8,0,0,2,0,0,1,0,0,3,0,5],[44,49,0.898,0.53125,0.29285,0.28571,0.35714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,16,0,0,3,0,0,3,0,0,0,0,0,4,0,6],[48,49,0.9796,0.45533,0.17289,0.28571,0.42857,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,11,0,0,5,0,0,4,0,0,0,0,1],[49,49,1.0,0.43303,0.20971,0.28571,0.28571,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,18,0,0,4,0,0,6,0,0,1,0,0,1,0,2]]},{"b":7,"e":1.0,"k":"rising","v":0.33035,"x":0.97768,"p":[[0,22,0.0,0.33035,0.09062,0.2857,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.49553,0.28791,0.28571,0.28571,0.60714,0.2857,1.0,0,7,0,0,0,0,0,0,18,0,0,3,0,0,3,0,0,1,0,0,0,0,7],[8,22,0.3636,0.53124,0.29717,0.28571,0.35714,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,16,0,0,3,0,0,3,0,0,2,0,0,0,0,8],[12,22,0.5455,0.60714,0.32341,0.28571,0.42857,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,12,0,0,6,0,0,1,0,0,0,0,0,1,0,12],[16,22,0.7273,0.52229,0.30222,0.28571,0.28571,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,18,0,0,1,0,0,3,0,0,2,0,0,0,0,8],[20,22,0.9091,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[22,22,1.0,0.87052,0.17629,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,4,0,18]]}]},{"i":"eb382f8b387fb46e","q":"Do there exist two real monic polynomials  $P(x)$  and  $Q(x)$  of degree 3,such that the roots of  $P(Q(X))$  are nine pairwise distinct nonnegative integers that add up to  $72$ ?\n(In a monic polynomial of degree 3, the coefficient of  $x^{3}$  is  $1$ .)","t":[{"b":4,"e":1.0,"k":"flat","v":0.88392,"x":1.0,"p":[[0,161,0.0,0.94196,0.19186,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[4,161,0.0248,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[8,161,0.0497,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,161,0.0745,0.92856,0.17499,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,27],[16,161,0.0994,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,161,0.1242,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,161,0.1491,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[28,161,0.1739,0.95534,0.12599,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[32,161,0.1988,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[36,161,0.2236,0.91962,0.16733,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[40,161,0.2484,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[44,161,0.2733,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[48,161,0.2981,0.93304,0.15561,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,27],[52,161,0.323,0.94195,0.15513,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[56,161,0.3478,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[60,161,0.3727,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[64,161,0.3975,0.94642,0.14177,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[68,161,0.4224,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[72,161,0.4472,0.97319,0.10384,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[76,161,0.472,0.95088,0.13179,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[80,161,0.4969,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[84,161,0.5217,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[88,161,0.5466,0.95981,0.12496,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[92,161,0.5714,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[96,161,0.5963,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[100,161,0.6211,0.95536,0.14032,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[104,161,0.646,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[108,161,0.6708,0.91071,0.17768,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,25],[112,161,0.6957,0.96874,0.12238,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[116,161,0.7205,0.88392,0.1871,0.67857,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,0,0,23],[120,161,0.7453,0.94642,0.14177,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[124,161,0.7702,0.95982,0.12492,1.0,1.0,1.0,0.5714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[128,161,0.795,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[132,161,0.8199,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[136,161,0.8447,0.96872,0.10565,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[140,161,0.8696,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[144,161,0.8944,0.92411,0.15966,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26],[148,161,0.9193,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[152,161,0.9441,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[156,161,0.9689,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[160,161,0.9938,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[161,161,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.71429,"k":"falling","v":0.66963,"x":1.0,"p":[[0,177,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,177,0.0226,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,177,0.0452,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[12,177,0.0678,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[16,177,0.0904,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[20,177,0.113,0.91963,0.1673,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[24,177,0.1356,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,177,0.1582,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,177,0.1808,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[36,177,0.2034,0.95982,0.12492,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,29],[40,177,0.226,0.96429,0.14286,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[44,177,0.2486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,177,0.2712,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,177,0.2938,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[56,177,0.3164,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[60,177,0.339,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[64,177,0.3616,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[68,177,0.3842,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,177,0.4068,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[76,177,0.4294,0.90624,0.1772,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,25],[80,177,0.452,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[84,177,0.4746,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[88,177,0.4972,0.91963,0.1673,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[92,177,0.5198,0.89284,0.1786,0.82143,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,23],[96,177,0.5424,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[100,177,0.565,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[104,177,0.5876,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[108,177,0.6102,0.91963,0.1673,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[112,177,0.6328,0.91964,0.16728,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[116,177,0.6554,0.92411,0.15966,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26],[120,177,0.678,0.92857,0.16752,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,27],[124,177,0.7006,0.9107,0.17037,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,25],[128,177,0.7232,0.90179,0.1729,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,24],[132,177,0.7458,0.91963,0.1673,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[136,177,0.7684,0.87947,0.18595,0.71429,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,0,0,22],[140,177,0.791,0.86159,0.19395,0.57143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,21],[144,177,0.8136,0.83036,0.20652,0.57143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,19],[148,177,0.8362,0.88837,0.18122,0.71429,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,0,0,23],[152,177,0.8588,0.89732,0.17941,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,24],[156,177,0.8814,0.87494,0.20133,0.57143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,23],[160,177,0.904,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[164,177,0.9266,0.875,0.18814,0.67857,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,0,0,22],[168,177,0.9492,0.83478,0.19274,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,0,0,18],[172,177,0.9718,0.71428,0.17127,0.57143,0.71429,0.74996,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,10,0,0,1,0,7],[176,177,0.9944,0.70088,0.13055,0.57143,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,17,0,0,0,0,4],[177,177,1.0,0.66963,0.09063,0.57143,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,19,0,0,0,0,1]]}]},{"i":"9e0ca4bc0f88f888","q":"Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v) \\in\\{1,2\\}$ and $f(e) \\in\\{1,2,3\\}$, respectively. Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$. We say that an assignment $f$ is cool if $S(u) \\neq S(v)$ for every pair $(u, v)$ of adjacent (i.e. connected by an edge) vertices in $G$. Prove that for every graph there exists a cool assignment.","t":[{"b":1,"e":0.57143,"k":"rising","v":0.04911,"x":0.31473,"p":[[0,25,0.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.10268,0.1394,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,15,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,25,0.32,0.12052,0.17533,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,14,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[12,25,0.48,0.15177,0.1747,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,14,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[16,25,0.64,0.20536,0.20497,0.10714,0.14286,0.32143,0.0,0.85714,8,0,0,8,0,15,0,0,1,0,0,5,0,0,2,0,0,0,0,0,1,0,0],[20,25,0.8,0.23213,0.17032,0.14286,0.14286,0.32143,0.0,0.57143,3,0,0,3,0,18,0,0,3,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[24,25,0.96,0.31473,0.23815,0.14286,0.14286,0.57143,0.07143,1.0,0,1,0,0,1,19,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1],[25,25,1.0,0.28563,0.19891,0.14286,0.14286,0.57143,0.14,0.57143,0,0,0,0,0,21,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.09821,"p":[[0,38,0.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,38,0.2105,0.0759,0.10092,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.09821,0.20341,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,38,0.4211,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6e333562bfad3581","q":"Consider the fourteen numbers,  $1^4,2^4,...,14^4$ . The smallest natural numebr  $n$  such that they leave distinct remainders when divided by  $n$  is:","t":[{"b":1,"e":0.2857,"k":"falling","v":0.18747,"x":0.82142,"p":[[0,119,0.0,0.73659,0.18596,0.67857,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,4,0,0,20,0,0],[4,119,0.0336,0.82142,0.16751,0.857,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,18,0,7],[8,119,0.0672,0.77677,0.15128,0.71429,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,19,0,2],[12,119,0.1008,0.77678,0.18877,0.82132,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,21,0,3],[16,119,0.1345,0.79018,0.18553,0.57143,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,1,0,0,13,0,8],[20,119,0.1681,0.73661,0.21461,0.57143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,4,0,0,11,0,6],[24,119,0.2017,0.79016,0.15967,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,20,0,3],[28,119,0.2353,0.8125,0.16536,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,16,0,7],[32,119,0.2689,0.70311,0.24098,0.57143,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,1,0,1,6,0,0,2,0,0,15,0,3],[36,119,0.3025,0.54463,0.34336,0.21429,0.71429,0.85714,0.0,0.85714,8,0,0,8,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,12,0,0],[40,119,0.3361,0.61606,0.35614,0.49967,0.85714,0.85714,0.0,1.0,7,3,0,7,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,14,0,3],[44,119,0.3697,0.46429,0.37115,0.0,0.5,0.85714,0.0,0.85714,10,0,0,10,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0,12,0,0],[48,119,0.4034,0.55357,0.35848,0.21429,0.71429,0.85714,0.0,1.0,8,1,0,8,0,0,0,0,2,0,0,1,0,0,3,0,0,4,0,0,13,0,1],[52,119,0.437,0.54462,0.36322,0.10714,0.71429,0.85714,0.0,1.0,8,2,0,8,0,1,0,0,1,0,0,1,0,0,4,0,0,4,0,0,11,0,2],[56,119,0.4706,0.48214,0.3859,0.0,0.57143,0.85714,0.0,0.85714,11,0,0,11,0,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,15,0,0],[60,119,0.5042,0.36607,0.37786,0.0,0.21429,0.71429,0.0,1.0,15,1,0,15,0,1,0,0,1,0,0,0,0,0,4,0,0,4,0,0,6,0,1],[64,119,0.5378,0.18747,0.30602,0.0,0.0,0.32143,0.0,1.0,22,1,0,22,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,2,0,1],[68,119,0.5714,0.20982,0.32534,0.0,0.0,0.42857,0.0,0.85714,21,0,0,21,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,0],[72,119,0.605,0.37499,0.37584,0.0,0.35716,0.75,0.0,0.85714,15,0,0,15,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,8,0,0],[76,119,0.6387,0.30134,0.35792,0.0,0.0,0.57143,0.0,1.0,17,1,0,17,0,0,0,0,1,0,1,4,0,0,2,0,0,0,0,0,6,0,1],[80,119,0.6723,0.22766,0.31714,0.0,0.0,0.57111,0.0,0.85714,20,0,0,20,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,3,0,0],[84,119,0.7059,0.4375,0.36932,0.0,0.57143,0.85714,0.0,0.85714,12,0,0,12,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,9,0,0],[88,119,0.7395,0.64284,0.22588,0.4286,0.64286,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,6,0,0,7,0,0,2,0,0,14,0,0],[92,119,0.7731,0.60712,0.26,0.57132,0.57143,0.85704,0.0,1.0,3,1,0,3,0,0,0,0,2,0,0,2,0,0,10,0,0,5,0,0,9,0,1],[96,119,0.8067,0.64284,0.22304,0.57132,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,3,0,0,8,0,0,6,0,0,10,0,1],[100,119,0.8403,0.60711,0.20824,0.42857,0.57143,0.857,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,4,0,0,7,0,0,6,0,0,9,0,0],[104,119,0.8739,0.62499,0.24157,0.4286,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,7,0,0,4,0,0,8,0,0,9,0,1],[108,119,0.9076,0.55133,0.22675,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,3,0,0,8,0,0,7,1,0,5,0,0,6,0,0],[112,119,0.9412,0.54908,0.23448,0.42857,0.4998,0.74996,0.0,0.85714,2,0,0,2,0,0,0,0,2,0,0,12,0,0,5,0,0,3,0,0,8,0,0],[116,119,0.9748,0.58033,0.24206,0.42857,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,7,0,0,5,0,0,4,0,0,10,0,0],[119,119,1.0,0.50893,0.22851,0.28571,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,8,0,0,6,0,0,4,0,0,4,0,1]]},{"b":7,"e":0.71429,"k":"falling","v":0.4464,"x":0.8348,"p":[[0,117,0.0,0.82142,0.11845,0.85711,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,23,0,3],[4,117,0.0342,0.80804,0.15815,0.82143,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,19,0,5],[8,117,0.0684,0.74107,0.22711,0.57143,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,0,0,0,6,0,0,5,0,0,13,0,5],[12,117,0.1026,0.74552,0.24416,0.57143,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,3,0,0,0,0,0,5,0,0,3,0,0,14,0,6],[16,117,0.1368,0.78124,0.16746,0.71429,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,21,0,2],[20,117,0.1709,0.75889,0.21562,0.57143,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,0,0,0,7,0,0,3,0,0,15,0,5],[24,117,0.2051,0.71875,0.21572,0.57143,0.78564,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,0,0,0,11,0,0,3,0,0,12,0,4],[28,117,0.2393,0.72319,0.23942,0.57143,0.85714,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,17,0,2],[32,117,0.2735,0.74549,0.16264,0.57143,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,0,0,0,21,0,0],[36,117,0.3077,0.75891,0.17291,0.67857,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,4,0,0,18,0,2],[40,117,0.3419,0.74998,0.18559,0.57143,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,3,0,0,16,0,3],[44,117,0.3761,0.69641,0.1948,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,3,0,0,13,0,2],[48,117,0.4103,0.71651,0.13658,0.57143,0.71429,0.85714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term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?","t":[{"b":0,"e":1.0,"k":"flat","v":0.69643,"x":0.9375,"p":[[0,103,0.0,0.83929,0.24157,0.71429,0.92857,1.0,0.0,1.0,1,16,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,16],[4,103,0.0388,0.9107,0.17408,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,21],[8,103,0.0777,0.84821,0.26229,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0,4,0,21],[12,103,0.1165,0.7991,0.26452,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,7,0,15],[16,103,0.1553,0.8124,0.24362,0.71429,0.85714,1.0,0.14,1.0,0,14,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,0,9,0,14],[20,103,0.1942,0.82143,0.25254,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,1,0,0,6,0,0,1,0,0,4,0,18],[24,103,0.233,0.84822,0.21706,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,0,7,0,17],[28,103,0.2718,0.80803,0.27574,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,18],[32,103,0.3107,0.79902,0.27421,0.67857,0.92857,1.0,0.14,1.0,0,16,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,2,0,0,6,0,16],[36,103,0.3495,0.81241,0.26373,0.67857,0.92857,1.0,0.14,1.0,0,16,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,1,0,0,7,0,16],[40,103,0.3883,0.74107,0.2911,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,2,0,0,2,0,0,3,0,0,4,0,0,5,0,13],[44,103,0.4272,0.69643,0.31693,0.39286,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,5,0,0,2,0,0,3,0,0,1,0,0,6,0,12],[48,103,0.466,0.89732,0.16458,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,8,0,19],[52,103,0.5049,0.84374,0.2212,0.7143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,0,0,6,0,17],[56,103,0.5437,0.86606,0.15544,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,12,0,14],[60,103,0.5825,0.83479,0.1754,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,10,0,13],[64,103,0.6214,0.83479,0.17168,0.82132,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,14,0,10],[68,103,0.6602,0.80802,0.18424,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,9,0,11],[72,103,0.699,0.82588,0.18117,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,8,0,13],[76,103,0.7379,0.83482,0.16409,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,10,0,12],[80,103,0.7767,0.82141,0.1713,0.57143,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,11,0,11],[84,103,0.8155,0.88393,0.18363,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,18],[88,103,0.8544,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[92,103,0.8932,0.92857,0.11845,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,20],[96,103,0.932,0.91071,0.13243,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,9,0,19],[100,103,0.9709,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[103,103,1.0,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15]]},{"b":5,"e":0.14286,"k":"falling","v":0.23188,"x":0.95536,"p":[[0,125,0.0,0.88838,0.09268,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,11],[4,125,0.032,0.95536,0.1448,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[8,125,0.064,0.86607,0.25985,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,5,0,22],[12,125,0.096,0.91517,0.1377,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[16,125,0.128,0.88392,0.19047,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,3,0,21],[20,125,0.16,0.87497,0.23355,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,0,2,0,23],[24,125,0.192,0.85256,0.23581,0.85714,1.0,1.0,0.14,1.0,0,18,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,18],[28,125,0.224,0.83481,0.2086,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,6,0,16],[32,125,0.256,0.82589,0.2618,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,6,0,18],[36,125,0.288,0.88392,0.22142,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,22],[40,125,0.32,0.81696,0.27487,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0,7,0,17],[44,125,0.352,0.79464,0.26229,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,0,0,0,2,0,0,3,0,0,8,0,14],[48,125,0.384,0.84372,0.24056,0.71429,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,0,4,0,19],[52,125,0.416,0.81249,0.24075,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,1,0,0,7,0,15],[56,125,0.448,0.76786,0.26904,0.53572,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,2,0,0,6,0,14],[60,125,0.48,0.74097,0.29779,0.53539,0.85714,1.0,0.14,1.0,0,14,0,0,0,3,0,0,2,0,0,3,0,0,2,0,0,4,0,0,4,0,14],[64,125,0.512,0.77221,0.28337,0.57143,0.85714,1.0,0.14,1.0,0,14,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,0,0,0,8,0,14],[68,125,0.544,0.81248,0.2156,0.67857,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,7,0,14],[72,125,0.576,0.80356,0.23624,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,3,0,0,5,0,15],[76,125,0.608,0.79464,0.25489,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,6,0,0,4,0,15],[80,125,0.64,0.79464,0.2549,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,9,0,13],[84,125,0.672,0.80804,0.23854,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,4,0,0,7,0,14],[88,125,0.704,0.74552,0.30668,0.5354,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,3,0,0,2,0,0,3,0,0,0,0,0,7,0,14],[92,125,0.736,0.80355,0.23353,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,0,8,0,13],[96,125,0.768,0.80802,0.23039,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,4,0,0,3,0,16],[100,125,0.8,0.82588,0.20121,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,2,0,17],[104,125,0.832,0.73214,0.32093,0.53569,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,3,0,0,1,0,0,3,0,0,2,0,0,4,0,15],[108,125,0.864,0.74107,0.28221,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,1,0,0,2,0,0,6,0,0,1,0,0,7,0,12],[112,125,0.896,0.62936,0.30708,0.42857,0.57143,1.0,0.14,1.0,0,9,0,0,0,5,0,0,2,0,0,4,0,0,6,0,0,3,0,0,3,0,9],[116,125,0.928,0.58036,0.36932,0.14286,0.57143,1.0,0.14286,1.0,0,11,0,0,0,9,0,0,5,0,0,1,0,0,2,0,0,1,0,0,3,0,11],[120,125,0.96,0.36607,0.23402,0.25,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,8,0,0,13,0,0,5,0,0,2,0,0,1,0,0,1,0,2],[124,125,0.992,0.36589,0.29881,0.14286,0.28571,0.42858,0.14,1.0,0,5,0,0,0,13,0,0,10,0,0,3,0,0,0,0,0,1,0,0,0,0,5],[125,125,1.0,0.23188,0.1548,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,21,0,0,6,0,0,3,0,0,0,0,0,2,0,0,0,0,0]]}]},{"i":"f42050f6b93f2be0","q":"**2.** Let  $n \\geq 3$  be an integer. Prove that for all integers  $k$ , with  $1 \\leq k \\leq \\binom{n}{2}$ , there exists a set  $A$  with  $n$  distinct positive integer elements such that the set  $B = \\{\\gcd(x, y): x, y \\in A, x \\neq y \\}$  (gotten from the greatest common divisor of all pairs of distinct elements from  $A$ ) contains exactly  $k$  distinct elements.","t":[{"b":4,"e":0.71429,"k":"rising","v":0.2142,"x":0.37499,"p":[[0,22,0.0,0.2142,0.175,0.14286,0.14286,0.14286,0.0,0.57143,3,0,3,3,0,22,0,0,0,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[4,22,0.1818,0.26784,0.20746,0.14286,0.14286,0.46418,0.14286,0.85714,0,0,0,0,0,23,0,0,0,0,0,1,0,0,7,0,0,0,0,0,1,0,0],[8,22,0.3636,0.28568,0.21423,0.14286,0.14286,0.571,0.14286,0.71429,0,0,0,0,0,22,0,0,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0],[12,22,0.5455,0.23212,0.16263,0.14286,0.14286,0.17857,0.14286,0.57143,0,0,0,0,0,24,0,0,1,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[16,22,0.7273,0.30346,0.21057,0.14286,0.14286,0.57143,0.14,0.71429,0,0,0,0,0,20,0,0,0,0,0,1,0,0,10,0,0,1,0,0,0,0,0],[20,22,0.9091,0.27675,0.19859,0.14286,0.14286,0.571,0.14286,0.57143,0,0,0,0,0,22,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0],[22,22,1.0,0.37499,0.25441,0.14286,0.14286,0.57143,0.14286,0.85714,0,0,0,0,0,17,0,0,0,0,0,0,0,0,9,0,0,5,0,0,1,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.18284,"x":0.24554,"p":[[0,26,0.0,0.23213,0.17403,0.14286,0.14286,0.32142,0.0,0.57143,2,0,1,2,0,21,0,0,1,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[4,26,0.1538,0.22759,0.17811,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0],[8,26,0.3077,0.20981,0.15558,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,27,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[12,26,0.4615,0.23214,0.18123,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,24,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[16,26,0.6154,0.24554,0.17941,0.14286,0.14286,0.21429,0.14286,0.57143,0,0,0,0,0,24,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[20,26,0.7692,0.18295,0.12495,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[24,26,0.9231,0.18284,0.12994,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[26,26,1.0,0.2141,0.15157,0.14286,0.14286,0.14292,0.14,0.57143,0,0,0,0,0,26,0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"24a8febb2bc44db2","q":"Find all functions  $ f:\\mathbb{R^{\\ast }}\\rightarrow \\mathbb{ R^{\\ast }}$  satisfying  $f(\\frac{f(x)}{f(y)})=\\frac{1}{y}f(f(x))$  for all  $x,y\\in \\mathbb{R^{\\ast }}$ and are strictly monotone in  $(0,+\\infty  )$","t":[{"b":3,"e":1.0,"k":"flat","v":0.81696,"x":0.98661,"p":[[0,44,0.0,0.81696,0.24804,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,16],[4,44,0.0909,0.9375,0.14698,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,26],[8,44,0.1818,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,44,0.2727,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[16,44,0.3636,0.91518,0.21975,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[20,44,0.4545,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[24,44,0.5455,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,44,0.6364,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[32,44,0.7273,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,44,0.8182,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[40,44,0.9091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,44,1.0,0.95536,0.14033,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,28]]},{"b":5,"e":0.71429,"k":"flat","v":0.83481,"x":0.98661,"p":[[0,174,0.0,0.87946,0.22047,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,22],[4,174,0.023,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,174,0.046,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[12,174,0.069,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,174,0.092,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[20,174,0.1149,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,174,0.1379,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[28,174,0.1609,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[32,174,0.1839,0.95534,0.09745,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[36,174,0.2069,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[40,174,0.2299,0.91964,0.15126,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,22],[44,174,0.2529,0.94643,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[48,174,0.2759,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[52,174,0.2989,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[56,174,0.3218,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[60,174,0.3448,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[64,174,0.3678,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,174,0.3908,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[72,174,0.4138,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[76,174,0.4368,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[80,174,0.4598,0.96428,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[84,174,0.4828,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[88,174,0.5057,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[92,174,0.5287,0.94196,0.11214,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,22],[96,174,0.5517,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[100,174,0.5747,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[104,174,0.5977,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[108,174,0.6207,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[112,174,0.6437,0.90178,0.18363,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,18],[116,174,0.6667,0.91518,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16],[120,174,0.6897,0.88392,0.14914,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,14,0,14],[124,174,0.7126,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[128,174,0.7356,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[132,174,0.7586,0.88392,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,9],[136,174,0.7816,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[140,174,0.8046,0.92411,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[144,174,0.8276,0.91964,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[148,174,0.8506,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[152,174,0.8736,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[156,174,0.8966,0.91518,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,17],[160,174,0.9195,0.89732,0.1439,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,12,0,16],[164,174,0.9425,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[168,174,0.9655,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[172,174,0.9885,0.88839,0.11143,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,19,0,11],[174,174,1.0,0.83481,0.13415,0.82132,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,18,0,6]]}]},{"i":"e699e2564ea54a82","q":"Are there positive integers  $a, b$  with  $b \\ge 2$  such that  $2^a + 1$  is divisible by  $2^b - 1$ ?","t":[{"b":0,"e":1.0,"k":"rising","v":0.8214,"x":1.0,"p":[[0,25,0.0,0.8214,0.22871,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,4,0,0,5,0,16],[4,25,0.16,0.89732,0.18292,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,4,0,22],[8,25,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,25,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.14286,"k":"falling","v":0.33034,"x":0.91518,"p":[[0,36,0.0,0.75001,0.28793,0.4286,1.0,1.0,0.143,1.0,0,17,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,0,0,0,1,0,17],[4,36,0.1111,0.91518,0.14664,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[8,36,0.2222,0.41963,0.26229,0.24999,0.42857,0.4642,0.14286,1.0,0,3,0,0,0,8,0,0,7,0,0,9,0,0,3,0,0,0,0,0,2,0,3],[12,36,0.3333,0.37947,0.17717,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,5,0,0,15,0,0,4,0,0,0,0,0,0,0,1],[16,36,0.4444,0.36606,0.17472,0.28571,0.28571,0.4286,0.14286,0.71429,0,0,0,0,0,6,0,0,12,0,0,8,0,0,2,0,0,4,0,0,0,0,0],[20,36,0.5556,0.3348,0.13646,0.24999,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,8,0,0,8,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[24,36,0.6667,0.37051,0.11765,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,11,0,0,14,0,0,4,0,0,0,0,0,0,0,0],[28,36,0.7778,0.35712,0.14722,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,8,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[32,36,0.8889,0.33034,0.12076,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,12,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[36,36,1.0,0.33929,0.13717,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,12,0,0,11,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"41e3fe60cd525f53","q":"Find all polynomials  $P(x)$  of the smallest possible degree with the following properties:\n\n(i) The leading coefficient is  $200$ ;\n(ii) The coefficient at the smallest non-vanishing power is  $2$ ;\n(iii) The sum of all the coefficients is  $4$ ;\n(iv)  $P(-1) = 0, P(2) = 6, P(3) = 8$ .","t":[{"b":0,"e":0.14286,"k":"falling","v":0.16518,"x":0.93304,"p":[[0,51,0.0,0.93304,0.18552,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[4,51,0.0784,0.64731,0.35712,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,6,0,0,2,0,0,3,0,0,2,0,0,3,0,0,2,0,13],[8,51,0.1569,0.6875,0.33586,0.39286,0.85714,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,3,0,0,3,0,0,2,0,0,1,0,0,5,0,13],[12,51,0.2353,0.64732,0.35172,0.28571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,7,0,0,2,0,0,3,0,0,4,0,0,0,0,0,3,0,13],[16,51,0.3137,0.42411,0.36331,0.14286,0.14286,0.75,0.0,1.0,1,7,0,1,0,16,0,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,7],[20,51,0.3922,0.48214,0.36553,0.14286,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,14,0,0,4,0,0,0,0,0,2,0,0,1,0,0,4,0,7],[24,51,0.4706,0.44643,0.3531,0.14286,0.28571,0.85714,0.0,1.0,2,5,0,2,0,11,0,0,5,0,0,3,0,0,0,0,0,1,0,0,5,0,5],[28,51,0.549,0.29018,0.26603,0.14286,0.14286,0.28571,0.14286,1.0,0,2,0,0,0,22,0,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[32,51,0.6275,0.42857,0.35535,0.14286,0.28571,0.85714,0.0,1.0,1,7,0,1,0,13,0,0,6,0,0,2,0,0,1,0,0,0,0,0,2,0,7],[36,51,0.7059,0.25446,0.25935,0.14286,0.14286,0.2857,0.0,1.0,3,2,0,3,0,19,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[40,51,0.7843,0.41518,0.35058,0.14286,0.14286,0.75,0.14286,1.0,0,6,0,0,0,18,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,6],[44,51,0.8627,0.20089,0.18161,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[48,51,0.9412,0.22768,0.20782,0.14286,0.14286,0.2857,0.14286,1.0,0,2,0,0,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[51,51,1.0,0.16518,0.08073,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,24,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"falling","v":0.23661,"x":0.92857,"p":[[0,75,0.0,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[4,75,0.0533,0.80357,0.28291,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,3,0,0,2,0,19],[8,75,0.1067,0.76784,0.30253,0.67846,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,0,10,0,13],[12,75,0.16,0.66518,0.34921,0.35714,0.78571,1.0,0.0,1.0,1,11,0,1,0,7,0,0,0,0,0,2,0,0,1,0,0,5,0,0,5,0,11],[16,75,0.2133,0.60268,0.34944,0.28571,0.64286,1.0,0.0,1.0,1,10,0,1,0,6,0,0,4,0,0,2,0,0,3,0,0,3,0,0,3,0,10],[20,75,0.2667,0.62054,0.38401,0.14286,0.78571,1.0,0.0,1.0,1,13,0,1,0,9,0,0,1,0,0,3,0,0,0,0,0,2,0,0,3,0,13],[24,75,0.32,0.55357,0.37754,0.14286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,11,0,0,4,0,0,2,0,0,0,0,0,1,0,0,4,0,10],[28,75,0.3733,0.6875,0.33586,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,2,0,0,5,0,0,1,0,0,2,0,0,3,0,14],[32,75,0.4267,0.65616,0.3368,0.28571,0.78571,1.0,0.14,1.0,0,11,0,0,0,5,0,0,5,0,0,2,0,0,1,0,0,3,0,0,5,0,11],[36,75,0.48,0.54909,0.38649,0.14286,0.49979,1.0,0.0,1.0,2,11,0,2,0,10,0,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,11],[40,75,0.5333,0.50446,0.36593,0.14289,0.35714,1.0,0.0,1.0,2,9,0,2,0,7,0,0,7,0,0,4,0,0,0,0,0,1,0,0,2,0,9],[44,75,0.5867,0.54017,0.37241,0.14286,0.49979,1.0,0.0,1.0,2,10,0,2,0,8,0,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,10],[48,75,0.64,0.36607,0.31931,0.14286,0.28571,0.42858,0.0,1.0,3,5,0,3,0,10,0,0,9,0,0,3,0,0,1,0,0,0,0,0,1,0,5],[52,75,0.6933,0.50447,0.37793,0.14286,0.28571,1.0,0.0,1.0,1,9,0,1,0,11,0,0,5,0,0,2,0,0,0,0,0,1,0,0,3,0,9],[56,75,0.7467,0.49999,0.3677,0.14286,0.28571,1.0,0.0,1.0,1,10,0,1,0,8,0,0,9,0,0,2,0,0,1,0,0,0,0,0,1,0,10],[60,75,0.8,0.57143,0.37287,0.14286,0.64286,0.89286,0.0,1.0,2,8,0,2,0,8,0,0,3,0,0,1,0,0,2,0,0,1,0,0,7,0,8],[64,75,0.8533,0.54909,0.36789,0.14286,0.42836,1.0,0.14286,1.0,0,9,0,0,0,10,0,0,6,0,0,0,0,0,1,0,0,2,0,0,4,0,9],[68,75,0.9067,0.41072,0.33645,0.14286,0.28571,0.57143,0.0,1.0,1,6,0,1,0,14,0,0,4,0,0,2,0,0,4,0,0,0,0,0,1,0,6],[72,75,0.96,0.33482,0.33237,0.14286,0.14286,0.46431,0.0,1.0,5,4,0,5,0,14,0,0,2,0,0,3,0,0,2,0,0,0,0,0,2,0,4],[75,75,1.0,0.23661,0.17353,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,18,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"2ec87f87548dc51e","q":"Find the maximum positive integer  $k$  such that for any positive integers  $m,n$  such that  $m^3+n^3>(m+n)^2$ , we have  $$ m^3+n^3\\geq (m+n)^2+k $$  *Proposed by Dorlir Ahmeti, Albania*","t":[{"b":2,"e":0.85714,"k":"flat","v":0.52232,"x":0.80804,"p":[[0,57,0.0,0.67409,0.34485,0.28571,0.78571,1.0,0.14286,1.0,0,14,0,0,0,6,0,0,3,0,0,1,0,0,4,0,0,2,0,0,2,0,14],[4,57,0.0702,0.58034,0.3272,0.24999,0.64271,0.89275,0.14286,1.0,0,8,0,0,0,8,0,0,3,0,0,1,0,0,4,0,0,7,0,0,1,0,8],[8,57,0.1404,0.59374,0.30117,0.28571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,6,0,0,4,0,0,2,0,0,1,0,0,10,0,0,4,0,5],[12,57,0.2105,0.59375,0.32948,0.25,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,3,0,0,0,0,0,5,0,0,4,0,0,5,0,7],[16,57,0.2807,0.625,0.32684,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,2,0,0,2,0,0,3,0,0,6,0,0,3,0,9],[20,57,0.3509,0.68303,0.27603,0.53571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,4,0,0,3,0,10],[24,57,0.4211,0.61598,0.31845,0.28571,0.57143,1.0,0.14,1.0,0,9,0,0,0,6,0,0,3,0,0,1,0,0,8,0,0,2,0,0,3,0,9],[28,57,0.4912,0.62945,0.30901,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,5,0,0,3,0,0,2,0,0,7,0,0,2,0,9],[32,57,0.5614,0.59366,0.27933,0.39286,0.57143,0.74996,0.14,1.0,0,5,0,0,0,5,0,0,3,0,0,2,0,0,7,0,0,7,0,0,3,0,5],[36,57,0.6316,0.6874,0.29562,0.39286,0.71429,1.0,0.14,1.0,0,10,0,0,0,2,0,0,6,0,0,1,0,0,3,0,0,5,0,0,5,0,10],[40,57,0.7018,0.52232,0.24643,0.28571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,4,0,0,2,0,0,7,0,0,10,0,0,2,0,1],[44,57,0.7719,0.55804,0.27976,0.28571,0.71429,0.75,0.14286,1.0,0,2,0,0,0,5,0,0,6,0,0,3,0,0,1,0,0,9,0,0,6,0,2],[48,57,0.8421,0.65625,0.21086,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,11,0,0,7,0,2],[52,57,0.9123,0.70536,0.16728,0.67857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,12,0,0,12,0,0],[56,57,0.9825,0.80804,0.10479,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,14,0,4],[57,57,1.0,0.78558,0.08759,0.71429,0.78571,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,15,0,1]]},{"b":3,"e":0.57143,"k":"flat","v":0.45089,"x":0.74553,"p":[[0,48,0.0,0.63839,0.35172,0.28571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,7,0,0,3,0,0,2,0,0,3,0,0,3,0,0,1,0,13],[4,48,0.0833,0.45089,0.28596,0.14286,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,10,0,0,5,0,0,2,0,0,9,0,0,0,0,0,3,0,3],[8,48,0.1667,0.54464,0.32031,0.24999,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,8,0,0,5,0,0,0,0,0,6,0,0,4,0,0,3,0,6],[12,48,0.25,0.5,0.35714,0.14286,0.28571,0.89286,0.14286,1.0,0,8,0,0,0,11,0,0,6,0,0,1,0,0,3,0,0,0,0,0,3,0,8],[16,48,0.3333,0.52668,0.31842,0.24999,0.57143,0.75,0.14,1.0,0,6,0,0,0,8,0,0,6,0,0,0,0,0,6,0,0,4,0,0,2,0,6],[20,48,0.4167,0.61161,0.32189,0.28571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,5,0,0,0,0,0,4,0,0,4,0,0,6,0,7],[24,48,0.5,0.65179,0.29001,0.53571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,5,0,0,2,0,0,1,0,0,5,0,0,6,0,0,7,0,6],[28,48,0.5833,0.62054,0.25657,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,3,0,0,7,0,0,3,0,0,9,0,3],[32,48,0.6667,0.65625,0.2547,0.53571,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0,12,0,0,6,0,4],[36,48,0.75,0.60712,0.27433,0.49968,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,5,0,0,3,0,0,0,0,0,8,0,0,7,0,0,5,0,4],[40,48,0.8333,0.74553,0.25688,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,7,0,0,7,0,10],[44,48,0.9167,0.71427,0.17497,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,11,0,0,10,0,2],[48,48,1.0,0.71427,0.13833,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,10,0,0,12,0,0]]}]},{"i":"c0955976f8f56b9f","q":"Find all positive integers  $n$  for which exist three nonzero integers  $x, y, z$  such that  $x+y+z=0$  and:\n\\[\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{n}\\]","t":[{"b":6,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,67,0.0,0.91518,0.21975,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[4,67,0.0597,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[8,67,0.1194,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[12,67,0.1791,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,67,0.2388,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,67,0.2985,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,67,0.3582,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,67,0.4179,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,67,0.4776,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,67,0.5373,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,67,0.597,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,67,0.6567,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,67,0.7164,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,67,0.7761,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,67,0.8358,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,67,0.8955,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,67,0.9552,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[67,67,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,70,0.0,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,27],[4,70,0.0571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,70,0.1143,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[12,70,0.1714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,70,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,70,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,70,0.3429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,70,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,70,0.4571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,70,0.5143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,70,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,70,0.6286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,70,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,70,0.7429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,70,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,70,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,70,0.9143,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,70,0.9714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[70,70,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d726010c35271a7e","q":"Find all pairs of natural numbers  $(a,n)$ ,  $a\\geq n \\geq 2,$  for which  $a^n+a-2$  is a power of  $2$ .","t":[{"b":3,"e":0.14286,"k":"falling","v":0.29019,"x":0.94196,"p":[[0,109,0.0,0.60713,0.3093,0.2857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,7,0,0,3,0,0,0,0,0,5,0,0,2,0,0,12,0,3],[4,109,0.0367,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[8,109,0.0734,0.83482,0.23987,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,5,0,0,7,0,16],[12,109,0.1101,0.88392,0.20652,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,17],[16,109,0.1468,0.83482,0.25281,0.85711,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,0,9,0,16],[20,109,0.1835,0.83928,0.17768,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,9,0,13],[24,109,0.2202,0.82589,0.2618,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,0,5,0,18],[28,109,0.2569,0.85268,0.24867,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,6,0,19],[32,109,0.2936,0.84373,0.23788,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,18],[36,109,0.3303,0.79464,0.27879,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,4,0,0,7,0,15],[40,109,0.367,0.78562,0.29034,0.71429,0.85714,1.0,0.14,1.0,0,15,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,5,0,0,6,0,15],[44,109,0.4037,0.7991,0.32115,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,20],[48,109,0.4404,0.75892,0.31224,0.64286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0,4,0,0,4,0,16],[52,109,0.4771,0.72767,0.31412,0.60714,0.85714,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,4,0,0,0,0,0,0,0,0,5,0,0,7,0,12],[56,109,0.5138,0.77679,0.30501,0.82143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0,10,0,14],[60,109,0.5505,0.74999,0.33694,0.53539,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,18],[64,109,0.5872,0.75445,0.28623,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,5,0,0,9,0,11],[68,109,0.6239,0.61604,0.35253,0.1429,0.78564,1.0,0.14286,1.0,0,9,0,0,0,9,0,0,1,0,0,3,0,0,2,0,0,1,0,0,7,0,9],[72,109,0.6606,0.79911,0.29636,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,7,0,17],[76,109,0.6972,0.67409,0.31387,0.39286,0.85707,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,4,0,0,9,0,8],[80,109,0.7339,0.75447,0.32778,0.5,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,4,0,0,0,0,0,1,0,0,1,0,0,6,0,16],[84,109,0.7706,0.60267,0.3189,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,3,0,0,4,0,0,4,0,0,3,0,0,4,0,8],[88,109,0.8073,0.57143,0.35534,0.1429,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,10,0,0,2,0,0,3,0,0,2,0,0,1,0,0,6,0,8],[92,109,0.844,0.54902,0.36972,0.14286,0.4286,1.0,0.14,1.0,0,11,0,0,0,10,0,0,4,0,0,4,0,0,0,0,0,2,0,0,1,0,11],[96,109,0.8807,0.71415,0.3312,0.39287,0.85714,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,3,0,0,1,0,0,2,0,0,1,0,0,7,0,13],[100,109,0.9174,0.7098,0.29556,0.42857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,3,0,0,3,0,0,2,0,0,3,0,0,8,0,10],[104,109,0.9541,0.39277,0.25514,0.14286,0.28571,0.57143,0.14,1.0,0,1,0,0,0,11,0,0,7,0,0,4,0,0,3,0,0,4,0,0,2,0,1],[108,109,0.9908,0.30348,0.24425,0.14286,0.14286,0.42857,0.14,1.0,0,2,0,0,0,18,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,2],[109,109,1.0,0.29019,0.20972,0.14286,0.1429,0.42858,0.14286,0.85714,0,0,0,0,0,18,0,0,5,0,0,4,0,0,1,0,0,3,0,0,1,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.63837,"x":0.96875,"p":[[0,142,0.0,0.63837,0.28344,0.28571,0.78564,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,7,0,0,0,0,0,2,0,0,4,0,0,14,0,2],[4,142,0.0282,0.92857,0.16751,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[8,142,0.0563,0.92855,0.16755,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,23],[12,142,0.0845,0.89062,0.20281,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,1,0,7,0,20],[16,142,0.1127,0.89285,0.09449,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,12],[20,142,0.1408,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,142,0.169,0.94195,0.10017,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[28,142,0.1972,0.87946,0.11356,0.82132,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,13],[32,142,0.2254,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[36,142,0.2535,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[40,142,0.2817,0.93301,0.12368,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[44,142,0.3099,0.89731,0.1439,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,11,0,17],[48,142,0.338,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,20],[52,142,0.3662,0.875,0.14173,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,8,0,15],[56,142,0.3944,0.90625,0.13175,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,12,0,17],[60,142,0.4225,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[64,142,0.4507,0.89285,0.12372,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,14,0,14],[68,142,0.4789,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[72,142,0.507,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[76,142,0.5352,0.91962,0.11267,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,12,0,18],[80,142,0.5634,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[84,142,0.5915,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[88,142,0.6197,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[92,142,0.6479,0.91739,0.11172,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,1,18],[96,142,0.6761,0.92409,0.10709,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[100,142,0.7042,0.89284,0.11849,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,14],[104,142,0.7324,0.9241,0.13355,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,21],[108,142,0.7606,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[112,142,0.7887,0.92409,0.12368,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21],[116,142,0.8169,0.90625,0.11633,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,17],[120,142,0.8451,0.92856,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[124,142,0.8732,0.90176,0.11542,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,16],[128,142,0.9014,0.87052,0.13536,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,9,0,14],[132,142,0.9296,0.89731,0.12492,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,13,0,15],[136,142,0.9577,0.87496,0.1325,0.85714,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,13,0,13],[140,142,0.9859,0.93302,0.10093,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[142,142,1.0,0.86607,0.11811,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,18,0,9]]}]},{"i":"d874871044edd8c7","q":"Find the number of  $(a_1,a_2, ... ,a_{2014})$  permutations of the  $(1,2, . . . ,2014)$  such that, for all  $1\\leq i<j\\leq2014$ ,  $i+a_i \\leq j+a_j$ .","t":[{"b":0,"e":0.14286,"k":"volatile","v":0.08482,"x":0.90179,"p":[[0,23,0.0,0.74554,0.23072,0.57143,0.57143,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,0,0,0,0,0,14],[4,23,0.1739,0.83482,0.20858,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,5,0,17],[8,23,0.3478,0.88839,0.19475,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0,2,0,23],[12,23,0.5217,0.90179,0.20652,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,1,0,25],[16,23,0.6957,0.61607,0.4,0.14286,0.57143,1.0,0.0,1.0,4,15,0,4,0,6,0,0,0,0,0,1,0,0,6,0,0,0,0,0,0,0,15],[20,23,0.8696,0.15179,0.22851,0.0,0.14286,0.14286,0.0,1.0,10,2,0,10,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[23,23,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"falling","v":0.28125,"x":0.86607,"p":[[0,70,0.0,0.83929,0.21943,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,1,0,20],[4,70,0.0571,0.86607,0.22286,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,0,3,0,21],[8,70,0.1143,0.82143,0.26726,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,0,0,0,8,0,0,1,0,0,1,0,20],[12,70,0.1714,0.85254,0.23559,0.67536,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,1,0,0,2,0,21],[16,70,0.2286,0.58482,0.41243,0.14286,0.57143,1.0,0.0,1.0,4,14,0,4,0,8,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,14],[20,70,0.2857,0.61161,0.42892,0.14286,1.0,1.0,0.0,1.0,3,17,0,3,0,10,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,17],[24,70,0.3429,0.34375,0.39263,0.14286,0.14286,0.67857,0.0,1.0,7,8,0,7,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,8],[28,70,0.4,0.55357,0.44857,0.14286,0.57143,1.0,0.0,1.0,4,16,0,4,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[32,70,0.4571,0.49554,0.42104,0.14286,0.14286,1.0,0.0,1.0,4,10,0,4,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,10],[36,70,0.5143,0.46875,0.4421,0.14286,0.14286,1.0,0.0,1.0,7,12,0,7,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,12],[40,70,0.5714,0.38393,0.41563,0.0,0.14286,1.0,0.0,1.0,9,9,0,9,0,11,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,9],[44,70,0.6286,0.36607,0.40396,0.14286,0.14286,0.85714,0.0,1.0,7,7,0,7,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,7],[48,70,0.6857,0.59375,0.44047,0.14286,0.85714,1.0,0.0,1.0,6,16,0,6,0,7,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,16],[52,70,0.7429,0.58036,0.44022,0.14286,0.85714,1.0,0.0,1.0,7,14,0,7,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,14],[56,70,0.8,0.50446,0.44605,0.0,0.5,1.0,0.0,1.0,10,13,0,10,0,4,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,13],[60,70,0.8571,0.54464,0.4141,0.14286,0.57143,1.0,0.0,1.0,5,13,0,5,0,8,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,13],[64,70,0.9143,0.28125,0.38546,0.0,0.14286,0.25,0.0,1.0,13,6,0,13,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,6],[68,70,0.9714,0.66071,0.36025,0.28571,0.85714,1.0,0.0,1.0,1,13,0,1,0,6,0,0,3,0,0,0,0,0,4,0,0,1,0,0,4,0,13],[70,70,1.0,0.37946,0.41743,0.0,0.14286,1.0,0.0,1.0,10,9,0,10,0,10,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9]]}]},{"i":"96e1d9f962fcd921","q":"Find the maximum possible length of a sequence consisting of non-zero integers, in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.27678,"x":0.70089,"p":[[0,28,0.0,0.39732,0.17762,0.28571,0.35714,0.42857,0.0,0.71429,1,0,1,1,0,1,0,0,14,0,0,10,0,0,0,0,0,6,0,0,0,0,0],[4,28,0.1429,0.70089,0.1488,0.57143,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,12,0,0,1,0,5],[8,28,0.2857,0.61607,0.23266,0.53572,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,0,0,0,4,0,0,6,0,0,13,0,0,2,0,3],[12,28,0.4286,0.55804,0.27516,0.39286,0.57143,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,1,0,0,2,0,0,7,0,0,9,0,0,4,0,2],[16,28,0.5714,0.45981,0.29823,0.14286,0.57143,0.71429,0.0,1.0,3,1,0,3,0,8,0,0,3,0,0,0,0,0,5,0,0,10,0,0,2,0,1],[20,28,0.7143,0.41964,0.24467,0.2857,0.35714,0.57143,0.0,0.85714,3,0,0,3,0,3,0,0,10,0,0,2,0,0,7,0,0,5,0,0,2,0,0],[24,28,0.8571,0.3125,0.23538,0.14286,0.2857,0.57143,0.0,0.85714,4,0,0,4,0,11,0,0,5,0,0,3,0,0,6,0,0,2,0,0,1,0,0],[28,28,1.0,0.27678,0.07936,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.12045,"x":0.58481,"p":[[0,137,0.0,0.34821,0.17474,0.2857,0.28571,0.42857,0.0,0.71429,2,0,1,2,0,2,0,0,16,0,0,8,0,0,0,0,0,4,0,0,0,0,0],[4,137,0.0292,0.58481,0.23517,0.5713,0.57143,0.71429,0.0,1.0,1,3,1,1,0,3,0,0,1,0,0,2,0,0,11,0,0,11,0,0,0,0,3],[8,137,0.0584,0.54018,0.25439,0.28571,0.57143,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,4,0,0,0,0,0,12,0,0,8,0,0,0,0,3],[12,137,0.0876,0.44195,0.30589,0.14286,0.57121,0.71429,0.0,1.0,3,3,0,3,0,10,0,0,0,0,0,2,0,0,8,0,0,6,0,0,0,0,3],[16,137,0.1168,0.35705,0.30311,0.14286,0.14286,0.71429,0.0,1.0,3,2,1,3,0,15,0,0,2,0,0,0,0,0,3,0,0,7,0,0,0,0,2],[20,137,0.146,0.32143,0.30514,0.14286,0.14286,0.57143,0.0,1.0,6,2,3,6,0,14,0,0,0,0,0,0,0,0,6,0,0,4,0,0,0,0,2],[24,137,0.1752,0.29456,0.22004,0.14286,0.14286,0.57143,0.14,0.71429,0,0,0,0,0,21,0,0,1,0,0,0,0,0,7,0,0,3,0,0,0,0,0],[28,137,0.2044,0.15598,0.16508,0.105,0.14286,0.14286,0.0,0.71429,8,0,3,8,0,20,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[32,137,0.2336,0.27679,0.25489,0.14286,0.14286,0.46429,0.0,1.0,4,1,2,4,0,17,0,0,2,0,0,1,0,0,4,0,0,3,0,0,0,0,1],[36,137,0.2628,0.28561,0.24226,0.14286,0.14286,0.57111,0.0,0.71429,1,0,0,1,0,22,0,0,0,0,0,0,0,0,3,0,0,6,0,0,0,0,0],[40,137,0.292,0.16518,0.15612,0.14286,0.14286,0.14286,0.0,0.71429,5,0,2,5,0,24,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[44,137,0.3212,0.19625,0.16662,0.14286,0.14286,0.14286,0.0,0.71429,3,0,1,3,0,22,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[48,137,0.3504,0.20981,0.22009,0.14286,0.14286,0.14286,0.0,0.71429,7,0,2,7,0,18,0,0,1,0,0,0,0,0,3,0,0,3,0,0,0,0,0],[52,137,0.3796,0.26777,0.23081,0.14286,0.14286,0.42857,0.0,0.85714,4,0,1,4,0,16,0,0,3,0,0,2,0,0,4,0,0,2,0,0,1,0,0],[56,137,0.4088,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,137,0.438,0.22759,0.1952,0.14286,0.14286,0.14287,0.0,0.71429,1,0,0,1,0,24,0,0,2,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[64,137,0.4672,0.14732,0.09771,0.14286,0.14286,0.14286,0.0,0.57143,4,0,2,4,0,25,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,137,0.4964,0.14723,0.10999,0.14286,0.14286,0.14286,0.0,0.71429,3,0,1,3,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,137,0.5255,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,1,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,137,0.5547,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,137,0.5839,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,137,0.6131,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,2,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,137,0.6423,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,137,0.6715,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,3,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,137,0.7007,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,137,0.7299,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,137,0.7591,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,137,0.7883,0.13393,0.09407,0.14286,0.14286,0.14286,0.0,0.57143,5,0,2,5,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[112,137,0.8175,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,137,0.8467,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,137,0.8759,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,137,0.9051,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,3,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,137,0.9343,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.14286,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,137,0.9635,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,137,0.9927,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[137,137,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4e24441f09795347","q":"For a prime number  $p$ , let  $GL_2(\\mathbb{Z}/p\\mathbb{Z})$  be the group of invertible  $2 \\times 2$  matrices of residues modulo  $p$ , and let  $S_p$  be the symmetric group (the group of all permutations) on  $p$  elements. Show that there is no injective group homomorphism  $\\phi : GL_2(\\mathbb{Z}/p\\mathbb{Z}) \\rightarrow S_p$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.14062,"x":0.14286,"p":[[0,15,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.14062,0.01243,0.14286,0.14286,0.14286,0.07143,0.14286,0,0,0,0,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.1383,"x":0.14723,"p":[[0,12,0.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c43e29525d424a83","q":"Find all prime numbers  $p \\geq 3$  such that the number  $1 + k(p - 1)$  is prime for every positive integer  $k \\leq \\frac{p - 1}{2}$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13831,"x":0.165,"p":[[0,38,0.0,0.14715,0.05631,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.14277,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.165,0.07246,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.08482,"x":0.15607,"p":[[0,59,0.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.13384,0.07086,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.15607,0.0655,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,59,0.4068,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.125,0.09943,0.10714,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.09152,0.09147,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,1,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.13616,0.08444,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,1,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.13616,0.07651,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,1,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9c84b07c60933c5c","q":"Find, with proof, all functions $f: \\mathbb{R} \\backslash\\{0\\} \\rightarrow \\mathbb{R}$ such that\n\n$$\nf(x)^{2}-f(y) f(z)=x(x+y+z)(f(x)+f(y)+f(z))\n$$\n\nfor all real $x, y, z$ such that $x y z=1$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.08482,"x":0.87053,"p":[[0,135,0.0,0.52228,0.26391,0.28571,0.571,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,4,0,0,5,0,0,8,0,0,5,0,0,2,0,3],[4,135,0.0296,0.79015,0.23418,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,3,0,0,7,0,13],[8,135,0.0593,0.87053,0.22689,0.82132,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,21],[12,135,0.0889,0.49999,0.3977,0.0,0.57143,0.85714,0.0,1.0,9,7,0,9,0,3,0,0,1,0,0,1,0,0,3,0,0,5,0,0,3,0,7],[16,135,0.1185,0.51786,0.42969,0.0,0.57143,1.0,0.0,1.0,10,11,0,10,0,3,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,11],[20,135,0.1481,0.60267,0.35487,0.2857,0.64286,1.0,0.0,1.0,3,9,0,3,0,3,0,0,5,0,0,1,0,0,4,0,0,2,0,0,5,0,9],[24,135,0.1778,0.49551,0.41184,0.10714,0.57121,0.85714,0.0,1.0,8,7,0,8,0,6,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,7],[28,135,0.2074,0.55803,0.41705,0.14286,0.71429,1.0,0.0,1.0,7,10,1,7,0,4,0,0,3,0,0,0,0,0,0,0,0,3,0,0,5,0,10],[32,135,0.237,0.59375,0.34646,0.28571,0.71429,0.85714,0.0,1.0,4,5,0,4,0,3,0,0,2,0,0,3,0,0,3,0,0,2,0,0,10,0,5],[36,135,0.2667,0.63837,0.39767,0.24999,0.85707,1.0,0.0,1.0,6,12,0,6,0,2,0,0,2,0,0,0,0,0,3,0,0,1,0,0,6,0,12],[40,135,0.2963,0.62947,0.40855,0.14289,0.85714,1.0,0.0,1.0,7,12,0,7,0,2,0,0,1,0,0,1,0,0,0,0,0,4,0,0,5,0,12],[44,135,0.3259,0.59375,0.38649,0.2857,0.71429,1.0,0.0,1.0,6,11,0,6,0,1,0,0,4,0,0,2,0,0,1,0,0,5,0,0,2,0,11],[48,135,0.3556,0.5848,0.36659,0.14286,0.64286,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,0,0,0,2,0,0,5,0,0,2,0,0,7,0,7],[52,135,0.3852,0.55357,0.39569,0.10714,0.71429,0.85714,0.0,1.0,8,6,0,8,0,1,0,0,4,0,0,0,0,0,1,0,0,3,0,0,9,0,6],[56,135,0.4148,0.47765,0.37047,0.0,0.57121,0.74996,0.0,1.0,9,5,0,9,0,2,0,0,1,0,0,2,0,0,6,0,0,4,0,0,3,0,5],[60,135,0.4444,0.64285,0.40877,0.24999,0.85707,1.0,0.0,1.0,7,14,0,7,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,3,0,14],[64,135,0.4741,0.5,0.35892,0.14286,0.50001,0.85714,0.0,1.0,7,5,0,7,0,2,0,0,2,0,0,5,0,0,4,0,0,2,0,0,5,0,5],[68,135,0.5037,0.54455,0.41109,0.14286,0.57141,1.0,0.0,1.0,6,10,0,6,0,6,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,10],[72,135,0.5333,0.42854,0.39446,0.0,0.49979,0.85704,0.0,1.0,11,5,0,11,0,4,0,0,0,0,0,1,0,0,5,0,0,2,0,0,4,0,5],[76,135,0.563,0.49107,0.38949,0.14286,0.42857,0.89286,0.0,1.0,7,8,0,7,0,3,0,0,5,0,0,3,0,0,1,0,0,2,0,0,3,0,8],[80,135,0.5926,0.45087,0.40423,0.0,0.28571,0.85714,0.0,1.0,10,7,0,10,0,2,0,0,5,0,0,1,0,0,2,0,0,1,0,0,4,0,7],[84,135,0.6222,0.57143,0.41802,0.14286,0.64286,1.0,0.0,1.0,7,12,0,7,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,0,3,0,12],[88,135,0.6519,0.62491,0.41622,0.14214,0.85714,1.0,0.0,1.0,7,14,0,7,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,14],[92,135,0.6815,0.54463,0.33963,0.28571,0.57121,0.85714,0.0,1.0,4,6,0,4,0,3,0,0,3,0,0,5,0,0,3,0,0,4,0,0,4,0,6],[96,135,0.7111,0.49552,0.40561,0.0,0.57121,0.89286,0.0,1.0,9,8,0,9,0,3,0,0,2,0,0,1,0,0,3,0,0,3,0,0,3,0,8],[100,135,0.7407,0.375,0.37415,0.0,0.14286,0.71429,0.0,1.0,9,5,0,9,0,8,0,0,2,0,0,1,0,0,3,0,0,2,0,0,2,0,5],[104,135,0.7704,0.40177,0.3787,0.0,0.28571,0.75,0.0,1.0,9,5,0,9,0,6,0,0,3,0,0,1,0,0,3,0,0,2,0,0,3,0,5],[108,135,0.8,0.24107,0.29974,0.0,0.14286,0.1786,0.0,1.0,10,1,0,10,0,14,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,1],[112,135,0.8296,0.34375,0.3533,0.14286,0.14286,0.60714,0.0,1.0,6,4,0,6,0,14,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,4],[116,135,0.8593,0.26786,0.36025,0.0,0.14286,0.35714,0.0,1.0,14,4,1,14,0,8,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,4],[120,135,0.8889,0.25893,0.35254,0.0,0.14286,0.17857,0.0,1.0,12,4,0,12,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,4],[124,135,0.9185,0.23214,0.30878,0.0,0.14286,0.32143,0.0,1.0,14,1,0,14,0,9,0,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,1],[128,135,0.9481,0.14732,0.27078,0.0,0.0,0.14286,0.0,1.0,17,2,0,17,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[132,135,0.9778,0.19642,0.2829,0.0,0.0,0.2857,0.0,0.85714,17,0,1,17,0,6,0,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,0],[135,135,1.0,0.08482,0.18851,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.4286,"k":"flat","v":0.50441,"x":0.89284,"p":[[0,259,0.0,0.50441,0.29009,0.24999,0.571,0.71429,0.0,1.0,2,2,0,2,0,6,0,0,3,0,0,3,0,0,5,0,0,8,0,0,3,0,2],[4,259,0.0154,0.86159,0.18382,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,5,0,18],[8,259,0.0309,0.83482,0.23987,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,6,0,17],[12,259,0.0463,0.79241,0.24444,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,0,1,0,5,0,15],[16,259,0.0618,0.84372,0.19682,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,6,0,16],[20,259,0.0772,0.84375,0.22689,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,6,0,17],[24,259,0.0927,0.88838,0.17764,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,6,0,20],[28,259,0.1081,0.89284,0.1557,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,7,0,19],[32,259,0.1236,0.79015,0.23144,0.57142,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,2,0,0,2,0,16],[36,259,0.139,0.70086,0.3059,0.571,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,3,0,0,1,0,0,5,0,0,2,0,0,9,0,9],[40,259,0.1544,0.80799,0.25411,0.57143,0.928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4,0.66961,0.22143,0.571,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,1,0,0,6,0,6],[224,259,0.8649,0.77677,0.27185,0.5354,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,0,7,0,14],[228,259,0.8803,0.7232,0.1954,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,5,0,0,11,0,4],[232,259,0.8958,0.72319,0.25739,0.57132,0.78564,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,5,0,0,7,0,9],[236,259,0.9112,0.56248,0.26228,0.42857,0.42859,0.85704,0.0,1.0,1,3,0,1,0,1,0,0,4,0,0,11,0,0,4,0,0,1,0,0,7,0,3],[240,259,0.9266,0.61157,0.22084,0.42857,0.4998,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,16,0,0,5,0,0,2,0,0,4,0,5],[244,259,0.9421,0.57584,0.16163,0.42857,0.5712,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,11,0,0,11,0,0,5,0,0,3,0,1],[248,259,0.9575,0.54911,0.25281,0.42857,0.57143,0.75,0.0,1.0,1,2,0,1,0,3,0,0,1,0,0,10,0,0,7,0,0,2,0,0,6,0,2],[252,259,0.973,0.64285,0.26,0.42857,0.64286,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,3,0,0,9,0,0,3,0,0,5,0,0,4,0,7],[256,259,0.9884,0.64284,0.272,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,8,0,0,5,0,0,0,0,0,7,0,7],[259,259,1.0,0.53122,0.27254,0.39286,0.571,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,3,0,0,7,0,0,7,0,0,4,0,0,2,0,4]]}]},{"i":"b214b2bce92432f1","q":"For  $n \\geq 3$  find the eigenvalues (with their multiplicities) of the  $n \\times n$  matrix  $$ \\begin{bmatrix}\n1 & 0 & 1 & 0 & 0 & 0 & \\dots & \\dots & 0 & 0 \n0 & 2 & 0 & 1 & 0 & 0 & \\dots & \\dots & 0 & 0 \n1 & 0 & 2 & 0 & 1 & 0 & \\dots & \\dots & 0 & 0 \n0 & 1 & 0 & 2 & 0 & 1 & \\dots & \\dots & 0 & 0 \n0 & 0 & 1 & 0 & 2 & 0 & \\dots & \\dots & 0 & 0 \n0 & 0 & 0 & 1 & 0 & 2 & \\dots & \\dots & 0 & 0 \n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots &  & \\vdots & \\vdots \n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots &  & \\ddots & \\vdots & \\vdots \n0 & 0 & 0 & 0 & 0 & 0 & \\dots & \\dots & 2 & 0 \n0 & 0 & 0 & 0 & 0 & 0 & \\dots & \\dots & 0 & 1\n\\end{bmatrix} $$","t":[{"b":2,"e":1.0,"k":"rising","v":0.74998,"x":0.99554,"p":[[0,93,0.0,0.74998,0.26245,0.57143,0.85714,1.0,0.0,1.0,1,11,1,1,0,1,0,0,0,0,0,3,0,0,6,0,0,3,0,0,7,0,11],[4,93,0.043,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,93,0.086,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[12,93,0.129,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,93,0.172,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[20,93,0.2151,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[24,93,0.2581,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,93,0.3011,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,93,0.3441,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,93,0.3871,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[40,93,0.4301,0.9375,0.17472,1.0,1.0,1.0,0.143,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[44,93,0.4731,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,93,0.5161,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[52,93,0.5591,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,93,0.6022,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,93,0.6452,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,93,0.6882,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[68,93,0.7312,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[72,93,0.7742,0.94641,0.1505,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[76,93,0.8172,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,93,0.8602,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[84,93,0.9032,0.95536,0.1357,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[88,93,0.9462,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[92,93,0.9892,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[93,93,1.0,0.91069,0.15471,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,23]]},{"b":4,"e":1.0,"k":"rising","v":0.75446,"x":0.97768,"p":[[0,52,0.0,0.75446,0.28623,0.67857,0.85714,1.0,0.0,1.0,2,10,1,2,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,0,11,0,10],[4,52,0.0769,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,52,0.1538,0.94643,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[12,52,0.2308,0.89732,0.18977,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,23],[16,52,0.3077,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[20,52,0.3846,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[24,52,0.4615,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[28,52,0.5385,0.91518,0.23381,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,27],[32,52,0.6154,0.93304,0.19228,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[36,52,0.6923,0.89732,0.18638,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,22],[40,52,0.7692,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[44,52,0.8462,0.9464,0.1276,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[48,52,0.9231,0.9107,0.2075,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[52,52,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]}]},{"i":"20675676d3312651","q":"For each positive integer  $n$ , define  $V_n=\\lfloor 2^n\\sqrt{2020}\\rfloor+\\lfloor 2^n\\sqrt{2021}\\rfloor$ . Prove that, in the sequence  $V_1,V_2,\\ldots,$  there are infinitely many odd integers, as well as infinitely many even integers.\n\n*Remark.*  $\\lfloor x\\rfloor$  is the largest integer that does not exceed the real number  $x$ .","t":[{"b":0,"e":0.2857,"k":"flat","v":0.51786,"x":0.81695,"p":[[0,26,0.0,0.51786,0.4414,0.0,0.5,1.0,0.0,1.0,10,11,0,10,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,11],[4,26,0.1538,0.81695,0.30144,0.82143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,5,0,19],[8,26,0.3077,0.72768,0.32608,0.57143,0.85714,1.0,0.0,1.0,4,12,0,4,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,6,0,12],[12,26,0.4615,0.66518,0.34554,0.39286,0.78571,1.0,0.0,1.0,4,10,0,4,0,0,0,0,4,0,0,2,0,0,1,0,0,5,0,0,6,0,10],[16,26,0.6154,0.6875,0.31428,0.39288,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,5,0,0,1,0,0,2,0,0,6,0,0,4,0,11],[20,26,0.7692,0.60713,0.32927,0.28571,0.71429,0.85714,0.0,1.0,4,4,0,4,0,1,0,0,5,0,0,0,0,0,2,0,0,7,0,0,9,0,4],[24,26,0.9231,0.71874,0.28004,0.57143,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,3,0,0,2,0,0,9,0,0,2,0,0,2,0,13],[26,26,1.0,0.65178,0.2922,0.39286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,6,0,0,3,0,0,2,0,0,9,0,0,0,0,10]]},{"b":7,"e":0.0,"k":"volatile","v":0.27232,"x":0.77232,"p":[[0,15,0.0,0.58036,0.44455,0.0,0.78571,1.0,0.0,1.0,9,15,0,9,0,1,0,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,15],[4,15,0.2667,0.6071,0.36944,0.28571,0.71429,0.89286,0.0,1.0,6,8,0,6,0,0,0,0,4,0,0,1,0,0,3,0,0,3,0,0,7,0,8],[8,15,0.5333,0.77232,0.31715,0.71429,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,0,3,0,17],[12,15,0.8,0.56696,0.4439,0.0,0.71429,1.0,0.0,1.0,11,12,0,11,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,12],[15,15,1.0,0.27232,0.39827,0.0,0.0,0.60714,0.0,1.0,21,4,0,21,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,4]]}]},{"i":"9478b287e05e208f","q":"For any point  $X$  inside an acute-angled triangle  $ABC$  we define   $$ f(X)=\\frac{AX}{A_1X}\\cdot \\frac{BX}{B_1X}\\cdot \\frac{CX}{C_1X} $$   where  $A_1, B_1$ , and  $C_1$  are the intersection points of the lines  $AX, BX,$  and  $CX$  with the sides  $BC, AC$ , and  $AB$ , respectively. Let  $H, I$ , and  $G$  be the orthocenter, the incenter, and the centroid of the triangle  $ABC$ , respectively. Prove that   $f(H) \\ge f(I) \\ge f(G)$  .\n\n(D. Bazylev)","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.8928,"p":[[0,82,0.0,0.65176,0.18878,0.57142,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,20,0,0,2,0,0,1,0,6],[4,82,0.0488,0.87946,0.21162,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,17],[8,82,0.0976,0.88391,0.18709,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,6,0,20],[12,82,0.1463,0.86606,0.21999,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,19],[16,82,0.1951,0.8303,0.24078,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,5,0,17],[20,82,0.2439,0.83924,0.23627,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,5,0,18],[24,82,0.2927,0.8125,0.33204,0.78561,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,22],[28,82,0.3415,0.8928,0.22311,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,2,0,24],[32,82,0.3902,0.85275,0.30197,0.85929,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,23],[36,82,0.439,0.87051,0.22692,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,4,0,21],[40,82,0.4878,0.82588,0.33643,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,23],[44,82,0.5366,0.77677,0.36411,0.57143,1.0,1.0,0.0,1.0,4,21,0,4,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,21],[48,82,0.5854,0.80802,0.3362,0.78561,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,21],[52,82,0.6341,0.82139,0.33696,0.82143,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,23],[56,82,0.6829,0.54464,0.43366,0.0,0.57143,1.0,0.0,1.0,10,13,0,10,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,0,0,0,13],[60,82,0.7317,0.21427,0.38298,0.0,0.0,0.14275,0.0,1.0,24,5,0,24,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,5],[64,82,0.7805,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[68,82,0.8293,0.20089,0.36397,0.0,0.0,0.10714,0.0,1.0,24,4,0,24,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[72,82,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.571,"k":"flat","v":0.60709,"x":0.93748,"p":[[0,70,0.0,0.60709,0.12878,0.57143,0.57143,0.57143,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,23,0,0,3,0,0,1,0,2],[4,70,0.0571,0.89282,0.20522,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,23],[8,70,0.1143,0.93748,0.1285,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,24],[12,70,0.1714,0.83481,0.26271,0.857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,4,0,0,1,0,0,0,0,0,6,0,19],[16,70,0.2286,0.85267,0.19063,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,7,0,16],[20,70,0.2857,0.90622,0.16219,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,22],[24,70,0.3429,0.93302,0.13827,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[28,70,0.4,0.808,0.2541,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,0,0,0,7,0,0,1,0,0,7,0,15],[32,70,0.4571,0.85707,0.19571,0.78561,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,0,0,0,6,0,18],[36,70,0.5143,0.84373,0.20629,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,4,0,18],[40,70,0.5714,0.89278,0.1638,0.85711,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,6,0,20],[44,70,0.6286,0.90624,0.14111,0.85713,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,9,0,19],[48,70,0.6857,0.82587,0.19476,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,0,10,0,13],[52,70,0.7429,0.89285,0.16751,0.85711,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,6,0,20],[56,70,0.8,0.90625,0.21312,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[60,70,0.8571,0.79018,0.23954,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,0,0,0,7,0,14],[64,70,0.9143,0.82586,0.21352,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,0,2,0,18],[68,70,0.9714,0.86599,0.1889,0.57143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,3,0,20],[70,70,1.0,0.74991,0.20833,0.5713,0.57143,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,1,0,0,2,0,12]]}]},{"i":"1a8d088c8750623c","q":"For each positive integer  $n>1$ , if  $n=p_1^{\\alpha_1}p_2^{\\alpha_2}\\cdots p_k^{\\alpha_k}$ ( $p_i$  are pairwise different prime numbers  and  $\\alpha_i$  are positive integers),  define  $f(n)$  as  $\\alpha_1+\\alpha_2+\\cdots+\\alpha_k$ . For  $n=1$ , let  $f(1)=0$ . Find all pairs of integer polynomials  $P(x)$  and  $Q(x)$  such that for any positive integer  $m$ ,  $f(P(m))=Q(f(m))$  holds.","t":[{"b":4,"e":0.4286,"k":"rising","v":0.33041,"x":0.88393,"p":[[0,37,0.0,0.33041,0.14035,0.25002,0.42857,0.42857,0.0,0.43,2,0,0,2,0,6,0,0,4,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.8348,0.20552,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,6,0,16],[8,37,0.2162,0.75445,0.2059,0.57143,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,5,0,0,9,0,8],[12,37,0.3243,0.88393,0.19045,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,2,0,22],[16,37,0.4324,0.75892,0.20653,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,6,0,0,9,0,8],[20,37,0.5405,0.76339,0.22192,0.57143,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,4,0,0,6,0,11],[24,37,0.6486,0.71427,0.24485,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,5,0,0,4,0,10],[28,37,0.7568,0.7321,0.20749,0.57132,0.78564,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,3,0,0,9,0,7],[32,37,0.8649,0.79016,0.20513,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,6,0,12],[36,37,0.973,0.60267,0.23887,0.42857,0.4286,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,17,0,0,3,0,0,3,0,0,1,0,7],[37,37,1.0,0.49996,0.18897,0.42857,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,17,0,0,7,0,0,0,0,0,2,0,2]]},{"b":7,"e":0.42857,"k":"flat","v":0.34822,"x":0.82143,"p":[[0,59,0.0,0.34822,0.12846,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,6,0,0,3,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.82143,0.19885,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,7,0,14],[8,59,0.1356,0.69196,0.25028,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,5,0,0,2,0,0,8,0,0,6,0,7],[12,59,0.2034,0.73213,0.22233,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,8,0,0,6,0,8],[16,59,0.2712,0.7589,0.21263,0.57132,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,6,0,0,6,0,10],[20,59,0.339,0.69643,0.22516,0.42857,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,3,0,0,9,0,6],[24,59,0.4068,0.65625,0.2547,0.42857,0.50001,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,15,0,0,1,0,0,2,0,0,5,0,8],[28,59,0.4746,0.56697,0.21865,0.42857,0.4286,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,17,0,0,2,0,0,3,0,0,6,0,2],[32,59,0.5424,0.66963,0.20959,0.42857,0.64286,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,5,0,0,6,0,5],[36,59,0.6102,0.65624,0.21975,0.4286,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,8,0,0,4,0,0,7,0,0,8,0,3],[40,59,0.678,0.58482,0.21829,0.42857,0.42859,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,14,0,0,3,0,0,3,0,0,7,0,2],[44,59,0.7458,0.58481,0.2,0.42857,0.42859,0.75,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,17,0,0,1,0,0,5,0,0,7,0,1],[48,59,0.8136,0.64731,0.23414,0.42857,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,14,0,0,2,0,0,1,0,0,10,0,4],[52,59,0.8814,0.56701,0.22439,0.42857,0.42857,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,16,0,0,0,0,0,2,0,0,9,0,1],[56,59,0.9492,0.56695,0.20666,0.42857,0.4286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,13,0,0,3,0,0,5,0,0,6,0,1],[59,59,1.0,0.46427,0.14285,0.42857,0.42857,0.4286,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,20,0,0,3,0,0,2,0,0,2,0,0]]}]},{"i":"15afbc7deae8bf88","q":"For each positive integer  $n$ , let  $f(n)$  denote the number of ways of representing  $n$  as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance,  $f(4)=4$ , because the number  $4$  can be represented in the following four ways: \\[4, 2+2, 2+1+1, 1+1+1+1.\\] Prove that, for any integer  $n \\geq 3$ , \\[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\\]","t":[{"b":0,"e":0.0,"k":"falling","v":0.02679,"x":0.32587,"p":[[0,42,0.0,0.32587,0.1347,0.14286,0.35714,0.42857,0.14286,0.571,0,0,0,0,0,9,0,0,7,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[4,42,0.0952,0.10714,0.14725,0.0,0.0,0.1786,0.0,0.42857,19,0,0,19,0,5,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.11161,0.1504,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.11161,0.15865,0.0,0.0,0.2857,0.0,0.42857,20,0,1,20,0,3,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.08483,0.13296,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,42,0.7619,0.05804,0.13296,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,42,0.8571,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.03125,"x":0.30804,"p":[[0,54,0.0,0.30804,0.18935,0.14286,0.28571,0.42857,0.0,0.71429,2,0,1,2,0,12,0,0,3,0,0,11,0,0,2,0,0,2,0,0,0,0,0],[4,54,0.0741,0.15616,0.19352,0.0,0.07,0.28571,0.0,0.71429,16,0,0,16,0,6,0,0,3,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[8,54,0.1481,0.08473,0.12803,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.08929,0.16269,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,54,0.2963,0.05803,0.11214,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.03125,0.07771,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.0758,0.1128,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.11607,0.10971,0.0,0.14286,0.1429,0.0,0.28571,13,0,0,13,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.17857,0.15152,0.0,0.14286,0.28571,0.0,0.42857,10,0,0,10,0,9,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.3036,0.13245,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,4,0,0,16,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[52,54,0.963,0.2857,0.20823,0.14286,0.14288,0.4286,0.0,0.71429,4,0,0,4,0,13,0,0,2,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[54,54,1.0,0.25,0.2342,0.0,0.2857,0.32143,0.0,1.0,9,1,0,9,0,6,0,0,9,0,0,4,0,0,2,0,0,1,0,0,0,0,1]]}]},{"i":"6cf3517156dbe204","q":"Find all real-coefficient polynomials  $f(x)$  which satisfy the following conditions:**i.**  $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \\cdots + a_{2n - 2}\nx^2 + a_{2n}, a_0 > 0$ ;**ii.**  $\\sum_{j=0}^n a_{2j} a_{2n - 2j} \\leq \\left(\n\\begin{array}{c}\n2n\nn\\end{array} \\right) a_0 a_{2n}$ ;**iii.** All the roots of  $f(x)$  are imaginary numbers with no real part.","t":[{"b":1,"e":0.28571,"k":"rising","v":0.76784,"x":1.0,"p":[[0,27,0.0,0.76784,0.19481,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,8,0,0,12,0,6],[4,27,0.1481,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,27,0.2963,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,27,0.4444,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,27,0.8889,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[27,27,1.0,0.95089,0.17354,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29]]},{"b":7,"e":0.28571,"k":"rising","v":0.67856,"x":1.0,"p":[[0,40,0.0,0.67856,0.20826,0.57142,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,9,0,0,6,0,4],[4,40,0.1,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,40,0.2,0.92857,0.20203,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[12,40,0.3,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,40,0.6,0.95089,0.14987,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[28,40,0.7,0.91518,0.22548,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[32,40,0.8,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,40,0.9,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[40,40,1.0,0.90179,0.22711,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,26]]}]},{"i":"e186a1fb3f5cc576","q":"Find all primes  $p$ , so that for every prime  $q<p$  and  $x\\in \\mathbb{Z}$  one has  $p\\nmid x^2-q$ .","t":[{"b":2,"e":0.14286,"k":"falling","v":0.16956,"x":0.54464,"p":[[0,58,0.0,0.37946,0.30433,0.14286,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,14,0,0,8,0,0,0,0,0,5,0,0,0,0,0,0,0,5],[4,58,0.069,0.54464,0.38206,0.14286,0.78571,0.85714,0.14286,1.0,0,6,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,6],[8,58,0.1379,0.4241,0.34531,0.14286,0.2143,0.74996,0.14286,1.0,0,5,0,0,0,16,0,0,4,0,0,1,0,0,0,0,0,3,0,0,3,0,5],[12,58,0.2069,0.35267,0.32337,0.14286,0.14286,0.32143,0.0,1.0,1,5,0,1,0,16,0,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,5],[16,58,0.2759,0.36607,0.33491,0.14286,0.14286,0.57141,0.14286,1.0,0,5,0,0,0,20,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,5],[20,58,0.3448,0.34812,0.32136,0.14286,0.14286,0.32143,0.14,1.0,0,4,0,0,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,0,3,0,4],[24,58,0.4138,0.22321,0.15126,0.14286,0.14286,0.2857,0.14286,0.85714,0,0,0,0,0,21,0,0,8,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[28,58,0.4828,0.20089,0.1551,0.14286,0.14286,0.17857,0.14286,1.0,0,1,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,58,0.5517,0.22759,0.21979,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,24,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[36,58,0.6207,0.17411,0.05906,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.17856,0.10096,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,25,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,58,0.7586,0.21875,0.09438,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,17,0,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,58,0.8276,0.21428,0.15567,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,58,0.8966,0.17411,0.06901,0.14286,0.14286,0.17857,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.16956,0.07527,0.14286,0.14286,0.1429,0.0,0.4286,1,0,0,1,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.20089,0.08645,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,21,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.16295,"x":0.4375,"p":[[0,55,0.0,0.39285,0.23689,0.24999,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,8,0,0,11,0,0,3,0,0,5,0,0,2,0,0,2,0,1],[4,55,0.0727,0.37053,0.34043,0.14286,0.14286,0.42857,0.0,1.0,1,5,0,1,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0,3,0,5],[8,55,0.1455,0.40615,0.35918,0.14286,0.2857,0.85704,0.0,1.0,1,7,0,1,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,2,0,7],[12,55,0.2182,0.4375,0.36235,0.14286,0.14288,0.85714,0.14286,1.0,0,6,0,0,0,17,0,0,2,0,0,2,0,0,0,0,0,1,0,0,4,0,6],[16,55,0.2909,0.26331,0.25032,0.14286,0.14286,0.28571,0.0,1.0,1,2,1,1,0,20,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[20,55,0.3636,0.18295,0.08176,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,25,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.23214,0.19805,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,23,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[28,55,0.5091,0.21875,0.1636,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,55,0.5818,0.20089,0.15093,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,26,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[36,55,0.6545,0.22768,0.17075,0.14286,0.14286,0.2857,0.14286,0.85714,0,0,0,0,0,22,0,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[40,55,0.7273,0.20973,0.13829,0.14286,0.14286,0.2857,0.14,0.85714,0,0,0,0,0,22,0,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[44,55,0.8,0.19643,0.07784,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.17402,0.0591,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.19641,0.11147,0.14286,0.14286,0.17857,0.14286,0.57143,0,0,0,0,0,24,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[55,55,1.0,0.16295,0.05426,0.14286,0.14286,0.14286,0.07143,0.28571,0,0,0,0,1,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"899d8f0fc8a60a31","q":"Find all positive integers  $a, b,c$  greater than  $1$ , such that  $ab + 1$  is divisible by  $c, bc + 1$  is divisible by  $a$  and  $ca + 1$  is divisible by  $b$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.25446,"x":0.33473,"p":[[0,71,0.0,0.33473,0.18775,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,16,0,0,4,0,0,0,0,0,5,0,0,0,0,0],[4,71,0.0563,0.32589,0.18638,0.14289,0.28571,0.42858,0.0,0.71429,1,0,0,1,0,8,0,0,14,0,0,2,0,0,4,0,0,3,0,0,0,0,0],[8,71,0.1127,0.25892,0.1357,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,8,0,0,19,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,71,0.169,0.29465,0.11811,0.25,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,8,0,0,16,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[16,71,0.2254,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,71,0.2817,0.2991,0.09006,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[24,71,0.338,0.29911,0.10926,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,24,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[28,71,0.3944,0.30357,0.08564,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,23,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[32,71,0.4507,0.29009,0.0838,0.28571,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,3,0,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,71,0.507,0.25446,0.08553,0.24999,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,7,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,71,0.5634,0.28571,0.06186,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.3125,0.08328,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[48,71,0.6761,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.29464,0.07087,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[56,71,0.7887,0.3125,0.09062,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,24,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[60,71,0.8451,0.28561,0.07997,0.2857,0.28571,0.28571,0.14,0.571,0,0,0,0,0,4,0,0,25,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,71,0.9014,0.30803,0.0724,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,26,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[68,71,0.9577,0.2857,0.07981,0.2857,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,4,0,0,25,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[71,71,1.0,0.31696,0.07771,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.27232,"x":0.54464,"p":[[0,63,0.0,0.34821,0.19865,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,20,0,0,1,0,0,3,0,0,1,0,0,1,0,1],[4,63,0.0635,0.35714,0.20202,0.24999,0.28571,0.4286,0.14286,0.857,0,0,0,0,0,8,0,0,13,0,0,4,0,0,2,0,0,4,0,0,1,0,0],[8,63,0.127,0.54464,0.28446,0.28571,0.57141,0.85714,0.14286,1.0,0,2,0,0,0,5,0,0,7,0,0,3,0,0,2,0,0,6,0,0,7,0,2],[12,63,0.1905,0.38837,0.24802,0.14286,0.28571,0.57111,0.14286,1.0,0,2,0,0,0,9,0,0,10,0,0,4,0,0,4,0,0,2,0,0,1,0,2],[16,63,0.254,0.32588,0.1085,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,20,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[20,63,0.3175,0.3259,0.08171,0.2857,0.28571,0.28579,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[24,63,0.381,0.33482,0.11071,0.2857,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,1,0,0,23,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[28,63,0.4444,0.29017,0.09771,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,25,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[32,63,0.5079,0.31696,0.07771,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[36,63,0.5714,0.30803,0.08073,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[40,63,0.6349,0.29464,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,63,0.6984,0.28123,0.09767,0.2857,0.28571,0.28571,0.0,0.571,1,0,0,1,0,4,0,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[48,63,0.7619,0.2767,0.04996,0.2857,0.28571,0.28571,0.14,0.4286,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,63,0.8254,0.28125,0.05629,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,63,0.8889,0.27232,0.07457,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,3,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.28571,0.06186,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"011b40d8ebe0baf4","q":"Given a convex quadrilateral  $ABCD$  with  $\\angle BCD=90^\\circ$ . Let  $E$  be the midpoint of  $AB$ . Prove that  $2EC \\leqslant AD+BD$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.30802,"x":0.60267,"p":[[0,26,0.0,0.35714,0.37796,0.0,0.28571,0.57143,0.0,1.0,12,7,0,12,0,0,0,0,10,0,0,1,0,0,2,0,0,0,0,0,0,0,7],[4,26,0.1538,0.60267,0.41455,0.2857,0.78564,1.0,0.0,1.0,7,14,0,7,0,0,0,0,6,0,0,0,0,0,2,0,0,1,0,0,2,0,14],[8,26,0.3077,0.48214,0.45281,0.0,0.35714,1.0,0.0,1.0,13,12,0,13,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,12],[12,26,0.4615,0.40625,0.29904,0.2857,0.28571,0.57143,0.0,1.0,3,4,0,3,0,2,0,0,18,0,0,0,0,0,2,0,0,1,0,0,2,0,4],[16,26,0.6154,0.41518,0.24317,0.28571,0.28571,0.71429,0.0,0.85714,3,0,0,3,0,2,0,0,13,0,0,1,0,0,4,0,0,8,0,0,1,0,0],[20,26,0.7692,0.37052,0.26209,0.2857,0.28571,0.32143,0.0,1.0,2,3,0,2,0,4,0,0,18,0,0,1,0,0,1,0,0,3,0,0,0,0,3],[24,26,0.9231,0.35266,0.15964,0.28571,0.28571,0.4642,0.0,0.71429,2,0,0,2,0,0,0,0,20,0,0,2,0,0,7,0,0,1,0,0,0,0,0],[26,26,1.0,0.30802,0.17167,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,3,0,0,19,0,0,2,0,0,3,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.06696,"x":0.54462,"p":[[0,136,0.0,0.54462,0.40158,0.28571,0.49979,1.0,0.0,1.0,7,11,0,7,0,0,0,0,8,0,0,1,0,0,1,0,0,2,0,0,2,0,11],[4,136,0.0294,0.50446,0.42706,0.0,0.28571,1.0,0.0,1.0,9,12,0,9,0,1,0,0,7,0,0,0,0,0,2,0,0,0,0,0,1,0,12],[8,136,0.0588,0.49107,0.45448,0.0,0.28571,1.0,0.0,1.0,12,13,0,12,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,13],[12,136,0.0882,0.47768,0.41895,0.0,0.35714,1.0,0.0,1.0,10,11,0,10,0,1,0,0,5,0,0,1,0,0,4,0,0,0,0,0,0,0,11],[16,136,0.1176,0.37054,0.37941,0.0,0.28571,0.60714,0.0,1.0,12,6,0,12,0,1,0,0,7,0,0,1,0,0,3,0,0,1,0,0,1,0,6],[20,136,0.1471,0.44195,0.4291,0.0,0.28571,1.0,0.0,1.0,12,10,0,12,0,0,0,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,10],[24,136,0.1765,0.44196,0.43501,0.0,0.28571,1.0,0.0,1.0,13,9,0,13,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,0,3,0,9],[28,136,0.2059,0.28571,0.40406,0.0,0.0,0.64286,0.0,1.0,19,5,0,19,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,5],[32,136,0.2353,0.50445,0.47243,0.0,0.71407,1.0,0.0,1.0,14,12,0,14,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,12],[36,136,0.2647,0.41071,0.40838,0.0,0.28571,0.89286,0.0,1.0,11,8,0,11,0,2,0,0,7,0,0,0,0,0,2,0,0,0,0,0,2,0,8],[40,136,0.2941,0.34821,0.41332,0.0,0.07143,0.74996,0.0,1.0,16,6,0,16,0,1,0,0,4,0,0,0,0,0,0,0,0,3,0,0,2,0,6],[44,136,0.3235,0.41963,0.39759,0.0,0.28571,0.78571,0.0,1.0,11,8,0,11,0,0,0,0,8,0,0,1,0,0,1,0,0,3,0,0,0,0,8],[48,136,0.3529,0.33482,0.40816,0.0,0.21428,0.67857,0.0,1.0,15,8,0,15,0,1,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,8],[52,136,0.3824,0.26339,0.37646,0.0,0.0,0.42857,0.0,1.0,19,5,0,19,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,5],[56,136,0.4118,0.43749,0.42698,0.0,0.28571,1.0,0.0,1.0,13,10,0,13,0,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0,0,0,10],[60,136,0.4412,0.24107,0.38703,0.0,0.0,0.28571,0.0,1.0,21,6,0,21,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[64,136,0.4706,0.18302,0.36809,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[68,136,0.5,0.17411,0.35126,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[72,136,0.5294,0.16518,0.30745,0.0,0.0,0.2857,0.0,1.0,23,2,0,23,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[76,136,0.5588,0.16518,0.29904,0.0,0.0,0.28571,0.0,1.0,21,3,0,21,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[80,136,0.5882,0.14732,0.30196,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[84,136,0.6176,0.12946,0.25343,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[88,136,0.6471,0.26339,0.38484,0.0,0.0,0.57143,0.0,1.0,20,5,0,20,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,5],[92,136,0.6765,0.16071,0.29179,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[96,136,0.7059,0.06696,0.20198,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[100,136,0.7353,0.14732,0.34531,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[104,136,0.7647,0.14286,0.26486,0.0,0.0,0.28571,0.0,1.0,22,2,0,22,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[108,136,0.7941,0.11161,0.28061,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[112,136,0.8235,0.09821,0.26351,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[116,136,0.8529,0.14286,0.23419,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,0,0,0,5,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[120,136,0.8824,0.28571,0.41802,0.0,0.0,0.71429,0.0,1.0,20,7,0,20,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,7],[124,136,0.9118,0.33482,0.40972,0.0,0.0,0.75,0.0,1.0,17,6,0,17,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,2,0,6],[128,136,0.9412,0.2232,0.29651,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,3,0,0,6,0,0,1,0,0,2,0,0,2,0,0,0,0,2],[132,136,0.9706,0.29464,0.38621,0.0,0.0,0.46429,0.0,1.0,17,6,0,17,0,1,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,6],[136,136,1.0,0.35714,0.3312,0.0,0.28571,0.60714,0.0,1.0,9,3,0,9,0,4,0,0,7,0,0,1,0,0,3,0,0,4,0,0,1,0,3]]}]},{"i":"229275468ce4a04c","q":"27. N5 (BUL) ${ }^{\\mathrm{IMO} 6}$ Let $a>b>c>d$ be positive integers and suppose  $$ a c+b d=(b+d+a-c)(b+d-a+c) . $$  Prove that $a b+c d$ is not prime.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.17411,"p":[[0,85,0.0,0.12482,0.05917,0.14286,0.14286,0.14286,0.0,0.2857,5,0,1,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,85,0.0471,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,85,0.0941,0.17411,0.10555,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,22,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,85,0.1412,0.15179,0.09407,0.14286,0.14286,0.14286,0.0,0.4286,4,0,0,4,0,24,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,85,0.1882,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,22,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,85,0.2353,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,85,0.2824,0.1025,0.0891,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,85,0.3294,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,85,0.3765,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,85,0.4235,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,19,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,85,0.4706,0.12491,0.10563,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,20,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,85,0.5176,0.10714,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,85,0.5647,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,85,0.6118,0.11161,0.08553,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,85,0.6588,0.10706,0.11291,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,85,0.7059,0.10259,0.08167,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,85,0.7529,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.2857,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,85,0.8,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,85,0.8471,0.11607,0.12596,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,15,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.09366,"x":0.1875,"p":[[0,137,0.0,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.14286,6,0,1,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,137,0.0292,0.16072,0.08564,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,137,0.0584,0.16072,0.07784,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,137,0.0876,0.1875,0.11538,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,23,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,137,0.1168,0.14286,0.09449,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,24,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,137,0.146,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,137,0.1752,0.13839,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,24,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,137,0.2044,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,137,0.2336,0.13839,0.10403,0.14286,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,21,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,137,0.2628,0.14732,0.10403,0.14286,0.14286,0.14287,0.0,0.42857,7,0,0,7,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,137,0.292,0.13393,0.10677,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,137,0.3212,0.13393,0.10677,0.10714,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,20,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,137,0.3504,0.16509,0.10173,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,24,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,137,0.3796,0.14286,0.07986,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,25,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,137,0.4088,0.14286,0.10102,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,22,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,137,0.438,0.16518,0.13415,0.14286,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,18,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[64,137,0.4672,0.14723,0.10999,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,137,0.4964,0.13393,0.10062,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,19,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,137,0.5255,0.16071,0.09942,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,137,0.5547,0.12946,0.09688,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,137,0.5839,0.14286,0.09449,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,21,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,137,0.6131,0.12054,0.08827,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,137,0.6423,0.14732,0.12619,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,16,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[92,137,0.6715,0.09366,0.11067,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,137,0.7007,0.16518,0.11904,0.14286,0.14286,0.1786,0.0,0.4286,6,0,0,6,0,18,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[100,137,0.7299,0.13393,0.11812,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,16,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[104,137,0.7591,0.13393,0.10677,0.10714,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,20,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[108,137,0.7883,0.16062,0.12754,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,18,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[112,137,0.8175,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,137,0.8467,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,17,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[120,137,0.8759,0.12054,0.1017,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,18,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,137,0.9051,0.13393,0.10062,0.14286,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,22,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[128,137,0.9343,0.12054,0.1017,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,18,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,137,0.9635,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.4286,9,0,0,9,0,14,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[136,137,0.9927,0.15616,0.11496,0.105,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,14,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[137,137,1.0,0.13393,0.09407,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0c6eb7eab3156fa","q":"26. S4 (NZL) Suppose that $x_{1}, x_{2}, x_{3}, \\ldots$ are positive real numbers for which  $$ x_{n}^{n}=\\sum_{j=0}^{n-1} x_{n}^{j} $$  for $n=1,2,3, \\ldots$ Prove that for all $n$,  $$ 2-\\frac{1}{2^{n-1}} \\leq x_{n}<2-\\frac{1}{2^{n}} $$","t":[{"b":1,"e":1.0,"k":"flat","v":0.95982,"x":0.99107,"p":[[0,44,0.0,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[4,44,0.0909,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,44,0.1818,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,44,0.2727,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,44,0.3636,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,44,0.4545,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,44,0.5455,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,44,0.6364,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,44,0.7273,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,44,0.8182,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,44,0.9091,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,44,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]},{"b":6,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,31,0.0,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[4,31,0.129,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,31,0.2581,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[12,31,0.3871,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,31,0.5161,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,31,0.6452,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c251403bcfd12cc2","q":"21. (NET 2) Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $120^{\\circ}$.","t":[{"b":4,"e":1.0,"k":"rising","v":0.76339,"x":0.98661,"p":[[0,120,0.0,0.76339,0.15407,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,18,0,0,3,0,7],[4,120,0.0333,0.92856,0.10715,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[8,120,0.0667,0.93304,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[12,120,0.1,0.94195,0.10635,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[16,120,0.1333,0.95087,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[20,120,0.1667,0.9375,0.17835,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[24,120,0.2,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[28,120,0.2333,0.91963,0.14701,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,23],[32,120,0.2667,0.89285,0.20203,0.85711,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,21],[36,120,0.3,0.90179,0.12595,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,19],[40,120,0.3333,0.9241,0.13356,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,22],[44,120,0.3667,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[48,120,0.4,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[52,120,0.4333,0.91517,0.17075,0.85714,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,22],[56,120,0.4667,0.91071,0.13717,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,20],[60,120,0.5,0.875,0.23623,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,21],[64,120,0.5333,0.89729,0.13943,0.82132,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,5,0,19],[68,120,0.5667,0.90625,0.20079,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[72,120,0.6,0.88839,0.198,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,20],[76,120,0.6333,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[80,120,0.6667,0.91071,0.1915,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,22],[84,120,0.7,0.94642,0.12246,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[88,120,0.7333,0.89284,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[92,120,0.7667,0.9464,0.12251,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[96,120,0.8,0.92411,0.16746,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[100,120,0.8333,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[104,120,0.8667,0.88839,0.25439,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,24],[108,120,0.9,0.87499,0.23351,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,19],[112,120,0.9333,0.90625,0.12682,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,19],[116,120,0.9667,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[120,120,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":7,"e":0.71429,"k":"flat","v":0.67408,"x":0.93303,"p":[[0,66,0.0,0.72767,0.26088,0.71429,0.71429,1.0,0.0,1.0,2,9,2,2,0,0,0,0,1,0,0,2,0,0,1,0,0,14,0,0,3,0,9],[4,66,0.0606,0.90177,0.12599,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,18],[8,66,0.1212,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[12,66,0.1818,0.91071,0.14617,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,21],[16,66,0.2424,0.88392,0.14034,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,6,0,17],[20,66,0.303,0.87946,0.16016,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,18],[24,66,0.3636,0.84373,0.2063,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,5,0,17],[28,66,0.4242,0.8348,0.27691,0.82132,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,5,0,19],[32,66,0.4848,0.88391,0.20027,0.82132,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,21],[36,66,0.5455,0.85268,0.19719,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,5,0,17],[40,66,0.6061,0.91071,0.12753,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,18],[44,66,0.6667,0.76334,0.23316,0.71429,0.857,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,8,0,9],[48,66,0.7273,0.74103,0.18711,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,8,0,0,8,0,6],[52,66,0.7879,0.77232,0.17807,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,12,0,0,6,0,8],[56,66,0.8485,0.80354,0.23354,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,2,0,0,7,0,14],[60,66,0.9091,0.82586,0.18469,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,4,0,14],[64,66,0.9697,0.74549,0.15867,0.71429,0.71429,0.85702,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,14,0,0,8,0,4],[66,66,1.0,0.67408,0.19639,0.571,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,10,0,0,3,0,5]]}]},{"i":"a3749d398f22da6a","q":"26. N7 (BRA) The sequence $a_{0}, a_{1}, a_{2}, \\ldots$ is defined as follows:  $$ a_{0}=2, \\quad a_{k+1}=2 a_{k}^{2}-1 \\quad \\text { for } k \\geq 0 $$  Prove that if an odd prime $p$ divides $a_{n}$, then $2^{n+3}$ divides $p^{2}-1$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.98661,"p":[[0,79,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,79,0.0506,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,79,0.1013,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[12,79,0.1519,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[16,79,0.2025,0.87947,0.23449,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,24],[20,79,0.2532,0.93304,0.18205,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[24,79,0.3038,0.89284,0.17499,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,1,0,22],[28,79,0.3544,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[32,79,0.4051,0.90625,0.18073,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,0,0,24],[36,79,0.4557,0.9375,0.16728,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[40,79,0.5063,0.92411,0.15966,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[44,79,0.557,0.90624,0.1772,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,24],[48,79,0.6076,0.79464,0.30501,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,3,0,0,1,0,0,5,0,0,1,0,19],[52,79,0.6582,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[56,79,0.7089,0.85714,0.26964,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,23],[60,79,0.7595,0.90624,0.21611,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,26],[64,79,0.8101,0.79464,0.29653,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,1,0,18],[68,79,0.8608,0.70089,0.38525,0.42857,1.0,1.0,0.0,1.0,4,18,0,4,0,3,0,0,0,0,0,3,0,0,1,0,0,3,0,0,0,0,18],[72,79,0.9114,0.38392,0.42474,0.0,0.07143,0.71429,0.0,1.0,16,7,0,16,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,0,0,0,7],[76,79,0.962,0.25446,0.40364,0.0,0.0,0.5,0.0,1.0,22,6,0,22,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,6],[79,79,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.7098,"x":0.97768,"p":[[0,31,0.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[4,31,0.129,0.97322,0.10971,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,31,0.2581,0.94195,0.11773,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[12,31,0.3871,0.92411,0.14279,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,24],[16,31,0.5161,0.90179,0.22428,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,26],[20,31,0.6452,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,31,0.7742,0.75445,0.26544,0.57143,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,2,0,0,0,0,16],[28,31,0.9032,0.80356,0.2468,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,9,0,0,0,0,17],[31,31,1.0,0.7098,0.25376,0.53539,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,5,0,0,1,0,0,11,0,0,3,0,9]]}]},{"i":"5cbc5f8dc0154cf9","q":"12. (SWE 6) Consider the two square matrices  $$ A=\\left[\\begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & -1 & -1 \\\\ 1 & -1 & -1 & 1 & 1 \\\\ 1 & -1 & -1 & -1 & 1 \\\\ 1 & 1 & -1 & 1 & -1 \\end{array}\\right] \\quad \\text { and } \\quad B=\\left[\\begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & -1 & -1 \\\\ 1 & 1 & -1 & 1 & -1 \\\\ 1 & -1 & -1 & 1 & 1 \\\\ 1 & -1 & 1 & -1 & 1 \\end{array}\\right] $$  with entries 1 and -1 . The following operations will be called elementary: (1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns.  Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.","t":[{"b":2,"e":1.0,"k":"rising","v":0.39286,"x":1.0,"p":[[0,93,0.0,0.39286,0.45175,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,10],[4,93,0.043,0.8125,0.37019,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,24],[8,93,0.086,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[12,93,0.129,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[16,93,0.172,0.75893,0.4141,0.75,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,23],[20,93,0.2151,0.76339,0.38234,0.64286,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,21],[24,93,0.2581,0.78125,0.37962,0.75,1.0,1.0,0.0,1.0,4,23,0,4,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,23],[28,93,0.3011,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[32,93,0.3441,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,93,0.3871,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,93,0.4301,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,93,0.4731,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[48,93,0.5161,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[52,93,0.5591,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[56,93,0.6022,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[60,93,0.6452,0.93304,0.2172,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[64,93,0.6882,0.91518,0.22548,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,27],[68,93,0.7312,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,93,0.7742,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[76,93,0.8172,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[80,93,0.8602,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[84,93,0.9032,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,93,0.9462,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[92,93,0.9892,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[93,93,1.0,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]},{"b":4,"e":1.0,"k":"rising","v":0.24107,"x":0.99107,"p":[[0,93,0.0,0.24107,0.40158,0.0,0.0,0.32143,0.0,1.0,22,6,1,22,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,6],[4,93,0.043,0.85268,0.3204,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,26],[8,93,0.086,0.74554,0.39727,0.39286,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,21],[12,93,0.129,0.82589,0.36375,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[16,93,0.172,0.82142,0.3677,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[20,93,0.2151,0.72766,0.40933,0.53539,1.0,1.0,0.0,1.0,7,20,0,7,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,20],[24,93,0.2581,0.82589,0.33069,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,22],[28,93,0.3011,0.85268,0.33784,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,26],[32,93,0.3441,0.90179,0.25614,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,27],[36,93,0.3871,0.86607,0.27418,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,24],[40,93,0.4301,0.91518,0.24707,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[44,93,0.4731,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,93,0.5161,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[52,93,0.5591,0.94196,0.17076,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[56,93,0.6022,0.91071,0.23623,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,27],[60,93,0.6452,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[64,93,0.6882,0.92411,0.24218,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[68,93,0.7312,0.91071,0.25191,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[72,93,0.7742,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[76,93,0.8172,0.92411,0.2172,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,27],[80,93,0.8602,0.95536,0.13092,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[84,93,0.9032,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[88,93,0.9462,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[92,93,0.9892,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[93,93,1.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]}]},{"i":"5509d5729ece56a8","q":"Find the sum of all integers  $n$  satisfying the following inequality:\n\\[\\frac{1}{4}<\\sin\\frac{\\pi}{n}<\\frac{1}{3}.\\]","t":[{"b":1,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,71,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,71,0.0563,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[8,71,0.1127,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[12,71,0.169,0.93304,0.14279,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,26],[16,71,0.2254,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,71,0.2817,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[24,71,0.338,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,71,0.3944,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[32,71,0.4507,0.88839,0.1665,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,2,0,21],[36,71,0.507,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,71,0.5634,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,71,0.6197,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,71,0.6761,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,71,0.7324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,71,0.7887,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,71,0.8451,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,71,0.9014,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,71,0.9577,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[71,71,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,54,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,54,0.0741,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,54,0.1481,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,54,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,54,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,54,0.3704,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,54,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,54,0.5185,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,54,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,54,0.6667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,54,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,54,0.8148,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,54,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,54,0.963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[54,54,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"015445b848a1a4e1","q":"Consider $0<\\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_{n}=\\{a \\in$ $A: a \\leq n\\}$. Assume that for every $n \\in \\mathbb{N}$, the multiset $A_{n}$ contains at most $n \\lambda$ numbers. Show that there are infinitely many $n \\in \\mathbb{N}$ for which the sum of the elements in $A_{n}$ is at most $\\frac{n(n+1)}{2} \\lambda$.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.14732,"x":0.43303,"p":[[0,92,0.0,0.34375,0.17807,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,8,0,0,8,0,0,9,0,0,4,0,0,2,0,0,0,0,0],[4,92,0.0435,0.35266,0.3194,0.14286,0.28571,0.57111,0.0,1.0,6,3,0,6,0,8,0,0,7,0,0,2,0,0,2,0,0,2,0,0,2,0,3],[8,92,0.087,0.29454,0.23133,0.14286,0.2857,0.4642,0.0,0.71429,6,0,0,6,0,9,0,0,5,0,0,4,0,0,5,0,0,3,0,0,0,0,0],[12,92,0.1304,0.3125,0.24598,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,10,0,0,6,0,0,1,0,0,1,0,0,3,0,0],[16,92,0.1739,0.29463,0.31121,0.0,0.14286,0.46418,0.0,1.0,11,1,0,11,0,6,0,0,4,0,0,3,0,0,2,0,0,2,0,0,3,0,1],[20,92,0.2174,0.41515,0.32801,0.14286,0.35714,0.71429,0.0,1.0,5,3,0,5,0,8,0,0,3,0,0,3,0,0,3,0,0,5,0,0,2,0,3],[24,92,0.2609,0.31247,0.31222,0.0,0.21428,0.57111,0.0,1.0,11,1,0,11,0,5,0,0,3,0,0,3,0,0,5,0,0,1,0,0,3,0,1],[28,92,0.3043,0.39287,0.31743,0.14286,0.28571,0.60714,0.0,1.0,4,4,0,4,0,8,0,0,6,0,0,5,0,0,1,0,0,3,0,0,1,0,4],[32,92,0.3478,0.32589,0.29066,0.14286,0.28571,0.46431,0.0,1.0,7,1,0,7,0,8,0,0,4,0,0,5,0,0,3,0,0,1,0,0,3,0,1],[36,92,0.3913,0.375,0.31894,0.14286,0.28571,0.5,0.0,1.0,4,3,0,4,0,10,0,0,5,0,0,5,0,0,0,0,0,2,0,0,3,0,3],[40,92,0.4348,0.43303,0.35442,0.14286,0.28571,0.85704,0.0,1.0,5,5,0,5,0,7,0,0,5,0,0,3,0,0,3,0,0,0,0,0,4,0,5],[44,92,0.4783,0.20527,0.21708,0.14286,0.14286,0.17857,0.0,0.85714,6,0,0,6,0,18,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,0],[48,92,0.5217,0.34375,0.28763,0.14286,0.28571,0.57143,0.0,1.0,6,1,0,6,0,8,0,0,5,0,0,3,0,0,4,0,0,3,0,0,2,0,1],[52,92,0.5652,0.28125,0.23551,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,9,0,0,7,0,0,4,0,0,4,0,0,1,0,0,0,0,1],[56,92,0.6087,0.37946,0.27573,0.14286,0.35714,0.57141,0.0,0.85714,5,0,0,5,0,5,0,0,6,0,0,7,0,0,3,0,0,1,0,0,5,0,0],[60,92,0.6522,0.30803,0.30328,0.0,0.2857,0.46429,0.0,1.0,10,2,0,10,0,5,0,0,5,0,0,4,0,0,3,0,0,2,0,0,1,0,2],[64,92,0.6957,0.34375,0.2942,0.14286,0.28571,0.42857,0.0,1.0,6,2,0,6,0,7,0,0,5,0,0,8,0,0,1,0,0,0,0,0,3,0,2],[68,92,0.7391,0.20973,0.23956,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,12,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,1],[72,92,0.7826,0.24999,0.25252,0.0,0.14286,0.42857,0.0,1.0,9,1,0,9,0,10,0,0,3,0,0,5,0,0,2,0,0,2,0,0,0,0,1],[76,92,0.8261,0.16517,0.20856,0.0,0.14286,0.2857,0.0,0.71429,14,0,0,14,0,9,0,0,5,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[80,92,0.8696,0.26338,0.2627,0.10714,0.14286,0.42857,0.0,0.85714,8,0,0,8,0,11,0,0,3,0,0,5,0,0,1,0,0,1,0,0,3,0,0],[84,92,0.913,0.16963,0.24335,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,6,0,0,3,0,0,2,0,0,1,0,0,2,0,0,1,0,0],[88,92,0.9565,0.14732,0.19061,0.0,0.14286,0.1786,0.0,0.71429,15,0,0,15,0,9,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[92,92,1.0,0.20536,0.16727,0.14286,0.14286,0.14292,0.0,0.71429,2,0,0,2,0,23,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.03125,"x":0.37054,"p":[[0,33,0.0,0.37054,0.21086,0.14286,0.42857,0.57141,0.0,0.85714,1,0,1,1,0,10,0,0,3,0,0,9,0,0,6,0,0,2,0,0,1,0,0],[4,33,0.1212,0.32585,0.25808,0.14286,0.28571,0.571,0.0,0.85714,7,0,0,7,0,6,0,0,5,0,0,5,0,0,4,0,0,4,0,0,1,0,0],[8,33,0.2424,0.32142,0.32536,0.10714,0.14286,0.57111,0.0,1.0,8,3,0,8,0,9,0,0,5,0,0,1,0,0,2,0,0,3,0,0,1,0,3],[12,33,0.3636,0.32589,0.26542,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,11,0,0,5,0,0,5,0,0,2,0,0,1,0,0,4,0,0],[16,33,0.4848,0.11161,0.20119,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[20,33,0.6061,0.08487,0.14234,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.09375,0.16982,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,4,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,33,0.8485,0.17857,0.30305,0.0,0.0,0.28571,0.0,1.0,20,2,0,20,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[32,33,0.9697,0.09821,0.19045,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,1,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[33,33,1.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c9b0cabec4b5b8f3","q":"Consider a set  $S = \\{a_1, \\ldots, a_{2024}\\}$  consisting of  $2024$  distinct positive integers that satisfies the following property:\n[center] \"For every positive integer  $m < 2024,$  the sum of no  $m$  distinct elements of  $S$  is a multiple of  $2024.$ \" [/center]\nProve  $a_1, \\ldots, a_{2024}$  all leave the same remainder when divided by  $2024.$  Justify your answer.","t":[{"b":2,"e":0.28571,"k":"rising","v":0.04464,"x":0.41966,"p":[[0,49,0.0,0.04464,0.17655,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[4,49,0.0816,0.39729,0.25437,0.21429,0.42857,0.57143,0.0,1.0,8,1,0,8,0,0,0,0,1,0,0,8,0,0,14,0,0,0,0,0,0,0,1],[8,49,0.1633,0.27679,0.28107,0.0,0.14285,0.57143,0.0,0.57143,16,0,0,16,0,0,0,0,1,0,0,0,0,0,15,0,0,0,0,0,0,0,0],[12,49,0.2449,0.30802,0.25531,0.0,0.35714,0.57143,0.0,0.57143,12,0,0,12,0,0,0,0,4,0,0,3,0,0,13,0,0,0,0,0,0,0,0],[16,49,0.3265,0.32589,0.27487,0.0,0.57141,0.57143,0.0,0.57143,13,0,0,13,0,0,0,0,1,0,0,1,0,0,17,0,0,0,0,0,0,0,0],[20,49,0.4082,0.21875,0.23686,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,3,0,0,3,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[24,49,0.4898,0.25447,0.28061,0.0,0.0,0.57143,0.0,0.57143,17,0,0,17,0,1,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0],[28,49,0.5714,0.23214,0.26184,0.0,0.0,0.57143,0.0,0.57143,17,0,0,17,0,1,0,0,1,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[32,49,0.6531,0.26339,0.28146,0.0,0.0,0.57143,0.0,0.57143,17,0,0,17,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[36,49,0.7347,0.24105,0.25859,0.0,0.07143,0.57111,0.0,0.57143,16,0,0,16,0,1,0,0,2,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[40,49,0.8163,0.26786,0.27837,0.0,0.07143,0.57143,0.0,0.57143,16,0,0,16,0,1,0,0,0,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[44,49,0.898,0.28122,0.2731,0.0,0.28571,0.57143,0.0,0.57143,15,0,0,15,0,0,0,0,2,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[48,49,0.9796,0.40625,0.19269,0.2857,0.5,0.57143,0.0,0.57143,3,0,0,3,0,2,0,0,8,0,0,3,0,0,16,0,0,0,0,0,0,0,0],[49,49,1.0,0.41966,0.17833,0.28571,0.42929,0.57143,0.0,0.57143,2,0,0,2,0,3,0,0,5,0,0,7,0,0,15,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"rising","v":0.04906,"x":0.5714,"p":[[0,48,0.0,0.04906,0.15392,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,48,0.0833,0.32589,0.2506,0.0,0.42857,0.57143,0.0,0.85714,9,0,0,9,0,3,0,0,3,0,0,6,0,0,10,0,0,0,0,0,1,0,0],[8,48,0.1667,0.31696,0.27833,0.0,0.42859,0.57143,0.0,0.85714,13,0,0,13,0,0,0,0,2,0,0,3,0,0,13,0,0,0,0,0,1,0,0],[12,48,0.25,0.37499,0.2714,0.0,0.57143,0.57143,0.0,0.57143,11,0,0,11,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0],[16,48,0.3333,0.43301,0.25872,0.39285,0.57121,0.57143,0.0,1.0,7,1,0,7,0,0,0,0,1,0,0,6,0,0,16,0,0,0,0,0,1,0,1],[20,48,0.4167,0.42857,0.23419,0.39286,0.57143,0.57143,0.0,0.57143,7,0,0,7,0,0,0,0,1,0,0,2,0,0,22,0,0,0,0,0,0,0,0],[24,48,0.5,0.53569,0.14285,0.57143,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,0,0,0,1,0,0,28,0,0,1,0,0,0,0,0],[28,48,0.5833,0.49552,0.18892,0.57143,0.57143,0.57143,0.0,0.57143,4,0,0,4,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0],[32,48,0.6667,0.55804,0.11495,0.57143,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,1,0,0],[36,48,0.75,0.5714,0.0001,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[40,48,0.8333,0.45982,0.21349,0.53572,0.57143,0.57143,0.0,0.57143,5,0,0,5,0,1,0,0,0,0,0,2,0,0,24,0,0,0,0,0,0,0,0],[44,48,0.9167,0.36607,0.22851,0.14286,0.5,0.57143,0.0,0.57143,5,0,0,5,0,6,0,0,3,0,0,2,0,0,16,0,0,0,0,0,0,0,0],[48,48,1.0,0.25,0.20203,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,4,0,0,5,0,0,10,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"1c98b1d3a4ddb4f6","q":"For which numbers  $n$  is it possible to put marks on a stick such that all distances  $1$  cm,  $2$  cm, . . . ,  $n$  cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?","t":[{"b":1,"e":0.14286,"k":"flat","v":0.17411,"x":0.24554,"p":[[0,34,0.0,0.20303,0.06949,0.14286,0.14286,0.2857,0.14,0.28571,0,0,0,0,0,18,0,1,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.24554,0.15663,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,20,0,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[8,34,0.2353,0.23214,0.20748,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,26,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[12,34,0.3529,0.24554,0.16458,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,20,0,0,5,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[16,34,0.4706,0.17411,0.06901,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,26,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.23214,0.15872,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,21,0,0,6,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[24,34,0.7059,0.19197,0.09852,0.14286,0.14286,0.14287,0.14286,0.4286,0,0,0,0,0,25,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.20313,0.11725,0.14286,0.14286,0.16075,0.14286,0.57143,0,0,0,0,0,24,0,1,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,34,0.9412,0.22321,0.13452,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,21,0,0,6,0,1,2,0,1,0,0,0,1,0,0,0,0,0],[34,34,1.0,0.20973,0.11148,0.14286,0.14286,0.2857,0.14,0.4286,0,0,0,0,0,23,0,0,2,0,2,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.1429,"k":"flat","v":0.16072,"x":0.24554,"p":[[0,52,0.0,0.19866,0.099,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,22,0,1,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,52,0.0769,0.2008,0.14668,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[8,52,0.1538,0.22759,0.17449,0.14286,0.14286,0.1786,0.14,0.85714,0,0,0,0,0,24,0,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[12,52,0.2308,0.19643,0.16656,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[16,52,0.3077,0.18304,0.12492,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,28,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,52,0.3846,0.24554,0.1439,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,19,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[24,52,0.4615,0.1942,0.09146,0.14286,0.14286,0.23214,0.14286,0.42857,0,0,0,0,0,23,0,1,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.19196,0.11071,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,26,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,52,0.6154,0.17402,0.07567,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,26,0,2,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.16072,0.06916,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.1942,0.14087,0.14286,0.14286,0.14287,0.14286,0.85714,0,0,0,0,0,26,0,0,4,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[44,52,0.8462,0.19643,0.10565,0.14286,0.14286,0.14287,0.14286,0.4286,0,0,0,0,0,25,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.16742,0.06341,0.14286,0.14286,0.1429,0.14286,0.42857,0,0,0,0,0,27,0,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"98390d711c8831e6","q":"Consider the complex numbers  $x,y,z$  such that $|x|=|y|=|z|=1$ . Define the number  $$ a=\\left (1+\\frac xy\\right )\\left (1+\\frac yz\\right )\\left (1+\\frac zx\\right ). $$   $\\textbf{(a)}$  Prove that  $a$  is a real number. $\\textbf{(b)}$  Find the minimal and maximal value  $a$  can achieve, when  $x,y,z$  vary subject to  $|x|=|y|=|z|=1$ .\n\n*(Stefan B\u0103l\u0103uc\u0103 & Vlad Robu)*","t":[{"b":3,"e":0.85714,"k":"flat","v":0.85714,"x":0.94196,"p":[[0,24,0.0,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[4,24,0.1667,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[8,24,0.3333,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[12,24,0.5,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[16,24,0.6667,0.86161,0.10403,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,23,0,6],[20,24,0.8333,0.85714,0.07143,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4],[24,24,1.0,0.875,0.07784,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,7]]},{"b":5,"e":1.0,"k":"flat","v":0.875,"x":0.92411,"p":[[0,38,0.0,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[4,38,0.1053,0.89732,0.08917,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12],[8,38,0.2105,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[12,38,0.3158,0.90178,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[16,38,0.4211,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[20,38,0.5263,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[24,38,0.6316,0.89732,0.10248,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,13],[28,38,0.7368,0.91518,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[32,38,0.8421,0.90178,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[36,38,0.9474,0.88839,0.09268,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,11],[38,38,1.0,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15]]}]},{"i":"133176b033e7d873","q":"In trapezoid  $ABCD$ ,  $AD$  is parallel to  $BC$ . Knowing that  $AB=AD+BC$ , prove that the bisector of  $\\angle A$  also bisects  $CD$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00884,"x":0.03125,"p":[[0,24,0.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04909,"p":[[0,85,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,85,0.0471,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,85,0.0941,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,85,0.1412,0.04909,0.16209,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[16,85,0.1882,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,85,0.2353,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,85,0.2824,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,85,0.3294,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,85,0.3765,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,85,0.4235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,85,0.4706,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,85,0.5176,0.04017,0.15659,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[48,85,0.5647,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,85,0.6118,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,85,0.6588,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,85,0.7059,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,85,0.7529,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,85,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,85,0.8471,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"05001f9c76b63cd3","q":"Find all real quadruples  $(a,b,c,d)$  satisfying the system of equations  $$ \\left\\{ \\begin{array}{ll}\nab+cd = 6 \nac + bd = 3 \nad + bc = 2 \na + b + c + d = 6.\n\\end{array} \\right.\t $$","t":[{"b":4,"e":0.57143,"k":"falling","v":0.55357,"x":0.78571,"p":[[0,86,0.0,0.77232,0.17445,0.57143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,5,0,0,12,0,6],[4,86,0.0465,0.78571,0.18558,0.57143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,6,0,11],[8,86,0.093,0.70087,0.16508,0.57143,0.57143,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,8,0,4],[12,86,0.1395,0.63838,0.13826,0.57143,0.57143,0.57143,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,2,0,0,2,0,3],[16,86,0.186,0.57589,0.04351,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0],[20,86,0.2326,0.57141,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[24,86,0.2791,0.59821,0.09062,0.57143,0.57143,0.57143,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,0,1,0,1],[28,86,0.3256,0.59375,0.08828,0.57143,0.57143,0.57143,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,0,1],[32,86,0.3721,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[36,86,0.4186,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[40,86,0.4651,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[44,86,0.5116,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[48,86,0.5581,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[52,86,0.6047,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[56,86,0.6512,0.57138,0.00025,0.57143,0.57143,0.57143,0.57,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[60,86,0.6977,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[64,86,0.7442,0.57143,0.0,0.57143,0.57143,0.57143,0.5714,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[68,86,0.7907,0.55357,0.09942,0.57143,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0],[72,86,0.8372,0.57143,0.0,0.57143,0.57143,0.57143,0.57143,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[76,86,0.8837,0.56694,0.02485,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0],[80,86,0.9302,0.57142,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[84,86,0.9767,0.57142,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[86,86,1.0,0.56697,0.02485,0.57143,0.57143,0.57143,0.4286,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.71875,"x":0.92857,"p":[[0,28,0.0,0.71875,0.15765,0.57143,0.64286,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,2,0,0,11,0,3],[4,28,0.1429,0.77232,0.17076,0.57143,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,1,0,0,17,0,4],[8,28,0.2857,0.78571,0.16751,0.57143,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,10,0,8],[12,28,0.4286,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[16,28,0.5714,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[20,28,0.7143,0.88839,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[24,28,0.8571,0.875,0.07784,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,7],[28,28,1.0,0.85268,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,2]]}]},{"i":"c6075aaea8e815ce","q":"In the triangle  $ABC$ ,  $AD$  is altitude,  $M$  is the midpoint of  $BC$ . It is known that  $\\angle BAD = \\angle DAM =  \\angle MAC$ . Find the values of the angles of the triangle  $ABC$","t":[{"b":6,"e":0.85714,"k":"flat","v":0.86161,"x":0.95089,"p":[[0,62,0.0,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[4,62,0.0645,0.87946,0.13415,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,17],[8,62,0.129,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[12,62,0.1935,0.875,0.13243,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,4,0,16],[16,62,0.2581,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[20,62,0.3226,0.87054,0.12556,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,14],[24,62,0.3871,0.90178,0.12595,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,19],[28,62,0.4516,0.91964,0.11811,0.85713,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[32,62,0.5161,0.87053,0.12037,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,9,0,13],[36,62,0.5806,0.89286,0.13363,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,19],[40,62,0.6452,0.89284,0.14289,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,1,0,20],[44,62,0.7097,0.86161,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,5,0,15],[48,62,0.7742,0.88393,0.12595,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,16],[52,62,0.8387,0.88393,0.14914,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,4,0,18],[56,62,0.9032,0.87054,0.15303,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,6,0,16],[60,62,0.9677,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[62,62,1.0,0.91071,0.10565,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17]]},{"b":7,"e":1.0,"k":"flat","v":0.73214,"x":0.98214,"p":[[0,74,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,74,0.0541,0.83034,0.17657,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,0,3,0,14],[8,74,0.1081,0.89286,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[12,74,0.1622,0.82589,0.15866,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,11,0,0,9,0,10],[16,74,0.2162,0.81696,0.19638,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,11,0,0,8,0,11],[20,74,0.2703,0.73214,0.2165,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,14,0,0,5,0,7],[24,74,0.3243,0.87054,0.14445,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,2,0,17],[28,74,0.3784,0.79911,0.18161,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,10,0,0,7,0,10],[32,74,0.4324,0.82142,0.17496,0.71429,0.85707,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,13,0,0,5,0,12],[36,74,0.4865,0.86607,0.15126,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,15],[40,74,0.5405,0.85268,0.14054,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,4,0,14],[44,74,0.5946,0.8616,0.16164,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,7,0,15],[48,74,0.6486,0.84375,0.18681,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,8,0,0,4,0,16],[52,74,0.7027,0.76339,0.18766,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,13,0,5],[56,74,0.7568,0.81697,0.1684,0.71429,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,4,0,12],[60,74,0.8108,0.82589,0.17762,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,5,0,13],[64,74,0.8649,0.84373,0.13536,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,6,0,12],[68,74,0.9189,0.88393,0.1357,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,5,0,17],[72,74,0.973,0.88839,0.20743,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,20],[74,74,1.0,0.87945,0.16796,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,7,0,18]]}]},{"i":"cbd428dba4ba64b4","q":"In the triangle  $ABC$  we have  $|AB|<|AC|$ . The bisectors of the angles at  $B$  and  $C$  meet  $AC$  and  $AB$  at  $D$  and  $E$  respectively.  $BD$  and  $CE$  intersect at the incenter  $I$  of  $\\triangle ABC$ .\n\nProve that  $\\angle BAC=60^\\circ$  if and only if  $|IE|=|ID|$","t":[{"b":0,"e":0.57143,"k":"flat","v":0.15179,"x":0.558,"p":[[0,251,0.0,0.39732,0.26663,0.24999,0.42857,0.57143,0.0,1.0,7,1,0,7,0,1,0,0,3,0,0,11,0,0,3,0,0,6,0,0,0,0,1],[4,251,0.0159,0.27677,0.28999,0.0,0.28571,0.57111,0.0,1.0,15,1,0,15,0,0,0,0,3,0,0,5,0,0,6,0,0,2,0,0,0,0,1],[8,251,0.0319,0.26784,0.28063,0.0,0.28571,0.42857,0.0,1.0,14,1,0,14,0,1,0,0,4,0,0,7,0,0,2,0,0,3,0,0,0,0,1],[12,251,0.0478,0.2857,0.25504,0.0,0.35714,0.4642,0.0,0.71429,13,0,0,13,0,0,0,0,3,0,0,8,0,0,6,0,0,2,0,0,0,0,0],[16,251,0.0637,0.21427,0.23688,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,3,0,0,3,0,0,7,0,0,2,0,0,2,0,0,0,0,0],[20,251,0.0797,0.21429,0.29451,0.0,0.0,0.42857,0.0,0.85714,20,0,0,20,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,1,0,0],[24,251,0.0956,0.32585,0.2556,0.0,0.28571,0.571,0.0,0.85714,9,0,0,9,0,2,0,0,6,0,0,5,0,0,7,0,0,2,0,0,1,0,0],[28,251,0.1116,0.27221,0.2609,0.0,0.28571,0.4642,0.0,0.85714,13,0,0,13,0,1,0,0,5,0,0,5,0,0,6,0,0,1,0,0,1,0,0],[32,251,0.1275,0.15179,0.24206,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,2,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[36,251,0.1434,0.2589,0.26104,0.0,0.2143,0.42857,0.0,1.0,12,1,0,12,0,4,0,0,4,0,0,6,0,0,4,0,0,1,0,0,0,0,1],[40,251,0.1594,0.22321,0.27649,0.0,0.0,0.42857,0.0,0.857,17,0,0,17,0,2,0,0,2,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[44,251,0.1753,0.24106,0.32228,0.0,0.0,0.57111,0.0,1.0,19,2,0,19,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,0,0,0,2],[48,251,0.1912,0.31694,0.31688,0.0,0.28571,0.4642,0.0,1.0,11,3,0,11,0,4,0,0,3,0,0,6,0,0,3,0,0,2,0,0,0,0,3],[52,251,0.2072,0.25004,0.31545,0.0,0.0,0.46536,0.0,1.0,18,1,0,18,0,0,0,0,3,0,0,3,0,0,1,0,0,6,0,0,0,0,1],[56,251,0.2231,0.16955,0.23809,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,2,0,0,4,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[60,251,0.239,0.20532,0.24462,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,0,0,0,3,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[64,251,0.255,0.22319,0.26468,0.0,0.0,0.571,0.0,0.71429,18,0,0,18,0,0,0,0,2,0,0,3,0,0,8,0,0,1,0,0,0,0,0],[68,251,0.2709,0.16518,0.23176,0.0,0.0,0.32143,0.0,0.71429,20,0,0,20,0,0,0,0,4,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[72,251,0.2869,0.1607,0.2569,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,0,0,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[76,251,0.3028,0.25445,0.25185,0.0,0.2857,0.42858,0.0,0.71429,15,0,0,15,0,0,0,0,2,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[80,251,0.3187,0.20982,0.22299,0.0,0.1429,0.42857,0.0,0.71429,15,0,0,15,0,2,0,0,4,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[84,251,0.3347,0.20089,0.22263,0.0,0.14288,0.32143,0.0,0.71429,15,0,0,15,0,2,0,0,7,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[88,251,0.3506,0.21874,0.29446,0.0,0.0,0.57111,0.0,0.71429,20,0,0,20,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,0,0,0,0],[92,251,0.3665,0.28124,0.29119,0.0,0.2857,0.4286,0.0,1.0,13,2,0,13,0,1,0,0,6,0,0,5,0,0,4,0,0,1,0,0,0,0,2],[96,251,0.3825,0.16963,0.23805,0.0,0.0,0.42857,0.0,0.71429,20,0,0,20,0,1,0,0,2,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[100,251,0.3984,0.20981,0.25748,0.0,0.0,0.42858,0.0,0.71429,18,0,0,18,0,1,0,0,1,0,0,6,0,0,4,0,0,2,0,0,0,0,0],[104,251,0.4143,0.33033,0.2586,0.0,0.35714,0.571,0.0,0.85714,9,0,0,9,0,2,0,0,5,0,0,7,0,0,5,0,0,3,0,0,1,0,0],[108,251,0.4303,0.20534,0.26709,0.0,0.0,0.42857,0.0,1.0,18,1,0,18,0,0,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,1],[112,251,0.4462,0.26786,0.28065,0.0,0.14286,0.57143,0.0,0.71429,16,0,0,16,0,0,0,0,1,0,0,5,0,0,7,0,0,3,0,0,0,0,0],[116,251,0.4622,0.24552,0.27018,0.0,0.07143,0.4642,0.0,0.71429,16,0,0,16,0,1,0,0,2,0,0,5,0,0,5,0,0,3,0,0,0,0,0],[120,251,0.4781,0.21874,0.25995,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,0,0,0,1,0,0,7,0,0,4,0,0,2,0,0,0,0,0],[124,251,0.494,0.27677,0.29436,0.0,0.2857,0.4286,0.0,1.0,14,2,0,14,0,0,0,0,6,0,0,5,0,0,4,0,0,1,0,0,0,0,2],[128,251,0.51,0.20987,0.23958,0.0,0.07143,0.4286,0.0,0.71429,16,0,0,16,0,2,0,0,4,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[132,251,0.5259,0.2723,0.29955,0.0,0.14286,0.57143,0.0,1.0,16,1,0,16,0,0,0,0,3,0,0,2,0,0,8,0,0,2,0,0,0,0,1],[136,251,0.5418,0.2366,0.28706,0.0,0.0,0.4286,0.0,1.0,17,1,0,17,0,1,0,0,2,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[140,251,0.5578,0.18304,0.25313,0.0,0.0,0.42857,0.0,0.71429,20,0,0,20,0,0,0,0,2,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[144,251,0.5737,0.17409,0.23885,0.0,0.0,0.28571,0.0,0.71429,19,0,0,19,0,1,0,0,5,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[148,251,0.5896,0.24543,0.27256,0.0,0.21285,0.42857,0.0,1.0,15,1,0,15,0,1,0,0,4,0,0,7,0,0,2,0,0,2,0,0,0,0,1],[152,251,0.6056,0.26339,0.27919,0.0,0.21428,0.42858,0.0,1.0,15,1,0,15,0,1,0,0,1,0,0,8,0,0,5,0,0,1,0,0,0,0,1],[156,251,0.6215,0.18302,0.22932,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,1,0,0,4,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[160,251,0.6375,0.22319,0.28104,0.0,0.0,0.571,0.0,0.71429,19,0,0,19,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,0,0,0,0],[164,251,0.6534,0.28122,0.24865,0.0,0.28571,0.42857,0.0,0.71429,12,0,0,12,0,1,0,0,4,0,0,9,0,0,3,0,0,3,0,0,0,0,0],[168,251,0.6693,0.24996,0.25749,0.0,0.2857,0.4642,0.0,0.71429,15,0,0,15,0,0,0,0,5,0,0,4,0,0,6,0,0,2,0,0,0,0,0],[172,251,0.6853,0.36605,0.30078,0.0,0.42857,0.57143,0.0,1.0,10,1,0,10,0,2,0,0,3,0,0,3,0,0,7,0,0,6,0,0,0,0,1],[176,251,0.7012,0.25445,0.24151,0.0,0.28571,0.4286,0.0,0.71429,14,0,0,14,0,0,0,0,4,0,0,8,0,0,5,0,0,1,0,0,0,0,0],[180,251,0.7171,0.36156,0.26237,0.0,0.4286,0.57143,0.0,0.71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,12,0,1,0,0,5,0,0,8,0,0,3,0,0,3,0,0,0,0,0],[108,184,0.587,0.29464,0.28558,0.0,0.28571,0.4643,0.0,1.0,13,1,0,13,0,0,0,0,5,0,0,6,0,0,5,0,0,1,0,0,1,0,1],[112,184,0.6087,0.26338,0.30536,0.0,0.07143,0.5711,0.0,1.0,16,2,0,16,0,1,0,0,2,0,0,4,0,0,7,0,0,0,0,0,0,0,2],[116,184,0.6304,0.35265,0.31537,0.0,0.42857,0.57143,0.0,1.0,12,2,0,12,0,1,0,0,1,0,0,5,0,0,8,0,0,3,0,0,0,0,2],[120,184,0.6522,0.23214,0.27606,0.0,0.07143,0.42857,0.0,0.85714,16,0,0,16,0,1,0,0,6,0,0,3,0,0,1,0,0,4,0,0,1,0,0],[124,184,0.6739,0.35268,0.2474,0.14289,0.42857,0.46431,0.0,0.85714,7,0,0,7,0,2,0,0,6,0,0,9,0,0,3,0,0,4,0,0,1,0,0],[128,184,0.6957,0.27677,0.28106,0.0,0.28571,0.4286,0.0,1.0,14,1,0,14,0,1,0,0,2,0,0,8,0,0,4,0,0,2,0,0,0,0,1],[132,184,0.7174,0.31693,0.26419,0.0,0.42857,0.4286,0.0,1.0,11,1,0,11,0,1,0,0,1,0,0,12,0,0,5,0,0,1,0,0,0,0,1],[136,184,0.7391,0.27231,0.28427,0.0,0.2143,0.57143,0.0,0.85714,14,0,0,14,0,2,0,0,4,0,0,2,0,0,6,0,0,3,0,0,1,0,0],[140,184,0.7609,0.27676,0.31527,0.0,0.07143,0.57143,0.0,1.0,16,1,0,16,0,2,0,0,1,0,0,1,0,0,7,0,0,4,0,0,0,0,1],[144,184,0.7826,0.29463,0.32129,0.0,0.21428,0.57111,0.0,1.0,15,1,0,15,0,1,0,0,2,0,0,5,0,0,3,0,0,3,0,0,2,0,1],[148,184,0.8043,0.29909,0.2681,0.0,0.35714,0.42858,0.0,1.0,12,1,0,12,0,0,0,0,4,0,0,10,0,0,3,0,0,2,0,0,0,0,1],[152,184,0.8261,0.30801,0.2651,0.0,0.42857,0.4642,0.0,0.71429,12,0,0,12,0,1,0,0,1,0,0,10,0,0,4,0,0,4,0,0,0,0,0],[156,184,0.8478,0.3124,0.2845,0.0,0.28571,0.4642,0.0,1.0,11,1,0,11,0,2,0,0,4,0,0,7,0,0,4,0,0,2,0,0,1,0,1],[160,184,0.8696,0.32589,0.30562,0.0,0.28586,0.57143,0.0,1.0,12,1,0,12,0,2,0,0,3,0,0,3,0,0,7,0,0,3,0,0,1,0,1],[164,184,0.8913,0.3214,0.27197,0.0,0.42857,0.57141,0.0,0.71429,12,0,0,12,0,0,0,0,3,0,0,6,0,0,7,0,0,4,0,0,0,0,0],[168,184,0.913,0.17857,0.22304,0.0,0.0,0.42857,0.0,0.71429,18,0,0,18,0,1,0,0,4,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[172,184,0.9348,0.20534,0.21702,0.0,0.14288,0.42857,0.0,0.71429,14,0,0,14,0,4,0,0,4,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[176,184,0.9565,0.33928,0.27606,0.0,0.42857,0.57143,0.0,1.0,11,1,0,11,0,0,0,0,2,0,0,9,0,0,7,0,0,2,0,0,0,0,1],[180,184,0.9783,0.22319,0.27646,0.0,0.0,0.571,0.0,0.71429,18,0,0,18,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,0,0,0,0],[184,184,1.0,0.1607,0.21351,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,2,0,0,7,0,0,1,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"4ba1a8146dbc4523","q":"In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it. We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of 2 greater than 1 (i.e. of the form $2^{n}$ for some integer $n \\geq 1$ ).","t":[{"b":1,"e":0.57143,"k":"flat","v":0.33929,"x":0.52231,"p":[[0,30,0.0,0.4374,0.19877,0.24999,0.49979,0.57143,0.14,0.71429,0,0,0,0,0,8,0,0,2,0,0,6,0,0,12,0,0,4,0,0,0,0,0],[4,30,0.1333,0.49097,0.25501,0.14286,0.57143,0.57143,0.14,1.0,0,2,0,0,0,9,0,0,1,0,0,1,0,0,14,0,0,4,0,0,1,0,2],[8,30,0.2667,0.52231,0.28259,0.2857,0.57143,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,5,0,0,2,0,0,8,0,0,6,0,0,1,0,4],[12,30,0.4,0.49107,0.27879,0.25,0.57143,0.71429,0.0,1.0,1,3,0,1,0,7,0,0,3,0,0,4,0,0,7,0,0,6,0,0,1,0,3],[16,30,0.5333,0.39731,0.22793,0.14286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,12,0,0,3,0,0,3,0,0,8,0,0,6,0,0,0,0,0],[20,30,0.6667,0.37946,0.22759,0.14286,0.35714,0.57143,0.14286,0.71429,0,0,0,0,0,14,0,0,2,0,0,1,0,0,11,0,0,4,0,0,0,0,0],[24,30,0.8,0.3973,0.19797,0.1429,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,10,0,0,3,0,0,5,0,0,12,0,0,2,0,0,0,0,0],[28,30,0.9333,0.34821,0.2141,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,14,0,0,4,0,0,4,0,0,6,0,0,4,0,0,0,0,0],[30,30,1.0,0.33929,0.20438,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,13,0,0,7,0,0,2,0,0,7,0,0,3,0,0,0,0,0]]},{"b":2,"e":0.71429,"k":"flat","v":0.44196,"x":0.45981,"p":[[0,10,0.0,0.44196,0.21829,0.14286,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,10,0,0,1,0,0,2,0,0,14,0,0,5,0,0,0,0,0],[4,10,0.4,0.45981,0.25688,0.14286,0.57121,0.60714,0.14286,1.0,0,2,0,0,0,9,0,0,4,0,0,2,0,0,9,0,0,6,0,0,0,0,2],[8,10,0.8,0.45087,0.18593,0.28571,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,6,0,0,4,0,0,13,0,0,4,0,0,0,0,0],[10,10,1.0,0.45981,0.20118,0.2857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,3,0,0,3,0,0,14,0,0,5,0,0,0,0,0]]}]},{"i":"396ce322a6d89afd","q":"In every vertex of a regular  $n$  -gon exactly one chip is placed. At each  $step$  one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by  $[\\frac{n}{2}]$   positions clockwise from its initial position.","t":[{"b":0,"e":0.4286,"k":"rising","v":0.42854,"x":0.7991,"p":[[0,40,0.0,0.42854,0.23689,0.28571,0.42857,0.57111,0.0,1.0,2,2,0,2,0,2,0,0,10,0,0,8,0,0,4,0,0,4,0,0,0,0,2],[4,40,0.1,0.77232,0.28983,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,0,0,0,4,0,0,0,0,0,6,0,0,4,0,15],[8,40,0.2,0.62039,0.22753,0.42857,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,9,0,0,3,0,0,12,0,0,3,0,3],[12,40,0.3,0.7991,0.21386,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,4,0,14],[16,40,0.4,0.77677,0.21411,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,9,0,0,7,0,10],[20,40,0.5,0.67854,0.22017,0.42857,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,8,0,0,3,0,7],[24,40,0.6,0.69192,0.24774,0.5354,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,7,0,0,7,0,0,4,0,0,5,0,8],[28,40,0.7,0.70979,0.19721,0.571,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,7,0,0,8,0,5],[32,40,0.8,0.77677,0.20807,0.67857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,6,0,0,8,0,10],[36,40,0.9,0.73214,0.21354,0.67857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,12,0,0,4,0,8],[40,40,1.0,0.6964,0.23353,0.571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,6,0,0,5,0,0,9,0,0,4,0,7]]},{"b":3,"e":0.42857,"k":"rising","v":0.49552,"x":0.84375,"p":[[0,34,0.0,0.49552,0.25996,0.28571,0.42857,0.57143,0.0,1.0,2,3,0,2,0,0,0,0,9,0,0,7,0,0,7,0,0,1,0,0,3,0,3],[4,34,0.1176,0.84375,0.23787,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,5,0,18],[8,34,0.2353,0.7723,0.23654,0.57132,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,7,0,0,0,0,15],[12,34,0.3529,0.75,0.28347,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,6,0,0,3,0,14],[16,34,0.4706,0.81697,0.1996,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,5,0,14],[20,34,0.5882,0.79018,0.22864,0.71429,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,9,0,0,1,0,15],[24,34,0.7059,0.79463,0.21111,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,5,0,13],[28,34,0.8235,0.74999,0.25755,0.57132,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,5,0,0,2,0,0,6,0,0,6,0,11],[32,34,0.9412,0.70087,0.20629,0.5354,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,10,0,0,3,0,7],[34,34,1.0,0.65176,0.24468,0.42857,0.71429,0.85704,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,8,0,0,2,0,0,9,0,0,5,0,5]]}]},{"i":"01de720454da37c1","q":"Let  $ \\mathcal G$  be the set of all finite groups with at least two elements. \r\n\r\na) Prove that if  $ G\\in \\mathcal G$ , then the number of morphisms  $ f: G\\to G$  is at most  $ \\sqrt [p]{n^n}$ , where  $ p$  is the largest prime divisor of  $ n$ , and  $ n$  is the number of elements in  $ G$ .\r\n\r\nb) Find all the groups in  $ \\mathcal G$  for which the inequality at point a) is an equality.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.12947,"x":0.69195,"p":[[0,32,0.0,0.69195,0.08829,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,22,0,0,3,0,0],[4,32,0.125,0.64731,0.16361,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,4,0,0,2,0,4],[8,32,0.25,0.60271,0.17395,0.5713,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,16,0,0,5,0,0,1,0,3],[12,32,0.375,0.5132,0.22005,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,3,0,0,5,0,0,13,0,0,4,0,0,2,0,1],[16,32,0.5,0.49106,0.1954,0.42857,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,2,0,0,10,0,0,11,0,0,3,0,0,1,0,1],[20,32,0.625,0.46874,0.19959,0.39286,0.57143,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,5,0,0,17,0,0,1,0,0,0,0,1],[24,32,0.75,0.40166,0.173,0.2857,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,6,0,0,3,0,0,10,0,0,12,0,0,0,0,0,0,0,0],[28,32,0.875,0.30356,0.2223,0.14286,0.28571,0.46418,0.0,0.71429,6,0,0,6,0,7,0,0,6,0,0,5,0,0,6,0,0,2,0,0,0,0,0],[32,32,1.0,0.12947,0.12555,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,0,10,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.56249,"x":0.69196,"p":[[0,31,0.0,0.69196,0.09522,0.67857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,20,0,0,4,0,0],[4,31,0.129,0.66963,0.19378,0.57143,0.57143,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,15,0,0,5,0,0,2,0,6],[8,31,0.2581,0.62497,0.15465,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,9,0,0,3,0,1],[12,31,0.3871,0.63834,0.15148,0.57143,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,14,0,0,12,0,0,1,0,2],[16,31,0.5161,0.59379,0.1077,0.57143,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,0,2,0,0,3,0,0],[20,31,0.6452,0.56249,0.14698,0.57132,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,3,0,0,18,0,0,6,0,0,0,0,1],[24,31,0.7742,0.56695,0.16164,0.5713,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,2,0,0,17,0,0,6,0,0,1,0,1],[28,31,0.9032,0.58021,0.06092,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0],[31,31,1.0,0.58033,0.03459,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0]]}]},{"i":"b5269ea6d1e3bcd9","q":"In an acute triangle  $ABC,$  let  $D$  be a point on the side  $BC.$  Let  $M_1, M_2, M_3, M_4, M_5$  be the midpoints of the line segments  $AD, AB, AC, BD, CD,$  respectively and  $O_1, O_2, O_3, O_4$  be the circumcenters of triangles  $ABD, ACD, M_1M_2M_4, M_1M_3M_5,$  respectively. If  $S$  and  $T$  are midpoints of the line segments  $AO_1$  and  $AO_2,$  respectively, prove that  $SO_3O_4T$  is an isosceles trapezoid.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.46425,"x":0.5223,"p":[[0,98,0.0,0.49997,0.07983,0.42857,0.571,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,14,0,0,17,0,0,0,0,0,0,0,0],[4,98,0.0408,0.47316,0.13566,0.42857,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,16,0,0,12,0,0,0,0,0,1,0,0],[8,98,0.0816,0.48218,0.09939,0.42857,0.4293,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,1,0,0,15,0,0,15,0,0,0,0,0,0,0,0],[12,98,0.1224,0.49554,0.11836,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,1,0,0,9,0,0,20,0,0,0,0,0,0,0,0],[16,98,0.1633,0.48213,0.11709,0.42857,0.4998,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,14,0,0,16,0,0,0,0,0,0,0,0],[20,98,0.2041,0.49997,0.08746,0.42857,0.5712,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,12,0,0,18,0,0,0,0,0,0,0,0],[24,98,0.2449,0.49558,0.07125,0.42857,0.4293,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,17,0,0,15,0,0,0,0,0,0,0,0],[28,98,0.2857,0.46426,0.14723,0.42857,0.4998,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,2,0,0,12,0,0,16,0,0,0,0,0,0,0,0],[32,98,0.3265,0.4955,0.07969,0.42857,0.4998,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0],[36,98,0.3673,0.5223,0.08457,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,1,0,0,0,0,0],[40,98,0.4082,0.49998,0.0714,0.42857,0.4998,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0],[44,98,0.449,0.50446,0.07973,0.42857,0.49999,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0],[48,98,0.4898,0.51782,0.06913,0.42857,0.57141,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0],[52,98,0.5306,0.49551,0.09436,0.42857,0.57121,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,3,0,0,11,0,0,18,0,0,0,0,0,0,0,0],[56,98,0.5714,0.49104,0.07931,0.42857,0.4286,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,16,0,0,15,0,0,0,0,0,0,0,0],[60,98,0.6122,0.48659,0.11768,0.42857,0.5712,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,13,0,0,17,0,0,0,0,0,0,0,0],[64,98,0.6531,0.48657,0.07868,0.42857,0.4286,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,17,0,0,14,0,0,0,0,0,0,0,0],[68,98,0.6939,0.48216,0.11706,0.42857,0.571,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,1,0,0,12,0,0,17,0,0,0,0,0,0,0,0],[72,98,0.7347,0.51782,0.07781,0.42857,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,10,0,0,21,0,0,0,0,0,0,0,0],[76,98,0.7755,0.46425,0.08743,0.42857,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,21,0,0,10,0,0,0,0,0,0,0,0],[80,98,0.8163,0.49548,0.07968,0.42857,0.4998,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0],[84,98,0.8571,0.48206,0.0927,0.42857,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,17,0,0,14,0,0,0,0,0,0,0,0],[88,98,0.898,0.48217,0.12747,0.42857,0.571,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,0,0,0,13,0,0,17,0,0,0,0,0,0,0,0],[92,98,0.9388,0.49103,0.07932,0.42857,0.42859,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,16,0,0,15,0,0,0,0,0,0,0,0],[96,98,0.9796,0.46428,0.10713,0.42857,0.42859,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,0,0,0,18,0,0,12,0,0,0,0,0,0,0,0],[98,98,1.0,0.48213,0.08562,0.42857,0.4286,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,16,0,0,14,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.18294,"x":0.48658,"p":[[0,100,0.0,0.46873,0.1299,0.42857,0.4286,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,1,0,0,14,0,0,15,0,0,0,0,0,0,0,0],[4,100,0.04,0.47765,0.10477,0.42857,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,11,0,0,16,0,0,0,0,0,0,0,0],[8,100,0.08,0.43749,0.1181,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,4,0,0,16,0,0,10,0,0,0,0,0,0,0,0],[12,100,0.12,0.44194,0.13995,0.28571,0.42859,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,8,0,0,9,0,0,14,0,0,0,0,0,0,0,0],[16,100,0.16,0.45079,0.11926,0.42857,0.42857,0.57143,0.14,0.57143,0,0,0,0,0,2,0,0,3,0,0,15,0,0,12,0,0,0,0,0,0,0,0],[20,100,0.2,0.44639,0.15043,0.39286,0.4998,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,4,0,0,8,0,0,16,0,0,0,0,0,0,0,0],[24,100,0.24,0.43299,0.14048,0.42857,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,4,0,0,3,0,0,13,0,0,12,0,0,0,0,0,0,0,0],[28,100,0.28,0.47767,0.11632,0.42857,0.5712,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,10,0,0,17,0,0,0,0,0,0,0,0],[32,100,0.32,0.48658,0.12297,0.42857,0.57141,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,2,0,0,9,0,0,19,0,0,0,0,0,0,0,0],[36,100,0.36,0.42409,0.14932,0.39286,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,3,0,0,12,0,0,12,0,0,0,0,0,0,0,0],[40,100,0.4,0.46426,0.13361,0.42857,0.49979,0.57143,0.14286,0.57143,0,0,0,0,0,3,0,0,2,0,0,11,0,0,16,0,0,0,0,0,0,0,0],[44,100,0.44,0.43746,0.13329,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,7,0,0,10,0,0,13,0,0,0,0,0,0,0,0],[48,100,0.48,0.46425,0.12368,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,12,0,0,13,0,0,1,0,0,0,0,0],[52,100,0.52,0.43748,0.12336,0.42857,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,3,0,0,2,0,0,17,0,0,10,0,0,0,0,0,0,0,0],[56,100,0.56,0.43304,0.14054,0.42857,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,3,0,0,13,0,0,12,0,0,0,0,0,0,0,0],[60,100,0.6,0.44639,0.11148,0.39286,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,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a triangle  $ABC$ , points  $D,E,F$  lie on sides  $BC,CA,AB$  respectively. Moreover, the radii of incircles of  $\\triangle AEF, \\triangle BFD, \\triangle CDE$  are equal to  $r$ . Denote by  $r_0$  and  $R$  the radii of incircles of  $\\triangle DEF$  and  $\\triangle ABC$  respectively. Prove that  $r+r_0=R$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.20536,"x":0.57589,"p":[[0,57,0.0,0.57589,0.35443,0.28571,0.57144,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,6,0,0,5,0,0,0,0,0,3,0,0,5,0,8],[4,57,0.0702,0.44194,0.32995,0.0,0.4286,0.71429,0.0,1.0,9,2,0,9,0,0,0,0,3,0,0,5,0,0,4,0,0,6,0,0,3,0,2],[8,57,0.1404,0.54006,0.3403,0.28571,0.57121,0.85714,0.0,1.0,5,6,0,5,0,2,0,0,2,0,0,5,0,0,6,0,0,2,0,0,4,0,6],[12,57,0.2105,0.51786,0.35129,0.2857,0.57143,0.85714,0.0,1.0,5,7,0,5,0,2,0,0,6,0,0,2,0,0,5,0,0,3,0,0,2,0,7],[16,57,0.2807,0.5,0.31135,0.28571,0.42859,0.75,0.0,1.0,5,1,0,5,0,2,0,0,3,0,0,7,0,0,1,0,0,6,0,0,7,0,1],[20,57,0.3509,0.49105,0.32524,0.28571,0.42859,0.74996,0.0,1.0,4,6,0,4,0,1,0,0,9,0,0,4,0,0,5,0,0,1,0,0,2,0,6],[24,57,0.4211,0.50444,0.26959,0.28571,0.571,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,7,0,0,4,0,0,8,0,0,4,0,0,2,0,3],[28,57,0.4912,0.49105,0.32524,0.25,0.57141,0.71429,0.0,1.0,6,3,0,6,0,2,0,0,4,0,0,1,0,0,7,0,0,6,0,0,3,0,3],[32,57,0.5614,0.49995,0.24483,0.28571,0.571,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,7,0,0,5,0,0,8,0,0,6,0,0,1,0,2],[36,57,0.6316,0.37052,0.29419,0.14286,0.28571,0.57143,0.0,1.0,7,2,0,7,0,3,0,0,8,0,0,3,0,0,5,0,0,3,0,0,1,0,2],[40,57,0.7018,0.45969,0.28964,0.2857,0.42859,0.60714,0.0,1.0,3,2,0,3,0,4,0,0,7,0,0,3,0,0,7,0,0,2,0,0,4,0,2],[44,57,0.7719,0.45086,0.30116,0.2857,0.49979,0.71429,0.0,1.0,5,3,0,5,0,2,0,0,8,0,0,1,0,0,6,0,0,7,0,0,0,0,3],[48,57,0.8421,0.30357,0.25939,0.0,0.28571,0.42857,0.0,1.0,9,1,0,9,0,1,0,0,11,0,0,6,0,0,1,0,0,2,0,0,1,0,1],[52,57,0.9123,0.28562,0.28798,0.0,0.28571,0.32143,0.0,1.0,10,2,0,10,0,3,0,0,11,0,0,3,0,0,0,0,0,2,0,0,1,0,2],[56,57,0.9825,0.2142,0.175,0.0,0.28571,0.28571,0.0,0.85714,9,0,0,9,0,3,0,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[57,57,1.0,0.20536,0.12846,0.10714,0.2857,0.28571,0.0,0.4286,8,0,0,8,0,3,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.31685,"x":0.52678,"p":[[0,83,0.0,0.48213,0.32878,0.1429,0.42857,0.75,0.0,1.0,3,4,0,3,0,6,0,0,5,0,0,4,0,0,2,0,0,4,0,0,4,0,4],[4,83,0.0482,0.36604,0.34242,0.0,0.28571,0.60714,0.0,1.0,11,3,0,11,0,1,0,0,6,0,0,3,0,0,3,0,0,3,0,0,2,0,3],[8,83,0.0964,0.47768,0.32264,0.28571,0.42859,0.75,0.0,1.0,6,3,0,6,0,0,0,0,7,0,0,4,0,0,5,0,0,2,0,0,5,0,3],[12,83,0.1446,0.46426,0.29665,0.28571,0.42859,0.71429,0.0,1.0,4,3,0,4,0,2,0,0,8,0,0,3,0,0,6,0,0,4,0,0,2,0,3],[16,83,0.1928,0.48659,0.32899,0.28571,0.49979,0.71429,0.0,1.0,5,5,0,5,0,2,0,0,6,0,0,3,0,0,6,0,0,3,0,0,2,0,5],[20,83,0.241,0.51116,0.29833,0.2857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,5,0,0,4,0,0,3,0,1,6,0,0,5,0,0,2,0,4],[24,83,0.2892,0.52678,0.33586,0.2857,0.57143,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,9,0,0,1,0,0,5,0,0,1,0,0,6,0,5],[28,83,0.3373,0.40179,0.37019,0.0,0.28571,0.71429,0.0,1.0,10,6,0,10,0,0,0,0,9,0,0,3,0,0,0,0,0,3,0,0,1,0,6],[32,83,0.3855,0.44194,0.27974,0.28571,0.42857,0.57143,0.0,1.0,5,3,0,5,0,0,0,0,7,0,0,8,0,0,6,0,0,2,0,0,1,0,3],[36,83,0.4337,0.39732,0.35845,0.0,0.28571,0.71429,0.0,1.0,10,4,0,10,0,1,0,0,7,0,0,2,0,0,3,0,0,2,0,0,3,0,4],[40,83,0.4819,0.38838,0.31182,0.14286,0.28571,0.60714,0.0,1.0,6,3,0,6,0,4,0,0,9,0,0,2,0,0,3,0,0,4,0,0,1,0,3],[44,83,0.5301,0.35268,0.2575,0.2857,0.28571,0.42858,0.0,1.0,6,2,0,6,0,1,0,0,12,0,0,6,0,0,3,0,0,2,0,0,0,0,2],[48,83,0.5783,0.41517,0.32996,0.14286,0.28571,0.60714,0.0,1.0,7,3,0,7,0,3,0,0,7,0,0,1,0,0,6,0,0,2,0,0,3,0,3],[52,83,0.6265,0.45981,0.28734,0.2857,0.57143,0.71429,0.0,1.0,4,2,0,4,0,2,0,0,9,0,0,0,0,0,8,0,0,5,0,0,2,0,2],[56,83,0.6747,0.50445,0.27195,0.28571,0.4286,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,8,0,0,7,0,0,4,0,0,4,0,0,3,0,3],[60,83,0.7229,0.38391,0.27764,0.2857,0.28571,0.57143,0.0,1.0,7,2,0,7,0,0,0,0,10,0,0,3,0,0,7,0,0,3,0,0,0,0,2],[64,83,0.7711,0.49998,0.30929,0.28571,0.42859,0.71429,0.0,1.0,3,5,0,3,0,2,0,0,8,0,0,4,0,0,5,0,0,3,0,0,2,0,5],[68,83,0.8193,0.45534,0.23538,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,0,0,0,12,0,0,5,0,0,7,0,0,2,0,0,3,0,1],[72,83,0.8675,0.40178,0.24073,0.28571,0.28571,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,14,0,0,3,0,0,3,0,0,5,0,0,1,0,1],[76,83,0.9157,0.40625,0.33141,0.14286,0.28571,0.60714,0.0,1.0,7,4,0,7,0,2,0,0,9,0,0,3,0,0,3,0,0,2,0,0,2,0,4],[80,83,0.9639,0.3838,0.18371,0.2857,0.42857,0.571,0.0,0.71429,2,0,0,2,0,3,0,0,10,0,0,7,0,0,8,0,0,2,0,0,0,0,0],[83,83,1.0,0.31685,0.17767,0.25,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,13,0,0,5,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"9a8a83668eb55152","q":"Given a real number  $ c > 0$ , a sequence  $ (x_n)$  of real numbers is defined by  $ x_{n \\plus{} 1} \\equal{} \\sqrt {c \\minus{} \\sqrt {c \\plus{} x_n}}$  for  $ n \\ge 0$ . Find all values of  $ c$  such that for each initial value  $ x_0$  in  $ (0, c)$ , the sequence  $ (x_n)$  is defined for all  $ n$  and has a finite limit  $ \\lim x_n$  when  $ n\\to \\plus{} \\infty$ .","t":[{"b":3,"e":0.85714,"k":"rising","v":0.67854,"x":0.93303,"p":[[0,17,0.0,0.67854,0.19886,0.571,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,8,0,0,8,0,3],[4,17,0.2353,0.83928,0.17405,0.82132,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,13,0,11],[8,17,0.4706,0.85268,0.1838,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,9,0,15],[12,17,0.7059,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[16,17,0.9412,0.88838,0.09933,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,18,0,11],[17,17,1.0,0.93303,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19]]},{"b":5,"e":0.57143,"k":"flat","v":0.58929,"x":0.87053,"p":[[0,24,0.0,0.66961,0.20341,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,5,0,0,10,0,2],[4,24,0.1667,0.84598,0.18394,0.82143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,10,1,13],[8,24,0.3333,0.79461,0.205,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,10,0,11],[12,24,0.5,0.87053,0.16888,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,1,0,19],[16,24,0.6667,0.71428,0.15152,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,20,0,0,2,0,4],[20,24,0.8333,0.58929,0.14617,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,13,0,0,0,0,1],[24,24,1.0,0.5937,0.12429,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,4,0,0,15,0,0,12,0,0,0,0,0]]}]},{"i":"0fb787433da5a372","q":"In the right triangle  $ABC$  with shorter side  $AC$  the hypotenuse  $AB$  has  length  $12$ . Denote  $T$  its centroid and  $D$  the feet of altitude from the vertex  $C$ . Determine the size of its inner angle at the vertex  $B$  for which the triangle  $DTC$  has the greatest possible area.","t":[{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,40,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,40,0.1,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,40,0.2,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,40,0.3,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,40,0.4,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,40,0.5,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,40,0.7,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,40,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,40,0.9,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.91518,"x":0.99554,"p":[[0,26,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,26,0.1538,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,26,0.3077,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,26,0.4615,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,26,0.6154,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[20,26,0.7692,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[24,26,0.9231,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[26,26,1.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25]]}]},{"i":"8111004c1ddcec19","q":"Consider an integer  $n \\geq 1$ ,  $a_1,a_2, \\ldots , a_n$  real numbers in  $[-1,1]$  satisfying \n\\begin{align*}a_1+a_2+\\ldots +a_n=0 \\end{align*}\nand a function  $f: [-1,1] \\mapsto \\mathbb{R}$  such\n\\begin{align*} \\mid f(x)-f(y) \\mid \\le \\mid x-y \\mid \\end{align*}\nfor every  $x,y \\in [-1,1]$ . Prove\n\\begin{align*} \\left| f(x) - \\frac{f(a_1) +f(a_2) + \\ldots + f(a_n)}{n}  \\right| \\le 1 \\end{align*}\nfor every  $x$   $\\in [-1,1]$ . For a given sequence  $a_1,a_2, \\ldots ,a_n$ , Find  $f$  and  $x$  so hat the equality holds.","t":[{"b":0,"e":1.0,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,10,0.0,0.89732,0.18977,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,18],[4,10,0.4,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[8,10,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[10,10,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":2,"e":1.0,"k":"flat","v":0.8616,"x":0.95982,"p":[[0,12,0.0,0.9241,0.14279,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,21],[4,12,0.3333,0.8616,0.21275,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,11,0,16],[8,12,0.6667,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[12,12,1.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]}]},{"i":"9ae163c5465caa70","q":"If  $x,y,z$  satisfies  $x^2+y^2+z^2=1$ , find the maximum possible value of   $$ (x^2-yz)(y^2-zx)(z^2-xy) $$","t":[{"b":1,"e":0.0,"k":"falling","v":0.03125,"x":0.3258,"p":[[0,75,0.0,0.3258,0.218,0.14286,0.14286,0.57143,0.14,0.71429,0,0,0,0,0,18,0,0,1,0,0,1,0,0,10,0,0,2,0,0,0,0,0],[4,75,0.0533,0.18304,0.1439,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[8,75,0.1067,0.23214,0.20438,0.14286,0.14286,0.17857,0.0,1.0,2,1,0,2,0,22,0,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,1],[12,75,0.16,0.31696,0.24932,0.14286,0.14286,0.57143,0.0,1.0,2,1,0,2,0,16,0,0,2,0,0,3,0,0,5,0,0,3,0,0,0,0,1],[16,75,0.2133,0.26777,0.24684,0.14286,0.14286,0.17857,0.0,1.0,1,1,0,1,0,23,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,1],[20,75,0.2667,0.23659,0.22189,0.14286,0.14286,0.1786,0.0,1.0,2,1,0,2,0,22,0,0,3,0,0,0,0,0,2,0,0,2,0,0,0,0,1],[24,75,0.32,0.24991,0.21729,0.14286,0.14286,0.35714,0.0,0.85714,3,0,0,3,0,20,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0],[28,75,0.3733,0.17848,0.20205,0.105,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,19,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[32,75,0.4267,0.27232,0.25344,0.14286,0.14286,0.57143,0.0,0.85714,6,0,0,6,0,15,0,0,1,0,0,0,0,0,7,0,0,2,0,0,1,0,0],[36,75,0.48,0.24999,0.24483,0.14286,0.14286,0.46418,0.0,0.71429,7,0,0,7,0,15,0,0,1,0,0,1,0,0,4,0,0,4,0,0,0,0,0],[40,75,0.5333,0.20982,0.22583,0.0,0.14286,0.21432,0.0,0.71429,9,0,0,9,0,15,0,0,0,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[44,75,0.5867,0.22321,0.2394,0.14286,0.14286,0.14287,0.0,0.857,7,0,0,7,0,18,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0],[48,75,0.64,0.24545,0.26785,0.14286,0.14286,0.14286,0.0,1.0,4,2,0,4,0,21,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,2],[52,75,0.6933,0.1875,0.18363,0.14286,0.14286,0.14286,0.0,0.71429,7,0,0,7,0,18,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[56,75,0.7467,0.21875,0.20511,0.14286,0.14286,0.21429,0.0,0.71429,6,0,0,6,0,18,0,0,0,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[60,75,0.8,0.20536,0.17835,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,21,0,0,1,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[64,75,0.8533,0.20536,0.17474,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,24,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[68,75,0.9067,0.14286,0.19885,0.0,0.14286,0.14286,0.0,0.85714,12,0,0,12,0,17,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[72,75,0.96,0.14286,0.22588,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,11,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[75,75,1.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.15179,"x":0.30357,"p":[[0,88,0.0,0.27223,0.20322,0.14286,0.14286,0.46429,0.14,0.71429,0,0,0,0,0,22,0,0,1,0,0,1,0,0,6,0,0,2,0,0,0,0,0],[4,88,0.0455,0.17857,0.12877,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,88,0.0909,0.25893,0.20652,0.14286,0.14286,0.35714,0.0,0.71429,1,0,0,1,0,22,0,0,1,0,0,0,0,0,6,0,0,2,0,0,0,0,0],[12,88,0.1364,0.25447,0.22794,0.14286,0.14286,0.1786,0.0,1.0,1,1,0,1,0,23,0,0,1,0,0,1,0,0,3,0,0,2,0,0,0,0,1],[16,88,0.1818,0.30357,0.25939,0.14286,0.14286,0.57143,0.0,1.0,1,1,0,1,0,21,0,0,0,0,0,0,0,0,5,0,0,4,0,0,0,0,1],[20,88,0.2273,0.24991,0.2083,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,1,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[24,88,0.2727,0.22321,0.19541,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,23,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[28,88,0.3182,0.30357,0.25692,0.14286,0.14286,0.57143,0.14286,1.0,0,1,0,0,0,22,0,0,1,0,0,0,0,0,3,0,0,5,0,0,0,0,1],[32,88,0.3636,0.22768,0.18851,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[36,88,0.4091,0.22759,0.1708,0.14286,0.14286,0.14287,0.14,0.71429,0,0,0,0,0,25,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[40,88,0.4545,0.22322,0.19865,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[44,88,0.5,0.26777,0.20444,0.14286,0.14286,0.32143,0.14,0.71429,0,0,0,0,0,22,0,0,2,0,0,1,0,0,4,0,0,3,0,0,0,0,0],[48,88,0.5455,0.21875,0.19227,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[52,88,0.5909,0.27677,0.21996,0.14286,0.14286,0.46418,0.14286,1.0,0,1,0,0,0,22,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,1],[56,88,0.6364,0.23661,0.19434,0.14286,0.14286,0.1429,0.14286,0.85714,0,0,0,0,0,25,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,0],[60,88,0.6818,0.16956,0.10377,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,88,0.7273,0.20518,0.14706,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[68,88,0.7727,0.23661,0.17717,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,25,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[72,88,0.8182,0.22759,0.16702,0.14286,0.14286,0.17857,0.14,0.71429,0,0,0,0,0,24,0,0,3,0,0,0,0,0,4,0,0,1,0,0,0,0,0],[76,88,0.8636,0.20973,0.14723,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,26,0,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[80,88,0.9091,0.21875,0.1988,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[84,88,0.9545,0.16964,0.10374,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[88,88,1.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de03f253bd5b8399","q":"If on  $ \\triangle ABC$ , trinagles  $ AEB$  and  $ AFC$  are constructed externally such that  $ \\angle AEB\\equal{}2 \\alpha$ ,  $ \\angle AFB\\equal{} 2 \\beta$ .\r $ AE\\equal{}EB$ ,  $ AF\\equal{}FC$ .\r\nCOnstructed externally on  $ BC$  is triangle  $ BDC$  with  $ \\angle DBC\\equal{} \\beta$  ,  $ \\angle BCD\\equal{} \\alpha$ .\r\nProve that 1.  $ DA$  is perpendicular to  $ EF$ .\r\n                2. If  $ T$  is the projection of  $ D$  on  $ BC$ , then prove that  $ \\frac{DA}{EF}\\equal{} 2 \\frac{DT}{BC}$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.13839,"x":0.23206,"p":[[0,58,0.0,0.14942,0.15607,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,1,10,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,58,0.069,0.20981,0.15963,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,17,0,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[8,58,0.1379,0.23206,0.16274,0.105,0.28571,0.28579,0.0,0.57143,8,0,0,8,0,4,0,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[12,58,0.2069,0.17856,0.14722,0.14286,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,15,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,58,0.2759,0.2008,0.12303,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,14,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,58,0.3448,0.20067,0.12827,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,58,0.4138,0.17384,0.17034,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,12,0,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[28,58,0.4828,0.1875,0.12595,0.14286,0.14288,0.28571,0.0,0.4286,6,0,0,6,0,13,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.19196,0.16602,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,15,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[36,58,0.6207,0.21427,0.17125,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[40,58,0.6897,0.14737,0.12119,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.16071,0.1171,0.14286,0.14286,0.1786,0.0,0.57143,6,0,0,6,0,18,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,58,0.8276,0.15169,0.12842,0.14214,0.14286,0.14286,0.0,0.571,7,0,0,7,0,20,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[52,58,0.8966,0.18747,0.15329,0.14286,0.14286,0.28571,0.0,0.571,7,0,0,7,0,14,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[56,58,0.9655,0.18295,0.22372,0.0,0.14286,0.1786,0.0,1.0,11,1,0,11,0,13,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[58,58,1.0,0.13839,0.145,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,10,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.11598,"x":0.23436,"p":[[0,51,0.0,0.11598,0.14031,0.0,0.07,0.1786,0.0,0.57143,16,0,1,16,0,8,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,51,0.0784,0.1875,0.1448,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,13,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,51,0.1569,0.22319,0.18184,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,11,0,0,7,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[12,51,0.2353,0.2009,0.15916,0.14286,0.14288,0.28571,0.0,0.57143,7,0,0,7,0,12,0,0,8,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,51,0.3137,0.17857,0.14726,0.0,0.14286,0.28571,0.0,0.4286,9,0,0,9,0,11,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,51,0.3922,0.16518,0.14773,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,11,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,51,0.4706,0.20536,0.18536,0.0,0.14288,0.42857,0.0,0.57143,10,0,0,10,0,9,0,0,4,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[28,51,0.549,0.19642,0.16653,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,12,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[32,51,0.6275,0.1741,0.15862,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,15,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[36,51,0.7059,0.20972,0.20199,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,15,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[40,51,0.7843,0.23436,0.17875,0.14286,0.2143,0.28571,0.0,0.57143,6,0,0,6,1,9,0,0,9,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[44,51,0.8627,0.14286,0.13832,0.0,0.14286,0.14287,0.0,0.57143,10,0,0,10,0,16,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,51,0.9412,0.14732,0.14053,0.0,0.14286,0.1429,0.0,0.5714,10,0,0,10,0,15,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[51,51,1.0,0.1517,0.1234,0.105,0.14286,0.1786,0.0,0.57143,8,0,0,8,0,16,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"bbf7561c6fd497a9","q":"Given an integer  $k$ .  $f(n)$  is defined on negative integer set and its values are integers.  $f(n)$  satisfies \\[ f(n)f(n+1)=(f(n)+n-k)^2, \\] for  $n=-2,-3,\\cdots$ . Find an expression of  $f(n)$ 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 $\\triangle ABC$ , side  $BC>AB$ . Point  $D$  lies on side  $AC$  such that  $\\angle ABD=\\angle CBD$ . Points  $Q,P$  lie on line  $BD$  such that  $AQ\\bot BD$  and  $CP\\bot BD$ .  $M,E$  are the midpoints of side  $AC$  and  $BC$  respectively. Circle  $O$  is the circumcircle of  $\\triangle PQM$  intersecting side  $AC$  at  $H$ . Prove that  $O,H,E,M$  lie on a circle.","t":[{"b":0,"e":0.0,"k":"flat","v":0.27237,"x":0.45759,"p":[[0,37,0.0,0.3258,0.1685,0.14289,0.28571,0.42857,0.0,0.57143,2,0,1,2,0,7,0,0,9,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[4,37,0.1081,0.4375,0.06121,0.42857,0.42857,0.4286,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0],[8,37,0.2162,0.4241,0.09769,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,23,0,0,3,0,0,1,0,0,0,0,0],[12,37,0.3243,0.39723,0.13726,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,3,0,0,3,0,0,21,0,0,3,0,0,1,0,0,0,0,0],[16,37,0.4324,0.40622,0.10775,0.39286,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,2,0,0,6,0,0,19,0,0,5,0,0,0,0,0,0,0,0],[20,37,0.5405,0.36161,0.14279,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,8,0,0,18,0,0,1,0,0,1,0,0,0,0,0],[24,37,0.6486,0.43304,0.12619,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,24,0,0,3,0,0,2,0,0,0,0,0],[28,37,0.7568,0.45759,0.06648,0.42857,0.42857,0.44645,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,23,0,1,7,0,0,0,0,0,0,0,0],[32,37,0.8649,0.44197,0.12037,0.42857,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,23,0,0,4,0,0,2,0,0,0,0,0],[36,37,0.973,0.36607,0.15124,0.28571,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,1,0,0,7,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[37,37,1.0,0.27237,0.18684,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,6,0,0,4,0,0,13,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.35268,"x":0.43749,"p":[[0,95,0.0,0.35719,0.13835,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,9,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[4,95,0.0421,0.42862,0.11845,0.42857,0.42857,0.42895,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,19,0,0,6,0,0,1,0,0,0,0,0],[8,95,0.0842,0.39956,0.10693,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,1,4,0,0,23,0,0,1,0,0,1,0,0,0,0,0],[12,95,0.1263,0.43749,0.07934,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,28,0,0,2,0,0,1,0,0,0,0,0],[16,95,0.1684,0.41963,0.1006,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,25,0,0,2,0,0,1,0,0,0,0,0],[20,95,0.2105,0.41518,0.09689,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,28,0,0,0,0,0,1,0,0,0,0,0],[24,95,0.2526,0.375,0.13243,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,3,0,0,1,0,0,25,0,0,1,0,0,0,0,0,0,0,0],[28,95,0.2947,0.39286,0.09449,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[32,95,0.3368,0.41969,0.07936,0.42857,0.42857,0.42857,0.1429,0.57143,0,0,0,0,0,1,0,0,3,0,0,25,0,0,3,0,0,0,0,0,0,0,0],[36,95,0.3789,0.40179,0.06622,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[40,95,0.4211,0.38392,0.14031,0.39286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,3,0,0,21,0,0,2,0,0,1,0,0,0,0,0],[44,95,0.4632,0.41964,0.06121,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,26,0,0,2,0,0,0,0,0,0,0,0],[48,95,0.5053,0.39732,0.12234,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,1,0,0,1,0,0,26,0,0,2,0,0,0,0,0,0,0,0],[52,95,0.5474,0.38838,0.1085,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,4,0,0,3,0,0,23,0,0,2,0,0,0,0,0,0,0,0],[56,95,0.5895,0.36607,0.12846,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,2,0,0,23,0,0,1,0,0,0,0,0,0,0,0],[60,95,0.6316,0.37947,0.09182,0.39286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,5,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[64,95,0.6737,0.35268,0.12869,0.2857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[68,95,0.7158,0.36157,0.12887,0.39286,0.42857,0.42857,0.0,0.43,2,0,0,2,0,3,0,0,3,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[72,95,0.7579,0.38398,0.10974,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,4,0,0,24,0,0,1,0,0,0,0,0,0,0,0],[76,95,0.8,0.37947,0.11071,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,5,0,0,23,0,0,1,0,0,0,0,0,0,0,0],[80,95,0.8421,0.39286,0.08748,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[84,95,0.8842,0.41072,0.05923,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[88,95,0.9263,0.39286,0.07986,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[92,95,0.9684,0.40179,0.07523,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[95,95,1.0,0.38393,0.11538,0.42857,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,1,0,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4874963fb3324175","q":"Let  $\\vartriangle ABC$  be a triangle and let every angle of  $\\vartriangle ABC$  be acute. Prove that   $$ R \\le \\frac{ AB + BC + CA}{4} $$   where  $R$  is the radius of the circumscribed circle.","t":[{"b":4,"e":0.0,"k":"flat","v":0.05357,"x":0.12054,"p":[[0,24,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.08911,0.06903,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.05357,"x":0.09822,"p":[[0,12,0.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"675ecfcda2675576","q":"In a quadrilateral  $ABCD$  diagonal  $AC$  is a bisector of  $\\angle BAD$  and  $\\angle ADC = \\angle ACB$ . The points  $X$  and  $Y$  are the feet of the perpendiculars from  $A$  to  $BC$  and  $CD$  respectively. Prove that the orthocenter of  $\\triangle AXY$  lies on the line  $BD$ 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triangle  $ABC$  we have  $|AB|=1$  and  $\\angle ABC=120^\\circ.$  The perpendicular line to  $AB$  at  $B$  meets  $AC$  at  $D$  such that  $|DC|=1$ . Find the length of  $AD$ .","t":[{"b":1,"e":1.0,"k":"falling","v":0.40179,"x":0.83929,"p":[[0,32,0.0,0.78125,0.22011,0.67857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,9,0,0,1,0,14],[4,32,0.125,0.81696,0.2321,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,6,0,0,0,0,18],[8,32,0.25,0.83929,0.22232,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,0,0,20],[12,32,0.375,0.49107,0.2141,0.42857,0.42857,0.42857,0.28571,1.0,0,4,0,0,0,0,0,0,6,0,0,20,0,0,0,0,0,2,0,0,0,0,4],[16,32,0.5,0.53571,0.27894,0.28571,0.42857,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,12,0,0,9,0,0,0,0,0,4,0,0,0,0,7],[20,32,0.625,0.5,0.28347,0.28571,0.42857,0.57143,0.14286,1.0,0,7,0,0,0,1,0,0,14,0,0,6,0,0,4,0,0,0,0,0,0,0,7],[24,32,0.75,0.40179,0.22142,0.28571,0.28571,0.42857,0.0,1.0,1,3,0,1,0,0,0,0,17,0,0,10,0,0,0,0,0,1,0,0,0,0,3],[28,32,0.875,0.57589,0.30407,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,11,0,0,9,0,0,0,0,0,2,0,0,0,0,10],[32,32,1.0,0.45089,0.25281,0.28571,0.28571,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,17,0,0,9,0,0,0,0,0,1,0,0,0,0,5]]},{"b":3,"e":1.0,"k":"flat","v":0.71429,"x":0.85714,"p":[[0,8,0.0,0.79911,0.22548,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,7,0,0,1,0,16],[4,8,0.5,0.71429,0.26726,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,13],[8,8,1.0,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16]]}]},{"i":"90de7c8d3a48842b","q":"In a scalene triangle  $ABC$  with  $\\angle A = 90^\\circ,$  the tangent line at  $A$  to its circumcircle meets line  $BC$  at  $M$  and the incircle touches  $AC$  at  $S$  and  $AB$  at  $R.$  \r\nThe lines  $RS$  and  $BC$  intersect at  $N,$  while the lines  $AM$  and  $SR$  intersect at  $U.$  \r\nProve that the triangle  $UMN$  is isosceles.","t":[{"b":2,"e":0.0,"k":"falling","v":0.2366,"x":0.49999,"p":[[0,48,0.0,0.49999,0.27664,0.28571,0.42857,0.85704,0.0,1.0,1,3,0,1,0,0,0,0,14,0,0,5,0,0,3,0,0,0,0,0,6,0,3],[4,48,0.0833,0.35714,0.2369,0.28571,0.28571,0.42857,0.0,1.0,4,2,0,4,0,0,0,0,18,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[8,48,0.1667,0.40625,0.24513,0.28571,0.28571,0.42857,0.0,1.0,1,4,0,1,0,1,0,0,17,0,0,8,0,0,1,0,0,0,0,0,0,0,4],[12,48,0.25,0.28125,0.2004,0.2857,0.28571,0.28571,0.0,1.0,6,1,0,6,0,0,0,0,22,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[16,48,0.3333,0.35268,0.26241,0.28571,0.28571,0.32143,0.0,1.0,4,3,0,4,0,1,0,0,19,0,0,3,0,0,1,0,0,0,0,0,1,0,3],[20,48,0.4167,0.33929,0.19805,0.28571,0.28571,0.42857,0.0,1.0,2,2,0,2,0,1,0,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,2],[24,48,0.5,0.28125,0.1838,0.2857,0.28571,0.28571,0.0,0.85714,6,0,0,6,0,1,0,0,18,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[28,48,0.5833,0.3125,0.14914,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[32,48,0.6667,0.30804,0.18935,0.28571,0.28571,0.28571,0.0,1.0,3,1,0,3,0,1,0,0,23,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[36,48,0.75,0.37052,0.22262,0.28571,0.28571,0.42857,0.0,1.0,1,3,0,1,0,1,0,0,21,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[40,48,0.8333,0.34821,0.26229,0.2857,0.28571,0.28571,0.0,1.0,4,3,0,4,0,1,0,0,20,0,0,2,0,0,1,0,0,0,0,0,1,0,3],[44,48,0.9167,0.2366,0.13175,0.2857,0.28571,0.28571,0.0,0.42857,7,0,0,7,0,0,0,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.2857,0.12369,0.2857,0.28571,0.28571,0.0,0.571,3,0,0,3,0,2,0,0,20,0,0,6,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.21876,"x":0.4732,"p":[[0,70,0.0,0.4732,0.27301,0.28571,0.35714,0.60682,0.0,1.0,1,4,0,1,0,0,0,0,15,0,0,7,0,0,1,0,0,1,0,0,3,0,4],[4,70,0.0571,0.31695,0.17027,0.28571,0.28571,0.32143,0.0,1.0,3,1,0,3,0,0,0,0,21,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[8,70,0.1143,0.26784,0.13713,0.2857,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,0,0,0,23,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,70,0.1714,0.29464,0.06121,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,70,0.2286,0.27232,0.1488,0.2857,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,1,0,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.21876,0.14279,0.0,0.28571,0.28571,0.0,0.4286,9,0,0,9,0,0,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.24107,0.11538,0.2857,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,2,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.27678,0.14698,0.2857,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,2,0,0,21,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[32,70,0.4571,0.28125,0.09095,0.28571,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,1,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.28571,0.08748,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.28571,0.06186,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.24553,0.12492,0.2857,0.28571,0.28571,0.0,0.4286,6,0,0,6,0,0,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.27232,0.10926,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,2,0,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.30357,0.16656,0.28571,0.28571,0.28571,0.0,1.0,3,1,0,3,0,0,0,0,24,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[60,70,0.8571,0.25893,0.13092,0.28571,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,1,0,0,22,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[64,70,0.9143,0.29464,0.06121,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.26339,0.11355,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,1,0,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.25893,0.12078,0.2857,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,0,0,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bb3f30d735370448","q":"Let  $a$  and  $b$  be positive integers, and let  $A$  and  $B$  be finite sets of integers satisfying\n(i)  $A$  and  $B$  are disjoint;\n(ii) if an integer  $i$  belongs to either to  $A$  or to  $B$ , then either  $i+a$  belongs to  $A$  or  $i-b$  belongs to  $B$ .\nProve that  $a\\left\\lvert A \\right\\rvert = b \\left\\lvert B \\right\\rvert$ .  (Here  $\\left\\lvert X \\right\\rvert$  denotes the number of elements in the set  $X$ .)","t":[{"b":0,"e":1.0,"k":"flat","v":0.70982,"x":0.95969,"p":[[0,53,0.0,0.70982,0.3668,0.42857,0.92857,1.0,0.0,1.0,4,16,2,4,0,1,0,0,1,0,0,4,0,0,2,0,0,0,0,0,4,0,16],[4,53,0.0755,0.88393,0.20652,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,18],[8,53,0.1509,0.875,0.1948,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,16],[12,53,0.2264,0.72768,0.33949,0.67857,0.85714,1.0,0.0,1.0,5,11,0,5,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,11],[16,53,0.3019,0.79462,0.20807,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,10,0,9],[20,53,0.3774,0.87498,0.13246,0.82132,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,10,0,14],[24,53,0.4528,0.79911,0.2985,0.85714,0.85714,1.0,0.0,1.0,2,14,0,2,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,14],[28,53,0.5283,0.95969,0.08955,1.0,1.0,1.0,0.71,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[32,53,0.6038,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[36,53,0.6792,0.92857,0.18211,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,23],[40,53,0.7547,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[44,53,0.8302,0.89732,0.20589,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,4,0,23],[48,53,0.9057,0.86606,0.21708,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,7,0,19],[52,53,0.9811,0.78125,0.29663,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,0,6,0,16],[53,53,1.0,0.81695,0.24547,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,3,0,17]]},{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.80804,"p":[[0,36,0.0,0.75447,0.33926,0.57143,0.92857,1.0,0.0,1.0,4,16,1,4,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,5,0,16],[4,36,0.1111,0.80804,0.31055,0.85714,0.92857,1.0,0.0,1.0,3,16,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,16],[8,36,0.2222,0.71873,0.26604,0.57142,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,4,0,0,1,0,0,4,0,0,7,0,0,6,0,9],[12,36,0.3333,0.52667,0.39039,0.14214,0.57143,0.89286,0.0,1.0,7,8,0,7,0,3,0,0,3,0,0,1,0,0,4,0,0,2,0,0,4,0,8],[16,36,0.4444,0.5848,0.3524,0.42857,0.71429,0.85714,0.0,1.0,7,5,0,7,0,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,7,0,5],[20,36,0.5556,0.48658,0.36571,0.0,0.57143,0.85714,0.0,1.0,10,3,0,10,0,0,0,0,1,0,0,2,0,0,6,0,0,4,0,0,6,0,3],[24,36,0.6667,0.31695,0.38586,0.0,0.0,0.60714,0.0,1.0,18,3,0,18,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,4,0,3],[28,36,0.7778,0.12054,0.24251,0.0,0.0,0.03572,0.0,1.0,24,1,0,24,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[32,36,0.8889,0.03124,0.12228,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63d6d3feded5a23b","q":"Let $A$ and $B$ be sets of positive integers with $|A| \\geq 2$ and $|B| \\geq 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $a b$ where $a \\in A$ and $b \\in B$. Prove that there exist pairwise distinct $x, y, z \\in S$ such that $x$ is a divisor of $y z$.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.23205,"x":0.36607,"p":[[0,44,0.0,0.23205,0.13249,0.14286,0.2857,0.28571,0.0,0.4286,4,0,1,4,0,10,0,0,12,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.33035,0.19376,0.24999,0.28571,0.42857,0.0,1.0,3,1,0,3,0,5,0,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,1],[8,44,0.1818,0.2633,0.12437,0.14286,0.28571,0.32143,0.0,0.4286,2,0,0,2,0,9,0,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.27232,0.14445,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,5,0,0,14,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[16,44,0.3636,0.35713,0.1956,0.25,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,10,0,0,9,0,0,2,0,0,2,0,0,0,0,1],[20,44,0.4545,0.30804,0.1992,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,9,0,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,1],[24,44,0.5455,0.32812,0.17934,0.28571,0.28571,0.42857,0.0,1.0,3,1,0,3,0,3,0,0,11,0,1,13,0,0,0,0,0,0,0,0,0,0,1],[28,44,0.6364,0.32155,0.18207,0.24999,0.28571,0.42857,0.0,1.0,2,1,0,2,0,6,0,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,1],[32,44,0.7273,0.33036,0.13092,0.25,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,7,0,0,5,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.35268,0.13356,0.2857,0.35714,0.42857,0.14286,0.85714,0,0,0,0,0,4,0,0,12,0,0,15,0,0,0,0,0,0,0,0,1,0,0],[40,44,0.9091,0.33036,0.12078,0.28571,0.35714,0.42857,0.0,0.4286,2,0,0,2,0,2,0,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.36607,0.14698,0.2857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.01786,"x":0.38839,"p":[[0,46,0.0,0.22768,0.14664,0.14286,0.14286,0.42857,0.0,0.4286,4,0,0,4,0,14,0,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.37044,0.25226,0.24999,0.28571,0.42857,0.0,1.0,1,3,0,1,0,7,0,0,11,0,0,9,0,0,0,0,0,0,0,0,1,0,3],[8,46,0.1739,0.33044,0.25382,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,9,0,0,6,0,0,0,0,0,0,0,0,2,0,2],[12,46,0.2609,0.29018,0.2004,0.14286,0.2143,0.42857,0.0,1.0,2,1,0,2,0,14,0,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,1],[16,46,0.3478,0.38839,0.25564,0.2857,0.28571,0.42857,0.0,1.0,2,2,0,2,0,4,0,0,12,0,0,9,0,0,0,0,0,0,0,0,3,0,2],[20,46,0.4348,0.34375,0.22829,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,8,0,0,9,0,0,8,0,0,1,0,0,2,0,0,1,0,1],[24,46,0.5217,0.30804,0.26513,0.14286,0.28571,0.42858,0.0,1.0,5,3,0,5,0,8,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,3],[28,46,0.6087,0.3125,0.22142,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,8,0,0,7,0,0,9,0,0,1,0,0,1,0,0,2,0,0],[32,46,0.6957,0.32134,0.20524,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,11,0,0,7,0,0,10,0,0,0,0,0,2,0,0,0,0,1],[36,46,0.7826,0.19615,0.20443,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,0,12,0,0,3,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[40,46,0.8696,0.25447,0.19475,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,6,0,0,12,0,0,0,0,0,1,0,0,0,0,0],[44,46,0.9565,0.20089,0.14664,0.10714,0.21428,0.28571,0.0,0.42857,8,0,0,8,0,8,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6fa4e91d47afd5ca","q":"Let $A, B, C, D$ be four points on the same circle. Denote $A^{\\prime}$ and $C^{\\prime}$ as the orthogonal projections of $A$ and $C$ onto $(B D)$, and $B^{\\prime}$ and $D^{\\prime}$ as the orthogonal projections of $B$ and $D$ onto $(A C)$. Show that $A^{\\prime}, B^{\\prime}, C^{\\prime}, D^{\\prime}$ are concyclic.","t":[{"b":3,"e":0.2857,"k":"flat","v":0.23205,"x":0.32142,"p":[[0,111,0.0,0.25446,0.15458,0.14289,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,21,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[4,111,0.036,0.26785,0.08564,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,4,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,111,0.0721,0.26785,0.11152,0.2857,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,3,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,111,0.1081,0.25446,0.16262,0.14286,0.28571,0.28571,0.0,1.0,3,1,0,3,0,6,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,111,0.1441,0.26785,0.04724,0.2857,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,111,0.1802,0.32142,0.17496,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,2,0,0,25,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[24,111,0.2162,0.28123,0.11561,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,4,0,0,25,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,111,0.2523,0.30804,0.15612,0.2857,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[32,111,0.2883,0.25446,0.07771,0.2857,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,111,0.3243,0.25447,0.11701,0.25,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,6,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,111,0.3604,0.27678,0.07087,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,111,0.3964,0.26339,0.11904,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,5,0,0,23,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,111,0.4324,0.25446,0.09268,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,111,0.4685,0.23205,0.0995,0.14286,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,7,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,111,0.5045,0.28571,0.13363,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,2,0,0,26,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[60,111,0.5405,0.26776,0.04748,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,111,0.5766,0.24553,0.08917,0.25,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,6,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,111,0.6126,0.24553,0.07349,0.24999,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,111,0.6486,0.25446,0.07771,0.2857,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,111,0.6847,0.29464,0.11811,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,26,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[80,111,0.7207,0.29016,0.14931,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,6,0,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[84,111,0.7568,0.25875,0.08353,0.2857,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,3,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.24107,0.13092,0.14289,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[92,111,0.8288,0.2634,0.11356,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,4,0,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[96,111,0.8649,0.27232,0.16888,0.14289,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,6,0,0,20,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[100,111,0.9009,0.24553,0.07349,0.25,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,111,0.9369,0.26339,0.13415,0.14286,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,7,0,0,20,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[108,111,0.973,0.26785,0.1171,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,4,0,0,24,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[111,111,1.0,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.16518,"x":0.30803,"p":[[0,70,0.0,0.30803,0.17896,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,4,0,0,24,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[4,70,0.0571,0.29464,0.13333,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,70,0.1143,0.25894,0.10972,0.25,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,7,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,70,0.1714,0.28558,0.09388,0.2857,0.28571,0.28571,0.0,0.71,1,0,0,1,0,1,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,70,0.2286,0.26786,0.15464,0.1429,0.28571,0.28571,0.0,1.0,1,1,0,1,0,8,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,70,0.2857,0.27232,0.10926,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,5,0,0,24,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,70,0.3429,0.24553,0.08171,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,5,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.28125,0.12619,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,7,0,0,23,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[32,70,0.4571,0.23214,0.08564,0.14289,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.20527,0.14262,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,9,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,70,0.5714,0.24544,0.13946,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,8,0,0,19,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[44,70,0.6286,0.24999,0.12372,0.24999,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,5,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,70,0.6857,0.24108,0.12078,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,9,0,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,70,0.7429,0.20526,0.09413,0.14286,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.22759,0.1002,0.14286,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,5,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.21866,0.10098,0.14286,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,7,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.24098,0.13099,0.14286,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,5,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[68,70,0.9714,0.18303,0.1197,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.16518,0.19597,0.0,0.14286,0.2857,0.0,0.85714,12,0,0,12,0,10,0,0,8,0,0,0,0,0,0,0,0,1,0,0,1,0,0]]}]},{"i":"ab5360b7ff6632be","q":"Let $A$ be a set of positive integers containing the number 1 and at least one more element. Given that for any two different elements $m, n$ of $A$ the number $\\frac{m+1}{(m+1, n+1)}$ is also an element of $A$, prove that $A$ coincides with the set of positive integers.","t":[{"b":2,"e":0.0,"k":"falling","v":0.03572,"x":0.683,"p":[[0,63,0.0,0.683,0.16652,0.67857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,18,0,0,4,0,2],[4,63,0.0635,0.46873,0.38997,0.0,0.64264,0.85704,0.0,1.0,10,4,0,10,0,3,0,0,1,0,0,1,0,0,1,0,0,7,0,0,5,0,4],[8,63,0.127,0.45982,0.3792,0.0,0.42857,0.75,0.0,1.0,9,6,0,9,0,2,0,0,3,0,0,4,0,0,1,0,0,5,0,0,2,0,6],[12,63,0.1905,0.40179,0.3787,0.0,0.28571,0.71429,0.0,1.0,11,5,0,11,0,2,0,0,4,0,0,3,0,0,1,0,0,4,0,0,2,0,5],[16,63,0.254,0.37946,0.40027,0.0,0.21429,0.75,0.0,1.0,15,3,0,15,0,1,0,0,2,0,0,0,0,0,1,0,0,5,0,0,5,0,3],[20,63,0.3175,0.32142,0.37114,0.0,0.14286,0.71429,0.0,1.0,14,4,0,14,0,4,0,0,3,0,0,0,0,0,1,0,0,6,0,0,0,0,4],[24,63,0.381,0.44195,0.37688,0.0,0.42857,0.71429,0.0,1.0,9,6,1,9,0,3,0,0,3,0,0,3,0,0,2,0,0,5,0,0,1,0,6],[28,63,0.4444,0.34822,0.36759,0.0,0.14288,0.71429,0.0,1.0,12,3,0,12,0,6,0,0,1,0,0,0,0,0,3,0,0,5,0,0,2,0,3],[32,63,0.5079,0.24541,0.27925,0.0,0.14286,0.42858,0.0,0.85714,14,0,0,14,0,4,0,0,3,0,0,6,0,0,0,0,0,3,0,0,2,0,0],[36,63,0.5714,0.29464,0.3173,0.0,0.2143,0.4286,0.0,1.0,13,2,0,13,0,3,0,0,3,0,0,6,0,0,1,0,0,3,0,0,1,0,2],[40,63,0.6349,0.26785,0.3025,0.0,0.14293,0.42857,0.0,1.0,14,1,0,14,0,3,0,0,3,0,0,6,0,0,1,0,0,2,0,0,2,0,1],[44,63,0.6984,0.29911,0.33761,0.0,0.14286,0.60714,0.0,1.0,14,2,0,14,0,4,0,0,1,0,0,4,0,0,1,0,0,5,0,0,1,0,2],[48,63,0.7619,0.28124,0.3184,0.0,0.14286,0.57111,0.0,1.0,14,1,0,14,0,5,0,0,0,0,0,4,0,0,2,0,0,5,0,0,1,0,1],[52,63,0.8254,0.08034,0.16338,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[56,63,0.8889,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.0491,0.11627,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[63,63,1.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.50437,"x":0.82586,"p":[[0,82,0.0,0.68755,0.18357,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,17,0,0,2,0,4],[4,82,0.0488,0.82586,0.17764,0.71429,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,5,0,13],[8,82,0.0976,0.77231,0.19841,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,3,0,9],[12,82,0.1463,0.72321,0.2765,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,2,0,0,2,0,0,1,0,0,12,0,0,3,0,10],[16,82,0.1951,0.57142,0.35535,0.39286,0.71429,0.85714,0.0,1.0,7,7,0,7,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,2,0,7],[20,82,0.2439,0.62052,0.32851,0.53539,0.71429,0.85704,0.0,1.0,5,6,0,5,0,1,0,0,1,0,0,1,0,0,2,0,0,13,0,0,3,0,6],[24,82,0.2927,0.71873,0.27776,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,0,0,0,2,0,0,13,0,0,2,0,10],[28,82,0.3415,0.61161,0.36811,0.39286,0.71429,0.85714,0.0,1.0,7,7,0,7,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,7,0,7],[32,82,0.3902,0.68302,0.32876,0.53539,0.78564,1.0,0.0,1.0,4,9,0,4,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,7,0,9],[36,82,0.439,0.61606,0.33776,0.42859,0.71429,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,1,0,0,2,0,0,3,0,0,11,0,0,1,0,8],[40,82,0.4878,0.59822,0.34706,0.39285,0.71429,0.85714,0.0,1.0,5,7,0,5,0,2,0,0,1,0,0,3,0,0,1,0,0,10,0,0,3,0,7],[44,82,0.5366,0.72321,0.29653,0.57142,0.78571,1.0,0.0,1.0,2,11,0,2,0,1,0,0,1,0,0,3,0,0,2,0,0,7,0,0,5,0,11],[48,82,0.5854,0.70982,0.25626,0.71429,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,2,0,0,1,0,0,1,0,0,14,0,0,6,0,6],[52,82,0.6341,0.68749,0.34707,0.57143,0.78571,1.0,0.0,1.0,5,10,0,5,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,0,6,0,10],[56,82,0.6829,0.54464,0.30606,0.28571,0.71429,0.75,0.0,1.0,3,2,0,3,0,4,0,0,2,0,0,5,0,0,1,0,0,9,0,0,6,0,2],[60,82,0.7317,0.67409,0.3159,0.53539,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,2,0,0,2,0,0,2,0,0,9,0,0,4,0,9],[64,82,0.7805,0.50437,0.32933,0.2857,0.42859,0.85714,0.0,1.0,4,4,0,4,0,3,0,0,4,0,0,8,0,0,1,0,0,2,0,0,6,0,4],[68,82,0.8293,0.68297,0.28737,0.571,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,3,0,0,0,0,0,4,0,0,9,0,0,6,0,7],[72,82,0.878,0.62047,0.27805,0.4286,0.71429,0.85704,0.0,1.0,2,5,0,2,0,1,0,0,3,0,0,3,0,0,6,0,0,8,0,0,4,0,5],[76,82,0.9268,0.62497,0.26667,0.5354,0.71429,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,2,0,0,3,0,0,6,0,0,11,0,0,2,0,5],[80,82,0.9756,0.69195,0.2578,0.67857,0.71429,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,3,0,0,1,0,0,9,0,0,12,0,3],[82,82,1.0,0.6696,0.18365,0.571,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,12,0,0,10,0,0]]}]},{"i":"abfa37604136e2a7","q":"Let $B$ be a set of integers either bounded below or bounded above. Then show that if $S$ tiles all other integers $\\mathbf{Z} \\backslash B$, then $S$ tiles all integers $\\mathbf{Z}$.","t":[{"b":3,"e":0.2857,"k":"flat","v":0.14268,"x":0.3884,"p":[[0,21,0.0,0.23214,0.23623,0.14286,0.14286,0.17857,0.0,1.0,3,1,0,3,0,21,0,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,1],[4,21,0.1905,0.3884,0.30143,0.14286,0.28571,0.4286,0.0,1.0,1,4,0,1,0,10,0,0,10,0,0,4,0,0,0,0,0,1,0,0,2,0,4],[8,21,0.381,0.36598,0.31938,0.14286,0.21428,0.57143,0.0,1.0,3,4,0,3,0,13,0,0,4,0,0,3,0,0,2,0,0,2,0,0,1,0,4],[12,21,0.5714,0.34826,0.28109,0.14286,0.2857,0.42895,0.0,1.0,3,3,0,3,0,11,0,0,5,0,0,6,0,0,2,0,0,2,0,0,0,0,3],[16,21,0.7619,0.20982,0.14279,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,13,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,21,0.9524,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.14268,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.15179,"x":0.51337,"p":[[0,21,0.0,0.25884,0.27769,0.14286,0.14286,0.14286,0.0,1.0,3,3,0,3,0,22,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,3],[4,21,0.1905,0.51337,0.30274,0.28571,0.4286,0.75,0.14286,1.0,0,6,0,0,0,7,0,0,4,0,0,7,0,0,5,0,0,1,0,0,2,0,6],[8,21,0.381,0.49097,0.29878,0.2857,0.42857,0.71429,0.0,1.0,1,4,0,1,0,6,0,0,6,0,0,5,0,0,4,0,0,3,0,0,3,0,4],[12,21,0.5714,0.38816,0.2505,0.14286,0.28571,0.4642,0.14,1.0,0,2,0,0,0,9,0,0,10,0,0,5,0,0,2,0,0,3,0,0,1,0,2],[16,21,0.7619,0.27232,0.21828,0.14286,0.14288,0.42857,0.0,1.0,2,1,0,2,0,16,0,0,5,0,0,6,0,0,1,0,0,0,0,0,1,0,1],[20,21,0.9524,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.15179,0.10677,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,29,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"d7abc3ebed4da1cb","q":"Let $ABC$ be a triangle. Let $P$ belong to the circumcircle. We know that the projections of $P$ onto $(BC)$, $(CA)$, and $(AB)$ are aligned on the line known as the Simson line. We assume that this line passes through the point diametrically opposite to $P$. Show that it also passes through the centroid of $ABC$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.12946,"x":0.95982,"p":[[0,50,0.0,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[4,50,0.08,0.71428,0.37458,0.28571,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,19],[8,50,0.16,0.53125,0.41685,0.10717,0.5,1.0,0.0,1.0,8,11,0,8,0,1,0,0,7,0,0,0,0,0,0,0,0,3,0,0,2,0,11],[12,50,0.24,0.37053,0.37773,0.0,0.2857,0.71429,0.0,1.0,10,7,0,10,0,2,0,0,11,0,0,0,0,0,0,0,0,2,0,0,0,0,7],[16,50,0.32,0.45982,0.41455,0.0,0.28571,1.0,0.0,1.0,10,10,0,10,0,1,0,0,7,0,0,0,0,0,2,0,0,2,0,0,0,0,10],[20,50,0.4,0.54464,0.3787,0.2857,0.28571,1.0,0.0,1.0,4,12,0,4,0,0,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,12],[24,50,0.48,0.42856,0.39123,0.0,0.28571,0.78571,0.0,1.0,10,8,0,10,0,0,0,0,9,0,0,1,0,0,1,0,0,3,0,0,0,0,8],[28,50,0.56,0.41071,0.37584,0.0,0.28571,0.74996,0.0,1.0,9,7,0,9,0,1,0,0,10,0,0,1,0,0,2,0,0,1,0,0,1,0,7],[32,50,0.64,0.38839,0.35035,0.10714,0.28571,0.57143,0.0,1.0,8,6,0,8,0,1,0,0,12,0,0,2,0,0,2,0,0,0,0,0,1,0,6],[36,50,0.72,0.20089,0.24186,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,3,0,0,11,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[40,50,0.8,0.21875,0.26721,0.0,0.2143,0.28571,0.0,1.0,14,1,0,14,0,2,0,0,12,0,0,0,0,0,1,0,0,0,0,0,2,0,1],[44,50,0.88,0.19196,0.20705,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,2,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[48,50,0.96,0.12946,0.18336,0.0,0.0,0.28571,0.0,0.71429,20,0,0,20,0,0,0,0,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[50,50,1.0,0.18303,0.19638,0.0,0.21428,0.28571,0.0,0.71429,15,0,0,15,0,1,0,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.05357,"x":0.8125,"p":[[0,54,0.0,0.8125,0.3719,0.96429,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,24],[4,54,0.0741,0.44196,0.37857,0.2857,0.28571,1.0,0.0,1.0,7,9,0,7,0,0,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,9],[8,54,0.1481,0.50446,0.39927,0.2857,0.28571,1.0,0.0,1.0,6,12,0,6,0,0,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,12],[12,54,0.2222,0.30357,0.31693,0.0,0.28571,0.28571,0.0,1.0,11,3,0,11,0,1,0,0,13,0,0,0,0,0,1,0,0,2,0,0,1,0,3],[16,54,0.2963,0.46427,0.35714,0.2857,0.28571,0.71429,0.0,1.0,7,7,0,7,0,0,0,0,10,0,0,0,0,0,5,0,0,3,0,0,0,0,7],[20,54,0.3704,0.31249,0.32817,0.0,0.28571,0.42857,0.0,1.0,12,3,0,12,0,0,0,0,11,0,0,2,0,0,1,0,0,1,0,0,2,0,3],[24,54,0.4444,0.2366,0.21312,0.0,0.28571,0.28571,0.0,1.0,10,1,0,10,0,0,0,0,19,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[28,54,0.5185,0.13839,0.19719,0.0,0.0,0.28571,0.0,0.71429,19,0,0,19,0,2,0,0,8,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[32,54,0.5926,0.17857,0.13832,0.0,0.2857,0.28571,0.0,0.28571,12,0,0,12,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.21428,0.15972,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,0,0,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[40,54,0.7407,0.13839,0.15355,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,1,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.16071,0.17768,0.0,0.14285,0.28571,0.0,0.71429,16,0,0,16,0,0,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,54,0.8889,0.16071,0.16656,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,3,0,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,54,0.963,0.14286,0.22868,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[54,54,1.0,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10556b2205cc2440","q":"Let $A B C D$ be a trapezium with $A B \\| C D, 2|A B|=|C D|$ and $B D \\perp B C$. Let $M$ be the midpoint of $C D$ and let $E$ be the intersection of $B C$ and $A D$. Let $O$ be the intersection of $A M$ and $B D$. Let $N$ be the intersection of $O E$ and $A B$.\n(a) Prove that $A B M D$ is a rhombus.\n(b) Prove that the line $D N$ goes through the midpoint of line segment $B E$.","t":[{"b":4,"e":0.42857,"k":"falling","v":0.43317,"x":0.82589,"p":[[0,32,0.0,0.82589,0.21048,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,9,0,0,4,0,15],[4,32,0.125,0.65623,0.09353,0.57143,0.64286,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,13,0,0,3,0,0],[8,32,0.25,0.67854,0.16753,0.57143,0.71429,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,9,0,0,5,0,3],[12,32,0.375,0.67409,0.13475,0.57143,0.57143,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,7,0,0,7,0,1],[16,32,0.5,0.63384,0.17836,0.42857,0.64271,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,8,0,0,7,0,0,10,0,0,4,0,2],[20,32,0.625,0.57142,0.13832,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,11,0,0,6,0,0,3,0,0],[24,32,0.75,0.55356,0.12242,0.53539,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,5,0,0,17,0,0,7,0,0,0,0,0],[28,32,0.875,0.61159,0.16065,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,4,0,0,1,0,0,13,0,0,10,0,0,4,0,0],[32,32,1.0,0.43317,0.12087,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,5,0,0,15,0,0,10,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"falling","v":0.62052,"x":0.83924,"p":[[0,44,0.0,0.83924,0.17411,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,4,0,15],[4,44,0.0909,0.68749,0.1154,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,16,0,0,6,0,0],[8,44,0.1818,0.7232,0.17836,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,8,0,0,9,0,4],[12,44,0.2727,0.72768,0.14445,0.67857,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,14,0,0,7,0,3],[16,44,0.3636,0.65179,0.1234,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,8,0,0,6,0,0],[20,44,0.4545,0.70981,0.14502,0.57143,0.71429,0.75,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,11,0,0,4,0,4],[24,44,0.5455,0.62052,0.12168,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,9,0,0,3,0,0],[28,44,0.6364,0.66518,0.11071,0.57143,0.64286,0.71429,0.5714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,12,0,0,3,0,1],[32,44,0.7273,0.62493,0.10568,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,8,0,0,3,0,0],[36,44,0.8182,0.67857,0.11294,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,13,0,0,4,0,1],[40,44,0.9091,0.6428,0.11848,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,8,0,0,3,0,1],[44,44,1.0,0.62942,0.08648,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,12,0,0,1,0,0]]}]},{"i":"996e183faa004d1d","q":"Let  $a, b, c$  be non-zero real numbers that satisfy  $\\frac{1}{abc} + \\frac{1}{a} + \\frac{1}{c} = \\frac{1}{b}$ . The expression  $\\frac{4}{a^2 + 1} + \\frac{4}{b^2 + 1} + \\frac{7}{c^2 + 1}$  has a maximum value  $M$ . Find the sum of the numerator and denominator of the reduced form of  $M$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.75893,"x":0.95089,"p":[[0,84,0.0,0.83929,0.19805,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,7,0,15],[4,84,0.0476,0.82589,0.21048,0.82143,0.85714,1.0,0.0,1.0,1,11,1,1,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,13,0,11],[8,84,0.0952,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[12,84,0.1429,0.75893,0.23808,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,0,17,0,5],[16,84,0.1905,0.89286,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[20,84,0.2381,0.875,0.17035,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,12,0,15],[24,84,0.2857,0.83036,0.15335,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,14,0,9],[28,84,0.3333,0.86161,0.16554,0.82143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,10,0,14],[32,84,0.381,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[36,84,0.4286,0.83482,0.18935,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,8,0,14],[40,84,0.4762,0.89286,0.12372,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,9,0,16],[44,84,0.5238,0.87054,0.16506,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,16],[48,84,0.5714,0.87946,0.17536,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,13,0,15],[52,84,0.619,0.875,0.21651,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,18],[56,84,0.6667,0.90625,0.1411,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,19],[60,84,0.7143,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[64,84,0.7619,0.88393,0.18707,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,6,0,19],[68,84,0.8095,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[72,84,0.8571,0.91964,0.11811,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,20],[76,84,0.9048,0.89286,0.18898,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,20],[80,84,0.9524,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[84,84,1.0,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25]]},{"b":6,"e":1.0,"k":"rising","v":0.76786,"x":1.0,"p":[[0,268,0.0,0.79018,0.22583,0.71429,0.85714,1.0,0.0,1.0,1,11,1,1,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,8,0,11],[4,268,0.0149,0.83036,0.23538,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,4,0,0,7,0,16],[8,268,0.0299,0.91071,0.13716,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,20],[12,268,0.0448,0.82589,0.19799,0.82143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,12,0,12],[16,268,0.0597,0.875,0.19804,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,9,0,18],[20,268,0.0746,0.86607,0.20806,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,4,0,0,5,0,19],[24,268,0.0896,0.88839,0.1665,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,8,0,18],[28,268,0.1045,0.87946,0.18935,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,19],[32,268,0.1194,0.86161,0.16554,0.82143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,10,0,14],[36,268,0.1343,0.90179,0.1448,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,19],[40,268,0.1493,0.87946,0.1992,0.85714,1.0,1.0,0.0,1.0,1,17,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,17],[44,268,0.1642,0.88839,0.13709,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,12,0,15],[48,268,0.1791,0.89286,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[52,268,0.194,0.85268,0.22442,0.85714,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,9,0,16],[56,268,0.209,0.83482,0.2055,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,8,0,15],[60,268,0.2239,0.90179,0.1448,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,11,0,17],[64,268,0.2388,0.80357,0.19805,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,11,0,10],[68,268,0.2537,0.85268,0.16554,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,13,0,12],[72,268,0.2687,0.88839,0.12745,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,13,0,14],[76,268,0.2836,0.83482,0.18595,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,8,0,14],[80,268,0.2985,0.88839,0.17029,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,11,0,17],[84,268,0.3134,0.83929,0.18123,0.82143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,12,0,12],[88,268,0.3284,0.84375,0.16115,0.82143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,13,0,11],[92,268,0.3433,0.83036,0.15335,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,14,0,9],[96,268,0.3582,0.88393,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,16,0,12],[100,268,0.3731,0.82589,0.18118,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,7,0,13],[104,268,0.3881,0.875,0.18814,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,12,0,16],[108,268,0.403,0.83482,0.15612,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,13,0,10],[112,268,0.4179,0.84375,0.18681,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,6,0,15],[116,268,0.4328,0.83929,0.15047,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,11,0,11],[120,268,0.4478,0.82143,0.16366,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,9,0,11],[124,268,0.4627,0.82143,0.14725,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,14,0,8],[128,268,0.4776,0.83036,0.15746,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,10,0,11],[132,268,0.4925,0.83482,0.16793,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,10,0,12],[136,268,0.5075,0.87054,0.13054,0.82143,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,11,0,13],[140,268,0.5224,0.82589,0.17399,0.67857,0.85714,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,7,0,13],[144,268,0.5373,0.80804,0.19434,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,10,0,11],[148,268,0.5522,0.76786,0.18814,0.57143,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,4,0,0,11,0,7],[152,268,0.5672,0.79911,0.18161,0.57143,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,5,0,12],[156,268,0.5821,0.79464,0.18536,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,12,0,9],[160,268,0.597,0.82589,0.14167,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,14,0,8],[164,268,0.6119,0.77679,0.18189,0.57143,0.78571,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,7,0,9],[168,268,0.6269,0.80357,0.17405,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,12,0,9],[172,268,0.6418,0.81696,0.17215,0.57143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,10,0,11],[176,268,0.6567,0.87054,0.15714,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,10,0,15],[180,268,0.6716,0.81696,0.18977,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,9,0,12],[184,268,0.6866,0.76786,0.20124,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,2,0,0,9,0,9],[188,268,0.7015,0.80804,0.16213,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,8,0,10],[192,268,0.7164,0.81696,0.17582,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,10,0,11],[196,268,0.7313,0.82143,0.19562,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,5,0,14],[200,268,0.7463,0.80804,0.15815,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,10,0,9],[204,268,0.7612,0.83036,0.16536,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,11,0,11],[208,268,0.7761,0.83034,0.18709,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,10,0,13],[212,268,0.791,0.90625,0.09852,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,15],[216,268,0.806,0.88839,0.11143,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,13],[220,268,0.8209,0.85268,0.12103,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,18,0,8],[224,268,0.8358,0.875,0.11152,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,18,0,10],[228,268,0.8507,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[232,268,0.8657,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[236,268,0.8806,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[240,268,0.8955,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[244,268,0.9104,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[248,268,0.9254,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[252,268,0.9403,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[256,268,0.9552,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[260,268,0.9701,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[264,268,0.9851,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[268,268,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"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 $\\triangle ABC$  be a triangle with  $\\angle BAC>90^{\\circ}$ ,  $AB=5$  and  $AC=7$ . Points  $D$  and  $E$  lie on segment  $BC$  such that  $BD=DE=EC$ . If  $\\angle BAC+\\angle DAE=180^{\\circ}$ , compute  $BC$ 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2,0.42857,0.71429,0.85714,0.0,1.0,1,7,1,1,0,2,0,0,1,0,0,6,0,0,4,0,0,7,1,0,3,0,7]]}]},{"i":"237def2ca419abd7","q":"Let $A B C$ be a triangle. Its excircles touch sides $B C, C A, A B$ at $D, E, F$. Prove that the perimeter of triangle $A B C$ is at most twice that of triangle $D E F$.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.12054,"x":0.2142,"p":[[0,43,0.0,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.17848,0.15155,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,21,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,43,0.186,0.16509,0.15613,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,22,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,43,0.2791,0.2142,0.1429,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,18,0,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[16,43,0.3721,0.174,0.17027,0.14214,0.14286,0.14286,0.0,0.71429,7,0,0,7,0,19,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[20,43,0.4651,0.18304,0.16457,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,21,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[24,43,0.5581,0.15179,0.12846,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,43,0.6512,0.16518,0.09523,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,43,0.7442,0.15179,0.12339,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,43,0.8372,0.16965,0.19045,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,19,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[40,43,0.9302,0.16518,0.13882,0.14286,0.14286,0.14287,0.0,0.57143,7,0,0,7,0,18,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[43,43,1.0,0.12054,0.08828,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04893,"x":0.21875,"p":[[0,61,0.0,0.13821,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.1875,0.15746,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,21,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[8,61,0.1311,0.16964,0.16917,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,18,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[12,61,0.1967,0.21875,0.19227,0.14286,0.14286,0.1786,0.0,0.71429,3,0,0,3,0,21,0,0,3,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[16,61,0.2623,0.16963,0.09735,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,27,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,61,0.3279,0.15625,0.12037,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,25,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,61,0.3934,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,61,0.459,0.17857,0.17496,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,20,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[32,61,0.5246,0.17848,0.1557,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,21,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[36,61,0.5902,0.17411,0.11143,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,25,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,61,0.6557,0.12947,0.12037,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,18,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.10705,0.14723,0.0,0.14143,0.14286,0.0,0.71429,15,0,0,15,0,14,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,61,0.7869,0.07142,0.11288,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,61,0.8525,0.04893,0.07646,0.0,0.0,0.14071,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,61,0.918,0.09374,0.1317,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[60,61,0.9836,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b86f917c774e41a4","q":"In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00447,"x":0.03571,"p":[[0,43,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,46,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"afccbec8f5dbb7b3","q":"Let  $n$  be a positive integer. Let  $S$  be a set of ordered pairs  $(x, y)$  such that  $1\\leq x \\leq n$  and  $0 \\leq y \\leq n$  in each pair, and there are no pairs  $(a, b)$  and  $(c, d)$  of different elements in  $S$  such that  $a^2+b^2$  divides both $ac+bd$  and  $ad - bc$ . In terms of  $n$ , determine the size of the largest possible set  $S$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.15607,"x":0.55355,"p":[[0,32,0.0,0.26784,0.1324,0.14286,0.2857,0.28571,0.0,0.571,3,0,3,3,0,6,0,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,32,0.125,0.55355,0.28291,0.39286,0.57143,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,1,0,0,4,0,0,5,0,0,9,0,0,3,0,3],[8,32,0.25,0.29463,0.20495,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,18,0,0,3,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[12,32,0.375,0.20536,0.13333,0.14286,0.14286,0.17857,0.14286,0.71429,0,0,0,0,0,24,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,32,0.5,0.23197,0.16665,0.14286,0.14286,0.2857,0.14,0.85714,0,0,0,0,0,22,0,0,5,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[20,32,0.625,0.22301,0.12344,0.14286,0.14286,0.2857,0.14,0.571,0,0,0,0,0,20,0,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[24,32,0.75,0.16955,0.07527,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.16955,"x":0.48661,"p":[[0,30,0.0,0.28125,0.11564,0.2857,0.28571,0.32143,0.0,0.4286,2,0,2,2,0,5,0,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.48661,0.29636,0.24999,0.42857,0.75,0.0,1.0,1,1,0,1,0,7,0,0,6,0,0,4,0,0,1,0,0,5,0,0,7,0,1],[8,30,0.2667,0.30354,0.19801,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,14,0,0,6,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[12,30,0.4,0.25445,0.1776,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,17,0,0,4,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[16,30,0.5333,0.26339,0.21162,0.14286,0.14286,0.32143,0.0,1.0,1,1,0,1,0,19,0,0,4,0,0,5,0,0,0,0,0,2,0,0,0,0,1],[20,30,0.6667,0.24108,0.21852,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,16,0,0,5,0,0,5,0,0,0,0,0,0,0,0,1,0,1],[24,30,0.8,0.19643,0.12752,0.14286,0.14286,0.2857,0.0,0.71429,3,0,0,3,0,17,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,30,0.9333,0.17411,0.11143,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,17,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.16955,0.07527,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"abb46796136a36f6","q":"Let  $n$  be a positive integer. Let  $S$  be a set of  $n$  positive integers such that the greatest common divisors of all nonempty sets of  $S$  are distinct. Determine the smallest possible number of distinct prime divisors of the product of the elements of  $S$ .","t":[{"b":0,"e":0.42857,"k":"rising","v":0.28125,"x":0.65624,"p":[[0,70,0.0,0.32143,0.27199,0.0,0.42857,0.42857,0.0,0.85714,12,0,5,12,0,0,0,0,0,0,0,14,0,0,1,0,0,4,0,0,1,0,0],[4,70,0.0571,0.65624,0.23921,0.42857,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,13,0,0,6,0,0,3,0,0,1,0,9],[8,70,0.1143,0.62938,0.26469,0.42857,0.42857,1.0,0.14,1.0,0,9,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,4,0,0,1,0,9],[12,70,0.1714,0.54911,0.21162,0.42857,0.42857,0.57143,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,20,0,0,4,0,0,2,0,0,0,0,5],[16,70,0.2286,0.54464,0.28669,0.42857,0.50001,0.71429,0.0,1.0,4,5,0,4,0,0,0,0,0,0,0,12,0,0,5,0,0,5,0,0,1,0,5],[20,70,0.2857,0.54017,0.26422,0.42857,0.42857,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,1,0,0,15,0,0,3,0,0,3,0,0,3,0,4],[24,70,0.3429,0.54909,0.30328,0.42857,0.42857,0.75,0.0,1.0,3,7,0,3,0,1,0,0,1,0,0,14,0,0,2,0,0,3,0,0,1,0,7],[28,70,0.4,0.5625,0.30291,0.42857,0.4286,0.85714,0.0,1.0,3,7,0,3,0,0,0,0,1,0,0,16,0,0,1,0,0,1,0,0,3,0,7],[32,70,0.4571,0.53124,0.29501,0.42857,0.42859,0.71429,0.0,1.0,4,6,0,4,0,0,0,0,1,0,0,13,0,0,4,0,0,4,0,0,0,0,6],[36,70,0.5143,0.49107,0.28333,0.42857,0.42857,0.60714,0.0,1.0,4,4,0,4,0,1,0,0,2,0,0,12,0,0,5,0,0,3,0,0,1,0,4],[40,70,0.5714,0.625,0.29827,0.42857,0.4286,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,15,0,0,1,0,0,3,0,0,1,0,10],[44,70,0.6286,0.52228,0.24642,0.42857,0.49979,0.60714,0.0,1.0,3,3,0,3,0,0,0,0,1,0,0,12,0,0,8,0,0,4,0,0,1,0,3],[48,70,0.6857,0.57142,0.27432,0.42857,0.57143,0.71429,0.0,1.0,3,5,0,3,0,0,0,0,1,0,0,10,0,0,5,0,0,7,0,0,1,0,5],[52,70,0.7429,0.49107,0.31529,0.28571,0.42857,0.71429,0.0,1.0,4,6,0,4,0,1,0,0,5,0,0,12,0,0,0,0,0,3,0,0,1,0,6],[56,70,0.8,0.30804,0.2969,0.0,0.42857,0.42857,0.0,1.0,12,2,0,12,0,2,0,0,1,0,0,12,0,0,0,0,0,3,0,0,0,0,2],[60,70,0.8571,0.28125,0.289,0.0,0.28571,0.42857,0.0,1.0,13,2,0,13,0,1,0,0,3,0,0,12,0,0,0,0,0,0,0,0,1,0,2],[64,70,0.9143,0.54017,0.19474,0.42857,0.42857,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,22,0,0,3,0,0,3,0,0,0,0,4],[68,70,0.9714,0.52679,0.1448,0.42857,0.42857,0.60714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,20,0,0,4,0,0,7,0,0,0,0,1],[70,70,1.0,0.49107,0.15126,0.42857,0.42857,0.4286,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,26,0,0,2,0,0,2,0,0,0,0,2]]},{"b":4,"e":0.857,"k":"flat","v":0.43749,"x":0.66964,"p":[[0,63,0.0,0.43749,0.22569,0.42857,0.42857,0.57111,0.0,1.0,4,1,1,4,0,0,0,0,3,0,0,16,0,0,4,0,0,3,0,0,1,0,1],[4,63,0.0635,0.66964,0.28669,0.42857,0.64286,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,12,0,0,2,0,0,4,0,0,0,0,12],[8,63,0.127,0.66071,0.29613,0.42857,0.64286,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,11,0,0,3,0,0,4,0,0,1,0,11],[12,63,0.1905,0.64283,0.2342,0.42857,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,1,0,0,0,0,9],[16,63,0.254,0.61161,0.27487,0.42857,0.4286,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,16,0,0,1,0,0,4,0,0,0,0,9],[20,63,0.3175,0.66964,0.26107,0.42857,0.5,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,3,0,0,1,0,11],[24,63,0.381,0.61607,0.2683,0.42857,0.4286,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,16,0,0,3,0,0,3,0,0,0,0,9],[28,63,0.4444,0.5625,0.25738,0.42857,0.42857,0.71429,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,18,0,0,2,0,0,3,0,0,1,0,6],[32,63,0.5079,0.61606,0.26592,0.42857,0.4998,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,14,0,0,3,0,0,4,0,0,1,0,8],[36,63,0.5714,0.62946,0.30275,0.42857,0.42857,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,15,0,0,1,0,0,3,0,0,0,0,11],[40,63,0.6349,0.52232,0.22759,0.42857,0.42857,0.46431,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,23,0,0,2,0,0,1,0,0,0,0,5],[44,63,0.6984,0.60714,0.24223,0.42857,0.42857,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,19,0,0,2,0,0,3,0,0,0,0,8],[48,63,0.7619,0.59375,0.22618,0.42857,0.42857,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,19,0,0,2,0,0,4,0,0,1,0,6],[52,63,0.8254,0.59374,0.23176,0.42857,0.42859,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,19,0,0,3,0,0,3,0,0,0,0,7],[56,63,0.8889,0.58481,0.23517,0.42857,0.42857,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,21,0,0,1,0,0,3,0,0,0,0,7],[60,63,0.9524,0.58929,0.22798,0.42857,0.42857,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,20,0,0,1,0,0,4,0,0,1,0,6],[63,63,1.0,0.55357,0.19804,0.42857,0.42857,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,21,0,0,2,0,0,5,0,0,0,0,4]]}]},{"i":"e8e181492861d1f8","q":"Let $\\left(P_{n}\\right)_{n \\in \\mathbb{N}}$ be a sequence of polynomials with integer coefficients. Suppose there exists a monic polynomial $Q$ with integer coefficients such that $P_{n+1}-P_{n}=Q$ for all $n \\geqslant 0$. Furthermore, suppose that for all $\\mathrm{n} \\geqslant 0, \\mathrm{P}_{\\mathrm{n}}$ has an integer root. Show that we are in one of the following two cases:\n\n- $P_{0}$ and $Q$ have a common integer root\n- There exists a polynomial with integer coefficients $R$ such that $\\mathrm{P}_{0}=R Q$ and the degree of $R$ is 1.","t":[{"b":2,"e":1.0,"k":"flat","v":0.85489,"x":0.95535,"p":[[0,33,0.0,0.87054,0.1488,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,8,0,15],[4,33,0.1212,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[8,33,0.2424,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[12,33,0.3636,0.92856,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20],[16,33,0.4848,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[20,33,0.6061,0.85714,0.11845,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,13,0,10],[24,33,0.7273,0.85714,0.11845,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,13,0,10],[28,33,0.8485,0.89285,0.11294,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,14],[32,33,0.9697,0.8616,0.09771,0.85711,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,17,0,8],[33,33,1.0,0.85489,0.11081,0.85711,0.85714,0.87501,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,18,1,7]]},{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.96875,"p":[[0,35,0.0,0.90623,0.13179,0.85711,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,19],[4,35,0.1143,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[8,35,0.2286,0.9442,0.099,0.91071,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,1,23],[12,35,0.3429,0.95313,0.08301,0.91074,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,1,23],[16,35,0.4571,0.91964,0.11811,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,20],[20,35,0.5714,0.90178,0.12595,0.85711,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,18],[24,35,0.6857,0.88839,0.15866,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,8,0,18],[28,35,0.8,0.91517,0.10631,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[32,35,0.9143,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[35,35,1.0,0.91728,0.11192,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,1,18]]}]},{"i":"8a3f8c048a450524","q":"Let  $n$  be a positive integer. There are  $n$  lamps, each with a switch that changes the lamp from on to off, or from off to on, each time it is pressed. The lamps are initially all off.\nYou are going to press the switches in a series of rounds. In the first round, you are going to press exactly  $1$  switch; in the second round, you are going to press exactly  $2$  switches; and so on, so that in the  $k$ th round you are going to press exactly  $k$  switches. In each round you will press each switch at most once. Your goal is to finish a round with all of the lamps switched on.\nDetermine for which  $n$  you can achieve this 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 $n$  be a positive integer and let  $d_{1},d_{2},,\\ldots ,d_{k}$  be its divisors, such that  $1=d_{1}<d_{2}<\\ldots <d_{k}=n$ . Find all values of  $n$  for which  $k\\geq 4$  and  $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.87946,"x":1.0,"p":[[0,96,0.0,0.87946,0.18935,0.82132,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,20],[4,96,0.0417,0.89286,0.22868,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,24],[8,96,0.0833,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,96,0.125,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,96,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,96,0.2083,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,96,0.25,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,96,0.2917,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,96,0.3333,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[36,96,0.375,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,96,0.4167,0.96428,0.10715,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[44,96,0.4583,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,96,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,96,0.5417,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[56,96,0.5833,0.97322,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[60,96,0.625,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,96,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,96,0.7083,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,96,0.75,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,96,0.7917,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[80,96,0.8333,0.96875,0.11142,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[84,96,0.875,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,96,0.9167,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,96,0.9583,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[96,96,1.0,0.87946,0.2372,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,22]]},{"b":7,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,216,0.0,0.89731,0.1964,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,23],[4,216,0.0185,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[8,216,0.037,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[12,216,0.0556,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,216,0.0741,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,216,0.0926,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[24,216,0.1111,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,216,0.1296,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,216,0.1481,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,216,0.1667,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[40,216,0.1852,0.91961,0.1334,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[44,216,0.2037,0.94194,0.15516,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[48,216,0.2222,0.91518,0.1551,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23],[52,216,0.2407,0.94643,0.15047,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[56,216,0.2593,0.95089,0.09853,0.96429,1.0,1.0,0.5714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[60,216,0.2778,0.89732,0.15663,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,20],[64,216,0.2963,0.95982,0.10852,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[68,216,0.3148,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[72,216,0.3333,0.92411,0.14718,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,24],[76,216,0.3519,0.93303,0.17122,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[80,216,0.3704,0.90625,0.22476,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,27],[84,216,0.3889,0.91518,0.16698,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,22],[88,216,0.4074,0.90625,0.18766,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,24],[92,216,0.4259,0.93304,0.18205,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[96,216,0.4444,0.91963,0.14261,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[100,216,0.463,0.91964,0.16728,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[104,216,0.4815,0.91964,0.18189,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,26],[108,216,0.5,0.89732,0.16065,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,20],[112,216,0.5185,0.90177,0.23809,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,25],[116,216,0.537,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[120,216,0.5556,0.91071,0.21354,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,25],[124,216,0.5741,0.9241,0.13825,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,23],[128,216,0.5926,0.87946,0.20238,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,21],[132,216,0.6111,0.86161,0.24868,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,0,0,23],[136,216,0.6296,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[140,216,0.6481,0.93304,0.13825,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,25],[144,216,0.6667,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,20],[148,216,0.6852,0.87946,0.24513,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,24],[152,216,0.7037,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[156,216,0.7222,0.87946,0.25281,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,24],[160,216,0.7407,0.8793,0.16811,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,19],[164,216,0.7593,0.91518,0.16698,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,25],[168,216,0.7778,0.90623,0.17356,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,3,0,23],[172,216,0.7963,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[176,216,0.8148,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[180,216,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[184,216,0.8519,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[188,216,0.8704,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[192,216,0.8889,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[196,216,0.9074,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[200,216,0.9259,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[204,216,0.9444,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[208,216,0.963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[212,216,0.9815,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[216,216,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"00008131a8de53b9","q":"Let $\\mathbb{Q}^{+}$denote the set of positive rational numbers. Show that there exists one and only one function $f: \\mathbb{Q}^{+} \\rightarrow \\mathbb{Q}^{+}$satisfying the following conditions:\n\n(i) If $0<q<\\frac{1}{2}$ then $f(q)=1+f\\left(\\frac{q}{1-2 q}\\right)$.\n\n(ii) If $1<q \\leq 2$ then $f(q)=1+f(q-1)$.\n\n(iii) $f(q) \\cdot f\\left(\\frac{1}{q}\\right)=1$ for all $q \\in \\mathbb{Q}^{+}$.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.7098,"x":0.91963,"p":[[0,81,0.0,0.7098,0.32041,0.42857,0.78571,1.0,0.0,1.0,1,15,1,1,0,2,0,0,3,0,0,3,0,0,4,0,0,3,0,0,1,0,15],[4,81,0.0494,0.86159,0.20039,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,19],[8,81,0.0988,0.86606,0.18879,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,2,0,20],[12,81,0.1481,0.87723,0.19824,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,1,0,2,0,0,3,0,21],[16,81,0.1975,0.87947,0.2055,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,1,0,23],[20,81,0.2469,0.85712,0.17861,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,0,0,19],[24,81,0.2963,0.90178,0.18013,0.85711,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,2,0,23],[28,81,0.3457,0.90177,0.15749,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,21],[32,81,0.3951,0.86831,0.18567,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,1,0,3,0,0,2,0,20],[36,81,0.4444,0.85495,0.22646,0.71429,1.0,1.0,0.14,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,1,0,2,0,20],[40,81,0.4938,0.77668,0.279,0.57132,1.0,1.0,0.14,1.0,0,17,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,5,0,0,1,0,17],[44,81,0.5432,0.91963,0.16345,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,24],[48,81,0.5926,0.83482,0.18935,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,4,0,16],[52,81,0.642,0.83486,0.21748,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,4,0,17],[56,81,0.6914,0.83482,0.17445,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,2,0,2,0,15],[60,81,0.7407,0.84371,0.21242,0.67857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,2,0,19],[64,81,0.7901,0.81695,0.22372,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,4,0,16],[68,81,0.8395,0.8125,0.24856,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,2,0,0,1,0,19],[72,81,0.8889,0.87054,0.19351,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,3,0,20],[76,81,0.9383,0.85488,0.18431,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,1,0,4,0,0,5,0,17],[80,81,0.9877,0.75433,0.23418,0.57143,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,2,2,0,3,0,12],[81,81,1.0,0.79461,0.23131,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,7,0,0,4,0,14]]},{"b":2,"e":0.71429,"k":"rising","v":0.66071,"x":0.93302,"p":[[0,48,0.0,0.66071,0.32683,0.42857,0.78564,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,3,0,0,7,0,0,1,0,0,1,0,0,4,0,12],[4,48,0.0833,0.93302,0.15149,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,26],[8,48,0.1667,0.89731,0.21202,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,23],[12,48,0.25,0.89286,0.19562,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[16,48,0.3333,0.84596,0.1941,0.69643,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,5,1,0,2,0,0,5,0,17],[20,48,0.4167,0.83926,0.19484,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,2,0,17],[24,48,0.5,0.86606,0.18537,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,3,0,19],[28,48,0.5833,0.87051,0.18338,0.82132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,6,0,18],[32,48,0.6667,0.85268,0.17308,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,8,0,15],[36,48,0.75,0.89286,0.15972,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,5,0,20],[40,48,0.8333,0.89728,0.16465,0.85711,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,21],[44,48,0.9167,0.84808,0.22859,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,4,0,19],[48,48,1.0,0.85714,0.19562,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,3,0,19]]}]},{"i":"7a1e3c8beab8539c","q":"Let  $z$  be a non-real complex number with  $z^{23}=1$ .  Compute \\[\\sum_{k=0}^{22}\\dfrac{1}{1+z^k+z^{2k}}.\\]","t":[{"b":2,"e":0.28571,"k":"falling","v":0.20089,"x":0.625,"p":[[0,73,0.0,0.5,0.44464,0.14286,0.14286,1.0,0.0,1.0,5,14,1,5,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[4,73,0.0548,0.51786,0.40208,0.14286,0.28571,1.0,0.14286,1.0,0,13,0,0,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[8,73,0.1096,0.625,0.3859,0.24999,0.78571,1.0,0.14286,1.0,0,16,0,0,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,16],[12,73,0.1644,0.42411,0.32632,0.14286,0.28571,0.42857,0.0,1.0,1,7,0,1,0,10,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,7],[16,73,0.2192,0.48214,0.31084,0.28571,0.42857,0.64286,0.14286,1.0,0,7,0,0,0,7,0,0,6,0,0,10,0,0,1,0,0,0,0,0,1,0,7],[20,73,0.274,0.33929,0.27606,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,15,0,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,4],[24,73,0.3288,0.29911,0.25843,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,17,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[28,73,0.3836,0.27232,0.21535,0.14286,0.14286,0.28571,0.14286,1.0,0,2,0,0,0,18,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[32,73,0.4384,0.24107,0.11538,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,17,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.34821,0.27185,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,14,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,4],[40,73,0.5479,0.30804,0.25532,0.14286,0.28571,0.42857,0.0,1.0,3,3,0,3,0,10,0,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[44,73,0.6027,0.25446,0.17029,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,17,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[48,73,0.6575,0.30357,0.17768,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,9,0,0,9,0,0,1,0,0,0,0,0,0,0,1],[52,73,0.7123,0.27232,0.11495,0.14286,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,12,0,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[56,73,0.7671,0.20089,0.08645,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.28125,0.21572,0.14286,0.14286,0.32143,0.14286,1.0,0,2,0,0,0,17,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[64,73,0.8767,0.23661,0.21609,0.14286,0.14286,0.17857,0.14286,1.0,0,2,0,0,0,24,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[68,73,0.9315,0.28125,0.19061,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,18,0,0,2,0,0,10,0,0,1,0,0,0,0,0,0,0,1],[72,73,0.9863,0.25,0.12877,0.14286,0.14286,0.42857,0.14286,0.4286,0,0,0,0,0,18,0,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.30357,0.1171,0.14286,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,9,0,0,10,0,0,13,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.4375,"x":1.0,"p":[[0,28,0.0,0.4375,0.37447,0.14286,0.21431,1.0,0.0,1.0,1,9,0,1,0,15,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,9],[4,28,0.1429,0.66518,0.39059,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,18],[8,28,0.2857,0.91964,0.24984,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[12,28,0.4286,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"62b9bcc2a98ffbf8","q":"Let  $x,y,z$  be three positive integers with  $\\gcd(x,y,z)=1$ . If\n\\[x\\mid yz(x+y+z),\\]\n\\[y\\mid xz(x+y+z),\\]\n\\[z\\mid xy(x+y+z),\\]\nand\n\\[x+y+z\\mid xyz,\\]\nshow that  $xyz(x+y+z)$  is a perfect square.\n\n*Proposed by usjl*","t":[{"b":3,"e":0.857,"k":"flat","v":0.69642,"x":0.94643,"p":[[0,140,0.0,0.79464,0.24468,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,2,0,16],[4,140,0.0286,0.84821,0.2549,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,2,0,21],[8,140,0.0571,0.94197,0.20781,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[12,140,0.0857,0.90625,0.24383,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,27],[16,140,0.1143,0.87054,0.24578,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,24],[20,140,0.1429,0.78125,0.36419,0.57143,1.0,1.0,0.0,1.0,3,22,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,22],[24,140,0.1714,0.82142,0.31745,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,23],[28,140,0.2,0.79008,0.29467,0.57143,1.0,1.0,0.14,1.0,0,19,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,0,0,0,2,0,19],[32,140,0.2286,0.8705,0.25598,0.96425,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,24],[36,140,0.2571,0.82142,0.31945,0.78571,1.0,1.0,0.0,1.0,1,23,0,1,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,23],[40,140,0.2857,0.77679,0.34615,0.64286,1.0,1.0,0.14286,1.0,0,21,0,0,0,6,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,21],[44,140,0.3143,0.83481,0.31565,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,24],[48,140,0.3429,0.80357,0.34764,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,24],[52,140,0.3714,0.74552,0.3707,0.49967,1.0,1.0,0.0,1.0,1,21,0,1,0,6,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,21],[56,140,0.4,0.83036,0.3102,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,23],[60,140,0.4286,0.86159,0.29772,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,26],[64,140,0.4571,0.80357,0.28516,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,1,0,0,0,0,0,5,0,0,2,0,0,2,0,19],[68,140,0.4857,0.79018,0.34623,0.57143,1.0,1.0,0.0,1.0,1,22,0,1,0,5,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,22],[72,140,0.5143,0.77679,0.35163,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,7,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,21],[76,140,0.5429,0.77232,0.37091,0.4643,1.0,1.0,0.14286,1.0,0,23,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,23],[80,140,0.5714,0.72768,0.38025,0.14286,1.0,1.0,0.14286,1.0,0,20,0,0,0,9,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,20],[84,140,0.6,0.69643,0.39407,0.14286,1.0,1.0,0.0,1.0,2,19,0,2,0,7,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,19],[88,140,0.6286,0.80354,0.32293,0.57132,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,22],[92,140,0.6571,0.78125,0.3214,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,20],[96,140,0.6857,0.69642,0.3567,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,7,0,0,2,0,0,0,0,0,4,0,0,1,0,0,2,0,16],[100,140,0.7143,0.85705,0.25779,0.82143,1.0,1.0,0.14,1.0,0,23,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,1,0,0,1,0,23],[104,140,0.7429,0.83927,0.32292,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,25],[108,140,0.7714,0.79911,0.3551,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,24],[112,140,0.8,0.75893,0.38538,0.46429,1.0,1.0,0.0,1.0,2,22,0,2,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,22],[116,140,0.8286,0.81696,0.32972,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,24],[120,140,0.8571,0.71429,0.37287,0.46429,1.0,1.0,0.0,1.0,1,19,0,1,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,19],[124,140,0.8857,0.70536,0.37954,0.14286,1.0,1.0,0.14286,1.0,0,19,0,0,0,9,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,19],[128,140,0.9143,0.8482,0.29439,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,24],[132,140,0.9429,0.93304,0.18205,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[136,140,0.9714,0.94643,0.17768,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[140,140,1.0,0.76786,0.25939,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,4,0,0,10,0,0,0,0,0,0,0,17]]},{"b":6,"e":1.0,"k":"rising","v":0.65179,"x":1.0,"p":[[0,112,0.0,0.76339,0.32851,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,4,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,19],[4,112,0.0357,0.87054,0.24053,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,23],[8,112,0.0714,0.8125,0.26351,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,1,0,0,8,0,0,1,0,0,0,0,20],[12,112,0.1071,0.65179,0.37786,0.14289,0.85714,1.0,0.14286,1.0,0,16,0,0,0,9,0,0,2,0,0,0,0,0,4,0,0,1,0,0,0,0,16],[16,112,0.1429,0.78125,0.31336,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,0,0,0,1,0,0,6,0,0,0,0,0,2,0,19],[20,112,0.1786,0.74107,0.3597,0.5,1.0,1.0,0.14286,1.0,0,20,0,0,0,7,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,20],[24,112,0.2143,0.79017,0.31942,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,21],[28,112,0.25,0.77679,0.35344,0.57143,1.0,1.0,0.14286,1.0,0,22,0,0,0,7,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,22],[32,112,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,112,0.3214,0.86159,0.28457,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,25],[40,112,0.3571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,112,0.3929,0.92857,0.21724,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[48,112,0.4286,0.87946,0.2969,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[52,112,0.4643,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[56,112,0.5,0.93304,0.18893,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[60,112,0.5357,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[64,112,0.5714,0.83036,0.30606,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,23],[68,112,0.6071,0.88393,0.27067,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,25],[72,112,0.6429,0.82589,0.29393,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,22],[76,112,0.6786,0.8125,0.3163,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,22],[80,112,0.7143,0.875,0.26426,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,25],[84,112,0.75,0.84375,0.27747,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,0,1,0,23],[88,112,0.7857,0.9375,0.2111,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[92,112,0.8214,0.88839,0.2618,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,26],[96,112,0.8571,0.94196,0.17807,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[100,112,0.8929,0.87054,0.25842,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,24],[104,112,0.9286,0.88839,0.23619,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,25],[108,112,0.9643,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[112,112,1.0,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26]]}]},{"i":"0b2b28bfd6928b86","q":"Let  $n,m$  be positive integers such that  $n<m$  and  $a_1, a_2, ..., a_m$  be different real numbers.\n(a) Find all polynomials  $P$  with real coefficients and degree at most  $n$  such that: $|P(a_i)-P(a_j)|=|a_i-a_j|$  for all  $i,j=\\{1, 2, ..., m\\}$  such that  $i<j$ .\n(b) If  $n,m\\ge 2$  does there exist a polynomial  $Q$  with real coefficients and degree  $n$  such that: $|Q(a_i)-Q(a_j)|<|a_i-a_j|$  for all  $i,j=\\{1, 2, ..., m\\}$  such that  $i<j$ Edit: See #3","t":[{"b":1,"e":0.0,"k":"flat","v":0.0625,"x":0.99554,"p":[[0,102,0.0,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22],[4,102,0.0392,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[8,102,0.0784,0.91071,0.13716,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[12,102,0.1176,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[16,102,0.1569,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,102,0.1961,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,102,0.2353,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,102,0.2745,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,102,0.3137,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,102,0.3529,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[40,102,0.3922,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[44,102,0.4314,0.17857,0.37457,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[48,102,0.4706,0.33482,0.46375,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,9],[52,102,0.5098,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[56,102,0.549,0.08482,0.26453,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[60,102,0.5882,0.21429,0.40564,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[64,102,0.6275,0.20536,0.39113,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[68,102,0.6667,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[72,102,0.7059,0.08929,0.27837,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[76,102,0.7451,0.20982,0.39927,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[80,102,0.7843,0.14732,0.34346,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[84,102,0.8235,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[88,102,0.8627,0.14286,0.33503,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,3],[92,102,0.902,0.14732,0.34346,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[96,102,0.9412,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[100,102,0.9804,0.20982,0.39767,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[102,102,1.0,0.79027,0.38464,0.85714,1.0,1.0,0.0,1.0,6,22,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,22]]},{"b":5,"e":0.57143,"k":"flat","v":0.85714,"x":0.95982,"p":[[0,14,0.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,14,0.2857,0.91518,0.12299,0.82143,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[8,14,0.5714,0.85714,0.12877,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,6,0,13],[12,14,0.8571,0.88837,0.11704,0.82143,0.85714,1.0,0.714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,15],[14,14,1.0,0.89719,0.11447,0.85714,0.92857,1.0,0.71,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16]]}]},{"i":"e4b6a6a4e3a93ce1","q":"Let $ABC$ be a triangle, $O$ the center of its circumcircle. Suppose that $\\widehat{C B A}=60^{\\circ}$ and $\\widehat{\\mathrm{CBO}}=45^{\\circ}$. Let $D$ be the point of intersection of the lines $(\\mathrm{AC})$ and $(B O)$. Show that $\\mathrm{AD}=\\mathrm{DO}$.","t":[{"b":2,"e":1.0,"k":"flat","v":0.90179,"x":0.99554,"p":[[0,19,0.0,0.92411,0.11285,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,18],[4,19,0.2105,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[8,19,0.4211,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,19,0.6316,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,19,0.8421,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[19,19,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":0.85714,"k":"flat","v":0.8616,"x":0.90624,"p":[[0,26,0.0,0.90624,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[4,26,0.1538,0.89732,0.12993,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,0,13],[8,26,0.3077,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[12,26,0.4615,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[16,26,0.6154,0.87499,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[20,26,0.7692,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[24,26,0.9231,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[26,26,1.0,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4]]}]},{"i":"67338f5f99427428","q":"Let $A B C$ be a triangle, point $P$ the midpoint of $B C$, and point $Q$ on segment $C A$ such that $|C Q|=2|Q A|$. Let $S$ be the intersection of $B Q$ and $A P$. Prove that $|A S|=|S P|$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.15625,"p":[[0,30,0.0,0.15625,0.22689,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,13,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[4,30,0.1333,0.08036,0.18189,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,30,0.2667,0.08481,0.14218,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,11,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,30,0.4,0.10268,0.19638,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,30,0.5333,0.09366,0.17715,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[20,30,0.6667,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03571,"x":0.12054,"p":[[0,33,0.0,0.0758,0.17848,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,33,0.1212,0.0625,0.12846,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,33,0.2424,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,33,0.3636,0.05357,0.11152,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,33,0.4848,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.11607,0.1729,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,33,0.7273,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.11607,0.1729,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,33,0.9697,0.11152,0.17398,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[33,33,1.0,0.12054,0.17169,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"b85c8ac6273e0150","q":"Let $a, b, c$ be positive real numbers such that $a^{2}+b^{2}+c^{2}+(a+b+c)^{2} \\leq 4$. Prove that  $$ \\frac{a b+1}{(a+b)^{2}}+\\frac{b c+1}{(b+c)^{2}}+\\frac{c a+1}{(c+a)^{2}} \\geq 3 $$","t":[{"b":1,"e":0.14286,"k":"falling","v":0.12946,"x":0.49107,"p":[[0,72,0.0,0.49107,0.33108,0.14286,0.42857,0.75,0.0,1.0,1,7,0,1,0,9,0,0,2,0,0,8,0,0,2,0,0,2,0,0,1,0,7],[4,72,0.0556,0.26339,0.2969,0.14286,0.14286,0.28571,0.0,1.0,6,3,0,6,0,17,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,3],[8,72,0.1111,0.25445,0.33641,0.10714,0.14286,0.14286,0.0,1.0,8,5,0,8,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[12,72,0.1667,0.24107,0.28669,0.14286,0.14286,0.14286,0.0,1.0,6,3,0,6,0,19,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[16,72,0.2222,0.23214,0.25191,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,25,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,2],[20,72,0.2778,0.36598,0.34436,0.14286,0.14286,0.5,0.0,1.0,2,6,0,2,0,17,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,6],[24,72,0.3333,0.39286,0.36422,0.14286,0.14286,0.71429,0.0,1.0,1,7,0,1,0,19,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,7],[28,72,0.3889,0.40625,0.37983,0.14286,0.14286,0.85714,0.0,1.0,4,6,0,4,0,15,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,6],[32,72,0.4444,0.3125,0.33776,0.14286,0.14286,0.17857,0.0,1.0,2,5,0,2,0,22,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[36,72,0.5,0.36607,0.36585,0.14286,0.14286,0.60714,0.0,1.0,3,7,0,3,0,18,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,7],[40,72,0.5556,0.32143,0.30514,0.14286,0.14286,0.46429,0.0,1.0,1,3,0,1,0,21,0,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,3],[44,72,0.6111,0.26339,0.25281,0.14286,0.14286,0.17857,0.0,1.0,1,2,0,1,0,23,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,2],[48,72,0.6667,0.38839,0.33165,0.14286,0.14286,0.71429,0.14286,1.0,0,5,0,0,0,19,0,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,5],[52,72,0.7222,0.33036,0.33396,0.14286,0.14286,0.60714,0.0,1.0,4,4,0,4,0,18,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,4],[56,72,0.7778,0.20536,0.22286,0.14286,0.14286,0.14286,0.0,1.0,3,2,0,3,0,24,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[60,72,0.8333,0.25,0.25,0.14286,0.14286,0.14287,0.0,1.0,1,2,0,1,0,24,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[64,72,0.8889,0.12946,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.15624,0.08261,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,29,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"falling","v":0.10268,"x":0.62054,"p":[[0,60,0.0,0.62054,0.35104,0.25,0.71429,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,2,0,0,3,0,0,2,0,0,3,0,0,3,0,11],[4,60,0.0667,0.5625,0.42699,0.14286,0.78571,1.0,0.0,1.0,3,13,0,3,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,13],[8,60,0.1333,0.49554,0.38463,0.14286,0.42857,1.0,0.0,1.0,2,9,0,2,0,13,0,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,9],[12,60,0.2,0.62054,0.37899,0.14286,0.71429,1.0,0.0,1.0,1,12,0,1,0,10,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,12],[16,60,0.2667,0.23661,0.28928,0.14286,0.14286,0.14286,0.0,1.0,7,2,0,7,0,19,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,2],[20,60,0.3333,0.21875,0.2923,0.0,0.14286,0.14286,0.0,1.0,9,3,0,9,0,17,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[24,60,0.4,0.16071,0.17768,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,23,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[28,60,0.4667,0.30804,0.33902,0.14286,0.14286,0.17857,0.0,1.0,2,6,0,2,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[32,60,0.5333,0.14286,0.17496,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,17,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[36,60,0.6,0.17858,0.19885,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,21,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,60,0.6667,0.21429,0.23419,0.14286,0.14286,0.14286,0.0,1.0,3,2,0,3,0,24,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[44,60,0.7333,0.14286,0.13832,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,20,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,60,0.8,0.10268,0.08171,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.10714,0.12877,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,60,0.9333,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d01790242c3fc3ac","q":"Let  $n>1$  be a positive integer and  $d_1<d_2<\\dots<d_m$  be its  $m$  positive divisors, including  $1$  and  $n$ . Lalo writes the following  $2m$  numbers on a board:\n\\[d_1,d_2\\dots, d_m,d_1+d_2,d_2+d_3,\\dots,d_{m-1}+d_m,N \\]\nwhere  $N$  is a positive integer. Afterwards, Lalo erases any number that is repeated (for example, if a number appears twice, he erases one of them). Finally, Lalo realizes that the numbers left on the board are exactly all the divisors of  $N$ . Find all possible values that  $n$  can take.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.14268,"x":0.45536,"p":[[0,78,0.0,0.14268,0.07143,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,78,0.0513,0.44641,0.31083,0.14286,0.28571,0.85714,0.14286,1.0,0,2,0,0,0,11,0,0,7,0,0,2,0,0,2,0,0,1,0,0,7,0,2],[8,78,0.1026,0.37938,0.30439,0.14286,0.14286,0.71429,0.14,1.0,0,1,0,0,0,18,0,0,2,0,0,1,0,0,2,0,0,3,0,0,5,0,1],[12,78,0.1538,0.45536,0.2911,0.14286,0.35714,0.71429,0.14286,1.0,0,1,0,0,0,10,0,0,6,0,0,3,0,0,2,0,0,4,0,0,6,0,1],[16,78,0.2051,0.41526,0.32619,0.14286,0.21431,0.74999,0.14286,1.0,0,2,0,0,0,16,0,0,4,0,0,0,0,0,1,0,0,3,0,0,6,0,2],[20,78,0.2564,0.29893,0.28437,0.14286,0.14286,0.28579,0.0,1.0,1,1,0,1,0,21,0,0,3,0,0,0,0,0,1,0,0,1,0,0,4,0,1],[24,78,0.3077,0.36142,0.2601,0.14286,0.28571,0.46429,0.14,0.85714,0,0,0,0,0,12,0,0,11,0,0,1,0,0,1,0,0,2,0,0,5,0,0],[28,78,0.359,0.37057,0.28984,0.14286,0.2857,0.57111,0.0,1.0,1,1,0,1,0,14,0,0,5,0,0,3,0,0,2,0,0,1,0,0,5,0,1],[32,78,0.4103,0.37053,0.28089,0.14286,0.2143,0.60714,0.14286,1.0,0,1,0,0,0,16,0,0,4,0,0,2,0,0,2,0,0,4,0,0,3,0,1],[36,78,0.4615,0.29464,0.23941,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,16,0,0,8,0,0,1,0,0,3,0,0,0,0,0,2,0,1],[40,78,0.5128,0.30357,0.23076,0.14286,0.21428,0.28571,0.14286,1.0,0,1,0,0,0,16,0,0,9,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[44,78,0.5641,0.27673,0.20813,0.14286,0.14286,0.42857,0.14,0.85714,0,0,0,0,0,20,0,0,3,0,0,4,0,0,3,0,0,0,0,0,2,0,0],[48,78,0.6154,0.22766,0.17442,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,20,0,0,8,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[52,78,0.6667,0.20982,0.1636,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,22,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[56,78,0.7179,0.22312,0.20809,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,25,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,0],[60,78,0.7692,0.25445,0.20741,0.14286,0.14286,0.2857,0.14286,0.85714,0,0,0,0,0,23,0,0,2,0,0,2,0,0,3,0,0,0,0,0,2,0,0],[64,78,0.8205,0.23643,0.19768,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,22,0,0,4,0,0,1,0,0,1,0,0,2,0,0,1,0,0],[68,78,0.8718,0.16965,0.08328,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,78,0.9231,0.18304,0.10853,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[76,78,0.9744,0.20089,0.09354,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[78,78,1.0,0.16956,0.08332,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.43,"k":"rising","v":0.14732,"x":0.48212,"p":[[0,77,0.0,0.14732,0.0563,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.43749,0.32524,0.14286,0.28571,0.85714,0.0,1.0,1,1,0,1,0,13,0,0,3,0,0,3,0,0,1,0,0,1,0,0,9,0,1],[8,77,0.1039,0.4107,0.29178,0.14286,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,14,0,0,0,0,0,5,0,0,3,0,0,3,0,0,6,0,0],[12,77,0.1558,0.42402,0.33792,0.14286,0.14288,0.75,0.14,1.0,0,3,0,0,0,17,0,0,2,0,0,1,0,0,0,0,0,4,0,0,5,0,3],[16,77,0.2078,0.31246,0.23803,0.14286,0.14286,0.4642,0.14286,1.0,0,1,0,0,0,18,0,0,4,0,0,2,0,0,5,0,0,1,0,0,1,0,1],[20,77,0.2597,0.41061,0.3269,0.14286,0.21429,0.64282,0.0,1.0,1,3,0,1,0,15,0,0,2,0,0,2,0,0,4,0,0,0,0,0,5,0,3],[24,77,0.3117,0.35696,0.26501,0.14286,0.2857,0.57143,0.0,0.85714,1,0,0,1,0,13,0,0,7,0,0,2,0,0,2,0,0,3,0,0,4,0,0],[28,77,0.3636,0.29008,0.2187,0.14286,0.14286,0.4642,0.0,0.85714,2,0,0,2,0,16,0,0,4,0,0,2,0,0,6,0,0,1,0,0,1,0,0],[32,77,0.4156,0.43294,0.29779,0.14286,0.35714,0.71429,0.14,0.85714,0,0,0,0,0,14,0,0,2,0,0,3,0,0,2,0,0,4,0,0,7,0,0],[36,77,0.4675,0.3482,0.25737,0.14286,0.21428,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,3,0,0,6,0,0,1,0,0,3,0,0,2,0,1],[40,77,0.5195,0.42856,0.31943,0.14286,0.28571,0.71429,0.0,1.0,1,2,0,1,0,14,0,0,2,0,0,1,0,0,4,0,0,3,0,0,5,0,2],[44,77,0.5714,0.42402,0.30831,0.14286,0.35714,0.60714,0.14,1.0,0,3,0,0,0,14,0,0,2,0,0,5,0,0,3,0,0,1,0,0,4,0,3],[48,77,0.6234,0.4017,0.31232,0.14286,0.2143,0.71429,0.14,1.0,0,2,0,0,0,16,0,0,4,0,0,0,0,0,2,0,0,4,0,0,4,0,2],[52,77,0.6753,0.43303,0.28004,0.14286,0.35714,0.71429,0.14286,1.0,0,1,0,0,0,11,0,0,5,0,0,4,0,0,2,0,0,5,0,0,4,0,1],[56,77,0.7273,0.35714,0.26726,0.14286,0.21429,0.46431,0.14286,1.0,0,1,0,0,0,16,0,0,3,0,0,5,0,0,2,0,0,2,0,0,3,0,1],[60,77,0.7792,0.3749,0.27843,0.14286,0.28571,0.46418,0.0,1.0,1,4,0,1,0,10,0,0,9,0,0,4,0,0,4,0,0,0,0,0,0,0,4],[64,77,0.8312,0.32588,0.18636,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,11,0,0,10,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[68,77,0.8831,0.39284,0.24743,0.14286,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,9,0,0,11,0,0,2,0,0,3,0,0,3,0,0,4,0,0],[72,77,0.9351,0.48212,0.25691,0.2857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,10,0,0,5,0,0,3,0,0,5,0,0,3,0,2],[76,77,0.987,0.43303,0.27311,0.2857,0.28586,0.57143,0.14286,1.0,0,3,0,0,0,7,0,0,10,0,0,5,0,0,4,0,0,0,0,0,3,0,3],[77,77,1.0,0.30804,0.17536,0.14286,0.2857,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,5,0,0,9,0,0,2,0,0,2,0,0,0,0,0]]}]},{"i":"793ec2b54a3e9687","q":"Let  $n>1$  be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by  $n$ . Prove that the number of consecutive beautiful numbers is less or equal to  $1+\\lfloor \\sqrt{3n} \\rfloor$ .","t":[{"b":0,"e":0.2857,"k":"rising","v":0.09376,"x":0.53122,"p":[[0,45,0.0,0.09376,0.09853,0.0,0.14286,0.14286,0.0,0.286,15,0,13,15,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.49105,0.25236,0.2857,0.42857,0.74996,0.14286,0.85714,0,0,0,0,0,4,0,0,9,0,0,5,0,0,5,0,0,1,0,0,8,0,0],[8,45,0.1778,0.41964,0.27879,0.14286,0.28571,0.71429,0.0,1.0,1,2,0,1,0,9,0,0,7,0,0,4,0,0,1,0,0,7,0,0,1,0,2],[12,45,0.2667,0.41961,0.27417,0.14286,0.42857,0.71429,0.0,1.0,3,1,0,3,0,6,0,0,6,0,0,5,0,0,3,0,0,6,0,0,2,0,1],[16,45,0.3556,0.53122,0.27486,0.28571,0.4286,0.74996,0.14286,1.0,0,4,0,0,0,4,0,0,6,0,0,7,0,0,5,0,0,2,0,0,4,0,4],[20,45,0.4444,0.35266,0.2201,0.14286,0.28571,0.46418,0.14286,0.85714,0,0,0,0,0,10,0,0,12,0,0,2,0,0,3,0,0,3,0,0,2,0,0],[24,45,0.5333,0.52676,0.28669,0.2857,0.4998,0.75,0.14286,1.0,0,3,0,0,0,5,0,0,8,0,0,3,0,0,3,0,0,5,0,0,5,0,3],[28,45,0.6222,0.38392,0.31427,0.14286,0.28571,0.60714,0.0,1.0,7,2,0,7,0,5,0,0,5,0,0,3,0,0,4,0,0,4,0,0,2,0,2],[32,45,0.7111,0.46862,0.28409,0.28571,0.42859,0.60714,0.0,1.0,1,3,0,1,0,6,0,0,7,0,0,4,0,0,6,0,0,2,0,0,3,0,3],[36,45,0.8,0.41962,0.30708,0.14286,0.28571,0.60714,0.0,1.0,2,3,0,2,0,9,0,0,7,0,0,2,0,0,4,0,0,2,0,0,3,0,3],[40,45,0.8889,0.37053,0.21683,0.2857,0.28571,0.42857,0.0,1.0,2,1,0,2,0,3,0,0,13,0,0,9,0,0,2,0,0,0,0,0,2,0,1],[44,45,0.9778,0.22758,0.15918,0.14286,0.1429,0.28571,0.0,0.57143,5,0,0,5,0,12,0,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[45,45,1.0,0.27228,0.17254,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,6,0,0,12,0,0,5,0,0,4,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.1429,"k":"rising","v":0.14286,"x":0.54006,"p":[[0,59,0.0,0.14286,0.10101,0.14286,0.14286,0.14286,0.0,0.42857,7,0,6,7,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.50891,0.29436,0.28571,0.42857,0.74996,0.14286,1.0,0,4,0,0,0,6,0,0,7,0,0,5,0,0,3,0,0,3,0,0,4,0,4],[8,59,0.1356,0.4732,0.24598,0.2857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,7,0,0,3,0,0,4,0,0,4,0,1],[12,59,0.2034,0.41964,0.27184,0.14289,0.35714,0.71429,0.0,1.0,1,1,0,1,0,8,0,0,7,0,0,7,0,0,0,0,0,4,0,0,4,0,1],[16,59,0.2712,0.49552,0.29011,0.28571,0.4998,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,7,0,0,2,0,0,3,0,0,7,0,0,4,0,2],[20,59,0.339,0.48661,0.31513,0.14286,0.42857,0.71429,0.0,1.0,1,4,0,1,0,9,0,0,4,0,0,3,0,0,2,0,0,7,0,0,2,0,4],[24,59,0.4068,0.44197,0.29957,0.14286,0.42857,0.71429,0.0,1.0,3,1,0,3,0,8,0,0,2,0,0,6,0,0,3,0,0,4,0,0,5,0,1],[28,59,0.4746,0.54006,0.30886,0.28571,0.4286,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,8,0,0,5,0,0,4,0,0,1,0,0,4,0,6],[32,59,0.5424,0.45983,0.30036,0.28571,0.35714,0.71429,0.0,1.0,3,2,0,3,0,4,0,0,9,0,0,2,0,0,3,0,0,5,0,0,4,0,2],[36,59,0.6102,0.41963,0.29219,0.14286,0.35714,0.60714,0.0,1.0,4,2,0,4,0,5,0,0,7,0,0,3,0,0,5,0,0,4,0,0,2,0,2],[40,59,0.678,0.39282,0.21125,0.2857,0.42857,0.571,0.0,0.71429,3,0,0,3,0,3,0,0,8,0,0,8,0,0,5,0,0,5,0,0,0,0,0],[44,59,0.7458,0.52678,0.27533,0.28571,0.42859,0.75,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,10,0,0,1,0,0,4,0,0,5,0,3],[48,59,0.8136,0.41946,0.24227,0.2857,0.28571,0.57143,0.14,1.0,0,1,0,0,0,7,0,0,10,0,0,4,0,0,4,0,0,4,0,0,2,0,1],[52,59,0.8814,0.49107,0.25985,0.2857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,4,0,0,5,0,0,6,0,0,6,0,0,2,0,2],[56,59,0.9492,0.35702,0.2625,0.14286,0.2857,0.42857,0.0,1.0,1,2,0,1,0,10,0,0,11,0,0,3,0,0,2,0,0,1,0,0,2,0,2],[59,59,1.0,0.41961,0.18874,0.2857,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,3,0,0,11,0,0,10,0,0,3,0,0,3,0,0,2,0,0]]}]},{"i":"1626cee0a5f553f5","q":"Let $A B C$ be an acute-angled triangle with altitudes $A D, B E$, and $C F$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that\n\n$$\n\\frac{A B \\cdot A C+B C \\cdot B A+C A \\cdot C B}{A H \\cdot A D+B H \\cdot B E+C H \\cdot C F} \\leq 2 .\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,35,0.0,0.82143,0.3312,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,23],[4,35,0.1143,0.83482,0.32559,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,25],[8,35,0.2286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,35,0.9143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.14286,"k":"falling","v":0.34812,"x":0.74107,"p":[[0,26,0.0,0.74107,0.3787,0.25,1.0,1.0,0.14286,1.0,0,21,0,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,21],[4,26,0.1538,0.50893,0.37785,0.14286,0.42856,1.0,0.14286,1.0,0,9,0,0,0,15,0,0,1,0,0,0,0,0,3,0,0,2,0,0,2,0,9],[8,26,0.3077,0.64723,0.40729,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,10,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,17],[12,26,0.4615,0.50437,0.39614,0.14286,0.2143,1.0,0.14,1.0,0,11,0,0,0,16,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,11],[16,26,0.6154,0.34812,0.33875,0.14286,0.14286,0.42852,0.14,1.0,0,4,0,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,4],[20,26,0.7692,0.50447,0.35711,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,13,0,0,1,0,0,4,0,0,2,0,0,2,0,0,2,0,8],[24,26,0.9231,0.51785,0.40049,0.14286,0.28574,1.0,0.0,1.0,1,10,0,1,0,15,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,10],[26,26,1.0,0.43746,0.29217,0.14286,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,13,0,0,2,0,0,3,0,0,5,0,0,4,0,0,3,0,2]]}]},{"i":"32bbb23cec11e0b9","q":"Let  $p$  be a fixed odd prime. A  $p$ -tuple  $(a_1,a_2,a_3,\\ldots,a_p)$  of integers is said to be *good* if \n\n\n\n- **(i)**  $0\\le a_i\\le p-1$  for all  $i$ , and\n- **(ii)**  $a_1+a_2+a_3+\\cdots+a_p$  is not divisible by  $p$ , and\n- **(iii)**  $a_1a_2+a_2a_3+a_3a_4+\\cdots+a_pa_1$  is divisible by  $p$ .\n\n\nDetermine the number of good  $p$ -tuples.","t":[{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.52232,"p":[[0,57,0.0,0.17857,0.27433,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[4,57,0.0702,0.20089,0.34967,0.0,0.0,0.14286,0.0,1.0,17,5,0,17,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,57,0.1404,0.44643,0.43995,0.0,0.14286,0.89286,0.0,1.0,10,8,0,10,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,8],[12,57,0.2105,0.36607,0.42699,0.0,0.14286,0.85714,0.0,1.0,14,6,0,14,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,6],[16,57,0.2807,0.52232,0.4596,0.0,0.85707,1.0,0.0,1.0,11,11,0,11,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,11],[20,57,0.3509,0.48214,0.44571,0.0,0.35716,1.0,0.0,1.0,11,10,0,11,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,10],[24,57,0.4211,0.42848,0.45881,0.0,0.14286,1.0,0.0,1.0,12,11,0,12,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,11],[28,57,0.4912,0.4642,0.44183,0.0,0.14286,1.0,0.0,1.0,10,9,0,10,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,9],[32,57,0.5614,0.30804,0.42274,0.0,0.0,0.85714,0.0,1.0,18,5,0,18,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,5],[36,57,0.6316,0.41518,0.44515,0.0,0.14286,1.0,0.0,1.0,12,9,0,12,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,9],[40,57,0.7018,0.30357,0.40524,0.0,0.14286,0.75,0.0,1.0,15,6,0,15,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,6],[44,57,0.7719,0.45982,0.45839,0.0,0.14288,1.0,0.0,1.0,13,9,0,13,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,9],[48,57,0.8421,0.26339,0.38318,0.0,0.07143,0.28571,0.0,1.0,16,5,0,16,0,8,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,5],[52,57,0.9123,0.17848,0.32342,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[56,57,0.9825,0.19643,0.36727,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[57,57,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02679,"x":0.35268,"p":[[0,86,0.0,0.13393,0.28557,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,86,0.0465,0.35268,0.41952,0.0,0.14286,0.85714,0.0,1.0,14,7,0,14,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,7],[8,86,0.093,0.35268,0.43739,0.0,0.14286,0.89286,0.0,1.0,15,8,0,15,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,8],[12,86,0.1395,0.33036,0.41869,0.0,0.14286,0.85714,0.0,1.0,14,6,0,14,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,6],[16,86,0.186,0.3125,0.42173,0.0,0.0,0.85714,0.0,1.0,17,6,0,17,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,6],[20,86,0.2326,0.19643,0.3531,0.0,0.0,0.14286,0.0,1.0,19,5,0,19,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[24,86,0.2791,0.12045,0.26027,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[28,86,0.3256,0.13393,0.2922,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,1],[32,86,0.3721,0.26339,0.37134,0.0,0.14286,0.28571,0.0,1.0,15,3,0,15,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,3],[36,86,0.4186,0.19643,0.31288,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,1],[40,86,0.4651,0.12946,0.27283,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[44,86,0.5116,0.18303,0.33356,0.0,0.0,0.14286,0.0,1.0,19,3,0,19,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[48,86,0.5581,0.2008,0.28764,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,14,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[52,86,0.6047,0.06696,0.2082,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[56,86,0.6512,0.14286,0.25254,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,10,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0],[60,86,0.6977,0.12052,0.20547,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,12,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[64,86,0.7442,0.14286,0.29233,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[68,86,0.7907,0.05357,0.15465,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[72,86,0.8372,0.07589,0.15561,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[76,86,0.8837,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,86,0.9302,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,86,0.9767,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[86,86,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"13e14a65c5bbc3de","q":"Let $\\mathcal{S}$ be a set consisting of $n \\geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\\mathcal{S}$. Prove that the elements of $\\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \\ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \\ldots, n-1$. (Ukraine)","t":[{"b":6,"e":0.28571,"k":"flat","v":0.08027,"x":0.25,"p":[[0,43,0.0,0.19196,0.13651,0.0,0.2857,0.28571,0.0,0.42857,9,0,2,9,0,5,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.09375,0.14987,0.0,0.0,0.17857,0.0,0.42857,22,0,0,22,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.08027,0.12335,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.14286,0.15152,0.0,0.14286,0.28571,0.0,0.4286,14,0,0,14,0,8,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.16509,0.1641,0.0,0.14286,0.28571,0.0,0.5714,12,0,0,12,0,9,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,43,0.4651,0.17858,0.17128,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,4,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,43,0.5581,0.16518,0.16793,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,7,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[28,43,0.6512,0.1875,0.14914,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,11,0,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,43,0.7442,0.19643,0.1171,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,12,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.16072,0.16656,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,9,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[40,43,0.9302,0.22768,0.15093,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,9,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[43,43,1.0,0.25,0.09449,0.14286,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,9,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.08036,"x":0.1741,"p":[[0,46,0.0,0.1741,0.12745,0.0,0.14286,0.28571,0.0,0.42857,9,0,1,9,0,8,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.15179,0.15947,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,9,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,46,0.1739,0.11159,0.15862,0.0,0.0,0.17857,0.0,0.571,19,0,0,19,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,46,0.2609,0.11161,0.15458,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,10,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,46,0.3478,0.10268,0.1931,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[20,46,0.4348,0.09813,0.12592,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.125,0.15047,0.0,0.07143,0.1786,0.0,0.4286,16,0,0,16,0,8,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.14732,0.18029,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,4,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[32,46,0.6957,0.12491,0.13715,0.0,0.14286,0.17857,0.0,0.57143,14,0,0,14,0,10,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,46,0.7826,0.12946,0.13997,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,7,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.1517,0.18878,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,4,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[44,46,0.9565,0.15183,0.1555,0.0,0.14286,0.28571,0.0,0.43,14,0,0,14,0,6,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.08036,0.10677,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a2aa30df1bd2cd6e","q":"Let $\\triangle A B C$ be an isosceles triangle with $|A B|=|A C|$. Let $D, E$ and $F$ be points on the respective line segments $B C, C A$ and $A B$ such that $|B F|=|B E|$ and such that $E D$ is the angle bisector of $\\angle B E C$.\nProve that $|B D|=|E F|$ if and only if $|A F|=|E C|$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.08027,"x":0.13822,"p":[[0,85,0.0,0.12938,0.08267,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,85,0.0471,0.13822,0.06666,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,85,0.0941,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,85,0.1412,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,85,0.1882,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,85,0.2353,0.12464,0.05912,0.14,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,85,0.2824,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,85,0.3294,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,85,0.3765,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,85,0.4235,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,85,0.4706,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,85,0.5176,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,85,0.5647,0.12492,0.04722,0.14286,0.14286,0.14286,0.0,0.143,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,85,0.6118,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,85,0.6588,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,85,0.7059,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,85,0.7529,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,85,0.8,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,85,0.8471,0.10259,0.07344,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.10714,"x":0.15179,"p":[[0,72,0.0,0.15179,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.12911,0.04154,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,72,0.3333,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,72,0.3889,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.13385,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,72,0.7778,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,72,0.8889,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.11134,0.05892,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6a9ef2968d8ae24c","q":"Let  $p=2017$  be a prime.  Find the remainder when \\[\\left\\lfloor\\dfrac{1^p}p\\right\\rfloor + \\left\\lfloor\\dfrac{2^p}p\\right\\rfloor+\\left\\lfloor\\dfrac{3^p}p\\right\\rfloor+\\cdots+\\left\\lfloor\\dfrac{2015^p}p\\right\\rfloor \\] is divided by  $p$ .  Here  $\\lfloor\\cdot\\rfloor$  denotes the greatest integer function.\n\n*Proposed by David Altizio*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.46875,"x":0.60713,"p":[[0,45,0.0,0.60713,0.24744,0.42859,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,6,0,0,9,0,0,2,0,0,9,0,2],[4,45,0.0889,0.55357,0.13716,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,16,0,0,1,0,0,2,0,1],[8,45,0.1778,0.52231,0.08458,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0],[12,45,0.2667,0.54464,0.10972,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,1,0,0,2,0,0],[16,45,0.3556,0.52679,0.0974,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,17,0,0,1,0,0,1,0,0],[20,45,0.4444,0.53571,0.08748,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,3,0,0,0,0,0],[24,45,0.5333,0.54463,0.05575,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0],[28,45,0.6222,0.5,0.15152,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,13,0,0,14,0,0,2,0,0,1,0,0],[32,45,0.7111,0.46875,0.10853,0.42857,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,0,0,0,19,0,0,12,0,0,0,0,0,0,0,0],[36,45,0.8,0.51339,0.1063,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,12,0,0,17,0,0,2,0,0,0,0,0],[40,45,0.8889,0.50446,0.07973,0.42857,0.5,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0],[44,45,0.9778,0.5089,0.09405,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,11,0,0,20,0,0,0,0,0,0,0,0],[45,45,1.0,0.52231,0.06784,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.14732,"x":0.61607,"p":[[0,51,0.0,0.61161,0.21793,0.42857,0.57143,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,9,0,0,8,0,0,3,0,0,9,0,1],[4,51,0.0784,0.59822,0.17655,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,4,0,0,4,0,2],[8,51,0.1569,0.54463,0.17655,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,10,0,0,15,0,0,1,0,0,3,0,1],[12,51,0.2353,0.5982,0.19045,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,9,0,0,13,0,0,2,0,0,5,0,2],[16,51,0.3137,0.58927,0.21651,0.42857,0.57143,0.75,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,7,0,0,11,0,0,3,0,0,7,0,1],[20,51,0.3922,0.60714,0.19562,0.42857,0.57143,0.75,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,7,0,0,10,0,0,5,0,0,8,0,0],[24,51,0.4706,0.56246,0.12339,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,19,0,0,2,0,0,1,0,1],[28,51,0.549,0.61607,0.1357,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,15,0,0,6,0,0,5,0,0],[32,51,0.6275,0.60268,0.19799,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,11,0,0,9,0,0,2,0,0,8,0,1],[36,51,0.7059,0.59375,0.17169,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,11,0,0,2,0,0,6,0,1],[40,51,0.7843,0.55357,0.14616,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,9,0,0,18,0,0,0,0,0,4,0,0],[44,51,0.8627,0.57588,0.16935,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,10,0,0,12,0,0,5,0,0,3,0,1],[48,51,0.9412,0.29911,0.20628,0.14286,0.14286,0.46429,0.14286,0.85714,0,0,0,0,0,19,0,0,1,0,0,4,0,0,7,0,0,0,0,0,1,0,0],[51,51,1.0,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"238736760992bdb1","q":"Let $a, b, c$ and $d$ be positive real numbers. Prove that\n\n$$\n\\frac{a-b}{b+c}+\\frac{b-c}{c+d}+\\frac{c-d}{d+a}+\\frac{d-a}{a+b} \\geq 0\n$$","t":[{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.84821,"p":[[0,47,0.0,0.84821,0.32328,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,24],[4,47,0.0851,0.29018,0.42481,0.0,0.0,0.67857,0.0,1.0,19,8,0,19,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,8],[8,47,0.1702,0.17857,0.35892,0.0,0.0,0.14286,0.0,1.0,23,5,0,23,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,47,0.2553,0.29911,0.44515,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[16,47,0.3404,0.23214,0.40838,0.0,0.0,0.14286,0.0,1.0,22,7,0,22,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[20,47,0.4255,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,47,0.5106,0.16964,0.3597,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[28,47,0.5957,0.19196,0.38896,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[32,47,0.6809,0.13393,0.32915,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[36,47,0.766,0.22768,0.41166,0.0,0.0,0.07143,0.0,1.0,24,7,0,24,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[40,47,0.8511,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.82143,"p":[[0,66,0.0,0.82143,0.3677,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[4,66,0.0606,0.21875,0.3813,0.0,0.0,0.14286,0.0,1.0,20,6,0,20,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[8,66,0.1212,0.23659,0.41589,0.0,0.0,0.14275,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7],[12,66,0.1818,0.14732,0.32632,0.0,0.0,0.14286,0.0,1.0,23,4,0,23,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,66,0.2424,0.22321,0.41178,0.0,0.0,0.03571,0.0,1.0,24,7,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[20,66,0.303,0.13393,0.32915,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,66,0.3636,0.08482,0.24186,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,66,0.4242,0.04463,0.15741,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[32,66,0.4848,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,66,0.5455,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,66,0.6061,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,66,0.6667,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,66,0.7273,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,66,0.7879,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,66,0.8485,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[60,66,0.9091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a0a79e5adfb3d23","q":"Point  $O$  is inside  $\\triangle ABC$ . The feet of perpendicular from  $O$  to  $BC,CA,AB$  are  $D,E,F$ . Perpendiculars from  $A$  and  $B$  respectively to  $EF$  and  $FD$  meet at  $P$ . Let  $H$  be the foot of perpendicular from  $P$  to  $AB$ . Prove that  $D,E,F,H$  are concyclic.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.19178,"x":0.62496,"p":[[0,83,0.0,0.5357,0.36246,0.2857,0.35714,1.0,0.0,1.0,3,10,0,3,0,2,0,0,11,0,0,2,0,0,1,0,0,2,0,0,1,0,10],[4,83,0.0482,0.62496,0.35311,0.42859,0.71429,1.0,0.0,1.0,5,9,0,5,0,2,0,0,0,0,0,2,0,0,4,0,0,7,0,0,3,0,9],[8,83,0.0964,0.48658,0.30062,0.28571,0.42857,0.71429,0.0,1.0,3,3,0,3,0,2,0,0,9,0,0,4,0,0,3,0,0,4,0,0,4,0,3],[12,83,0.1446,0.46863,0.28183,0.2857,0.42859,0.71429,0.0,1.0,4,1,0,4,0,2,0,0,6,0,0,5,0,0,5,0,0,5,0,0,4,0,1],[16,83,0.1928,0.50892,0.3173,0.25,0.50001,0.71429,0.0,1.0,3,3,0,3,0,5,0,0,4,0,0,4,0,0,1,0,0,8,0,0,4,0,3],[20,83,0.241,0.45981,0.26662,0.2857,0.28571,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,14,0,0,4,0,0,2,0,0,3,0,0,4,0,2],[24,83,0.2892,0.38837,0.31385,0.1429,0.28571,0.57143,0.0,1.0,6,3,0,6,0,4,0,0,9,0,0,2,0,0,4,0,0,2,0,0,2,0,3],[28,83,0.3373,0.33905,0.22525,0.1429,0.28571,0.4286,0.0,1.0,3,1,0,3,0,7,0,0,9,0,0,6,0,0,4,0,0,2,0,0,0,0,1],[32,83,0.3855,0.44194,0.29091,0.25,0.35714,0.71429,0.0,1.0,1,3,0,1,0,7,0,0,8,0,0,6,0,0,1,0,0,3,0,0,3,0,3],[36,83,0.4337,0.4955,0.32138,0.14286,0.4286,0.74996,0.0,1.0,1,5,0,1,0,9,0,0,2,0,0,6,0,0,3,0,0,3,0,0,3,0,5],[40,83,0.4819,0.41297,0.31631,0.14286,0.35714,0.71429,0.0,1.0,4,4,0,4,1,5,0,0,6,0,0,7,0,0,0,0,0,4,0,0,1,0,4],[44,83,0.5301,0.40614,0.25289,0.25,0.35714,0.571,0.0,1.0,1,2,0,1,0,7,0,0,8,0,0,7,0,0,3,0,0,3,0,0,1,0,2],[48,83,0.5783,0.36582,0.27426,0.14214,0.28571,0.57111,0.0,0.857,6,0,0,6,0,5,0,0,7,0,0,2,0,0,5,0,0,5,0,0,2,0,0],[52,83,0.6265,0.29009,0.24873,0.14286,0.2857,0.2857,0.0,1.0,5,2,0,5,0,7,0,0,14,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[56,83,0.6747,0.28561,0.23958,0.14286,0.21435,0.42858,0.0,0.857,6,0,0,6,0,10,0,0,5,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[60,83,0.7229,0.29006,0.28458,0.0,0.2857,0.42857,0.0,1.0,9,2,0,9,0,5,0,0,9,0,0,3,0,0,2,0,0,1,0,0,1,0,2],[64,83,0.7711,0.22319,0.21993,0.0,0.2857,0.28571,0.0,0.85714,11,0,0,11,0,4,0,0,12,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[68,83,0.8193,0.19178,0.13657,0.14214,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,13,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[72,83,0.8675,0.28117,0.24091,0.14286,0.21435,0.28571,0.0,1.0,4,1,0,4,0,12,0,0,9,0,0,2,0,0,0,0,0,4,0,0,0,0,1],[76,83,0.9157,0.20072,0.17812,0.14214,0.14288,0.2857,0.0,0.85714,6,0,0,6,0,15,0,0,7,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[80,83,0.9639,0.30345,0.2594,0.14286,0.2857,0.42858,0.0,1.0,5,1,0,5,0,9,0,0,9,0,0,3,0,0,2,0,0,1,0,0,2,0,1],[83,83,1.0,0.20518,0.13811,0.14286,0.14288,0.2857,0.0,0.71429,4,0,0,4,0,14,0,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"falling","v":0.29018,"x":0.58482,"p":[[0,59,0.0,0.58482,0.36133,0.2857,0.42857,1.0,0.0,1.0,1,13,0,1,0,3,0,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,13],[4,59,0.0678,0.52226,0.31052,0.25004,0.571,0.71429,0.0,1.0,3,3,0,3,0,5,0,0,2,0,0,4,0,0,4,0,0,7,0,0,4,0,3],[8,59,0.1356,0.55357,0.31083,0.2857,0.57144,0.85704,0.0,1.0,1,4,0,1,0,5,0,0,5,0,0,5,0,0,0,0,0,6,0,0,6,0,4],[12,59,0.2034,0.55353,0.29394,0.42857,0.571,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,1,0,0,8,0,0,2,0,0,8,0,0,3,0,4],[16,59,0.2712,0.54016,0.33068,0.2857,0.57121,0.85714,0.0,1.0,3,5,0,3,0,4,0,0,4,0,0,4,0,0,3,0,0,4,0,0,5,0,5],[20,59,0.339,0.50216,0.25966,0.2857,0.5355,0.71407,0.0,1.0,1,3,0,1,0,4,0,0,5,0,0,5,0,1,7,0,0,5,0,0,1,0,3],[24,59,0.4068,0.52226,0.25902,0.42857,0.42857,0.57143,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,11,0,0,7,0,0,0,0,0,3,0,4],[28,59,0.4746,0.4954,0.29457,0.2857,0.42857,0.75,0.0,1.0,2,2,0,2,0,5,0,0,4,0,0,7,0,0,3,0,0,3,0,0,6,0,2],[32,59,0.5424,0.47082,0.24419,0.28571,0.49979,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,5,0,0,5,0,0,5,1,0,8,0,0,2,0,0],[36,59,0.6102,0.482,0.29404,0.2857,0.4286,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,4,0,0,7,0,0,3,0,0,6,0,0,2,0,3],[40,59,0.678,0.54015,0.27602,0.28571,0.42857,0.74996,0.14286,1.0,0,4,0,0,0,4,0,0,5,0,0,9,0,0,2,0,0,4,0,0,4,0,4],[44,59,0.7458,0.45077,0.24779,0.2857,0.42857,0.71407,0.0,1.0,1,1,0,1,0,4,0,0,8,0,0,9,0,0,1,0,0,5,0,0,3,0,1],[48,59,0.8136,0.47767,0.33428,0.2857,0.42857,0.71429,0.0,1.0,5,6,0,5,0,2,0,0,6,0,0,6,0,0,3,0,0,3,0,0,1,0,6],[52,59,0.8814,0.44194,0.25342,0.28571,0.42857,0.571,0.0,1.0,3,2,0,3,0,2,0,0,5,0,0,13,0,0,3,0,0,2,0,0,2,0,2],[56,59,0.9492,0.29018,0.2382,0.14286,0.2857,0.42857,0.0,1.0,6,2,0,6,0,5,0,0,12,0,0,6,0,0,1,0,0,0,0,0,0,0,2],[59,59,1.0,0.36607,0.25487,0.14286,0.28571,0.46525,0.0,1.0,3,1,0,3,0,7,0,0,8,0,0,6,0,0,3,0,0,2,0,0,2,0,1]]}]},{"i":"4712bb34683a8413","q":"Point  $O$  is a center of circumcircle of acute triangle  $ABC$ , bisector of angle  $BAC$  cuts side  $BC$  in point  $D$ . Let  $M$  be a point such that,  $MC \\perp BC$  and  $MA \\perp AD$ . Lines  $BM$  and  $OA$  intersect in point  $P$ . Show that circle of center in point  $P$  passing through a point  $A$  is tangent to line  $BC$ .","t":[{"b":1,"e":0.85714,"k":"rising","v":0.44194,"x":0.86607,"p":[[0,101,0.0,0.44194,0.32802,0.24999,0.28571,0.60714,0.0,1.0,4,6,0,4,0,4,0,0,9,0,0,3,0,0,4,0,0,2,0,0,0,0,6],[4,101,0.0396,0.86607,0.17835,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,5,0,18],[8,101,0.0792,0.76339,0.28033,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,5,0,0,3,0,15],[12,101,0.1188,0.77678,0.2285,0.71429,0.85707,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,9,0,9],[16,101,0.1584,0.71427,0.25254,0.57132,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,5,0,0,3,0,0,7,0,0,8,0,7],[20,101,0.198,0.84374,0.22408,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,18],[24,101,0.2376,0.67857,0.32341,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,2,0,0,4,0,0,5,0,0,0,0,0,6,0,0,1,0,13],[28,101,0.2772,0.77677,0.20807,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,12,0,0,3,0,11],[32,101,0.3168,0.78121,0.27905,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,0,2,0,16],[36,101,0.3564,0.7991,0.23107,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,10,0,0,3,0,14],[40,101,0.396,0.70535,0.26949,0.53571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,4,0,0,1,0,0,10,0,0,4,0,9],[44,101,0.4356,0.71427,0.35175,0.53539,0.85714,1.0,0.0,1.0,3,15,0,3,0,3,0,0,0,0,0,2,0,0,1,0,0,6,0,0,2,0,15],[48,101,0.4752,0.77228,0.27167,0.57143,0.857,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,2,0,0,4,0,0,4,0,0,5,0,14],[52,101,0.5149,0.71428,0.34069,0.42859,0.85707,1.0,0.0,1.0,3,14,0,3,0,2,0,0,0,0,0,4,0,0,0,0,0,6,0,0,3,0,14],[56,101,0.5545,0.81696,0.24545,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,8,0,0,5,0,15],[60,101,0.5941,0.70535,0.31122,0.42857,0.857,1.0,0.0,1.0,2,10,0,2,0,1,0,0,3,0,0,3,0,0,0,0,0,6,0,0,7,0,10],[64,101,0.6337,0.77679,0.31931,0.71429,1.0,1.0,0.0,1.0,2,17,0,2,0,2,0,0,0,0,0,3,0,0,0,0,0,4,0,0,4,0,17],[68,101,0.6733,0.85268,0.23003,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,3,0,19],[72,101,0.7129,0.8348,0.20552,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,15],[76,101,0.7525,0.78568,0.24746,0.571,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,4,0,0,4,0,15],[80,101,0.7921,0.81694,0.22935,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,4,0,15],[84,101,0.8317,0.81249,0.25111,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,3,0,0,0,0,0,7,0,0,4,0,16],[88,101,0.8713,0.85268,0.20972,0.71429,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,0,6,0,17],[92,101,0.9109,0.82587,0.21352,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,4,0,16],[96,101,0.9505,0.73214,0.28065,0.53572,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,3,0,0,4,0,0,3,0,0,4,0,0,5,0,12],[100,101,0.9901,0.82139,0.25756,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,2,0,18],[101,101,1.0,0.70978,0.32633,0.5354,0.85714,1.0,0.0,1.0,2,12,0,2,0,3,0,0,0,0,0,3,0,0,3,0,0,3,0,0,6,0,12]]},{"b":7,"e":0.71429,"k":"rising","v":0.28572,"x":0.94641,"p":[[0,140,0.0,0.28572,0.26,0.14286,0.2143,0.42857,0.0,1.0,6,2,0,6,0,10,0,0,6,0,0,5,0,0,2,0,0,1,0,0,0,0,2],[4,140,0.0286,0.87946,0.17896,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,18],[8,140,0.0571,0.83479,0.23452,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,2,0,18],[12,140,0.0857,0.80356,0.26184,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,5,0,14],[16,140,0.1143,0.89284,0.15974,0.82132,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,20],[20,140,0.1429,0.94641,0.1057,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[24,140,0.1714,0.89272,0.15168,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,19],[28,140,0.2,0.84821,0.20806,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,8,0,0,4,0,17],[32,140,0.2286,0.83927,0.24937,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,19],[36,140,0.2571,0.86161,0.25626,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,6,0,20],[40,140,0.2857,0.83482,0.21162,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,4,0,15],[44,140,0.3143,0.84808,0.23681,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,18],[48,140,0.3429,0.88392,0.21852,0.82132,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,22],[52,140,0.3714,0.87945,0.18249,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,3,0,20],[56,140,0.4,0.875,0.16269,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,1,0,19],[60,140,0.4286,0.88392,0.20341,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,5,0,20],[64,140,0.4571,0.80354,0.22235,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,6,0,0,5,0,14],[68,140,0.4857,0.79911,0.22548,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,13],[72,140,0.5143,0.8281,0.20037,0.71429,0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 $K$  and  $L$  on the sides  $AB$  and  $BC$  of parallelogram  $ABCD$  are such that  $\\angle AKD = \\angle CLD$ . Prove that the circumcenter of triangle  $BKL$  is equidistant from  $A$  and  $C$ .\n*Proposed by I.I.Bogdanov*","t":[{"b":5,"e":0.28571,"k":"rising","v":0.23652,"x":0.41514,"p":[[0,165,0.0,0.23652,0.15412,0.14214,0.2857,0.32143,0.0,0.4286,7,0,0,7,0,5,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[4,165,0.0242,0.32143,0.13832,0.24999,0.42857,0.42857,0.0,0.42857,2,0,0,2,0,6,0,0,6,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[8,165,0.0485,0.29462,0.18873,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,7,0,0,6,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[12,165,0.0727,0.29018,0.14934,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,7,0,0,9,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[16,165,0.097,0.29464,0.17105,0.24999,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,2,0,0,10,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[20,165,0.1212,0.30802,0.2116,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,3,0,0,7,0,0,10,0,0,3,0,0,2,0,0,0,0,0],[24,165,0.1455,0.26786,0.17768,0.14286,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,7,0,0,6,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[28,165,0.1697,0.27232,0.20935,0.0,0.28571,0.42857,0.0,0.57143,10,0,0,10,0,2,0,0,5,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[32,165,0.1939,0.25001,0.19562,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,6,0,0,3,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[36,165,0.2182,0.30356,0.17766,0.24999,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,2,0,0,9,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[40,165,0.2424,0.27229,0.17622,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[44,165,0.2667,0.26786,0.18123,0.14286,0.2857,0.42857,0.0,0.57143,7,0,0,7,0,5,0,0,7,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[48,165,0.2909,0.35713,0.14724,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,3,0,0,8,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[52,165,0.3152,0.32143,0.15567,0.28571,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,10,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[56,165,0.3394,0.30803,0.18935,0.25,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,3,0,0,12,0,0,8,0,0,2,0,0,2,0,0,0,0,0],[60,165,0.3636,0.2991,0.17983,0.14289,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,3,0,0,8,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[64,165,0.3879,0.33036,0.16146,0.28571,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,2,0,0,9,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[68,165,0.4121,0.29911,0.16115,0.24999,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,3,0,0,9,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[72,165,0.4364,0.29463,0.16725,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,9,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[76,165,0.4606,0.26339,0.15612,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,7,0,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[80,165,0.4848,0.34375,0.14664,0.2857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,4,0,0,8,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[84,165,0.5091,0.26786,0.15047,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,5,0,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[88,165,0.5333,0.34374,0.17805,0.28571,0.42857,0.4286,0.0,0.57143,4,0,0,4,0,3,0,0,7,0,0,12,0,0,6,0,0,0,0,0,0,0,0],[92,165,0.5576,0.28572,0.15568,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,9,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[96,165,0.5818,0.28571,0.15152,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,5,0,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[100,165,0.6061,0.27678,0.15542,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,7,0,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[104,165,0.6303,0.35268,0.17491,0.2857,0.35714,0.4286,0.0,0.71429,3,0,0,3,0,2,0,0,11,0,0,11,0,0,3,0,0,2,0,0,0,0,0],[108,165,0.6545,0.27666,0.15947,0.14286,0.28571,0.42857,0.0,0.571,3,0,0,3,0,9,0,0,10,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[112,165,0.6788,0.30356,0.15462,0.2857,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,3,0,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[116,165,0.703,0.35268,0.14279,0.2857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,6,0,0,5,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[120,165,0.7273,0.28125,0.16164,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,6,0,0,6,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[124,165,0.7515,0.33036,0.11538,0.2857,0.35714,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,11,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[128,165,0.7758,0.29017,0.14051,0.14286,0.28571,0.42857,0.0,0.571,1,0,0,1,0,11,0,0,7,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[132,165,0.8,0.33482,0.16602,0.2857,0.35714,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,9,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[136,165,0.8242,0.33033,0.14475,0.24999,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,7,0,0,8,0,0,13,0,0,3,0,0,0,0,0,0,0,0],[140,165,0.8485,0.30803,0.13883,0.24999,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,10,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[144,165,0.8727,0.28116,0.13601,0.25,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,5,0,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[148,165,0.897,0.33918,0.13254,0.28571,0.35714,0.42857,0.0,0.57143,2,0,0,2,0,2,0,0,12,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[152,165,0.9212,0.32811,0.13698,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,5,0,1,11,0,0,11,0,0,3,0,0,0,0,0,0,0,0],[156,165,0.9455,0.35267,0.17489,0.25,0.42857,0.4286,0.0,0.71429,2,0,0,2,0,6,0,0,6,0,0,12,0,0,5,0,0,1,0,0,0,0,0],[160,165,0.9697,0.34819,0.15122,0.24999,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,7,0,0,5,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[164,165,0.9939,0.37497,0.11706,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,13,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[165,165,1.0,0.41514,0.13529,0.42857,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,2,0,0,3,0,0,20,0,0,5,0,0,1,0,0,0,0,0]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trinomials with positive leading coefficients are arranged in the squares of a  $6 \\times 6$  table. Their  $108$   coefficients are all integers from  $-60$  to  $47$  (each number is used once). Prove that at least in one column the sum of all trinomials has a real root.\n\n*K. Kokhas & F. Petrov*","t":[{"b":0,"e":1.0,"k":"rising","v":0.61605,"x":1.0,"p":[[0,76,0.0,0.61605,0.29759,0.42857,0.71429,0.85704,0.0,1.0,2,6,0,2,0,2,0,0,2,0,0,6,0,0,2,0,0,8,0,0,4,0,6],[4,76,0.0526,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,76,0.1053,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,76,0.1579,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[16,76,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,76,0.2632,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,76,0.3158,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,76,0.3684,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[32,76,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,76,0.4737,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,76,0.5263,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[44,76,0.5789,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[48,76,0.6316,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[52,76,0.6842,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,76,0.7368,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,76,0.7895,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[64,76,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,76,0.8947,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,76,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,76,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.85714,"k":"rising","v":0.6473,"x":0.96874,"p":[[0,36,0.0,0.6473,0.3174,0.53539,0.71429,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,1,0,0,2,0,0,3,0,0,7,0,0,9,0,5],[4,36,0.1111,0.95536,0.14914,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[8,36,0.2222,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[12,36,0.3333,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[16,36,0.4444,0.95981,0.12993,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[20,36,0.5556,0.93303,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[24,36,0.6667,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[28,36,0.7778,0.91071,0.11152,0.85711,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[32,36,0.8889,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[36,36,1.0,0.92857,0.10714,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21]]}]},{"i":"31b09468c587bd72","q":"Prove that for any pair of positive integers $k$ and $n$ there exist $k$ positive integers $m_{1}, m_{2}, \\ldots, m_{k}$ such that  $$ 1+\\frac{2^{k}-1}{n}=\\left(1+\\frac{1}{m_{1}}\\right)\\left(1+\\frac{1}{m_{2}}\\right) \\cdots\\left(1+\\frac{1}{m_{k}}\\right) . $$  (Japan)","t":[{"b":1,"e":0.57143,"k":"rising","v":0.15177,"x":0.51783,"p":[[0,68,0.0,0.34812,0.32531,0.105,0.14286,0.57143,0.0,1.0,8,3,6,8,0,9,0,0,1,0,0,0,0,0,10,0,0,0,0,0,1,0,3],[4,68,0.0588,0.26785,0.3229,0.10714,0.14286,0.24992,0.0,1.0,8,4,0,8,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[8,68,0.1176,0.24554,0.28174,0.10714,0.14286,0.25011,0.0,1.0,8,2,0,8,0,16,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,2],[12,68,0.1765,0.36159,0.32533,0.14286,0.14286,0.57143,0.0,1.0,4,4,0,4,0,14,0,0,1,0,0,1,0,0,7,0,0,0,0,0,1,0,4],[16,68,0.2353,0.24108,0.2911,0.14286,0.14286,0.1429,0.0,1.0,6,3,0,6,0,20,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[20,68,0.2941,0.23214,0.26904,0.10714,0.14286,0.1786,0.0,1.0,8,2,0,8,0,16,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,0,2],[24,68,0.3529,0.16954,0.19376,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,23,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[28,68,0.4118,0.15177,0.14694,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,22,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[32,68,0.4706,0.16955,0.16147,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[36,68,0.5294,0.40176,0.29327,0.14286,0.571,0.57143,0.0,1.0,4,1,0,4,0,10,0,0,1,0,0,0,0,0,12,0,0,1,0,0,3,0,1],[40,68,0.5882,0.35714,0.32733,0.14286,0.14286,0.57143,0.0,1.0,6,4,0,6,0,11,0,0,2,0,0,0,0,0,8,0,0,1,0,0,0,0,4],[44,68,0.6471,0.35712,0.23417,0.14286,0.35714,0.57143,0.0,0.71429,3,0,0,3,0,11,0,0,2,0,0,1,0,0,13,0,0,2,0,0,0,0,0],[48,68,0.7059,0.29017,0.25872,0.14286,0.14286,0.57143,0.0,0.85714,7,0,0,7,0,12,0,0,1,0,0,0,0,0,10,0,0,1,0,0,1,0,0],[52,68,0.7647,0.36602,0.24723,0.14286,0.42836,0.57143,0.0,1.0,2,1,0,2,0,13,0,0,1,0,0,0,0,0,15,0,0,0,0,0,0,0,1],[56,68,0.8235,0.42408,0.27544,0.14286,0.42836,0.57143,0.0,1.0,1,2,0,1,0,10,0,0,5,0,0,0,0,0,11,0,0,1,0,0,2,0,2],[60,68,0.8824,0.44639,0.25937,0.25,0.57121,0.57143,0.0,1.0,4,2,0,4,0,4,0,0,2,0,0,3,0,0,16,0,0,1,0,0,0,0,2],[64,68,0.9412,0.4375,0.20497,0.28571,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,4,0,0,2,0,0,2,0,0,21,0,0,0,0,0,0,0,0],[68,68,1.0,0.51783,0.14616,0.5713,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,0,0,0,25,0,0,0,0,0,1,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.0,"x":0.45979,"p":[[0,130,0.0,0.45979,0.36022,0.14286,0.571,0.64286,0.0,1.0,3,7,1,3,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,7],[4,130,0.0308,0.23648,0.25655,0.0,0.14286,0.571,0.0,0.85714,11,0,0,11,0,11,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0],[8,130,0.0615,0.16518,0.2372,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,130,0.0923,0.1741,0.18805,0.0,0.14286,0.1429,0.0,0.57143,10,0,0,10,0,15,0,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[16,130,0.1231,0.21427,0.26485,0.0,0.14286,0.17857,0.0,1.0,11,1,0,11,0,13,0,0,1,0,0,0,0,0,5,0,0,0,0,0,1,0,1],[20,130,0.1538,0.15179,0.21998,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,15,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[24,130,0.1846,0.23214,0.28959,0.10714,0.14286,0.14287,0.0,1.0,8,3,0,8,0,17,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[28,130,0.2154,0.16518,0.26513,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,1],[32,130,0.2462,0.13393,0.13803,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,18,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,130,0.2769,0.10713,0.15148,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,13,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,130,0.3077,0.11161,0.17762,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,130,0.3385,0.16517,0.18592,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,17,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[48,130,0.3692,0.16072,0.17768,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,17,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[52,130,0.4,0.16517,0.21754,0.0,0.14286,0.14286,0.0,0.85714,13,0,0,13,0,13,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[56,130,0.4308,0.14286,0.15152,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,20,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[60,130,0.4615,0.15622,0.19013,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,15,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[64,130,0.4923,0.14285,0.16363,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,17,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[68,130,0.5231,0.13838,0.15351,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,19,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[72,130,0.5538,0.11161,0.17399,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,130,0.5846,0.08929,0.11152,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,130,0.6154,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[84,130,0.6462,0.0892,0.0691,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,130,0.6769,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,130,0.7077,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,130,0.7385,0.125,0.20438,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[100,130,0.7692,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,130,0.8,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,130,0.8308,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,130,0.8615,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[116,130,0.8923,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[120,130,0.9231,0.08036,0.17835,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[124,130,0.9538,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,130,0.9846,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[130,130,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a471031c4d2081b8","q":"Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x, y)$ by either $(y+1, x)$ or $(x-1, x)$. Prove that she can perform only finitely many such iterations.","t":[{"b":3,"e":0.57143,"k":"flat","v":0.40167,"x":0.80355,"p":[[0,47,0.0,0.50446,0.38296,0.14286,0.28571,1.0,0.14286,1.0,0,11,0,0,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,11],[4,47,0.0851,0.80355,0.25693,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,7,0,0,5,0,15],[8,47,0.1702,0.75446,0.33926,0.5,1.0,1.0,0.14286,1.0,0,18,0,0,0,5,0,0,3,0,0,0,0,0,1,0,0,2,0,0,3,0,18],[12,47,0.2553,0.7991,0.2942,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,17],[16,47,0.3404,0.7142,0.34641,0.42857,0.92857,1.0,0.14,1.0,0,16,0,0,0,7,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,16],[20,47,0.4255,0.73661,0.30328,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,1,0,0,1,0,0,0,0,0,6,0,0,8,0,11],[24,47,0.5106,0.62053,0.34369,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,4,0,0,2,0,0,1,0,0,3,0,0,6,0,9],[28,47,0.5957,0.56241,0.34441,0.14286,0.57143,0.89286,0.14,1.0,0,8,0,0,0,9,0,0,3,0,0,3,0,0,3,0,0,2,0,0,4,0,8],[32,47,0.6809,0.64732,0.33309,0.39286,0.78564,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,1,0,0,4,0,0,1,0,0,3,0,0,7,0,9],[36,47,0.766,0.40167,0.32037,0.14286,0.2857,0.60682,0.14,1.0,0,4,0,0,0,15,0,0,5,0,0,2,0,0,2,0,0,1,0,0,3,0,4],[40,47,0.8511,0.64286,0.32537,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,7,0,0,1,0,0,1,0,0,4,0,0,6,0,9],[44,47,0.9362,0.59821,0.33012,0.24999,0.78571,0.85714,0.14286,1.0,0,4,0,0,0,8,0,0,3,0,0,2,0,0,1,0,0,2,0,0,12,0,4],[47,47,1.0,0.65162,0.28105,0.39286,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,1,0,0,2,0,0,6,0,0,12,0,3]]},{"b":5,"e":0.71429,"k":"flat","v":0.5714,"x":0.79911,"p":[[0,60,0.0,0.62044,0.38245,0.14289,0.78564,1.0,0.0,1.0,1,13,0,1,0,8,0,0,3,0,0,2,0,0,0,0,0,2,0,0,3,0,13],[4,60,0.0667,0.79911,0.24707,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,5,0,0,7,0,14],[8,60,0.1333,0.74106,0.31224,0.5,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,5,0,0,0,0,0,1,0,0,2,0,0,8,0,13],[12,60,0.2,0.75445,0.24546,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,7,0,0,7,0,10],[16,60,0.2667,0.74552,0.25187,0.67857,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,6,0,0,10,0,8],[20,60,0.3333,0.75892,0.27303,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,2,0,0,2,0,0,4,0,0,9,0,11],[24,60,0.4,0.68746,0.2683,0.53539,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,2,0,0,3,0,0,3,0,0,6,0,0,9,0,6],[28,60,0.4667,0.64725,0.25998,0.571,0.71429,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,0,0,0,4,0,0,8,0,0,6,0,0,7,0,4],[32,60,0.5333,0.61154,0.31183,0.28571,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,6,0,0,3,0,0,2,0,0,6,0,0,1,0,0,8,0,6],[36,60,0.6,0.74985,0.25508,0.67536,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,0,13,0,7],[40,60,0.6667,0.70981,0.27544,0.57143,0.85707,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,0,0,0,3,0,0,4,0,0,3,0,0,11,0,7],[44,60,0.7333,0.5714,0.29233,0.28571,0.57143,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,4,0,0,4,0,0,4,0,0,6,0,0,5,0,4],[48,60,0.8,0.70086,0.22121,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,4,0,0,11,0,4],[52,60,0.8667,0.69193,0.26989,0.57132,0.857,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,2,0,0,4,0,0,3,0,0,14,0,4],[56,60,0.9333,0.6607,0.22799,0.57132,0.71429,0.74996,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,5,0,0,5,0,0,12,0,0,4,0,4],[60,60,1.0,0.57807,0.1959,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,9,0,1,10,0,0,4,0,0,5,0,1]]}]},{"i":"a9618546fd573e8b","q":"Show that the number  $3$  can be written in a infinite number of different ways as the sum of the cubes of four integers.","t":[{"b":2,"e":0.2857,"k":"rising","v":0.16964,"x":0.47322,"p":[[0,43,0.0,0.16964,0.35792,0.0,0.0,0.0,0.0,1.0,26,4,5,26,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,4],[4,43,0.093,0.47322,0.46898,0.0,0.35714,1.0,0.0,1.0,14,13,0,14,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,13],[8,43,0.186,0.22309,0.30708,0.0,0.0,0.42858,0.0,1.0,18,2,0,18,0,2,0,0,2,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[12,43,0.2791,0.30802,0.3236,0.0,0.35714,0.42858,0.0,1.0,14,3,0,14,0,0,0,0,2,0,0,10,0,0,2,0,0,0,0,0,1,0,3],[16,43,0.3721,0.41518,0.42612,0.0,0.35714,1.0,0.0,1.0,14,9,0,14,0,0,0,0,2,0,0,5,0,0,0,0,0,1,0,0,1,0,9],[20,43,0.4651,0.35268,0.43739,0.0,0.0,0.89286,0.0,1.0,18,8,0,18,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,8],[24,43,0.5581,0.40625,0.42724,0.0,0.28571,1.0,0.0,1.0,14,9,0,14,0,0,0,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,9],[28,43,0.6512,0.31249,0.38205,0.0,0.14286,0.4642,0.0,1.0,15,6,0,15,0,3,0,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,6],[32,43,0.7442,0.29911,0.42462,0.0,0.0,0.57145,0.0,1.0,20,8,0,20,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,8],[36,43,0.8372,0.31696,0.39727,0.0,0.0,0.42857,0.0,1.0,17,7,0,17,0,0,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,7],[40,43,0.9302,0.21875,0.3273,0.0,0.0,0.32143,0.0,1.0,20,2,0,20,0,0,0,0,4,0,0,2,0,0,0,0,0,3,0,0,1,0,2],[43,43,1.0,0.39731,0.34576,0.0,0.42857,0.60682,0.0,1.0,10,5,0,10,0,1,0,0,2,0,0,10,0,0,1,0,0,3,0,0,0,0,5]]},{"b":4,"e":0.0,"k":"flat","v":0.17411,"x":0.5,"p":[[0,103,0.0,0.22321,0.41178,0.0,0.0,0.03571,0.0,1.0,24,7,3,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,103,0.0388,0.29464,0.43144,0.0,0.0,0.85704,0.0,1.0,21,7,0,21,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,7],[8,103,0.0777,0.20982,0.38957,0.0,0.0,0.07143,0.0,1.0,24,6,4,24,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[12,103,0.1165,0.25446,0.40363,0.0,0.0,0.42857,0.0,1.0,22,6,2,22,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,6],[16,103,0.1553,0.5,0.44892,0.0,0.57143,1.0,0.0,1.0,13,11,0,13,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,11],[20,103,0.1942,0.42409,0.4731,0.0,0.0,1.0,0.0,1.0,17,12,0,17,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,12],[24,103,0.233,0.17411,0.31488,0.0,0.0,0.2857,0.0,1.0,22,3,0,22,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[28,103,0.2718,0.37945,0.42198,0.0,0.21428,0.89286,0.0,1.0,15,8,0,15,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,8],[32,103,0.3107,0.25,0.37287,0.0,0.0,0.50002,0.0,1.0,21,3,0,21,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,3],[36,103,0.3495,0.24999,0.36594,0.0,0.0,0.42857,0.0,1.0,19,3,0,19,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0,3,0,3],[40,103,0.3883,0.38842,0.38998,0.0,0.42857,0.64254,0.0,1.0,13,7,0,13,0,0,0,0,2,0,0,8,0,0,1,0,0,0,0,0,1,0,7],[44,103,0.4272,0.25893,0.39194,0.0,0.0,0.42857,0.0,1.0,20,6,0,20,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,6],[48,103,0.466,0.33035,0.38204,0.0,0.14285,0.46431,0.0,1.0,16,5,0,16,0,0,0,0,1,0,0,7,0,0,1,0,0,0,0,0,2,0,5],[52,103,0.5049,0.32143,0.41803,0.0,0.0,0.75,0.0,1.0,18,7,0,18,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,7],[56,103,0.5437,0.47766,0.43537,0.0,0.42857,1.0,0.0,1.0,13,10,0,13,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,2,0,10],[60,103,0.5825,0.30804,0.39465,0.0,0.0,0.4286,0.0,1.0,18,6,0,18,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,6],[64,103,0.6214,0.25893,0.38206,0.0,0.0,0.42857,0.0,1.0,19,5,0,19,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,5],[68,103,0.6602,0.35266,0.36417,0.0,0.28571,0.60714,0.0,1.0,14,4,0,14,0,0,0,0,3,0,0,4,0,0,3,0,0,3,0,0,1,0,4],[72,103,0.699,0.21875,0.3214,0.0,0.0,0.42857,0.0,1.0,19,3,0,19,0,1,0,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,3],[76,103,0.7379,0.33478,0.38398,0.0,0.14286,0.571,0.0,1.0,16,6,0,16,0,0,0,0,1,0,0,5,0,0,4,0,0,0,0,0,0,0,6],[80,103,0.7767,0.36161,0.36243,0.0,0.35714,0.60714,0.0,1.0,13,5,0,13,0,0,0,0,3,0,0,7,0,0,1,0,0,3,0,0,0,0,5],[84,103,0.8155,0.34822,0.3443,0.0,0.35714,0.4286,0.0,1.0,12,4,0,12,0,1,0,0,3,0,0,9,0,0,0,0,0,2,0,0,1,0,4],[88,103,0.8544,0.40623,0.32947,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,1,0,0,3,0,0,9,0,0,1,0,0,4,0,0,2,0,3],[92,103,0.8932,0.37053,0.31311,0.0,0.42857,0.4643,0.0,1.0,10,3,0,10,0,0,0,0,4,0,0,10,0,0,1,0,0,4,0,0,0,0,3],[96,103,0.932,0.37944,0.34182,0.0,0.42857,0.571,0.0,1.0,11,4,0,11,0,1,0,0,1,0,0,10,0,0,2,0,0,2,0,0,1,0,4],[100,103,0.9709,0.25446,0.29609,0.0,0.14285,0.42857,0.0,1.0,16,2,0,16,0,0,0,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,2],[103,103,1.0,0.28569,0.26723,0.0,0.35714,0.42858,0.0,1.0,13,1,0,13,0,0,0,0,3,0,0,10,0,0,4,0,0,1,0,0,0,0,1]]}]},{"i":"38cecfb517e9c5d0","q":"Let  $p$  and  $k$  be positive integers such that  $p$  is prime and  $k>1$ . Prove that there is at most one pair  $(x,y)$  of positive integers such that $x^k+px=y^k$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.82588,"x":0.97545,"p":[[0,11,0.0,0.82588,0.21646,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,6,0,14],[4,11,0.3636,0.93304,0.17491,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[8,11,0.7273,0.94642,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[11,11,1.0,0.97545,0.0524,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,1,26]]},{"b":5,"e":0.71429,"k":"falling","v":0.70087,"x":0.88839,"p":[[0,33,0.0,0.88839,0.14279,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,1,0,3,1,18],[4,33,0.1212,0.84598,0.20053,0.71429,0.85714,1.0,0.07143,1.0,0,15,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,6,0,15],[8,33,0.2424,0.77456,0.25007,0.62511,0.85714,1.0,0.21429,1.0,0,13,0,0,0,0,0,1,2,0,0,3,0,1,1,1,0,4,0,0,6,0,13],[12,33,0.3636,0.8482,0.1426,0.71429,0.85707,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,3,0,14],[16,33,0.4848,0.74551,0.12235,0.71429,0.71429,0.857,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,0,6,0,3],[20,33,0.6061,0.73437,0.12529,0.71429,0.71429,0.857,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,17,0,0,6,1,2],[24,33,0.7273,0.74329,0.13348,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,16,0,0,8,1,2],[28,33,0.8485,0.70087,0.10327,0.67857,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,18,0,0,6,0,0],[32,33,0.9697,0.70981,0.12105,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,18,0,0,6,0,1],[33,33,1.0,0.71413,0.12373,0.71321,0.71429,0.74996,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,17,0,0,7,0,1]]}]},{"i":"f0c455650ae3d307","q":"On a board there are  $n$  nails, each two connected by a rope. Each rope is colored in one of  $n$  given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.\na) Can  $n$  be  $6$  ?\nb) Can  $n$  be  $7$  ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.82142,"x":0.98214,"p":[[0,33,0.0,0.82142,0.21129,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,2,0,17],[4,33,0.1212,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[8,33,0.2424,0.93304,0.22583,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[12,33,0.3636,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[16,33,0.4848,0.93304,0.2082,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[20,33,0.6061,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,33,0.7273,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,33,0.8485,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,33,0.9697,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[33,33,1.0,0.9375,0.1234,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25]]},{"b":3,"e":1.0,"k":"rising","v":0.76786,"x":1.0,"p":[[0,19,0.0,0.76786,0.30462,0.57143,0.92857,1.0,0.0,1.0,3,16,1,3,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,2,0,16],[4,19,0.2105,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,19,0.4211,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,19,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,19,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8a5fd52e4ce9d9ee","q":"Let $\\triangle A B C$ be a triangle. Let $M$ be the midpoint of $B C$ and let $D$ be a point on the interior of side $A B$. The intersection of $A M$ and $C D$ we call $E$. Suppose that $|A D|=|D E|$. Prove that $|A B|=|C E|$.","t":[{"b":6,"e":0.0,"k":"falling","v":0.01786,"x":0.27229,"p":[[0,18,0.0,0.27229,0.35056,0.0,0.14286,0.571,0.0,1.0,14,4,0,14,0,7,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,4],[4,18,0.2222,0.18304,0.29501,0.0,0.0,0.1786,0.0,1.0,18,2,0,18,0,6,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,2],[8,18,0.4444,0.14732,0.2382,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[12,18,0.6667,0.08036,0.21706,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[16,18,0.8889,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.05357,"x":0.92855,"p":[[0,36,0.0,0.22321,0.31731,0.0,0.14286,0.25,0.0,1.0,15,3,0,15,0,9,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,3],[4,36,0.1111,0.3884,0.41069,0.0,0.14288,0.67857,0.0,1.0,13,8,0,13,0,4,0,0,0,0,0,1,0,0,6,0,0,0,0,0,0,0,8],[8,36,0.2222,0.05357,0.14617,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,36,0.3333,0.44195,0.22405,0.28571,0.57121,0.57143,0.0,1.0,3,1,0,3,0,1,0,0,8,0,0,3,0,0,15,0,0,0,0,0,1,0,1],[16,36,0.4444,0.4598,0.15864,0.42857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,2,0,0,15,0,0,12,0,0,0,0,0,1,0,0],[20,36,0.5556,0.5268,0.12076,0.42859,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,9,0,0,20,0,0,0,0,0,0,0,1],[24,36,0.6667,0.82589,0.23887,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,0,0,0,1,0,20],[28,36,0.7778,0.82143,0.27433,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,3,0,0,5,0,0,0,0,0,1,0,21],[32,36,0.8889,0.92855,0.15978,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[36,36,1.0,0.92411,0.18205,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,27]]}]},{"i":"b47110da99c80191","q":"Let $a, b, c$ be lengths of triangle sides, $p=\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{a}$ and $q=\\frac{a}{c}+\\frac{c}{b}+\\frac{b}{a}$.\n\nProve that $|p-q|<1$.","t":[{"b":3,"e":1.0,"k":"rising","v":0.82589,"x":1.0,"p":[[0,16,0.0,0.83929,0.17768,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,6,0,14],[4,16,0.25,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[8,16,0.5,0.87499,0.19805,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,18],[12,16,0.75,0.82589,0.18466,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,13,0,0,1,0,15],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,11,0.0,0.875,0.20438,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,19],[4,11,0.3636,0.89286,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[8,11,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[11,11,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e232429852cddf88","q":"On a circle with diameter  $AC$ , let  $B$  be an arbitrary point distinct from  $A$  and  $C$ . Points  $M, N$  are the midpoints of chords  $AB, BC$ , and points  $P, Q$  are the midpoints of smaller arcs restricted by these chords. Lines  $AQ$  and  $BC$  meet at point  $K$ , and lines  $CP$  and  $AB$  meet at point  $L$ . Prove that lines  $MQ, NP$  and  $KL$  concur.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.77227,"x":0.92411,"p":[[0,125,0.0,0.79016,0.2624,0.71429,0.85714,1.0,0.0,1.0,2,10,1,2,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,13,0,10],[4,125,0.032,0.92411,0.13825,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,22],[8,125,0.064,0.85267,0.24868,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,17],[12,125,0.096,0.85267,0.20666,0.82132,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,15],[16,125,0.128,0.84373,0.22121,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,0,0,6,0,17],[20,125,0.16,0.85714,0.19562,0.82132,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,8,0,16],[24,125,0.192,0.8482,0.19543,0.82132,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,9,0,15],[28,125,0.224,0.86158,0.16167,0.85708,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,13,0,13],[32,125,0.256,0.8973,0.17584,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,19],[36,125,0.288,0.8616,0.17671,0.71429,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,15],[40,125,0.32,0.84812,0.1891,0.82132,0.85714,1.0,0.14,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,11,0,13],[44,125,0.352,0.90178,0.11538,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,16],[48,125,0.384,0.875,0.1948,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,18],[52,125,0.416,0.86604,0.19545,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,18],[56,125,0.448,0.90623,0.13651,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,18],[60,125,0.48,0.82588,0.2362,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,14],[64,125,0.512,0.9107,0.13247,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,20],[68,125,0.544,0.91075,0.14162,0.85714,1.0,1.0,0.43,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[72,125,0.576,0.85713,0.17857,0.857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,11,0,14],[76,125,0.608,0.89732,0.1439,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,18],[80,125,0.64,0.82587,0.19801,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,5,0,0,10,0,12],[84,125,0.672,0.87054,0.15714,0.82143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,8,0,16],[88,125,0.704,0.78122,0.20201,0.67857,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,12,0,8],[92,125,0.736,0.82812,0.20034,0.857,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,2,1,0,16,0,9],[96,125,0.768,0.7991,0.18161,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,8,0,0,13,0,7],[100,125,0.8,0.82576,0.14177,0.71429,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,13,0,8],[104,125,0.832,0.77227,0.23923,0.71429,0.85707,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,0,10,0,9],[108,125,0.864,0.79463,0.17835,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,6,0,0,15,0,6],[112,125,0.896,0.78123,0.18207,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,4,0,0,18,0,4],[116,125,0.928,0.83936,0.18814,0.82132,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,0,10],[120,125,0.96,0.83035,0.12595,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,17,0,6],[124,125,0.992,0.79459,0.13809,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,19,0,3],[125,125,1.0,0.7723,0.17077,0.71429,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,7,0,0,20,0,1]]},{"b":6,"e":0.85714,"k":"flat","v":0.66964,"x":0.9241,"p":[[0,134,0.0,0.70087,0.29093,0.53539,0.857,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,0,9,0,8],[4,134,0.0299,0.88392,0.18364,0.857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,8,0,18],[8,134,0.0597,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[12,134,0.0896,0.84373,0.1968,0.82143,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,12,0,12],[16,134,0.1194,0.7723,0.31106,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,0,0,0,2,0,0,1,0,0,4,0,0,6,0,15],[20,134,0.1493,0.86158,0.16937,0.85711,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,12,0,14],[24,134,0.1791,0.90165,0.12616,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,17],[28,134,0.209,0.85714,0.26244,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,5,0,21],[32,134,0.2388,0.84372,0.26091,0.82132,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,19],[36,134,0.2687,0.86607,0.19865,0.85711,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,9,0,16],[40,134,0.2985,0.9241,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[44,134,0.3284,0.84374,0.1689,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,7,0,14],[48,134,0.3582,0.86159,0.17674,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,12,0,14],[52,134,0.3881,0.88393,0.20652,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,18],[56,134,0.4179,0.875,0.17035,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,4,0,18],[60,134,0.4478,0.78568,0.24225,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,0,8,0,11],[64,134,0.4776,0.79012,0.23691,0.57132,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,4,0,0,5,0,14],[68,134,0.5075,0.7991,0.24186,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,9,0,12],[72,134,0.5373,0.8616,0.1838,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,5,0,17],[76,134,0.5672,0.7857,0.26727,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,11,0,11],[80,134,0.597,0.83927,0.21652,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,7,0,15],[84,134,0.6269,0.81249,0.25616,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,0,9,0,14],[88,134,0.6567,0.83478,0.2205,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,15],[92,134,0.6866,0.79017,0.23141,0.67857,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,12],[96,134,0.7164,0.80355,0.17036,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,10,0,9],[100,134,0.7463,0.71424,0.24224,0.57132,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,2,0,0,2,0,0,5,0,0,10,0,0,4,0,8],[104,134,0.7761,0.7857,0.17498,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,11,0,0,6,0,9],[108,134,0.806,0.78578,0.27436,0.71429,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,13],[112,134,0.8358,0.80355,0.21944,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,6,0,0,10,0,11],[116,134,0.8657,0.80353,0.15875,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,15,0,6],[120,134,0.8955,0.74107,0.32031,0.71429,0.85712,1.0,0.0,1.0,3,13,0,3,0,1,0,0,1,0,0,1,0,0,1,0,0,7,0,0,5,0,13],[124,134,0.9254,0.66964,0.27067,0.42859,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,3,0,0,4,0,0,2,0,0,10,0,0,4,0,7],[128,134,0.9552,0.75444,0.19962,0.57143,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,6,0,0,8,0,8],[132,134,0.9851,0.7723,0.17446,0.71429,0.857,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,10,0,0,12,0,5],[134,134,1.0,0.72314,0.15548,0.57142,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,12,0,0,6,0,4]]}]},{"i":"94e1d9d0dd7d6415","q":"Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\\operatorname{gcd}(a+i, b+j)>1$ for all $i, j \\in\\{0,1, \\ldots, n\\}$, then  $$ \\min \\{a, b\\}>(c n)^{n / 2} $$","t":[{"b":5,"e":0.14286,"k":"rising","v":0.09375,"x":0.65625,"p":[[0,61,0.0,0.09375,0.22477,0.0,0.0,0.0,0.0,1.0,26,1,2,26,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[4,61,0.0656,0.52677,0.34707,0.14286,0.57121,0.85714,0.0,1.0,4,4,0,4,0,5,0,0,3,0,0,3,0,0,2,0,0,4,0,0,7,0,4],[8,61,0.1311,0.46875,0.31791,0.24999,0.5,0.71429,0.0,1.0,6,1,0,6,0,2,0,0,5,0,0,3,0,0,4,0,0,5,0,0,6,0,1],[12,61,0.1967,0.65625,0.32115,0.42859,0.78571,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,1,0,0,2,0,0,2,0,0,5,0,0,10,0,6],[16,61,0.2623,0.50893,0.32328,0.28571,0.57143,0.71429,0.0,1.0,6,3,0,6,0,0,0,0,5,0,0,4,0,0,3,0,0,7,0,0,4,0,3],[20,61,0.3279,0.50445,0.27429,0.28571,0.4998,0.71429,0.0,1.0,2,1,0,2,0,4,0,0,4,0,0,6,0,0,4,0,0,6,0,0,5,0,1],[24,61,0.3934,0.54015,0.29175,0.28571,0.57121,0.71429,0.0,1.0,2,4,0,2,0,2,0,0,7,0,0,3,0,0,5,0,0,6,0,0,3,0,4],[28,61,0.459,0.55357,0.27141,0.28571,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,7,0,0,2,0,0,2,0,0,10,0,0,6,0,1],[32,61,0.5246,0.45536,0.2976,0.24999,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,4,0,0,7,0,0,2,0,0,3,0,0,6,0,0,6,0,0],[36,61,0.5902,0.50446,0.34066,0.24999,0.42857,0.85714,0.0,1.0,5,3,0,5,0,3,0,0,4,0,0,5,0,0,2,0,0,2,0,0,8,0,3],[40,61,0.6557,0.48212,0.27374,0.28571,0.571,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,7,0,0,2,0,0,7,0,0,6,0,0,2,0,2],[44,61,0.7213,0.45087,0.31563,0.24999,0.42859,0.71429,0.0,1.0,6,2,0,6,0,2,0,0,6,0,0,3,0,0,5,0,0,4,0,0,4,0,2],[48,61,0.7869,0.37491,0.24945,0.24999,0.28571,0.46431,0.0,0.85714,3,0,0,3,0,5,0,0,11,0,0,5,0,0,1,0,0,4,0,0,3,0,0],[52,61,0.8525,0.37945,0.23036,0.2857,0.28571,0.57111,0.0,1.0,2,1,0,2,0,5,0,0,11,0,0,5,0,0,5,0,0,2,0,0,1,0,1],[56,61,0.918,0.29911,0.17261,0.24999,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,4,0,0,14,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[60,61,0.9836,0.39284,0.20515,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,12,0,0,2,0,0,7,0,0,5,0,0,0,0,0],[61,61,1.0,0.38392,0.18706,0.2857,0.28586,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,12,0,0,5,0,0,8,0,0,1,0,0,1,0,0]]},{"b":7,"e":0.0,"k":"rising","v":0.08482,"x":0.52679,"p":[[0,63,0.0,0.08482,0.25719,0.0,0.0,0.0,0.0,1.0,28,2,2,28,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,63,0.0635,0.50445,0.33309,0.25,0.49979,0.75,0.0,1.0,4,5,0,4,0,4,0,0,4,0,0,4,0,0,4,0,0,4,0,0,3,0,5],[8,63,0.127,0.48659,0.28315,0.28571,0.57121,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,7,0,0,2,0,0,8,0,0,4,0,0,2,0,3],[12,63,0.1905,0.49107,0.30291,0.28571,0.57143,0.71429,0.0,1.0,5,1,0,5,0,1,0,0,7,0,0,2,0,0,2,0,0,10,0,0,4,0,1],[16,63,0.254,0.5,0.26726,0.28571,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,5,0,0,4,0,0,4,0,0,8,0,0,5,0,0],[20,63,0.3175,0.52232,0.30849,0.28571,0.57143,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,4,0,0,3,0,0,4,0,0,7,0,0,4,0,3],[24,63,0.381,0.50446,0.3194,0.28571,0.42857,0.85704,0.0,1.0,4,3,0,4,0,3,0,0,4,0,0,6,0,0,3,0,0,3,0,0,6,0,3],[28,63,0.4444,0.52679,0.27994,0.28571,0.57143,0.85714,0.0,0.85714,2,0,0,2,0,2,0,0,8,0,0,3,0,0,3,0,0,5,0,0,9,0,0],[32,63,0.5079,0.43304,0.29555,0.14286,0.35714,0.60714,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,1,0,0,7,0,0,2,0,0,5,0,1],[36,63,0.5714,0.45088,0.28817,0.24999,0.42857,0.71429,0.0,1.0,3,1,0,3,0,5,0,0,6,0,0,5,0,0,2,0,0,6,0,0,4,0,1],[40,63,0.6349,0.41071,0.29613,0.14286,0.28571,0.71429,0.0,1.0,3,1,0,3,0,6,0,0,10,0,0,3,0,0,0,0,0,4,0,0,5,0,1],[44,63,0.6984,0.45534,0.28445,0.24999,0.42859,0.71429,0.0,1.0,3,1,0,3,0,5,0,0,6,0,0,3,0,0,4,0,0,7,0,0,3,0,1],[48,63,0.7619,0.43308,0.33213,0.14286,0.28571,0.71429,0.0,1.0,5,4,0,5,0,4,0,0,8,0,0,4,0,0,1,0,0,3,0,0,3,0,4],[52,63,0.8254,0.4107,0.28514,0.14289,0.28571,0.71429,0.0,1.0,3,1,0,3,0,8,0,0,6,0,0,1,0,0,5,0,0,6,0,0,2,0,1],[56,63,0.8889,0.29911,0.24317,0.14286,0.21428,0.57143,0.0,0.71429,5,0,0,5,0,11,0,0,6,0,0,1,0,0,4,0,0,5,0,0,0,0,0],[60,63,0.9524,0.33929,0.22232,0.14286,0.35714,0.42857,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,13,0,0,0,0,0,1,0,0,1,0,1],[63,63,1.0,0.27231,0.16885,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,8,0,0,11,0,0,5,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"3a32351361284351","q":"On a whiteboard, the numbers  $1$  to  $9$  are written. Players  $A$  and  $B$  take turns, and  $A$  is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses.\nDetermine whether any of the players has a winning strategy, and explain why.","t":[{"b":0,"e":0.0,"k":"falling","v":0.0759,"x":0.93304,"p":[[0,152,0.0,0.60714,0.36246,0.42857,0.42859,1.0,0.0,1.0,4,11,0,4,0,2,0,0,0,0,0,11,0,0,0,0,0,0,0,0,4,0,11],[4,152,0.0263,0.86607,0.23941,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,2,0,22],[8,152,0.0526,0.88391,0.22713,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,22],[12,152,0.0789,0.88834,0.228,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,23],[16,152,0.1053,0.90625,0.25657,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,26],[20,152,0.1316,0.89732,0.20896,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,24],[24,152,0.1579,0.85268,0.28456,0.82143,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[28,152,0.1842,0.86159,0.23553,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,20],[32,152,0.2105,0.75446,0.36984,0.64286,1.0,1.0,0.0,1.0,5,20,0,5,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,20],[36,152,0.2368,0.84374,0.22408,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,7,0,16],[40,152,0.2632,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,26],[44,152,0.2895,0.81696,0.26782,0.71429,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,6,0,16],[48,152,0.3158,0.83482,0.32362,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,23],[52,152,0.3421,0.79463,0.32131,0.71429,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,16],[56,152,0.3684,0.78571,0.33312,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,19],[60,152,0.3947,0.7857,0.31135,0.53571,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,5,0,0,2,0,0,0,0,0,3,0,19],[64,152,0.4211,0.82142,0.29451,0.85714,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,16],[68,152,0.4474,0.81244,0.29979,0.67846,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,20],[72,152,0.4737,0.87947,0.23719,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,23],[76,152,0.5,0.71875,0.3754,0.53571,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,17],[80,152,0.5263,0.74107,0.36147,0.64286,0.92857,1.0,0.0,1.0,5,16,0,5,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,5,0,16],[84,152,0.5526,0.77232,0.33093,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,18],[88,152,0.5789,0.79018,0.36243,0.71429,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,21],[92,152,0.6053,0.76339,0.37391,0.53571,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,21],[96,152,0.6316,0.7857,0.37116,0.57143,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,23],[100,152,0.6579,0.65625,0.39907,0.2857,0.85714,1.0,0.0,1.0,6,15,0,6,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,0,2,0,15],[104,152,0.6842,0.74553,0.38752,0.64286,1.0,1.0,0.0,1.0,6,19,0,6,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,19],[108,152,0.7105,0.74999,0.36771,0.67857,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,16],[112,152,0.7368,0.73651,0.38164,0.71429,1.0,1.0,0.0,1.0,5,17,0,5,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,17],[116,152,0.7632,0.69196,0.41205,0.35714,1.0,1.0,0.0,1.0,7,17,0,7,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,17],[120,152,0.7895,0.52237,0.44119,0.0,0.57143,1.0,0.0,1.0,12,12,0,12,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,1,0,12],[124,152,0.8158,0.57589,0.41724,0.10714,0.71429,1.0,0.0,1.0,8,12,0,8,0,2,0,0,0,0,0,5,0,0,0,0,0,2,0,0,3,0,12],[128,152,0.8421,0.45536,0.45237,0.0,0.42859,1.0,0.0,1.0,15,10,0,15,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,10],[132,152,0.8684,0.5267,0.44677,0.0,0.71429,1.0,0.0,1.0,11,11,0,11,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,11],[136,152,0.8947,0.67857,0.39448,0.42857,0.85714,1.0,0.0,1.0,7,14,0,7,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,14],[140,152,0.9211,0.49102,0.4431,0.0,0.64214,1.0,0.0,1.0,13,9,0,13,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,9],[144,152,0.9474,0.62487,0.45699,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,17],[148,152,0.9737,0.36161,0.46837,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,9],[152,152,1.0,0.0759,0.23415,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1]]},{"b":3,"e":0.0,"k":"falling","v":0.07143,"x":0.86606,"p":[[0,152,0.0,0.6741,0.30977,0.42857,0.85714,1.0,0.0,1.0,2,9,1,2,0,0,0,0,2,0,0,10,0,0,0,0,0,0,0,0,9,0,9],[4,152,0.0263,0.80804,0.31055,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,20],[8,152,0.0526,0.73214,0.34947,0.42857,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,4,0,16],[12,152,0.0789,0.80357,0.34022,0.71429,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,21],[16,152,0.1053,0.80804,0.32461,0.78571,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,3,0,21],[20,152,0.1316,0.81697,0.30771,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,21],[24,152,0.1579,0.76339,0.34921,0.64286,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,0,4,0,18],[28,152,0.1842,0.80804,0.33428,0.82143,1.0,1.0,0.0,1.0,2,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cell of a  $(2m +1) \\times (2n +1)$  board are painted in two colors - white and black. The unit cell of a row (column) is called *dominant* on the row (the column) if more than half of the cells that row (column) have the same color as this cell.  Prove that at least  $m + n-1$  cells on the board are dominant in both their row and column.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.33929,"x":0.59818,"p":[[0,38,0.0,0.59818,0.30606,0.39286,0.57143,0.85714,0.0,1.0,2,6,2,2,0,2,0,0,4,0,0,4,0,0,5,0,0,4,0,0,5,0,6],[4,38,0.1053,0.47766,0.249,0.28571,0.42857,0.60682,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,13,0,0,1,0,0,3,0,0,2,0,3],[8,38,0.2105,0.49107,0.27879,0.28571,0.42857,0.60714,0.14286,1.0,0,5,0,0,0,6,0,0,4,0,0,11,0,0,3,0,0,2,0,0,1,0,5],[12,38,0.3158,0.50893,0.22851,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,17,0,0,1,0,0,5,0,0,1,0,3],[16,38,0.4211,0.37945,0.21608,0.14286,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,10,0,0,4,0,0,11,0,0,4,0,0,1,0,0,1,0,1],[20,38,0.5263,0.47545,0.29753,0.25,0.42857,0.71429,0.0,1.0,1,5,0,1,0,7,0,0,4,0,0,8,0,0,3,0,0,3,1,0,0,0,5],[24,38,0.6316,0.39286,0.26486,0.14286,0.42857,0.42858,0.14286,1.0,0,3,0,0,0,12,0,0,3,0,0,10,0,0,1,0,0,3,0,0,0,0,3],[28,38,0.7368,0.33929,0.20124,0.14286,0.35714,0.42857,0.14286,0.85714,0,0,0,0,0,13,0,0,3,0,0,11,0,0,3,0,0,0,0,0,2,0,0],[32,38,0.8421,0.45089,0.2699,0.24999,0.42857,0.46431,0.14286,1.0,0,3,0,0,0,8,0,0,3,0,0,13,0,0,1,0,0,1,0,0,3,0,3],[36,38,0.9474,0.3482,0.21704,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,11,0,0,6,0,0,11,0,0,2,0,0,0,0,0,0,0,2],[38,38,1.0,0.38839,0.21498,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,2,0,0,14,0,0,1,0,0,4,0,0,0,0,1]]},{"b":2,"e":0.42857,"k":"flat","v":0.47321,"x":0.65177,"p":[[0,62,0.0,0.61594,0.26608,0.42857,0.57121,0.85714,0.14,1.0,0,6,0,0,0,3,0,0,1,0,0,9,0,0,5,0,0,4,0,0,4,0,6],[4,62,0.0645,0.60714,0.29667,0.42857,0.50001,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,3,0,0,10,0,0,3,0,0,2,0,0,2,0,9],[8,62,0.129,0.53571,0.29451,0.39286,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,5,0,0,3,0,0,13,0,0,0,0,0,2,0,0,3,0,6],[12,62,0.1935,0.625,0.28738,0.42857,0.50001,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,1,0,0,12,0,0,1,0,0,3,0,0,4,0,8],[16,62,0.2581,0.56249,0.30501,0.42857,0.57121,0.75,0.0,1.0,1,6,0,1,0,6,0,0,0,0,0,8,0,0,3,0,0,6,0,0,2,0,6],[20,62,0.3226,0.55355,0.28738,0.42857,0.42857,0.74996,0.14286,1.0,0,6,0,0,0,5,0,0,2,0,0,11,0,0,2,0,0,4,0,0,2,0,6],[24,62,0.3871,0.57142,0.27199,0.42857,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,13,0,0,2,0,0,5,0,0,0,0,7],[28,62,0.4516,0.63392,0.27418,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,10,0,0,3,0,0,4,0,0,3,0,8],[32,62,0.5161,0.53572,0.26486,0.42857,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,4,0,0,2,0,0,13,0,0,4,0,0,2,0,0,2,0,5],[36,62,0.5806,0.47321,0.30813,0.28571,0.42857,0.60714,0.14286,1.0,0,7,0,0,0,7,0,0,7,0,0,9,0,0,1,0,0,1,0,0,0,0,7],[40,62,0.6452,0.50893,0.29437,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,7,0,0,3,0,0,10,0,0,1,0,0,4,0,0,2,0,5],[44,62,0.7097,0.48647,0.25458,0.39286,0.42857,0.71107,0.0,1.0,1,3,0,1,0,4,0,0,3,0,0,13,0,0,2,0,0,5,0,0,1,0,3],[48,62,0.7742,0.65177,0.2878,0.42857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,7,0,0,7,0,0,3,0,0,0,0,11],[52,62,0.8387,0.52229,0.32064,0.2857,0.42857,0.85704,0.0,1.0,2,6,0,2,0,5,0,0,3,0,0,8,0,0,3,0,0,2,0,0,3,0,6],[56,62,0.9032,0.61607,0.31428,0.42857,0.57143,1.0,0.0,1.0,2,10,0,2,0,2,0,0,1,0,0,9,0,0,3,0,0,5,0,0,0,0,10],[60,62,0.9677,0.64732,0.31741,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,4,0,0,7,0,0,3,0,0,1,0,0,2,0,12],[62,62,1.0,0.50888,0.21996,0.28571,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,8,0,0,10,0,0,4,0,0,5,0,0,2,0,2]]}]},{"i":"3e507fdd62208b04","q":"Triangle $A B C$ is inscribed in circle $\\Omega$. The interior angle bisector of angle $A$ intersects side $B C$ and $\\Omega$ at $D$ and $L$ (other than $A$ ), respectively. Let $M$ be the midpoint of side $B C$. The circumcircle of triangle $A D M$ intersects sides $A B$ and $A C$ again at $Q$ and $P$ (other than $A$ ), respectively. Let $N$ be the midpoint of segment $P Q$, and let $H$ be the foot of the perpendicular from $L$ to line $N D$. Prove that line $M L$ is tangent to the circumcircle of triangle $H M N$.","t":[{"b":4,"e":0.28571,"k":"flat","v":0.04911,"x":0.34365,"p":[[0,49,0.0,0.34365,0.36923,0.0,0.14288,0.60714,0.0,1.0,12,5,2,12,0,5,0,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,5],[4,49,0.0816,0.21429,0.29014,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,1,0,0,5,0,0,2,0,0,1,0,0,4,0,0,0,0,1],[8,49,0.1633,0.08481,0.1466,0.0,0.0,0.17857,0.0,0.571,23,0,0,23,0,1,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,49,0.2449,0.05804,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.11607,0.19704,0.0,0.0,0.17857,0.0,0.71429,21,0,0,21,0,3,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[24,49,0.4898,0.11161,0.15865,0.0,0.0,0.17857,0.0,0.71429,18,0,0,18,0,6,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,49,0.5714,0.09821,0.16917,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[32,49,0.6531,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.10715,0.19233,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,5,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[40,49,0.8163,0.15179,0.23128,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,4,0,0,4,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[44,49,0.898,0.14723,0.22156,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,8,0,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[48,49,0.9796,0.20535,0.2394,0.0,0.07143,0.32164,0.0,0.71429,16,0,0,16,0,1,0,0,7,0,0,4,0,0,1,0,0,3,0,0,0,0,0],[49,49,1.0,0.19643,0.19805,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,5,0,0,10,0,0,3,0,0,0,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.03571,"x":0.35267,"p":[[0,33,0.0,0.35267,0.34438,0.0,0.28571,0.57111,0.0,1.0,9,4,1,9,0,5,0,0,6,0,0,3,0,0,2,0,0,1,0,0,2,0,4],[4,33,0.1212,0.28125,0.3164,0.0,0.14286,0.71429,0.0,0.85714,14,0,0,14,0,4,0,0,4,0,0,0,0,0,0,0,0,9,0,0,1,0,0],[8,33,0.2424,0.23214,0.28291,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,7,0,0,5,0,0,1,0,0,0,0,0,5,0,0,0,0,1],[12,33,0.3636,0.23661,0.30433,0.0,0.0,0.42857,0.0,1.0,17,1,0,17,0,1,0,0,5,0,0,3,0,0,0,0,0,4,0,0,1,0,1],[16,33,0.4848,0.10705,0.19883,0.0,0.0,0.17642,0.0,0.71429,23,0,0,23,0,1,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[20,33,0.6061,0.13839,0.2435,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,3,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[24,33,0.7273,0.13839,0.24086,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,1,0,0,5,0,0,0,0,0,0,0,0,4,0,0,0,0,0],[28,33,0.8485,0.10268,0.20589,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,2,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[32,33,0.9697,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.06696,0.11836,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cd07da294286cf11","q":"Prove that  $2^{2^{n}}+2^{2^{{n-1}}}+1$  has at least  $n$  distinct prime divisors.","t":[{"b":4,"e":1.0,"k":"rising","v":0.83928,"x":1.0,"p":[[0,28,0.0,0.83928,0.25692,0.78561,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,5,0,19],[4,28,0.1429,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,28,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,28,0.4286,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,28,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,28,0.7143,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"rising","v":0.81696,"x":0.98884,"p":[[0,28,0.0,0.81696,0.22654,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,1,0,0,3,0,17],[4,28,0.1429,0.87497,0.18476,0.67857,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,2,0,21],[8,28,0.2857,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,28,0.4286,0.97321,0.05578,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,28,0.5714,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[20,28,0.7143,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,28,0.8571,0.98884,0.0362,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,29],[28,28,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"2a75fe8a5d3363d6","q":"Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are  equal?\n\n\n*Proposed by Belarus for the 1999th IMO*","t":[{"b":5,"e":1.0,"k":"falling","v":0.31696,"x":0.61606,"p":[[0,23,0.0,0.52231,0.22191,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,18,0,0,2,0,0,5,0,0,1,0,3],[4,23,0.1739,0.61606,0.24598,0.42857,0.42857,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,3,0,0,1,0,8],[8,23,0.3478,0.4509,0.26027,0.14289,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,9,0,0,0,0,0,14,0,0,2,0,0,3,0,0,1,0,3],[12,23,0.5217,0.45982,0.30249,0.14286,0.42857,0.50002,0.14286,1.0,0,6,0,0,0,10,0,0,1,0,0,13,0,0,0,0,0,2,0,0,0,0,6],[16,23,0.6957,0.50447,0.25997,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,0,0,0,15,0,0,2,0,0,4,0,0,1,0,4],[20,23,0.8696,0.33482,0.2412,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,17,0,0,0,0,0,10,0,0,0,0,0,3,0,0,1,0,1],[23,23,1.0,0.31696,0.23072,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,17,0,0,3,0,0,7,0,0,0,0,0,4,0,0,0,0,1]]},{"b":6,"e":0.42857,"k":"flat","v":0.45982,"x":0.62499,"p":[[0,31,0.0,0.54464,0.22141,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,22,0,0,0,0,0,3,0,0,2,0,4],[4,31,0.129,0.52231,0.25407,0.42857,0.42857,0.60714,0.14286,1.0,0,5,0,0,0,4,0,0,0,0,0,18,0,0,2,0,0,2,0,0,1,0,5],[8,31,0.2581,0.62499,0.25939,0.42857,0.42857,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,19,0,0,2,0,0,1,0,0,0,0,10],[12,31,0.3871,0.56697,0.22441,0.42857,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,19,0,0,1,0,0,6,0,0,0,0,5],[16,31,0.5161,0.53125,0.17581,0.42857,0.42857,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,6,0,0,1,0,2],[20,31,0.6452,0.48214,0.11152,0.42857,0.42857,0.42858,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0],[24,31,0.7742,0.46429,0.11845,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,29,0,0,0,0,0,2,0,0,0,0,1],[28,31,0.9032,0.45982,0.15458,0.42857,0.42857,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,27,0,0,1,0,0,1,0,0,1,0,1],[31,31,1.0,0.46874,0.08916,0.42857,0.42857,0.42858,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,3,0,0,3,0,0,0,0,0]]}]},{"i":"f5be1fa0e553d35f","q":"Two intersecting circles  $\\omega_1$  and  $\\omega_2$  are given.Lines  $AB,CD$  are common tangents of  $\\omega_1,\\omega_2$ ( $A,C \\in \\omega_1 ,B,D \\in \\omega_2$ )\nLet  $M$  be the midpoint of  $AB$ .Tangents through  $M$  to  $\\omega_1$  and  $\\omega_2$ (other than  $AB$ ) intersect  $CD$  at  $X,Y$ .Let  $I$  be the incenter of  $MXY$ .Prove that  $IC=ID$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.22739,"x":0.5914,"p":[[0,43,0.0,0.22739,0.16027,0.14214,0.2857,0.28571,0.0,0.71429,5,0,3,5,0,10,0,0,13,0,1,0,0,1,1,0,0,1,0,0,0,0,0],[4,43,0.093,0.5914,0.32999,0.28571,0.571,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,8,0,0,2,0,1,2,0,0,2,0,0,5,0,8],[8,43,0.186,0.45533,0.273,0.2857,0.35714,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,12,0,0,4,0,0,3,0,0,5,0,0,1,0,3],[12,43,0.2791,0.50884,0.33691,0.25,0.42857,0.89286,0.14,1.0,0,8,0,0,0,8,0,0,6,0,0,7,0,0,0,0,0,1,0,0,2,0,8],[16,43,0.3721,0.42855,0.33263,0.14289,0.28571,0.6425,0.0,1.0,4,5,0,4,0,5,0,0,9,0,0,3,0,2,1,0,0,0,0,0,3,0,5],[20,43,0.4651,0.47983,0.33428,0.25,0.42857,0.75,0.0,1.0,3,6,0,3,0,5,0,0,6,0,0,6,0,1,0,0,0,3,0,0,2,0,6],[24,43,0.5581,0.37265,0.29442,0.14286,0.28571,0.5,0.0,1.0,3,4,0,3,0,9,0,0,7,0,0,3,0,3,2,0,0,1,0,0,0,0,4],[28,43,0.6512,0.39723,0.31495,0.14286,0.28571,0.46431,0.0,1.0,2,5,0,2,0,9,0,0,9,0,0,4,0,0,1,0,0,1,0,0,1,0,5],[32,43,0.7442,0.36605,0.25172,0.1429,0.28571,0.44643,0.14286,1.0,0,3,0,0,0,10,0,0,11,0,0,3,0,2,2,0,0,1,0,0,0,0,3],[36,43,0.8372,0.39722,0.25634,0.2857,0.28571,0.42857,0.0,1.0,1,4,0,1,0,4,0,0,14,0,0,6,0,2,1,0,0,0,0,0,0,0,4],[40,43,0.9302,0.2699,0.19386,0.14286,0.2857,0.32143,0.0,1.0,3,1,0,3,0,11,0,0,10,0,0,5,0,1,1,0,0,0,0,0,0,0,1],[43,43,1.0,0.23884,0.15731,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,10,0,0,14,0,0,1,0,1,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.25875,"x":0.64284,"p":[[0,53,0.0,0.25875,0.2186,0.14286,0.21435,0.28571,0.0,1.0,3,1,2,3,0,13,0,0,12,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[4,53,0.0755,0.51783,0.31288,0.2857,0.42857,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,12,0,0,4,0,0,3,0,0,1,0,0,2,0,7],[8,53,0.1509,0.5357,0.3481,0.28571,0.42857,1.0,0.0,1.0,1,10,0,1,0,6,0,0,6,0,0,6,0,0,1,0,0,2,0,0,0,0,10],[12,53,0.2264,0.64284,0.369,0.28571,0.714,1.0,0.0,1.0,2,15,0,2,0,3,0,0,5,0,0,3,0,2,1,0,0,0,0,0,1,0,15],[16,53,0.3019,0.48863,0.33287,0.14286,0.42857,0.75,0.0,1.0,1,7,0,1,0,8,0,0,6,0,0,3,0,1,3,0,0,2,0,0,1,0,7],[20,53,0.3774,0.44845,0.25161,0.28571,0.42857,0.571,0.14,1.0,0,3,0,0,0,5,0,0,9,0,0,6,0,3,4,0,0,0,0,0,2,0,3],[24,53,0.4528,0.36152,0.26004,0.14286,0.28571,0.42857,0.0,1.0,2,3,0,2,0,7,0,0,11,0,0,7,0,0,0,0,0,2,0,0,0,0,3],[28,53,0.5283,0.38828,0.25509,0.2857,0.28571,0.4642,0.0,1.0,2,2,0,2,0,5,0,0,11,0,2,4,0,0,2,0,0,3,0,0,1,0,2],[32,53,0.6038,0.46647,0.30302,0.2857,0.42857,0.571,0.0,1.0,1,6,0,1,0,5,0,0,9,0,0,6,0,1,3,0,0,0,0,0,1,0,6],[36,53,0.6792,0.36827,0.28406,0.14289,0.28571,0.51775,0.0,1.0,4,2,0,4,0,7,0,0,8,0,0,4,0,1,2,0,0,2,0,0,2,0,2],[40,53,0.7547,0.41508,0.29536,0.24999,0.42857,0.57111,0.0,1.0,4,4,0,4,0,4,0,0,7,0,0,8,0,0,2,0,0,3,0,0,0,0,4],[44,53,0.8302,0.51775,0.30992,0.2857,0.46429,0.75,0.0,1.0,2,6,0,2,0,2,0,0,9,0,0,3,0,4,1,0,0,3,0,0,2,0,6],[48,53,0.9057,0.47971,0.34561,0.14286,0.42857,0.85704,0.0,1.0,2,6,0,2,0,9,0,0,4,0,0,3,0,1,2,0,0,2,0,0,3,0,6],[52,53,0.9811,0.41487,0.29551,0.14289,0.28571,0.571,0.0,1.0,1,4,0,1,0,9,0,0,8,0,0,4,0,0,3,0,0,2,0,0,1,0,4],[53,53,1.0,0.39953,0.2599,0.24999,0.42857,0.44645,0.0,1.0,2,3,0,2,0,6,0,0,7,0,0,9,0,1,2,0,0,2,0,0,0,0,3]]}]},{"i":"32f2768ee35a164f","q":"Two real sequence  $ \\{x_{n}\\}$  and  $ \\{y_{n}\\}$  satisfies following recurrence formula;\r\n\r $ x_{0}\\equal{} 1$ ,  $ y_{0}\\equal{} 2007$ \r $ x_{n\\plus{}1}\\equal{} x_{n}\\minus{}(x_{n}y_{n}\\plus{}x_{n\\plus{}1}y_{n\\plus{}1}\\minus{}2)(y_{n}\\plus{}y_{n\\plus{}1})$ ,\r $ y_{n\\plus{}1}\\equal{} y_{n}\\minus{}(x_{n}y_{n}\\plus{}x_{n\\plus{}1}y_{n\\plus{}1}\\minus{}2)(x_{n}\\plus{}x_{n\\plus{}1})$ \r\n\r\nThen show that for all nonnegative integer  $ n$ ,  $ {x_{n}}^{2}\\leq 2007$ 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28,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,18,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[84,141,0.5957,0.34821,0.11259,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,17,0,0,13,0,0,0,0,0,1,0,0,0,0,0],[88,141,0.6241,0.32589,0.10853,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,3,0,0,14,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[92,141,0.6525,0.35268,0.11285,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,16,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[96,141,0.6809,0.31695,0.09929,0.2857,0.28571,0.42857,0.0,0.571,1,0,0,1,0,1,0,0,21,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[100,141,0.7092,0.31696,0.08553,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[104,141,0.7376,0.33034,0.14478,0.2857,0.42857,0.42857,0.0,0.571,4,0,0,4,0,0,0,0,11,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[108,141,0.766,0.34375,0.07016,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[112,141,0.7943,0.32143,0.09449,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[116,141,0.8227,0.34375,0.07017,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[120,141,0.8511,0.32143,0.07986,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[124,141,0.8794,0.33035,0.12595,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,20,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[128,141,0.9078,0.33482,0.09182,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[132,141,0.9362,0.34374,0.11767,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,17,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[136,141,0.9645,0.35268,0.13356,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,18,0,0,11,0,0,1,0,0,0,0,0,1,0,0],[140,141,0.9929,0.34821,0.15947,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,0,0,0,15,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[141,141,1.0,0.29464,0.11259,0.28571,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,0,0,0,26,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"40f7132b973e6031","q":"We have $n \\geq 2$ lamps $L_{1}, \\ldots, L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows:  - if the lamp $L_{i}$ and its neighbours (only one neighbour for $i=1$ or $i=n$, two neighbours for other $i$ ) are in the same state, then $L_{i}$ is switched off; - otherwise, $L_{i}$ is switched on.  Initially all the lamps are off except the leftmost one which is on.  (a) Prove that there are infinitely many integers $n$ for which all the lamps will eventually be off.  (b) Prove that there are infinitely many integers $n$ for which the lamps will never be all off.  (France)","t":[{"b":0,"e":0.28571,"k":"rising","v":0.29019,"x":0.8705,"p":[[0,52,0.0,0.29019,0.09771,0.2857,0.28571,0.28571,0.0,0.57143,1,0,1,1,0,2,0,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,52,0.0769,0.74104,0.22428,0.57143,0.857,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,3,0,0,13,0,6],[8,52,0.1538,0.82142,0.23419,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,14],[12,52,0.2308,0.8705,0.14451,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,15],[16,52,0.3077,0.78122,0.24481,0.71429,0.85712,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,4,0,0,11,0,10],[20,52,0.3846,0.75443,0.1864,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,12,0,0,8,0,6],[24,52,0.4615,0.78124,0.24218,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,4,0,0,8,0,12],[28,52,0.5385,0.69194,0.31158,0.571,0.71429,1.0,0.0,1.0,2,11,0,2,0,2,0,0,2,0,0,1,0,0,3,0,0,9,0,0,2,0,11],[32,52,0.6154,0.7857,0.23147,0.57143,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,3,0,0,8,0,12],[36,52,0.6923,0.68737,0.28907,0.57132,0.78564,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,3,0,0,0,0,0,5,0,0,5,0,0,10,0,6],[40,52,0.7692,0.73658,0.29258,0.57132,0.85707,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,3,0,0,4,0,0,5,0,0,5,0,12],[44,52,0.8462,0.72765,0.27977,0.571,0.85707,1.0,0.0,1.0,1,10,0,1,0,0,0,0,5,0,0,1,0,0,2,0,0,6,0,0,7,0,10],[48,52,0.9231,0.76337,0.28929,0.67857,0.85707,1.0,0.0,1.0,2,13,0,2,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,0,6,0,13],[52,52,1.0,0.73209,0.31289,0.5713,0.857,1.0,0.0,1.0,3,12,0,3,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,6,0,12]]},{"b":5,"e":0.2857,"k":"rising","v":0.31696,"x":0.79907,"p":[[0,50,0.0,0.31696,0.13236,0.2857,0.28571,0.28571,0.0,0.71429,1,0,1,1,0,1,0,0,25,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[4,50,0.08,0.74997,0.22305,0.57132,0.85707,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,4,0,0,10,0,8],[8,50,0.16,0.79907,0.23655,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,9,0,12],[12,50,0.24,0.7232,0.28558,0.57143,0.85714,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,10,0,8],[16,50,0.32,0.74552,0.27371,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,1,0,0,3,0,0,6,0,0,5,0,12],[20,50,0.4,0.70978,0.25377,0.571,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,6,0,0,7,0,8],[24,50,0.48,0.65619,0.26694,0.571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,3,0,0,2,0,0,6,0,0,8,0,0,6,0,5],[28,50,0.56,0.71867,0.24354,0.57075,0.78564,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,4,0,0,8,0,8],[32,50,0.64,0.66513,0.29367,0.571,0.71429,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,1,0,0,1,0,0,6,0,0,6,0,0,9,0,5],[36,50,0.72,0.62497,0.27837,0.42857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,3,0,0,5,0,0,5,0,0,8,0,4],[40,50,0.8,0.64266,0.28345,0.571,0.71429,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,0,0,0,7,0,0,5,0,0,9,0,4],[44,50,0.88,0.71427,0.19885,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,10,0,0,9,0,4],[48,50,0.96,0.76333,0.20399,0.57132,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,1,0,0,13,0,7],[50,50,1.0,0.77226,0.1745,0.57143,0.85707,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,3,0,0,13,0,6]]}]},{"i":"a081c081b738fa75","q":"There is a positive integer  $A$ . Two operations are allowed: increasing this number by  $9$  and deleting a digit equal to  $1$  from any position. Is it always possible to obtain  $A+1$  by applying these operations several times?","t":[{"b":0,"e":0.14286,"k":"flat","v":0.22322,"x":0.54455,"p":[[0,35,0.0,0.53571,0.27199,0.28571,0.57143,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,5,0,0,1,0,0,7,0,0,11,0,0,0,0,3],[4,35,0.1143,0.28125,0.30823,0.10714,0.14286,0.28571,0.0,1.0,8,3,0,8,0,11,0,0,6,0,0,0,0,0,1,0,0,3,0,0,0,0,3],[8,35,0.2286,0.22322,0.23402,0.14286,0.14286,0.28571,0.0,1.0,6,2,0,6,0,14,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[12,35,0.3429,0.375,0.36202,0.0,0.21431,0.71429,0.0,1.0,9,4,0,9,0,7,0,0,3,0,0,1,0,0,1,0,0,6,0,0,1,0,4],[16,35,0.4571,0.40179,0.36672,0.14286,0.28571,0.71429,0.0,1.0,6,7,1,6,0,8,0,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,7],[20,35,0.5714,0.36607,0.35524,0.14286,0.21428,0.71429,0.0,1.0,7,5,0,7,0,9,0,0,4,0,0,3,0,0,0,0,0,3,0,0,1,0,5],[24,35,0.6857,0.54455,0.38215,0.14286,0.64286,1.0,0.0,1.0,3,10,0,3,0,7,0,0,5,0,0,0,0,0,1,0,0,5,0,0,1,0,10],[28,35,0.8,0.40177,0.31831,0.14286,0.28571,0.71429,0.0,1.0,2,4,0,2,0,12,0,0,5,0,0,2,0,0,1,0,0,6,0,0,0,0,4],[32,35,0.9143,0.36607,0.36759,0.14286,0.14286,0.71429,0.0,1.0,7,5,0,7,0,10,0,0,5,0,0,0,0,0,0,0,0,3,0,0,2,0,5],[35,35,1.0,0.44196,0.39018,0.14286,0.14288,0.89286,0.0,1.0,3,8,0,3,0,14,0,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,8]]},{"b":5,"e":0.71429,"k":"flat","v":0.22764,"x":0.4732,"p":[[0,27,0.0,0.4732,0.25614,0.24999,0.57143,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,5,0,0,0,0,0,8,0,0,10,0,0,0,0,1],[4,27,0.1481,0.22764,0.23914,0.14286,0.14286,0.2857,0.0,0.85714,7,0,0,7,0,16,0,0,3,0,0,0,0,0,2,0,0,3,0,0,1,0,0],[8,27,0.2963,0.28125,0.2461,0.14286,0.14288,0.42857,0.0,0.71429,6,0,0,6,0,11,0,0,6,0,0,2,0,0,1,0,0,6,0,0,0,0,0],[12,27,0.4444,0.43747,0.3368,0.14286,0.28571,0.71429,0.0,1.0,4,5,0,4,0,8,0,0,5,0,0,0,0,0,5,0,0,5,0,0,0,0,5],[16,27,0.5926,0.4508,0.40272,0.14286,0.2857,1.0,0.0,1.0,5,10,0,5,0,9,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,10],[20,27,0.7407,0.34821,0.2878,0.14286,0.2857,0.57143,0.0,1.0,5,2,0,5,0,8,0,0,8,0,0,1,0,0,3,0,0,5,0,0,0,0,2],[24,27,0.8889,0.27679,0.25238,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,6,0,0,0,0,0,2,0,0,4,0,0,0,0,1],[27,27,1.0,0.42857,0.32143,0.14286,0.35714,0.71429,0.0,1.0,4,3,0,4,0,8,0,0,4,0,0,2,0,0,4,0,0,5,0,0,2,0,3]]}]},{"i":"cd6538013a544111","q":"Solve the equation  $p^3-q^3=pq^3-1$  in primes  $p,q$ .","t":[{"b":3,"e":0.28571,"k":"flat","v":0.30803,"x":0.49107,"p":[[0,61,0.0,0.43303,0.30615,0.28571,0.28571,0.32143,0.0,1.0,1,7,0,1,0,1,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[4,61,0.0656,0.38839,0.24284,0.28571,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[8,61,0.1311,0.46875,0.29717,0.28571,0.28571,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,2,0,6],[12,61,0.1967,0.43303,0.28004,0.28571,0.28571,0.28571,0.2857,1.0,0,5,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[16,61,0.2623,0.43303,0.28231,0.2857,0.28571,0.35714,0.14286,1.0,0,6,0,0,0,1,0,0,23,0,0,0,0,0,2,0,0,0,0,0,0,0,6],[20,61,0.3279,0.49107,0.31932,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[24,61,0.3934,0.38839,0.24284,0.28571,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[28,61,0.459,0.34375,0.18509,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[32,61,0.5246,0.4375,0.2878,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[36,61,0.5902,0.46875,0.30143,0.28571,0.28571,0.64286,0.2857,1.0,0,7,0,0,0,0,0,0,23,0,0,0,0,0,1,0,0,0,0,0,1,0,7],[40,61,0.6557,0.4375,0.2878,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[44,61,0.7213,0.34375,0.17807,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[48,61,0.7869,0.35714,0.20825,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[52,61,0.8525,0.47768,0.31055,0.28571,0.28571,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[56,61,0.918,0.3616,0.19557,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[60,61,0.9836,0.36607,0.22851,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[61,61,1.0,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.2857,"k":"falling","v":0.28571,"x":0.50893,"p":[[0,66,0.0,0.49553,0.33499,0.28571,0.28571,1.0,0.1429,1.0,0,9,0,0,0,3,0,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,9],[4,66,0.0606,0.48661,0.31311,0.28571,0.28571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0,1,0,8],[8,66,0.1212,0.50893,0.33108,0.28571,0.28571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[12,66,0.1818,0.44196,0.27976,0.28571,0.28571,0.35714,0.2857,1.0,0,5,0,0,0,0,0,0,24,0,0,0,0,0,1,0,0,0,0,0,2,0,5],[16,66,0.2424,0.50446,0.32534,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[20,66,0.303,0.43303,0.28231,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[24,66,0.3636,0.38839,0.23753,0.2857,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,4],[28,66,0.4242,0.41964,0.2788,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[32,66,0.4848,0.44196,0.28652,0.28571,0.28571,0.32143,0.2857,1.0,0,6,0,0,0,0,0,0,24,0,0,1,0,0,0,0,0,0,0,0,1,0,6],[36,66,0.5455,0.41516,0.26088,0.2857,0.28571,0.28571,0.2857,1.0,0,5,0,0,0,0,0,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[40,66,0.6061,0.38839,0.24546,0.28571,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,1,0,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[44,66,0.6667,0.41965,0.27879,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[48,66,0.7273,0.36607,0.21706,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[52,66,0.7879,0.36159,0.21123,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[56,66,0.8485,0.35714,0.20825,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[60,66,0.9091,0.31251,0.12595,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[64,66,0.9697,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a7c54ce9f3125362","q":"Some sides and diagonals of a regular  $ n$ -gon form a connected path that visits each vertex exactly once. A *parallel pair* of edges is a pair of two different parallel edges of the path. Prove that\r\n(a) if  $ n$  is even, there is at least one *parallel pair*.\r\n(b) if  $ n$  is odd, there can't be one single *parallel pair*.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.12499,"x":0.35268,"p":[[0,48,0.0,0.35268,0.27196,0.14286,0.28571,0.42857,0.0,1.0,3,3,2,3,0,8,0,0,9,0,0,6,0,0,1,0,0,2,0,0,0,0,3],[4,48,0.0833,0.33909,0.28527,0.14286,0.2143,0.57111,0.0,1.0,3,3,0,3,0,13,0,0,4,0,0,3,0,0,5,0,0,1,0,0,0,0,3],[8,48,0.1667,0.2409,0.20661,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,22,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[12,48,0.25,0.27677,0.22284,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,15,0,0,4,0,0,5,0,0,3,0,0,1,0,0,0,0,1],[16,48,0.3333,0.19188,0.20707,0.105,0.14286,0.1786,0.0,1.0,8,1,0,8,0,16,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[20,48,0.4167,0.20081,0.2522,0.0,0.14286,0.28571,0.0,1.0,12,2,0,12,0,9,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[24,48,0.5,0.15177,0.17101,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,13,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[28,48,0.5833,0.23215,0.26184,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,12,0,0,1,0,0,4,0,0,1,0,0,3,0,0,0,0,1],[32,48,0.6667,0.12499,0.15868,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,15,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[36,48,0.75,0.24544,0.20591,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,21,0,0,5,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[40,48,0.8333,0.17848,0.13366,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[44,48,0.9167,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.16956,0.10974,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.16071,"x":0.26768,"p":[[0,12,0.0,0.26768,0.19491,0.14286,0.14286,0.32143,0.0,1.0,1,1,1,1,0,16,0,0,7,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[4,12,0.3333,0.24552,0.25561,0.10714,0.14286,0.32143,0.0,1.0,8,1,0,8,0,12,0,0,4,0,0,3,0,0,2,0,0,1,0,0,1,0,1],[8,12,0.6667,0.25874,0.22997,0.14286,0.14286,0.32143,0.0,1.0,3,1,0,3,0,18,0,0,3,0,0,3,0,0,2,0,0,2,0,0,0,0,1],[12,12,1.0,0.16071,0.29397,0.0,0.0,0.14286,0.0,1.0,19,3,0,19,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"36727c59c730a925","q":"Prove that the sequence  $a_{n}=\\lfloor n\\sqrt 2 \\rfloor+\\lfloor n\\sqrt 3 \\rfloor$  contains infintely many even and infinitely many odd numbers.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09363,"p":[[0,39,0.0,0.08929,0.20748,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[4,39,0.1026,0.09363,0.20697,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[8,39,0.2051,0.02232,0.10171,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,39,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.07143,0.17857,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[20,39,0.5128,0.07142,0.18207,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[24,39,0.6154,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,39,0.7179,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[32,39,0.8205,0.03125,0.15038,0.0,0.0,0.0,0.0,0.857,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,39,0.9231,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[39,39,1.0,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.0,"x":0.05354,"p":[[0,31,0.0,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.05354,0.16648,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[12,31,0.3871,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,31,0.6452,0.04464,0.1448,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,31,0.7742,0.03571,0.13832,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,31,0.9032,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"160bfa83f387da00","q":"Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?\n\n*Proposed by A. Golovanov*","t":[{"b":2,"e":0.0,"k":"falling","v":0.08929,"x":0.84819,"p":[[0,34,0.0,0.33464,0.34748,0.14286,0.14286,0.28571,0.0,1.0,1,5,0,1,0,23,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,5],[4,34,0.1176,0.70535,0.30291,0.42859,0.85714,1.0,0.0,1.0,1,9,0,1,0,3,0,0,1,0,0,4,0,0,1,0,0,4,0,0,9,0,9],[8,34,0.2353,0.7232,0.34058,0.53539,0.85714,1.0,0.0,1.0,3,13,0,3,0,2,0,0,0,0,0,3,0,0,3,0,0,0,0,0,8,0,13],[12,34,0.3529,0.78121,0.24222,0.67856,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,5,0,0,7,0,12],[16,34,0.4706,0.7857,0.23959,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,5,0,0,4,0,14],[20,34,0.5882,0.77678,0.24468,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,1,0,0,10,0,11],[24,34,0.7059,0.84819,0.21112,0.82132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,8,0,16],[28,34,0.8235,0.72319,0.29221,0.57132,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,2,0,0,4,0,0,5,0,0,7,0,10],[32,34,0.9412,0.45536,0.40632,0.0,0.42859,0.85714,0.0,1.0,11,6,0,11,0,3,0,0,0,0,0,3,0,0,1,0,0,4,0,0,4,0,6],[34,34,1.0,0.08929,0.27837,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2]]},{"b":5,"e":0.57143,"k":"rising","v":0.28097,"x":0.83928,"p":[[0,26,0.0,0.28097,0.32051,0.14286,0.14286,0.14286,0.0,1.0,2,5,0,2,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[4,26,0.1538,0.83928,0.23891,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,18],[8,26,0.3077,0.64286,0.31135,0.39286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,3,0,0,3,0,0,2,0,0,7,0,0,3,0,9],[12,26,0.4615,0.61597,0.2563,0.42857,0.57143,0.85714,0.14,1.0,0,3,0,0,0,3,0,0,2,0,0,6,0,0,7,0,0,2,0,0,9,0,3],[16,26,0.6154,0.83478,0.179,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,12,0,12],[20,26,0.7692,0.81258,0.14483,0.71429,0.85714,0.85786,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,13,0,7],[24,26,0.9231,0.8214,0.10717,0.857,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,24,0,2],[26,26,1.0,0.79462,0.12851,0.71429,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,18,0,3]]}]},{"i":"14d66f142d750c9f","q":"The Baltic Sea has 2016 harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages $C_{1}-C_{2}-\\cdots-C_{1062}$ where all the harbours $C_{1}, \\ldots, C_{1062}$ are distinct. Prove that there exist two disjoint sets $A$ and $B$ of 477 harbours each, such that there is no harbour in $A$ with a direct ferry connection to a harbour in $B$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.12946,"p":[[0,71,0.0,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.12945,0.20313,0.0,0.0,0.14287,0.0,0.71429,19,0,0,19,0,6,0,0,3,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[8,71,0.1127,0.12946,0.16887,0.0,0.0,0.17857,0.0,0.5714,17,0,0,17,0,7,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,71,0.169,0.11607,0.18707,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[16,71,0.2254,0.07589,0.18205,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[20,71,0.2817,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,71,0.338,0.05348,0.11145,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,71,0.4507,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.06696,0.13825,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,71,0.5634,0.04911,0.13651,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,71,0.6197,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,71,0.7324,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,71,0.7887,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.08482,"x":0.26784,"p":[[0,50,0.0,0.08928,0.11152,0.0,0.0,0.14286,0.0,0.28571,18,0,2,18,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,9,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,50,0.16,0.18301,0.25808,0.0,0.0,0.42857,0.0,0.71429,19,0,1,19,0,3,0,0,1,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[12,50,0.24,0.13393,0.19212,0.0,0.0,0.14287,0.0,0.71429,17,0,0,17,0,8,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[16,50,0.32,0.13821,0.19392,0.0,0.0,0.17857,0.0,0.71429,17,0,0,17,0,7,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[20,50,0.4,0.12054,0.17169,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,8,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[24,50,0.48,0.11606,0.15331,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[28,50,0.56,0.125,0.18814,0.0,0.0,0.17857,0.0,0.71429,18,0,0,18,0,6,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[32,50,0.64,0.15168,0.15944,0.0,0.14286,0.2857,0.0,0.571,13,0,0,13,0,9,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,50,0.72,0.25893,0.20652,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,8,0,0,7,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[40,50,0.8,0.24553,0.22934,0.0,0.21428,0.28571,0.0,0.71429,9,0,0,9,0,7,0,0,9,0,0,2,0,0,1,0,0,4,0,0,0,0,0],[44,50,0.88,0.25447,0.17028,0.14286,0.14293,0.32143,0.0,0.71429,2,0,0,2,0,15,0,0,7,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[48,50,0.96,0.26784,0.15869,0.14286,0.2857,0.32143,0.0,0.71429,1,0,0,1,0,14,0,0,9,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[50,50,1.0,0.23661,0.15815,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,15,0,0,11,0,0,2,0,0,0,0,0,2,0,0,0,0,0]]}]},{"i":"cba4923cf4aff297","q":"A special number is a positive integer $n$ for which there exist positive integers $a, b, c$ and $d$ with\n\n$$\nn=\\frac{a^{3}+2 b^{3}}{c^{3}+2 d^{3}}\n$$\n\nProve that:\n(a) there are infinitely many special numbers;\n(b) 2014 is not a special number.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.71875,"x":0.93303,"p":[[0,40,0.0,0.91964,0.15947,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[4,40,0.1,0.85714,0.24223,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,20],[8,40,0.2,0.85713,0.19563,0.82143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,16],[12,40,0.3,0.93303,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[16,40,0.4,0.79464,0.28107,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,17],[20,40,0.5,0.79018,0.3009,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,0,0,0,3,0,0,3,0,0,3,0,18],[24,40,0.6,0.80356,0.21943,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,0,8,0,12],[28,40,0.7,0.80803,0.16982,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,11,0,9],[32,40,0.8,0.71875,0.145,0.57143,0.71429,0.75,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,14,0,0,4,0,4],[36,40,0.9,0.74551,0.15461,0.57143,0.71429,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,11,0,0,5,0,6],[40,40,1.0,0.80356,0.15467,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,8,0,9]]},{"b":5,"e":1.0,"k":"flat","v":0.85714,"x":0.95536,"p":[[0,48,0.0,0.92411,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,26],[4,48,0.0833,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,25],[8,48,0.1667,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[12,48,0.25,0.89284,0.14728,0.82143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,19],[16,48,0.3333,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[20,48,0.4167,0.93303,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[24,48,0.5,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[28,48,0.5833,0.91964,0.12846,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,21],[32,48,0.6667,0.91071,0.13716,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[36,48,0.75,0.91517,0.14223,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[40,48,0.8333,0.85714,0.14286,0.82143,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,12,0,12],[44,48,0.9167,0.88393,0.18013,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,3,0,20],[48,48,1.0,0.91964,0.14698,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,21]]}]},{"i":"e06988b0bb34cd0f","q":"A sequence of real numbers  $u_1, u_2, u_3, \\dots$  is determined by  $u_1$  and the following recurrence relation for  $n \\geq 1$ :\n\\[4u_{n+1} = \\sqrt[3]{ 64u_n + 15.}\\]\nDescribe, with proof, the behavior of  $u_n$  as  $n \\to \\infty.$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.35714,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[8,28,0.2857,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,28,0.4286,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,28,0.5714,0.89285,0.26,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,26],[20,28,0.7143,0.85268,0.27545,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,6,0,21],[24,28,0.8571,0.83928,0.29613,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,23],[28,28,1.0,0.35714,0.28571,0.14289,0.28571,0.28571,0.0,1.0,1,3,0,1,0,9,0,0,16,0,0,0,0,0,0,0,0,0,0,0,3,0,3]]},{"b":5,"e":0.14286,"k":"falling","v":0.30357,"x":0.99107,"p":[[0,43,0.0,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,43,0.093,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,43,0.186,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,43,0.2791,0.9375,0.15542,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,23],[16,43,0.3721,0.83929,0.25191,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,7,0,18],[20,43,0.4651,0.87054,0.18681,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,16,0,13],[24,43,0.5581,0.83929,0.25937,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,10,0,17],[28,43,0.6512,0.83929,0.28291,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,8,0,19],[32,43,0.7442,0.8125,0.26107,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,5,0,0,9,0,14],[36,43,0.8372,0.63839,0.37625,0.28571,0.85714,1.0,0.0,1.0,2,12,0,2,0,4,0,0,7,0,0,0,0,0,0,0,0,1,0,0,6,0,12],[40,43,0.9302,0.62054,0.36,0.25,0.85712,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,5,0,0,0,0,0,0,0,0,2,0,0,8,0,9],[43,43,1.0,0.30357,0.26426,0.14286,0.28571,0.28571,0.0,1.0,2,3,0,2,0,11,0,0,15,0,0,0,0,0,0,0,0,0,0,0,1,0,3]]}]},{"i":"b065edf469dcb66b","q":"A triple of positive integers $(a, b, c)$ is called quasi-Pythagorean if there exists a triangle with lengths of the sides $a, b, c$ and the angle opposite to the side $c$ equal to $120^{\\circ}$. Prove that if $(a, b, c)$ is a quasi-Pythagorean triple then $c$ has a prime divisor greater than 5 .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.83482,"x":0.97321,"p":[[0,42,0.0,0.87052,0.25844,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,24],[4,42,0.0952,0.9241,0.14279,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,21],[8,42,0.1905,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,42,0.2857,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[16,42,0.381,0.90625,0.16982,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,20],[20,42,0.4762,0.87946,0.19269,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,6,0,19],[24,42,0.5714,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[28,42,0.6667,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[32,42,0.7619,0.84821,0.17474,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,13],[36,42,0.8571,0.89283,0.11849,0.85711,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,15],[40,42,0.9524,0.8973,0.10855,0.85711,0.85714,1.0,0.714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,15],[42,42,1.0,0.83482,0.19269,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,8,0,12]]},{"b":7,"e":0.85714,"k":"flat","v":0.8348,"x":0.96875,"p":[[0,54,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,54,0.0741,0.92857,0.17857,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[8,54,0.1481,0.9375,0.14699,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[12,54,0.2222,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[16,54,0.2963,0.96873,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[20,54,0.3704,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[24,54,0.4444,0.90625,0.16982,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,20],[28,54,0.5185,0.90625,0.11071,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[32,54,0.5926,0.87946,0.16016,0.85711,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,16],[36,54,0.6667,0.87946,0.09523,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,10],[40,54,0.7407,0.88838,0.10555,0.857,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,13],[44,54,0.8148,0.84821,0.12339,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,11,0,10],[48,54,0.8889,0.8348,0.12431,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,10,0,9],[52,54,0.963,0.90178,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,12],[54,54,1.0,0.88838,0.09269,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,11]]}]},{"i":"2b76b005848cbc45","q":"A positive integer  $N$  is *interoceanic* if its prime factorization   $$ N=p_1^{x_1}p_2^{x_2}\\cdots p_k^{x_k} $$  satisfies   $$ x_1+x_2+\\dots +x_k=p_1+p_2+\\cdots +p_k. $$  Find all interoceanic numbers less than 2020.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.75,"x":0.8482,"p":[[0,100,0.0,0.75,0.07986,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,4,0,2],[4,100,0.04,0.8482,0.11812,0.71429,0.85714,1.0,0.714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,10,0,10],[8,100,0.08,0.81696,0.11971,0.71429,0.71429,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,7,0,8],[12,100,0.12,0.83022,0.12608,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,9,0,9],[16,100,0.16,0.81696,0.10248,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,13,0,5],[20,100,0.2,0.82143,0.10714,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,12,0,6],[24,100,0.24,0.81237,0.09076,0.71429,0.85714,0.85714,0.71,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,16,0,3],[28,100,0.28,0.82142,0.11294,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,13,0,6],[32,100,0.32,0.8125,0.0974,0.71429,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,14,0,4],[36,100,0.36,0.82589,0.10555,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,13,0,6],[40,100,0.4,0.8125,0.11538,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,11,0,6],[44,100,0.44,0.80357,0.10564,0.71429,0.71429,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,10,0,5],[48,100,0.48,0.8125,0.11538,0.71429,0.85707,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,11,0,6],[52,100,0.52,0.78112,0.09448,0.71429,0.71429,0.85714,0.71,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,3],[56,100,0.56,0.80356,0.10565,0.71429,0.71429,0.85714,0.714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,10,0,5],[60,100,0.6,0.82589,0.08552,0.71429,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,19,0,3],[64,100,0.64,0.79464,0.11811,0.71429,0.71429,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,4,0,7],[68,100,0.68,0.83035,0.11538,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,10,0,8],[72,100,0.72,0.81696,0.10853,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,11,0,6],[76,100,0.76,0.79911,0.09354,0.71429,0.78571,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,13,0,3],[80,100,0.8,0.82142,0.10101,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,14,0,5],[84,100,0.84,0.80803,0.10479,0.71429,0.78564,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,11,0,5],[88,100,0.88,0.78571,0.10102,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,8,0,4],[92,100,0.92,0.82142,0.10101,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,14,0,5],[96,100,0.96,0.76786,0.10564,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,5,0,4],[100,100,1.0,0.81696,0.10248,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,13,0,5]]},{"b":6,"e":0.85714,"k":"flat","v":0.76325,"x":0.84821,"p":[[0,107,0.0,0.77232,0.08645,0.71429,0.71429,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,9,0,2],[4,107,0.0374,0.82143,0.19233,0.71429,0.85714,1.0,0.0,1.0,1,11,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,7,0,11],[8,107,0.0748,0.78133,0.27663,0.71429,0.85714,1.0,0.0,1.0,3,11,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,11],[12,107,0.1121,0.82103,0.18921,0.71429,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,9,0,10],[16,107,0.1495,0.84821,0.11259,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,9],[20,107,0.1869,0.78571,0.17496,0.71429,0.85714,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,14,0,4],[24,107,0.2243,0.81249,0.18707,0.71429,0.85707,1.0,0.0,1.0,1,9,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,9,0,9],[28,107,0.2617,0.82129,0.08765,0.71429,0.85714,0.85714,0.71,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,3],[32,107,0.2991,0.76325,0.16606,0.71429,0.71429,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,10,0,3],[36,107,0.3364,0.79463,0.09406,0.71429,0.857,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,15,0,2],[40,107,0.3738,0.76339,0.16602,0.71429,0.71429,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,10,0,3],[44,107,0.4112,0.76772,0.1666,0.71429,0.71429,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,11,0,3],[48,107,0.4486,0.7991,0.08645,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,18,0,1],[52,107,0.486,0.79018,0.09439,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,11,0,3],[56,107,0.5234,0.79464,0.08702,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,17,0,1],[60,107,0.5607,0.79018,0.08737,0.71429,0.71429,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,13,0,2],[64,107,0.5981,0.8125,0.10374,0.71429,0.85707,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,12,0,5],[68,107,0.6355,0.77679,0.07936,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,12,0,1],[72,107,0.6729,0.83022,0.10389,0.71429,0.85714,0.85714,0.71,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,14,0,6],[76,107,0.7103,0.79017,0.07973,0.71429,0.78564,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,15,0,1],[80,107,0.7477,0.78558,0.10111,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,17,0,1],[84,107,0.785,0.7991,0.10012,0.71429,0.85707,0.85714,0.5714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,14,0,3],[88,107,0.8224,0.81249,0.10971,0.71429,0.85707,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,13,0,5],[92,107,0.8598,0.76339,0.10479,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,10,0,2],[96,107,0.8972,0.78125,0.10705,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,11,0,3],[100,107,0.9346,0.82129,0.11857,0.71429,0.85707,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,11,0,7],[104,107,0.972,0.81696,0.09606,0.71429,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,15,0,4],[107,107,1.0,0.78125,0.07128,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,15,0,0]]}]},{"i":"0890d732058759a4","q":"Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\\ell$ in the plane which is horizontal, vertical, or has slope +1 or -1 , she tells Bob the number of points of $S$ that lie on $\\ell$. Bob wins if he can then determine the set $S$. Prove that if Alice picks $S$ to be of the form  $$ S=\\left\\{(x, y) \\in \\mathbb{Z}^{2} \\mid m \\leq x^{2}+y^{2} \\leq n\\right\\} $$  for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.0133,"x":0.125,"p":[[0,52,0.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.08927,0.14169,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,52,0.1538,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,52,0.2308,0.125,0.17768,0.0,0.0,0.1786,0.0,0.71429,18,0,0,18,0,6,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,52,0.3077,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,52,0.3846,0.06241,0.11252,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.05358,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.04464,0.11538,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.0758,0.13351,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.04009,0.10845,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,52,0.7692,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.04464,0.11538,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00884,"x":0.10714,"p":[[0,64,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.10714,0.15153,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,9,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.07589,0.15146,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,64,0.3125,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.05804,0.12807,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.05357,0.13243,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,64,0.5625,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.03572,0.09449,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.10259,0.1758,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[56,64,0.875,0.07588,0.15961,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[60,64,0.9375,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b3a0e7aec6583477","q":"A round-robin tournament among  $2n$  teams lasted for  $2n-1$  days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the  $n$  games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?","t":[{"b":1,"e":1.0,"k":"flat","v":0.68749,"x":0.97768,"p":[[0,30,0.0,0.85714,0.20825,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,5,0,18],[4,30,0.1333,0.70534,0.24207,0.57132,0.78571,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,2,0,0,9,0,7],[8,30,0.2667,0.76338,0.21903,0.57143,0.78564,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,7,0,0,6,0,10],[12,30,0.4,0.68749,0.27535,0.42857,0.78571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,7,0,0,2,0,0,6,0,0,1,0,0,7,0,9],[16,30,0.5333,0.69194,0.25534,0.571,0.78571,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,6,0,0,1,0,0,8,0,0,1,0,0,9,0,7],[20,30,0.6667,0.79016,0.24482,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,3,0,0,8,0,13],[24,30,0.8,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[28,30,0.9333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[30,30,1.0,0.95535,0.13571,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27]]},{"b":4,"e":1.0,"k":"flat","v":0.558,"x":0.87945,"p":[[0,34,0.0,0.87945,0.16412,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,10,0,16],[4,34,0.1176,0.57587,0.27078,0.28571,0.57121,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,13,0,0,1,0,0,3,0,0,6,0,0,5,0,4],[8,34,0.2353,0.558,0.27283,0.28571,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,8,0,0,2,0,0,8,0,0,1,0,0,7,0,3],[12,34,0.3529,0.73647,0.23177,0.57143,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,0,0,0,8,0,0,3,0,0,9,0,8],[16,34,0.4706,0.72321,0.25238,0.57143,0.78571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,5,0,0,7,0,9],[20,34,0.5882,0.78125,0.2172,0.67857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,5,0,0,9,0,10],[24,34,0.7059,0.77665,0.2171,0.57143,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,4,0,0,9,0,10],[28,34,0.8235,0.63391,0.26229,0.39286,0.57143,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,7,0,0,1,0,0,8,0,0,3,0,0,7,0,5],[32,34,0.9412,0.77679,0.26229,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,3,0,0,6,0,14],[34,34,1.0,0.73659,0.24252,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,7,0,0,5,0,10]]}]},{"i":"e70c2efa62a337d7","q":"A rectangle  $ABCD$  is inscribed in a circle with centre  $O$ . The exterior bisectors of  $\\angle ABD$  and  $\\angle ADB$  intersect at  $P$ ; those of  $\\angle DAB$  and  $\\angle DBA$  intersect at  $Q$ ; those of  $\\angle ACD$  and  $\\angle ADC$  intersect at  $R$ ; and those of  $\\angle DAC$  and  $\\angle DCA$  intersect at  $S$ . Prove that  $P,Q,R$ , and  $S$  are concyclic.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.12938,"x":0.18749,"p":[[0,56,0.0,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.18286,0.1348,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,56,0.1429,0.165,0.10174,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,56,0.2143,0.15625,0.10326,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,56,0.2857,0.15616,0.06548,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.14259,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.16063,0.06918,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.15616,0.10927,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,28,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,56,0.6429,0.18749,0.17652,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,27,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[40,56,0.7143,0.167,0.0922,0.14286,0.14286,0.14286,0.07,0.571,0,0,0,0,1,28,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,56,0.7857,0.12938,0.05185,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,2,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13375,"x":0.18295,"p":[[0,82,0.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,82,0.0488,0.18295,0.13477,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,82,0.0976,0.17411,0.11701,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,82,0.1463,0.15616,0.0969,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,28,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,82,0.1951,0.16956,0.10376,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,82,0.2439,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,82,0.2927,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,82,0.3415,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,82,0.3902,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,82,0.439,0.1383,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,1,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,82,0.4878,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,82,0.5366,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,82,0.5854,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,82,0.6341,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,82,0.6829,0.1425,0.00094,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,82,0.7317,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,82,0.7805,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,82,0.8293,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,82,0.878,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.14732,0.0563,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"696209ee1b2c8c05","q":"All the positive divisors of a positive integer $n$ are stored into an array in increasing order. Mary has to write a program which decides for an arbitrarily chosen divisor $d>1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\\lceil k / 2\\rceil$ divisors of $n$ : If a divisor of $d$ greater than 1 is found among them, then $d$ is composite, otherwise d is prime. Is Mary right?","t":[{"b":5,"e":0.42857,"k":"flat","v":0.41067,"x":0.62938,"p":[[0,23,0.0,0.50445,0.37961,0.14286,0.28571,1.0,0.0,1.0,2,10,1,2,0,9,0,0,6,0,0,1,0,0,2,0,0,1,0,0,1,0,10],[4,23,0.1739,0.62938,0.33298,0.39286,0.71429,1.0,0.0,1.0,2,10,0,2,0,3,0,0,3,0,0,4,0,0,2,0,0,6,0,0,2,0,10],[8,23,0.3478,0.54911,0.30537,0.39286,0.4286,0.85714,0.0,1.0,1,6,0,1,0,5,0,0,2,0,0,9,0,0,3,0,0,3,0,0,3,0,6],[12,23,0.5217,0.62053,0.30641,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,1,0,0,6,0,8],[16,23,0.6957,0.52677,0.27534,0.42857,0.42857,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,3,0,0,14,0,0,3,0,0,2,0,0,2,0,5],[20,23,0.8696,0.42409,0.18379,0.28571,0.42857,0.57111,0.0,1.0,2,1,0,2,0,0,0,0,8,0,0,13,0,0,7,0,0,1,0,0,0,0,1],[23,23,1.0,0.41067,0.20121,0.2857,0.35714,0.57036,0.14286,1.0,0,2,0,0,0,3,0,0,13,0,0,7,0,0,7,0,0,0,0,0,0,0,2]]},{"b":6,"e":0.0,"k":"falling","v":0.40625,"x":0.81236,"p":[[0,44,0.0,0.59375,0.39304,0.2857,0.85714,1.0,0.0,1.0,4,11,0,4,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,0,6,0,11],[4,44,0.0909,0.81236,0.22148,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,8,0,0,7,0,13],[8,44,0.1818,0.61161,0.34853,0.2857,0.71429,1.0,0.0,1.0,1,9,0,1,0,6,0,0,4,0,0,2,0,0,1,0,0,4,0,0,5,0,9],[12,44,0.2727,0.58036,0.27185,0.28571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,6,0,0,4,0,0,5,0,0,5,0,0,5,0,4],[16,44,0.3636,0.62944,0.34968,0.39286,0.71429,1.0,0.0,1.0,3,10,0,3,0,3,0,0,2,0,0,4,0,0,2,0,0,4,0,0,4,0,10],[20,44,0.4545,0.64732,0.3499,0.28571,0.85707,1.0,0.0,1.0,1,12,0,1,0,4,0,0,4,0,0,5,0,0,1,0,0,0,0,0,5,0,12],[24,44,0.5455,0.64738,0.34996,0.2857,0.78564,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,6,0,0,2,0,0,2,0,0,1,0,0,3,0,13],[28,44,0.6364,0.61607,0.29545,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,4,0,0,7,0,0,1,0,0,5,0,0,6,0,6],[32,44,0.7273,0.47766,0.34368,0.14286,0.35714,0.85714,0.0,1.0,3,4,0,3,0,7,0,0,6,0,0,2,0,0,2,0,0,2,0,0,6,0,4],[36,44,0.8182,0.45517,0.32639,0.14286,0.42857,0.71429,0.0,1.0,3,4,0,3,0,7,0,0,5,0,0,5,0,0,1,0,0,4,0,0,3,0,4],[40,44,0.9091,0.60268,0.32485,0.28571,0.71429,0.85714,0.0,1.0,1,6,0,1,0,5,0,0,4,0,0,3,0,0,1,0,0,5,0,0,7,0,6],[44,44,1.0,0.40625,0.37815,0.10714,0.28571,0.85714,0.0,1.0,8,7,0,8,0,4,0,0,7,0,0,4,0,0,0,0,0,0,0,0,2,0,7]]}]},{"i":"a19a1d5b0610a583","q":"4. (AUS 1) ${ }^{\\mathrm{IMO} 2}$ Each of the numbers in the set $N=\\{1,2,3, \\ldots, n-1\\}$, where $n \\geq 3$, is colored with one of two colors, say red or black, so that: (i) $i$ and $n-i$ always receive the same color, and (ii) for some $j \\in N$, relatively prime to $n, i$ and $|j-i|$ receive the same color for all $i \\in N, i \\neq j$. Prove that all numbers in $N$ must receive the same color.","t":[{"b":4,"e":1.0,"k":"flat","v":0.76785,"x":0.9732,"p":[[0,72,0.0,0.84818,0.24987,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,21],[4,72,0.0556,0.93749,0.12344,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25],[8,72,0.1111,0.91518,0.17445,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,24],[12,72,0.1667,0.86159,0.23553,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,21],[16,72,0.2222,0.90624,0.18074,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,0,0,24],[20,72,0.2778,0.88839,0.20743,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,2,0,23],[24,72,0.3333,0.83036,0.26592,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,20],[28,72,0.3889,0.875,0.20748,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,0,2,0,21],[32,72,0.4444,0.88839,0.20743,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,3,0,23],[36,72,0.5,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[40,72,0.5556,0.90178,0.19045,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,22],[44,72,0.6111,0.90625,0.17354,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,4,0,22],[48,72,0.6667,0.91963,0.1285,0.85711,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[52,72,0.7222,0.875,0.24418,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,23],[56,72,0.7778,0.87945,0.24774,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,25],[60,72,0.8333,0.76785,0.31084,0.42859,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0,2,0,0,4,0,17],[64,72,0.8889,0.95089,0.14111,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[68,72,0.9444,0.9732,0.08335,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[72,72,1.0,0.81246,0.25617,0.71421,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,0,2,0,18]]},{"b":7,"e":1.0,"k":"flat","v":0.90179,"x":0.96875,"p":[[0,26,0.0,0.90179,0.20341,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,23],[4,26,0.1538,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,26,0.3077,0.91964,0.17473,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[12,26,0.4615,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[16,26,0.6154,0.90179,0.18013,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,1,0,23],[20,26,0.7692,0.91964,0.20183,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,26],[24,26,0.9231,0.91071,0.14617,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,22],[26,26,1.0,0.91518,0.18851,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,25]]}]},{"i":"63b4b03c2aabe09c","q":"6. (IRE) (a) Let $n$ be a positive integer. Prove that there exist distinct positive integers $x, y, z$ such that  $$ x^{n-1}+y^{n}=z^{n+1} . $$  (b) Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or to $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers $x, y, z$ such that  $$ x^{a}+y^{b}=z^{c} . $$  Original formulation: Let $a, b, c, n$ be positive integers such that $n$ is odd and $a c$ is relatively prime to $2 b$. Prove that there exist distinct positive integers $x, y, z$ such that (i) $x^{a}+y^{b}=z^{c}$, and (ii) $x y z$ is relatively prime to $n$.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.14732,"x":0.44642,"p":[[0,41,0.0,0.44642,0.41457,0.0,0.35714,1.0,0.0,1.0,9,9,5,9,0,6,0,0,1,0,0,3,0,0,2,0,0,0,0,0,2,0,9],[4,41,0.0976,0.38828,0.31189,0.14286,0.28571,0.4642,0.0,1.0,3,5,0,3,0,9,0,0,6,0,0,6,0,0,2,0,0,1,0,0,0,0,5],[8,41,0.1951,0.27232,0.17627,0.14286,0.2857,0.32143,0.0,0.71429,4,0,0,4,0,8,0,0,12,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[12,41,0.2927,0.27231,0.20933,0.14286,0.2143,0.42857,0.0,0.85714,4,0,0,4,0,12,0,0,7,0,0,4,0,0,3,0,0,1,0,0,1,0,0],[16,41,0.3902,0.37054,0.28762,0.14286,0.28571,0.42858,0.0,1.0,3,2,0,3,0,10,0,0,4,0,0,8,0,0,1,0,0,1,0,0,3,0,2],[20,41,0.4878,0.34372,0.25964,0.14286,0.2857,0.4286,0.0,1.0,2,2,0,2,0,11,0,0,7,0,0,5,0,0,3,0,0,1,0,0,1,0,2],[24,41,0.5854,0.31696,0.30248,0.14286,0.2143,0.42857,0.0,1.0,6,3,0,6,0,10,0,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,3],[28,41,0.6829,0.24997,0.23416,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,9,0,0,8,0,0,1,0,0,3,0,0,2,0,0,1,0,0],[32,41,0.7805,0.21429,0.22867,0.0,0.14288,0.28571,0.0,1.0,9,1,0,9,0,10,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[36,41,0.878,0.20536,0.14698,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,15,0,0,11,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[40,41,0.9756,0.19641,0.16653,0.14286,0.14286,0.2857,0.0,0.71429,7,0,0,7,0,13,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[41,41,1.0,0.14732,0.14053,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,17,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.1384,"x":0.41962,"p":[[0,48,0.0,0.36607,0.34981,0.14286,0.2857,0.32143,0.0,1.0,4,7,2,4,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[4,48,0.0833,0.30357,0.25441,0.14286,0.2857,0.42858,0.0,1.0,4,2,0,4,0,11,0,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,2],[8,48,0.1667,0.28562,0.21135,0.14286,0.2143,0.42857,0.0,1.0,1,1,0,1,0,15,0,0,7,0,0,6,0,0,1,0,0,0,0,0,1,0,1],[12,48,0.25,0.30804,0.32165,0.14286,0.14286,0.42857,0.0,1.0,7,4,0,7,0,11,0,0,4,0,0,4,0,0,1,0,0,0,0,0,1,0,4],[16,48,0.3333,0.34821,0.305,0.14286,0.2143,0.57111,0.0,1.0,4,3,0,4,0,12,0,0,4,0,0,3,0,0,3,0,0,2,0,0,1,0,3],[20,48,0.4167,0.41962,0.33868,0.14286,0.28571,0.71429,0.0,1.0,4,4,0,4,0,10,0,0,3,0,0,2,0,0,3,0,0,4,0,0,2,0,4],[24,48,0.5,0.26328,0.30953,0.0,0.14286,0.42858,0.0,1.0,11,3,0,11,0,9,0,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,3],[28,48,0.5833,0.22303,0.18541,0.14286,0.14286,0.2857,0.0,0.85714,4,0,0,4,0,17,0,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[32,48,0.6667,0.21875,0.12869,0.14286,0.14295,0.28571,0.0,0.57143,3,0,0,3,0,14,0,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,48,0.75,0.2723,0.21826,0.14286,0.2143,0.32143,0.0,1.0,3,1,0,3,0,13,0,0,8,0,0,5,0,0,1,0,0,0,0,0,1,0,1],[40,48,0.8333,0.21867,0.23418,0.0,0.1429,0.28571,0.0,0.85714,9,0,0,9,0,11,0,0,7,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[44,48,0.9167,0.1384,0.14934,0.0,0.14286,0.2857,0.0,0.4286,14,0,0,14,0,9,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.16072,0.16656,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,12,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"2e050619aedda24e","q":"8. (HUN 1) (a) Let $(m, k)=1$. Prove that there exist integers $a_{1}, a_{2}, \\ldots, a_{m}$ and $b_{1}, b_{2}, \\ldots, b_{k}$ such that each product $a_{i} b_{j}(i=1,2, \\ldots, m ; j=$ $1,2, \\ldots, k)$ gives a different residue when divided by $m k$. (b) Let $(m, k)>1$. Prove that for any integers $a_{1}, a_{2}, \\ldots, a_{m}$ and $b_{1}, b_{2}$, $\\ldots, b_{k}$ there must be two products $a_{i} b_{j}$ and $a_{s} b_{t}((i, j) \\neq(s, t))$ that give the same residue when divided by $m k$.","t":[{"b":3,"e":0.2857,"k":"flat","v":0.33039,"x":0.52229,"p":[[0,71,0.0,0.33039,0.20653,0.21427,0.42857,0.4286,0.0,0.57143,8,0,8,8,0,0,0,0,4,0,0,14,0,0,6,0,0,0,0,0,0,0,0],[4,71,0.0563,0.52229,0.14986,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,7,0,0,16,0,0,3,0,0,0,0,1],[8,71,0.1127,0.51329,0.11764,0.42859,0.5714,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,5,0,0,20,0,0,2,0,0,0,0,0],[12,71,0.169,0.46422,0.11288,0.42857,0.4286,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,10,0,0,15,0,0,0,0,0,0,0,0],[16,71,0.2254,0.49104,0.12332,0.42857,0.571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,8,0,0,16,0,0,2,0,0,0,0,0],[20,71,0.2817,0.49101,0.12842,0.42857,0.571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,9,0,0,14,0,0,3,0,0,0,0,0],[24,71,0.338,0.48655,0.14659,0.39293,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,7,0,0,14,0,0,2,0,0,1,0,0],[28,71,0.3944,0.42408,0.17305,0.2857,0.42857,0.57141,0.0,1.0,1,1,0,1,0,0,0,0,12,0,0,8,0,0,10,0,0,0,0,0,0,0,1],[32,71,0.4507,0.47763,0.12678,0.42857,0.4998,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,9,0,0,14,0,0,2,0,0,0,0,0],[36,71,0.507,0.44194,0.15708,0.28571,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,5,0,0,9,0,0,4,0,0,0,0,0],[40,71,0.5634,0.41961,0.16726,0.28571,0.28571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,17,0,0,3,0,0,11,0,0,0,0,0,0,0,1],[44,71,0.6197,0.45084,0.12926,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,8,0,0,13,0,0,1,0,0,0,0,0],[48,71,0.6761,0.45083,0.13408,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,6,0,0,14,0,0,1,0,0,0,0,0],[52,71,0.7324,0.42853,0.12368,0.28571,0.42857,0.571,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,11,0,0,9,0,0,1,0,0,0,0,0],[56,71,0.7887,0.40177,0.19375,0.28571,0.28571,0.4286,0.2857,1.0,0,2,0,0,0,0,0,0,20,0,0,5,0,0,4,0,0,1,0,0,0,0,2],[60,71,0.8451,0.47316,0.19043,0.28571,0.4286,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,12,0,0,5,0,0,12,0,0,1,0,0,0,0,2],[64,71,0.9014,0.43302,0.13591,0.2857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,9,0,0,9,0,0,2,0,0,0,0,0],[68,71,0.9577,0.39285,0.12875,0.2857,0.28586,0.4642,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,7,0,0,7,0,0,1,0,0,0,0,0],[71,71,1.0,0.4509,0.15609,0.28571,0.42857,0.571,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,12,0,0,7,0,0,1,0,0,2,0,0]]},{"b":6,"e":0.571,"k":"flat","v":0.3482,"x":0.50887,"p":[[0,53,0.0,0.35709,0.20198,0.2857,0.42857,0.4642,0.0,0.71429,6,0,6,6,0,0,0,0,7,0,0,11,0,0,7,0,0,1,0,0,0,0,0],[4,53,0.0755,0.47771,0.15405,0.42857,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,6,0,0,8,0,0,14,0,0,3,0,0,0,0,0],[8,53,0.1509,0.49103,0.12337,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,10,0,0,16,0,0,0,0,0,1,0,0],[12,53,0.2264,0.43295,0.16157,0.28571,0.42859,0.57141,0.0,0.57143,2,0,0,2,0,0,0,0,8,0,0,7,0,0,15,0,0,0,0,0,0,0,0],[16,53,0.3019,0.43305,0.1359,0.28571,0.42857,0.57141,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,9,0,0,9,0,0,2,0,0,0,0,0],[20,53,0.3774,0.47319,0.15744,0.28571,0.4286,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,8,0,0,13,0,0,1,0,0,0,0,1],[24,53,0.4528,0.50887,0.1554,0.39286,0.57121,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,4,0,0,15,0,0,4,0,0,1,0,0],[28,53,0.5283,0.43746,0.12842,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,9,0,0,11,0,0,1,0,0,0,0,0],[32,53,0.6038,0.4107,0.14172,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,1,0,0,13,0,0,9,0,0,7,0,0,2,0,0,0,0,0],[36,53,0.6792,0.38832,0.12483,0.2857,0.35714,0.4642,0.14286,0.57143,0,0,0,0,0,1,0,0,15,0,0,8,0,0,8,0,0,0,0,0,0,0,0],[40,53,0.7547,0.38835,0.11419,0.28571,0.35714,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[44,53,0.8302,0.44201,0.17627,0.28571,0.42859,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,11,0,0,6,0,0,13,0,0,0,0,0,0,0,1],[48,53,0.9057,0.37498,0.11147,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[52,53,0.9811,0.36604,0.09402,0.2857,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,17,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[53,53,1.0,0.3482,0.11809,0.2857,0.28571,0.42858,0.0,0.57143,1,0,0,1,0,1,0,0,16,0,0,11,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"085646fa0579ec2f","q":"9. (POL 2) Let $a, b, c$ be positive numbers with $\\sqrt{a}+\\sqrt{b}+\\sqrt{c}=\\frac{\\sqrt{3}}{2}$. Prove that the system of equations  $$ \\begin{aligned} & \\sqrt{y-a}+\\sqrt{z-a}=1, \\\\ & \\sqrt{z-b}+\\sqrt{x-b}=1 \\\\ & \\sqrt{x-c}+\\sqrt{y-c}=1 \\end{aligned} $$  has exactly one solution $(x, y, z)$ in real numbers.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.16518,"x":0.3125,"p":[[0,118,0.0,0.2366,0.10479,0.24999,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,4,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,118,0.0339,0.24552,0.13936,0.14286,0.28571,0.28571,0.0,0.571,3,0,0,3,0,11,0,0,11,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,118,0.0678,0.25447,0.14167,0.14289,0.28571,0.32143,0.0,0.4286,5,0,0,5,0,5,0,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,118,0.1017,0.25893,0.15746,0.14286,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,5,0,0,10,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,118,0.1356,0.26331,0.15206,0.14286,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,9,0,0,12,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[20,118,0.1695,0.29018,0.14054,0.24999,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,6,0,0,15,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[24,118,0.2034,0.26339,0.15612,0.14286,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,4,0,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[28,118,0.2373,0.2634,0.15613,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,7,0,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[32,118,0.2712,0.28124,0.121,0.14286,0.28571,0.42857,0.14286,0.571,0,0,0,0,0,11,0,0,12,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[36,118,0.3051,0.28125,0.13592,0.14286,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,9,0,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[40,118,0.339,0.25446,0.14167,0.14286,0.28571,0.32143,0.0,0.57143,3,0,0,3,0,10,0,0,11,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[44,118,0.3729,0.26339,0.14773,0.14286,0.28571,0.42857,0.0,0.42857,5,0,0,5,0,5,0,0,12,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[48,118,0.4068,0.28125,0.14934,0.24999,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,3,0,0,12,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[52,118,0.4407,0.17857,0.16751,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,7,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,118,0.4746,0.27232,0.1488,0.14286,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,7,0,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[60,118,0.5085,0.2857,0.13829,0.24999,0.28571,0.42857,0.0,0.571,3,0,0,3,0,5,0,0,14,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[64,118,0.5424,0.21875,0.12869,0.14286,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,9,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[68,118,0.5763,0.30357,0.12242,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,4,0,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[72,118,0.6102,0.26339,0.12428,0.14286,0.28571,0.28571,0.0,0.4286,3,0,0,3,0,6,0,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[76,118,0.6441,0.24554,0.12492,0.14286,0.28571,0.28571,0.0,0.4286,3,0,0,3,0,9,0,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[80,118,0.678,0.26339,0.14773,0.14286,0.28571,0.42857,0.0,0.42857,4,0,0,4,0,8,0,0,9,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[84,118,0.7119,0.28125,0.13592,0.14286,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,6,0,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[88,118,0.7458,0.3125,0.1448,0.2857,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,5,0,0,12,0,0,12,0,0,0,0,0,1,0,0,0,0,0],[92,118,0.7797,0.24554,0.13475,0.14286,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,8,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[96,118,0.8136,0.25,0.16752,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,5,0,0,10,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[100,118,0.8475,0.27679,0.13333,0.14286,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,9,0,0,10,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[104,118,0.8814,0.26339,0.13415,0.14286,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,8,0,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[108,118,0.9153,0.28125,0.12619,0.24999,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,6,0,0,16,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[112,118,0.9492,0.29463,0.1634,0.14289,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,7,0,0,14,0,0,7,0,0,1,0,0,0,0,0,1,0,0],[116,118,0.9831,0.19647,0.15878,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,9,0,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[118,118,1.0,0.16518,0.13415,0.10714,0.14286,0.2857,0.0,0.4286,8,0,0,8,0,15,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.04018,"x":0.35268,"p":[[0,212,0.0,0.21428,0.11294,0.14286,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,4,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,212,0.0189,0.28124,0.11561,0.25,0.28571,0.28571,0.0,0.571,1,0,0,1,0,7,0,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,212,0.0377,0.31696,0.14167,0.28571,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,1,0,0,16,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[12,212,0.0566,0.3125,0.12079,0.28571,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,3,0,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[16,212,0.0755,0.31697,0.11701,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,5,0,0,12,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[20,212,0.0943,0.30357,0.12242,0.25,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,7,0,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[24,212,0.1132,0.33482,0.1411,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,5,0,0,6,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[28,212,0.1321,0.30357,0.13243,0.2857,0.28571,0.42857,0.0,0.42857,3,0,0,3,0,3,0,0,13,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[32,212,0.1509,0.28571,0.12877,0.2857,0.28571,0.42857,0.0,0.42857,3,0,0,3,0,4,0,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[36,212,0.1698,0.30357,0.15047,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,4,0,0,13,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[40,212,0.1887,0.35268,0.11285,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,8,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[44,212,0.2075,0.28572,0.15153,0.2857,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,2,0,0,14,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[48,212,0.2264,0.29911,0.10326,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[52,212,0.2453,0.26339,0.12931,0.24999,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,4,0,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,212,0.2642,0.26786,0.12242,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,2,0,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[60,212,0.283,0.3125,0.20958,0.25,0.28571,0.42857,0.0,1.0,4,1,0,4,0,4,0,0,13,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[64,212,0.3019,0.26786,0.14174,0.14289,0.28571,0.32143,0.0,0.57143,4,0,0,4,0,5,0,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[68,212,0.3208,0.25893,0.12595,0.24999,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,4,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[72,212,0.3396,0.25446,0.11143,0.24999,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,5,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,212,0.3585,0.28571,0.11294,0.24999,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,7,0,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[80,212,0.3774,0.26339,0.15198,0.14286,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,3,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[84,212,0.3962,0.27232,0.13533,0.24999,0.28571,0.42857,0.0,0.42857,4,0,0,4,0,4,0,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[88,212,0.4151,0.28125,0.13115,0.2857,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,2,0,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[92,212,0.434,0.29911,0.13054,0.2857,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,3,0,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[96,212,0.4528,0.29464,0.11259,0.2857,0.28571,0.42857,0.0,0.42857,2,0,0,2,0,3,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[100,212,0.4717,0.29018,0.12103,0.24999,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,7,0,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[104,212,0.4906,0.27232,0.13533,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,3,0,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[108,212,0.5094,0.25,0.14725,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,7,0,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[112,212,0.5283,0.33482,0.11633,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,13,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[116,212,0.5472,0.24553,0.15663,0.14286,0.2857,0.42857,0.0,0.42857,7,0,0,7,0,4,0,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[120,212,0.566,0.22768,0.15093,0.14286,0.28571,0.28571,0.0,0.4286,7,0,0,7,0,6,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[124,212,0.5849,0.30802,0.1477,0.2857,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,12,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[128,212,0.6038,0.27232,0.13534,0.2857,0.28571,0.32143,0.0,0.4286,5,0,0,5,0,1,0,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[132,212,0.6226,0.29915,0.12042,0.28571,0.28571,0.42857,0.0,0.43,2,0,0,2,0,4,0,0,15,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[136,212,0.6415,0.27678,0.13333,0.2857,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,3,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[140,212,0.6604,0.32143,0.11294,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,13,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[144,212,0.6792,0.20982,0.15561,0.10714,0.21428,0.28571,0.0,0.4286,8,0,0,8,0,8,0,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[148,212,0.6981,0.24107,0.15746,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,5,0,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[152,212,0.717,0.19644,0.15047,0.0,0.21428,0.28571,0.0,0.4286,9,0,0,9,0,7,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[156,212,0.7358,0.19197,0.14555,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,13,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[160,212,0.7547,0.21875,0.15146,0.14286,0.1429,0.32143,0.0,0.42857,6,0,0,6,0,11,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[164,212,0.7736,0.17411,0.16263,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,10,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[168,212,0.7925,0.19196,0.16602,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,6,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[172,212,0.8113,0.17411,0.16262,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,7,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[176,212,0.8302,0.14286,0.16751,0.0,0.07143,0.2857,0.0,0.42857,16,0,0,16,0,6,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[180,212,0.8491,0.14733,0.17308,0.0,0.07143,0.28571,0.0,0.4286,16,0,0,16,0,6,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[184,212,0.8679,0.14286,0.16367,0.0,0.07143,0.28571,0.0,0.4286,16,0,0,16,0,5,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[188,212,0.8868,0.13393,0.14258,0.0,0.14286,0.1429,0.0,0.4286,13,0,0,13,0,12,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[192,212,0.9057,0.13393,0.15947,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,4,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[196,212,0.9245,0.11161,0.1461,0.0,0.0,0.17857,0.0,0.42857,18,0,0,18,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[200,212,0.9434,0.125,0.15047,0.0,0.14286,0.14287,0.0,0.57143,15,0,0,15,0,10,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[204,212,0.9623,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[208,212,0.9811,0.125,0.15465,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,9,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[212,212,1.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"937964e4ded8c402","q":"Determine all positive integers  $n$  such that it is possible to fill the  $n \\times n$  table with numbers  $1, 2$  and  $-3$  so that the sum of the numbers in each row and each column is  $0$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.75444,"x":0.99107,"p":[[0,55,0.0,0.75444,0.16068,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,16,0,0,5,0,6],[4,55,0.0727,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,55,0.1455,0.89731,0.17943,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,23],[12,55,0.2182,0.91517,0.16701,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,24],[16,55,0.2909,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[20,55,0.3636,0.92856,0.15153,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[24,55,0.4364,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[28,55,0.5091,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[32,55,0.5818,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,55,0.6545,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[40,55,0.7273,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,55,0.8,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[48,55,0.8727,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[52,55,0.9455,0.92411,0.18553,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,26],[55,55,1.0,0.875,0.17035,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,1,0,19]]},{"b":6,"e":0.71429,"k":"flat","v":0.71429,"x":0.99107,"p":[[0,86,0.0,0.71429,0.13832,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,22,0,0,4,0,2],[4,86,0.0465,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,86,0.093,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[12,86,0.1395,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[16,86,0.186,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[20,86,0.2326,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[24,86,0.2791,0.92411,0.13825,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,23],[28,86,0.3256,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[32,86,0.3721,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,86,0.4186,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[40,86,0.4651,0.97991,0.0697,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,1,29],[44,86,0.5116,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[48,86,0.5581,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[52,86,0.6047,0.90179,0.16146,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,3,0,21],[56,86,0.6512,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[60,86,0.6977,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[64,86,0.7442,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,86,0.7907,0.9375,0.16728,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[72,86,0.8372,0.95536,0.14032,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[76,86,0.8837,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[80,86,0.9302,0.9375,0.14698,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,25],[84,86,0.9767,0.89732,0.12492,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,18],[86,86,1.0,0.84375,0.18336,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,9,0,13]]}]},{"i":"18b67dba2e52a9b5","q":"Determine all the complex numbers  $w = a + bi$  with  $a, b \\in \\mathbb{R}$ , such that there exists a polinomial  $p(z)$  whose coefficients are real and positive such that  $p(w) = 0.$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.4017,"x":0.71428,"p":[[0,45,0.0,0.49104,0.2394,0.28571,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,12,0,0,5,0,0,8,0,0,2,0,0,0,0,4],[4,45,0.0889,0.70088,0.18337,0.57143,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,14,0,0,3,0,5],[8,45,0.1778,0.71428,0.17857,0.57143,0.71429,0.74996,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,14,0,0,2,0,6],[12,45,0.2667,0.63393,0.23941,0.57143,0.57143,0.71429,0.1429,1.0,0,6,0,0,0,1,0,0,5,0,0,1,0,0,10,0,0,8,0,0,1,0,6],[16,45,0.3556,0.59371,0.1992,0.571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,3,0,0,16,0,0,4,0,0,2,0,3],[20,45,0.4444,0.60713,0.25254,0.57132,0.71429,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,0,0,0,8,0,0,12,0,0,1,0,4],[24,45,0.5333,0.50438,0.25513,0.28571,0.57141,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,8,0,0,3,0,0,5,0,0,9,0,0,1,0,2],[28,45,0.6222,0.55801,0.21829,0.39286,0.57143,0.60714,0.2857,1.0,0,4,0,0,0,0,0,0,8,0,0,3,0,0,13,0,0,4,0,0,0,0,4],[32,45,0.7111,0.4017,0.25871,0.14286,0.5,0.57143,0.0,0.85714,4,0,0,4,0,7,0,0,3,0,0,2,0,0,10,0,0,5,0,0,1,0,0],[36,45,0.8,0.45979,0.20432,0.28571,0.49979,0.60714,0.0,0.71429,1,0,0,1,0,2,0,0,10,0,0,3,0,0,8,0,0,8,0,0,0,0,0],[40,45,0.8889,0.44195,0.22688,0.2857,0.42859,0.71429,0.0,0.71429,1,0,0,1,0,6,0,0,6,0,0,4,0,0,6,0,0,9,0,0,0,0,0],[44,45,0.9778,0.46871,0.20894,0.28571,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,5,0,0,7,0,0,3,0,0,8,0,0,9,0,0,0,0,0],[45,45,1.0,0.4374,0.23413,0.25001,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,7,0,0,5,0,0,5,0,0,4,0,0,10,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.53124,"x":0.74552,"p":[[0,62,0.0,0.53124,0.21793,0.28571,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,10,0,0,4,0,0,9,0,0,6,0,0,0,0,3],[4,62,0.0645,0.74552,0.2012,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,16,0,0,1,0,9],[8,62,0.129,0.69643,0.21053,0.57143,0.71429,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,12,0,0,1,0,7],[12,62,0.1935,0.62945,0.2621,0.53539,0.71429,0.71429,0.0,1.0,1,6,1,1,0,2,0,0,2,0,0,3,0,0,7,0,0,10,0,0,1,0,6],[16,62,0.2581,0.57141,0.25254,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,4,0,0,7,0,0,7,0,0,7,0,0,0,0,5],[20,62,0.3226,0.66516,0.19759,0.57143,0.71429,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,1,0,0,10,0,0,14,0,0,1,0,4],[24,62,0.3871,0.63838,0.21424,0.5354,0.64286,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,10,0,0,1,0,5],[28,62,0.4516,0.57589,0.20038,0.5354,0.57143,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,3,0,0,16,0,0,4,0,0,1,0,3],[32,62,0.5161,0.61159,0.20589,0.57143,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,2,0,0,11,0,0,12,0,0,0,0,3],[36,62,0.5806,0.64732,0.1636,0.57143,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,14,0,0,2,0,2],[40,62,0.6452,0.57141,0.19885,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,2,0,0,10,0,0,10,0,0,2,0,1],[44,62,0.7097,0.53125,0.16064,0.42857,0.57143,0.71429,0.2857,0.857,0,0,0,0,0,0,0,0,6,0,0,7,0,0,10,0,0,8,0,0,1,0,0],[48,62,0.7742,0.65183,0.20801,0.57143,0.57143,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,3,0,0,12,0,0,6,0,0,3,0,5],[52,62,0.8387,0.63392,0.21706,0.5354,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,5,0,0,9,0,0,9,0,0,1,0,5],[56,62,0.9032,0.55354,0.21943,0.42857,0.5714,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,7,0,0,8,0,0,8,0,0,0,0,3],[60,62,0.9677,0.61607,0.19045,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,13,0,0,3,0,1],[62,62,1.0,0.5982,0.23266,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,7,0,0,5,0,0,7,0,0,3,0,4]]}]},{"i":"ef3f999b057527b3","q":"Determine all infinite sequences  $a_1, a_2, \\ldots$  of positive integers, such that\n\\[2024\\left(a_{n+1}^3 + 1\\right) = (2024a_na_{n+1} + 1)(a_{n+2} + 2023)\\]\nfor any positive integer  $n$ .","t":[{"b":4,"e":0.14286,"k":"falling","v":0.14723,"x":0.69195,"p":[[0,129,0.0,0.49999,0.2945,0.28571,0.42857,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,5,0,0,7,0,0,3,0,0,5,0,0,2,0,4],[4,129,0.031,0.64732,0.18893,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,16,0,0,1,0,3],[8,129,0.062,0.625,0.21651,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,2,0,0,4,0,0,20,0,0,0,0,2],[12,129,0.093,0.69195,0.22048,0.67857,0.71429,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,0,0,0,5,0,0,17,0,0,2,0,5],[16,129,0.124,0.54464,0.27765,0.28571,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,6,0,0,4,0,0,0,0,0,3,0,0,12,0,0,6,0,0],[20,129,0.155,0.62054,0.24383,0.5,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,5,0,0,0,0,0,1,0,0,19,0,0,1,0,3],[24,129,0.186,0.63393,0.28107,0.53571,0.71429,0.75,0.14286,1.0,0,6,0,0,0,6,0,0,0,0,0,2,0,0,4,0,0,12,0,0,2,0,6],[28,129,0.2171,0.65179,0.23941,0.71429,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,20,0,0,2,0,3],[32,129,0.2481,0.54017,0.26901,0.25,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,2,0,0,1,0,0,4,0,0,13,0,0,3,0,1],[36,129,0.2791,0.62487,0.25934,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,1,0,0,1,0,0,3,0,0,15,0,0,5,0,2],[40,129,0.3101,0.51784,0.30252,0.14286,0.64286,0.71429,0.0,1.0,1,2,0,1,0,10,0,0,0,0,0,1,0,0,4,0,0,11,0,0,3,0,2],[44,129,0.3411,0.58034,0.26949,0.28571,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,3,0,0,1,0,0,3,0,0,13,0,0,4,0,2],[48,129,0.3721,0.56697,0.28679,0.25001,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,8,0,0,1,0,0,3,0,0,1,0,0,12,0,0,5,0,2],[52,129,0.4031,0.49545,0.29022,0.14286,0.71429,0.71429,0.0,1.0,1,1,0,1,0,9,0,0,3,0,0,1,0,0,1,0,0,14,0,0,2,0,1],[56,129,0.4341,0.53125,0.26782,0.14286,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,8,0,0,0,0,0,1,0,0,5,0,0,14,0,0,3,0,0],[60,129,0.4651,0.53125,0.29931,0.14286,0.71429,0.71429,0.0,1.0,1,1,0,1,0,9,0,0,1,0,0,1,0,0,2,0,0,12,0,0,5,0,1],[64,129,0.4961,0.50896,0.30079,0.14286,0.57143,0.71429,0.0,1.0,1,3,0,1,0,9,0,0,2,0,0,1,0,0,4,0,0,11,0,0,1,0,3],[68,129,0.5271,0.51784,0.27374,0.14286,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,10,0,0,0,0,0,3,0,0,2,0,0,13,0,0,4,0,0],[72,129,0.5581,0.42411,0.31029,0.14286,0.21428,0.71429,0.14286,1.0,0,1,0,0,0,16,0,0,1,0,0,2,0,0,1,0,0,6,0,0,5,0,1],[76,129,0.5891,0.51337,0.27166,0.14286,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,10,0,0,1,0,0,0,0,0,6,0,0,11,0,0,4,0,0],[80,129,0.6202,0.41964,0.29867,0.14286,0.28571,0.71429,0.0,1.0,1,2,0,1,0,13,0,0,3,0,0,1,0,0,3,0,0,8,0,0,1,0,2],[84,129,0.6512,0.49098,0.28791,0.14286,0.57143,0.71429,0.14,1.0,0,1,0,0,0,11,0,0,2,0,0,1,0,0,3,0,0,11,0,0,3,0,1],[88,129,0.6822,0.43304,0.30615,0.14286,0.35714,0.71429,0.0,1.0,1,2,0,1,0,13,0,0,2,0,0,2,0,0,2,0,0,8,0,0,2,0,2],[92,129,0.7132,0.3125,0.24598,0.14286,0.14286,0.46429,0.14286,0.85714,0,0,0,0,0,20,0,0,2,0,0,2,0,0,1,0,0,6,0,0,1,0,0],[96,129,0.7442,0.37045,0.28547,0.14286,0.14286,0.71429,0.0,1.0,1,1,0,1,0,17,0,0,0,0,0,2,0,0,3,0,0,7,0,0,1,0,1],[100,129,0.7752,0.38384,0.29118,0.14286,0.14286,0.71429,0.14,1.0,0,1,0,0,0,18,0,0,1,0,0,0,0,0,2,0,0,9,0,0,1,0,1],[104,129,0.8062,0.37045,0.2646,0.14286,0.2143,0.71429,0.14,0.85714,0,0,0,0,0,16,0,0,4,0,0,0,0,0,2,0,0,9,0,0,1,0,0],[108,129,0.8372,0.4375,0.2878,0.14286,0.57143,0.71429,0.0,1.0,1,1,0,1,0,13,0,0,1,0,0,0,0,0,4,0,0,12,0,0,0,0,1],[112,129,0.8682,0.34375,0.30064,0.14286,0.14286,0.46429,0.0,1.0,1,3,0,1,0,18,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,3],[116,129,0.8992,0.38384,0.29553,0.14286,0.2143,0.71429,0.14,1.0,0,2,0,0,0,16,0,0,4,0,0,1,0,0,2,0,0,5,0,0,2,0,2],[120,129,0.9302,0.30357,0.25692,0.14286,0.14286,0.32143,0.0,1.0,1,1,0,1,0,18,0,0,5,0,0,1,0,0,1,0,0,4,0,0,1,0,1],[124,129,0.9612,0.19179,0.14561,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[128,129,0.9922,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[129,129,1.0,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.16063,"x":0.68304,"p":[[0,135,0.0,0.41955,0.28116,0.14286,0.42857,0.71429,0.0,0.85714,4,0,1,4,0,5,0,0,6,0,0,5,0,0,2,0,0,6,0,0,4,0,0],[4,135,0.0296,0.68304,0.17762,0.57143,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,15,0,0,4,0,3],[8,135,0.0593,0.56696,0.24868,0.42857,0.71429,0.71429,0.0,1.0,1,1,0,1,0,5,0,0,1,0,0,2,0,0,4,0,0,17,0,0,1,0,1],[12,135,0.0889,0.5625,0.26711,0.39286,0.71429,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,3,0,0,3,0,0,14,0,0,2,0,2],[16,135,0.1185,0.56696,0.24868,0.39286,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,2,0,0,1,0,0,6,0,0,13,0,0,3,0,1],[20,135,0.1481,0.54897,0.25773,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,3,0,0,4,0,0,11,0,0,1,0,3],[24,135,0.1778,0.49107,0.31529,0.14286,0.5,0.71429,0.14286,1.0,0,3,0,0,0,12,0,0,2,0,0,2,0,0,1,0,0,9,0,0,3,0,3],[28,135,0.2074,0.4375,0.32721,0.14286,0.5,0.71429,0.0,0.85714,5,0,0,5,0,9,0,0,1,0,0,1,0,0,1,0,0,10,0,0,5,0,0],[32,135,0.237,0.49545,0.30521,0.14286,0.57143,0.71429,0.0,1.0,3,1,0,3,0,7,0,0,2,0,0,1,0,0,4,0,0,10,0,0,4,0,1],[36,135,0.2667,0.40625,0.27919,0.14286,0.28571,0.71429,0.0,1.0,1,1,0,1,0,12,0,0,4,0,0,3,0,0,1,0,0,9,0,0,1,0,1],[40,135,0.2963,0.30356,0.23621,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,20,0,0,2,0,0,3,0,0,1,0,0,5,0,0,1,0,0],[44,135,0.3259,0.47768,0.27107,0.14286,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,11,0,0,2,0,0,1,0,0,3,0,0,13,0,0,2,0,0],[48,135,0.3556,0.48661,0.27862,0.14286,0.64286,0.71429,0.14286,0.85714,0,0,0,0,0,11,0,0,2,0,0,1,0,0,2,0,0,13,0,0,3,0,0],[52,135,0.3852,0.47312,0.27545,0.14286,0.57143,0.71429,0.14,0.85714,0,0,0,0,0,12,0,0,0,0,0,3,0,0,3,0,0,11,0,0,3,0,0],[56,135,0.4148,0.42402,0.25133,0.14286,0.42857,0.60714,0.14,1.0,0,1,0,0,0,11,0,0,4,0,0,2,0,0,7,0,0,7,0,0,0,0,1],[60,135,0.4444,0.37929,0.29381,0.14286,0.14286,0.71429,0.0,1.0,1,1,0,1,0,17,0,0,0,0,0,2,0,0,1,0,0,9,0,0,1,0,1],[64,135,0.4741,0.41963,0.27649,0.14286,0.28571,0.71429,0.14286,0.85714,0,0,0,0,0,14,0,0,3,0,0,0,0,0,3,0,0,10,0,0,2,0,0],[68,135,0.5037,0.44643,0.28065,0.14286,0.57143,0.71429,0.0,1.0,1,1,0,1,0,11,0,0,3,0,0,0,0,0,4,0,0,12,0,0,0,0,1],[72,135,0.5333,0.37054,0.23652,0.14286,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,14,0,0,2,0,0,7,0,0,2,0,0,6,0,0,1,0,0],[76,135,0.563,0.51338,0.24185,0.28571,0.64286,0.71429,0.14286,0.85714,0,0,0,0,0,7,0,0,3,0,0,3,0,0,3,0,0,15,0,0,1,0,0],[80,135,0.5926,0.37043,0.22556,0.14286,0.28571,0.57143,0.14,0.71429,0,0,0,0,0,12,0,0,7,0,0,1,0,0,6,0,0,6,0,0,0,0,0],[84,135,0.6222,0.36152,0.26248,0.14286,0.21428,0.71429,0.14,0.85714,0,0,0,0,0,16,0,0,5,0,0,0,0,0,1,0,0,9,0,0,1,0,0],[88,135,0.6519,0.47321,0.27534,0.14286,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,10,0,0,3,0,0,3,0,0,3,0,0,11,0,0,0,0,2],[92,135,0.6815,0.41509,0.24846,0.14286,0.42857,0.71429,0.14,0.85714,0,0,0,0,0,12,0,0,3,0,0,3,0,0,5,0,0,8,0,0,1,0,0],[96,135,0.7111,0.33929,0.2714,0.14286,0.14286,0.60714,0.0,0.85714,1,0,0,1,0,17,0,0,3,0,0,2,0,0,1,0,0,5,0,0,3,0,0],[100,135,0.7407,0.32133,0.23963,0.14286,0.14286,0.57111,0.14,0.85714,0,0,0,0,0,18,0,0,4,0,0,1,0,0,3,0,0,5,0,0,1,0,0],[104,135,0.7704,0.39732,0.23887,0.14286,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,13,0,0,2,0,0,4,0,0,5,0,0,8,0,0,0,0,0],[108,135,0.8,0.37045,0.23116,0.14286,0.28571,0.57143,0.14,0.85714,0,0,0,0,0,12,0,0,6,0,0,5,0,0,2,0,0,6,0,0,1,0,0],[112,135,0.8296,0.50893,0.29001,0.14286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,3,0,0,4,0,0,0,0,0,11,0,0,3,0,2],[116,135,0.8593,0.35266,0.25995,0.14286,0.14286,0.60714,0.14286,0.85714,0,0,0,0,0,18,0,0,2,0,0,0,0,0,4,0,0,7,0,0,1,0,0],[120,135,0.8889,0.35268,0.27196,0.14286,0.21429,0.60714,0.0,1.0,1,1,0,1,0,15,0,0,5,0,0,1,0,0,2,0,0,6,0,0,1,0,1],[124,135,0.9185,0.25893,0.18707,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,20,0,0,6,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[128,135,0.9481,0.16063,0.06918,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,135,0.9778,0.16509,0.05191,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[135,135,1.0,0.19197,0.07668,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f910c81eb3f8dc92","q":"Determine all positive real  $M$  such that for any positive reals  $a,b,c$ , at least one of  $a + \\dfrac{M}{ab}, b + \\dfrac{M}{bc}, c + \\dfrac{M}{ca}$  is greater than or equal to  $1+M$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.58929,"x":0.9107,"p":[[0,135,0.0,0.67856,0.19886,0.57143,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,1,0,0,1,0,8],[4,135,0.0296,0.85714,0.20203,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,4,0,19],[8,135,0.0593,0.80804,0.21902,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,0,0,0,2,0,17],[12,135,0.0889,0.84821,0.17835,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,16],[16,135,0.1185,0.63837,0.2172,0.53539,0.57143,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,11,0,0,6,0,0,1,0,6],[20,135,0.1481,0.59821,0.24856,0.42857,0.57143,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,6,0,0,2,0,0,14,0,0,1,0,0,2,0,6],[24,135,0.1778,0.63393,0.2765,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,9,0,0,7,0,0,3,0,0,1,0,9],[28,135,0.2074,0.69643,0.21651,0.57143,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,2,0,0,4,0,8],[32,135,0.237,0.60714,0.20516,0.53571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,5,0,0,14,0,0,3,0,0,4,0,3],[36,135,0.2667,0.58929,0.25191,0.42857,0.57143,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,4,0,0,6,0,0,9,0,0,3,0,0,3,0,5],[40,135,0.2963,0.69643,0.21651,0.57143,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,13,0,0,4,0,0,3,0,8],[44,135,0.3259,0.67411,0.24283,0.57143,0.57143,1.0,0.143,1.0,0,9,0,0,0,1,0,0,1,0,0,5,0,0,12,0,0,2,0,0,2,0,9],[48,135,0.3556,0.77232,0.18851,0.57143,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,4,0,11],[52,135,0.3852,0.79018,0.20198,0.57143,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,3,0,14],[56,135,0.4148,0.77231,0.24708,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,2,0,0,2,0,15],[60,135,0.4444,0.67857,0.21429,0.57143,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,2,0,0,16,0,0,1,0,0,4,0,7],[64,135,0.4741,0.79911,0.20159,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,0,4,0,14],[68,135,0.5037,0.7857,0.19234,0.57143,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,2,0,13],[72,135,0.5333,0.77679,0.20806,0.57143,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,3,0,13],[76,135,0.563,0.77232,0.21086,0.57143,0.78571,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,5,0,0,5,0,11],[80,135,0.5926,0.79017,0.20513,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,2,0,0,3,0,14],[84,135,0.6222,0.66517,0.17718,0.57143,0.57143,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,3,0,0,3,0,5],[88,135,0.6519,0.73214,0.21943,0.57143,0.57143,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,1,0,0,4,0,10],[92,135,0.6815,0.69642,0.20125,0.57143,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,4,0,0,3,0,7],[96,135,0.7111,0.76338,0.20706,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,5,0,0,4,0,11],[100,135,0.7407,0.77679,0.21706,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,0,0,0,3,0,14],[104,135,0.7704,0.73661,0.18935,0.57143,0.64286,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,0,6,0,8],[108,135,0.8,0.73661,0.20858,0.57143,0.57143,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,2,0,0,0,0,12],[112,135,0.8296,0.77232,0.20473,0.57143,0.78571,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,4,0,12],[116,135,0.8593,0.84375,0.20628,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,4,0,18],[120,135,0.8889,0.86161,0.17307,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,5,0,17],[124,135,0.9185,0.83482,0.19269,0.57143,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,3,0,17],[128,135,0.9481,0.9107,0.16271,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,23],[132,135,0.9778,0.86161,0.17672,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,7,0,16],[135,135,1.0,0.84375,0.16115,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,10,0,12]]},{"b":6,"e":0.85714,"k":"flat","v":0.60714,"x":0.91518,"p":[[0,121,0.0,0.60714,0.18211,0.57143,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,4,0,0,21,0,0,1,0,0,1,0,4],[4,121,0.0331,0.79911,0.20473,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,2,0,0,6,0,13],[8,121,0.0661,0.84375,0.18681,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,3,0,17],[12,121,0.0992,0.89732,0.14827,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,19],[16,121,0.1322,0.91518,0.14664,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,21],[20,121,0.1653,0.82143,0.22016,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,9,0,13],[24,121,0.1983,0.81696,0.1931,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,4,0,0,7,0,13],[28,121,0.2314,0.875,0.16656,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,6,0,18],[32,121,0.2645,0.86161,0.19061,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,3,0,19],[36,121,0.2975,0.88393,0.19045,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,4,0,21],[40,121,0.3306,0.79908,0.20475,0.57143,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,5,0,0,3,0,14],[44,121,0.3636,0.87054,0.19351,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,19],[48,121,0.3967,0.73661,0.29474,0.57143,0.78571,1.0,0.0,1.0,2,14,0,2,0,0,0,0,2,0,0,1,0,0,7,0,0,4,0,0,2,0,14],[52,121,0.4298,0.79909,0.23654,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,3,0,0,3,0,16],[56,121,0.4628,0.79018,0.23415,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,2,0,0,2,0,16],[60,121,0.4959,0.79017,0.22585,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,4,0,0,4,0,14],[64,121,0.5289,0.76786,0.25692,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,3,0,0,2,0,15],[68,121,0.562,0.76786,0.24419,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,8,0,0,4,0,0,4,0,13],[72,121,0.595,0.69643,0.27837,0.53571,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,4,0,0,7,0,0,3,0,0,3,0,11],[76,121,0.6281,0.73659,0.25282,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,5,0,0,3,0,12],[80,121,0.6612,0.80804,0.20394,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,5,0,14],[84,121,0.6942,0.67856,0.24744,0.57143,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,4,0,0,12,0,0,3,0,0,2,0,9],[88,121,0.7273,0.70982,0.24868,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,4,0,0,2,0,0,9,0,0,3,0,0,4,0,10],[92,121,0.7603,0.74107,0.24338,0.53571,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,3,0,0,4,0,12],[96,121,0.7934,0.76786,0.20124,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,3,0,0,7,0,10],[100,121,0.8264,0.77677,0.24729,0.67857,0.78571,1.0,0.0,1.0,1,13,0,1,0,1,0,0,0,0,0,0,0,0,6,0,0,8,0,0,3,0,13],[104,121,0.8595,0.74106,0.23267,0.57143,0.64286,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,2,0,0,1,0,13],[108,121,0.8926,0.76786,0.20124,0.57143,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,5,0,11],[112,121,0.9256,0.78124,0.19881,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,0,3,0,12],[116,121,0.9587,0.8214,0.18902,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,5,0,14],[120,121,0.9917,0.72321,0.2111,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,5,0,0,5,0,8],[121,121,1.0,0.63393,0.18189,0.53572,0.64286,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,10,0,0,4,0,2]]}]},{"i":"bcaf03b377a26d06","q":"Find the smallest constant M, so that for any real numbers  $a_1, a_2, \\dots a_{2023} \\in [4, 6]$  and  $b_1, b_2, \\dots b_{2023} \\in [9, 12] $  following inequality holds:  $$  \\sqrt{a_1^2 + a_2^2 + \\dots + a_{2023}^2} \\cdot  \\sqrt{b_1^2 + b_2^2 + \\dots + b_{2023}^2} \\leq M \\cdot \\left ( a_1 b_1 + a_2 b_2 + \\dots  + a_{2023} b_{2023} \\right)  $$  *Proposed by Zaza Meliqidze, Georgia*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.38393,"x":0.58929,"p":[[0,84,0.0,0.38393,0.18363,0.28571,0.42857,0.42857,0.0,0.71429,4,0,0,4,0,0,0,0,6,0,0,17,0,0,2,0,0,3,0,0,0,0,0],[4,84,0.0476,0.46875,0.14827,0.42857,0.5,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,1,0,0,12,0,0,15,0,0,1,0,0,0,0,0],[8,84,0.0952,0.52229,0.11632,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,9,0,0,18,0,0,3,0,0,0,0,0],[12,84,0.1429,0.51337,0.15509,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,3,0,0,8,0,0,16,0,0,3,0,0,1,0,0],[16,84,0.1905,0.54463,0.13092,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,9,0,0,16,0,0,3,0,0,2,0,0],[20,84,0.2381,0.53125,0.10853,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,11,0,0,17,0,0,2,0,0,1,0,0],[24,84,0.2857,0.52232,0.1411,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,7,0,0,20,0,0,3,0,0,0,0,0],[28,84,0.3333,0.49554,0.15561,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,10,0,0,14,0,0,4,0,0,0,0,0],[32,84,0.381,0.58929,0.10564,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,0,8,0,0,1,0,0],[36,84,0.4286,0.54017,0.09268,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,20,0,0,1,0,0,1,0,0],[40,84,0.4762,0.54908,0.08072,0.5354,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,21,0,0,3,0,0,0,0,0],[44,84,0.5238,0.4866,0.10629,0.42857,0.4998,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,12,0,0,15,0,0,1,0,0,0,0,0],[48,84,0.5714,0.52231,0.1107,0.42857,0.57121,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,14,0,0,12,0,0,5,0,0,0,0,0],[52,84,0.619,0.53125,0.13475,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,7,0,0,16,0,0,4,0,0,1,0,0],[56,84,0.6667,0.52231,0.14555,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,11,0,0,12,0,0,5,0,0,1,0,0],[60,84,0.7143,0.54464,0.11538,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,2,0,0,0,0,1],[64,84,0.7619,0.53125,0.11426,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,7,0,0,18,0,0,4,0,0,0,0,0],[68,84,0.8095,0.46429,0.15568,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,14,0,0,8,0,0,1,0,0,2,0,0],[72,84,0.8571,0.4732,0.09739,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,17,0,0,11,0,0,1,0,0,0,0,0],[76,84,0.9048,0.54464,0.0974,0.42859,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,8,0,0,19,0,0,4,0,0,0,0,0],[80,84,0.9524,0.49554,0.10092,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,12,0,0,16,0,0,1,0,0,0,0,0],[84,84,1.0,0.49997,0.11844,0.42857,0.49979,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,13,0,0,14,0,0,1,0,0,1,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.42411,"x":0.58034,"p":[[0,104,0.0,0.42411,0.15355,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,10,0,0,10,0,0,7,0,0,3,0,0,0,0,0],[4,104,0.0385,0.45536,0.20341,0.39286,0.42857,0.57143,0.0,1.0,2,1,0,2,0,0,0,0,6,0,0,14,0,0,5,0,0,3,0,0,1,0,1],[8,104,0.0769,0.51786,0.13243,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,14,0,0,2,0,0,0,0,1],[12,104,0.1154,0.54464,0.15746,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,10,0,0,14,0,0,4,0,0,1,0,1],[16,104,0.1538,0.52232,0.14987,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,9,0,0,13,0,0,5,0,0,1,0,0],[20,104,0.1923,0.53125,0.15663,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,10,0,0,15,0,0,4,0,0,0,0,1],[24,104,0.2308,0.5,0.15152,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,8,0,0,16,0,0,2,0,0,1,0,0],[28,104,0.2692,0.54018,0.1461,0.42859,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,7,0,0,18,0,0,4,0,0,1,0,0],[32,104,0.3077,0.48213,0.12752,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,11,0,0,12,0,0,3,0,0,0,0,0],[36,104,0.3462,0.52231,0.11632,0.42857,0.57121,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,13,0,0,3,0,0,1,0,0],[40,104,0.3846,0.56696,0.12619,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,18,0,0,3,0,0,1,0,1],[44,104,0.4231,0.50893,0.10062,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,12,0,0,16,0,0,2,0,0,0,0,0],[48,104,0.4615,0.58034,0.15126,0.53539,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,18,0,0,3,0,0,1,0,2],[52,104,0.5,0.54911,0.15612,0.42857,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,10,0,0,12,0,0,7,0,0,0,0,1],[56,104,0.5385,0.54016,0.11142,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,17,0,0,1,0,0,2,0,0],[60,104,0.5769,0.51786,0.13243,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,9,0,0,15,0,0,3,0,0,1,0,0],[64,104,0.6154,0.49105,0.13802,0.42857,0.571,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,12,0,0,16,0,0,0,0,0,1,0,0],[68,104,0.6538,0.53571,0.14286,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,12,0,0,14,0,0,4,0,0,0,0,1],[72,104,0.6923,0.46874,0.11424,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,15,0,0,12,0,0,1,0,0,0,0,0],[76,104,0.7308,0.57589,0.145,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,16,0,0,4,0,0,2,0,1],[80,104,0.7692,0.52232,0.12169,0.42857,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,13,0,0,16,0,0,1,0,0,0,0,1],[84,104,0.8077,0.54462,0.12595,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,9,0,0,15,0,0,5,0,0,1,0,0],[88,104,0.8462,0.56697,0.13114,0.4286,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,16,0,0,4,0,0,1,0,1],[92,104,0.8846,0.46875,0.13474,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,13,0,0,12,0,0,2,0,0,0,0,0],[96,104,0.9231,0.53124,0.14826,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,8,0,0,15,0,0,4,0,0,0,0,1],[100,104,0.9615,0.52229,0.09851,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,12,0,0,16,0,0,3,0,0,0,0,0],[104,104,1.0,0.51339,0.1063,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,9,0,0,18,0,0,2,0,0,0,0,0]]}]},{"i":"369472b2e35ecb69","q":"Fix integers  $n\\ge k\\ge 2$ . We call a collection of integral valued coins  $n-diverse$  if no value occurs in it more than  $n$  times. Given such a collection, a number  $S$  is  $n-reachable$  if that collection contains  $n$  coins whose sum of values equals  $S$ . Find the least positive integer  $D$  such that for any  $n$ -diverse collection of  $D$  coins there are at least  $k$  numbers that are  $n$ -reachable.\n\n*Proposed by Alexandar Ivanov, Bulgaria.*","t":[{"b":1,"e":1.0,"k":"rising","v":0.35714,"x":0.99554,"p":[[0,62,0.0,0.35714,0.33503,0.14286,0.14286,0.28571,0.14286,1.0,0,6,0,0,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[4,62,0.0645,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,62,0.129,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,62,0.1935,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[16,62,0.2581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,62,0.3226,0.94196,0.17807,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[24,62,0.3871,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,62,0.4516,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,62,0.5161,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,62,0.5806,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,62,0.6452,0.97322,0.14913,1.0,1.0,1.0,0.1429,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,62,0.7097,0.95089,0.17354,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[48,62,0.7742,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[52,62,0.8387,0.82143,0.32341,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,24],[56,62,0.9032,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[60,62,0.9677,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[62,62,1.0,0.94643,0.15047,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27]]},{"b":7,"e":1.0,"k":"rising","v":0.39723,"x":0.99107,"p":[[0,32,0.0,0.39723,0.39733,0.14286,0.14286,0.89286,0.0,1.0,5,8,0,5,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,8],[4,32,0.125,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,32,0.25,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[12,32,0.375,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,32,0.5,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,32,0.625,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,32,0.75,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,32,0.875,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,32,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"2e6abe7af05fc932","q":"For a given positive integer  $n(\\ge 2)$ , find maximum positive integer  $A$  such that there exists  $P \\in \\mathbb{Z}[x]$  with degree  $n$  that satisfies the following two conditions.\n\n\n- For any  $1 \\le k \\le A$ , it satisfies that  $A \\mid P(k)$ , and\n- $P(0)= 0$  and the coefficient of the first term of  $P$  is  $1$ , which means that  $P(x)$  is in the following form where  $c_2, c_3, \\cdots, c_n$  are all integers and  $c_n \\neq 0$ .  $$ P(x) = c_nx^n + c_{n-1}x^{n-1}+\\dots+c_2x^2+x $$","t":[{"b":4,"e":0.14286,"k":"falling","v":0.05357,"x":0.64732,"p":[[0,19,0.0,0.64732,0.3273,0.39286,0.71429,1.0,0.0,1.0,3,10,1,3,0,0,0,0,5,0,0,2,0,0,4,0,0,5,0,0,3,0,10],[4,19,0.2105,0.24998,0.35353,0.0,0.0,0.60682,0.0,1.0,19,1,0,19,0,2,0,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,1],[8,19,0.4211,0.0625,0.19541,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[12,19,0.6316,0.06696,0.19226,0.0,0.0,0.0,0.0,0.857,26,0,0,26,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,19,0.8421,0.05357,0.1171,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[19,19,1.0,0.08482,0.22548,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1]]},{"b":6,"e":0.14286,"k":"falling","v":0.02232,"x":0.62947,"p":[[0,19,0.0,0.62947,0.37603,0.25,0.71429,1.0,0.0,1.0,3,12,0,3,0,5,0,0,2,0,0,2,0,0,1,0,0,4,0,0,3,0,12],[4,19,0.2105,0.20089,0.2621,0.0,0.14286,0.1786,0.0,0.85714,13,0,0,13,0,11,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0],[8,19,0.4211,0.11161,0.23347,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[12,19,0.6316,0.09821,0.19377,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,19,0.8421,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"915ad12dfa1d1393","q":"For any integer  $r \\geq 1$ , determine the smallest integer  $h(r) \\geq 1$  such that for any partition of the set  $\\{1, 2, \\cdots, h(r)\\}$  into  $r$  classes, there are integers  $a \\geq  0 \\ ; 1 \\leq x \\leq y$ , such that  $a + x, a + y, a + x + y$  belong to the same class.\n\n*Proposed by Romania*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.01339,"x":0.4732,"p":[[0,35,0.0,0.4732,0.30185,0.14286,0.57143,0.71429,0.0,1.0,1,3,1,1,0,11,0,0,1,0,0,0,0,0,10,0,0,4,0,0,2,0,3],[4,35,0.1143,0.3482,0.37785,0.10714,0.14286,0.64286,0.0,1.0,8,5,0,8,0,13,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,5],[8,35,0.2286,0.24999,0.27892,0.14286,0.14286,0.21429,0.0,1.0,7,1,0,7,0,17,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,1],[12,35,0.3429,0.33034,0.31223,0.14286,0.14286,0.57143,0.0,1.0,7,3,0,7,0,11,0,0,1,0,0,1,0,0,8,0,0,1,0,0,0,0,3],[16,35,0.4571,0.30804,0.32948,0.0,0.14286,0.57143,0.0,1.0,12,2,0,12,0,6,0,0,0,0,0,5,0,0,3,0,0,2,0,0,2,0,2],[20,35,0.5714,0.31244,0.306,0.0,0.14286,0.571,0.0,1.0,10,2,0,10,0,7,0,0,0,0,0,6,0,0,5,0,0,1,0,0,1,0,2],[24,35,0.6857,0.06696,0.14279,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.08035,0.15943,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,35,0.9143,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"volatile","v":0.01339,"x":0.48646,"p":[[0,24,0.0,0.48646,0.30494,0.14286,0.57121,0.71429,0.0,1.0,3,3,3,3,0,7,0,0,2,0,0,0,0,0,9,0,0,7,0,0,1,0,3],[4,24,0.1667,0.35714,0.3863,0.0,0.14286,0.85704,0.0,1.0,10,4,1,10,0,9,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,4],[8,24,0.3333,0.17857,0.22868,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,17,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[12,24,0.5,0.45525,0.38874,0.105,0.49979,0.85714,0.0,1.0,8,5,0,8,0,7,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,5],[16,24,0.6667,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.04911,0.15407,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,24,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3769341207d27887","q":"For each positive integer  $m$  let  $t_m$  be the smallest positive integer not dividing  $m$ . Prove that there are infinitely many positive integers which can not be represented in the form  $m + t_m$ .\n\n*(A. Golovanov)*","t":[{"b":2,"e":0.0,"k":"volatile","v":0.0,"x":0.49093,"p":[[0,10,0.0,0.49093,0.35873,0.14286,0.42859,0.85714,0.0,1.0,6,6,1,6,0,4,0,0,1,0,0,7,0,0,2,0,0,3,0,0,3,0,6],[4,10,0.4,0.10714,0.25,0.0,0.0,0.0,0.0,1.0,25,1,1,25,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.01339,"x":0.37499,"p":[[0,41,0.0,0.37499,0.36726,0.0,0.21428,0.74996,0.0,1.0,10,3,0,10,0,6,0,0,2,0,0,2,0,0,3,0,0,1,0,0,5,0,3],[4,41,0.0976,0.20526,0.30917,0.0,0.0,0.32143,0.0,1.0,19,2,0,19,0,2,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,2],[8,41,0.1951,0.20536,0.36059,0.0,0.0,0.21429,0.0,1.0,23,3,0,23,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,3],[12,41,0.2927,0.0982,0.19701,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[16,41,0.3902,0.07143,0.19562,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,41,0.4878,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,41,0.5854,0.06249,0.13798,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,41,0.6829,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.08036,0.23403,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[36,41,0.878,0.08482,0.21975,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[40,41,0.9756,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de5b17a479c6ce41","q":"For each positive integer $n$, let $\\operatorname{rad}(n)$ denote the product of the distinct prime factors of $n$. Show that there exist integers $a, b>1$ such that $\\operatorname{gcd}(a, b)=1$ and $$ \\operatorname{rad}(a b(a+b))<\\frac{a+b}{2024^{2024}} $$ For example, $\\operatorname{rad}(20)=\\operatorname{rad}\\left(2^{2} \\cdot 5\\right)=2 \\cdot 5=10$ and $\\operatorname{rad}(18)=\\operatorname{rad}\\left(2 \\cdot 3^{2}\\right)=2 \\cdot 3=6$.","t":[{"b":5,"e":0.42857,"k":"falling","v":0.26339,"x":0.42857,"p":[[0,23,0.0,0.42857,0.45175,0.0,0.2143,1.0,0.0,1.0,12,12,6,12,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[4,23,0.1739,0.29464,0.23128,0.14286,0.2857,0.42857,0.0,1.0,7,1,0,7,0,4,0,0,10,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[8,23,0.3478,0.26787,0.20124,0.14286,0.28571,0.28579,0.0,1.0,4,1,0,4,0,9,0,0,12,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[12,23,0.5217,0.26339,0.17169,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[16,23,0.6957,0.28116,0.19068,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,12,0,0,8,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[20,23,0.8696,0.29017,0.18027,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,7,0,0,5,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[23,23,1.0,0.26786,0.17767,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,12,0,0,6,0,0,10,0,0,0,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.27679,"x":0.54465,"p":[[0,57,0.0,0.5,0.4432,0.0,0.50001,1.0,0.0,1.0,12,11,10,12,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,11],[4,57,0.0702,0.45312,0.34004,0.14286,0.32136,0.71429,0.0,1.0,4,6,0,4,0,5,0,0,7,0,1,3,0,0,2,0,0,3,0,0,1,0,6],[8,57,0.1404,0.27679,0.18536,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,5,0,0,10,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[12,57,0.2105,0.30134,0.22849,0.14286,0.28571,0.42857,0.0,0.7857,6,0,0,6,0,7,0,0,6,0,0,8,0,0,1,0,0,3,1,0,0,0,0],[16,57,0.2807,0.3817,0.27871,0.14286,0.42857,0.46431,0.0,1.0,4,3,0,4,0,6,0,1,4,0,0,9,0,0,3,0,0,2,0,0,0,0,3],[20,57,0.3509,0.29911,0.21535,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,6,0,0,9,0,0,9,0,0,1,0,0,1,0,0,0,0,1],[24,57,0.4211,0.33036,0.19377,0.14289,0.35714,0.42857,0.0,0.71429,4,0,0,4,0,5,0,0,7,0,0,11,0,0,3,0,0,2,0,0,0,0,0],[28,57,0.4912,0.36607,0.2922,0.14286,0.28571,0.46431,0.0,1.0,4,3,0,4,0,9,0,0,4,0,0,7,0,0,3,0,0,1,0,0,1,0,3],[32,57,0.5614,0.33928,0.18814,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,4,0,0,11,0,0,9,0,0,2,0,0,3,0,0,0,0,0],[36,57,0.6316,0.51338,0.31106,0.28571,0.42857,0.75,0.0,1.0,3,6,0,3,0,1,0,0,7,0,0,8,0,0,3,0,0,2,0,0,2,0,6],[40,57,0.7018,0.51338,0.22829,0.39286,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,9,0,0,6,0,0,5,0,0,2,0,2],[44,57,0.7719,0.45982,0.23072,0.28571,0.42857,0.57143,0.0,1.0,2,2,0,2,0,2,0,0,5,0,0,11,0,0,6,0,0,4,0,0,0,0,2],[48,57,0.8421,0.35262,0.15554,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,4,0,0,8,0,0,13,0,0,5,0,0,0,0,0,0,0,0],[52,57,0.9123,0.38393,0.17655,0.28571,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,1,0,0,11,0,0,13,0,0,2,0,0,2,0,0,1,0,0],[56,57,0.9825,0.54465,0.2271,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,11,0,0,6,0,0,7,0,0,2,0,2],[57,57,1.0,0.48657,0.21085,0.42857,0.42859,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,4,0,0,11,0,0,8,0,0,3,0,0,2,0,1]]}]},{"i":"74c4304e12105e4b","q":"Let  $ p$  be a prime number and  $ f$  an integer polynomial of degree  $ d$  such that  $ f(0) = 0,f(1) = 1$  and  $ f(n)$  is congruent to  $ 0$  or  $ 1$  modulo  $ p$  for every integer  $ n$ . Prove that  $ d\\geq p - 1$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.15176,"x":0.32588,"p":[[0,12,0.0,0.32588,0.3916,0.0,0.14286,0.60714,0.0,1.0,14,6,1,14,0,4,0,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,6],[4,12,0.3333,0.22768,0.3551,0.0,0.0,0.28571,0.0,1.0,19,4,0,19,0,3,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,4],[8,12,0.6667,0.15176,0.19206,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,4,0,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,12,1.0,0.24104,0.25611,0.0,0.21428,0.42857,0.0,1.0,13,1,0,13,0,3,0,0,6,0,0,5,0,0,3,0,0,1,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.75893,"p":[[0,69,0.0,0.35268,0.36767,0.0,0.28571,0.57141,0.0,1.0,12,6,0,12,0,1,0,0,7,0,0,3,0,0,2,0,0,1,0,0,0,0,6],[4,69,0.058,0.65625,0.45156,0.0,1.0,1.0,0.0,1.0,9,18,0,9,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,18],[8,69,0.1159,0.65179,0.45728,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,18],[12,69,0.1739,0.75893,0.40475,0.75,1.0,1.0,0.0,1.0,6,22,0,6,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,22],[16,69,0.2319,0.29018,0.44676,0.0,0.0,1.0,0.0,1.0,22,9,0,22,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[20,69,0.2899,0.125,0.28065,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[24,69,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0ec42c9a1153e90","q":"For each positive integer  $n$  let  $a_n$  be the least positive integer multiple of  $23$  such that  $a_n\\equiv1\\pmod{2^n}$ . Find the number of positive integers  $n$  less than or equal to  $1000$  that satisfy  $a_n=a_{n+1}$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,42,0.0,0.89732,0.22371,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,23],[4,42,0.0952,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[8,42,0.1905,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,42,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,42,0.381,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,42,0.4762,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,42,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,42,0.6667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,42,0.7619,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,42,0.8571,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,42,0.9524,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[42,42,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":2,"e":1.0,"k":"flat","v":0.64732,"x":0.99554,"p":[[0,107,0.0,0.92857,0.22588,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[4,107,0.0374,0.94643,0.14617,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[8,107,0.0748,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,107,0.1121,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[16,107,0.1495,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[20,107,0.1869,0.93304,0.18205,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[24,107,0.2243,0.91517,0.19186,1.0,1.0,1.0,0.286,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,25],[28,107,0.2617,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[32,107,0.2991,0.90179,0.24074,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,26],[36,107,0.3364,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[40,107,0.3738,0.92409,0.16749,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,25],[44,107,0.4112,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[48,107,0.4486,0.81696,0.31387,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,4,0,21],[52,107,0.486,0.86607,0.27418,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[56,107,0.5234,0.79911,0.3251,0.60714,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,22],[60,107,0.5607,0.64732,0.37284,0.28571,0.78571,1.0,0.0,1.0,2,15,0,2,0,2,0,0,9,0,0,0,0,0,1,0,0,2,0,0,1,0,15],[64,107,0.5981,0.69196,0.36089,0.28571,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,6,0,0,1,0,0,1,0,0,1,0,0,4,0,15],[68,107,0.6355,0.91963,0.19214,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,27],[72,107,0.6729,0.87946,0.18249,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,2,0,21],[76,107,0.7103,0.80357,0.24679,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,4,0,0,2,0,17],[80,107,0.7477,0.93304,0.14718,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,25],[84,107,0.785,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[88,107,0.8224,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[92,107,0.8598,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[96,107,0.8972,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[100,107,0.9346,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,107,0.972,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[107,107,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"2cb8c2b3824b4cd0","q":"Let  $ m$  be the number of solutions in positive integers to the equation  $ 4x\\plus{}3y\\plus{}2z\\equal{}2009$ , and let  $ n$  be the number of solutions in positive integers to the equation  $ 4x\\plus{}3y\\plus{}2z\\equal{}2000$ . Find the remainder when  $ m\\minus{}n$  is divided by  $ 1000$ .","t":[{"b":3,"e":0.14,"k":"flat","v":0.12045,"x":0.38839,"p":[[0,142,0.0,0.12045,0.07237,0.14214,0.14286,0.14286,0.0,0.28571,7,0,1,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,142,0.0282,0.38384,0.35976,0.14286,0.14286,0.85714,0.14,1.0,0,6,0,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,6],[8,142,0.0563,0.32134,0.32932,0.14286,0.14286,0.2857,0.14,1.0,0,6,0,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[12,142,0.0845,0.27678,0.26949,0.14286,0.14286,0.2857,0.14286,1.0,0,2,0,0,0,23,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,2],[16,142,0.1127,0.29911,0.29744,0.14286,0.14286,0.28571,0.14286,1.0,0,4,0,0,0,22,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,4],[20,142,0.1408,0.38839,0.32972,0.14286,0.28571,0.75,0.0,1.0,2,3,0,2,0,13,0,0,6,0,0,2,0,0,0,0,0,1,0,0,5,0,3],[24,142,0.169,0.33929,0.32488,0.14286,0.14286,0.39285,0.14286,1.0,0,3,0,0,0,22,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,3],[28,142,0.1972,0.20089,0.20782,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,142,0.2254,0.29464,0.29001,0.14286,0.14286,0.2857,0.0,1.0,1,2,0,1,0,21,0,0,4,0,0,0,0,0,0,0,0,1,0,0,3,0,2],[36,142,0.2535,0.25893,0.25862,0.14286,0.14286,0.17857,0.14286,1.0,0,2,0,0,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[40,142,0.2817,0.20964,0.19234,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[44,142,0.3099,0.29018,0.28679,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[48,142,0.338,0.24107,0.21261,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,23,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[52,142,0.3662,0.24107,0.24856,0.14286,0.14286,0.14287,0.14286,1.0,0,3,0,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[56,142,0.3944,0.22759,0.17449,0.14286,0.14286,0.2857,0.14,0.85714,0,0,0,0,0,21,0,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[60,142,0.4225,0.21875,0.2082,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[64,142,0.4507,0.20982,0.1636,0.14286,0.14286,0.17857,0.14286,1.0,0,1,0,0,0,24,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[68,142,0.4789,0.17857,0.15567,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[72,142,0.507,0.2767,0.27884,0.14286,0.14286,0.2857,0.14,1.0,0,4,0,0,0,22,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[76,142,0.5352,0.21429,0.20825,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[80,142,0.5634,0.2408,0.22723,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,1],[84,142,0.5915,0.17411,0.05906,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,142,0.6197,0.18741,0.15748,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[92,142,0.6479,0.21876,0.2082,0.14286,0.14286,0.14289,0.14286,1.0,0,2,0,0,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[96,142,0.6761,0.16509,0.08075,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,24,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,142,0.7042,0.20527,0.13337,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[104,142,0.7324,0.17402,0.08556,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[108,142,0.7606,0.22768,0.20782,0.14286,0.14286,0.2857,0.14286,1.0,0,2,0,0,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[112,142,0.7887,0.17402,0.05911,0.14286,0.14286,0.14287,0.14,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,142,0.8169,0.16964,0.07523,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[120,142,0.8451,0.19197,0.07667,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,142,0.8732,0.20527,0.15546,0.14286,0.14286,0.2857,0.14,1.0,0,1,0,0,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[128,142,0.9014,0.16947,0.06629,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,142,0.9296,0.17384,0.06914,0.14286,0.14286,0.17857,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,142,0.9577,0.18304,0.15251,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[140,142,0.9859,0.16955,0.06625,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[142,142,1.0,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.12491,"x":0.44643,"p":[[0,91,0.0,0.12491,0.09277,0.105,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,91,0.044,0.44187,0.37012,0.14286,0.14286,0.85714,0.14,1.0,0,7,0,0,0,17,0,0,3,0,0,0,0,0,1,0,0,1,0,0,3,0,7],[8,91,0.0879,0.44643,0.37754,0.14286,0.21429,0.89286,0.14286,1.0,0,8,0,0,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,3,0,8],[12,91,0.1319,0.35705,0.3175,0.14286,0.14286,0.75,0.14,1.0,0,1,0,0,0,20,0,0,3,0,0,0,0,0,0,0,0,1,0,0,7,0,1],[16,91,0.1758,0.43295,0.37716,0.14286,0.14286,0.85714,0.14,1.0,0,7,0,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,7],[20,91,0.2198,0.36607,0.35163,0.14286,0.14286,0.53571,0.14286,1.0,0,6,0,0,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,6],[24,91,0.2637,0.30339,0.31702,0.14286,0.14286,0.2857,0.0,1.0,1,4,0,1,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[28,91,0.3077,0.27232,0.27049,0.14286,0.14286,0.17857,0.14286,1.0,0,2,0,0,0,24,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,2],[32,91,0.3516,0.33929,0.34395,0.14286,0.14286,0.32143,0.14286,1.0,0,6,0,0,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,6],[36,91,0.3956,0.2008,0.19189,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[40,91,0.4396,0.20982,0.1636,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,22,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[44,91,0.4835,0.2008,0.21089,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,91,0.5275,0.1875,0.15335,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,91,0.5714,0.23661,0.25155,0.14286,0.14286,0.14287,0.0,1.0,1,3,0,1,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[56,91,0.6154,0.31687,0.33457,0.14286,0.14286,0.2857,0.0,1.0,1,6,0,1,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[60,91,0.6593,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,91,0.7033,0.17411,0.15458,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,91,0.7473,0.19634,0.21356,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[72,91,0.7912,0.2008,0.21089,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[76,91,0.8352,0.2008,0.21389,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[80,91,0.8791,0.19643,0.19805,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[84,91,0.9231,0.17848,0.1288,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[88,91,0.967,0.17411,0.13236,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[91,91,1.0,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f4122e8531956c58","q":"Let  $0<a_1<a_2<\\cdots <a_n$  be real numbers. Prove that\n\\[\\left (\\frac{1}{1+a_1}+\\frac{1}{1+a_2}+\\cdots +\\frac{1}{1+a_n}\\right )^2 \\leq \\frac{1}{a_1}+\\frac{1}{a_2-a_1}+\\cdots +\\frac{1}{a_n-a_{n-1}}.\\]","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,67,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,67,0.2985,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,67,0.8955,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.0,"x":0.12947,"p":[[0,61,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,61,0.1311,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,61,0.1967,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,61,0.2623,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,61,0.3279,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,61,0.3934,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,61,0.459,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,61,0.5246,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,61,0.6557,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,61,0.7869,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,61,0.918,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,61,0.9836,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.12947,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"73862337f8b3e592","q":"Let  $A$ ,  $B$ ,  $C$ ,  $D$  be four points in space not all lying on the same plane. The segments  $AB$ ,  $BC$ ,  $CD$ , and  $DA$  are tangent to the same sphere. Prove that their four points of tangency are coplanar.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.14723,"p":[[0,53,0.0,0.14723,0.24611,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,53,0.0755,0.08036,0.24206,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,53,0.1509,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,53,0.2264,0.10268,0.2402,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,53,0.3019,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,53,0.3774,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,53,0.4528,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,53,0.5283,0.07589,0.24218,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,53,0.6038,0.10268,0.24804,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0],[36,53,0.6792,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,53,0.7547,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,53,0.8302,0.13839,0.3164,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[48,53,0.9057,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,53,0.9811,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.14286,"k":"flat","v":0.00893,"x":0.29008,"p":[[0,97,0.0,0.14286,0.28347,0.0,0.0,0.14286,0.0,1.0,18,3,2,18,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,97,0.0412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,97,0.0825,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,97,0.1237,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,97,0.1649,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,97,0.2062,0.05804,0.17807,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,97,0.2474,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,97,0.2887,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,97,0.3299,0.04009,0.08908,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,97,0.3711,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,97,0.4124,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,97,0.4536,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,97,0.4948,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,97,0.5361,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,97,0.5773,0.06696,0.15561,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[60,97,0.6186,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,97,0.6598,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,97,0.701,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,97,0.7423,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,97,0.7835,0.08482,0.09354,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,97,0.8247,0.10705,0.07139,0.0,0.14286,0.14286,0.0,0.2857,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,97,0.866,0.12491,0.17034,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[88,97,0.9072,0.17839,0.13837,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,22,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[92,97,0.9485,0.19642,0.21941,0.14286,0.14286,0.1429,0.0,1.0,7,1,0,7,0,19,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[96,97,0.9897,0.29008,0.22446,0.14286,0.14288,0.28571,0.14,1.0,0,1,0,0,0,18,0,0,7,0,0,0,0,0,5,0,0,0,0,0,1,0,1],[97,97,1.0,0.18295,0.13477,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,25,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"e4689386175ccfcb","q":"Let  $ABC$  be a triangle and  $A',B',C'$  be the midpoints of  $BC,CA,AB$  respectively. Let  $P$  and  $P'$  be points in plane such that  $PA=P'A',PB=P'B',PC=P'C'$ . Prove that all  $PP'$  pass through a fixed point.","t":[{"b":3,"e":0.42857,"k":"falling","v":0.6428,"x":0.83929,"p":[[0,82,0.0,0.83929,0.22517,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,4,0,18],[4,82,0.0488,0.72764,0.23789,0.571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,9,0,0,3,0,10],[8,82,0.0976,0.79015,0.17855,0.71429,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,11,0,0,5,0,10],[12,82,0.1463,0.79463,0.21111,0.67857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,7,0,12],[16,82,0.1951,0.808,0.20397,0.71429,0.85714,1.0,0.0,1.0,1,9,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,12,0,9],[20,82,0.2439,0.82141,0.1713,0.71429,0.85707,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,4,0,13],[24,82,0.2927,0.7677,0.21359,0.67536,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,9,0,0,5,0,10],[28,82,0.3415,0.76785,0.21353,0.67857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,9,0,0,4,0,11],[32,82,0.3902,0.73203,0.23375,0.57143,0.71429,1.0,0.14,1.0,0,9,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,9,0,0,4,0,9],[36,82,0.439,0.81247,0.1938,0.71429,0.85707,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,5,0,13],[40,82,0.4878,0.78567,0.16754,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,10,0,0,6,0,9],[44,82,0.5366,0.69192,0.12932,0.71421,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,19,0,0,6,0,0],[48,82,0.5854,0.73198,0.19483,0.71429,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,13,0,0,10,0,3],[52,82,0.6341,0.71872,0.13116,0.71421,0.71429,0.74996,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,17,0,0,6,0,2],[56,82,0.6829,0.74328,0.18023,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,1,9,0,0,5,0,0,9,0,6],[60,82,0.7317,0.73213,0.18124,0.67857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,13,0,0,6,0,5],[64,82,0.7805,0.77228,0.1592,0.67857,0.78564,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,10,0,6],[68,82,0.8293,0.75444,0.09608,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,10,0,1],[72,82,0.878,0.66966,0.13563,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,17,0,0,3,0,1],[76,82,0.9268,0.68075,0.15874,0.57132,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,9,1,0,11,0,0,4,0,3],[80,82,0.9756,0.67406,0.17583,0.571,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,11,0,0,3,0,4],[82,82,1.0,0.6428,0.19235,0.571,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,6,0,0,12,0,0,7,0,0,1,0,5]]},{"b":7,"e":0.71429,"k":"falling","v":0.60713,"x":0.87944,"p":[[0,78,0.0,0.81247,0.24075,0.71429,0.85714,1.0,0.0,1.0,1,13,1,1,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,9,0,13],[4,78,0.0513,0.87054,0.21829,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,20],[8,78,0.1026,0.87944,0.17901,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,1,0,21],[12,78,0.1538,0.83481,0.19599,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,3,0,16],[16,78,0.2051,0.83918,0.22261,0.71429,1.0,1.0,0.14,1.0,0,17,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,7,0,0,4,0,17],[20,78,0.2564,0.74996,0.26728,0.571,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,3,0,0,5,0,13],[24,78,0.3077,0.80354,0.18817,0.71429,0.85707,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,8,0,0,5,0,12],[28,78,0.359,0.79906,0.23109,0.67836,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,0,6,0,14],[32,78,0.4103,0.72325,0.27426,0.571,0.78571,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,3,0,0,5,0,0,4,0,0,5,0,11],[36,78,0.4615,0.81243,0.21858,0.67846,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,5,0,15],[40,78,0.5128,0.75446,0.27254,0.57143,0.857,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,1,0,0,4,0,0,5,0,0,3,0,14],[44,78,0.5641,0.71426,0.27434,0.57132,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,2,0,0,8,0,0,5,0,0,0,0,13],[48,78,0.6154,0.78569,0.22304,0.57143,0.857,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,6,0,0,3,0,14],[52,78,0.6667,0.71873,0.29121,0.42857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,2,0,0,6,0,12],[56,78,0.7179,0.81246,0.24602,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,2,0,0,4,0,17],[60,78,0.7692,0.70533,0.2695,0.5354,0.71429,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,5,0,0,2,0,0,2,0,0,9,0,0,3,0,10],[64,78,0.8205,0.74551,0.23621,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,6,0,0,4,0,11],[68,78,0.8718,0.73212,0.2594,0.57132,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,7,0,0,4,0,11],[72,78,0.9231,0.70089,0.31412,0.39286,0.85707,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,6,0,0,3,0,0,1,0,0,3,0,0,4,0,13],[76,78,0.9744,0.67408,0.33548,0.28571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,15],[78,78,1.0,0.60713,0.33882,0.28571,0.57121,1.0,0.0,1.0,1,11,0,1,0,2,0,0,9,0,0,3,0,0,2,0,0,2,0,0,2,0,11]]}]},{"i":"427a457ea36b0ea5","q":"Let  $ABCD$  be a convex quadrilateral such that  $BC=AD$ . Let  $M$  and  $N$  be the midpoints of  $AB$  and  $CD$ , respectively. The lines  $AD$  and  $BC$  meet the line  $MN$  at  $P$  and  $Q$ , respectively. Prove that  $CQ=DP$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.15848,"x":0.49552,"p":[[0,58,0.0,0.28345,0.18767,0.14286,0.28571,0.375,0.0,0.71429,6,0,0,6,0,4,0,0,10,0,4,3,0,1,3,0,0,1,0,0,0,0,0],[4,58,0.069,0.30795,0.31366,0.0,0.2857,0.46429,0.0,1.0,9,3,0,9,0,6,0,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,3],[8,58,0.1379,0.43973,0.31497,0.1429,0.32143,0.71429,0.0,1.0,2,5,0,2,1,6,0,0,7,0,1,4,0,0,2,0,0,3,1,0,0,0,5],[12,58,0.2069,0.42856,0.35129,0.14286,0.42857,0.58929,0.0,1.0,6,7,0,6,1,3,0,0,5,0,0,7,0,0,2,1,0,0,0,0,0,0,7],[16,58,0.2759,0.49552,0.32924,0.28571,0.42857,0.75,0.0,1.0,3,7,0,3,0,2,0,0,10,0,0,4,0,0,3,0,0,2,0,0,1,0,7],[20,58,0.3448,0.4933,0.34273,0.2857,0.35714,0.87501,0.0,1.0,3,6,0,3,1,2,0,0,10,0,0,3,0,0,2,0,0,2,0,0,1,2,6],[24,58,0.4138,0.39509,0.33833,0.14286,0.28571,0.50002,0.0,1.0,5,6,0,5,1,4,0,0,10,0,0,4,0,0,0,0,0,2,0,0,0,0,6],[28,58,0.4828,0.30129,0.30608,0.12464,0.2857,0.28571,0.0,1.0,7,4,0,7,1,6,0,0,11,0,0,2,0,0,0,0,0,1,0,0,0,0,4],[32,58,0.5517,0.30578,0.31351,0.05325,0.2857,0.42857,0.0,1.0,8,3,0,8,2,4,0,1,8,0,0,3,0,0,1,0,0,0,0,0,2,0,3],[36,58,0.6207,0.25446,0.26901,0.0,0.24999,0.28571,0.0,1.0,9,1,0,9,1,5,0,1,10,0,1,1,0,0,0,0,0,1,0,0,1,1,1],[40,58,0.6897,0.28124,0.30564,0.0,0.2857,0.28571,0.0,1.0,9,4,0,9,2,3,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[44,58,0.7586,0.25669,0.25496,0.125,0.21428,0.28571,0.0,1.0,7,2,0,7,1,8,0,0,10,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[48,58,0.8276,0.27893,0.30746,0.0,0.21428,0.32143,0.0,1.0,9,3,0,9,3,4,0,0,8,0,0,2,0,1,0,1,0,0,0,0,1,0,3],[52,58,0.8966,0.21652,0.27287,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,6,0,0,8,0,0,1,0,0,0,1,0,1,0,0,0,0,2],[56,58,0.9655,0.21426,0.13241,0.14286,0.2857,0.28571,0.0,0.571,5,0,0,5,2,6,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[58,58,1.0,0.15848,0.16534,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,6,0,1,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.22312,"x":0.5067,"p":[[0,116,0.0,0.26116,0.20871,0.10714,0.28571,0.32143,0.0,1.0,8,1,0,8,0,2,0,1,13,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[4,116,0.0345,0.47096,0.37408,0.23213,0.35714,0.89275,0.0,1.0,7,8,0,7,1,0,0,0,8,0,0,2,0,0,4,0,0,1,0,0,1,0,8],[8,116,0.069,0.38841,0.31588,0.14289,0.28571,0.60714,0.0,1.0,4,4,0,4,0,7,0,0,9,0,0,3,0,0,1,0,0,3,0,0,1,0,4],[12,116,0.1034,0.5067,0.3613,0.24999,0.35714,1.0,0.0,1.0,3,9,0,3,0,5,0,0,8,0,0,2,0,0,1,1,0,3,0,0,0,0,9],[16,116,0.1379,0.45982,0.35488,0.28571,0.28571,0.71429,0.0,1.0,7,6,0,7,0,0,0,0,10,0,0,2,0,0,1,0,0,5,0,0,1,0,6],[20,116,0.1724,0.37499,0.37754,0.0,0.28571,0.71429,0.0,1.0,11,6,0,11,0,2,0,0,7,0,0,2,0,0,1,0,0,2,0,0,1,0,6],[24,116,0.2069,0.3482,0.32524,0.10714,0.28571,0.46418,0.0,1.0,8,5,0,8,0,2,0,0,13,0,0,1,0,0,3,0,0,0,0,0,0,0,5],[28,116,0.2414,0.34375,0.36045,0.0,0.2857,0.46431,0.0,1.0,10,6,0,10,0,4,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,6],[32,116,0.2759,0.37723,0.31655,0.14286,0.28571,0.57143,0.0,1.0,5,4,0,5,1,4,0,0,11,0,0,2,0,0,2,0,0,2,0,0,1,0,4],[36,116,0.3103,0.3058,0.29733,0.10714,0.2857,0.42858,0.0,1.0,8,3,0,8,0,7,0,0,6,0,0,5,0,0,2,0,0,0,1,0,0,0,3],[40,116,0.3448,0.25446,0.22227,0.10714,0.28571,0.28571,0.0,1.0,8,1,0,8,0,4,0,0,15,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[44,116,0.3793,0.30561,0.24117,0.14,0.28571,0.42857,0.0,1.0,5,1,0,5,2,3,0,0,12,0,1,5,0,0,0,0,0,2,0,0,1,0,1],[48,116,0.4138,0.35937,0.30435,0.24999,0.28571,0.32143,0.0,1.0,4,4,0,4,3,1,0,0,16,0,0,1,0,0,1,0,0,1,0,0,1,0,4],[52,116,0.4483,0.3817,0.35964,0.125,0.28571,0.57143,0.0,1.0,7,7,0,7,1,3,0,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,7],[56,116,0.4828,0.43521,0.36789,0.1429,0.28571,0.78571,0.0,1.0,6,8,0,6,1,2,0,0,10,0,0,2,0,0,2,0,0,1,0,0,0,0,8],[60,116,0.5172,0.3125,0.31225,0.0,0.2857,0.32143,0.0,1.0,9,4,0,9,0,3,0,0,12,0,0,2,0,0,1,0,0,1,0,0,0,0,4],[64,116,0.5517,0.38169,0.33005,0.14286,0.28571,0.60714,0.0,1.0,6,5,0,6,1,2,0,0,13,0,0,1,0,0,1,0,0,3,0,0,0,0,5],[68,116,0.5862,0.40625,0.3409,0.14286,0.28571,0.71429,0.0,1.0,7,4,0,7,0,3,0,0,9,0,0,2,0,0,1,0,0,4,0,0,2,0,4],[72,116,0.6207,0.42854,0.33265,0.24999,0.28571,0.64254,0.0,1.0,2,6,0,2,2,4,0,0,12,0,0,3,0,0,1,0,0,0,0,0,2,0,6],[76,116,0.6552,0.31697,0.28287,0.14286,0.28571,0.28571,0.0,1.0,5,4,0,5,0,5,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[80,116,0.6897,0.27897,0.23516,0.125,0.2857,0.28571,0.0,1.0,6,1,0,6,2,2,0,1,16,0,0,0,0,0,1,0,0,3,0,0,0,0,1],[84,116,0.7241,0.37277,0.33251,0.125,0.28571,0.46429,0.0,1.0,7,5,0,7,1,3,0,0,8,0,0,5,0,0,1,0,0,2,0,0,0,0,5],[88,116,0.7586,0.37723,0.33609,0.14286,0.28571,0.60714,0.0,1.0,7,5,0,7,0,4,0,0,9,0,1,2,0,0,1,0,0,3,0,0,0,0,5],[92,116,0.7931,0.45537,0.35298,0.2857,0.28571,0.83929,0.0,1.0,3,8,0,3,1,3,0,0,13,0,0,2,0,0,0,0,0,1,1,0,0,0,8],[96,116,0.8276,0.32366,0.27663,0.125,0.2857,0.42857,0.0,1.0,7,2,0,7,1,1,0,0,14,0,0,3,0,0,1,0,0,2,0,0,1,0,2],[100,116,0.8621,0.36382,0.27626,0.2857,0.28571,0.42857,0.0,1.0,4,3,0,4,1,1,0,0,17,0,0,2,0,0,2,0,0,1,0,0,1,0,3],[104,116,0.8966,0.36597,0.26955,0.2857,0.28571,0.42857,0.0,1.0,3,3,0,3,0,4,0,0,16,0,0,2,0,0,1,0,0,3,0,0,0,0,3],[108,116,0.931,0.40624,0.27689,0.2857,0.28571,0.57143,0.0,1.0,3,3,0,3,0,2,0,0,15,0,0,3,0,0,2,0,0,3,0,0,1,0,3],[112,116,0.9655,0.45757,0.26836,0.28571,0.28571,0.57143,0.21429,1.0,0,5,0,0,0,0,0,1,20,0,0,0,0,0,4,0,0,2,0,0,0,0,5],[116,116,1.0,0.22312,0.23944,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,8,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"6a491be888e1c344","q":"Let  $ABCD$  be a convex quadrilateral such that the triangle  $ABD$  is equilateral and the triangle  $BCD$  is isosceles, with  $\\angle C = 90^o$ . If  $E$  is the midpoint of the side  $AD$ , determine the measure of the angle  $\\angle CED$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.77679,"x":1.0,"p":[[0,51,0.0,0.875,0.29179,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,26],[4,51,0.0784,0.87054,0.30589,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[8,51,0.1569,0.80804,0.35465,0.89286,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,24],[12,51,0.2353,0.77679,0.37617,0.60714,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,23],[16,51,0.3137,0.80804,0.3126,0.57143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,22],[20,51,0.3922,0.80357,0.33072,0.67857,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,22],[24,51,0.4706,0.86607,0.27418,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,25],[28,51,0.549,0.86161,0.24868,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,2,0,0,1,0,23],[32,51,0.6275,0.81249,0.30398,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,5,0,0,0,0,0,2,0,0,2,0,0,0,0,22],[36,51,0.7059,0.86161,0.31841,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,26],[40,51,0.7843,0.89284,0.26246,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,26],[44,51,0.8627,0.90625,0.22192,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,27],[48,51,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[51,51,1.0,0.95534,0.14917,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29]]},{"b":5,"e":0.0,"k":"flat","v":0.71425,"x":0.88392,"p":[[0,83,0.0,0.88392,0.25364,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,25],[4,83,0.0482,0.83034,0.2933,0.67857,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,4,0,0,0,0,0,3,0,0,1,0,0,0,0,23],[8,83,0.0964,0.82141,0.32144,0.82132,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,22],[12,83,0.1446,0.82588,0.28062,0.67857,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,0,0,21],[16,83,0.1928,0.80804,0.33619,0.71429,1.0,1.0,0.0,1.0,4,22,0,4,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,22],[20,83,0.241,0.81249,0.2867,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,2,0,19],[24,83,0.2892,0.85713,0.29668,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,23],[28,83,0.3373,0.87946,0.30952,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[32,83,0.3855,0.78124,0.3273,0.67857,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,5,0,0,0,0,0,1,0,0,2,0,0,3,0,19],[36,83,0.4337,0.83926,0.30464,0.71429,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,23],[40,83,0.4819,0.86604,0.25493,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,23],[44,83,0.5301,0.83928,0.29827,0.85714,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,7,0,20],[48,83,0.5783,0.86161,0.2461,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,7,0,20],[52,83,0.6265,0.8214,0.28124,0.67846,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,21],[56,83,0.6747,0.88391,0.24858,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,24],[60,83,0.7229,0.85714,0.23145,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,4,0,0,2,0,21],[64,83,0.7711,0.83926,0.27609,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,0,3,0,21],[68,83,0.8193,0.86607,0.24984,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,22],[72,83,0.8675,0.83036,0.26591,0.85714,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,9,0,16],[76,83,0.9157,0.79017,0.2879,0.82132,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,10,0,14],[80,83,0.9639,0.71425,0.26002,0.571,0.85714,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,1,0,0,4,0,0,4,0,0,3,0,0,12,0,6],[83,83,1.0,0.7991,0.16311,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,15,0,6]]}]},{"i":"3bf080683ab8d32e","q":"Let  $ABC$  be a triangle. Point  $E,F$  moves on the opposite ray of  $BA,CA$  such that  $BF=CE$ . Let  $M,N$  be the midpoint of  $BE,CF$ .  $BF$  cuts  $CE$  at  $D$ a) Suppost that  $I$  is the circumcenter of  $(DBE)$  and  $J$  is the circumcenter of  $(DCF)$ , Prove that  $MN \\parallel IJ$ b) Let  $K$  be the midpoint of  $MN$  and  $H$  be the orthocenter of triangle  $AEF$ . Prove that when  $E$  varies on the opposite ray of  $BA$ ,  $HK$  go through a fixed 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 $ABC$  be a triangle with orthocenter  $H,$  and let  $M$  be the midpoint of  $\\overline{BC}.$  Suppose that  $P$  and  $Q$  are distinct points on the circle with diameter  $\\overline{AH},$  different from  $A,$  such that  $M$  lies on line  $PQ.$  Prove that the orthocenter of  $\\triangle APQ$  lies on the circumcircle of  $\\triangle ABC.$  \n*Proposed by Michael Ren*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.03125,"x":0.29015,"p":[[0,80,0.0,0.29015,0.29118,0.0,0.14286,0.571,0.0,1.0,10,1,0,10,0,8,0,0,1,0,0,4,0,0,6,0,0,0,0,0,2,0,1],[4,80,0.05,0.23661,0.31866,0.0,0.14286,0.1786,0.0,1.0,12,2,0,12,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,2],[8,80,0.1,0.14286,0.19233,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,11,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[12,80,0.15,0.11607,0.20958,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[16,80,0.2,0.0892,0.17765,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[20,80,0.25,0.08929,0.11152,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,80,0.3,0.12054,0.17169,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[28,80,0.35,0.11607,0.20341,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,80,0.4,0.10268,0.20277,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[36,80,0.45,0.09813,0.16916,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,80,0.5,0.09803,0.19041,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,80,0.55,0.06697,0.14719,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,80,0.6,0.10714,0.23958,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[52,80,0.65,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,80,0.7,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,80,0.75,0.06249,0.15943,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[64,80,0.8,0.05804,0.1551,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[68,80,0.85,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,80,0.9,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,80,0.95,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,80,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.04911,"x":0.33035,"p":[[0,71,0.0,0.33035,0.3203,0.0,0.2857,0.57143,0.0,1.0,11,1,1,11,0,4,0,0,3,0,0,3,0,0,6,0,0,0,0,0,4,0,1],[4,71,0.0563,0.1517,0.25739,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,5,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0],[8,71,0.1127,0.125,0.21651,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[12,71,0.169,0.08031,0.13811,0.0,0.0,0.14286,0.0,0.43,22,0,0,22,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,71,0.2254,0.13839,0.20973,0.0,0.0,0.17857,0.0,0.85714,19,0,0,19,0,5,0,0,1,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[20,71,0.2817,0.12937,0.19349,0.0,0.07,0.14286,0.0,0.857,16,0,0,16,0,11,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[24,71,0.338,0.10705,0.21723,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[28,71,0.3944,0.07134,0.13358,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,7,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,71,0.4507,0.12054,0.21162,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[36,71,0.507,0.07143,0.13363,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,7,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,71,0.5634,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.06679,0.07957,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.0758,0.09432,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.10713,0.18554,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[56,71,0.7887,0.06697,0.13356,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[60,71,0.8451,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.11598,0.16916,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,10,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[68,71,0.9577,0.12054,0.24513,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0],[71,71,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c20ffa7ec06c5e30","q":"Let  $ABCD$  be a cyclic quadrilateral and let  $K,L,M,N,S,T$  the midpoints of  $AB, BC, CD, AD, AC, BD$  respectively.  Prove that the circumcenters of  $KLS, LMT, MNS, NKT$  form a cyclic quadrilateral which is similar to  $ABCD$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,36,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,36,0.1111,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,36,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,36,0.3333,0.94197,0.17075,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[16,36,0.4444,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,36,0.5556,0.93304,0.20198,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[24,36,0.6667,0.90625,0.1411,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,20],[28,36,0.7778,0.89286,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[32,36,0.8889,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[36,36,1.0,0.86607,0.14258,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,16]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,40,0.0,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,40,0.1,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,40,0.2,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,40,0.3,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,40,0.4,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,40,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,40,0.6,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,40,0.7,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,40,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,40,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"0abc33b37d61b999","q":"The fuction  $f(0,\\infty)\\rightarrow\\mathbb{R}$  verifies  $f(x)+f(y)=2f(\\sqrt{xy}), \\forall x,y>0$ . Show that for every positive integer  $n>2$  the following relation takes place   $$ f(x_1)+f(x_2)+\\ldots+f(x_n)=nf(\\sqrt[n]{x_1x_2\\ldots x_n}), $$   for every positive integers  $x_1,x_2,\\ldots,x_n$ .","t":[{"b":6,"e":0.28571,"k":"flat","v":0.76783,"x":0.95982,"p":[[0,21,0.0,0.88839,0.26422,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,25],[4,21,0.1905,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[8,21,0.381,0.87946,0.1992,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,19],[12,21,0.5714,0.90178,0.1729,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,6,0,21],[16,21,0.7619,0.85265,0.23282,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,3,0,0,1,0,21],[20,21,0.9524,0.83481,0.19922,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,4,0,16],[21,21,1.0,0.76783,0.25941,0.5354,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,4,0,0,1,0,16]]},{"b":7,"e":1.0,"k":"flat","v":0.88838,"x":1.0,"p":[[0,19,0.0,0.91964,0.19865,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,25],[4,19,0.2105,0.92857,0.16752,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[8,19,0.4211,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[12,19,0.6316,0.88838,0.16265,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,6,0,19],[16,19,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5a17e09df7df5638","q":"For any positive integer  $n$ , let  $a_n=\\sum_{k=1}^{\\infty}[\\frac{n+2^{k-1}}{2^k}]$ , where  $[x]$  is the largest integer that is equal or less than  $x$ . Determine the value of  $a_{2015}$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.63839,"x":0.86159,"p":[[0,75,0.0,0.72767,0.25596,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,1,0,0,14,0,0,1,0,0,1,0,13],[4,75,0.0533,0.73211,0.24938,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,0,0,0,15,0,0,0,0,0,3,0,12],[8,75,0.1067,0.86159,0.20974,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,1,0,0,5,0,19],[12,75,0.16,0.67409,0.27948,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,2,0,0,0,0,0,14,0,0,1,0,0,1,0,11],[16,75,0.2133,0.76784,0.2442,0.57143,0.85707,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,8,0,0,3,0,0,6,0,12],[20,75,0.2667,0.73212,0.25941,0.57143,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,3,0,0,10,0,0,1,0,0,4,0,12],[24,75,0.32,0.70089,0.27516,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,0,0,0,0,0,0,11,0,0,3,0,0,4,0,10],[28,75,0.3733,0.63839,0.3369,0.35714,0.57143,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,0,0,0,1,0,0,8,0,0,1,0,0,3,0,11],[32,75,0.4267,0.67856,0.29451,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,2,0,0,2,0,11],[36,75,0.48,0.70536,0.28107,0.57143,0.64286,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,1,0,0,4,0,11],[40,75,0.5333,0.74997,0.25256,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,9,0,0,2,0,0,5,0,12],[44,75,0.5867,0.70087,0.28204,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,4,0,11],[48,75,0.64,0.75892,0.24075,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,1,0,0,9,0,0,1,0,0,9,0,10],[52,75,0.6933,0.70982,0.24086,0.57143,0.64286,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,1,0,0,13,0,0,4,0,0,2,0,10],[56,75,0.7467,0.7723,0.23653,0.57143,0.85707,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,10,0,0,2,0,0,3,0,14],[60,75,0.8,0.74552,0.27372,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,3,0,14],[64,75,0.8533,0.7232,0.2765,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,0,0,0,2,0,0,10,0,0,2,0,0,2,0,13],[68,75,0.9067,0.7723,0.27401,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,1,0,0,7,0,0,2,0,0,4,0,15],[72,75,0.96,0.73658,0.19599,0.57143,0.57143,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,3,0,10],[75,75,1.0,0.70088,0.25595,0.57143,0.57143,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,0,0,0,0,0,0,14,0,0,1,0,0,5,0,9]]},{"b":4,"e":0.57143,"k":"flat","v":0.63393,"x":0.80804,"p":[[0,68,0.0,0.71874,0.2612,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,1,0,0,1,0,13],[4,68,0.0588,0.80804,0.22192,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,4,0,0,4,0,15],[8,68,0.1176,0.76339,0.2412,0.57143,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,12,0,0,2,0,0,2,0,14],[12,68,0.1765,0.72767,0.27517,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,0,0,0,1,0,0,12,0,0,0,0,0,3,0,13],[16,68,0.2353,0.75446,0.25812,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,2,0,0,10,0,0,0,0,0,5,0,13],[20,68,0.2941,0.75,0.24222,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,0,0,0,11,0,0,4,0,0,3,0,12],[24,68,0.3529,0.65177,0.28558,0.57143,0.57143,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,0,0,0,0,0,0,15,0,0,0,0,0,3,0,9],[28,68,0.4118,0.6875,0.29545,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,0,0,0,3,0,0,10,0,0,2,0,0,0,0,13],[32,68,0.4706,0.68302,0.30249,0.57132,0.64286,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,0,0,0,2,0,0,9,0,0,1,0,0,4,0,11],[36,68,0.5294,0.76338,0.2276,0.57143,0.85712,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,12,0,0,1,0,0,5,0,12],[40,68,0.5882,0.66515,0.30433,0.57132,0.57143,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,1,0,0,0,0,0,12,0,0,2,0,0,0,0,12],[44,68,0.6471,0.70981,0.3103,0.57143,0.78571,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,0,0,0,1,0,0,9,0,0,1,0,0,2,0,14],[48,68,0.7059,0.70534,0.30292,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,0,0,0,0,0,0,10,0,0,2,0,0,2,0,13],[52,68,0.7647,0.70979,0.25875,0.57143,0.64286,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,0,0,0,13,0,0,2,0,0,4,0,10],[56,68,0.8235,0.6473,0.24996,0.57143,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,0,0,0,0,0,0,15,0,0,2,0,0,6,0,5],[60,68,0.8824,0.77676,0.2447,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,0,0,0,10,0,0,1,0,0,6,0,13],[64,68,0.9412,0.68304,0.25688,0.57143,0.57143,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,2,0,0,1,0,10],[68,68,1.0,0.63393,0.23941,0.57143,0.57143,0.75,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,1,0,0,19,0,0,1,0,0,1,0,7]]}]},{"i":"38e60e8298cfe5af","q":"Let  $ABC$  be a triangle with integral side lengths such that  $\\angle A=3\\angle B$ . Find the minimum value of its perimeter.","t":[{"b":2,"e":0.571,"k":"falling","v":0.49106,"x":0.95089,"p":[[0,117,0.0,0.82143,0.25,0.71429,1.0,1.0,0.0,1.0,1,17,1,1,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,17],[4,117,0.0342,0.91516,0.18512,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[8,117,0.0684,0.91741,0.15476,0.91071,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,1,23],[12,117,0.1026,0.88839,0.18808,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,5,0,21],[16,117,0.1368,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,27],[20,117,0.1709,0.88393,0.20958,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,0,0,24],[24,117,0.2051,0.82143,0.22304,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,5,0,16],[28,117,0.2393,0.87052,0.2063,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,3,0,21],[32,117,0.2735,0.84375,0.22406,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,2,0,20],[36,117,0.3077,0.83036,0.23808,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,4,0,18],[40,117,0.3419,0.70536,0.2878,0.42859,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,9,0,0,3,0,0,2,0,0,1,0,14],[44,117,0.3761,0.82589,0.22794,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,1,0,19],[48,117,0.4103,0.7991,0.24964,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,0,4,0,15],[52,117,0.4444,0.67857,0.24744,0.42857,0.64286,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,3,0,0,5,0,8],[56,117,0.4786,0.73213,0.2594,0.57143,0.64286,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,1,0,0,1,0,14],[60,117,0.5128,0.66515,0.25408,0.42857,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,7,0,0,9,0,0,2,0,0,1,0,10],[64,117,0.547,0.63837,0.26723,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,10,0,0,7,0,0,2,0,0,0,0,10],[68,117,0.5812,0.71875,0.2912,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,8,0,0,3,0,0,4,0,0,1,0,14],[72,117,0.6154,0.72768,0.27283,0.4286,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,8,0,0,3,0,0,4,0,0,1,0,14],[76,117,0.6496,0.72759,0.27302,0.42857,0.78571,1.0,0.14,1.0,0,13,0,0,0,1,0,0,1,0,0,9,0,0,1,0,0,4,0,0,3,0,13],[80,117,0.6838,0.68749,0.21853,0.57132,0.64286,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,6,0,0,9,0,0,6,0,0,2,0,8],[84,117,0.7179,0.66517,0.2276,0.42859,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,10,0,0,3,0,0,2,0,8],[88,117,0.7521,0.66503,0.2734,0.42857,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,9,0,0,4,0,0,3,0,0,1,0,11],[92,117,0.7863,0.71427,0.25755,0.57132,0.64286,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,3,0,0,0,0,13],[96,117,0.8205,0.74997,0.26002,0.57143,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,4,0,0,0,0,15],[100,117,0.8547,0.57142,0.26486,0.42857,0.4998,0.71429,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,11,0,0,5,0,0,4,0,0,1,0,6],[104,117,0.8889,0.66065,0.27609,0.4286,0.57143,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,5,0,0,8,0,0,3,0,0,3,0,9],[108,117,0.9231,0.60268,0.24152,0.42857,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,9,0,0,7,0,0,5,0,0,1,0,6],[112,117,0.9573,0.57589,0.28231,0.42857,0.57143,0.78571,0.14286,1.0,0,8,0,0,0,3,0,0,4,0,0,8,0,0,7,0,0,2,0,0,0,0,8],[116,117,0.9915,0.49106,0.14257,0.42857,0.4286,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,12,0,0,8,0,0,6,0,0,0,0,0],[117,117,1.0,0.49108,0.17104,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,7,0,0,13,0,0,5,0,0,6,0,0,0,0,1]]},{"b":4,"e":0.28571,"k":"falling","v":0.22321,"x":0.95089,"p":[[0,258,0.0,0.92411,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21],[4,258,0.0155,0.94196,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[8,258,0.031,0.95089,0.13176,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[12,258,0.0465,0.80804,0.28928,0.64286,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,5,0,0,0,0,0,2,0,0,2,0,20],[16,258,0.062,0.89284,0.1786,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,3,0,22],[20,258,0.0775,0.85714,0.2369,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,0,4,0,21],[24,258,0.093,0.86607,0.23941,0.89286,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,24],[28,258,0.1085,0.86159,0.2328,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3,0,21],[32,258,0.124,0.86594,0.22579,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,5,0,20],[36,258,0.1395,0.87946,0.21162,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,1,0,23],[40,258,0.155,0.85267,0.24612,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,1,0,0,2,0,22],[44,258,0.1705,0.90625,0.19103,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[48,258,0.186,0.77232,0.29203,0.53572,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,1,0,0,0,0,19],[52,258,0.2016,0.82587,0.26425,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0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,1],[236,258,0.9147,0.37054,0.27632,0.14286,0.28571,0.57143,0.0,1.0,3,3,0,3,0,7,0,0,10,0,0,3,0,0,4,0,0,2,0,0,0,0,3],[240,258,0.9302,0.37052,0.2168,0.2857,0.35729,0.42857,0.0,1.0,2,2,0,2,0,4,0,0,10,0,0,11,0,0,3,0,0,0,0,0,0,0,2],[244,258,0.9457,0.3482,0.21107,0.14286,0.28571,0.4642,0.0,0.857,2,0,0,2,0,8,0,0,8,0,0,6,0,0,5,0,0,2,0,0,1,0,0],[248,258,0.9612,0.29465,0.18189,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,6,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[252,258,0.9767,0.38835,0.2179,0.24999,0.42857,0.571,0.0,1.0,1,1,0,1,0,7,0,0,7,0,0,8,0,0,5,0,0,3,0,0,0,0,1],[256,258,0.9922,0.22321,0.16342,0.14286,0.14286,0.28571,0.0,0.57143,6,0,0,6,0,11,0,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[258,258,1.0,0.25,0.17496,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,7,0,0,12,0,0,3,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"e20b0e1ac465e81d","q":"The natural numbers  $n$  and  $z$  are relatively prime and greater than  $1$ . For  $k = 0, 1, 2,..., n - 1$  let  $s(k) = 1 + z + z^2 + ...+ z^k.$  Prove that:\na) At least one of the numbers  $s(k)$  is divisible by  $n$ .\nb) If  $n$  and  $z - 1$  are also coprime, then already one of the numbers  $s(k)$  with  $k = 0,1, 2,..., n- 2$  is divisible by  $n$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.63402,"x":0.96875,"p":[[0,16,0.0,0.63402,0.31129,0.42857,0.42857,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,13,0,0,0,0,0,1,0,0,1,0,12],[4,16,0.25,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,16,0.5,0.90625,0.1411,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,22],[12,16,0.75,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,16,1.0,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,26]]},{"b":4,"e":0.71429,"k":"rising","v":0.59821,"x":0.99107,"p":[[0,20,0.0,0.59821,0.28221,0.42857,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,7,0,0,11,0,0,2,0,0,2,0,0,1,0,9],[4,20,0.2,0.94197,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[8,20,0.4,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,20,0.6,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,20,0.8,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,20,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"a9fcfd4521ce82a4","q":"Let  $ABCD$  be a convex quadrilateral with  $AB=2, AD=7,$  and  $CD=3$  such that the bisectors of acute angles  $\\angle{DAB}$  and  $\\angle{ADC}$  intersect at the midpoint of  $\\overline{BC}.$  Find the square of the area of  $ABCD.$","t":[{"b":2,"e":0.14286,"k":"falling","v":0.19196,"x":0.53571,"p":[[0,220,0.0,0.47319,0.16534,0.42857,0.57143,0.57143,0.0,0.57143,2,0,2,2,0,1,0,0,3,0,0,5,0,0,21,0,0,0,0,0,0,0,0],[4,220,0.0182,0.48661,0.13296,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,11,0,0,15,0,0,2,0,0,0,0,0],[8,220,0.0364,0.46875,0.13474,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,1,0,0,13,0,0,14,0,0,1,0,0,0,0,0],[12,220,0.0545,0.49552,0.1556,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,1,0,0,9,0,0,18,0,0,2,0,0,0,0,0],[16,220,0.0727,0.50891,0.10676,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,7,0,0,22,0,0,0,0,0,0,0,0],[20,220,0.0909,0.49552,0.11284,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,11,0,0,17,0,0,1,0,0,0,0,0],[24,220,0.1091,0.52679,0.10374,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,9,0,0,18,0,0,3,0,0,0,0,0],[28,220,0.1273,0.52676,0.09739,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,20,0,0,2,0,0,0,0,0],[32,220,0.1455,0.53571,0.11294,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,3,0,0,24,0,0,2,0,0,0,0,0],[36,220,0.1636,0.53125,0.08171,0.42859,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,8,0,0,22,0,0,1,0,0,0,0,0],[40,220,0.1818,0.52232,0.09181,0.4286,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,21,0,0,1,0,0,0,0,0],[44,220,0.2,0.53125,0.12993,0.53571,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,4,0,0,20,0,0,4,0,0,0,0,0],[48,220,0.2182,0.49999,0.11844,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,11,0,0,16,0,0,2,0,0,0,0,0],[52,220,0.2364,0.45536,0.14914,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,5,0,0,10,0,0,14,0,0,1,0,0,0,0,0],[56,220,0.2545,0.42414,0.16553,0.39286,0.42859,0.57143,0.0,0.57143,2,0,0,2,0,2,0,0,4,0,0,11,0,0,13,0,0,0,0,0,0,0,0],[60,220,0.2727,0.48214,0.1171,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,9,0,0,18,0,0,0,0,0,0,0,0],[64,220,0.2909,0.47321,0.12079,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,9,0,0,17,0,0,0,0,0,0,0,0],[68,220,0.3091,0.47319,0.12593,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,10,0,0,15,0,0,1,0,0,0,0,0],[72,220,0.3273,0.45981,0.18116,0.39286,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,4,0,0,8,0,0,12,0,0,4,0,0,0,0,0],[76,220,0.3455,0.5,0.15152,0.42859,0.57143,0.57143,0.0,0.71429,1,0,1,1,0,1,0,0,3,0,0,5,0,0,20,0,0,2,0,0,0,0,0],[80,220,0.3636,0.49108,0.09407,0.42857,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,12,0,0,17,0,0,0,0,0,0,0,0],[84,220,0.3818,0.50445,0.12363,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,8,0,0,18,0,0,2,0,0,0,0,0],[88,220,0.4,0.4866,0.11213,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,13,0,0,15,0,0,1,0,0,0,0,0],[92,220,0.4182,0.51339,0.1063,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,9,0,0,20,0,0,1,0,0,0,0,0],[96,220,0.4364,0.51339,0.11769,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,8,0,0,17,0,0,3,0,0,0,0,0],[100,220,0.4545,0.46872,0.11423,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,15,0,0,12,0,0,1,0,0,0,0,0],[104,220,0.4727,0.4107,0.17404,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,6,0,0,8,0,0,10,0,0,2,0,0,0,0,0],[108,220,0.4909,0.43746,0.15944,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,5,0,0,10,0,0,11,0,0,2,0,0,0,0,0],[112,220,0.5091,0.36594,0.18543,0.25,0.42857,0.571,0.0,0.57143,3,0,0,3,0,5,0,0,4,0,0,11,0,0,9,0,0,0,0,0,0,0,0],[116,220,0.5273,0.43303,0.14934,0.39286,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,5,0,0,11,0,0,13,0,0,0,0,0,0,0,0],[120,220,0.5455,0.4375,0.20806,0.28571,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,3,0,0,3,0,0,6,0,0,14,0,0,3,0,0,0,0,0],[124,220,0.5636,0.44196,0.1488,0.39286,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,6,0,0,11,0,0,12,0,0,1,0,0,0,0,0],[128,220,0.5818,0.42856,0.17127,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,4,0,0,9,0,0,13,0,0,1,0,0,0,0,0],[132,220,0.6,0.41963,0.12844,0.39286,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,3,0,0,5,0,0,15,0,0,9,0,0,0,0,0,0,0,0],[136,220,0.6182,0.39731,0.17028,0.28571,0.42857,0.57111,0.0,0.71429,1,0,0,1,0,4,0,0,7,0,0,11,0,0,7,0,0,2,0,0,0,0,0],[140,220,0.6364,0.36606,0.17833,0.24999,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,10,0,0,3,0,0,10,0,0,1,0,0,0,0,0],[144,220,0.6545,0.35714,0.15152,0.2857,0.28571,0.46431,0.0,0.57143,1,0,0,1,0,3,0,0,15,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[148,220,0.6727,0.28116,0.13124,0.14286,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,10,0,0,17,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[152,220,0.6909,0.24107,0.13092,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,9,0,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[156,220,0.7091,0.25893,0.16536,0.14286,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,6,0,0,15,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[160,220,0.7273,0.22322,0.15126,0.14286,0.2857,0.28571,0.0,0.57143,5,0,1,5,0,10,0,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[164,220,0.7455,0.23661,0.13651,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,10,0,0,16,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[168,220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$a, b, c$ be positive real numbers. Prove that  $$ \\frac{(2 a+b+c)^{2}}{2 a^{2}+(b+c)^{2}}+\\frac{(2 b+c+a)^{2}}{2 b^{2}+(c+a)^{2}}+\\frac{(2 c+a+b)^{2}}{2 c^{2}+(a+b)^{2}} \\leq 8 $$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.19196,"x":0.64286,"p":[[0,76,0.0,0.64286,0.41188,0.2857,1.0,1.0,0.0,1.0,3,18,0,3,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[4,76,0.0526,0.55804,0.41857,0.14286,0.28571,1.0,0.0,1.0,1,15,0,1,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[8,76,0.1053,0.50446,0.38957,0.14286,0.28571,1.0,0.14286,1.0,0,11,0,0,0,13,0,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,11],[12,76,0.1579,0.52232,0.40972,0.14286,0.28571,1.0,0.0,1.0,1,13,0,1,0,13,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,13],[16,76,0.2105,0.45983,0.36897,0.14286,0.28571,1.0,0.14286,1.0,0,9,0,0,0,14,0,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,9],[20,76,0.2632,0.33036,0.30186,0.14286,0.14286,0.28571,0.14286,1.0,0,4,0,0,0,19,0,0,6,0,0,0,0,0,0,0,0,3,0,0,0,0,4],[24,76,0.3158,0.27232,0.28428,0.14286,0.14286,0.2857,0.0,1.0,1,4,0,1,0,22,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[28,76,0.3684,0.3259,0.34299,0.14286,0.14286,0.28571,0.0,1.0,2,6,0,2,0,20,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[32,76,0.4211,0.37947,0.32263,0.14286,0.1429,0.60714,0.14286,1.0,0,5,0,0,0,17,0,0,5,0,0,0,0,0,2,0,0,3,0,0,0,0,5],[36,76,0.4737,0.29911,0.32803,0.14286,0.14286,0.2857,0.0,1.0,4,4,0,4,0,19,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,4],[40,76,0.5263,0.30804,0.32948,0.14286,0.14286,0.28571,0.0,1.0,4,5,0,4,0,17,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,5],[44,76,0.5789,0.40179,0.38205,0.14286,0.2143,1.0,0.0,1.0,3,9,0,3,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[48,76,0.6316,0.33928,0.33645,0.14286,0.14286,0.35714,0.0,1.0,3,5,0,3,0,16,0,0,5,0,0,0,0,0,1,0,0,1,0,0,1,0,5],[52,76,0.6842,0.33929,0.30462,0.14286,0.14286,0.35714,0.14286,1.0,0,4,0,0,0,19,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,4],[56,76,0.7368,0.26786,0.23076,0.14286,0.14288,0.28571,0.0,1.0,1,2,0,1,0,17,0,0,10,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[60,76,0.7895,0.28125,0.28679,0.14286,0.14286,0.28571,0.0,1.0,4,3,0,4,0,16,0,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,3],[64,76,0.8421,0.3125,0.28221,0.14286,0.14286,0.35714,0.0,1.0,3,2,0,3,0,15,0,0,6,0,0,0,0,0,1,0,0,5,0,0,0,0,2],[68,76,0.8947,0.21429,0.21724,0.14286,0.14286,0.2857,0.0,1.0,3,2,0,3,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[72,76,0.9474,0.19196,0.14987,0.14286,0.14286,0.14287,0.0,0.71429,2,0,0,2,0,23,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[76,76,1.0,0.20089,0.16698,0.14286,0.14286,0.1786,0.0,0.85714,2,0,0,2,0,22,0,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0]]},{"b":4,"e":0.2857,"k":"falling","v":0.13393,"x":0.63607,"p":[[0,88,0.0,0.63607,0.42092,0.14289,1.0,1.0,0.0,1.0,3,17,0,3,1,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,17],[4,88,0.0455,0.48659,0.37603,0.14286,0.28571,1.0,0.0,1.0,1,9,0,1,0,12,0,0,5,0,0,0,0,0,3,0,0,0,0,0,2,0,9],[8,88,0.0909,0.36607,0.33491,0.14286,0.14286,0.39286,0.14286,1.0,0,6,0,0,0,18,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,6],[12,88,0.1364,0.42857,0.37965,0.14286,0.14286,1.0,0.14286,1.0,0,9,0,0,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[16,88,0.1818,0.35714,0.35535,0.14286,0.14286,0.39286,0.14286,1.0,0,7,0,0,0,22,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[20,88,0.2273,0.45982,0.39243,0.14286,0.2857,1.0,0.0,1.0,2,10,0,2,0,13,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,10],[24,88,0.2727,0.31697,0.32288,0.14286,0.14286,0.28571,0.0,1.0,2,5,0,2,0,19,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,5],[28,88,0.3182,0.38391,0.36146,0.14286,0.14286,0.71429,0.0,1.0,1,7,0,1,0,19,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,7],[32,88,0.3636,0.32589,0.30771,0.14286,0.14286,0.28571,0.0,1.0,2,4,0,2,0,16,0,0,7,0,0,0,0,0,0,0,0,3,0,0,0,0,4],[36,88,0.4091,0.41518,0.38193,0.14286,0.14286,1.0,0.0,1.0,1,9,0,1,0,17,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[40,88,0.4545,0.36607,0.34981,0.14286,0.14286,0.71429,0.0,1.0,2,6,0,2,0,17,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,6],[44,88,0.5,0.51786,0.40524,0.14286,0.28571,1.0,0.0,1.0,2,12,0,2,0,12,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,12],[48,88,0.5455,0.34821,0.3387,0.14286,0.14286,0.35714,0.0,1.0,1,6,0,1,0,19,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[52,88,0.5909,0.29464,0.29867,0.14286,0.14286,0.28571,0.0,1.0,2,4,0,2,0,19,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[56,88,0.6364,0.42411,0.3416,0.14286,0.21428,0.71429,0.14286,1.0,0,5,0,0,0,16,0,0,4,0,0,0,0,0,1,0,0,4,0,0,2,0,5],[60,88,0.6818,0.2633,0.30956,0.14214,0.14286,0.28571,0.0,1.0,7,4,0,7,0,14,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[64,88,0.7273,0.3125,0.33586,0.14286,0.14286,0.28571,0.0,1.0,5,5,0,5,0,15,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[68,88,0.7727,0.32589,0.33737,0.14286,0.14286,0.35714,0.0,1.0,5,5,0,5,0,14,0,0,5,0,0,0,0,0,1,0,0,2,0,0,0,0,5],[72,88,0.8182,0.25,0.29233,0.14286,0.14286,0.28571,0.0,1.0,7,3,0,7,0,15,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[76,88,0.8636,0.30357,0.31084,0.14286,0.14286,0.28571,0.0,1.0,3,4,0,3,0,18,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[80,88,0.9091,0.17857,0.23146,0.0,0.14286,0.14286,0.0,1.0,9,2,0,9,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[84,88,0.9545,0.20089,0.16698,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,13,0,0,11,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[88,88,1.0,0.13393,0.09407,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"72069ba3d71f9bd7","q":"Two circles  $S_1$  and  $S_2$  with centers  $O_1$  and  $O_2$  respectively intersect at  $A$  and  $B$ . The tangents at  $A$  to  $S_1$  and  $S_2$  meet segments  $BO_2$  and  $BO_1$  at  $K$  and  $L$  respectively. Show that  $KL \\parallel O_1O_2.$","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.07589,"p":[[0,72,0.0,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,72,0.3333,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,72,0.3889,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,72,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.07589,0.24218,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,72,0.6111,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.05357,0.19805,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[56,72,0.7778,0.07143,0.24223,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[60,72,0.8333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,72,0.8889,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,72,0.9444,0.0133,0.04137,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.11161,"p":[[0,60,0.0,0.11161,0.24675,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[4,60,0.0667,0.08036,0.20183,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[8,60,0.1333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,60,0.3333,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,60,0.4,0.09375,0.2412,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,60,0.4667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,60,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,60,0.9333,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,60,1.0,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cc06ce276a8c9136","q":"Two infinite arithmetic sequences with positive integers are given:  $$ a_1<a_2<a_3<\\cdots  ;   b_1<b_2<b_3<\\cdots $$  It is known that there are infinitely many pairs of positive integers  $(i,j)$  for which  $i\\leq j\\leq i+2021$  and  $a_i$  divides  $b_j$ . Prove that for every positive integer  $i$  there exists a positive integer  $j$  such that  $a_i$  divides  $b_j$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.95535,"x":1.0,"p":[[0,57,0.0,0.95535,0.13092,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[4,57,0.0702,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,57,0.1404,0.96652,0.13241,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,29],[12,57,0.2105,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,57,0.2807,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,57,0.3509,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,57,0.4211,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,57,0.4912,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,57,0.5614,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[36,57,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,57,0.7018,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,57,0.7719,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,57,0.8421,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[52,57,0.9123,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,57,0.9825,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[57,57,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.28571,"k":"flat","v":0.9375,"x":1.0,"p":[[0,47,0.0,0.9375,0.18877,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[4,47,0.0851,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,47,0.1702,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[12,47,0.2553,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,47,0.3404,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,47,0.4255,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,47,0.5106,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,47,0.5957,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,47,0.6809,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,47,0.766,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,47,0.8511,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,47,0.9362,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[47,47,1.0,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]}]},{"i":"eb4e6a9b3dcdb134","q":"Two congruent equilateral triangles  $ABC$  and  $A'B'C'$  in the plane are given. Show that the midpoints of the segments  $AA',BB', CC'$  either are collinear or form an equilateral triangle.","t":[{"b":0,"e":1.0,"k":"rising","v":0.75893,"x":1.0,"p":[[0,26,0.0,0.75893,0.18013,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,16,0,0,1,0,9],[4,26,0.1538,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,26,0.3077,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,26,0.4615,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,26,0.6154,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,26,0.7692,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,26,0.9231,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"rising","v":0.83036,"x":1.0,"p":[[0,27,0.0,0.83036,0.15335,0.71429,0.71429,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,0,0,14],[4,27,0.1481,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"e1a1c4dc95154402","q":"There is infinite sequence of composite numbers  $a_1,a_2,...,$  where  $a_{n+1}=a_n-p_n+\\frac{a_n}{p_n}$  ;  $p_n$  is smallest prime divisor of  $a_n$ . It is known, that  $37|a_n$  for every  $n$ .\nFind possible values of  $a_1$","t":[{"b":1,"e":1.0,"k":"flat","v":0.88839,"x":0.99553,"p":[[0,77,0.0,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[4,77,0.0519,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,77,0.1039,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[12,77,0.1558,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,77,0.2078,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,77,0.2597,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[24,77,0.3117,0.95536,0.15746,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[28,77,0.3636,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[32,77,0.4156,0.94197,0.15915,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[36,77,0.4675,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[40,77,0.5195,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,77,0.5714,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,77,0.6234,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,77,0.6753,0.94195,0.14226,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,26],[56,77,0.7273,0.94195,0.18853,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[60,77,0.7792,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[64,77,0.8312,0.94643,0.17405,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[68,77,0.8831,0.91964,0.16727,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,24],[72,77,0.9351,0.88839,0.23346,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,23],[76,77,0.987,0.93303,0.13825,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[77,77,1.0,0.90176,0.1692,0.85711,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,21]]},{"b":5,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,57,0.0,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[4,57,0.0702,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,57,0.1404,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,57,0.2105,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,57,0.2807,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,57,0.3509,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,57,0.4211,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,57,0.4912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,57,0.5614,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[36,57,0.6316,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[40,57,0.7018,0.95536,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[44,57,0.7719,0.96652,0.09614,1.0,1.0,1.0,0.5,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,27],[48,57,0.8421,0.93304,0.14279,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[52,57,0.9123,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[56,57,0.9825,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[57,57,1.0,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]}]},{"i":"af76843d3a870c89","q":"We wish to write  $n$  distinct real numbers  $(n\\geq3)$  on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of  $n$  such that this is possible.","t":[{"b":5,"e":1.0,"k":"flat","v":0.96429,"x":0.98661,"p":[[0,34,0.0,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[4,34,0.1176,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,34,0.2353,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,34,0.3529,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,34,0.4706,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,34,0.5882,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,34,0.7059,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,34,0.8235,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,34,0.9412,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[34,34,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.95536,"x":0.99554,"p":[[0,22,0.0,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[4,22,0.1818,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[8,22,0.3636,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,22,0.9091,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[22,22,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"1418e2b727ef81e3","q":"We call a set of positive integers *suitable* if none of its elements is coprime to the sum of all\nelements of that set. Given a real number  $\\varepsilon \\in (0,1)$ , prove that, for all large enough positive\nintegers  $N$ , there exists a suitable set of size at least  $\\varepsilon N$ , each element of which is at most  $N$ .","t":[{"b":5,"e":1.0,"k":"falling","v":0.62497,"x":0.94196,"p":[[0,26,0.0,0.89732,0.12993,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,9,0,17],[4,26,0.1538,0.86606,0.22572,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,0,3,0,21],[8,26,0.3077,0.91071,0.16269,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,4,0,22],[12,26,0.4615,0.94196,0.1551,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[16,26,0.6154,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[20,26,0.7692,0.92411,0.1636,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,25],[24,26,0.9231,0.72322,0.21706,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,6,0,0,4,0,9],[26,26,1.0,0.62497,0.18472,0.571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,6,0,0,12,0,0,8,0,0,2,0,3]]},{"b":6,"e":1.0,"k":"flat","v":0.86606,"x":0.94643,"p":[[0,36,0.0,0.86606,0.15949,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,4,0,17],[4,36,0.1111,0.92411,0.14279,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,24],[8,36,0.2222,0.90624,0.16985,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,24],[12,36,0.3333,0.94643,0.12242,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[16,36,0.4444,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[20,36,0.5556,0.91964,0.17835,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,26],[24,36,0.6667,0.91964,0.15126,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,24],[28,36,0.7778,0.90624,0.16605,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,3,0,23],[32,36,0.8889,0.93304,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,23],[36,36,1.0,0.94642,0.14177,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28]]}]},{"i":"a58273799f4858b4","q":"We are given  $1978$  sets of size  $40$  each. The size of the intersection of any two sets is exactly  $1$ . Prove that all the sets have a common element.","t":[{"b":0,"e":0.0,"k":"falling","v":0.06695,"x":0.42409,"p":[[0,38,0.0,0.38393,0.29545,0.0,0.42857,0.57143,0.0,1.0,10,1,0,10,0,0,0,0,3,0,0,4,0,0,10,0,0,3,0,0,1,0,1],[4,38,0.1053,0.35268,0.31132,0.0,0.42857,0.57143,0.0,1.0,11,2,0,11,0,2,0,0,1,0,0,6,0,0,8,0,0,1,0,0,1,0,2],[8,38,0.2105,0.26339,0.2699,0.0,0.28571,0.46429,0.0,0.85714,15,0,0,15,0,0,0,0,3,0,0,6,0,0,6,0,0,1,0,0,1,0,0],[12,38,0.3158,0.375,0.31288,0.0,0.42857,0.57143,0.0,1.0,10,3,0,10,0,0,0,0,5,0,0,5,0,0,7,0,0,2,0,0,0,0,3],[16,38,0.4211,0.40179,0.3102,0.0,0.42857,0.57143,0.0,1.0,9,2,0,9,0,0,0,0,5,0,0,4,0,0,8,0,0,2,0,0,2,0,2],[20,38,0.5263,0.42409,0.34531,0.10714,0.42857,0.57143,0.0,1.0,8,5,0,8,0,2,0,0,4,0,0,6,0,0,5,0,0,0,0,0,2,0,5],[24,38,0.6316,0.40622,0.28816,0.21427,0.42857,0.57143,0.0,1.0,8,2,0,8,0,0,0,0,3,0,0,9,0,0,7,0,0,2,0,0,1,0,2],[28,38,0.7368,0.2098,0.33307,0.0,0.0,0.57111,0.0,1.0,22,2,0,22,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,1,0,2],[32,38,0.8421,0.23661,0.26633,0.0,0.14285,0.42857,0.0,1.0,16,1,0,16,0,0,0,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,1],[36,38,0.9474,0.14732,0.21866,0.0,0.0,0.17857,0.0,0.71429,19,0,0,19,0,5,0,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[38,38,1.0,0.06695,0.21716,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.0,"k":"falling","v":0.00893,"x":0.57143,"p":[[0,87,0.0,0.29464,0.2765,0.0,0.35714,0.57143,0.0,0.71429,14,0,0,14,0,1,0,0,1,0,0,2,0,0,13,0,0,1,0,0,0,0,0],[4,87,0.046,0.46875,0.31387,0.21429,0.57143,0.60714,0.0,1.0,8,3,0,8,0,0,0,0,1,0,0,4,0,0,11,0,0,4,0,0,1,0,3],[8,87,0.092,0.50893,0.34799,0.21429,0.57143,0.71429,0.0,1.0,8,6,0,8,0,0,0,0,1,0,0,3,0,0,10,0,0,3,0,0,1,0,6],[12,87,0.1379,0.41518,0.29312,0.0,0.57143,0.57143,0.0,1.0,9,2,0,9,0,0,0,0,1,0,0,5,0,0,13,0,0,2,0,0,0,0,2],[16,87,0.1839,0.45088,0.36615,0.0,0.57121,0.71429,0.0,1.0,9,6,0,9,0,2,0,0,3,0,0,1,0,0,7,0,0,4,0,0,0,0,6],[20,87,0.2299,0.37947,0.33619,0.0,0.42857,0.57143,0.0,1.0,11,4,0,11,0,1,0,0,2,0,0,5,0,0,8,0,0,1,0,0,0,0,4],[24,87,0.2759,0.57143,0.30093,0.57143,0.57143,0.71429,0.0,1.0,5,6,0,5,0,0,0,0,1,0,0,0,0,0,16,0,0,4,0,0,0,0,6],[28,87,0.3218,0.50893,0.38122,0.0,0.57143,0.89286,0.0,1.0,9,8,0,9,0,0,0,0,2,0,0,3,0,0,6,0,0,3,0,0,1,0,8],[32,87,0.3678,0.48214,0.36025,0.0,0.57143,0.71429,0.0,1.0,9,6,0,9,0,1,0,0,0,0,0,4,0,0,7,0,0,5,0,0,0,0,6],[36,87,0.4138,0.56249,0.28107,0.42857,0.57143,0.60714,0.0,1.0,3,6,0,3,0,1,0,0,2,0,0,3,0,0,15,0,0,2,0,0,0,0,6],[40,87,0.4598,0.37497,0.28063,0.0,0.42857,0.57141,0.0,1.0,9,2,0,9,0,0,0,0,3,0,0,9,0,0,8,0,0,1,0,0,0,0,2],[44,87,0.5057,0.3125,0.3102,0.0,0.28571,0.57143,0.0,1.0,13,2,0,13,0,1,0,0,4,0,0,3,0,0,7,0,0,2,0,0,0,0,2],[48,87,0.5517,0.37054,0.2809,0.0,0.57143,0.57143,0.0,0.71429,11,0,0,11,0,0,0,0,1,0,0,3,0,0,13,0,0,4,0,0,0,0,0],[52,87,0.5977,0.2857,0.27892,0.0,0.28571,0.57143,0.0,1.0,13,1,0,13,0,2,0,0,3,0,0,3,0,0,10,0,0,0,0,0,0,0,1],[56,87,0.6437,0.26339,0.29038,0.0,0.0,0.57143,0.0,0.71429,17,0,0,17,0,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,0,0,0],[60,87,0.6897,0.25892,0.29973,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,1,0,0,2,0,0,6,0,0,3,0,0,2,0,0,1,0,1],[64,87,0.7356,0.19642,0.26903,0.0,0.0,0.42857,0.0,0.85714,19,0,0,19,0,2,0,0,1,0,0,3,0,0,5,0,0,1,0,0,1,0,0],[68,87,0.7816,0.27676,0.26468,0.0,0.28571,0.57143,0.0,0.71429,14,0,0,14,0,0,0,0,5,0,0,1,0,0,11,0,0,1,0,0,0,0,0],[72,87,0.8276,0.26786,0.31491,0.0,0.07143,0.57143,0.0,1.0,16,2,0,16,0,2,0,0,1,0,0,3,0,0,7,0,0,1,0,0,0,0,2],[76,87,0.8736,0.20534,0.3029,0.0,0.0,0.46418,0.0,1.0,21,1,0,21,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,0,0,1],[80,87,0.9195,0.09821,0.15746,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[84,87,0.9655,0.13393,0.26471,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[87,87,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c421e0024b2552ae","q":"A set \\(A\\) of real numbers is framed when it is bounded and, for all \\(a, b \\in A\\), not necessarily distinct, \\((a-b)^{2} \\in A\\). What is the smallest real number that belongs to some framed set?","t":[{"b":0,"e":0.14286,"k":"falling","v":0.33482,"x":0.70088,"p":[[0,17,0.0,0.62052,0.36876,0.28571,0.57121,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,15],[4,17,0.2353,0.61607,0.34151,0.28571,0.64286,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,8,0,0,3,0,0,1,0,0,3,0,0,1,0,12],[8,17,0.4706,0.70088,0.33381,0.28571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,5,0,0,1,0,0,1,0,0,5,0,0,1,0,15],[12,17,0.7059,0.66964,0.33586,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,7,0,0,3,0,0,0,0,0,4,0,0,1,0,14],[16,17,0.9412,0.34821,0.26229,0.14286,0.28571,0.28571,0.0,1.0,1,3,0,1,0,8,0,0,16,0,0,2,0,0,0,0,0,1,0,0,1,0,3],[17,17,1.0,0.33482,0.27804,0.14286,0.2857,0.32143,0.14286,1.0,0,3,0,0,0,15,0,0,9,0,0,3,0,0,0,0,0,0,0,0,2,0,3]]},{"b":5,"e":0.28571,"k":"falling","v":0.23214,"x":0.72768,"p":[[0,31,0.0,0.59822,0.35792,0.25001,0.71429,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,5,0,0,1,0,0,1,0,0,3,0,0,4,0,10],[4,31,0.129,0.64732,0.33309,0.28571,0.78564,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,9,0,0,4,0,0,0,0,0,1,0,0,4,0,12],[8,31,0.2581,0.63838,0.3369,0.39286,0.71429,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,2,0,0,6,0,0,1,0,0,2,0,0,4,0,11],[12,31,0.3871,0.66071,0.3549,0.28571,0.71429,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,6,0,0,2,0,0,0,0,0,4,0,0,0,0,15],[16,31,0.5161,0.72768,0.35778,0.28571,1.0,1.0,0.14286,1.0,0,19,0,0,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,19],[20,31,0.6452,0.61607,0.37701,0.14286,0.78571,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,3,0,0,3,0,0,0,0,0,1,0,0,3,0,13],[24,31,0.7742,0.50893,0.35881,0.14286,0.42857,1.0,0.0,1.0,1,9,0,1,0,9,0,0,5,0,0,4,0,0,1,0,0,2,0,0,1,0,9],[28,31,0.9032,0.45089,0.37983,0.14286,0.28571,1.0,0.0,1.0,2,9,0,2,0,13,0,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,9],[31,31,1.0,0.23214,0.18471,0.14286,0.14288,0.28571,0.0,1.0,4,1,0,4,0,13,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"41c235f581db9256","q":"Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.\n\nNote: Two integers are relatively prime if no positive integer other than  $1$  divides both numbers.\n\n[i]Proposed by ABCDE and Tristan Shin","t":[{"b":3,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,14,0.0,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[4,14,0.2857,0.97322,0.1259,1.0,1.0,1.0,0.286,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,14,0.5714,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,14,0.8571,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[14,14,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.71429,"k":"flat","v":0.97768,"x":1.0,"p":[[0,6,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,6,0.6667,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[6,6,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"e1396cb36823257b","q":"Let  $AXYZB$  be a convex pentagon inscribed in a semicircle with diameter  $AB$ , and let  $K$  be the foot of the altitude from  $Y$  to  $AB$ . Let  $O$  denote the midpoint of  $AB$  and  $L$  be the intersection of  $XZ$  with  $YO$ . Select a point  $M$  on line  $KL$  with  $MA=MB$  , and finally, let  $I$  be the reflection of  $O$  across  $XZ$ .\nProve that if quadrilateral  $XKOZ$  is cyclic then so is quadrilateral  $YOMI$ .\n\n*Proposed by Evan Chen*","t":[{"b":1,"e":0.14286,"k":"falling","v":0.09374,"x":0.87946,"p":[[0,55,0.0,0.6696,0.21853,0.57143,0.71429,0.85704,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,13,0,0,7,0,2],[4,55,0.0727,0.81246,0.20344,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,14,0,9],[8,55,0.1455,0.83928,0.13243,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,15,0,8],[12,55,0.2182,0.83927,0.15467,0.71429,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,11,0,11],[16,55,0.2909,0.87946,0.11356,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,14,0,12],[20,55,0.3636,0.79016,0.2743,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,1,0,0,0,0,0,1,0,0,6,0,0,8,0,13],[24,55,0.4364,0.74542,0.27853,0.67857,0.85707,1.0,0.0,1.0,2,9,0,2,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,0,10,0,9],[28,55,0.5091,0.64285,0.34993,0.39287,0.71429,1.0,0.0,1.0,4,9,0,4,0,2,0,0,2,0,0,2,0,0,1,0,0,7,0,0,5,0,9],[32,55,0.5818,0.74549,0.23075,0.57143,0.85707,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,0,10,0,7],[36,55,0.6545,0.75889,0.2156,0.71429,0.857,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,11,0,6],[40,55,0.7273,0.65177,0.30291,0.57143,0.71429,0.85714,0.0,1.0,4,4,0,4,0,1,0,0,0,0,0,1,0,0,5,0,0,8,0,0,9,0,4],[44,55,0.8,0.58925,0.34579,0.28571,0.71429,0.85704,0.0,1.0,5,6,0,5,0,2,0,0,2,0,0,1,0,0,4,0,0,7,0,0,5,0,6],[48,55,0.8727,0.43303,0.36331,0.0,0.42859,0.85704,0.0,1.0,10,1,0,10,0,2,0,0,3,0,0,2,0,0,2,0,0,4,0,0,8,0,1],[52,55,0.9455,0.35712,0.3273,0.0,0.35714,0.71429,0.0,0.85714,12,0,0,12,0,2,0,0,2,0,0,3,0,0,3,0,0,7,0,0,3,0,0],[55,55,1.0,0.09374,0.22189,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.3839,"x":0.83479,"p":[[0,58,0.0,0.73211,0.13246,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,13,0,0,11,0,1],[4,58,0.069,0.82587,0.17766,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,10,0,11],[8,58,0.1379,0.83479,0.19923,0.82132,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,13,0,11],[12,58,0.2069,0.82142,0.18898,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,0,7,0,12],[16,58,0.2759,0.82587,0.19801,0.82132,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,0,14,0,10],[20,58,0.3448,0.80804,0.18423,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,0,13,0,8],[24,58,0.4138,0.71873,0.23003,0.57132,0.78564,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,7,0,0,11,0,5],[28,58,0.4828,0.82587,0.15461,0.71429,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,11,0,10],[32,58,0.5517,0.76784,0.17768,0.71429,0.78564,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,10,0,0,10,0,6],[36,58,0.6207,0.7991,0.17445,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,11,0,8],[40,58,0.6897,0.71425,0.23147,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,12,0,0,8,0,5],[44,58,0.7586,0.68747,0.29546,0.5354,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,1,0,0,2,0,0,2,0,0,7,0,0,7,0,8],[48,58,0.8276,0.71868,0.24872,0.571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,1,0,0,7,0,0,7,0,0,3,0,10],[52,58,0.8966,0.65623,0.29421,0.42859,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,2,0,0,5,0,0,3,0,0,6,0,0,6,0,7],[56,58,0.9655,0.49104,0.32524,0.24999,0.57121,0.71429,0.0,1.0,6,3,0,6,0,2,0,0,4,0,0,2,0,0,4,0,0,9,0,0,2,0,3],[58,58,1.0,0.3839,0.33203,0.0,0.35714,0.71429,0.0,1.0,10,1,0,10,0,3,0,0,3,0,0,2,0,0,4,0,0,6,0,0,3,0,1]]}]},{"i":"232d38bbf4b1e37d","q":"Julian and Johan are playing a game with an even number, say $2 n$, of cards ( $n \\in \\mathbb{Z}_{>0}$ ). On each card, there is a positive integer. The cards are shuffled and laid out in a row on the table with the numbers visible. A player whose turn it is may take either the leftmost card or the rightmost card. The players take turns alternately.\nJohan starts, so Julian picks the last card. Johan's score is the sum of the numbers on the $n$ cards he has picked, and the same goes for Julian. Prove that Johan can always achieve a score that is at least as high as Julian's.","t":[{"b":2,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,10,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,10,0.4,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,10,0.8,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[10,10,1.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,9,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,9,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,9,0.8889,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[9,9,1.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30]]}]},{"i":"87fae07eb85e4ba3","q":"Let  $G$  be a graph on  $n\\geq 6$  vertices and every vertex is of degree at least 3. If  $C_{1}, C_{2}, \\dots, C_{k}$  are all the cycles in  $G$ , determine all possible values of  $\\gcd(|C_{1}|, |C_{2}|, \\dots, |C_{k}|)$  where  $|C|$  denotes the number of vertices in the cycle  $C$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.29679,"x":0.5983,"p":[[0,38,0.0,0.3125,0.10374,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[4,38,0.1053,0.55811,0.24327,0.39286,0.4286,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,10,0,0,2,0,0,4,0,0,5,0,3],[8,38,0.2105,0.57584,0.25874,0.39286,0.4998,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,8,0,0,5,0,0,3,0,0,2,0,6],[12,38,0.3158,0.5983,0.27287,0.28893,0.571,0.85704,0.2857,1.0,0,7,0,0,0,0,0,0,9,0,0,6,0,0,3,0,0,5,0,0,2,0,7],[16,38,0.4211,0.54909,0.23986,0.42857,0.4286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,12,0,0,3,0,0,3,0,0,3,0,4],[20,38,0.5263,0.50885,0.25249,0.2857,0.42857,0.71429,0.14,1.0,0,4,0,0,0,1,0,0,11,0,0,8,0,0,2,0,0,5,0,0,1,0,4],[24,38,0.6316,0.433,0.2157,0.2857,0.42857,0.571,0.0,1.0,1,1,0,1,0,2,0,0,11,0,0,8,0,0,4,0,0,4,0,0,1,0,1],[28,38,0.7368,0.45531,0.2065,0.28571,0.42857,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,9,0,0,3,0,0,2,0,0,3,0,1],[32,38,0.8421,0.3705,0.14217,0.2857,0.28571,0.4286,0.14286,0.71429,0,0,0,0,0,3,0,0,14,0,0,10,0,0,3,0,0,2,0,0,0,0,0],[36,38,0.9474,0.39718,0.17035,0.2857,0.42857,0.4286,0.14,0.857,0,0,0,0,0,4,0,0,10,0,0,11,0,0,4,0,0,2,0,0,1,0,0],[38,38,1.0,0.29679,0.11036,0.25,0.2857,0.42857,0.14,0.5,0,0,0,0,0,8,0,0,14,0,0,9,0,1,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.2991,"x":0.70533,"p":[[0,37,0.0,0.2991,0.10326,0.2857,0.28571,0.28571,0.1429,0.85714,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,37,0.1081,0.70533,0.2695,0.53539,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,5,0,0,3,0,11],[8,37,0.2162,0.60264,0.26662,0.28571,0.57143,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,4,0,0,6,0,0,3,0,0,4,0,6],[12,37,0.3243,0.59371,0.22618,0.42857,0.57121,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,9,0,0,5,0,0,5,0,0,5,0,3],[16,37,0.4324,0.52675,0.24073,0.28571,0.42857,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,7,0,0,5,0,0,1,0,0,6,0,2],[20,37,0.5405,0.49104,0.24983,0.2857,0.42859,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,12,0,0,7,0,0,6,0,0,0,0,0,2,0,4],[24,37,0.6486,0.47763,0.1842,0.28571,0.4286,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,8,0,0,10,0,0,8,0,0,3,0,0,1,0,1],[28,37,0.7568,0.37926,0.13678,0.2857,0.42857,0.4286,0.14,0.71429,0,0,0,0,0,3,0,0,12,0,0,11,0,0,5,0,0,1,0,0,0,0,0],[32,37,0.8649,0.36158,0.16742,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,1,0,0,19,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[36,37,0.973,0.45086,0.20233,0.28571,0.42857,0.571,0.14286,1.0,0,1,0,0,0,1,0,0,13,0,0,7,0,0,6,0,0,2,0,0,2,0,1],[37,37,1.0,0.41962,0.16724,0.28571,0.42857,0.4286,0.0,0.857,1,0,0,1,0,0,0,0,11,0,0,13,0,0,3,0,0,3,0,0,1,0,0]]}]},{"i":"5b06a87ff26b87d3","q":"A triangle is called a parabolic triangle if its vertices lie on a parabola $y=x^{2}$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $\\left(2^{n} m\\right)^{2}$ \u3002","t":[{"b":3,"e":0.71429,"k":"flat","v":0.84372,"x":0.98661,"p":[[0,65,0.0,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[4,65,0.0615,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[8,65,0.1231,0.93304,0.15146,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[12,65,0.1846,0.96427,0.08754,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,65,0.2462,0.95533,0.10981,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[20,65,0.3077,0.87942,0.20244,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,2,0,22],[24,65,0.3692,0.87944,0.20862,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,3,0,22],[28,65,0.4308,0.87945,0.2086,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,20],[32,65,0.4923,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[36,65,0.5538,0.91518,0.1984,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,3,0,25],[40,65,0.6154,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[44,65,0.6769,0.90176,0.18368,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,3,0,23],[48,65,0.7385,0.9241,0.15561,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,23],[52,65,0.8,0.84372,0.21833,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,5,0,18],[56,65,0.8615,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,65,0.9231,0.94197,0.11214,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,22],[64,65,0.9846,0.94196,0.13767,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[65,65,1.0,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24]]},{"b":4,"e":1.0,"k":"flat","v":0.85265,"x":1.0,"p":[[0,79,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,79,0.0506,0.96426,0.09456,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,79,0.1013,0.9375,0.19212,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[12,79,0.1519,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,79,0.2025,0.91518,0.21086,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,26],[20,79,0.2532,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[24,79,0.3038,0.92411,0.19228,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[28,79,0.3544,0.8839,0.18017,0.85711,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,20],[32,79,0.4051,0.93302,0.15969,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[36,79,0.4557,0.88392,0.22142,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[40,79,0.5063,0.85265,0.25126,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,1,0,0,2,0,22],[44,79,0.557,0.94195,0.12303,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[48,79,0.6076,0.89731,0.20279,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,1,0,24],[52,79,0.6582,0.9375,0.15947,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,25],[56,79,0.7089,0.91516,0.20475,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,26],[60,79,0.7595,0.89731,0.17584,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,22],[64,79,0.8101,0.90624,0.18768,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,24],[68,79,0.8608,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,79,0.9114,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,79,0.962,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[79,79,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9470e650d48dd1cb","q":"In trapezium $A B C D$ is $A B \\| C D$. Let $M$ be the midpoint of diagonal $A C$. Assume that triangles $A B M$ and $A C D$ have the same area. Prove that $D M \\| B C$.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.59821,"x":0.72321,"p":[[0,42,0.0,0.64284,0.26001,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,11,0,0,4,0,0,5,0,0,2,0,8],[4,42,0.0952,0.66518,0.18423,0.67857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,20,0,0,2,0,2],[8,42,0.1905,0.59821,0.2299,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,2,0,0,0,0,0,2,0,0,22,0,0,0,0,1],[12,42,0.2857,0.69195,0.11905,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,27,0,0,2,0,0],[16,42,0.381,0.70089,0.05486,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,0,1,0,0],[20,42,0.4762,0.69643,0.06916,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,1,0,0],[24,42,0.5714,0.72321,0.13803,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,23,0,0,4,0,2],[28,42,0.6667,0.71427,0.0875,0.71429,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,0,0,0,2],[32,42,0.7619,0.70982,0.12619,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,25,0,0,3,0,1],[36,42,0.8571,0.72321,0.11812,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,24,0,0,3,0,2],[40,42,0.9524,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[42,42,1.0,0.70534,0.03463,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.64286,"x":0.70088,"p":[[0,13,0.0,0.64286,0.27199,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,3,0,0,4,0,0,3,0,0,10,0,0,2,0,7],[4,13,0.3077,0.64732,0.16745,0.67857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,23,0,0,0,0,1],[8,13,0.6154,0.70088,0.08268,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,24,0,0,1,0,1],[12,13,0.9231,0.69643,0.08564,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,26,0,0,0,0,1],[13,13,1.0,0.70088,0.04168,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0]]}]},{"i":"f40d8cf498abffd5","q":"Let  $ ABC$  be an acute-angled triangle. The lines  $ L_{A}$ ,  $ L_{B}$  and  $ L_{C}$  are constructed through the vertices  $ A$ ,  $ B$  and  $ C$  respectively according the following prescription: Let  $ H$  be the foot of the altitude drawn from the vertex  $ A$  to the side  $ BC$ ; let  $ S_{A}$  be the circle with diameter  $ AH$ ; let  $ S_{A}$  meet the sides  $ AB$  and  $ AC$  at  $ M$  and  $ N$  respectively, where  $ M$  and  $ N$  are distinct from  $ A$ ; then let  $ L_{A}$  be the line through  $ A$  perpendicular to  $ MN$ . The lines  $ L_{B}$  and  $ L_{C}$  are constructed similarly. Prove that the lines  $ L_{A}$ ,  $ L_{B}$  and  $ L_{C}$  are concurrent.","t":[{"b":4,"e":0.14286,"k":"falling","v":0.30357,"x":0.86161,"p":[[0,49,0.0,0.86161,0.16554,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,4,0,16],[4,49,0.0816,0.71428,0.23145,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,10,0,0,3,0,9],[8,49,0.1633,0.62947,0.28539,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,7,0,0,0,0,0,8,0,0,2,0,8],[12,49,0.2449,0.70982,0.27078,0.42859,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,10,0,0,0,0,0,6,0,0,2,0,12],[16,49,0.3265,0.69642,0.26427,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,8,0,0,2,0,0,8,0,0,2,0,10],[20,49,0.4082,0.75892,0.3163,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,3,0,0,3,0,0,0,0,0,4,0,0,1,0,18],[24,49,0.4898,0.70089,0.27976,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,7,0,0,0,0,0,6,0,0,5,0,10],[28,49,0.5714,0.62499,0.27607,0.42857,0.49979,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,11,0,0,1,0,0,4,0,0,3,0,8],[32,49,0.6531,0.36598,0.28565,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,9,0,0,6,0,0,3,0,0,3,0,0,2,0,0,5,0,0],[36,49,0.7347,0.48214,0.3531,0.14286,0.35714,0.85714,0.0,1.0,2,6,0,2,0,9,0,0,5,0,0,3,0,0,1,0,0,2,0,0,4,0,6],[40,49,0.8163,0.39275,0.288,0.14286,0.42857,0.57111,0.0,1.0,4,2,0,4,0,7,0,0,4,0,0,8,0,0,2,0,0,3,0,0,2,0,2],[44,49,0.898,0.34374,0.31914,0.14286,0.21431,0.57143,0.0,1.0,6,1,0,6,0,10,0,0,5,0,0,2,0,0,2,0,0,0,0,0,6,0,1],[48,49,0.9796,0.30357,0.30252,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,7,0,0],[49,49,1.0,0.30802,0.27688,0.14286,0.14286,0.57111,0.0,0.85714,4,0,0,4,0,15,0,0,3,0,0,1,0,0,4,0,0,1,0,0,4,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.63393,"x":0.85268,"p":[[0,78,0.0,0.85268,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,8,0,13],[4,78,0.0513,0.70089,0.24578,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,8,0,0,2,0,10],[8,78,0.1026,0.66518,0.30011,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,3,0,0,8,0,0,1,0,0,4,0,0,4,0,10],[12,78,0.1538,0.63839,0.26241,0.42857,0.42859,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,15,0,0,1,0,0,3,0,0,2,0,9],[16,78,0.2051,0.75,0.24223,0.64286,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,10,0,0,2,0,12],[20,78,0.2564,0.70089,0.23517,0.42857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,5,0,0,6,0,8],[24,78,0.3077,0.63393,0.25985,0.42857,0.71429,0.75,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,11,0,0,0,0,0,10,0,0,1,0,7],[28,78,0.359,0.78125,0.22583,0.67857,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,8,0,0,2,0,14],[32,78,0.4103,0.64286,0.20516,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,12,0,0,1,0,5],[36,78,0.4615,0.72768,0.2212,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,9,0,0,4,0,9],[40,78,0.5128,0.74107,0.23808,0.42857,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,8,0,0,2,0,12],[44,78,0.5641,0.76786,0.21354,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,10,0,0,4,0,11],[48,78,0.6154,0.75,0.25505,0.42857,0.78564,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,5,0,0,2,0,14],[52,78,0.6667,0.67856,0.25253,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,8,0,0,2,0,9],[56,78,0.7179,0.72321,0.17105,0.71429,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,21,0,0,0,0,6],[60,78,0.7692,0.66076,0.15458,0.67856,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,22,0,0,0,0,2],[64,78,0.8205,0.70536,0.11259,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,27,0,0,0,0,2],[68,78,0.8718,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[72,78,0.9231,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[76,78,0.9744,0.71428,5e-05,0.71429,0.71429,0.71429,0.714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[78,78,1.0,0.71415,0.00075,0.71429,0.71429,0.71429,0.71,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0]]}]},{"i":"6dc4e57f743f9f1a","q":"Given that  $a_1, a_2, \\dots, a_{10}$  are positive real numbers, determine the smallest possible value of \\[\\sum \\limits_{i = 1}^{10} \\left\\lfloor \\frac{7a_i}{a_i+a_{i+1}}\\right\\rfloor\\] where we define  $a_{11} = a_1$ .\n\n*Proposed by Sutanay Bhattacharya*","t":[{"b":2,"e":0.0,"k":"falling","v":0.06697,"x":0.75892,"p":[[0,83,0.0,0.61161,0.32972,0.28571,0.85714,0.85714,0.0,1.0,4,2,3,4,0,0,0,0,7,0,0,0,0,0,1,0,0,3,0,0,15,0,2],[4,83,0.0482,0.6517,0.31544,0.28571,0.85714,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,7,0,0,1,0,0,0,0,0,1,0,0,17,0,3],[8,83,0.0964,0.75892,0.2299,0.85714,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,24,0,2],[12,83,0.1446,0.58928,0.29827,0.28571,0.64286,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,15,0,0,1,0,0,0,0,0,0,0,0,13,0,3],[16,83,0.1928,0.60268,0.30249,0.28571,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,13,0,0,1,0,0,0,0,0,0,0,0,14,0,3],[20,83,0.241,0.59376,0.30327,0.28571,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,0,15,0,2],[24,83,0.2892,0.625,0.34022,0.28571,0.85714,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,10,0,0,0,0,0,0,0,0,2,0,0,10,0,7],[28,83,0.3373,0.58928,0.3067,0.28571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,11,0,0,0,0,0,1,0,0,2,0,0,12,0,3],[32,83,0.3855,0.55357,0.31288,0.28571,0.5,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,14,0,0,0,0,0,0,0,0,1,0,0,14,0,1],[36,83,0.4337,0.48214,0.3549,0.2857,0.28571,0.85714,0.0,1.0,5,3,0,5,0,2,0,0,11,0,0,1,0,0,0,0,0,0,0,0,10,0,3],[40,83,0.4819,0.46428,0.33311,0.28571,0.28586,0.85714,0.0,1.0,6,1,0,6,0,0,0,0,11,0,0,2,0,0,1,0,0,1,0,0,10,0,1],[44,83,0.5301,0.52231,0.35823,0.28571,0.64264,0.85714,0.0,1.0,6,2,0,6,0,1,0,0,8,0,0,0,0,0,1,0,0,2,0,0,12,0,2],[48,83,0.5783,0.52679,0.34523,0.28571,0.35714,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,11,0,0,3,0,0,0,0,0,1,0,0,5,0,7],[52,83,0.6265,0.5,0.3677,0.24999,0.28571,0.85714,0.0,1.0,6,2,0,6,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0,13,0,2],[56,83,0.6747,0.55356,0.34764,0.2857,0.71407,0.85714,0.0,1.0,3,3,0,3,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,0,13,0,3],[60,83,0.7229,0.63392,0.33491,0.28571,0.85714,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,7,0,0,0,0,0,1,0,0,1,0,0,15,0,4],[64,83,0.7711,0.38839,0.33548,0.14286,0.28571,0.85714,0.0,1.0,7,2,0,7,0,2,0,0,13,0,0,1,0,0,0,0,0,0,0,0,7,0,2],[68,83,0.8193,0.52679,0.36147,0.2857,0.4286,0.85714,0.0,1.0,5,5,0,5,0,1,0,0,9,0,0,2,0,0,1,0,0,0,0,0,9,0,5],[72,83,0.8675,0.49552,0.38296,0.0,0.57121,0.85714,0.0,1.0,9,3,0,9,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,0,10,0,3],[76,83,0.9157,0.375,0.37415,0.0,0.28571,0.85714,0.0,1.0,11,2,0,11,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,0,8,0,2],[80,83,0.9639,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[83,83,1.0,0.09375,0.13175,0.0,0.0,0.1786,0.0,0.42857,20,0,0,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.53569,"x":0.87509,"p":[[0,99,0.0,0.53569,0.30929,0.28571,0.5712,0.85714,0.0,1.0,2,3,1,2,0,2,0,0,10,0,0,1,0,0,4,0,0,2,0,0,8,0,3],[4,99,0.0404,0.61607,0.3223,0.28571,0.78571,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,12,0,0,2,0,0,0,0,0,1,0,0,9,0,7],[8,99,0.0808,0.69642,0.28291,0.28571,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,0,18,0,4],[12,99,0.1212,0.78571,0.24743,0.85714,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,18,0,8],[16,99,0.1616,0.72768,0.27516,0.39286,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,1,0,0,0,0,0,1,0,0,15,0,7],[20,99,0.202,0.625,0.33456,0.28571,0.85714,0.85714,0.0,1.0,1,5,0,1,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0,14,0,5],[24,99,0.2424,0.78571,0.21724,0.85714,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,23,0,3],[28,99,0.2828,0.7991,0.21975,0.85711,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,19,0,7],[32,99,0.3232,0.74107,0.24598,0.64286,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,4,0,0,0,0,0,2,0,0,17,0,5],[36,99,0.3636,0.76339,0.23854,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,3,0,0,15,0,7],[40,99,0.404,0.76784,0.23892,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,2,0,0,16,0,7],[44,99,0.4444,0.8214,0.18561,0.85714,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,20,0,7],[48,99,0.4848,0.87509,0.16655,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,14,0,14],[52,99,0.5253,0.84375,0.20935,0.85714,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,16,0,12],[56,99,0.5657,0.72321,0.2878,0.39286,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,7,0,0,1,0,0,1,0,0,0,0,0,14,0,8],[60,99,0.6061,0.79018,0.23686,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,3,0,0,13,0,10],[64,99,0.6465,0.76339,0.26392,0.74989,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,15,0,9],[68,99,0.6869,0.83929,0.22799,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,10,0,15],[72,99,0.7273,0.81696,0.21498,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,18,0,9],[76,99,0.7677,0.82588,0.23888,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,12,0,14],[80,99,0.8081,0.67411,0.27947,0.28571,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,10,0,0,1,0,0,0,0,0,2,0,0,15,0,4],[84,99,0.8485,0.62945,0.25966,0.28571,0.78571,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,11,0,0,0,0,0,2,0,0,3,0,0,16,0,0],[88,99,0.8889,0.63839,0.28568,0.28571,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,0,0,0,15,0,3],[92,99,0.9293,0.71875,0.27545,0.28571,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,18,0,5],[96,99,0.9697,0.55803,0.29093,0.28571,0.28571,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,14,0,1],[99,99,1.0,0.63392,0.2922,0.28571,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,15,0,3]]}]},{"i":"f70d5b403a6a8e70","q":"In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from   $1$  to  $n$  and let  $a_i$  be the number of friends of the  $i^{\\text{th}}$  resident. Suppose that\n\\[ \\sum_{i=1}^{n}a_i^2=n^2-n \\]\nLet  $k$  be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of  $k.$","t":[{"b":0,"e":1.0,"k":"rising","v":0.50445,"x":0.98214,"p":[[0,39,0.0,0.50445,0.31539,0.28571,0.4286,0.71429,0.0,1.0,4,4,0,4,0,2,0,0,6,0,0,5,0,0,1,0,0,8,0,0,2,0,4],[4,39,0.1026,0.95535,0.14033,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[8,39,0.2051,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,39,0.3077,0.89286,0.2342,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,24],[16,39,0.4103,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,39,0.5128,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[24,39,0.6154,0.92857,0.12877,0.96425,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,24],[28,39,0.7179,0.86607,0.17474,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,2,0,18],[32,39,0.8205,0.90179,0.19704,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,24],[36,39,0.9231,0.82588,0.24676,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,4,0,17],[39,39,1.0,0.75892,0.22429,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,13,0,0,0,0,12]]},{"b":5,"e":1.0,"k":"rising","v":0.50442,"x":1.0,"p":[[0,61,0.0,0.50442,0.26482,0.39286,0.571,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,2,0,0,7,0,0,4,0,0,10,0,0,2,0,1],[4,61,0.0656,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,61,0.1311,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,61,0.1967,0.9375,0.16728,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[16,61,0.2623,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,61,0.3279,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,61,0.3934,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[28,61,0.459,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[32,61,0.5246,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[36,61,0.5902,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,61,0.6557,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,61,0.7213,0.94642,0.11714,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[48,61,0.7869,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[52,61,0.8525,0.94642,0.11714,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[56,61,0.918,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[60,61,0.9836,0.93304,0.13356,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,24],[61,61,1.0,0.9375,0.15126,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,26]]}]},{"i":"0542da2c818ad482","q":"Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by  $2?$","t":[{"b":4,"e":0.57143,"k":"flat","v":0.50446,"x":0.80357,"p":[[0,40,0.0,0.73212,0.20126,0.57143,0.78571,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,12,0,0,2,0,0,10,0,6],[4,40,0.1,0.65625,0.31311,0.57143,0.85714,0.85714,0.0,1.0,5,3,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,14,0,3],[8,40,0.2,0.70534,0.27419,0.57143,0.85714,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,13,0,5],[12,40,0.3,0.80357,0.23891,0.85711,0.85714,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,17,0,8],[16,40,0.4,0.60714,0.35892,0.53571,0.64286,0.85714,0.0,1.0,7,6,0,7,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,9,0,6],[20,40,0.5,0.52678,0.38372,0.0,0.64286,0.85714,0.0,1.0,9,3,0,9,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,0,11,0,3],[24,40,0.6,0.50446,0.39927,0.0,0.64286,0.85714,0.0,1.0,11,3,0,11,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,11,0,3],[28,40,0.7,0.56249,0.39759,0.0,0.71429,0.85714,0.0,1.0,10,5,0,10,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,10,0,5],[32,40,0.8,0.74999,0.20825,0.57143,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,7,0,0,3,0,0,17,0,3],[36,40,0.9,0.76786,0.17768,0.57143,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,1,0,0,15,0,5],[40,40,1.0,0.61605,0.13571,0.57143,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,0,2,0,0,6,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.7366,"p":[[0,36,0.0,0.7366,0.18595,0.57143,0.78564,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,3,0,0,11,0,5],[4,36,0.1111,0.72768,0.25092,0.57143,0.78571,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,0,10,0,6],[8,36,0.2222,0.37946,0.40186,0.0,0.14285,0.85714,0.0,1.0,16,2,0,16,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,8,0,2],[12,36,0.3333,0.45536,0.40317,0.0,0.5,0.85714,0.0,1.0,12,2,0,12,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,12,0,2],[16,36,0.4444,0.37946,0.39383,0.0,0.28571,0.85704,0.0,1.0,16,1,0,16,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,9,0,1],[20,36,0.5556,0.36607,0.40554,0.0,0.07143,0.85714,0.0,1.0,16,3,0,16,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,0,7,0,3],[24,36,0.6667,0.28125,0.35262,0.0,0.0,0.57143,0.0,1.0,18,1,0,18,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,5,0,1],[28,36,0.7778,0.15179,0.3008,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0],[32,36,0.8889,0.09375,0.22476,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0],[36,36,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bdedd9b62f7c733d","q":"Let $ABC$ be a non-isosceles triangle. Let $\\omega$ be the incircle and $I$ its center. We denote $M, N, P$ as the points of tangency of $\\omega$ with the sides $[BC], [CA], [AB]$. Let $J$ be the intersection point between $(MN)$ and $(IC)$. The line $(PJ)$ intersects $\\omega$ again at $K$. Show that\na) $CKIP$ is cyclic;\nb) $(CI)$ is the bisector of $\\widehat{PC K}$.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.08036,"x":0.4329,"p":[[0,30,0.0,0.4329,0.27299,0.28571,0.42857,0.57143,0.0,1.0,3,2,1,3,0,3,0,0,8,0,0,8,0,0,3,0,0,2,0,0,3,0,2],[4,30,0.1333,0.37051,0.26208,0.2857,0.28571,0.42858,0.0,1.0,3,3,0,3,0,2,0,0,18,0,0,2,0,0,3,0,0,0,0,0,1,0,3],[8,30,0.2667,0.16509,0.13884,0.0,0.2143,0.28571,0.0,0.42857,12,0,0,12,0,4,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.24551,0.17577,0.10714,0.28571,0.28571,0.0,0.71429,8,0,0,8,0,1,0,0,19,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[16,30,0.5333,0.25884,0.23812,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,4,0,0,13,0,0,0,0,0,4,0,0,1,0,0,0,0,1],[20,30,0.6667,0.20088,0.25717,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,1,0,0,10,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[24,30,0.8,0.125,0.12242,0.0,0.14286,0.1786,0.0,0.4286,13,0,0,13,0,11,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.12044,0.11902,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,12,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.32132,"x":1.0,"p":[[0,90,0.0,0.44646,0.28064,0.2857,0.28571,0.57143,0.0,1.0,1,4,1,1,0,4,0,0,12,0,0,4,0,0,5,0,0,0,0,0,2,0,4],[4,90,0.0444,0.40619,0.28368,0.2857,0.28571,0.57025,0.0,1.0,2,3,0,2,0,5,0,0,13,0,0,3,0,0,2,0,0,2,0,0,2,0,3],[8,90,0.0889,0.45089,0.32165,0.2857,0.28571,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,16,0,0,1,0,0,1,0,0,0,0,0,4,0,5],[12,90,0.1333,0.3973,0.31889,0.14289,0.28571,0.60682,0.0,1.0,3,4,0,3,0,6,0,0,14,0,0,0,0,0,1,0,0,1,0,0,3,0,4],[16,90,0.1778,0.32132,0.28575,0.14286,0.2857,0.28571,0.0,1.0,5,2,0,5,0,7,0,0,13,0,0,1,0,0,1,0,0,0,0,0,3,0,2],[20,90,0.2222,0.92411,0.18552,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,26],[24,90,0.2667,0.78115,0.33518,0.5354,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,21],[28,90,0.3111,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[32,90,0.3556,0.95089,0.18074,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[36,90,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,90,0.4444,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[44,90,0.4889,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,90,0.5333,0.9108,0.21649,0.965,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,24],[52,90,0.5778,0.93749,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[56,90,0.6222,0.90625,0.1411,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,20],[60,90,0.6667,0.92857,0.13363,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,23],[64,90,0.7111,0.89732,0.14827,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,10,0,17],[68,90,0.7556,0.89732,0.1931,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,22],[72,90,0.8,0.88392,0.13092,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,7,0,16],[76,90,0.8444,0.93304,0.13825,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[80,90,0.8889,0.76337,0.13176,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,0,13,0,2],[84,90,0.9333,0.70089,0.15303,0.71429,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,23,0,0,1,0,3],[88,90,0.9778,0.70075,0.06546,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,1,0,0],[90,90,1.0,0.68735,0.08326,0.71429,0.71429,0.71429,0.42857,0.8571,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,26,0,0,1,0,0]]}]},{"i":"89151bf7415aa0bc","q":"Let $ABC$ be a non-right triangle such that $AB < AC$. We denote $H$ as the projection of $A$ onto $(BC)$, and $E, F$ as the projections of $H$ onto $(AB)$ and $(AC)$, respectively. The line $(EF)$ intersects $(BC)$ at point $D$. Consider the semicircle with diameter $[CD]$ located in the same half-plane delimited by $(CD)$ as $A$. Let $K$ be the point on this semicircle that projects onto $B$. Show that $(DK)$ is tangent to the circle $KEF$.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.21865,"x":0.53124,"p":[[0,100,0.0,0.53124,0.32583,0.2857,0.57121,0.78571,0.0,1.0,2,8,0,2,0,4,0,0,6,0,0,3,0,0,7,0,0,2,0,0,0,0,8],[4,100,0.04,0.40622,0.20548,0.2857,0.42857,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,11,0,0,5,0,0,10,0,0,1,0,0,0,0,1],[8,100,0.08,0.40177,0.26591,0.14286,0.57121,0.57143,0.0,1.0,6,1,0,6,0,3,0,0,5,0,0,1,0,0,13,0,0,3,0,0,0,0,1],[12,100,0.12,0.38382,0.25119,0.2857,0.28571,0.57143,0.0,1.0,4,2,0,4,0,3,0,0,10,0,0,4,0,0,8,0,0,1,0,0,0,0,2],[16,100,0.16,0.35265,0.24737,0.14286,0.35714,0.57143,0.0,1.0,6,1,0,6,0,4,0,0,6,0,0,5,0,0,9,0,0,1,0,0,0,0,1],[20,100,0.2,0.33926,0.23074,0.10714,0.35714,0.57143,0.0,0.57143,8,0,0,8,0,1,0,0,7,0,0,3,0,0,13,0,0,0,0,0,0,0,0],[24,100,0.24,0.27231,0.27515,0.0,0.2857,0.35704,0.0,1.0,10,2,0,10,0,5,0,0,9,0,0,0,0,0,6,0,0,0,0,0,0,0,2],[28,100,0.28,0.30804,0.18249,0.14286,0.28571,0.4286,0.0,0.57143,4,0,0,4,0,5,0,0,12,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[32,100,0.32,0.29911,0.21237,0.14286,0.28571,0.46429,0.0,0.71429,7,0,0,7,0,3,0,0,11,0,0,3,0,0,7,0,0,1,0,0,0,0,0],[36,100,0.36,0.21865,0.19558,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,4,0,0,10,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[40,100,0.4,0.25445,0.25933,0.0,0.2143,0.46418,0.0,1.0,12,1,0,12,0,4,0,0,6,0,0,2,0,0,7,0,0,0,0,0,0,0,1],[44,100,0.44,0.2856,0.19235,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,7,0,0,9,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[48,100,0.48,0.366,0.17465,0.2857,0.28571,0.57111,0.0,0.57143,2,0,0,2,0,3,0,0,13,0,0,3,0,0,11,0,0,0,0,0,0,0,0],[52,100,0.52,0.33035,0.18362,0.2857,0.28571,0.4642,0.0,0.71429,3,0,0,3,0,4,0,0,14,0,0,3,0,0,7,0,0,1,0,0,0,0,0],[56,100,0.56,0.31693,0.15861,0.2857,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,4,0,0,18,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[60,100,0.6,0.25893,0.1448,0.25,0.28571,0.28571,0.0,0.57143,5,0,0,5,0,3,0,0,19,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[64,100,0.64,0.28571,0.17496,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,6,0,0,13,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[68,100,0.68,0.25437,0.15047,0.24999,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,2,0,0,19,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[72,100,0.72,0.27677,0.17833,0.14286,0.2857,0.32143,0.0,0.57143,6,0,0,6,0,3,0,0,15,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[76,100,0.76,0.23214,0.16656,0.10714,0.2857,0.28571,0.0,0.57143,8,0,0,8,0,4,0,0,14,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[80,100,0.8,0.28571,0.19233,0.24999,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,1,0,0,15,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[84,100,0.84,0.31683,0.17369,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,2,0,0,20,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[88,100,0.88,0.33033,0.14029,0.2857,0.28571,0.28571,0.14286,0.857,0,0,0,0,0,2,0,0,25,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[92,100,0.92,0.29464,0.13803,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,0,0,0,25,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[96,100,0.96,0.28125,0.08364,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,2,0,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[100,100,1.0,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.01786,"x":0.45538,"p":[[0,81,0.0,0.45538,0.30811,0.24999,0.42929,0.57143,0.0,1.0,5,4,1,5,0,3,0,0,4,0,0,5,0,0,8,0,0,2,0,0,1,0,4],[4,81,0.0494,0.28123,0.19717,0.21427,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,0,0,0,15,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[8,81,0.0988,0.27228,0.2153,0.0,0.28571,0.46418,0.0,0.57143,9,0,0,9,0,3,0,0,10,0,0,2,0,0,8,0,0,0,0,0,0,0,0],[12,81,0.1481,0.27678,0.17835,0.14286,0.28571,0.32143,0.0,0.57143,6,0,0,6,0,3,0,0,15,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[16,81,0.1975,0.26338,0.21459,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,9,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[20,81,0.2469,0.2366,0.21312,0.0,0.2857,0.32143,0.0,0.71429,11,0,0,11,0,3,0,0,10,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[24,81,0.2963,0.27232,0.18681,0.14286,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,3,0,0,17,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[28,81,0.3457,0.32588,0.20587,0.24999,0.28571,0.42857,0.0,1.0,3,1,0,3,0,5,0,0,14,0,0,3,0,0,6,0,0,0,0,0,0,0,1],[32,81,0.3951,0.2857,0.23689,0.0,0.28571,0.57143,0.0,0.71429,10,0,0,10,0,2,0,0,9,0,0,1,0,0,9,0,0,1,0,0,0,0,0],[36,81,0.4444,0.3125,0.21852,0.24999,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,1,0,0,14,0,0,1,0,0,7,0,0,2,0,0,0,0,0],[40,81,0.4938,0.31248,0.14911,0.2857,0.28571,0.32143,0.0,0.57143,3,0,0,3,0,1,0,0,20,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[44,81,0.5432,0.18304,0.17941,0.0,0.14288,0.28571,0.0,0.57143,13,0,0,13,0,4,0,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[48,81,0.5926,0.19192,0.21003,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,7,0,0,6,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[52,81,0.642,0.13839,0.20666,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,2,0,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[56,81,0.6914,0.16964,0.22428,0.0,0.07143,0.2857,0.0,1.0,16,1,0,16,0,3,0,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[60,81,0.7407,0.11161,0.18466,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[64,81,0.7901,0.10713,0.15563,0.0,0.0,0.2857,0.0,0.571,20,0,0,20,0,3,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[68,81,0.8395,0.14286,0.22303,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,0,0,0,0],[72,81,0.8889,0.14277,0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$ABC$ be a triangle and $I$ the center of its inscribed circle. The perpendicular bisector of $[\\mathrm{BC}]$ intersects $(A \\mathrm{I})$ at $S$ and $(BI)$ at $T$. Show that $C, I, S$ and $T$ are concyclic.","t":[{"b":5,"e":0.14286,"k":"falling","v":0.16518,"x":0.54464,"p":[[0,67,0.0,0.54464,0.30606,0.42857,0.42859,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,2,0,0,10,0,0,6,0,0,0,0,0,2,0,7],[4,67,0.0597,0.31697,0.30458,0.14286,0.28571,0.42857,0.0,1.0,7,4,0,7,0,7,0,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,4],[8,67,0.1194,0.33481,0.30848,0.14286,0.2143,0.57111,0.0,1.0,5,4,0,5,0,11,0,0,5,0,0,2,0,0,5,0,0,0,0,0,0,0,4],[12,67,0.1791,0.32142,0.34992,0.0,0.14286,0.5711,0.0,1.0,9,5,0,9,0,10,0,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,5],[16,67,0.2388,0.20982,0.24739,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[20,67,0.2985,0.16518,0.20858,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,6,0,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[24,67,0.3582,0.23213,0.24677,0.0,0.1429,0.28571,0.0,1.0,9,2,0,9,0,8,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[28,67,0.4179,0.17411,0.1665,0.0,0.14286,0.28571,0.0,0.57143,12,0,0,12,0,7,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,67,0.4776,0.22768,0.23107,0.0,0.14286,0.32143,0.0,1.0,10,1,0,10,0,8,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[36,67,0.5373,0.17411,0.13236,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.22768,0.23107,0.0,0.21428,0.28571,0.0,1.0,10,1,0,10,0,6,0,0,11,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[44,67,0.6567,0.16964,0.1448,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,9,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,67,0.7164,0.1875,0.14913,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,5,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,67,0.7761,0.1741,0.13235,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.21875,0.14279,0.14286,0.14288,0.28571,0.0,0.57143,4,0,0,4,0,13,0,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[60,67,0.8955,0.25884,0.19383,0.14286,0.2857,0.28571,0.0,0.85714,6,0,0,6,0,5,0,0,16,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[64,67,0.9552,0.22321,0.13803,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,11,0,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[67,67,1.0,0.22768,0.07873,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,11,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"falling","v":0.125,"x":0.63389,"p":[[0,108,0.0,0.63389,0.26712,0.42857,0.57143,1.0,0.0,1.0,1,9,1,1,0,0,0,0,3,0,0,5,0,0,13,0,0,0,0,0,1,0,9],[4,108,0.037,0.26784,0.29826,0.0,0.14286,0.42858,0.0,1.0,11,3,0,11,0,6,0,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,3],[8,108,0.0741,0.13393,0.13803,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,10,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,108,0.1111,0.17411,0.19799,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,10,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[16,108,0.1481,0.20534,0.29218,0.0,0.14286,0.2857,0.0,1.0,14,3,0,14,0,8,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[20,108,0.1852,0.16072,0.16269,0.0,0.14286,0.1786,0.0,0.71429,10,0,0,10,0,14,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[24,108,0.2222,0.16518,0.16015,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,10,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,108,0.2593,0.1517,0.19542,0.0,0.14286,0.2857,0.0,1.0,13,1,0,13,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,108,0.2963,0.15624,0.21533,0.0,0.14286,0.1786,0.0,1.0,14,1,0,14,0,10,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[36,108,0.3333,0.20535,0.19541,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,10,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[40,108,0.3704,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,108,0.4074,0.20536,0.2141,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,11,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[48,108,0.4444,0.14286,0.11294,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,12,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,108,0.4815,0.19196,0.14987,0.0,0.2143,0.28571,0.0,0.57143,9,0,0,9,0,7,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[56,108,0.5185,0.13393,0.12339,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,11,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,108,0.5556,0.18304,0.21199,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,9,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[64,108,0.5926,0.16518,0.20238,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,5,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[68,108,0.6296,0.18302,0.11966,0.14286,0.14286,0.2857,0.0,0.571,5,0,0,5,0,15,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,108,0.6667,0.19196,0.18766,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,16,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[76,108,0.7037,0.13384,0.15125,0.0,0.14286,0.17857,0.0,0.71429,13,0,0,13,0,11,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[80,108,0.7407,0.125,0.11152,0.0,0.14286,0.17857,0.0,0.28571,12,0,0,12,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,108,0.7778,0.14724,0.11565,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,14,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,108,0.8148,0.12946,0.19678,0.0,0.0,0.2857,0.0,1.0,17,1,0,17,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[92,108,0.8519,0.14286,0.11294,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,108,0.8889,0.15179,0.10677,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,14,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,108,0.9259,0.19634,0.11714,0.14286,0.14288,0.28571,0.0,0.42857,5,0,0,5,0,12,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[104,108,0.963,0.1875,0.09061,0.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$I$ be the center of the incircle of a triangle $ABC$. Let $D$ be the point diametrically opposite to $A$ on the circumcircle. Suppose that the point $E$ on the ray $[BA)$ and the point $F$ on the ray $[CA)$ satisfy the condition\n\n$$\nBE=CF=\\frac{AB+BC+CA}{2}.\n$$\n\nShow that $(EF) \\perp (DI)$.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.67409,"x":0.94196,"p":[[0,90,0.0,0.79464,0.27185,0.71429,0.85714,1.0,0.0,1.0,2,15,2,2,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,3,0,15],[4,90,0.0444,0.94196,0.1851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[8,90,0.0889,0.84374,0.18681,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,0,7,0,14],[12,90,0.1333,0.89286,0.2369,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,24],[16,90,0.1778,0.91071,0.17035,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,3,0,23],[20,90,0.2222,0.82588,0.27372,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,4,0,18],[24,90,0.2667,0.90625,0.15407,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,21],[28,90,0.3111,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[32,90,0.3556,0.91515,0.14229,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,7,0,21],[36,90,0.4,0.72767,0.38193,0.53539,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,5,0,17],[40,90,0.4444,0.77231,0.30901,0.67857,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,6,0,15],[44,90,0.4889,0.74106,0.38704,0.57132,1.0,1.0,0.0,1.0,6,19,1,6,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,19],[48,90,0.5333,0.79911,0.30693,0.67857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,18],[52,90,0.5778,0.79907,0.33287,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,20],[56,90,0.6222,0.91516,0.14668,0.85711,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[60,90,0.6667,0.78125,0.35532,0.71429,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,20],[64,90,0.7111,0.79911,0.31106,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,1,0,19],[68,90,0.7556,0.67409,0.35934,0.42857,0.85714,1.0,0.0,1.0,5,12,0,5,0,0,0,0,1,0,0,3,0,0,5,0,0,0,0,0,6,0,12],[72,90,0.8,0.80807,0.32061,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,21],[76,90,0.8444,0.82588,0.29825,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,21],[80,90,0.8889,0.82142,0.26246,0.71429,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,16],[84,90,0.9333,0.86159,0.25627,0.85714,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,20],[88,90,0.9778,0.88839,0.21349,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[90,90,1.0,0.87943,0.1434,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,11,0,15]]},{"b":6,"e":1.0,"k":"rising","v":0.76782,"x":0.97768,"p":[[0,139,0.0,0.7991,0.29203,0.71429,0.85714,1.0,0.0,1.0,3,15,3,3,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,6,0,15],[4,139,0.0288,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[8,139,0.0576,0.90177,0.14483,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,21],[12,139,0.0863,0.90625,0.17717,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,23],[16,139,0.1151,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[20,139,0.1439,0.92411,0.20511,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[24,139,0.1727,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[28,139,0.2014,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[32,139,0.2302,0.94643,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[36,139,0.259,0.9375,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[40,139,0.2878,0.94642,0.11156,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[44,139,0.3165,0.91964,0.18536,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,26],[48,139,0.3453,0.90177,0.22713,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,25],[52,139,0.3741,0.92411,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,21],[56,139,0.4029,0.88839,0.21349,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[60,139,0.4317,0.87946,0.2055,0.85711,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,19],[64,139,0.4604,0.76782,0.33647,0.57132,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3,0,18],[68,139,0.4892,0.82142,0.28349,0.78571,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,0,5,0,19],[72,139,0.518,0.86158,0.23006,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,2,0,21],[76,139,0.5468,0.88391,0.2096,0.82132,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,21],[80,139,0.5755,0.84372,0.22409,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,17],[84,139,0.6043,0.90622,0.21613,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,25],[88,139,0.6331,0.87499,0.21945,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,20],[92,139,0.6619,0.90625,0.21902,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,25],[96,139,0.6906,0.85712,0.23423,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,1,0,21],[100,139,0.7194,0.90176,0.1766,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,22],[104,139,0.7482,0.86161,0.21572,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,18],[108,139,0.777,0.9107,0.16271,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,22],[112,139,0.8058,0.95087,0.09858,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[116,139,0.8345,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[120,139,0.8633,0.91071,0.23077,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[124,139,0.8921,0.96428,0.08749,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[128,139,0.9209,0.95981,0.1143,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[132,139,0.9496,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[136,139,0.9784,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[139,139,1.0,0.95979,0.11435,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28]]}]},{"i":"7677ad29952071d3","q":"Let $A B C D$ be an isosceles trapezoid with $A B \\| C D$ and $\\overline{B C}=\\overline{A D}$. The line parallel to $A D$ through $B$ meets the perpendicular to $A D$ through $D$ at point $X$. Furthermore, the line parallel to $B D$ through $A$ meets the perpendicular to $B D$ through $D$ at point $Y$. Prove that the points $C, X, D$ and $Y$ lie on a common circle.","t":[{"b":2,"e":1.0,"k":"rising","v":0.62495,"x":0.96427,"p":[[0,65,0.0,0.75444,0.30145,0.57143,0.92857,1.0,0.0,1.0,2,16,1,2,0,1,0,0,0,0,0,1,0,0,9,0,0,1,0,0,2,0,16],[4,65,0.0615,0.72309,0.27671,0.571,0.64286,1.0,0.14,1.0,0,14,0,0,0,2,0,0,1,0,0,3,0,0,10,0,0,1,0,0,1,0,14],[8,65,0.1231,0.77228,0.26456,0.57132,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,0,0,0,1,0,17],[12,65,0.1846,0.83927,0.21355,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,2,0,19],[16,65,0.2462,0.65625,0.30275,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,2,0,0,7,0,0,6,0,0,1,0,0,1,0,12],[20,65,0.3077,0.68746,0.30186,0.42859,0.64286,1.0,0.0,1.0,1,12,0,1,0,2,0,0,1,0,0,5,0,0,7,0,0,1,0,0,3,0,12],[24,65,0.3692,0.62495,0.228,0.5354,0.57143,0.64286,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,6,0,0,16,0,0,0,0,0,1,0,7],[28,65,0.4308,0.6473,0.29878,0.42857,0.57143,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,7,0,0,6,0,0,2,0,0,1,0,11],[32,65,0.4923,0.65618,0.2471,0.571,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,1,0,0,17,0,0,0,0,0,1,0,9],[36,65,0.5538,0.65622,0.31514,0.42857,0.57143,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,1,0,0,4,0,0,7,0,0,2,0,0,1,0,12],[40,65,0.6154,0.6607,0.29179,0.57132,0.57143,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,1,0,0,2,0,0,12,0,0,1,0,0,1,0,11],[44,65,0.6769,0.68749,0.24599,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,15,0,0,1,0,0,0,0,11],[48,65,0.7385,0.67409,0.29502,0.53539,0.57143,1.0,0.0,1.0,1,11,0,1,0,2,0,0,1,0,0,4,0,0,10,0,0,0,0,0,3,0,11],[52,65,0.8,0.67409,0.24805,0.5713,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,4,0,0,11,0,0,3,0,0,2,0,9],[56,65,0.8615,0.6428,0.25755,0.571,0.57143,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,0,0,0,2,0,0,16,0,0,1,0,0,3,0,7],[60,65,0.9231,0.89284,0.1786,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,3,0,22],[64,65,0.9846,0.96427,0.07146,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[65,65,1.0,0.94196,0.1984,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,28]]},{"b":6,"e":0.14286,"k":"falling","v":0.20088,"x":0.76783,"p":[[0,95,0.0,0.70087,0.28204,0.42859,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,4,0,0,4,0,0,5,0,0,5,0,0,0,0,13],[4,95,0.0421,0.74106,0.26107,0.57143,0.857,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,0,0,0,4,0,13],[8,95,0.0842,0.72767,0.26812,0.57143,0.71421,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,6,0,0,9,0,0,0,0,0,3,0,13],[12,95,0.1263,0.68746,0.27303,0.571,0.57143,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,4,0,0,13,0,0,1,0,0,0,0,12],[16,95,0.1684,0.76783,0.26668,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,4,0,0,9,0,0,0,0,0,2,0,16],[20,95,0.2105,0.62052,0.28033,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,4,0,0,12,0,0,1,0,0,2,0,8],[24,95,0.2526,0.66071,0.26904,0.42859,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,6,0,0,11,0,0,0,0,0,2,0,10],[28,95,0.2947,0.6607,0.26184,0.57132,0.57143,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,5,0,0,13,0,0,1,0,0,2,0,9],[32,95,0.3368,0.72322,0.21706,0.57143,0.64286,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,3,0,0,3,0,10],[36,95,0.3789,0.54908,0.22047,0.42857,0.57143,0.57143,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,9,0,0,17,0,0,0,0,0,0,0,4],[40,95,0.4211,0.69642,0.27375,0.5354,0.57143,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,6,0,0,10,0,0,0,0,0,2,0,12],[44,95,0.4632,0.61603,0.27302,0.5354,0.57143,0.78571,0.0,1.0,2,8,0,2,0,1,0,0,0,0,0,5,0,0,14,0,0,2,0,0,0,0,8],[48,95,0.5053,0.51326,0.27873,0.42857,0.571,0.57143,0.0,1.0,3,5,0,3,0,2,0,0,2,0,0,7,0,0,12,0,0,1,0,0,0,0,5],[52,95,0.5474,0.42854,0.28569,0.14286,0.571,0.57143,0.0,1.0,5,2,0,5,0,4,0,0,4,0,0,2,0,0,13,0,0,0,0,0,2,0,2],[56,95,0.5895,0.56244,0.24726,0.42859,0.57143,0.60714,0.0,1.0,2,3,0,2,0,2,0,0,0,0,0,5,0,0,15,0,0,2,0,0,3,0,3],[60,95,0.6316,0.43746,0.26947,0.25,0.571,0.57143,0.0,1.0,6,1,0,6,0,2,0,0,1,0,0,6,0,0,13,0,0,1,0,0,2,0,1],[64,95,0.6737,0.5848,0.26332,0.42857,0.57143,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,2,0,0,8,0,0,11,0,0,0,0,0,4,0,5],[68,95,0.7158,0.39274,0.25512,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,3,0,0,2,0,0,8,0,0,11,0,0,0,0,0,1,0,1],[72,95,0.7579,0.46873,0.29063,0.14289,0.571,0.57143,0.0,1.0,4,4,0,4,0,5,0,0,0,0,0,4,0,0,15,0,0,0,0,0,0,0,4],[76,95,0.8,0.35267,0.25996,0.10714,0.42857,0.57143,0.0,1.0,8,1,0,8,0,3,0,0,2,0,0,8,0,0,9,0,0,1,0,0,0,0,1],[80,95,0.8421,0.51768,0.3068,0.25,0.571,0.57143,0.0,1.0,3,5,0,3,0,5,0,0,1,0,0,2,0,0,14,0,0,0,0,0,2,0,5],[84,95,0.8842,0.44186,0.2483,0.28571,0.5705,0.57143,0.0,1.0,4,2,0,4,0,3,0,0,2,0,0,6,0,0,15,0,0,0,0,0,0,0,2],[88,95,0.9263,0.20088,0.21973,0.0,0.14286,0.42857,0.0,0.57143,13,0,0,13,0,8,0,0,2,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[92,95,0.9684,0.26331,0.24776,0.0,0.14286,0.57143,0.0,0.71429,11,0,0,11,0,7,0,0,0,0,0,5,0,0,8,0,0,1,0,0,0,0,0],[95,95,1.0,0.26786,0.27837,0.0,0.28571,0.42857,0.0,1.0,14,2,0,14,0,0,0,0,3,0,0,12,0,0,1,0,0,0,0,0,0,0,2]]}]},{"i":"d29900837667f6ce","q":"Let $S$ be a finite set. For a positive integer $n$, we say that a function $f: S \\rightarrow S$ is an $n$-th power if there exists some function $g: S \\rightarrow S$ such that\n\n$$\nf(x)=\\underbrace{g(g(\\ldots g(x) \\ldots))}_{g \\text { applied } n \\text { times }}\n$$\n\nfor each $x \\in S$.\nSuppose that a function $f: S \\rightarrow S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x))=f(x)$ for each $x \\in S$ ?","t":[{"b":3,"e":0.42857,"k":"falling","v":0.58927,"x":0.83473,"p":[[0,21,0.0,0.74105,0.32624,0.5354,0.85714,1.0,0.0,1.0,2,15,1,2,0,1,0,0,3,0,0,2,0,0,2,0,0,2,0,0,5,0,15],[4,21,0.1905,0.79464,0.36932,0.78571,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,23],[8,21,0.381,0.60713,0.3993,0.10714,0.71429,1.0,0.0,1.0,8,11,0,8,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,0,2,0,11],[12,21,0.5714,0.75,0.39286,0.64286,1.0,1.0,0.0,1.0,5,20,0,5,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,20],[16,21,0.7619,0.83473,0.28394,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,21],[20,21,0.9524,0.64285,0.41187,0.21429,0.85707,1.0,0.0,1.0,8,14,0,8,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,3,0,14],[21,21,1.0,0.58927,0.37585,0.42857,0.57143,1.0,0.0,1.0,7,10,0,7,0,0,0,0,0,0,0,6,0,0,4,0,0,2,0,0,3,0,10]]},{"b":7,"e":1.0,"k":"rising","v":0.6071,"x":0.92411,"p":[[0,40,0.0,0.6071,0.34441,0.53539,0.71414,0.89286,0.0,1.0,6,8,1,6,0,0,0,0,1,0,0,1,0,0,7,0,0,7,0,0,2,0,8],[4,40,0.1,0.74999,0.3896,0.57132,1.0,1.0,0.0,1.0,6,21,0,6,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,21],[8,40,0.2,0.83033,0.29113,0.78561,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,3,0,21],[12,40,0.3,0.81696,0.31386,0.82132,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,21],[16,40,0.4,0.8616,0.27078,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,21],[20,40,0.5,0.87945,0.22901,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,23],[24,40,0.6,0.75893,0.34151,0.53572,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,0,0,0,4,0,0,2,0,0,2,0,0,1,0,19],[28,40,0.7,0.92411,0.21719,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,28],[32,40,0.8,0.91516,0.18512,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,25],[36,40,0.9,0.8616,0.30406,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[40,40,1.0,0.85713,0.30723,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,23]]}]},{"i":"27f3843add75c9fa","q":"Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as  $\\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .\n\n*Proposed by Nathan Xiong*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.32143,"x":0.86607,"p":[[0,46,0.0,0.86607,0.31122,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,46,0.087,0.78125,0.3694,0.67857,1.0,1.0,0.14286,1.0,0,23,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[8,46,0.1739,0.66071,0.41304,0.14286,1.0,1.0,0.14286,1.0,0,19,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[12,46,0.2609,0.58929,0.42069,0.14286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,15],[16,46,0.3478,0.75893,0.38538,0.14286,1.0,1.0,0.14286,1.0,0,23,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[20,46,0.4348,0.80804,0.35285,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[24,46,0.5217,0.78571,0.35174,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,22],[28,46,0.6087,0.79911,0.3551,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,24],[32,46,0.6957,0.51786,0.41304,0.14286,0.14286,1.0,0.14286,1.0,0,13,0,0,0,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,13],[36,46,0.7826,0.61607,0.42021,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,17],[40,46,0.8696,0.34821,0.34244,0.14286,0.14286,0.57143,0.14286,1.0,0,6,0,0,0,23,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,6],[44,46,0.9565,0.32143,0.34069,0.14286,0.14286,0.14286,0.14286,1.0,0,6,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[46,46,1.0,0.86161,0.27078,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,23]]},{"b":2,"e":1.0,"k":"flat","v":0.81696,"x":1.0,"p":[[0,19,0.0,0.87054,0.28428,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[4,19,0.2105,0.81696,0.34669,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[8,19,0.4211,0.94196,0.20782,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[12,19,0.6316,0.88839,0.25439,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,25],[16,19,0.8421,0.91518,0.24964,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5d97582c68afd11a","q":"Let $\\alpha, \\beta, \\gamma$ be the angles of a triangle opposite to the sides $a, b, c$ respectively. Prove the inequality\n\n$$\n2\\left(\\cos ^{2} \\alpha+\\cos ^{2} \\beta+\\cos ^{2} \\gamma\\right) \\geq \\frac{a^{2}}{b^{2}+c^{2}}+\\frac{b^{2}}{a^{2}+c^{2}}+\\frac{c^{2}}{a^{2}+b^{2}}\n$$","t":[{"b":0,"e":0.57143,"k":"flat","v":0.20536,"x":0.38837,"p":[[0,108,0.0,0.20536,0.07935,0.14286,0.1429,0.28571,0.0,0.28571,1,0,0,1,0,16,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,108,0.037,0.34375,0.25966,0.2857,0.28571,0.28571,0.0,1.0,1,4,0,1,0,6,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[8,108,0.0741,0.26339,0.13415,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,13,0,0,14,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[12,108,0.1111,0.33482,0.2412,0.14286,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,9,0,0,17,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[16,108,0.1481,0.30357,0.24157,0.14286,0.28571,0.28571,0.0,1.0,2,3,0,2,0,8,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[20,108,0.1852,0.33036,0.23538,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,10,0,0,12,0,0,4,0,0,1,0,0,2,0,0,0,0,2],[24,108,0.2222,0.32143,0.27199,0.14286,0.21428,0.28571,0.14286,1.0,0,3,0,0,0,16,0,0,9,0,0,2,0,0,0,0,0,1,0,0,1,0,3],[28,108,0.2593,0.28572,0.18557,0.14286,0.28571,0.28571,0.0,1.0,1,1,0,1,0,11,0,0,13,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[32,108,0.2963,0.2991,0.25342,0.14286,0.2857,0.28571,0.0,1.0,2,2,0,2,0,11,0,0,15,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[36,108,0.3333,0.32589,0.21198,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,9,0,0,17,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[40,108,0.3704,0.32588,0.24803,0.14286,0.2857,0.42857,0.0,1.0,2,2,0,2,0,11,0,0,9,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[44,108,0.4074,0.2633,0.1294,0.14286,0.28571,0.28571,0.14,0.71429,0,0,0,0,0,12,0,0,16,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[48,108,0.4444,0.29009,0.20979,0.14286,0.2857,0.28571,0.14,1.0,0,1,0,0,0,15,0,0,11,0,0,2,0,0,0,0,0,3,0,0,0,0,1],[52,108,0.4815,0.25446,0.20118,0.14286,0.21428,0.28571,0.0,1.0,2,1,0,2,0,14,0,0,12,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[56,108,0.5185,0.38837,0.28399,0.24999,0.28571,0.42858,0.0,1.0,1,4,0,1,0,7,0,0,14,0,0,3,0,0,1,0,0,1,0,0,1,0,4],[60,108,0.5556,0.35267,0.26722,0.14286,0.2857,0.28571,0.14286,1.0,0,4,0,0,0,9,0,0,17,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[64,108,0.5926,0.32589,0.1794,0.1429,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,9,0,0,13,0,0,6,0,0,1,0,0,2,0,0,1,0,0],[68,108,0.6296,0.29018,0.16554,0.24999,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,8,0,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[72,108,0.6667,0.35268,0.25501,0.24999,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,8,0,0,18,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[76,108,0.7037,0.33036,0.17655,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,20,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[80,108,0.7407,0.31248,0.18361,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,20,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[84,108,0.7778,0.34374,0.21085,0.24999,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,15,0,0,4,0,0,1,0,0,2,0,0,1,0,1],[88,108,0.8148,0.33472,0.22765,0.14289,0.28571,0.32143,0.14,1.0,0,2,0,0,0,10,0,0,14,0,0,3,0,0,1,0,0,2,0,0,0,0,2],[92,108,0.8519,0.32143,0.19233,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,19,0,0,3,0,0,0,0,0,1,0,0,1,0,1],[96,108,0.8889,0.2991,0.13054,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,6,0,0,20,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[100,108,0.9259,0.30357,0.10564,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,25,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[104,108,0.963,0.32589,0.21793,0.24999,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,8,0,0,19,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[108,108,1.0,0.31695,0.18464,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,20,0,0,1,0,0,1,0,0,2,0,0,0,0,1]]},{"b":6,"e":0.28571,"k":"flat","v":0.22768,"x":0.42856,"p":[[0,147,0.0,0.22768,0.11214,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,147,0.0272,0.32143,0.2369,0.14286,0.28571,0.32143,0.0,1.0,2,2,0,2,0,8,0,0,14,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[8,147,0.0544,0.28571,0.16366,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,10,0,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[12,147,0.0816,0.24552,0.10244,0.14286,0.2857,0.28571,0.14286,0.571,0,0,0,0,0,13,0,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,147,0.1088,0.29911,0.23517,0.14286,0.2857,0.28571,0.14286,1.0,0,2,0,0,0,15,0,0,12,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[20,147,0.1361,0.25893,0.09061,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,147,0.1633,0.375,0.26666,0.2857,0.28571,0.42857,0.0,1.0,1,4,0,1,0,6,0,0,15,0,0,5,0,0,0,0,0,1,0,0,0,0,4],[28,147,0.1905,0.375,0.26184,0.25,0.28571,0.32144,0.14286,1.0,0,3,0,0,0,8,0,0,16,0,0,1,0,0,0,0,0,4,0,0,0,0,3],[32,147,0.2177,0.25446,0.18808,0.14286,0.2857,0.28571,0.0,1.0,3,1,0,3,0,10,0,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[36,147,0.2449,0.33482,0.26633,0.14286,0.28571,0.28571,0.0,1.0,1,3,0,1,0,10,0,0,15,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[40,147,0.2721,0.375,0.22799,0.2857,0.28571,0.46429,0.14286,1.0,0,2,0,0,0,7,0,0,14,0,0,3,0,0,4,0,0,2,0,0,0,0,2],[44,147,0.2993,0.35268,0.28118,0.14286,0.28571,0.28571,0.14286,1.0,0,4,0,0,0,12,0,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,4],[48,147,0.3265,0.23661,0.08459,0.14286,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,147,0.3537,0.33482,0.19434,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,16,0,0,6,0,0,0,0,0,1,0,0,1,0,1],[56,147,0.381,0.42856,0.30093,0.14289,0.28571,0.71429,0.0,1.0,1,4,0,1,0,9,0,0,8,0,0,3,0,0,1,0,0,6,0,0,0,0,4],[60,147,0.4082,0.35266,0.22298,0.2857,0.28571,0.32143,0.14286,1.0,0,2,0,0,0,7,0,0,17,0,0,2,0,0,2,0,0,2,0,0,0,0,2],[64,147,0.4354,0.39723,0.28743,0.24999,0.28571,0.42858,0.14,1.0,0,5,0,0,0,8,0,0,14,0,0,3,0,0,1,0,0,1,0,0,0,0,5],[68,147,0.4626,0.2366,0.08458,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,147,0.4898,0.40625,0.27918,0.14296,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,9,0,0,11,0,0,3,0,0,2,0,0,2,0,0,2,0,3],[76,147,0.517,0.27232,0.11495,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,10,0,0,17,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[80,147,0.5442,0.27678,0.19212,0.14286,0.2857,0.28571,0.0,1.0,1,1,0,1,0,11,0,0,17,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[84,147,0.5714,0.37054,0.274,0.14286,0.28571,0.32143,0.14286,1.0,0,4,0,0,0,9,0,0,15,0,0,2,0,0,0,0,0,2,0,0,0,0,4],[88,147,0.5986,0.38393,0.27994,0.14286,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,9,0,0,13,0,0,4,0,0,0,0,0,1,0,0,1,0,4],[92,147,0.6259,0.30357,0.19804,0.14289,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,9,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[96,147,0.6531,0.34375,0.25218,0.14286,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,10,0,0,15,0,0,2,0,0,0,0,0,2,0,0,0,0,3],[100,147,0.6803,0.39286,0.26726,0.2857,0.28571,0.46429,0.14286,1.0,0,4,0,0,0,7,0,0,15,0,0,2,0,0,3,0,0,1,0,0,0,0,4],[104,147,0.7075,0.2633,0.20243,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,13,0,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[108,147,0.7347,0.35714,0.27199,0.14286,0.28571,0.42857,0.0,1.0,2,2,0,2,0,8,0,0,13,0,0,2,0,0,2,0,0,0,0,0,3,0,2],[112,147,0.7619,0.30357,0.22232,0.14286,0.2857,0.32143,0.0,1.0,1,1,0,1,0,13,0,0,10,0,0,3,0,0,2,0,0,1,0,0,1,0,1],[116,147,0.7891,0.36161,0.21124,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,4,0,0,19,0,0,5,0,0,1,0,0,0,0,0,1,0,2],[120,147,0.8163,0.32588,0.18636,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,18,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[124,147,0.8435,0.33036,0.21852,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,10,0,0,13,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[128,147,0.8707,0.27232,0.11495,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,8,0,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[132,147,0.898,0.36607,0.23941,0.2857,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,7,0,0,15,0,0,5,0,0,1,0,0,1,0,0,0,0,3],[136,147,0.9252,0.34821,0.2257,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,7,0,0,18,0,0,2,0,0,0,0,0,3,0,0,0,0,2],[140,147,0.9524,0.33482,0.19759,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,7,0,0,17,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[144,147,0.9796,0.33036,0.19377,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,4,0,0,21,0,0,2,0,0,0,0,0,3,0,0,0,0,1],[147,147,1.0,0.34375,0.20782,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,6,0,0,20,0,0,1,0,0,0,0,0,2,0,0,3,0,0]]}]},{"i":"556d44c05fc9ec6d","q":"Let  $ ABC$  be a triangle with circumradius  $ R$ , perimeter  $ P$  and area  $ K$ . Determine the maximum value of:  $ \\frac{KP}{R^3}$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.65179,"x":1.0,"p":[[0,36,0.0,0.84821,0.24727,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,4,0,0,0,0,22],[4,36,0.1111,0.84375,0.25345,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,3,0,0,0,0,22],[8,36,0.2222,0.71875,0.27313,0.42859,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,4,0,0,0,0,14],[12,36,0.3333,0.73661,0.30537,0.42857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,7,0,0,4,0,0,0,0,0,4,0,0,0,0,17],[16,36,0.4444,0.65179,0.32915,0.28571,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,12,0,0,3,0,0,0,0,0,3,0,0,0,0,14],[20,36,0.5556,0.76786,0.33072,0.28571,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,21],[24,36,0.6667,0.69196,0.34276,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,2,0,0,0,0,17],[28,36,0.7778,0.69196,0.34277,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,2,0,0,0,0,17],[32,36,0.8889,0.8125,0.29974,0.64286,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,22],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,10,0.0,0.86161,0.24087,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,0,0,0,23],[4,10,0.4,0.89286,0.22304,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,25],[8,10,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[10,10,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"e52eeba20faf8d46","q":"Let  $ A'\\neq A $  be the intersection of the bisector of  $ \\angle BAC $  with the circumcircle of the triangle  $ ABC. $ Prove that  $ AA'>\\frac{AB+AC}{2}. $","t":[{"b":0,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,17,0.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[4,17,0.2353,0.91518,0.17445,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,23],[8,17,0.4706,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,17,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,17,0.9412,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[17,17,1.0,0.98214,0.04725,1.0,1.0,1.0,0.8571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":4,"e":0.14286,"k":"falling","v":0.35267,"x":0.95982,"p":[[0,66,0.0,0.90179,0.20652,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,24],[4,66,0.0606,0.91964,0.17835,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,25],[8,66,0.1212,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[12,66,0.1818,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[16,66,0.2424,0.91517,0.19516,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[20,66,0.303,0.85714,0.19562,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,4,0,17],[24,66,0.3636,0.79911,0.31514,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,19],[28,66,0.4242,0.83929,0.21354,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,3,0,16],[32,66,0.4848,0.74554,0.27371,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,14,0,0,4,0,10],[36,66,0.5455,0.75446,0.29717,0.71429,0.85707,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,11,0,0,6,0,11],[40,66,0.6061,0.69196,0.36789,0.71429,0.71429,1.0,0.0,1.0,6,13,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,2,0,13],[44,66,0.6667,0.62945,0.41011,0.14286,0.78571,1.0,0.0,1.0,6,14,0,6,0,3,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,14],[48,66,0.7273,0.65625,0.42085,0.14286,1.0,1.0,0.0,1.0,6,17,0,6,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,17],[52,66,0.7879,0.53571,0.3977,0.14286,0.57143,1.0,0.0,1.0,7,10,0,7,0,3,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,10],[56,66,0.8485,0.60268,0.41301,0.14286,0.71429,1.0,0.0,1.0,7,13,0,7,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,0,2,0,13],[60,66,0.9091,0.41969,0.31731,0.14286,0.42857,0.71429,0.0,1.0,4,4,0,4,0,9,0,0,0,0,0,8,0,0,2,0,0,5,0,0,0,0,4],[64,66,0.9697,0.35267,0.22582,0.14286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,8,0,0,2,0,0,14,0,0,3,0,0,0,0,0,1,0,1],[66,66,1.0,0.51337,0.27399,0.28571,0.4998,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,3,0,0,6,0,0,5,0,0,4,0,0,5,0,2]]}]},{"i":"6ee618ffe57d79e9","q":"Let  $ ABC $  be an acute triangle having  $ AB<AC, $  let  $ M $  be the midpoint of the segment  $ BC, D$  be a point on the segment  $ AM, E $  be a point on the segment  $ BD $  and  $ F $  on the line  $ AB $  such that  $ EF $  is parallel to  $ BC, $  and such that  $ AE $  and  $ DF $  pass through the orthocenter of  $ ABC. $ Prove that the interior bisectors of  $ \\angle BAC $  and  $ \\angle BDC, $  together with  $ BC $  are concurrent.\n\n\n*Vlad Robu*","t":[{"b":4,"e":0.14286,"k":"flat","v":0.01786,"x":0.10715,"p":[[0,96,0.0,0.10715,0.09449,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,96,0.0417,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,96,0.0833,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,96,0.125,0.06688,0.09432,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,96,0.1667,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,96,0.2083,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,96,0.25,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,96,0.2917,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,96,0.3333,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,96,0.375,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,96,0.4167,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,96,0.4583,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,96,0.5,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,96,0.5417,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,96,0.5833,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,96,0.625,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,96,0.6667,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,96,0.7083,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,96,0.75,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,96,0.7917,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,96,0.8333,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,96,0.875,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,96,0.9167,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,96,0.9583,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,96,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03125,"x":0.125,"p":[[0,107,0.0,0.125,0.1171,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,16,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,107,0.0374,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,107,0.0748,0.06232,0.10048,0.0,0.0,0.14071,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,107,0.1121,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,107,0.1495,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,107,0.1869,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,107,0.2243,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,107,0.2617,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,107,0.2991,0.08929,0.10564,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,107,0.3364,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,107,0.3738,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,107,0.4112,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,107,0.4486,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,107,0.486,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,107,0.5234,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,107,0.5607,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,107,0.5981,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,107,0.6355,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,107,0.6729,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,107,0.7103,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,107,0.7477,0.12491,0.09941,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,19,0,0,3,0,0,1,0,0,0,0,0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and Laban take turns playing a game. Each of them starts with the list of square numbers  $1, 4, 9, \\dots, 2021^2$ , and there is a whiteboard in front of them with the number  $0$  on it. Jacob chooses a number  $x^2$  from his list, removes it from his list, and replaces the number  $W$  on the whiteboard with  $W + x^2$ . Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by  $4$  after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number  $K$  such that Jacob can guarantee to get at least  $K$  sheep by the end of the game, no matter how Laban plays?","t":[{"b":5,"e":0.14286,"k":"flat","v":0.14714,"x":0.54017,"p":[[0,102,0.0,0.17411,0.1665,0.10714,0.14286,0.17857,0.0,0.71429,8,0,8,8,0,16,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,102,0.0392,0.45089,0.28146,0.14286,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,12,0,0,3,0,0,2,0,0,2,0,0,9,0,0,4,0,0],[8,102,0.0784,0.52232,0.30641,0.14286,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,11,0,0,2,0,0,0,0,0,1,0,0,11,0,0,6,0,1],[12,102,0.1176,0.47991,0.30527,0.14286,0.42857,0.85714,0.14286,1.0,0,1,0,0,0,10,0,0,5,0,0,3,0,0,2,1,0,1,0,0,9,0,1],[16,102,0.1569,0.45536,0.28221,0.14286,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,6,0,0,5,0,0,1,0,0,6,0,0,3,0,2],[20,102,0.1961,0.54017,0.28287,0.25001,0.57143,0.75,0.14286,1.0,0,1,0,0,0,8,0,0,2,0,0,3,0,0,4,0,0,7,0,0,7,0,1],[24,102,0.2353,0.42848,0.29459,0.14286,0.28571,0.71429,0.14,0.85714,0,0,0,0,0,13,0,0,5,0,0,1,0,0,1,0,0,6,0,0,6,0,0],[28,102,0.2745,0.39732,0.3108,0.14286,0.14288,0.71429,0.14286,1.0,0,1,0,0,0,17,0,0,2,0,0,2,0,0,1,0,0,3,0,0,6,0,1],[32,102,0.3137,0.43295,0.29348,0.14286,0.35714,0.71429,0.14,1.0,0,1,0,0,0,14,0,0,2,0,0,2,0,0,2,0,0,8,0,0,3,0,1],[36,102,0.3529,0.41509,0.2932,0.14286,0.35714,0.71429,0.14,1.0,0,1,0,0,0,15,0,0,1,0,0,4,0,0,2,0,0,5,0,0,4,0,1],[40,102,0.3922,0.29911,0.25843,0.14286,0.14286,0.32143,0.14286,1.0,0,1,0,0,0,21,0,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,1],[44,102,0.4314,0.33463,0.27119,0.14286,0.14286,0.60682,0.14,0.85714,0,0,0,0,0,20,0,0,1,0,0,2,0,0,1,0,0,5,0,0,3,0,0],[48,102,0.4706,0.31687,0.28293,0.14286,0.14286,0.60714,0.14,0.85714,0,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0],[52,102,0.5098,0.27643,0.22307,0.14286,0.14286,0.32143,0.14,0.85714,0,0,0,0,0,21,0,0,3,0,0,3,0,0,1,0,0,2,0,0,2,0,0],[56,102,0.549,0.2634,0.21756,0.14286,0.14286,0.28571,0.14286,0.85714,0,0,0,0,0,22,0,0,3,0,0,3,0,0,0,0,0,2,0,0,2,0,0],[60,102,0.5882,0.29446,0.27428,0.14286,0.14286,0.46429,0.0,0.85714,2,0,0,2,0,21,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0],[64,102,0.6275,0.30348,0.24942,0.14286,0.14286,0.46429,0.14,0.85714,0,0,0,0,0,21,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,0],[68,102,0.6667,0.20527,0.16731,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,27,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[72,102,0.7059,0.25875,0.24083,0.14286,0.14286,0.2857,0.0,0.85714,1,0,0,1,0,22,0,0,4,0,0,0,0,0,0,0,0,2,0,0,3,0,0],[76,102,0.7451,0.23661,0.19759,0.14286,0.14286,0.14287,0.14286,0.85714,0,0,0,0,0,25,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0],[80,102,0.7843,0.20974,0.17855,0.14286,0.14286,0.1429,0.14,0.85714,0,0,0,0,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[84,102,0.8235,0.27223,0.26577,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,24,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0],[88,102,0.8627,0.20527,0.1513,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[92,102,0.902,0.16955,0.10974,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[96,102,0.9412,0.18751,0.12596,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[100,102,0.9804,0.17402,0.09271,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[102,102,1.0,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.16062,"x":0.57143,"p":[[0,206,0.0,0.16062,0.17406,0.0,0.14286,0.14286,0.0,0.71429,9,0,9,9,0,17,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[4,206,0.0194,0.50883,0.30091,0.14286,0.57121,0.85714,0.14,0.85714,0,0,0,0,0,11,0,0,1,0,0,3,0,0,2,0,0,6,0,0,9,0,0],[8,206,0.0388,0.51786,0.34764,0.14286,0.71429,0.85714,0.0,1.0,2,2,0,2,0,10,0,0,2,0,0,1,0,0,0,0,0,5,0,0,10,0,2],[12,206,0.0583,0.57143,0.29233,0.25,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,7,0,0,1,0,0,2,0,0,2,0,0,9,0,0,10,0,0],[16,206,0.0777,0.54009,0.29405,0.2857,0.71429,0.75,0.14,1.0,0,2,0,0,0,7,0,0,5,0,0,3,0,0,0,0,0,9,0,0,6,0,2],[20,206,0.0971,0.55343,0.25683,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,8,0,0,2,0,0,4,0,0,9,0,0,4,0,2],[24,206,0.1165,0.56249,0.30501,0.25,0.71429,0.85714,0.0,1.0,2,2,0,2,0,6,0,0,1,0,0,2,0,0,4,0,0,8,0,0,7,0,2],[28,206,0.1359,0.4375,0.26948,0.14286,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,8,0,0,6,0,0,5,0,0,2,0,0,5,0,0,5,0,0],[32,206,0.1553,0.49543,0.27672,0.24999,0.49979,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,3,0,0,5,0,0,3,0,0,8,0,0,4,0,1],[36,206,0.1748,0.45089,0.30116,0.14286,0.42857,0.71429,0.0,1.0,1,2,0,1,0,10,0,0,4,0,0,4,0,0,2,0,0,5,0,0,4,0,2],[40,206,0.1942,0.49554,0.2525,0.2857,0.5,0.71429,0.14286,0.85714,0,0,0,0,0,7,0,0,4,0,0,5,0,0,3,0,0,9,0,0,4,0,0],[44,206,0.2136,0.45982,0.27833,0.14286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,9,0,0,6,0,0,3,0,0,4,0,0,4,0,0,5,0,1],[48,206,0.233,0.45982,0.27137,0.14286,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,9,0,0,5,0,0,5,0,0,2,0,0,5,0,0,6,0,0],[52,206,0.2524,0.45089,0.23987,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,8,0,0,6,0,0,2,0,0,8,0,0,1,0,1],[56,206,0.2718,0.375,0.26666,0.14286,0.28571,0.60714,0.0,1.0,2,1,0,2,0,11,0,0,4,0,0,6,0,0,1,0,0,6,0,0,1,0,1],[60,206,0.2913,0.40597,0.28396,0.14286,0.28571,0.60714,0.14,0.85714,0,0,0,0,0,13,0,0,5,0,0,3,0,0,3,0,0,1,0,0,7,0,0],[64,206,0.3107,0.37268,0.28057,0.14286,0.28571,0.60714,0.0,0.85714,2,0,0,2,1,11,0,0,5,0,0,2,0,0,3,0,0,4,0,0,4,0,0],[68,206,0.3301,0.35268,0.2448,0.14286,0.2857,0.42858,0.0,0.85714,1,0,0,1,0,12,0,0,6,0,0,6,0,0,1,0,0,3,0,0,3,0,0],[72,206,0.3495,0.40625,0.27689,0.14286,0.28571,0.60714,0.0,0.85714,1,0,0,1,0,10,0,0,7,0,0,3,0,0,3,0,0,2,0,0,6,0,0],[76,206,0.3689,0.44196,0.27976,0.24999,0.35714,0.71429,0.0,1.0,2,1,0,2,0,6,0,0,8,0,0,3,0,0,2,0,0,7,0,0,3,0,1],[80,206,0.3883,0.38391,0.24597,0.14286,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,13,0,0,4,0,0,4,0,0,4,0,0,5,0,0,2,0,0],[84,206,0.4078,0.33929,0.20438,0.14289,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,7,0,0,10,0,0,8,0,0,1,0,0,3,0,0,1,0,0],[88,206,0.4272,0.48214,0.25692,0.28571,0.5,0.71429,0.14286,1.0,0,1,0,0,0,7,0,0,6,0,0,3,0,0,4,0,0,9,0,0,2,0,1],[92,206,0.4466,0.34813,0.25746,0.14286,0.28571,0.57143,0.0,1.0,1,1,0,1,0,14,0,0,5,0,0,3,0,0,3,0,0,4,0,0,1,0,1],[96,206,0.466,0.40624,0.21161,0.2857,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,11,0,0,5,0,0,3,0,0,6,0,0,1,0,0],[100,206,0.4854,0.29455,0.19872,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,12,0,0,7,0,0,8,0,0,0,0,0,2,0,0,1,0,0],[104,206,0.5049,0.33037,0.18012,0.14286,0.28571,0.42858,0.14286,0.85714,0,0,0,0,0,10,0,0,10,0,0,7,0,0,3,0,0,1,0,0,1,0,0],[108,206,0.5243,0.39284,0.24484,0.14286,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,10,0,0,7,0,0,6,0,0,3,0,0,3,0,0,2,0,1],[112,206,0.5437,0.36604,0.19861,0.14286,0.35714,0.46418,0.14286,0.857,0,0,0,0,0,10,0,0,6,0,0,8,0,0,5,0,0,2,0,0,1,0,0],[116,206,0.5631,0.37946,0.27574,0.14286,0.28571,0.60714,0.0,0.85714,3,0,0,3,0,9,0,0,6,0,0,4,0,0,2,0,0,4,0,0,4,0,0],[120,206,0.5825,0.36607,0.23128,0.14289,0.28571,0.4286,0.0,1.0,1,1,0,1,0,8,0,0,10,0,0,6,0,0,2,0,0,3,0,0,1,0,1],[124,206,0.6019,0.37937,0.26156,0.14286,0.28586,0.57111,0.0,1.0,2,1,0,2,0,9,0,0,6,0,0,6,0,0,4,0,0,1,0,0,3,0,1],[128,206,0.6214,0.41509,0.24846,0.28571,0.28571,0.46429,0.14,0.85714,0,0,0,0,0,7,0,0,10,0,0,7,0,0,1,0,0,1,0,0,6,0,0],[132,206,0.6408,0.41964,0.26949,0.14286,0.28571,0.71429,0.0,1.0,1,1,0,1,0,9,0,0,7,0,0,2,0,0,4,0,0,6,0,0,2,0,1],[136,206,0.6602,0.24107,0.1357,0.14286,0.14288,0.28571,0.0,0.57143,1,0,0,1,0,16,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[140,206,0.6796,0.28572,0.19885,0.14286,0.28571,0.32143,0.0,0.71429,2,0,0,2,0,13,0,0,9,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[144,206,0.699,0.28125,0.16935,0.14286,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,13,0,0,10,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[148,206,0.7184,0.33027,0.24863,0.14286,0.2857,0.4286,0.0,0.85714,2,0,0,2,0,13,0,0,5,0,0,5,0,0,2,0,0,2,0,0,3,0,0],[152,206,0.7379,0.26786,0.18472,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,13,0,0,10,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[156,206,0.7573,0.25884,0.16152,0.14286,0.2857,0.28571,0.0,0.85714,1,0,0,1,0,14,0,0,11,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[160,206,0.7767,0.25446,0.17762,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,16,0,0,10,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[164,206,0.7961,0.25436,0.16266,0.14286,0.2143,0.28571,0.0,0.71429,2,0,0,2,0,14,0,0,9,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[168,206,0.8155,0.19187,0.11075,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,0,15,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[172,206,0.835,0.23219,0.12765,0.14286,0.21428,0.28571,0.0,0.71429,1,0,0,1,0,15,0,0,13,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[176,206,0.8544,0.26786,0.14617,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,14,0,0,12,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[180,206,0.8738,0.23661,0.16982,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,19,0,0,8,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[184,206,0.8932,0.17411,0.09268,0.14286,0.14286,0.14287,0.0,0.42857,2,0,0,2,0,23,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[188,206,0.9126,0.20973,0.08744,0.14286,0.14288,0.28571,0.0,0.42857,1,0,0,1,0,16,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[192,206,0.932,0.19625,0.11718,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,18,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[196,206,0.9515,0.23215,0.13716,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,17,0,0,9,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[200,206,0.9709,0.24107,0.18013,0.14286,0.14286,0.2857,0.14286,0.85714,0,0,0,0,0,20,0,0,9,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[204,206,0.9903,0.2143,0.0945,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[206,206,1.0,0.20982,0.10092,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,18,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"668f1af7d00e9461","q":"Let $P_{1}, P_{2}, \\ldots, P_{1993}=P_{0}$ be distinct points in the $x y$-plane with the following properties:\n(i) both coordinates of $P_{i}$ are integers, for $i=1,2, \\ldots, 1993$;\n(ii) there is no point other than $P_{i}$ and $P_{i+1}$ on the line segment joining $P_{i}$ with $P_{i+1}$ whose coordinates are both integers, for $i=0,1, \\ldots, 1992$.\n\nProve that for some $i, 0 \\leq i \\leq 1992$, there exists a point $Q$ with coordinates $\\left(q_{x}, q_{y}\\right)$ on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2 q_{x}$ and $2 q_{y}$ are odd integers.","t":[{"b":0,"e":1.0,"k":"flat","v":0.79463,"x":1.0,"p":[[0,43,0.0,0.87054,0.18336,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,3,0,19],[4,43,0.093,0.79463,0.1819,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,4,0,11],[8,43,0.186,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[12,43,0.2791,0.94196,0.15093,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,27],[16,43,0.3721,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,43,0.4651,0.91964,0.18877,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,25],[24,43,0.5581,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,43,0.6512,0.97322,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[32,43,0.7442,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,43,0.8372,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[40,43,0.9302,0.95535,0.11539,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[43,43,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":7,"e":1.0,"k":"flat","v":0.78124,"x":0.875,"p":[[0,22,0.0,0.87054,0.19351,0.82143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,5,0,19],[4,22,0.1818,0.875,0.16656,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,4,0,18],[8,22,0.3636,0.84372,0.19682,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,6,0,16],[12,22,0.5455,0.78124,0.20821,0.71429,0.85714,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,8,0,0,7,0,10],[16,22,0.7273,0.84375,0.16506,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,5,0,14],[20,22,0.9091,0.82141,0.16368,0.71429,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,7,0,11],[22,22,1.0,0.8616,0.19061,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,7,0,16]]}]},{"i":"9e86f308875e7d54","q":"Let  $ ABC$  be a triangle with integer side lengths, and let its incircle touch  $ BC$  at  $ D$  and  $ AC$  at  $ E$ . If  $ |AD^2\\minus{}BE^2| \\le 2$ , show that  $ AC\\equal{}BC$ 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7,0.40177,0.21557,0.28571,0.28571,0.42858,0.0,1.0,1,2,0,1,0,1,0,0,16,0,0,7,0,0,4,0,0,0,0,0,1,0,2],[108,249,0.4337,0.34374,0.18849,0.28571,0.28571,0.42857,0.0,1.0,3,1,0,3,0,0,0,0,18,0,0,7,0,0,2,0,0,1,0,0,0,0,1],[112,249,0.4498,0.46428,0.29015,0.28571,0.28571,0.60714,0.14286,1.0,0,6,0,0,0,1,0,0,20,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[116,249,0.4659,0.40623,0.22898,0.28571,0.28571,0.46418,0.14286,1.0,0,3,0,0,0,2,0,0,19,0,0,3,0,0,4,0,0,1,0,0,0,0,3],[120,249,0.4819,0.40177,0.2271,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,2,0,0,19,0,0,4,0,0,3,0,0,1,0,0,0,0,3],[124,249,0.498,0.37054,0.19186,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,22,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[128,249,0.5141,0.38393,0.22142,0.28571,0.28571,0.42857,0.0,1.0,1,3,0,1,0,0,0,0,21,0,0,5,0,0,2,0,0,0,0,0,0,0,3],[132,249,0.5301,0.35269,0.25996,0.28571,0.28571,0.42857,0.0,1.0,5,3,0,5,0,0,0,0,17,0,0,5,0,0,1,0,0,1,0,0,0,0,3],[136,249,0.5462,0.43748,0.29,0.2857,0.28571,0.46418,0.0,1.0,1,6,0,1,0,2,0,0,17,0,0,4,0,0,2,0,0,0,0,0,0,0,6],[140,249,0.5622,0.38391,0.2511,0.28571,0.28571,0.42857,0.0,1.0,1,4,0,1,0,2,0,0,20,0,0,4,0,0,1,0,0,0,0,0,0,0,4],[144,249,0.5783,0.37501,0.22232,0.28571,0.28571,0.32164,0.0,1.0,1,3,0,1,0,0,0,0,23,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[148,249,0.5944,0.41518,0.28428,0.28571,0.28571,0.42858,0.0,1.0,1,5,0,1,0,3,0,0,17,0,0,5,0,0,0,0,0,0,0,0,1,0,5],[152,249,0.6104,0.36161,0.15146,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,20,0,0,7,0,0,3,0,0,0,0,0,0,0,1],[156,249,0.6265,0.30802,0.20549,0.2857,0.28571,0.28571,0.0,1.0,2,2,0,2,0,4,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[160,249,0.6426,0.41517,0.20935,0.28571,0.28571,0.46428,0.14286,1.0,0,2,0,0,0,1,0,0,18,0,0,5,0,0,3,0,0,3,0,0,0,0,2],[164,249,0.6586,0.37946,0.22192,0.2857,0.28571,0.42857,0.0,1.0,1,2,0,1,0,2,0,0,18,0,0,6,0,0,1,0,0,1,0,0,1,0,2],[168,249,0.6747,0.36607,0.17474,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,22,0,0,8,0,0,0,0,0,0,0,0,0,0,2],[172,249,0.6908,0.36164,0.23415,0.28571,0.28571,0.32143,0.0,1.0,2,3,0,2,0,1,0,0,21,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[176,249,0.7068,0.46428,0.22304,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,7,0,0,3,0,0,3,0,0,2,0,2],[180,249,0.7229,0.37946,0.20395,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,22,0,0,4,0,0,2,0,0,0,0,0,1,0,2],[184,249,0.739,0.40178,0.21852,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,18,0,0,8,0,0,0,0,0,2,0,0,1,0,2],[188,249,0.755,0.33927,0.1847,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,2,0,0,22,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[192,249,0.7711,0.36607,0.19212,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,18,0,0,8,0,0,0,0,0,0,0,0,2,0,1],[196,249,0.7871,0.42856,0.25253,0.28571,0.28571,0.42858,0.14286,1.0,0,4,0,0,0,2,0,0,17,0,0,6,0,0,2,0,0,0,0,0,1,0,4],[200,249,0.8032,0.375,0.22232,0.28571,0.28571,0.32143,0.14286,1.0,0,3,0,0,0,2,0,0,22,0,0,4,0,0,0,0,0,1,0,0,0,0,3],[204,249,0.8193,0.39284,0.19232,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,18,0,0,8,0,0,2,0,0,1,0,0,0,0,2],[208,249,0.8353,0.41071,0.21651,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,19,0,0,5,0,0,2,0,0,2,0,0,1,0,2],[212,249,0.8514,0.36161,0.18205,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,22,0,0,6,0,0,1,0,0,0,0,0,0,0,2],[216,249,0.8675,0.35712,0.18896,0.28571,0.28571,0.42857,0.0,1.0,1,1,0,1,0,2,0,0,19,0,0,5,0,0,3,0,0,0,0,0,1,0,1],[220,249,0.8835,0.39285,0.21427,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,20,0,0,6,0,0,2,0,0,0,0,0,0,0,3],[224,249,0.8996,0.31248,0.09736,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,0,0,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[228,249,0.9157,0.41964,0.22286,0.2857,0.28571,0.4286,0.2857,1.0,0,3,0,0,0,0,0,0,20,0,0,5,0,0,2,0,0,2,0,0,0,0,3],[232,249,0.9317,0.38396,0.18707,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,21,0,0,7,0,0,1,0,0,1,0,0,0,0,2],[236,249,0.9478,0.40178,0.25111,0.28571,0.28571,0.42857,0.0,1.0,1,4,0,1,0,1,0,0,20,0,0,3,0,0,3,0,0,0,0,0,0,0,4],[240,249,0.9639,0.42857,0.24744,0.28571,0.28571,0.46429,0.2857,1.0,0,2,0,0,0,0,0,0,23,0,0,1,0,0,1,0,0,1,0,0,4,0,2],[244,249,0.9799,0.34373,0.15508,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,1,0,0,21,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[248,249,0.996,0.27232,0.09689,0.28571,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,3,0,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[249,249,1.0,0.29019,0.05629,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"82bc105aba5046a3","q":"16. (GDR 1) Let $F(n)$ be the set of polynomials $P(x)=a_{0}+a_{1} x+\\cdots+a_{n} x^{n}$, with $a_{0}, a_{1}, \\ldots, a_{n} \\in \\mathbb{R}$ and $0 \\leq a_{0}=a_{n} \\leq a_{1}=a_{n-1} \\leq \\cdots \\leq a_{[n / 2]}=$ $a_{[(n+1) / 2]}$. Prove that if $f \\in F(m)$ and $g \\in F(n)$, then $f g \\in F(m+n)$.","t":[{"b":0,"e":0.2857,"k":"rising","v":0.26338,"x":0.83481,"p":[[0,33,0.0,0.26338,0.25779,0.10714,0.2143,0.32143,0.0,1.0,8,2,0,8,0,8,0,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,2],[4,33,0.1212,0.83481,0.23989,0.78571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,0,0,0,6,0,18],[8,33,0.2424,0.79911,0.30485,0.53571,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,22],[12,33,0.3636,0.62945,0.28316,0.39286,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,8,0,0,4,0,0,8,0,0,1,0,0,1,0,10],[16,33,0.4848,0.62498,0.2896,0.28571,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,10,0,0,1,0,0,9,0,0,1,0,0,1,0,10],[20,33,0.6061,0.65177,0.23674,0.5713,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,4,0,0,3,0,0,13,0,0,3,0,0,1,0,8],[24,33,0.7273,0.51786,0.21354,0.28571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,10,0,0,5,0,0,11,0,0,1,0,0,3,0,2],[28,33,0.8485,0.44191,0.20931,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,12,0,0,2,0,0,13,0,0,0,0,0,0,0,2],[32,33,0.9697,0.38838,0.16456,0.28571,0.35714,0.57143,0.0,0.57143,1,0,0,1,0,3,0,0,12,0,0,4,0,0,12,0,0,0,0,0,0,0,0],[33,33,1.0,0.43748,0.14257,0.28571,0.49979,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,12,0,0,3,0,0,16,0,0,0,0,0,0,0,0]]},{"b":1,"e":1.0,"k":"rising","v":0.33482,"x":0.94643,"p":[[0,42,0.0,0.33482,0.36528,0.0,0.2143,0.57141,0.0,1.0,11,6,0,11,0,5,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,6],[4,42,0.0952,0.89731,0.22086,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,25],[8,42,0.1905,0.80357,0.25442,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,2,0,0,2,0,18],[12,42,0.2857,0.68303,0.26663,0.57142,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,3,0,0,1,0,11],[16,42,0.381,0.75893,0.26592,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,1,0,0,9,0,0,1,0,0,1,0,16],[20,42,0.4762,0.63837,0.26723,0.42857,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,7,0,0,3,0,0,10,0,0,0,0,0,4,0,8],[24,42,0.5714,0.66964,0.28669,0.42857,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,7,0,0,3,0,0,8,0,0,1,0,0,1,0,12],[28,42,0.6667,0.72765,0.3121,0.39286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,7,0,0,1,0,0,4,0,0,1,0,0,2,0,16],[32,42,0.7619,0.59375,0.27689,0.28571,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,11,0,0,0,0,0,11,0,0,1,0,0,1,0,8],[36,42,0.8571,0.69643,0.26904,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,0,0,0,13,0,0,1,0,0,1,0,12],[40,42,0.9524,0.94196,0.16698,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,28],[42,42,1.0,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27]]}]},{"i":"fa9638e50bcbebaf","q":"You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n \u00d7 n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?","t":[{"b":4,"e":0.71429,"k":"flat","v":0.79909,"x":0.99107,"p":[[0,10,0.0,0.79909,0.31107,0.67857,1.0,1.0,0.0,1.0,1,19,1,1,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,19],[4,10,0.4,0.93304,0.17122,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,27],[8,10,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[10,10,1.0,0.93304,0.15561,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,26]]},{"b":7,"e":1.0,"k":"flat","v":0.82143,"x":0.99554,"p":[[0,76,0.0,0.82143,0.28794,0.53572,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,22],[4,76,0.0526,0.87945,0.22901,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,23],[8,76,0.1053,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,76,0.1579,0.95089,0.15815,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,29],[16,76,0.2105,0.92857,0.17857,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,27],[20,76,0.2632,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,76,0.3158,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[28,76,0.3684,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[32,76,0.4211,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[36,76,0.4737,0.92412,0.17119,1.0,1.0,1.0,0.286,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,26],[40,76,0.5263,0.91517,0.14226,0.82143,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,23],[44,76,0.5789,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,76,0.6316,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[52,76,0.6842,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[56,76,0.7368,0.94195,0.12303,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[60,76,0.7895,0.94643,0.12753,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[64,76,0.8421,0.91071,0.18123,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,24],[68,76,0.8947,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[72,76,0.9474,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[76,76,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]}]},{"i":"52716520d2dfe709","q":"A circle  $\\omega$  passes through the two vertices  $B$  and  $C$  of a triangle  $ABC.$  Furthermore,  $\\omega$   intersects segment  $AC$  in  $D\\ne C$  and segment  $AB$  in  $E\\ne B.$  On the ray from  $B$  through  $D$  lies a point  $K$  such that  $|BK| = |AC|,$  and on the ray from  $C$   through  $E$  lies a point  $L$  such that  $|CL| = |AB|.$  Show that the circumcentre  $O$  of triangle  $AKL$  lies on  $\\omega$ 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 singers take part in a festival. The organiser wants to plan  $m$  concerts. For every concert there are  $4$  singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises  $m$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.50893,"x":1.0,"p":[[0,15,0.0,0.50893,0.4935,0.0,0.64286,1.0,0.0,1.0,15,16,0,15,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,15,0.2667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,15,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,15,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[15,15,1.0,0.95982,0.1394,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29]]},{"b":3,"e":1.0,"k":"rising","v":0.58482,"x":0.99107,"p":[[0,19,0.0,0.58482,0.48624,0.0,1.0,1.0,0.0,1.0,13,18,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,18],[4,19,0.2105,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,19,0.4211,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,19,0.6316,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,19,0.8421,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[19,19,1.0,0.93304,0.15561,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,26]]}]},{"i":"4314c686fb547cff","q":"12. C7 (BRA) Prove that for any integer $n \\geq 2$, there exists a set of $2^{n-1}$ points in the plane such that no 3 lie on a line and no $2 n$ are the vertices of a convex $2 n$-gon.","t":[{"b":1,"e":0.4286,"k":"rising","v":0.08025,"x":0.62052,"p":[[0,25,0.0,0.08025,0.13324,0.0,0.0,0.14286,0.0,0.571,21,0,4,21,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.55804,0.31412,0.35714,0.64286,0.85714,0.0,1.0,2,4,0,2,0,6,0,0,0,0,0,6,0,0,2,0,0,7,0,0,5,0,4],[8,25,0.32,0.49994,0.28569,0.28571,0.49979,0.71429,0.0,1.0,2,3,0,2,0,5,0,0,3,0,0,6,0,0,5,0,0,6,0,0,2,0,3],[12,25,0.48,0.47096,0.24927,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,3,0,1,8,0,0,4,0,0,6,0,0,1,0,2],[16,25,0.64,0.62052,0.24901,0.42857,0.64286,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,6,0,0,5,0,0,8,0,0,3,0,5],[20,25,0.8,0.5982,0.2299,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,9,0,0,1,0,0,11,0,0,3,0,3],[24,25,0.96,0.41963,0.23673,0.24999,0.42859,0.60714,0.0,1.0,1,1,0,1,0,7,0,0,5,0,0,9,0,0,2,0,0,7,0,0,0,0,1],[25,25,1.0,0.38836,0.19956,0.24999,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,8,0,0,7,0,0,8,0,0,5,0,0,3,0,0,1,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.08474,"x":0.77228,"p":[[0,27,0.0,0.08474,0.1121,0.0,0.0,0.14286,0.0,0.42857,18,0,4,18,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.64284,0.27433,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,4,0,0,3,0,0,6,0,0,9,0,4],[8,27,0.2963,0.58477,0.27283,0.42857,0.571,0.85704,0.0,1.0,1,4,0,1,0,2,0,0,3,0,0,8,0,0,5,0,0,3,0,0,6,0,4],[12,27,0.4444,0.72543,0.17325,0.57143,0.71429,0.85714,0.357,1.0,0,4,0,0,0,0,0,0,0,0,1,3,0,0,5,0,0,11,0,0,8,0,4],[16,27,0.5926,0.61602,0.21853,0.42859,0.57143,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,3,0,0,6,0,0,7,0,0,8,0,0,4,0,3],[20,27,0.7407,0.68735,0.24619,0.571,0.71429,0.85714,0.14,1.0,0,4,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,8,0,0,10,0,4],[24,27,0.8889,0.77228,0.21978,0.57143,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,5,0,0,6,0,11],[27,27,1.0,0.75442,0.21796,0.57143,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,6,0,0,9,0,8]]}]},{"i":"19c3f922642c973b","q":"7. (FRA 2) Let $a$ be the greatest positive root of the equation $x^{3}-3 x^{2}+1=$ 0 . Show that $\\left[a^{1788}\\right]$ and $\\left[a^{1988}\\right]$ are both divisible by 17 . ( $[x]$ denotes the integer part of $x$.)","t":[{"b":1,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,35,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,35,0.3429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,35,0.4571,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,35,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,35,0.6857,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,35,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,35,0.9143,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[35,35,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,28,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,28,0.1429,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"2b221542c8f87384","q":"A competition involving  $n\\ge 2$  players was held over  $k$  days. In each day, the players received scores of  $1,2,3,\\ldots , n$  points with no players receiving the same score. At the end of the  $k$  days, it was found that each player had exactly  $26$  points in total. Determine all pairs  $(n,k)$  for which this is possible.","t":[{"b":1,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,99,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,99,0.0404,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,99,0.0808,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,99,0.1212,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,99,0.1616,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,99,0.202,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[24,99,0.2424,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,99,0.2828,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[32,99,0.3232,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,99,0.3636,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,99,0.404,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,99,0.4444,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,99,0.4848,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[52,99,0.5253,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[56,99,0.5657,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[60,99,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,99,0.6465,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,99,0.6869,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,99,0.7273,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[76,99,0.7677,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[80,99,0.8081,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,99,0.8485,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[88,99,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,99,0.9293,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[96,99,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[99,99,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,50,0.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[4,50,0.08,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,50,0.16,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,50,0.24,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[16,50,0.32,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,50,0.4,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[24,50,0.48,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[28,50,0.56,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[32,50,0.64,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[36,50,0.72,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,50,0.8,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[44,50,0.88,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[48,50,0.96,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[50,50,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2de48214f27b8a09","q":"4. (GBR 1a) ${ }^{\\mathrm{IMO} 6}$ For all positive integral $n$, $u_{n+1}=u_{n}\\left(u_{n-1}^{2}-2\\right)-u_{1}$, $u_{0}=2$, and $u_{1}=5 / 2$. Prove that  $$ 3 \\log _{2}\\left[u_{n}\\right]=2^{n}-(-1)^{n} $$  where $[x]$ is the integral part of $x$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,60,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,60,0.0667,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[8,60,0.1333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,60,0.2,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,60,0.2667,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,60,0.3333,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,60,0.4,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,60,0.4667,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[32,60,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,60,0.6,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,60,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,60,0.7333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,60,0.8,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,60,0.8667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,60,0.9333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,60,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,115,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,115,0.0348,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,115,0.0696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,115,0.1043,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,115,0.1391,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,115,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,115,0.2087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,115,0.2435,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,115,0.2783,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,115,0.313,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,115,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,115,0.3826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,115,0.4174,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,115,0.4522,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,115,0.487,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,115,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,115,0.5565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,115,0.5913,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,115,0.6261,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,115,0.6609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,115,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,115,0.7304,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,115,0.7652,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,115,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,115,0.8348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,115,0.8696,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[104,115,0.9043,0.9933,0.03728,1.0,1.0,1.0,0.78571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,31],[108,115,0.9391,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[112,115,0.9739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[115,115,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"2680c2d33aeda138","q":"11. C5 (ROM) Every point with integer coordinates in the plane is the center of a disk with radius $1 / 1000$. (a) Prove that there exists an equilateral triangle whose vertices lie in different disks. (b) Prove that every equilateral triangle with vertices in different disks has side length greater than 96.","t":[{"b":5,"e":0.57143,"k":"rising","v":0.36151,"x":0.89284,"p":[[0,63,0.0,0.36151,0.29883,0.105,0.35714,0.57143,0.0,1.0,8,2,2,8,0,5,0,0,3,0,0,3,0,0,8,0,0,3,0,0,0,0,2],[4,63,0.0635,0.89284,0.15975,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,2,0,21],[8,63,0.127,0.86159,0.20974,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,3,0,20],[12,63,0.1905,0.85713,0.17499,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,4,0,17],[16,63,0.254,0.86607,0.19865,0.82143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,6,0,18],[20,63,0.3175,0.79015,0.21722,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,9,0,10],[24,63,0.381,0.74107,0.25364,0.57142,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,6,0,0,3,0,12],[28,63,0.4444,0.74554,0.23619,0.57143,0.78571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,7,0,0,7,0,9],[32,63,0.5079,0.74107,0.22711,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,13,0,0,3,0,9],[36,63,0.5714,0.73658,0.22049,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,7,0,0,3,0,10],[40,63,0.6349,0.70533,0.23942,0.5713,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,6,0,0,5,0,8],[44,63,0.6984,0.70086,0.1968,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,6,0,0,5,0,6],[48,63,0.7619,0.71874,0.22443,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,0,0,0,9,0,0,8,0,0,4,0,8],[52,63,0.8254,0.67391,0.24584,0.57132,0.71429,0.85714,0.14,1.0,0,7,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,6,0,0,4,0,7],[56,63,0.8889,0.73214,0.20748,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,11,0,0,2,0,9],[60,63,0.9524,0.68301,0.2012,0.57143,0.71429,0.74999,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,9,0,0,2,0,6],[63,63,1.0,0.54463,0.21852,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,4,0,0,9,0,0,8,0,0,1,0,2]]},{"b":7,"e":1.0,"k":"rising","v":0.40179,"x":0.9062,"p":[[0,74,0.0,0.40179,0.30606,0.14286,0.42857,0.60714,0.0,1.0,7,2,2,7,0,5,0,0,2,0,0,3,0,0,7,0,0,6,0,0,0,0,2],[4,74,0.0541,0.82143,0.23957,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,17],[8,74,0.1081,0.86158,0.18384,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,3,0,18],[12,74,0.1622,0.85713,0.17859,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,7,0,16],[16,74,0.2162,0.83926,0.20752,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,7,0,16],[20,74,0.2703,0.76786,0.30671,0.67857,0.92857,1.0,0.0,1.0,2,16,0,2,0,2,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,16],[24,74,0.3243,0.87499,0.17405,0.82132,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,7,0,17],[28,74,0.3784,0.75445,0.24546,0.57143,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,5,0,0,6,0,11],[32,74,0.4324,0.86607,0.17835,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,2,0,19],[36,74,0.4865,0.83034,0.18015,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,9,0,12],[40,74,0.5405,0.8482,0.21708,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,6,0,0,0,0,20],[44,74,0.5946,0.85266,0.2487,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,6,0,19],[48,74,0.6486,0.82142,0.2113,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,0,4,0,15],[52,74,0.7027,0.9062,0.15416,0.82132,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,22],[56,74,0.7568,0.85268,0.18723,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,4,0,17],[60,74,0.8108,0.85268,0.16164,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,5,0,15],[64,74,0.8649,0.86607,0.14698,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,9,0,14],[68,74,0.9189,0.80802,0.20706,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,2,0,0,6,0,14],[72,74,0.973,0.89283,0.13835,0.82132,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,18],[74,74,1.0,0.79015,0.1923,0.71429,0.85714,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,0,11,0,8]]}]},{"i":"4ffcb6ba42614d38","q":"15. (USS) ${ }^{\\mathrm{IMO} 5}$ Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?","t":[{"b":2,"e":0.14286,"k":"falling","v":0.15616,"x":0.83036,"p":[[0,66,0.0,0.64732,0.38793,0.25,0.71429,1.0,0.0,1.0,6,12,1,6,0,2,0,0,1,0,0,0,0,0,1,0,0,7,0,0,3,0,12],[4,66,0.0606,0.70089,0.31412,0.67857,0.71429,1.0,0.0,1.0,2,11,0,2,0,3,0,0,1,0,0,0,0,0,2,0,0,11,0,0,2,0,11],[8,66,0.1212,0.83036,0.29329,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,20],[12,66,0.1818,0.78571,0.26726,0.71429,0.78571,1.0,0.0,1.0,1,15,0,1,0,2,0,0,0,0,0,0,0,0,2,0,0,11,0,0,1,0,15],[16,66,0.2424,0.77232,0.32705,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,1,0,0,0,0,0,1,0,0,5,0,0,2,0,18],[20,66,0.303,0.69196,0.29474,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,1,0,0,1,0,0,2,0,0,11,0,0,2,0,10],[24,66,0.3636,0.73661,0.35554,0.67857,0.92857,1.0,0.0,1.0,2,16,0,2,0,5,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,16],[28,66,0.4242,0.6383,0.35184,0.25,0.71429,1.0,0.0,1.0,1,10,0,1,0,7,0,0,2,0,0,1,0,0,0,0,0,7,0,0,4,0,10],[32,66,0.4848,0.59375,0.35375,0.14286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,11,0,0,0,0,0,1,0,0,1,0,0,8,0,0,2,0,9],[36,66,0.5455,0.76339,0.32065,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,16],[40,66,0.6061,0.49999,0.38465,0.14286,0.35714,1.0,0.0,1.0,2,9,0,2,0,12,0,0,2,0,0,1,0,0,2,0,0,2,0,0,2,0,9],[44,66,0.6667,0.50437,0.39775,0.14286,0.5,0.89286,0.0,1.0,4,8,0,4,0,11,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,8],[48,66,0.7273,0.49554,0.35353,0.14286,0.57143,0.71429,0.0,1.0,1,6,0,1,0,13,0,0,1,0,0,1,0,0,0,0,0,9,0,0,1,0,6],[52,66,0.7879,0.51785,0.33455,0.14286,0.64286,0.71429,0.0,1.0,1,5,0,1,0,10,0,0,3,0,0,0,0,0,2,0,0,9,0,0,2,0,5],[56,66,0.8485,0.40177,0.38038,0.14286,0.14286,0.78571,0.0,1.0,2,8,0,2,0,18,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,8],[60,66,0.9091,0.24107,0.25111,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[64,66,0.9697,0.15616,0.06548,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"falling","v":0.16518,"x":0.62054,"p":[[0,28,0.0,0.62054,0.3001,0.5,0.71429,0.75,0.14286,1.0,0,7,0,0,0,7,0,0,1,0,0,0,0,0,5,0,0,11,0,0,1,0,7],[4,28,0.1429,0.58929,0.37923,0.14286,0.71429,1.0,0.0,1.0,2,11,0,2,0,9,0,0,1,0,0,0,0,0,2,0,0,6,0,0,1,0,11],[8,28,0.2857,0.61152,0.42152,0.14286,0.78571,1.0,0.0,1.0,5,15,0,5,0,6,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,15],[12,28,0.4286,0.32589,0.32189,0.14286,0.14286,0.39285,0.0,1.0,3,4,0,3,0,17,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4],[16,28,0.5714,0.28125,0.29555,0.14286,0.14286,0.28571,0.0,1.0,4,3,0,4,0,18,0,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,3],[20,28,0.7143,0.26777,0.26188,0.14286,0.14288,0.32143,0.0,1.0,4,2,0,4,0,17,0,0,3,0,0,3,0,0,1,0,0,2,0,0,0,0,2],[24,28,0.8571,0.2008,0.20161,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,21,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[28,28,1.0,0.16518,0.08073,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6777013b985769fe","q":"A natural number  $k$  is said  $n$ -squared if by colouring the squares of a  $2n \\times k$  chessboard, in any manner, with  $n$  different colours, we can find  $4$  separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of  $n$ , the smallest natural  $k$  that is  $n$ -squared.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.76786,"x":0.95094,"p":[[0,45,0.0,0.76786,0.36202,0.67857,1.0,1.0,0.0,1.0,5,19,1,5,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,19],[4,45,0.0889,0.95089,0.18073,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[8,45,0.1778,0.9241,0.18893,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[12,45,0.2667,0.8348,0.30539,0.71429,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,22],[16,45,0.3556,0.7857,0.26964,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,5,0,0,3,0,16],[20,45,0.4444,0.89283,0.2202,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,23],[24,45,0.5333,0.933,0.14725,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,25],[28,45,0.6222,0.9107,0.15468,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,22],[32,45,0.7111,0.95094,0.11613,1.0,1.0,1.0,0.43,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[36,45,0.8,0.89731,0.19313,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,19],[40,45,0.8889,0.87945,0.16018,0.82132,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,18],[44,45,0.9778,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[45,45,1.0,0.88392,0.12078,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,14,0,13]]},{"b":7,"e":0.85714,"k":"flat","v":0.74551,"x":0.95982,"p":[[0,28,0.0,0.84375,0.28873,0.85714,1.0,1.0,0.0,1.0,2,21,2,2,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,21],[4,28,0.1429,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[8,28,0.2857,0.9241,0.22864,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[12,28,0.4286,0.88839,0.23619,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,22],[16,28,0.5714,0.94196,0.14223,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[20,28,0.7143,0.875,0.25692,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,23],[24,28,0.8571,0.86159,0.15767,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,8,0,15],[28,28,1.0,0.74551,0.16651,0.71429,0.857,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,10,0,0,17,0,0]]}]},{"i":"56d6710beb4bcc0d","q":"20. G6 (IND) Let $A B C$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose $A P, B P, C P$ meet the sides $B C, C A, A B$ (or extensions thereof) in $D, E, F$, respectively. Suppose further that the areas of triangles $P B D, P C E, P A F$ are all equal. Prove that each of these areas is equal to the area of triangle $A B C$ itself.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.19107,"x":0.56692,"p":[[0,105,0.0,0.19107,0.17604,0.0,0.21428,0.28571,0.0,0.71429,11,0,10,11,0,5,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,105,0.0381,0.56692,0.23002,0.39288,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,1,0,0,10,0,0,9,0,0,2,0,2],[8,105,0.0762,0.49106,0.2111,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,9,0,0,3,0,0,8,0,0,1,0,1],[12,105,0.1143,0.44182,0.20944,0.28571,0.4286,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,3,0,0,7,0,0,9,0,0,6,0,0,0,0,0],[16,105,0.1524,0.42395,0.19039,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,3,0,0,12,0,0,7,0,0,4,0,0,5,0,0,1,0,0],[20,105,0.1905,0.4241,0.20971,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,8,0,0,2,0,0,9,0,0,6,0,0,0,0,0],[24,105,0.2286,0.40624,0.18934,0.2857,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,5,0,0,9,0,0,10,0,0,3,0,0,4,0,0,1,0,0],[28,105,0.2667,0.45978,0.19472,0.28571,0.42859,0.60714,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,4,0,0,7,0,0,8,0,0,0,0,0],[32,105,0.3048,0.41065,0.16262,0.28571,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,4,0,0,8,0,0,11,0,0,6,0,0,3,0,0,0,0,0],[36,105,0.3429,0.46422,0.18554,0.28571,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,8,0,0,6,0,0,8,0,0,7,0,0,0,0,0],[40,105,0.381,0.44636,0.20434,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,6,0,0,7,0,0,7,0,0,7,0,0,0,0,0],[44,105,0.419,0.48204,0.22245,0.28571,0.49979,0.71429,0.14,1.0,0,1,0,0,0,2,0,0,13,0,0,1,0,0,5,0,0,10,0,0,0,0,1],[48,105,0.4571,0.45084,0.17893,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,5,0,0,8,0,0,6,0,0,0,0,0],[52,105,0.4952,0.40623,0.21754,0.28571,0.28571,0.57143,0.14286,0.857,0,0,0,0,0,7,0,0,10,0,0,4,0,0,4,0,0,6,0,0,1,0,0],[56,105,0.5333,0.45981,0.19144,0.28571,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,2,0,0,12,0,0,3,0,0,7,0,0,8,0,0,0,0,0],[60,105,0.5714,0.33926,0.17764,0.2857,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,7,0,0,16,0,0,3,0,0,2,0,0,4,0,0,0,0,0],[64,105,0.6095,0.39276,0.24231,0.14286,0.35714,0.57143,0.0,1.0,1,1,0,1,0,9,0,0,6,0,0,6,0,0,3,0,0,6,0,0,0,0,1],[68,105,0.6476,0.41509,0.24588,0.2857,0.28571,0.71429,0.0,1.0,2,1,0,2,0,4,0,0,11,0,0,4,0,0,2,0,0,8,0,0,0,0,1],[72,105,0.6857,0.43736,0.23137,0.24999,0.4286,0.57143,0.14,1.0,0,1,0,0,0,8,0,0,5,0,0,5,0,0,7,0,0,6,0,0,0,0,1],[76,105,0.7238,0.404,0.17731,0.28571,0.28571,0.57111,0.14286,0.85714,0,0,0,0,0,2,0,0,16,0,0,4,0,1,5,0,0,3,0,0,1,0,0],[80,105,0.7619,0.42405,0.20662,0.2857,0.35714,0.57143,0.14286,0.857,0,0,0,0,0,5,0,0,11,0,0,3,0,0,7,0,0,5,0,0,1,0,0],[84,105,0.8,0.40179,0.24598,0.14289,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,9,0,0,8,0,0,5,0,0,4,0,0,4,0,0,0,0,2],[88,105,0.8381,0.39278,0.15962,0.28571,0.28586,0.571,0.14286,0.71429,0,0,0,0,0,3,0,0,14,0,0,5,0,0,8,0,0,2,0,0,0,0,0],[92,105,0.8762,0.41513,0.19995,0.28571,0.42857,0.5711,0.14286,1.0,0,1,0,0,0,5,0,0,10,0,0,5,0,0,9,0,0,2,0,0,0,0,1],[96,105,0.9143,0.39731,0.20742,0.28571,0.28571,0.5711,0.14286,1.0,0,1,0,0,0,5,0,0,13,0,0,5,0,0,4,0,0,4,0,0,0,0,1],[100,105,0.9524,0.37945,0.16981,0.28571,0.35714,0.57111,0.14286,0.71429,0,0,0,0,0,6,0,0,10,0,0,7,0,0,7,0,0,2,0,0,0,0,0],[104,105,0.9905,0.29464,0.13803,0.25002,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,8,0,0,19,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[105,105,1.0,0.27223,0.12046,0.14286,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,11,0,0,15,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.28125,"x":0.59807,"p":[[0,54,0.0,0.28125,0.19061,0.24999,0.28571,0.42857,0.0,0.71429,7,0,7,7,0,1,0,0,15,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[4,54,0.0741,0.49104,0.18876,0.28571,0.49979,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,8,0,0,6,0,0,6,0,0,10,0,0,0,0,0],[8,54,0.1481,0.52674,0.17654,0.39286,0.57121,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,5,0,0,10,0,0,7,0,0,2,0,0],[12,54,0.2222,0.41518,0.19352,0.28571,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,15,0,0,5,0,0,3,0,0,7,0,0,0,0,0],[16,54,0.2963,0.44638,0.17031,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,10,0,0,7,0,0,8,0,0,5,0,0,0,0,0],[20,54,0.3704,0.45084,0.17893,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,7,0,0,8,0,0,4,0,0,1,0,0],[24,54,0.4444,0.54017,0.20119,0.28571,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,5,0,0,5,0,0,11,0,0,1,0,1],[28,54,0.5185,0.39282,0.21721,0.2857,0.28571,0.571,0.14286,1.0,0,1,0,0,0,6,0,0,13,0,0,4,0,0,3,0,0,5,0,0,0,0,1],[32,54,0.5926,0.46421,0.18894,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,2,0,0,9,0,0,7,0,0,9,0,0,4,0,0,0,0,1],[36,54,0.6667,0.53569,0.22303,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,5,0,0,5,0,0,7,0,0,10,0,0,0,0,2],[40,54,0.7407,0.53567,0.20824,0.28571,0.571,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,9,0,0,5,0,0,8,0,0,7,0,0,1,0,2],[44,54,0.8148,0.46872,0.20586,0.2857,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,9,0,0,5,0,0,8,0,0,6,0,0,0,0,1],[48,54,0.8889,0.59807,0.24355,0.42857,0.64286,0.71429,0.14,1.0,0,3,0,0,0,3,0,0,4,0,0,2,0,0,7,0,0,10,0,0,3,0,3],[52,54,0.963,0.5714,0.22867,0.42859,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,5,0,0,7,0,0,10,0,0,2,0,2],[54,54,1.0,0.58478,0.23517,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,8,0,0,4,0,0,7,0,0,4,0,3]]}]},{"i":"b2c929bee1b5282e","q":"A number n is called charming when  $ 4k^2 + n $  is a prime number for every  $ 0 \\leq k <n $  integer, find all charming numbers.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.26776,"x":0.56697,"p":[[0,85,0.0,0.29465,0.04971,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,85,0.0471,0.45979,0.26662,0.28571,0.28571,0.57143,0.0,1.0,1,5,1,1,0,0,0,0,17,0,0,2,0,0,7,0,0,0,0,0,0,0,5],[8,85,0.0941,0.50893,0.30917,0.28571,0.28571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,20,0,0,0,0,0,3,0,0,0,0,0,1,0,8],[12,85,0.1412,0.37945,0.20393,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,25,0,0,1,0,0,3,0,0,0,0,0,1,0,2],[16,85,0.1882,0.3616,0.1749,0.2857,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,26,0,0,0,0,0,4,0,0,0,0,0,1,0,1],[20,85,0.2353,0.37054,0.1885,0.28571,0.28571,0.28579,0.2857,1.0,0,2,0,0,0,0,0,0,25,0,0,1,0,0,4,0,0,0,0,0,0,0,2],[24,85,0.2824,0.38838,0.18637,0.2857,0.28571,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,20,0,0,1,0,0,8,0,0,1,0,0,0,0,1],[28,85,0.3294,0.37946,0.2412,0.28571,0.28571,0.28571,0.0,1.0,1,3,1,1,0,1,0,0,23,0,0,1,0,0,2,0,0,0,0,0,1,0,3],[32,85,0.3765,0.36607,0.16728,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,0,0,0,5,0,0,1,0,0,0,0,1],[36,85,0.4235,0.35713,0.16365,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,1],[40,85,0.4706,0.39284,0.21427,0.28571,0.28571,0.57141,0.14286,1.0,0,2,0,0,0,3,0,0,19,0,0,0,0,0,7,0,0,1,0,0,0,0,2],[44,85,0.5176,0.35714,0.19885,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,3,0,0,22,0,0,1,0,0,4,0,0,0,0,0,0,0,2],[48,85,0.5647,0.3348,0.13173,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,24,0,0,0,0,0,5,0,0,1,0,0,0,0,0],[52,85,0.6118,0.3259,0.1525,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,26,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[56,85,0.6588,0.32589,0.19638,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,5,0,0,23,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[60,85,0.7059,0.26776,0.10575,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,6,0,0,23,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[64,85,0.7529,0.3348,0.16212,0.2857,0.28571,0.28571,0.0,1.0,1,1,1,1,0,0,0,0,26,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[68,85,0.8,0.34375,0.13767,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,27,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[72,85,0.8471,0.34375,0.12299,0.2857,0.28571,0.28579,0.14286,0.57143,0,0,0,0,0,1,0,0,24,0,0,0,0,0,7,0,0,0,0,0,0,0,0],[76,85,0.8941,0.49552,0.24739,0.28571,0.42836,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,10,0,0,1,0,0,1,0,4],[80,85,0.9412,0.51339,0.25218,0.28571,0.5,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,14,0,0,2,0,0,9,0,0,2,0,0,0,0,5],[84,85,0.9882,0.56697,0.20037,0.53575,0.57143,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,2,0,0,19,0,0,1,0,0,0,0,4],[85,85,1.0,0.5222,0.1912,0.39286,0.57143,0.57143,0.14,1.0,0,2,0,0,0,1,0,0,7,0,0,2,0,0,19,0,0,0,0,0,1,0,2]]},{"b":5,"e":0.14286,"k":"falling","v":0.17393,"x":0.43303,"p":[[0,54,0.0,0.33482,0.10479,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[4,54,0.0741,0.36606,0.12844,0.28571,0.28571,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[8,54,0.1481,0.43303,0.24868,0.2857,0.28571,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,22,0,0,1,0,0,3,0,0,2,0,0,0,0,4],[12,54,0.2222,0.38392,0.19377,0.2857,0.28571,0.35714,0.2857,1.0,0,2,0,0,0,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,2],[16,54,0.2963,0.33034,0.17653,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,25,0,0,0,0,0,2,0,0,0,0,0,1,0,1],[20,54,0.3704,0.32143,0.16752,0.2857,0.28571,0.28571,0.0,1.0,1,1,1,1,0,3,0,0,22,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[24,54,0.4444,0.35268,0.14719,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[28,54,0.5185,0.42857,0.26964,0.2857,0.28571,0.57143,0.14286,1.0,0,5,0,0,0,3,0,0,18,0,0,2,0,0,4,0,0,0,0,0,0,0,5],[32,54,0.5926,0.35705,0.24231,0.2857,0.28571,0.28571,0.14,1.0,0,3,0,0,0,5,0,0,22,0,0,0,0,0,1,0,0,0,0,0,1,0,3],[36,54,0.6667,0.25437,0.10565,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,54,0.7407,0.24982,0.15984,0.14286,0.28571,0.28571,0.0,1.0,1,1,0,1,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,54,0.8148,0.3616,0.25996,0.24999,0.28571,0.32143,0.0,1.0,1,3,0,1,0,7,0,0,16,0,0,1,0,0,3,0,0,0,0,0,1,0,3],[48,54,0.8889,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.18723,0.0664,0.14286,0.14286,0.2857,0.14,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.17393,0.05915,0.14286,0.14286,0.14287,0.14,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"45392a84ba621fe5","q":"A sequence of integers $a_{0}, a_{1}, a_{2}, \\ldots$ is called kawaii, if $a_{0}=0, a_{1}=1$, and, for any positive integer $n$, we have  $$ \\left(a_{n+1}-3 a_{n}+2 a_{n-1}\\right)\\left(a_{n+1}-4 a_{n}+3 a_{n-1}\\right)=0 . $$  An integer is called kawaii if it belongs to a kawaii sequence. Suppose that two consecutive positive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 3 divides $m$, and that $m / 3$ is kawaii. (China)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.26338,"x":0.43748,"p":[[0,42,0.0,0.26338,0.17166,0.14286,0.14286,0.42858,0.0,0.57143,1,0,0,1,0,18,0,0,3,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[4,42,0.0952,0.34375,0.22263,0.14286,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,5,0,0,3,0,0,9,0,0,1,0,0,1,0,0],[8,42,0.1905,0.33926,0.20435,0.14286,0.28571,0.5711,0.0,0.71429,1,0,0,1,0,12,0,0,5,0,0,4,0,0,8,0,0,2,0,0,0,0,0],[12,42,0.2857,0.43748,0.20805,0.2857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,7,0,0,4,0,0,8,0,0,7,0,0,5,0,0,1,0,0],[16,42,0.381,0.3214,0.20822,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,5,0,0,8,0,0,5,0,0,2,0,0,0,0,0],[20,42,0.4762,0.39286,0.24484,0.14286,0.42859,0.57143,0.0,0.85714,3,0,0,3,0,7,0,0,4,0,0,6,0,0,7,0,0,3,0,0,2,0,0],[24,42,0.5714,0.33033,0.21849,0.14286,0.28571,0.57111,0.0,0.71429,3,0,0,3,0,11,0,0,3,0,0,5,0,0,8,0,0,2,0,0,0,0,0],[28,42,0.6667,0.29902,0.20635,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,12,0,0,5,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[32,42,0.7619,0.37051,0.19513,0.14286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,5,0,0,7,0,0,10,0,0,1,0,0,0,0,0],[36,42,0.8571,0.35714,0.19562,0.14286,0.28571,0.57143,0.0,0.57143,1,0,0,1,0,10,0,0,6,0,0,2,0,0,13,0,0,0,0,0,0,0,0],[40,42,0.9524,0.29914,0.1902,0.14286,0.28571,0.4286,0.0,0.57143,4,0,0,4,0,9,0,0,5,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[42,42,1.0,0.26339,0.17536,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,10,0,0,4,0,0,11,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.2142,"x":0.47772,"p":[[0,70,0.0,0.2142,0.1473,0.14286,0.14286,0.17857,0.0,0.57143,1,0,0,1,0,23,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[4,70,0.0571,0.47772,0.2639,0.25,0.5005,0.71429,0.14286,1.0,0,2,0,0,0,8,0,0,4,0,0,4,0,0,7,0,0,5,0,0,2,0,2],[8,70,0.1143,0.38839,0.21498,0.14286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,9,0,0,4,0,0,6,0,0,9,0,0,2,0,0,1,0,0],[12,70,0.1714,0.38392,0.2271,0.14286,0.35716,0.57143,0.0,0.71429,1,0,0,1,0,11,0,0,4,0,0,1,0,0,11,0,0,4,0,0,0,0,0],[16,70,0.2286,0.36161,0.22583,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,9,0,0,6,0,0,3,0,0,11,0,0,0,0,0,0,0,1],[20,70,0.2857,0.39286,0.19233,0.1429,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,4,0,0,8,0,0,8,0,0,3,0,0,0,0,0],[24,70,0.3429,0.36158,0.20195,0.14286,0.42857,0.57111,0.0,0.71429,1,0,0,1,0,11,0,0,2,0,0,8,0,0,8,0,0,2,0,0,0,0,0],[28,70,0.4,0.30789,0.21162,0.14286,0.14286,0.571,0.0,0.71429,1,0,0,1,0,16,0,0,4,0,0,1,0,0,8,0,0,2,0,0,0,0,0],[32,70,0.4571,0.39286,0.26964,0.14286,0.28571,0.57143,0.0,1.0,1,1,0,1,0,12,0,0,4,0,0,2,0,0,8,0,0,1,0,0,3,0,1],[36,70,0.5143,0.26784,0.16266,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,5,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[40,70,0.5714,0.28124,0.18027,0.14286,0.14286,0.42857,0.14286,0.57143,0,0,0,0,0,19,0,0,2,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[44,70,0.6286,0.30348,0.17413,0.14286,0.2857,0.42857,0.14,0.57143,0,0,0,0,0,15,0,0,5,0,0,5,0,0,7,0,0,0,0,0,0,0,0],[48,70,0.6857,0.37054,0.18851,0.14286,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,11,0,0,4,0,0,4,0,0,13,0,0,0,0,0,0,0,0],[52,70,0.7429,0.33036,0.19377,0.14286,0.2857,0.57143,0.0,0.57143,1,0,0,1,0,13,0,0,3,0,0,5,0,0,10,0,0,0,0,0,0,0,0],[56,70,0.8,0.37937,0.19114,0.14286,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,9,0,0,3,0,0,6,0,0,13,0,0,0,0,0,0,0,0],[60,70,0.8571,0.3973,0.20431,0.14286,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,9,0,0,3,0,0,3,0,0,15,0,0,1,0,0,0,0,0],[64,70,0.9143,0.37945,0.20704,0.14286,0.4286,0.57143,0.0,0.57143,2,0,0,2,0,9,0,0,2,0,0,4,0,0,15,0,0,0,0,0,0,0,0],[68,70,0.9714,0.39277,0.17509,0.24999,0.42857,0.57143,0.14,0.57143,0,0,0,0,0,8,0,0,5,0,0,6,0,0,13,0,0,0,0,0,0,0,0],[70,70,1.0,0.38392,0.18362,0.14286,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,10,0,0,3,0,0,6,0,0,13,0,0,0,0,0,0,0,0]]}]},{"i":"4af6c0a992b6a597","q":"7. G1 (BUL) ${ }^{\\mathrm{IMO} 1}$ Let $A, B, C$, and $D$ be distinct points on a line, in that order. The circles with diameters $A C$ and $B D$ intersect at $X$ and $Y . O$ is an arbitrary point on the line $X Y$ but not on $A D$. $C O$ intersects the circle with diameter $A C$ again at $M$, and $B O$ intersects the other circle again at $N$. Prove that the lines $A M, D N$, and $X Y$ are concurrent.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.04464,"x":0.42411,"p":[[0,49,0.0,0.42411,0.44102,0.0,0.2857,1.0,0.0,1.0,14,11,0,14,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,11],[4,49,0.0816,0.16518,0.33141,0.0,0.0,0.14286,0.0,1.0,23,4,0,23,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[8,49,0.1633,0.08482,0.22548,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,49,0.2449,0.08033,0.1672,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,49,0.3265,0.08036,0.20183,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,49,0.4082,0.15625,0.25595,0.0,0.0,0.28571,0.0,1.0,21,1,0,21,0,0,0,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[24,49,0.4898,0.04464,0.14914,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,49,0.5714,0.09821,0.22142,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[32,49,0.6531,0.0625,0.15947,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,49,0.7347,0.11161,0.17029,0.0,0.0,0.28571,0.0,0.4286,22,0,0,22,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.21427,0.24998,0.0,0.14285,0.28571,0.0,0.71429,16,0,0,16,0,0,0,0,9,0,0,2,0,0,1,0,0,4,0,0,0,0,0],[44,49,0.898,0.26786,0.24679,0.0,0.28571,0.42857,0.0,0.71429,12,0,0,12,0,1,0,0,7,0,0,7,0,0,1,0,0,4,0,0,0,0,0],[48,49,0.9796,0.29911,0.25344,0.0,0.28571,0.42857,0.0,0.71429,11,0,0,11,0,0,0,0,7,0,0,8,0,0,1,0,0,5,0,0,0,0,0],[49,49,1.0,0.32586,0.248,0.0,0.42857,0.4642,0.0,0.71429,9,0,0,9,0,2,0,0,4,0,0,9,0,0,4,0,0,4,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.04464,"x":0.37054,"p":[[0,43,0.0,0.37054,0.43428,0.0,0.14285,1.0,0.0,1.0,16,9,0,16,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,9],[4,43,0.093,0.10268,0.2237,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,43,0.186,0.04464,0.12596,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.04912,0.14557,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,43,0.3721,0.12054,0.21756,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,1,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[20,43,0.4651,0.0625,0.13333,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.11607,0.21852,0.0,0.0,0.2857,0.0,1.0,23,1,0,23,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[32,43,0.7442,0.0758,0.12864,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.08482,0.18161,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,43,0.9302,0.0625,0.19541,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[43,43,1.0,0.09375,0.21011,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"5dddfaafd2292866","q":"$a_1,a_2,\\cdots ,a_n\\geq 0,\\lambda\\in [0,1],A=\\dfrac{1}{n}\\sum_{i=1}^n,$  in  $a_i,a_1,a_2,\\cdots a_n,$  there\u2019re exactly  $m$  numbers not smaller than  $\\lambda A.$ Prove that   $$ m\\sum_{i=1}^na_i^2\\geq (1-\\lambda)^2\\left(\\sum_{i=1}^na_i\\right)^2 $$  *Proposed by Luo ZhenHua*","t":[{"b":4,"e":0.57143,"k":"flat","v":0.60265,"x":0.91964,"p":[[0,37,0.0,0.64732,0.42706,0.14286,0.92857,1.0,0.0,1.0,7,16,0,7,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,16],[4,37,0.1081,0.80357,0.26666,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,0,1,0,18],[8,37,0.2162,0.86607,0.22851,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,3,0,21],[12,37,0.3243,0.91964,0.16728,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[16,37,0.4324,0.7991,0.24448,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,2,0,0,0,0,18],[20,37,0.5405,0.77232,0.25966,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,0,3,0,15],[24,37,0.6486,0.80354,0.21946,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,3,0,0,2,0,16],[28,37,0.7568,0.64283,0.23146,0.571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,3,0,0,12,0,0,3,0,0,5,0,5],[32,37,0.8649,0.67857,0.24484,0.57143,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,3,0,0,12,0,0,3,0,0,2,0,9],[36,37,0.973,0.60265,0.18467,0.57143,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,2,0,0,15,0,0,9,0,0,1,0,2],[37,37,1.0,0.65623,0.14223,0.57143,0.57143,0.71429,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,7,0,0,2,0,3]]},{"b":5,"e":0.1429,"k":"falling","v":0.14732,"x":0.94643,"p":[[0,234,0.0,0.64286,0.42106,0.32143,1.0,1.0,0.0,1.0,8,17,0,8,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,17],[4,234,0.0171,0.86607,0.20806,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,2,0,21],[8,234,0.0342,0.76339,0.33808,0.57143,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,1,0,0,1,0,0,3,0,0,4,0,0,0,0,19],[12,234,0.0513,0.87499,0.26185,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,25],[16,234,0.0684,0.83927,0.27376,0.67857,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,21],[20,234,0.0855,0.86161,0.28901,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[24,234,0.1026,0.71875,0.34899,0.57143,0.85714,1.0,0.0,1.0,4,15,0,4,0,1,0,0,1,0,0,0,0,0,4,0,0,5,0,0,2,0,15],[28,234,0.1197,0.78124,0.31133,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,1,0,0,1,0,0,4,0,0,4,0,0,0,0,19],[32,234,0.1368,0.76338,0.33997,0.57143,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,19],[36,234,0.1538,0.83482,0.33333,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,24],[40,234,0.1709,0.86607,0.30917,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,26],[44,234,0.188,0.94643,0.17035,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[48,234,0.2051,0.84375,0.33571,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,25],[52,234,0.2222,0.91071,0.21943,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[56,234,0.2393,0.89732,0.25564,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[60,234,0.2564,0.69643,0.39245,0.46429,1.0,1.0,0.0,1.0,4,18,0,4,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,18],[64,234,0.2735,0.79018,0.34253,0.57143,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,21],[68,234,0.2906,0.73214,0.3567,0.57143,1.0,1.0,0.0,1.0,4,17,0,4,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,0,2,0,17],[72,234,0.3077,0.70534,0.33109,0.53539,0.78571,1.0,0.0,1.0,3,15,0,3,0,0,0,0,2,0,0,3,0,0,6,0,0,2,0,0,1,0,15],[76,234,0.3248,0.67856,0.37458,0.42857,0.85714,1.0,0.0,1.0,4,14,0,4,0,3,0,0,0,0,0,3,0,0,2,0,0,2,0,0,4,0,14],[80,234,0.3419,0.65177,0.38122,0.39286,0.71429,1.0,0.0,1.0,5,14,0,5,0,2,0,0,1,0,0,2,0,0,3,0,0,4,0,0,1,0,14],[84,234,0.359,0.73214,0.29397,0.57143,0.78571,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,2,0,0,8,0,0,2,0,0,1,0,15],[88,234,0.3761,0.58482,0.37348,0.14286,0.57143,1.0,0.0,1.0,5,10,0,5,0,4,0,0,0,0,0,2,0,0,6,0,0,3,0,0,2,0,10],[92,234,0.3932,0.63393,0.37617,0.28571,0.64286,1.0,0.0,1.0,5,13,0,5,0,1,0,0,3,0,0,1,0,0,6,0,0,1,0,0,2,0,13],[96,234,0.4103,0.58481,0.39181,0.14286,0.57143,1.0,0.0,1.0,4,13,0,4,0,6,0,0,1,0,0,1,0,0,6,0,0,1,0,0,0,0,13],[100,234,0.4274,0.65176,0.32916,0.42859,0.64286,1.0,0.0,1.0,3,12,0,3,0,1,0,0,2,0,0,3,0,0,7,0,0,4,0,0,0,0,12],[104,234,0.4444,0.58481,0.36659,0.14286,0.64286,1.0,0.0,1.0,4,9,0,4,0,5,0,0,1,0,0,1,0,0,5,0,0,4,0,0,3,0,9],[108,234,0.4615,0.56694,0.3838,0.25,0.57143,1.0,0.0,1.0,6,11,0,6,0,2,0,0,3,0,0,1,0,0,7,0,0,1,0,0,1,0,11],[112,234,0.4786,0.49552,0.41185,0.0,0.57121,1.0,0.0,1.0,9,10,0,9,0,3,0,0,2,0,0,1,0,0,5,0,0,1,0,0,1,0,10],[116,234,0.4957,0.51338,0.40225,0.10714,0.57121,1.0,0.0,1.0,8,9,0,8,0,4,0,0,0,0,0,3,0,0,3,0,0,3,0,0,2,0,9],[120,234,0.5128,0.59822,0.37362,0.39286,0.57143,1.0,0.0,1.0,5,11,0,5,0,2,0,0,1,0,0,6,0,0,3,0,0,1,0,0,3,0,11],[124,234,0.5299,0.67857,0.36246,0.42857,0.78571,1.0,0.0,1.0,4,16,0,4,0,1,0,0,1,0,0,3,0,0,7,0,0,0,0,0,0,0,16],[128,234,0.547,0.63392,0.41178,0.14286,0.78571,1.0,0.0,1.0,7,15,0,7,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,0,1,0,15],[132,234,0.5641,0.61607,0.43805,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,17],[136,234,0.5812,0.58929,0.40838,0.14286,0.71429,1.0,0.0,1.0,6,13,0,6,0,4,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1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how much time will the particle first return to the origin?\n\n(A)  $\\text{0.785 s}$ (B)  $\\text{1.26 s}$ (C)  $\\text{1.57 s}$ (D)  $\\text{2.00 s}$ (E)  $\\text{3.14 s}$","t":[{"b":3,"e":0.0,"k":"volatile","v":0.0,"x":0.74997,"p":[[0,13,0.0,0.74997,0.19887,0.57143,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,17,0,3],[4,13,0.3077,0.18303,0.29501,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,3,0,1],[8,13,0.6154,0.20089,0.23381,0.0,0.2857,0.28571,0.0,1.0,15,1,0,15,0,0,0,0,13,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[12,13,0.9231,0.125,0.30878,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.63393,"x":0.89732,"p":[[0,9,0.0,0.77679,0.16342,0.57143,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,19,0,3],[4,9,0.4444,0.63393,0.37447,0.28571,0.85714,1.0,0.0,1.0,4,10,0,4,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,9,0,10],[8,9,0.8889,0.73661,0.30745,0.78571,0.85714,0.85714,0.0,1.0,3,7,0,3,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,17,0,7],[9,9,1.0,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9]]}]},{"i":"d4981998e11281f8","q":"An *up-right path* from  $(a, b) \\in \\mathbb{R}^2$  to  $(c, d) \\in \\mathbb{R}^2$  is a finite sequence  $(x_1, y_z), \\dots, (x_k, y_k)$  of points in  $ \\mathbb{R}^2 $  such that  $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$ , and for each  $1 \\le i < k$  we have that either  $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$  or  $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$ . Two up-right paths are said to intersect if they share any point.\n\nFind the number of pairs  $(A, B)$  where  $A$  is an up-right path from  $(0, 0)$  to  $(4, 4)$ ,  $B$  is an up-right path from  $(2, 0)$  to  $(6, 4)$ , and  $A$  and  $B$  do not intersect.","t":[{"b":1,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,24,0.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,24,0.1667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,24,0.3333,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,24,0.5,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,24,1.0,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]},{"b":3,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,29,0.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[4,29,0.1379,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,29,0.2759,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,29,0.4138,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,29,0.9655,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[29,29,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"40a0f9d849b511d3","q":"Determine the gcd of all numbers of the form  $p^8-1$ , with p a prime above 5.","t":[{"b":5,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,34,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,34,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,34,0.2353,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,34,0.3529,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,34,0.4706,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,34,0.5882,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[34,34,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.83035,"x":1.0,"p":[[0,25,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,25,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,25,0.32,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,25,0.48,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,25,0.64,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,25,0.8,0.93302,0.1071,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[24,25,0.96,0.84821,0.11259,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,9],[25,25,1.0,0.83035,0.12595,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,12,0,8]]}]},{"i":"237c9f7a89c6c47e","q":"BLR   Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that  $$ f(x-f(y))>y f(x)+x $$","t":[{"b":4,"e":0.42857,"k":"rising","v":0.13393,"x":0.41517,"p":[[0,46,0.0,0.13393,0.11811,0.0,0.14286,0.17857,0.0,0.42857,11,0,2,11,0,13,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.27233,0.16888,0.14286,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,12,0,0,13,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[8,46,0.1739,0.29911,0.2212,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,6,0,0,8,0,0,7,0,0,1,0,0,4,0,0,0,0,0],[12,46,0.2609,0.35265,0.19877,0.14286,0.28571,0.4642,0.0,0.71429,2,0,0,2,0,7,0,0,8,0,0,7,0,0,5,0,0,3,0,0,0,0,0],[16,46,0.3478,0.3482,0.22569,0.14286,0.28571,0.46418,0.0,0.85714,4,0,0,4,0,5,0,0,9,0,0,6,0,0,4,0,0,3,0,0,1,0,0],[20,46,0.4348,0.3125,0.24074,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,7,0,0,10,0,0,4,0,0,0,0,0,5,0,0,1,0,0],[24,46,0.5217,0.29911,0.20316,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,14,0,0,7,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[28,46,0.6087,0.32141,0.14283,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,15,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[32,46,0.6957,0.39732,0.19475,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,12,0,0,6,0,0,4,0,0,4,0,0,1,0,0],[36,46,0.7826,0.30795,0.20245,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,10,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[40,46,0.8696,0.38829,0.1997,0.28571,0.28571,0.5711,0.14,1.0,0,1,0,0,0,6,0,0,11,0,0,6,0,0,6,0,0,2,0,0,0,0,1],[44,46,0.9565,0.33482,0.14987,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,16,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[46,46,1.0,0.41517,0.1726,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,8,0,0,8,0,0,10,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.16509,"x":0.375,"p":[[0,33,0.0,0.16509,0.12933,0.14214,0.14286,0.2857,0.0,0.57143,7,0,1,7,0,16,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.375,0.23077,0.14286,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,10,0,0,4,0,0,3,0,0,4,0,0,2,0,0],[8,33,0.2424,0.27231,0.19676,0.14286,0.21428,0.42857,0.0,0.85714,3,0,0,3,0,13,0,0,6,0,0,7,0,0,1,0,0,1,0,0,1,0,0],[12,33,0.3636,0.34821,0.23673,0.14286,0.28571,0.57143,0.0,0.85714,5,0,0,5,0,5,0,0,7,0,0,6,0,0,5,0,0,3,0,0,1,0,0],[16,33,0.4848,0.25893,0.21261,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,11,0,0,5,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[20,33,0.6061,0.19634,0.20441,0.0,0.14286,0.32143,0.0,0.71429,12,0,0,12,0,8,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[24,33,0.7273,0.26339,0.24251,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,3,0,0,5,0,0,9,0,0,3,0,0,0,0,0,0,0,1],[28,33,0.8485,0.26338,0.23174,0.0,0.2857,0.42857,0.0,0.71429,10,0,0,10,0,5,0,0,4,0,0,9,0,0,1,0,0,3,0,0,0,0,0],[32,33,0.9697,0.23661,0.22192,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,6,0,0,3,0,0,9,0,0,1,0,0,2,0,0,0,0,0],[33,33,1.0,0.19197,0.18073,0.0,0.14288,0.42857,0.0,0.4286,13,0,0,13,0,4,0,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eaaaf66fa6b6ab58","q":"Consider a set  $X$  with  $|X| = n\\geq 1$  elements. A family  $\\mathcal{F}$  of distinct subsets of  $X$  is said to have property  $\\mathcal{P}$  if there exist  $A,B \\in \\mathcal{F}$  so that  $A\\subset B$  and  $|B\\setminus A| = 1$ .\n\ni) Determine the least value  $m$ , so that any family  $\\mathcal{F}$  with  $|\\mathcal{F}| > m$  has property  $\\mathcal{P}$ .\nii) Describe all families  $\\mathcal{F}$  with  $|\\mathcal{F}| = m$ , and not having property  $\\mathcal{P}$ .\n\n(*Dan Schwarz*)","t":[{"b":1,"e":1.0,"k":"flat","v":0.90623,"x":0.99554,"p":[[0,23,0.0,0.90623,0.15818,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[4,23,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,23,0.3478,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,23,0.5217,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,23,0.6957,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[20,23,0.8696,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[23,23,1.0,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19]]},{"b":6,"e":1.0,"k":"flat","v":0.8571,"x":0.96875,"p":[[0,8,0.0,0.91964,0.15947,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,24],[4,8,0.5,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[8,8,1.0,0.8571,0.19567,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,2,0,19]]}]},{"i":"86ae5f2b760746d2","q":"An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.","t":[{"b":2,"e":1.0,"k":"flat","v":0.85714,"x":0.98214,"p":[[0,18,0.0,0.85714,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,7,0,15],[4,18,0.2222,0.92854,0.11299,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[8,18,0.4444,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[12,18,0.6667,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,18,0.8889,0.95981,0.08173,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[18,18,1.0,0.88839,0.12234,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,16]]},{"b":5,"e":0.71429,"k":"flat","v":0.79463,"x":0.92857,"p":[[0,17,0.0,0.82141,0.18213,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,9,0,0,5,0,13],[4,17,0.2353,0.90179,0.15745,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,21],[8,17,0.4706,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[12,17,0.7059,0.8616,0.13115,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,11,0,12],[16,17,0.9412,0.80802,0.12684,0.71429,0.78564,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,9,0,7],[17,17,1.0,0.79463,0.14257,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,5,0,8]]}]},{"i":"d09b164b926f8aad","q":"Determine all values of  $n$  such that it is possible to divide a triangle in  $n$  smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.","t":[{"b":2,"e":1.0,"k":"rising","v":0.72768,"x":0.99107,"p":[[0,59,0.0,0.72768,0.24836,0.57143,0.71429,1.0,0.0,1.0,1,9,1,1,0,1,0,0,0,0,0,3,0,0,4,0,0,10,0,0,4,0,9],[4,59,0.0678,0.83022,0.16927,0.71429,0.78571,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,1,0,15],[8,59,0.1356,0.91517,0.14667,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[12,59,0.2034,0.85267,0.15767,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,2,0,16],[16,59,0.2712,0.86606,0.17105,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,4,0,18],[20,59,0.339,0.90625,0.1411,0.82143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,3,0,21],[24,59,0.4068,0.91964,0.11811,0.85713,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[28,59,0.4746,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[32,59,0.5424,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,59,0.6102,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,59,0.678,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,59,0.7458,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[48,59,0.8136,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,59,0.8814,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[56,59,0.9492,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[59,59,1.0,0.90625,0.11633,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,18]]},{"b":3,"e":0.71429,"k":"falling","v":0.59816,"x":0.90625,"p":[[0,43,0.0,0.81695,0.25061,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,16],[4,43,0.093,0.90625,0.16602,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,0,0,24],[8,43,0.186,0.90177,0.17293,0.92857,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,24],[12,43,0.2791,0.88826,0.1588,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,20],[16,43,0.3721,0.74109,0.18007,0.57143,0.71429,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,10,0,0,3,0,8],[20,43,0.4651,0.74552,0.16264,0.57143,0.71429,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,13,0,0,3,0,7],[24,43,0.5581,0.71427,0.15569,0.57143,0.71429,0.71429,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,12,0,0,1,0,6],[28,43,0.6512,0.71872,0.18031,0.57143,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,13,0,0,1,0,7],[32,43,0.7442,0.66951,0.12591,0.57143,0.64071,0.71429,0.5714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,13,0,0,0,0,3],[36,43,0.8372,0.60713,0.08749,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,8,0,0,1,0,0],[40,43,0.9302,0.61158,0.1025,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,19,0,0,10,0,0,1,0,0],[43,43,1.0,0.59816,0.06624,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,7,0,0,0,0,0]]}]},{"i":"7238e879bae3688a","q":"Bessie and her  $2015$  bovine buddies work at the Organic Milk Organization, for a total of  $2016$  workers. They have a hierarchy of bosses, where obviously no cow is its own boss. In other words, for some pairs of employees  $(A, B)$ ,  $B$  is the boss of  $A$ . This relationship satisfies an obvious condition: if  $B$  is the boss of  $A$  and  $C$  is the boss of  $B$ , then  $C$  is also a boss of  $A$ . Business has been slow, so Bessie hires an outside organizational company to partition the company into some number of groups. To promote growth, every group is one of two forms. Either no one in the group is the boss of another in the group, or for every pair of cows in the group, one is the boss of the other. Let  $G$  be the minimum number of groups needed in such a partition. Find the maximum value of  $G$  over all possible company structures.\n\n*Proposed by Yang Liu*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.14723,"x":0.28125,"p":[[0,50,0.0,0.16071,0.23623,0.0,0.0,0.14286,0.0,0.71429,17,0,16,17,0,8,0,0,1,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[4,50,0.08,0.19625,0.09289,0.14286,0.14286,0.2857,0.14,0.5714,0,0,0,0,0,22,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,50,0.16,0.19187,0.11075,0.14286,0.14286,0.17857,0.0,0.57143,1,0,0,1,0,23,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,50,0.24,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.17402,0.0591,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.18527,0.07755,0.14286,0.14286,0.1786,0.14286,0.42857,0,0,0,0,0,24,0,0,6,0,1,1,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.19643,0.10564,0.14286,0.14286,0.17857,0.14286,0.57143,0,0,0,0,0,24,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,50,0.72,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.18304,0.08171,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,25,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.27679,0.12846,0.24999,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,8,0,0,21,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[50,50,1.0,0.28125,0.09771,0.24999,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,8,0,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.1651,"x":0.25892,"p":[[0,34,0.0,0.19643,0.27837,0.0,0.14286,0.14286,0.0,1.0,13,1,12,13,0,13,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,1],[4,34,0.1176,0.1651,0.05193,0.14286,0.14286,0.14286,0.14,0.286,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.19196,0.11633,0.14286,0.14286,0.2857,0.0,0.71429,1,0,0,1,0,22,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,34,0.3529,0.17858,0.07986,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,26,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.21875,0.15561,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,23,0,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[20,34,0.5882,0.19188,0.09186,0.14286,0.14286,0.2857,0.14,0.57143,0,0,0,0,0,23,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,34,0.7059,0.19643,0.12752,0.14286,0.14286,0.14287,0.14286,0.71429,0,0,0,0,0,25,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,34,0.8235,0.24999,0.10097,0.14286,0.28571,0.28571,0.0,0.571,1,0,0,1,0,9,0,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,34,0.9412,0.23661,0.08459,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"50ee9dad744d9a98","q":"Find all  $f:\\mathbb{Q}\\to \\mathbb{Q}$  satisfying  $$ f(a) + b^3 f\\left(\\dfrac{1}{b}\\right) = b^3 + f(b^2+a) $$  for all  $a,b \\in \\mathbb{Q}$  and  $b\\ne 0$ .\n\n*Proposed by Cody Johnson*","t":[{"b":1,"e":1.0,"k":"flat","v":0.86998,"x":0.99219,"p":[[0,46,0.0,0.86998,0.20459,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,1,2,1,0,4,0,0,3,0,20],[4,46,0.087,0.95201,0.17663,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[8,46,0.1739,0.98382,0.0429,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,46,0.2609,0.99163,0.0325,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,46,0.3478,0.96429,0.10052,1.0,1.0,1.0,0.5,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,3,0,27],[20,46,0.4348,0.99219,0.03026,1.0,1.0,1.0,0.875,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,46,0.5217,0.98326,0.04441,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,46,0.6087,0.94921,0.13797,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,3,0,26],[32,46,0.6957,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,46,0.7826,0.9827,0.04585,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,46,0.8696,0.94531,0.13825,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,4,0,25],[44,46,0.9565,0.95311,0.10153,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,4,0,25],[46,46,1.0,0.9202,0.20987,0.96875,1.0,1.0,0.125,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,24]]},{"b":3,"e":1.0,"k":"flat","v":0.85826,"x":0.99219,"p":[[0,147,0.0,0.85826,0.17666,0.71429,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,1,2,1,0,6,0,0,4,0,17],[4,147,0.0272,0.9827,0.04585,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,147,0.0544,0.99219,0.03026,1.0,1.0,1.0,0.875,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,147,0.0816,0.97824,0.06198,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,147,0.1088,0.96985,0.08423,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,147,0.1361,0.976,0.05899,1.0,1.0,1.0,0.75,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4,0,27],[24,147,0.1633,0.99163,0.0325,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,147,0.1905,0.97098,0.05499,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,147,0.2177,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,147,0.2449,0.98326,0.05699,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,147,0.2721,0.9654,0.06438,0.98214,1.0,1.0,0.75,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6,1,24],[44,147,0.2993,0.95535,0.08794,0.96875,1.0,1.0,0.625,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,6,0,24],[48,147,0.3265,0.97935,0.05737,1.0,1.0,1.0,0.75,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,0,28],[52,147,0.3537,0.95815,0.0783,0.96875,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,6,0,24],[56,147,0.381,0.93583,0.16402,0.96875,1.0,1.0,0.125,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,5,0,24],[60,147,0.4082,0.96987,0.09662,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,29],[64,147,0.4354,0.96596,0.10558,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[68,147,0.4626,0.97377,0.0653,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[72,147,0.4898,0.98717,0.05362,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,147,0.517,0.97545,0.06002,1.0,1.0,1.0,0.75,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4,0,27],[80,147,0.5442,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[84,147,0.5714,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[88,147,0.5986,0.97154,0.06232,1.0,1.0,1.0,0.75,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5,0,26],[92,147,0.6259,0.97991,0.07154,1.0,1.0,1.0,0.625,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2,0,29],[96,147,0.6531,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[100,147,0.6803,0.97879,0.0494,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[104,147,0.7075,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[108,147,0.7347,0.98326,0.06499,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,30],[112,147,0.7619,0.98772,0.03826,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[116,147,0.7891,0.98717,0.03999,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[120,147,0.8163,0.9721,0.06136,1.0,1.0,1.0,0.75,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5,0,26],[124,147,0.8435,0.96596,0.07581,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,4,0,26],[128,147,0.8707,0.97545,0.06002,1.0,1.0,1.0,0.75,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4,0,27],[132,147,0.898,0.95312,0.10896,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[136,147,0.9252,0.97433,0.05357,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[140,147,0.9524,0.96205,0.10656,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[144,147,0.9796,0.97935,0.05736,1.0,1.0,1.0,0.75,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,0,28],[147,147,1.0,0.96093,0.08628,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,3,0,26]]}]},{"i":"5e4856b4faa5c1d3","q":"Find all finite sets  $M \\subset R, |M|  \\ge  2$ , satisfying the following condition: \n*for all  $a, b \\in  M, a \\ne b$ , the number  $a^3 - \\frac{4}{9}b$  also belongs to  $M$ .*\n(I. Voronovich)","t":[{"b":1,"e":0.0,"k":"falling","v":0.03125,"x":0.54909,"p":[[0,94,0.0,0.3258,0.11984,0.2857,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,4,0,0,13,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[4,94,0.0426,0.54909,0.18249,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,5,0,0,16,0,0,2,0,0,3,0,1],[8,94,0.0851,0.51786,0.22232,0.42857,0.57143,0.60714,0.0,1.0,2,1,0,2,0,0,0,0,4,0,0,9,0,0,9,0,0,4,0,0,3,0,1],[12,94,0.1277,0.51339,0.14223,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,9,0,0,18,0,0,3,0,0,0,0,0],[16,94,0.1702,0.54908,0.19596,0.571,0.57143,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,2,0,0,3,0,0,19,0,0,2,0,0,4,0,0],[20,94,0.2128,0.39277,0.23429,0.24999,0.42857,0.57143,0.0,0.71429,6,0,0,6,0,2,0,0,3,0,0,7,0,0,11,0,0,3,0,0,0,0,0],[24,94,0.2553,0.32588,0.23209,0.10714,0.42857,0.57143,0.0,0.71429,8,0,0,8,0,3,0,0,4,0,0,7,0,0,9,0,0,1,0,0,0,0,0],[28,94,0.2979,0.44196,0.23517,0.28571,0.4286,0.57143,0.0,0.85714,4,0,0,4,0,2,0,0,3,0,0,8,0,0,10,0,0,3,0,0,2,0,0],[32,94,0.3404,0.32143,0.27199,0.0,0.28571,0.57143,0.0,0.85714,10,0,0,10,0,2,0,0,5,0,0,6,0,0,5,0,0,2,0,0,2,0,0],[36,94,0.383,0.29464,0.27417,0.0,0.28571,0.46431,0.0,0.85714,12,0,0,12,0,2,0,0,3,0,0,7,0,0,5,0,0,1,0,0,2,0,0],[40,94,0.4255,0.23662,0.2412,0.0,0.14286,0.42857,0.0,0.71429,12,0,0,12,0,6,0,0,4,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[44,94,0.4681,0.24106,0.2511,0.0,0.14286,0.57111,0.0,0.71429,14,0,0,14,0,3,0,0,4,0,0,2,0,0,8,0,0,1,0,0,0,0,0],[48,94,0.5106,0.25893,0.2299,0.0,0.28571,0.42857,0.0,0.71429,11,0,0,11,0,4,0,0,4,0,0,7,0,0,5,0,0,1,0,0,0,0,0],[52,94,0.5532,0.10714,0.16366,0.0,0.0,0.14287,0.0,0.71429,19,0,0,19,0,6,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,94,0.5957,0.18749,0.2586,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,2,0,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[60,94,0.6383,0.23213,0.25441,0.0,0.14286,0.42857,0.0,0.85714,14,0,0,14,0,4,0,0,3,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[64,94,0.6809,0.14732,0.19393,0.0,0.14286,0.17857,0.0,0.85714,15,0,0,15,0,9,0,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[68,94,0.7234,0.24545,0.20901,0.105,0.14286,0.42857,0.0,0.71429,8,0,0,8,0,9,0,0,5,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[72,94,0.766,0.21875,0.23415,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,7,0,0,4,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[76,94,0.8085,0.22321,0.2141,0.0,0.21428,0.42857,0.0,0.71429,12,0,0,12,0,4,0,0,7,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[80,94,0.8511,0.17411,0.21048,0.0,0.07143,0.28571,0.0,0.71429,16,0,0,16,0,4,0,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[84,94,0.8936,0.30786,0.24264,0.14,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,6,0,0,4,0,0,10,0,0,2,0,0,1,0,0,2,0,0],[88,94,0.9362,0.16521,0.20861,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,3,0,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[92,94,0.9787,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[94,94,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.08482,"x":0.57141,"p":[[0,84,0.0,0.30804,0.13882,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,5,0,0,13,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[4,84,0.0476,0.53567,0.13831,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,7,0,0,17,0,0,4,0,0,1,0,0],[8,84,0.0952,0.57141,0.20825,0.42857,0.57143,0.60714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,6,0,0,15,0,0,3,0,0,3,0,2],[12,84,0.1429,0.48214,0.18814,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,4,0,0,7,0,0,15,0,0,2,0,0,0,0,1],[16,84,0.1905,0.4933,0.21381,0.42857,0.5,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,1,0,0,11,0,0,9,0,0,5,1,0,0,0,1],[20,84,0.2381,0.40615,0.23996,0.28571,0.4286,0.57143,0.0,0.85714,6,0,0,6,0,1,0,0,4,0,0,6,0,0,12,0,0,2,0,0,1,0,0],[24,84,0.2857,0.51339,0.2693,0.42857,0.57143,0.71429,0.0,1.0,5,1,0,5,0,0,0,0,1,0,0,7,0,0,8,0,0,7,0,0,3,0,1],[28,84,0.3333,0.375,0.24679,0.14289,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,2,0,0,3,0,0,8,0,0,9,0,0,2,0,0,1,0,0],[32,84,0.381,0.39286,0.26726,0.24999,0.42857,0.57143,0.0,1.0,6,1,0,6,0,2,0,0,6,0,0,6,0,0,8,0,0,1,0,0,2,0,1],[36,84,0.4286,0.32143,0.22304,0.14286,0.42857,0.4286,0.0,0.71429,7,0,0,7,0,4,0,0,4,0,0,10,0,0,5,0,0,2,0,0,0,0,0],[40,84,0.4762,0.30356,0.25189,0.0,0.35714,0.57111,0.0,0.85714,11,0,0,11,0,1,0,0,4,0,0,7,0,0,8,0,0,0,0,0,1,0,0],[44,84,0.5238,0.36161,0.2172,0.2857,0.42857,0.4286,0.0,0.85714,5,0,0,5,0,2,0,0,7,0,0,11,0,0,4,0,0,2,0,0,1,0,0],[48,84,0.5714,0.28572,0.24223,0.0,0.42857,0.4286,0.0,0.71429,12,0,0,12,0,1,0,0,2,0,0,10,0,0,6,0,0,1,0,0,0,0,0],[52,84,0.619,0.26339,0.22899,0.0,0.35714,0.42857,0.0,0.71429,12,0,0,12,0,2,0,0,2,0,0,12,0,0,3,0,0,1,0,0,0,0,0],[56,84,0.6667,0.29465,0.24468,0.0,0.42857,0.4286,0.0,0.85714,12,0,0,12,0,0,0,0,1,0,0,14,0,0,4,0,0,0,0,0,1,0,0],[60,84,0.7143,0.29009,0.20362,0.105,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,2,0,0,8,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[64,84,0.7619,0.32143,0.26964,0.0,0.35714,0.46429,0.0,1.0,9,1,0,9,0,4,0,0,3,0,0,8,0,0,4,0,0,3,0,0,0,0,1],[68,84,0.8095,0.29911,0.27747,0.0,0.28571,0.42858,0.0,1.0,12,1,0,12,0,0,0,0,6,0,0,7,0,0,4,0,0,1,0,0,1,0,1],[72,84,0.8571,0.22312,0.24471,0.0,0.07,0.42857,0.0,0.71429,16,0,0,16,0,1,0,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[76,84,0.9048,0.25893,0.21261,0.0,0.28571,0.42857,0.0,0.57143,11,0,0,11,0,2,0,0,5,0,0,10,0,0,4,0,0,0,0,0,0,0,0],[80,84,0.9524,0.18304,0.21793,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,1,0,0,6,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[84,84,1.0,0.08482,0.20159,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"6ead9233f7d60f44","q":"Each cell of a  $100\\times 100$  grid is colored with one of  $101$  colors. A cell is *diverse* if, among the  $199$  cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.","t":[{"b":2,"e":0.28571,"k":"rising","v":0.12937,"x":0.3616,"p":[[0,53,0.0,0.12937,0.10925,0.0,0.14286,0.17857,0.0,0.28571,11,0,7,11,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.14715,0.06668,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,53,0.1509,0.18294,0.07354,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,53,0.2264,0.19625,0.1276,0.14286,0.14286,0.17857,0.0,0.71429,1,0,0,1,0,23,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[16,53,0.3019,0.17411,0.08552,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,19,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,53,0.3774,0.1875,0.11538,0.14286,0.14286,0.1786,0.0,0.57143,2,0,0,2,0,22,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,53,0.4528,0.16071,0.05922,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,53,0.5283,0.2633,0.13423,0.14286,0.2857,0.28571,0.14,0.71429,0,0,0,0,0,13,0,0,14,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[32,53,0.6038,0.29902,0.15723,0.14286,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,12,0,0,10,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[36,53,0.6792,0.28125,0.15765,0.14286,0.28571,0.32143,0.0,0.71429,2,0,0,2,0,9,0,0,13,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[40,53,0.7547,0.28125,0.15355,0.14286,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,11,0,0,12,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[44,53,0.8302,0.3616,0.15561,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,3,0,0,17,0,0,7,0,0,3,0,0,1,0,0,1,0,0],[48,53,0.9057,0.31696,0.11701,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,15,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[52,53,0.9811,0.3125,0.12079,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,16,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[53,53,1.0,0.34375,0.08645,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.08929,"x":0.20089,"p":[[0,84,0.0,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,8,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,84,0.0476,0.18295,0.06429,0.14286,0.14286,0.28571,0.14,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,84,0.0952,0.1875,0.0974,0.14286,0.14286,0.14287,0.14286,0.57143,0,0,0,0,0,25,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,84,0.1429,0.16518,0.06298,0.14286,0.14286,0.14287,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,84,0.1905,0.19643,0.08564,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,19,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,84,0.2381,0.17411,0.07771,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,21,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,84,0.2857,0.19643,0.07784,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,84,0.3333,0.19187,0.08464,0.14286,0.14286,0.2857,0.14,0.42857,0,0,0,0,0,23,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,84,0.381,0.18304,0.06423,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,84,0.4286,0.16518,0.0724,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,84,0.4762,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,84,0.5238,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,84,0.5714,0.19643,0.06916,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,84,0.619,0.18751,0.08329,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,84,0.6667,0.1875,0.08328,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,84,0.7143,0.17384,0.0592,0.14286,0.14286,0.14287,0.14,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,84,0.7619,0.18295,0.07354,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,84,0.8095,0.20089,0.07016,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,84,0.8571,0.19197,0.07667,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,84,0.9048,0.19634,0.09284,0.14286,0.14286,0.2857,0.14,0.57143,0,0,0,0,0,22,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,84,0.9524,0.17411,0.06901,0.14286,0.14286,0.1786,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,84,1.0,0.19643,0.07784,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,21,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"435da175181bda69","q":"At a chess tournament, every pair of contestants played each other at most once. If any two con-\ntestants,  $A$  and  $B$ , failed to play each other, then exactly two other contestants,  $C$  and  $D$ , played\nagainst both  $A$  and  $B$  during the tournament. Moreover, no  $4$  contestants played exactly  $5$  games\nbetween them. Prove that every contestant played the same number of games.\n\n*Authored by Mirko Petrushevski*","t":[{"b":0,"e":0.571,"k":"flat","v":0.36149,"x":0.63839,"p":[[0,159,0.0,0.36607,0.2257,0.28571,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,3,0,0,18,0,0,2,0,0,0,0,0,5,0,0,2,0,0],[4,159,0.0252,0.58034,0.27418,0.28571,0.64286,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,9,0,0,1,0,0,4,0,0,6,0,0,8,0,2],[8,159,0.0503,0.63839,0.2624,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,5,0,0,2,0,0,5,0,0,12,0,2],[12,159,0.0755,0.61607,0.25111,0.42857,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,5,0,0,2,0,0,5,0,0,13,0,0],[16,159,0.1006,0.56692,0.2435,0.39286,0.57121,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,5,0,0,5,0,0,5,0,0,9,0,0],[20,159,0.1258,0.52902,0.28619,0.28571,0.5,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,9,0,0,4,0,0,1,0,0,5,1,0,9,0,0],[24,159,0.1509,0.53125,0.29716,0.39286,0.42859,0.85714,0.0,0.85714,3,0,0,3,0,3,0,0,2,0,0,9,0,0,2,0,0,1,0,0,12,0,0],[28,159,0.1761,0.61383,0.26174,0.42857,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,6,0,0,0,0,0,5,1,0,13,0,0],[32,159,0.2013,0.49554,0.26483,0.28571,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,6,0,0,6,0,0,3,0,0,6,0,0,6,0,0],[36,159,0.2264,0.57813,0.22042,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,1,3,0,0,10,0,0,1,0,0,10,0,0,5,0,1],[40,159,0.2516,0.56472,0.26022,0.41068,0.42859,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,1,9,0,0,1,0,0,2,0,0,12,0,0],[44,159,0.2767,0.51785,0.27606,0.28571,0.5,0.85714,0.0,0.85714,1,0,0,1,0,3,0,0,9,0,0,3,0,0,4,0,0,2,0,0,10,0,0],[48,159,0.3019,0.46875,0.2506,0.28571,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,2,0,0,9,0,0,6,0,0,2,0,0,7,0,0,4,0,0],[52,159,0.327,0.52678,0.27591,0.28571,0.57143,0.73214,0.0,0.85714,2,0,0,2,0,3,0,0,5,0,1,4,0,0,2,0,0,7,1,0,7,0,0],[56,159,0.3522,0.42857,0.27894,0.14289,0.28571,0.71429,0.0,0.85714,1,0,0,1,0,8,0,0,8,0,0,4,0,0,2,0,0,2,0,0,7,0,0],[60,159,0.3774,0.45971,0.25698,0.28571,0.42857,0.60714,0.0,1.0,2,1,0,2,0,4,0,0,6,0,0,6,0,0,6,0,0,4,0,0,3,0,1],[64,159,0.4025,0.49321,0.25226,0.28571,0.42857,0.75,0.0,0.85714,1,0,0,1,0,3,0,0,6,0,0,9,0,1,3,0,0,1,0,0,8,0,0],[68,159,0.4277,0.43741,0.27194,0.2857,0.28571,0.75,0.0,0.85714,1,0,0,1,0,5,0,0,11,0,0,6,0,0,0,0,0,1,0,0,8,0,0],[72,159,0.4528,0.43295,0.26189,0.28571,0.28571,0.58932,0.0,0.85714,2,0,0,2,0,3,0,1,11,0,0,3,0,0,4,1,0,1,0,0,6,0,0],[76,159,0.478,0.44643,0.26666,0.2857,0.42857,0.60714,0.0,0.85714,2,0,0,2,0,4,0,0,8,0,0,7,0,0,3,0,0,1,0,0,7,0,0],[80,159,0.5031,0.52231,0.29366,0.28571,0.57143,0.75,0.0,0.85714,3,0,0,3,0,4,0,0,3,0,0,4,0,0,3,0,0,7,0,0,8,0,0],[84,159,0.5283,0.48205,0.26195,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,5,0,0,5,0,0,8,0,0,3,0,0,3,0,0,7,0,0],[88,159,0.5535,0.52228,0.23037,0.39286,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,7,0,0,7,0,0,4,0,0,6,0,0],[92,159,0.5786,0.49105,0.26228,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,4,0,0,7,0,0,6,0,0,3,0,0,4,0,0,7,0,0],[96,159,0.6038,0.46205,0.2829,0.2857,0.35714,0.74996,0.0,0.85714,2,0,0,2,0,3,0,1,10,0,0,4,0,0,1,0,0,3,0,0,8,0,0],[100,159,0.6289,0.38606,0.21795,0.2857,0.32143,0.42857,0.0,0.85714,2,0,0,2,0,3,0,0,11,0,1,9,0,0,1,0,0,2,0,0,3,0,0],[104,159,0.6541,0.48204,0.25454,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,8,0,0,8,0,0,2,0,0,3,0,0,7,0,0],[108,159,0.6792,0.44641,0.23351,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,5,0,0,10,0,0,5,0,0,4,0,0,4,0,0,4,0,0],[112,159,0.7044,0.45758,0.25244,0.28571,0.4286,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,6,0,0,7,0,0,3,1,0,5,0,0,4,0,0],[116,159,0.7296,0.57366,0.26874,0.42857,0.64286,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,4,0,0,7,0,1,1,0,0,5,0,0,11,0,0],[120,159,0.7547,0.43303,0.24609,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,6,0,0,9,0,0,3,0,0,2,0,0,5,0,0],[124,159,0.7799,0.47543,0.27641,0.2857,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,8,0,0,3,0,0,4,1,0,3,0,0,7,0,0],[128,159,0.805,0.49552,0.23141,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,8,0,0,9,0,0,4,0,0,3,0,0,6,0,0],[132,159,0.8302,0.47988,0.23705,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,9,0,1,7,0,0,3,0,0,4,0,0,4,0,1],[136,159,0.8553,0.51785,0.21943,0.28571,0.4286,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,9,0,0,5,0,0,2,0,0,7,0,0],[140,159,0.8805,0.41518,0.17984,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,2,0,0,12,0,0,12,0,0,1,0,0,3,0,0,2,0,0],[144,159,0.9057,0.45982,0.19475,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,13,0,0,2,0,0,4,0,0,3,0,0],[148,159,0.9308,0.50223,0.22972,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,1,6,0,0,2,0,0,5,0,0,6,0,0],[152,159,0.956,0.39284,0.22303,0.2857,0.28571,0.4642,0.0,0.85714,2,0,0,2,0,3,0,0,12,0,0,7,0,0,3,0,0,2,0,0,3,0,0],[156,159,0.9811,0.4107,0.17033,0.28571,0.35714,0.42858,0.14286,0.85714,0,0,0,0,0,1,0,0,15,0,0,9,0,0,2,0,0,4,0,0,1,0,0],[159,159,1.0,0.36149,0.15876,0.28571,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,3,0,0,18,0,2,2,0,0,4,0,0,3,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.36161,"x":0.62945,"p":[[0,68,0.0,0.36161,0.2448,0.2857,0.28571,0.46429,0.0,0.85714,3,0,0,3,0,4,0,0,15,0,0,2,0,0,3,0,0,1,0,0,4,0,0],[4,68,0.0588,0.52677,0.27993,0.28571,0.42857,0.85714,0.0,0.85714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the 24 points in four-dimensional space whose coordinates are the various permutations of 1, 2, 3, and 4.  Among these 24 points, how many subsets of 4 form the vertices of a square?","t":[{"b":1,"e":0.2857,"k":"falling","v":0.18732,"x":0.56249,"p":[[0,71,0.0,0.35715,0.21129,0.14286,0.42857,0.46431,0.0,0.71429,3,0,3,3,0,8,0,0,2,0,0,11,0,0,5,0,0,3,0,0,0,0,0],[4,71,0.0563,0.56249,0.28107,0.2857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,6,0,0,3,0,0,3,0,0,5,0,0,10,0,0,0,0,5],[8,71,0.1127,0.51785,0.24418,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,5,0,0,4,0,0,5,0,0,11,0,0,0,0,2],[12,71,0.169,0.38817,0.21515,0.14286,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,11,0,0,4,0,0,4,0,0,10,0,0,2,0,0,1,0,0],[16,71,0.2254,0.35703,0.22021,0.14286,0.35714,0.4642,0.14,1.0,0,1,0,0,0,13,0,0,3,0,0,8,0,0,5,0,0,2,0,0,0,0,1],[20,71,0.2817,0.27674,0.15959,0.14286,0.2143,0.42857,0.14,0.57143,0,0,0,0,0,16,0,0,7,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[24,71,0.338,0.26327,0.16015,0.14286,0.14286,0.42858,0.14,0.57143,0,0,0,0,0,19,0,0,3,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[28,71,0.3944,0.29008,0.22728,0.14286,0.1429,0.57111,0.0,1.0,1,1,0,1,0,19,0,0,2,0,0,1,0,0,8,0,0,0,0,0,0,0,1],[32,71,0.4507,0.37051,0.23918,0.14286,0.42857,0.57143,0.0,1.0,1,1,0,1,0,13,0,0,1,0,0,5,0,0,9,0,0,2,0,0,0,0,1],[36,71,0.507,0.34813,0.21714,0.14286,0.28574,0.57143,0.14,0.85714,0,0,0,0,0,16,0,0,0,0,0,4,0,0,11,0,0,0,0,0,1,0,0],[40,71,0.5634,0.30793,0.18601,0.14286,0.21429,0.42858,0.14,0.71429,0,0,0,0,0,16,0,0,3,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[44,71,0.6197,0.30801,0.22331,0.14286,0.2143,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,7,0,0,2,0,0,5,0,0,0,0,0,1,0,1],[48,71,0.6761,0.25426,0.17035,0.14286,0.14286,0.32143,0.14,0.57143,0,0,0,0,0,21,0,0,3,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[52,71,0.7324,0.30802,0.20236,0.14286,0.14288,0.4642,0.14286,0.85714,0,0,0,0,0,17,0,0,3,0,0,4,0,0,7,0,0,0,0,0,1,0,0],[56,71,0.7887,0.24982,0.17507,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,21,0,0,5,0,0,0,0,0,5,0,0,1,0,0,0,0,0],[60,71,0.8451,0.25427,0.18817,0.14286,0.14286,0.32142,0.14,0.85714,0,0,0,0,0,22,0,0,2,0,0,3,0,0,4,0,0,0,0,0,1,0,0],[64,71,0.9014,0.21427,0.16749,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,26,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[68,71,0.9577,0.21875,0.12364,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,22,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[71,71,1.0,0.18732,0.10979,0.14286,0.14286,0.14287,0.14,0.57143,0,0,0,0,0,26,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"flat","v":0.33017,"x":0.51784,"p":[[0,113,0.0,0.36606,0.22569,0.14286,0.42857,0.57143,0.0,0.71429,5,0,5,5,0,5,0,0,2,0,0,10,0,0,7,0,0,3,0,0,0,0,0],[4,113,0.0354,0.5089,0.24467,0.28571,0.5712,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,3,0,0,5,0,0,8,0,0,7,0,0,1,0,2],[8,113,0.0708,0.51784,0.24156,0.28571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,7,0,0,2,0,0,3,0,0,6,0,0,13,0,0,0,0,1],[12,113,0.1062,0.44195,0.26572,0.14286,0.42859,0.60714,0.14286,1.0,0,2,0,0,0,12,0,0,0,0,0,5,0,0,7,0,0,6,0,0,0,0,2],[16,113,0.1416,0.36606,0.2285,0.14286,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,14,0,0,2,0,0,6,0,0,5,0,0,4,0,0,1,0,0],[20,113,0.177,0.33017,0.19393,0.14286,0.28571,0.57111,0.0,0.71429,1,0,0,1,0,12,0,0,5,0,0,5,0,0,8,0,0,1,0,0,0,0,0],[24,113,0.2124,0.36149,0.25755,0.14286,0.2857,0.57143,0.14,1.0,0,2,0,0,0,15,0,0,4,0,0,2,0,0,7,0,0,2,0,0,0,0,2],[28,113,0.2478,0.3616,0.19223,0.14286,0.28571,0.571,0.14286,0.71429,0,0,0,0,0,10,0,0,7,0,0,6,0,0,6,0,0,3,0,0,0,0,0],[32,113,0.2832,0.41068,0.24155,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,9,0,0,8,0,0,2,0,0,7,0,0,4,0,0,1,0,1],[36,113,0.3186,0.49099,0.18185,0.42857,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,0,0,0,4,0,0,18,0,0,4,0,0,0,0,0],[40,113,0.354,0.49097,0.23125,0.2857,0.571,0.57143,0.14286,1.0,0,2,0,0,0,6,0,0,4,0,0,2,0,0,14,0,0,4,0,0,0,0,2],[44,113,0.3894,0.4328,0.2754,0.14286,0.42857,0.71429,0.14,1.0,0,1,0,0,0,13,0,0,2,0,0,2,0,0,4,0,0,9,0,0,1,0,1],[48,113,0.4248,0.46427,0.26963,0.14286,0.4998,0.60714,0.14286,1.0,0,2,0,0,0,10,0,0,2,0,0,4,0,0,8,0,0,4,0,0,2,0,2],[52,113,0.4602,0.39283,0.24998,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,12,0,0,6,0,0,1,0,0,6,0,0,6,0,0,0,0,1],[56,113,0.4956,0.41951,0.24476,0.25,0.28571,0.57143,0.14,1.0,0,2,0,0,0,8,0,0,9,0,0,1,0,0,9,0,0,3,0,0,0,0,2],[60,113,0.531,0.44181,0.25351,0.14286,0.571,0.60714,0.0,0.85714,1,0,0,1,0,10,0,0,1,0,0,3,0,0,9,0,0,6,0,0,2,0,0],[64,113,0.5664,0.40623,0.20235,0.25,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,7,0,0,3,0,0,10,0,0,4,0,0,0,0,0],[68,113,0.6018,0.41055,0.22522,0.14289,0.42857,0.571,0.0,0.85714,1,0,0,1,0,8,0,0,4,0,0,6,0,0,9,0,0,2,0,0,2,0,0],[72,113,0.6372,0.41963,0.21997,0.14286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,10,0,0,2,0,0,7,0,0,6,0,0,7,0,0,0,0,0],[76,113,0.6726,0.38833,0.24015,0.14286,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,14,0,0,1,0,0,2,0,0,12,0,0,2,0,0,0,0,1],[80,113,0.708,0.45082,0.22615,0.25,0.571,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,2,0,0,4,0,0,14,0,0,3,0,0,0,0,1],[84,113,0.7434,0.45081,0.21456,0.25,0.571,0.57143,0.14286,1.0,0,1,0,0,0,8,0,0,2,0,0,4,0,0,15,0,0,2,0,0,0,0,1],[88,113,0.7788,0.39731,0.21645,0.14286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,11,0,0,3,0,0,4,0,0,11,0,0,2,0,0,1,0,0],[92,113,0.8142,0.41065,0.18117,0.14286,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,9,0,0,1,0,0,7,0,0,15,0,0,0,0,0,0,0,0],[96,113,0.8496,0.36163,0.20197,0.14286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,14,0,0,0,0,0,6,0,0,11,0,0,1,0,0,0,0,0],[100,113,0.885,0.4062,0.1893,0.14286,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,3,0,0,5,0,0,14,0,0,1,0,0,0,0,0],[104,113,0.9204,0.49998,0.1428,0.4286,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,1,0,0,7,0,0,19,0,0,2,0,0,0,0,0],[108,113,0.9558,0.49537,0.15577,0.42857,0.571,0.57143,0.14,0.85714,0,0,0,0,0,4,0,0,0,0,0,7,0,0,20,0,0,0,0,0,1,0,0],[112,113,0.9912,0.44639,0.16266,0.39286,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,8,0,0,15,0,0,1,0,0,0,0,0],[113,113,1.0,0.49995,0.12369,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,11,0,0,17,0,0,0,0,0,1,0,0]]}]},{"i":"33e9f1745a7561b8","q":"Determine all integers $n \\geq 2$ with the following property:\nFor any not necessarily distinct integers $m_{1}, m_{2}, \\ldots, m_{n}$, whose sum is not divisible by $n$, there exists an index $i(1 \\leq i \\leq n)$, such that none of the numbers\n\n$$\nm_{i}, m_{i}+m_{i+1}, m_{i}+m_{i+1}+m_{i+2}, \\ldots, m_{i}+m_{i+1}+\\ldots+m_{i+n-1}\n$$\n\nis divisible by $n$. (Here, $m_{i}=m_{i-n}$ for $i>n$.)","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.30804,"p":[[0,72,0.0,0.12946,0.13054,0.0,0.14286,0.2857,0.0,0.28571,15,0,0,15,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.22767,0.25217,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,3,0,0,5,0,0,5,0,0,4,0,0,0,0,0,0,0,1],[8,72,0.1111,0.20088,0.24962,0.0,0.0,0.32143,0.0,1.0,17,1,0,17,0,0,0,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[12,72,0.1667,0.21874,0.26481,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,2,0,0,8,0,0,2,0,0,3,0,0,0,0,0,1,0,1],[16,72,0.2222,0.26337,0.28369,0.0,0.28571,0.42857,0.0,1.0,14,2,0,14,0,0,0,0,7,0,0,5,0,0,4,0,0,0,0,0,0,0,2],[20,72,0.2778,0.22768,0.30485,0.0,0.0,0.42858,0.0,1.0,17,2,0,17,0,3,0,0,2,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[24,72,0.3333,0.22768,0.25966,0.0,0.21428,0.42857,0.0,1.0,15,1,0,15,0,1,0,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,1],[28,72,0.3889,0.24554,0.30978,0.0,0.14288,0.32143,0.0,1.0,15,3,0,15,0,2,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[32,72,0.4444,0.29018,0.33784,0.0,0.14288,0.46431,0.0,1.0,15,3,0,15,0,2,0,0,2,0,0,5,0,0,3,0,0,1,0,0,1,0,3],[36,72,0.5,0.22768,0.27166,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,2,0,0,8,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[40,72,0.5556,0.25006,0.18562,0.0,0.2857,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,12,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[44,72,0.6111,0.23213,0.28958,0.0,0.14286,0.32143,0.0,1.0,15,2,0,15,0,3,0,0,6,0,0,2,0,0,3,0,0,1,0,0,0,0,2],[48,72,0.6667,0.24554,0.25564,0.0,0.2857,0.42857,0.0,1.0,14,1,0,14,0,0,0,0,7,0,0,7,0,0,2,0,0,1,0,0,0,0,1],[52,72,0.7222,0.22768,0.2809,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,4,0,0,4,0,0,3,0,0,3,0,0,1,0,0,1,0,1],[56,72,0.7778,0.30804,0.31564,0.0,0.28571,0.46431,0.0,1.0,13,3,0,13,0,0,0,0,6,0,0,5,0,0,4,0,0,1,0,0,0,0,3],[60,72,0.8333,0.10714,0.17128,0.0,0.0,0.2857,0.0,0.57143,22,0,0,22,0,0,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[64,72,0.8889,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.12945,"x":0.65175,"p":[[0,86,0.0,0.13392,0.13803,0.0,0.07143,0.2857,0.0,0.28571,16,0,0,16,0,2,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,86,0.0465,0.25446,0.26663,0.0,0.2857,0.42857,0.0,1.0,14,1,0,14,0,0,0,0,7,0,0,6,0,0,2,0,0,2,0,0,0,0,1],[8,86,0.093,0.20533,0.23398,0.0,0.07145,0.32143,0.0,0.71429,16,0,0,16,0,1,0,0,7,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[12,86,0.1395,0.16515,0.21456,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,4,0,0,5,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[16,86,0.186,0.25883,0.31022,0.0,0.2143,0.32143,0.0,1.0,14,3,0,14,0,2,0,0,8,0,0,2,0,0,2,0,0,1,0,0,0,0,3],[20,86,0.2326,0.25455,0.25445,0.0,0.2857,0.42893,0.0,1.0,12,1,0,12,0,2,0,0,9,0,0,3,0,0,4,0,0,1,0,0,0,0,1],[24,86,0.2791,0.21874,0.21122,0.0,0.2857,0.42857,0.0,0.57143,14,0,0,14,0,0,0,0,9,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[28,86,0.3256,0.20088,0.15912,0.0,0.2857,0.28571,0.0,0.571,10,0,0,10,0,4,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,86,0.3721,0.13839,0.16935,0.0,0.0,0.28571,0.0,0.57143,17,0,0,17,0,4,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,86,0.4186,0.27232,0.17627,0.24999,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,1,0,0,15,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[40,86,0.4651,0.25893,0.20341,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,5,0,0,10,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[44,86,0.5116,0.24107,0.21261,0.0,0.2857,0.42857,0.0,0.71429,11,0,0,11,0,3,0,0,8,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[48,86,0.5581,0.31695,0.32484,0.0,0.28571,0.57143,0.0,1.0,13,3,0,13,0,0,0,0,7,0,0,2,0,0,5,0,0,2,0,0,0,0,3],[52,86,0.6047,0.43302,0.32238,0.24999,0.35714,0.60714,0.0,1.0,5,4,0,5,0,3,0,0,8,0,0,5,0,0,3,0,0,1,0,0,3,0,4],[56,86,0.6512,0.45089,0.29039,0.2857,0.50001,0.71429,0.0,1.0,5,2,0,5,0,2,0,0,7,0,0,2,0,0,5,0,0,9,0,0,0,0,2],[60,86,0.6977,0.39731,0.25934,0.2857,0.35714,0.57143,0.0,1.0,4,1,0,4,0,3,0,0,9,0,0,6,0,0,4,0,0,3,0,0,2,0,1],[64,86,0.7442,0.65175,0.27881,0.4286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,5,0,0,8,0,0,3,0,0,1,0,10],[68,86,0.7907,0.60932,0.29881,0.42857,0.57143,0.85714,0.0,1.0,2,6,0,2,1,0,0,0,4,0,0,4,0,0,6,0,0,4,0,0,5,0,6],[72,86,0.8372,0.29903,0.22959,0.0,0.28571,0.4642,0.0,0.71429,10,0,0,10,0,0,0,0,8,0,0,6,0,0,7,0,0,1,0,0,0,0,0],[76,86,0.8837,0.23197,0.23083,0.0,0.14293,0.42857,0.0,0.85714,11,0,0,11,0,6,0,0,6,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[80,86,0.9302,0.20078,0.23105,0.0,0.07,0.42857,0.0,0.71429,16,0,0,16,0,3,0,0,2,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[84,86,0.9767,0.12945,0.19676,0.0,0.0,0.17857,0.0,0.57143,20,0,0,20,0,4,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[86,86,1.0,0.2343,0.21598,0.0,0.24999,0.42857,0.0,0.71429,11,0,0,11,0,4,0,1,6,0,0,7,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"cc6fb5877d36df3b","q":"Let  $ABC$  be an acute triangle with orthocenter  $H$ , circumcenter  $O$ , and circumcircle  $\\Omega$ . Points  $E$  and  $F$  are the feet of the altitudes from  $B$  to  $AC$ , and from  $C$  to  $AB$ , respectively. Let line  $AH$  intersect  $\\Omega$  again at  $D$ . The circumcircle of  $DEF$  intersects  $\\Omega$  again at  $X$ , and  $AX$  intersects  $BC$  at  $I$ . The circumcircle of  $IEF$  intersects  $BC$  again at  $G$ . If  $M$  is the midpoint of  $BC$ , prove that lines  $MX$  and  $OG$  intersect on  $\\Omega$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.00446,"x":0.12501,"p":[[0,44,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.12501,0.16656,0.0,0.07143,0.14292,0.0,0.71429,16,0,0,16,0,9,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[8,44,0.1818,0.12053,0.1433,0.0,0.14286,0.14286,0.0,0.571,14,0,0,14,0,13,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,44,0.2727,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.07148,0.14736,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,44,0.6364,0.11607,0.20341,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,44,0.7273,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,9,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[36,44,0.8182,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.02233,0.08074,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.21427,"p":[[0,61,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.21427,0.29013,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,9,0,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,1],[8,61,0.1311,0.16518,0.24772,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,12,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0],[12,61,0.1967,0.14277,0.19233,0.0,0.14286,0.14286,0.0,0.85714,12,0,0,12,0,16,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[16,61,0.2623,0.18749,0.25612,0.0,0.14286,0.2857,0.0,1.0,15,1,0,15,0,7,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,1],[20,61,0.3279,0.16954,0.27764,0.0,0.14143,0.14286,0.0,1.0,15,2,0,15,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[24,61,0.3934,0.12054,0.23176,0.0,0.0,0.14286,0.0,0.85714,19,0,1,19,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[28,61,0.459,0.06695,0.11279,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,61,0.5246,0.11152,0.17027,0.0,0.07,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[36,61,0.5902,0.09375,0.16602,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,61,0.6557,0.10705,0.23957,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,61,0.7213,0.12945,0.18333,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,13,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[48,61,0.7869,0.07134,0.11289,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.11608,0.16536,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,10,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[56,61,0.918,0.06697,0.18893,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[60,61,0.9836,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6f76f9d3871a0452","q":"Consider the sequence of rational numbers defned by $x_{1}=\\frac{4}{3}$ and $x_{n+1}=\\frac{x_{n}^{2}}{x_{n}^{2}-x_{n}+1}, n \\geq 1$.\nShow that the numerator of the lowest term expression of each sum $\\sum_{k=1}^{n} x_{k}$ is a perfect square.","t":[{"b":1,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,53,0.0,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[4,53,0.0755,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[8,53,0.1509,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[12,53,0.2264,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,53,0.3019,0.95535,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[20,53,0.3774,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,53,0.4528,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,53,0.5283,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,53,0.6038,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,53,0.6792,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,53,0.7547,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,53,0.8302,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,53,0.9057,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,53,0.9811,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[53,53,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.8616,"x":0.99554,"p":[[0,205,0.0,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[4,205,0.0195,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,205,0.039,0.94196,0.12301,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[12,205,0.0585,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[16,205,0.078,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[20,205,0.0976,0.95535,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[24,205,0.1171,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[28,205,0.1366,0.94643,0.11152,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,24],[32,205,0.1561,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[36,205,0.1756,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[40,205,0.1951,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[44,205,0.2146,0.92411,0.17122,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,25],[48,205,0.2341,0.95534,0.10976,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[52,205,0.2537,0.90625,0.19434,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,25],[56,205,0.2732,0.9107,0.17037,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,24],[60,205,0.2927,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,26],[64,205,0.3122,0.9107,0.17037,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,24],[68,205,0.3317,0.87946,0.17536,0.82143,1.0,1.0,0.5714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,4,0,20],[72,205,0.3512,0.92409,0.15564,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,2,0,25],[76,205,0.3707,0.88839,0.17399,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,4,0,21],[80,205,0.3902,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[84,205,0.4098,0.8616,0.19061,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,6,0,18],[88,205,0.4293,0.88839,0.18466,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,3,0,22],[92,205,0.4488,0.87946,0.1992,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,20],[96,205,0.4683,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,205,0.4878,0.92411,0.14719,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,23],[104,205,0.5073,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[108,205,0.5268,0.95535,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[112,205,0.5463,0.91518,0.14664,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[116,205,0.5659,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[120,205,0.5854,0.95534,0.11543,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[124,205,0.6049,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[128,205,0.6244,0.93304,0.17852,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[132,205,0.6439,0.91517,0.17445,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,22],[136,205,0.6634,0.93749,0.13337,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[140,205,0.6829,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[144,205,0.7024,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[148,205,0.722,0.94196,0.10631,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[152,205,0.7415,0.93302,0.12873,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[156,205,0.761,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[160,205,0.7805,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[164,205,0.8,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[168,205,0.8195,0.95089,0.16602,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[172,205,0.839,0.91964,0.20183,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,26],[176,205,0.8585,0.91518,0.1984,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[180,205,0.878,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[184,205,0.8976,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[188,205,0.9171,0.92411,0.1636,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,24],[192,205,0.9366,0.88392,0.17657,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,20],[196,205,0.9561,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[200,205,0.9756,0.91963,0.1673,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[204,205,0.9951,0.89732,0.15663,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,20],[205,205,1.0,0.87054,0.16506,0.85714,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,16]]}]},{"i":"7aef7ae1570b85eb","q":"Given an acute triangle  $ABC$ .  $\\Gamma _{B}$  is a circle that passes through  $AB$ , tangent to  $AC$  at  $A$  and centered at  $O_{B}$ . Define  $\\Gamma_C$  and  $O_C$  the same way. Let the altitudes of  $\\triangle ABC$  from  $B$  and  $C$  meets the circumcircle of  $\\triangle ABC$  at  $X$  and  $Y$ , respectively. Prove that  $A$ , the midpoint of  $XY$  and the midpoint of  $O_{B}O_{C}$  is collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.38829,"p":[[0,159,0.0,0.16518,0.26751,0.0,0.0,0.1786,0.0,1.0,18,2,0,18,0,6,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[4,159,0.0252,0.18304,0.29717,0.0,0.0,0.32143,0.0,1.0,21,2,0,21,0,1,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,2],[8,159,0.0503,0.35268,0.35352,0.0,0.35714,0.57143,0.0,1.0,13,4,0,13,0,0,0,0,3,0,0,7,0,0,3,0,0,0,0,0,2,0,4],[12,159,0.0755,0.38829,0.36642,0.0,0.28571,0.60714,0.0,1.0,10,6,0,10,0,1,0,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,6],[16,159,0.1006,0.31697,0.35667,0.0,0.14286,0.57143,0.0,1.0,16,3,0,16,0,0,0,0,1,0,0,5,0,0,4,0,0,1,0,0,2,0,3],[20,159,0.1258,0.23214,0.32291,0.0,0.0,0.42857,0.0,1.0,19,2,0,19,0,0,0,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,2],[24,159,0.1509,0.25445,0.36374,0.0,0.0,0.4642,0.0,1.0,20,4,0,20,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,4],[28,159,0.1761,0.11607,0.25862,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[32,159,0.2013,0.07143,0.19885,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[36,159,0.2264,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,159,0.2516,0.10268,0.29284,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[44,159,0.2767,0.10714,0.25505,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[48,159,0.3019,0.08929,0.23623,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0],[52,159,0.327,0.11161,0.25935,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0],[56,159,0.3522,0.06696,0.2082,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[60,159,0.3774,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[64,159,0.4025,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,159,0.4277,0.07589,0.21124,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[72,159,0.4528,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,159,0.478,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[80,159,0.5031,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[84,159,0.5283,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,159,0.5535,0.04911,0.17353,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[92,159,0.5786,0.0625,0.19541,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[96,159,0.6038,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[100,159,0.6289,0.10713,0.24998,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0],[104,159,0.6541,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[108,159,0.6792,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[112,159,0.7044,0.0625,0.2141,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[116,159,0.7296,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[120,159,0.7547,0.11161,0.26901,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,1],[124,159,0.7799,0.06697,0.20198,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[128,159,0.805,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[132,159,0.8302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,159,0.8553,0.09375,0.24643,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[140,159,0.8805,0.04902,0.17349,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[144,159,0.9057,0.08481,0.18847,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[148,159,0.9308,0.10268,0.26301,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[152,159,0.956,0.05357,0.16269,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[156,159,0.9811,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[159,159,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.38393,"p":[[0,68,0.0,0.18747,0.31016,0.0,0.0,0.17857,0.0,1.0,20,1,0,20,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,1],[4,68,0.0588,0.38393,0.40159,0.0,0.28571,0.75,0.0,1.0,13,7,0,13,0,2,0,0,2,0,0,5,0,0,0,0,0,2,0,0,1,0,7],[8,68,0.1176,0.2857,0.37456,0.0,0.0,0.60682,0.0,1.0,18,4,0,18,0,0,0,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,4],[12,68,0.1765,0.27232,0.356,0.0,0.0,0.42857,0.0,1.0,17,5,0,17,0,0,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,5],[16,68,0.2353,0.30801,0.36263,0.0,0.14285,0.4642,0.0,1.0,16,5,0,16,0,0,0,0,2,0,0,6,0,0,3,0,0,0,0,0,0,0,5],[20,68,0.2941,0.33927,0.33454,0.0,0.28571,0.57111,0.0,1.0,12,2,0,12,0,1,0,0,5,0,0,5,0,0,3,0,0,0,0,0,4,0,2],[24,68,0.3529,0.27231,0.3357,0.0,0.07145,0.42858,0.0,1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an integer  $n \\geqslant 2$ . Suppose there is a point  $P$  inside a convex cyclic  $2n$ -gon  $A_1 \\ldots A_{2n}$  satisfying   $$ \\angle PA_1A_2 = \\angle PA_2A_3 = \\ldots = \\angle PA_{2n}A_1, $$  prove that   $$  \\prod_{i=1}^{n} \\left|A_{2i - 1}A_{2i} \\right| = \\prod_{i=1}^{n} \\left|A_{2i}A_{2i+1} \\right|, $$  where  $A_{2n + 1} = A_1$ .","t":[{"b":4,"e":0.57143,"k":"rising","v":0.35713,"x":0.80803,"p":[[0,143,0.0,0.35713,0.29013,0.0,0.42857,0.60714,0.0,0.85714,9,0,0,9,0,4,0,0,2,0,0,5,0,0,4,0,0,7,0,0,1,0,0],[4,143,0.028,0.71873,0.26842,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,3,0,0,6,0,0,5,0,0,5,0,10],[8,143,0.0559,0.69195,0.2699,0.42857,0.78564,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,6,0,0,5,0,0,1,0,0,7,0,9],[12,143,0.0839,0.73214,0.29827,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,4,0,0,4,0,0,3,0,0,6,0,12],[16,143,0.1119,0.75447,0.23483,0.57143,0.85714,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,5,0,0,9,0,9],[20,143,0.1399,0.74107,0.25614,0.53571,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,5,0,0,4,0,12],[24,143,0.1678,0.67855,0.27664,0.5354,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,0,5,0,8],[28,143,0.1958,0.72776,0.23792,0.5354,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,7,0,0,3,0,0,6,0,0,6,0,9],[32,143,0.2238,0.73202,0.23101,0.57143,0.78564,0.85714,0.14,1.0,0,7,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,5,0,0,9,0,7],[36,143,0.2517,0.74107,0.23538,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,4,0,0,6,0,10],[40,143,0.2797,0.80803,0.22191,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,1,0,0,9,0,13],[44,143,0.3077,0.69193,0.229,0.5354,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,7,0,0,4,0,0,6,0,0,10,0,4],[48,143,0.3357,0.74996,0.22307,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,6,0,0,5,0,10],[52,143,0.3636,0.71874,0.2461,0.53571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,6,0,0,1,0,0,8,0,0,8,0,7],[56,143,0.3916,0.76785,0.2519,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,6,0,0,6,0,12],[60,143,0.4196,0.73215,0.24156,0.57143,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,4,0,0,4,0,0,5,0,0,10,0,7],[64,143,0.4476,0.69197,0.29904,0.4286,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,4,0,0,3,0,0,4,0,0,4,0,0,4,0,11],[68,143,0.4755,0.741,0.22432,0.571,0.78564,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,6,0,0,8,0,8],[72,143,0.5035,0.72768,0.27283,0.53571,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,3,0,12],[76,143,0.5315,0.64732,0.24739,0.42859,0.71429,0.85704,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,6,0,0,5,0,0,9,0,0,4,0,5],[80,143,0.5594,0.76339,0.20705,0.67857,0.78564,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,8,0,0,7,0,9],[84,143,0.5874,0.71425,0.24487,0.57132,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,3,0,0,8,0,0,6,0,0,4,0,9],[88,143,0.6154,0.70533,0.21411,0.57132,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,6,0,0,7,0,6],[92,143,0.6434,0.7366,0.2627,0.57143,0.857,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,4,0,0,5,0,0,4,0,0,7,0,10],[96,143,0.6713,0.70534,0.25985,0.57132,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,3,0,0,6,0,9],[100,143,0.6993,0.73659,0.26272,0.57132,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,5,0,0,9,0,9],[104,143,0.7273,0.77678,0.22286,0.67857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,5,0,0,9,0,10],[108,143,0.7552,0.72767,0.25345,0.57143,0.78571,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,5,0,0,7,0,9],[112,143,0.7832,0.62052,0.26633,0.42857,0.57143,0.74996,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,6,0,0,6,0,0,7,0,0,1,0,7],[116,143,0.8112,0.72325,0.242,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,3,0,0,6,0,0,8,0,0,4,0,9],[120,143,0.8392,0.70536,0.25738,0.67857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,11,0,0,6,0,7],[124,143,0.8671,0.79464,0.2257,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,6,0,12],[128,143,0.8951,0.7723,0.1851,0.67857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,9,0,0,6,0,9],[132,143,0.9231,0.70536,0.20497,0.57142,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,8,0,0,4,0,7],[136,143,0.951,0.74548,0.18813,0.57132,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,8,0,0,7,0,7],[140,143,0.979,0.73214,0.17034,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,10,0,0,5,0,6],[143,143,1.0,0.78125,0.18552,0.67857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,10,0,0,3,0,11]]},{"b":7,"e":0.71429,"k":"rising","v":0.44192,"x":0.85265,"p":[[0,186,0.0,0.44192,0.21533,0.24999,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,2,0,0,6,0,0,10,0,0,6,0,0,0,0,0],[4,186,0.0215,0.74107,0.27765,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,1,0,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,11],[8,186,0.043,0.7366,0.2699,0.42859,0.78571,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,2,0,0,6,0,0,2,0,0,5,0,0,3,0,13],[12,186,0.0645,0.78125,0.23954,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,3,0,0,5,0,14],[16,186,0.086,0.76337,0.25156,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,4,0,13],[20,186,0.1075,0.82143,0.20516,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,8,0,13],[24,186,0.129,0.80356,0.23076,0.67857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,3,0,0,7,0,14],[28,186,0.1505,0.83926,0.21652,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,6,0,17],[32,186,0.172,0.69197,0.32164,0.42857,0.78571,1.0,0.0,1.0,2,11,0,2,0,2,0,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,11],[36,186,0.1935,0.7991,0.22829,0.71429,0.9285,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,8,0,0,1,0,16],[40,186,0.2151,0.72317,0.28782,0.571,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,3,0,0,5,0,0,5,0,0,2,0,13],[44,186,0.2366,0.76337,0.25906,0.5354,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,6,0,0,2,0,0,3,0,0,6,0,13],[48,186,0.2581,0.74552,0.27138,0.5354,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,3,0,0,3,0,14],[52,186,0.2796,0.79911,0.20782,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,4,0,0,6,0,13],[56,186,0.3011,0.76784,0.26666,0.71429,0.85707,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,8,0,11],[60,186,0.3226,0.69645,0.29175,0.42965,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,0,0,0,6,0,0,5,0,0,3,0,0,4,0,11],[64,186,0.3441,0.72321,0.26471,0.53571,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,5,0,0,2,0,0,7,0,0,5,0,10],[68,186,0.3656,0.70536,0.27879,0.42857,0.78571,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,6,0,0,2,0,0,4,0,0,6,0,10],[72,186,0.3871,0.82588,0.24417,0.82132,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,10,0,14],[76,186,0.4086,0.69195,0.24252,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,6,0,0,3,0,9],[80,186,0.4301,0.68748,0.30397,0.42859,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,6,0,0,6,0,0,3,0,0,2,0,12],[84,186,0.4516,0.70534,0.26229,0.4286,0.71429,1.0,0.143,1.0,0,10,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,3,0,0,5,0,10],[88,186,0.4731,0.77678,0.23402,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,4,0,0,5,0,13],[92,186,0.4946,0.69642,0.29613,0.5354,0.71429,1.0,0.0,1.0,2,11,0,2,0,1,0,0,0,0,0,5,0,0,5,0,0,5,0,0,3,0,11],[96,186,0.5161,0.85265,0.1731,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,7,0,15],[100,186,0.5376,0.70088,0.24837,0.57132,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,4,0,0,4,0,0,8,0,0,6,0,7],[104,186,0.5591,0.80804,0.23038,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,2,0,0,10,0,13],[108,186,0.5806,0.82143,0.19562,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,6,0,14],[112,186,0.6022,0.80355,0.22799,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,4,0,0,8,0,13],[116,186,0.6237,0.75008,0.24748,0.57132,0.85714,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,5,0,0,6,0,11],[120,186,0.6452,0.7232,0.25739,0.53539,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,7,0,0,7,0,0,3,0,0,1,0,13],[124,186,0.6667,0.79908,0.22265,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,13],[128,186,0.6882,0.80357,0.25191,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,6,0,15],[132,186,0.7097,0.80357,0.25939,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,3,0,0,0,0,0,7,0,0,5,0,15],[136,186,0.7312,0.79463,0.20497,0.67857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,11,0,10],[140,186,0.7527,0.75446,0.30978,0.67857,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,0,8,0,13],[144,186,0.7742,0.75221,0.22588,0.57132,0.85707,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,5,0,1,4,0,0,4,0,0,7,0,10],[148,186,0.7957,0.74104,0.2867,0.571,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,4,0,0,3,0,0,3,0,0,8,0,11],[152,186,0.8172,0.74105,0.27994,0.42859,0.85707,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,7,0,0,4,0,0,1,0,0,4,0,14],[156,186,0.8387,0.71427,0.272,0.42859,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,7,0,0,2,0,0,6,0,0,5,0,10],[160,186,0.8602,0.84375,0.17985,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,6,0,15],[164,186,0.8817,0.78125,0.2448,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,5,0,0,6,0,13],[168,186,0.9032,0.72764,0.24055,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,6,0,9],[172,186,0.9247,0.72768,0.25594,0.57143,0.78564,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,0,8,0,8],[176,186,0.9462,0.71427,0.2113,0.57132,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,6,0,0,8,0,6],[180,186,0.9677,0.74553,0.23618,0.67857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,9,0,0,6,0,9],[184,186,0.9892,0.76338,0.22478,0.57143,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,0,4,0,12],[186,186,1.0,0.75445,0.23754,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,7,0,0,3,0,12]]}]},{"i":"83c97c0ae27f4a42","q":"Determine all the functions  $f:\\mathbb R\\mapsto\\mathbb R$  satisfies the equation  $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\\,\\forall a,b\\in\\mathbb R $","t":[{"b":1,"e":1.0,"k":"rising","v":0.38831,"x":0.96875,"p":[[0,396,0.0,0.38831,0.22094,0.28571,0.28571,0.4286,0.14,1.0,0,1,0,0,0,6,0,0,12,0,0,8,0,0,1,0,0,2,0,0,2,0,1],[4,396,0.0101,0.91964,0.16342,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,21],[8,396,0.0202,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[12,396,0.0303,0.91516,0.16312,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,8,0,21],[16,396,0.0404,0.85714,0.26,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,22],[20,396,0.0505,0.91517,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[24,396,0.0606,0.91517,0.19517,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,25],[28,396,0.0707,0.92855,0.12881,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,9,0,21],[32,396,0.0808,0.95536,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[36,396,0.0909,0.82141,0.24486,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,8,0,15],[40,396,0.101,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[44,396,0.1111,0.86607,0.18189,0.85711,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,9,0,16],[48,396,0.1212,0.84372,0.21833,0.85711,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,10,0,15],[52,396,0.1313,0.82587,0.20745,0.82132,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,11,0,13],[56,396,0.1414,0.9375,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[60,396,0.1515,0.86606,0.21112,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,7,0,19],[64,396,0.1616,0.75445,0.28846,0.57132,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,3,0,0,3,0,0,2,0,0,7,0,13],[68,396,0.1717,0.87053,0.18681,0.82132,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,18],[72,396,0.1818,0.80348,0.26448,0.71429,0.85714,1.0,0.14,1.0,0,15,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,2,0,0,8,0,15],[76,396,0.1919,0.84822,0.23673,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,19],[80,396,0.202,0.90179,0.20652,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,24],[84,396,0.2121,0.87945,0.1992,0.85711,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,7,0,19],[88,396,0.2222,0.83482,0.23175,0.82132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,15],[92,396,0.2323,0.88383,0.19412,0.82132,1.0,1.0,0.14,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,20],[96,396,0.2424,0.89286,0.21428,0.85714,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,22],[100,396,0.2525,0.90175,0.14486,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,19],[104,396,0.2626,0.84375,0.26088,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,20],[108,396,0.2727,0.84373,0.17263,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,15,0,11],[112,396,0.2828,0.85265,0.19395,0.857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,10,0,15],[116,396,0.2929,0.875,0.15465,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,13,0,14],[120,396,0.303,0.80801,0.23314,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,11,0,12],[124,396,0.3131,0.81696,0.28174,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,5,0,18],[128,396,0.3232,0.75444,0.24547,0.67825,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,5,0,0,10,0,9],[132,396,0.3333,0.71427,0.32538,0.42857,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,4,0,0,3,0,0,2,0,0,0,0,0,7,0,13],[136,396,0.3434,0.84375,0.20935,0.82143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,8,0,16],[140,396,0.3535,0.79909,0.24449,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,7,0,14],[144,396,0.3636,0.81249,0.25364,0.857,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,12,0,13],[148,396,0.3737,0.76783,0.27143,0.57132,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,3,0,0,7,0,13],[152,396,0.3838,0.67408,0.26544,0.42857,0.78571,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,6,0,0,3,0,0,2,0,0,10,0,6],[156,396,0.3939,0.71874,0.28231,0.42857,0.857,1.0,0.0,1.0,1,10,0,1,0,0,0,0,4,0,0,4,0,0,1,0,0,5,0,0,7,0,10],[160,396,0.404,0.84821,0.19541,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,13,0,13],[164,396,0.4141,0.80356,0.23623,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,0,10,0,12],[168,396,0.4242,0.75446,0.23753,0.67857,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,0,11,0,8],[172,396,0.4343,0.72321,0.26471,0.42859,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,4,0,0,1,0,0,2,0,0,13,0,7],[176,396,0.4444,0.84821,0.21705,0.85714,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,13,0,14],[180,396,0.4545,0.69642,0.27606,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i":"d18d3b140685ec16","q":"Given an integer  $ k > 1.$  We call a  $ k \\minus{}$ digits decimal integer  $ a_{1}a_{2}\\cdots a_{k}$  is  $ p \\minus{}$ monotonic, if for each of integers  $ i$  satisfying  $ 1\\le i\\le k \\minus{} 1,$  when  $ a_{i}$  is an odd number,  $ a_{i} > a_{i \\plus{} 1};$  when  $ a_{i}$  is an even number,  $ a_{i}<a_{i \\plus{} 1}.$  Find the number of  $ p \\minus{}$ monotonic  $ k \\minus{}$ digits integers.","t":[{"b":0,"e":0.0,"k":"falling","v":0.12054,"x":0.99553,"p":[[0,135,0.0,0.62052,0.25407,0.57142,0.57143,0.71429,0.0,1.0,1,6,1,1,0,3,0,0,0,0,0,0,0,0,17,0,0,4,0,0,1,0,6],[4,135,0.0296,0.65179,0.2111,0.57143,0.57143,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,19,0,0,3,0,0,0,0,7],[8,135,0.0593,0.79911,0.27632,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,0,0,0,8,0,0,0,0,0,3,0,18],[12,135,0.0889,0.80356,0.21055,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,1,0,0,2,0,16],[16,135,0.1185,0.78125,0.28344,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,2,0,17],[20,135,0.1481,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,135,0.1778,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,135,0.2074,0.92411,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,26],[32,135,0.237,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,135,0.2667,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[40,135,0.2963,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[44,135,0.3259,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,135,0.3556,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[52,135,0.3852,0.73214,0.25191,0.57143,0.64286,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,13,0,0,1,0,0,3,0,12],[56,135,0.4148,0.70534,0.23129,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,18,0,0,2,0,0,1,0,10],[60,135,0.4444,0.75445,0.22935,0.57143,0.57143,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,15,0,0,0,0,0,1,0,14],[64,135,0.4741,0.67857,0.29233,0.57143,0.57143,1.0,0.0,1.0,1,11,0,1,0,2,0,0,2,0,0,1,0,0,11,0,0,2,0,0,2,0,11],[68,135,0.5037,0.83482,0.25281,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,2,0,0,0,0,21],[72,135,0.5333,0.80804,0.23854,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,1,0,17],[76,135,0.563,0.61607,0.4307,0.10714,0.85714,1.0,0.0,1.0,8,15,0,8,0,2,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,15],[80,135,0.5926,0.68304,0.3792,0.42857,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,0,1,0,16],[84,135,0.6222,0.70088,0.37349,0.57132,0.92857,1.0,0.0,1.0,4,16,0,4,0,3,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,16],[88,135,0.6519,0.64732,0.42406,0.21429,1.0,1.0,0.0,1.0,8,17,0,8,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,17],[92,135,0.6815,0.5625,0.40867,0.10714,0.57143,1.0,0.0,1.0,8,12,0,8,0,1,0,0,1,0,0,5,0,0,3,0,0,0,0,0,2,0,12],[96,135,0.7111,0.66964,0.41255,0.14286,1.0,1.0,0.0,1.0,6,17,0,6,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,17],[100,135,0.7407,0.6875,0.39357,0.39286,1.0,1.0,0.0,1.0,5,17,0,5,0,2,0,0,1,0,0,1,0,0,4,0,0,0,0,0,2,0,17],[104,135,0.7704,0.73661,0.37983,0.53571,1.0,1.0,0.0,1.0,4,20,0,4,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,20],[108,135,0.8,0.52677,0.41255,0.14286,0.4998,1.0,0.0,1.0,7,12,0,7,0,5,0,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,12],[112,135,0.8296,0.62945,0.40699,0.32143,0.64286,1.0,0.0,1.0,8,15,0,8,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,0,0,15],[116,135,0.8593,0.67411,0.40757,0.28571,1.0,1.0,0.0,1.0,6,17,0,6,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,17],[120,135,0.8889,0.76786,0.31693,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,2,0,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,17],[124,135,0.9185,0.61605,0.41716,0.0,0.71429,1.0,0.0,1.0,9,14,0,9,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,1,0,14],[128,135,0.9481,0.24107,0.37532,0.0,0.0,0.35714,0.0,1.0,20,5,0,20,0,2,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,5],[132,135,0.9778,0.24107,0.35254,0.0,0.0,0.46429,0.0,1.0,18,4,0,18,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,0,0,0,4],[135,135,1.0,0.12054,0.25781,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,2]]},{"b":6,"e":0.14286,"k":"falling","v":0.15848,"x":0.82142,"p":[[0,176,0.0,0.82142,0.24745,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,1,0,19],[4,176,0.0227,0.62054,0.23313,0.57143,0.57143,0.75,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,1,0,0,18,0,0,2,0,0,4,0,4],[8,176,0.0455,0.63838,0.13356,0.57143,0.57143,0.57143,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,1,0,0,4,0,2],[12,176,0.0682,0.57587,0.17672,0.57143,0.57143,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,3,0,0,23,0,0,0,0,0,2,0,2],[16,176,0.0909,0.63391,0.17835,0.57143,0.57143,0.74996,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,0,0,0,21,0,0,1,0,0,6,0,2],[20,176,0.1136,0.62054,0.19434,0.57143,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,2,0,0,21,0,0,0,0,0,3,0,4],[24,176,0.1364,0.59819,0.18363,0.57143,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,2,0,0,22,0,0,0,0,0,4,0,2],[28,176,0.1591,0.59372,0.17169,0.57143,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,23,0,0,2,0,0,2,0,2],[32,176,0.1818,0.59149,0.10601,0.57143,0.57143,0.57143,0.357,1.0,0,1,0,0,0,0,0,0,0,0,1,2,0,0,23,0,0,5,0,0,0,0,1],[36,176,0.2045,0.57143,0.21724,0.57143,0.57143,0.57143,0.0,1.0,2,3,0,2,0,1,0,0,0,0,0,1,0,0,23,0,0,1,0,0,1,0,3],[40,176,0.2273,0.65624,0.19516,0.57143,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,20,0,0,3,0,0,1,0,6],[44,176,0.25,0.529,0.20583,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,4,0,1,0,0,0,4,0,0,19,0,0,1,0,0,1,0,2],[48,176,0.2727,0.51339,0.23107,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,7,0,0,0,0,0,2,0,0,18,0,0,2,0,0,1,0,2],[52,176,0.2955,0.49999,0.17127,0.42859,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,0,0,0,4,0,0,21,0,0,1,0,0,1,0,0],[56,176,0.3182,0.62498,0.23352,0.57143,0.57143,0.64286,0.14286,1.0,0,6,0,0,0,3,0,0,0,0,0,1,0,0,20,0,0,0,0,0,2,0,6],[60,176,0.3409,0.63393,0.25489,0.57143,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,1,0,0,1,0,0,18,0,0,0,0,0,1,0,8],[64,176,0.3636,0.51786,0.25191,0.53571,0.57143,0.57143,0.0,1.0,2,3,0,2,0,4,0,0,1,0,0,1,0,0,20,0,0,0,0,0,1,0,3],[68,176,0.3864,0.58482,0.23517,0.57143,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,0,0,0,18,0,0,3,0,0,3,0,3],[72,176,0.4091,0.63839,0.23686,0.57143,0.57143,0.75,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,0,0,0,20,0,0,1,0,0,1,0,7],[76,176,0.4318,0.49554,0.30301,0.14286,0.57143,0.57143,0.0,1.0,1,5,0,1,0,10,0,0,0,0,0,0,0,0,14,0,0,2,0,0,0,0,5],[80,176,0.4545,0.54908,0.2055,0.57132,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,0,0,0,1,0,0,19,0,0,4,0,0,2,0,1],[84,176,0.4773,0.61606,0.29329,0.53539,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,0,0,0,2,0,0,12,0,0,2,0,0,2,0,8],[88,176,0.5,0.53124,0.21199,0.57143,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,0,0,0,0,0,0,21,0,0,2,0,0,2,0,1],[92,176,0.5227,0.55803,0.26088,0.39288,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,5,0,0,3,0,0,1,0,0,14,0,0,4,0,0,0,0,5],[96,176,0.5455,0.54464,0.2683,0.42857,0.57143,0.57143,0.14286,1.0,0,5,0,0,0,7,0,0,0,0,0,2,0,0,17,0,0,0,0,0,1,0,5],[100,176,0.5682,0.55357,0.23891,0.57143,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,6,0,0,0,0,0,0,0,0,20,0,0,1,0,0,2,0,3],[104,176,0.5909,0.48213,0.27606,0.14286,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,10,0,0,1,0,0,1,0,0,15,0,0,1,0,0,0,0,4],[108,176,0.6136,0.50893,0.32915,0.14286,0.57143,0.64286,0.14286,1.0,0,7,0,0,0,12,0,0,0,0,0,1,0,0,11,0,0,0,0,0,1,0,7],[112,176,0.6364,0.35268,0.3174,0.14286,0.14286,0.57143,0.14286,1.0,0,5,0,0,0,20,0,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,5],[116,176,0.6591,0.33929,0.31288,0.14286,0.14286,0.46431,0.14286,1.0,0,5,0,0,0,20,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,5],[120,176,0.6818,0.5267,0.36508,0.14286,0.5,1.0,0.14,1.0,0,10,0,0,0,12,0,0,2,0,0,2,0,0,5,0,0,0,0,0,1,0,10],[124,176,0.7045,0.39732,0.36023,0.14286,0.14286,0.64286,0.0,1.0,1,7,0,1,0,18,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,7],[128,176,0.7273,0.38838,0.32533,0.14286,0.14286,0.57143,0.14286,1.0,0,5,0,0,0,17,0,0,3,0,0,2,0,0,3,1,0,0,0,0,0,1,5],[132,176,0.75,0.3884,0.31791,0.14286,0.21431,0.57143,0.0,1.0,1,5,0,1,0,15,0,0,3,0,0,2,0,0,5,0,0,1,0,0,0,0,5],[136,176,0.7727,0.40624,0.34646,0.14286,0.14288,0.57143,0.14286,1.0,0,7,0,0,0,18,0,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,7],[140,176,0.7955,0.40177,0.33775,0.14286,0.14286,0.57143,0.14286,1.0,0,6,0,0,0,18,0,0,0,0,0,4,0,0,3,0,0,0,0,0,1,0,6],[144,176,0.8182,0.43304,0.35081,0.14286,0.14288,0.85714,0.14286,1.0,0,6,0,0,0,17,0,0,0,0,0,4,0,0,2,0,0,0,0,0,3,0,6],[148,176,0.8409,0.41518,0.33571,0.14286,0.2143,0.60714,0.14286,1.0,0,6,0,0,0,16,0,0,2,0,0,4,0,0,2,0,0,1,0,0,1,0,6],[152,176,0.8636,0.30801,0.2651,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,20,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,3],[156,176,0.8864,0.34374,0.27165,0.14286,0.14286,0.5711,0.14286,1.0,0,3,0,0,0,18,0,0,1,0,0,4,0,0,6,0,0,0,0,0,0,0,3],[160,176,0.9091,0.30356,0.28958,0.14286,0.14286,0.32143,0.14286,1.0,0,4,0,0,0,22,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,4],[164,176,0.9318,0.32589,0.30354,0.14286,0.14286,0.42858,0.14286,1.0,0,4,0,0,0,21,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,4],[168,176,0.9545,0.31688,0.2874,0.14286,0.14286,0.42857,0.14,1.0,0,4,0,0,0,20,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,4],[172,176,0.9773,0.20518,0.16734,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,26,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[176,176,1.0,0.15848,0.087,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,1,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fdf75131ecb3618d","q":"Given a convex $n$-gon, $n \\geq 4$, at most how many diagonals can be drawn such that each drawn diagonal intersects every other drawn diagonal either in the interior of the $n$-gon or at a vertex? Prove your answer.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.05804,"x":0.375,"p":[[0,84,0.0,0.21866,0.2988,0.0,0.0,0.46429,0.0,0.85714,18,0,18,18,0,3,0,0,2,0,0,1,0,0,3,0,0,3,0,0,2,0,0],[4,84,0.0476,0.25,0.20825,0.14286,0.14286,0.32143,0.0,1.0,3,1,0,3,0,17,0,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[8,84,0.0952,0.21429,0.17857,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,17,0,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[12,84,0.1429,0.16964,0.10374,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,23,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,84,0.1905,0.125,0.1171,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,21,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,84,0.2381,0.13831,0.13114,0.14214,0.14286,0.14286,0.0,0.71429,7,0,0,7,0,23,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,84,0.2857,0.2008,0.22263,0.14286,0.14286,0.14287,0.0,1.0,5,1,0,5,0,22,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[28,84,0.3333,0.24552,0.20894,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,20,0,0,0,0,0,4,0,0,2,0,0,3,0,0,0,0,0],[32,84,0.381,0.21875,0.22583,0.14286,0.14286,0.32142,0.0,1.0,7,1,0,7,0,16,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[36,84,0.4286,0.375,0.30252,0.14286,0.28571,0.57143,0.0,1.0,3,3,0,3,0,12,0,0,2,0,0,6,0,0,2,0,0,3,0,0,1,0,3],[40,84,0.4762,0.375,0.28515,0.14286,0.2857,0.71429,0.0,1.0,2,1,0,2,0,13,0,0,4,0,0,2,0,0,1,0,0,8,0,0,1,0,1],[44,84,0.5238,0.31696,0.27833,0.14286,0.14286,0.46429,0.0,1.0,3,1,0,3,0,16,0,0,2,0,0,3,0,0,2,0,0,3,0,0,2,0,1],[48,84,0.5714,0.27683,0.24987,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,12,0,0,4,0,0,4,0,0,2,0,0,2,0,0,2,0,0],[52,84,0.619,0.25,0.26,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,10,0,0,1,0,0,4,0,0,5,0,0,1,0,0,0,0,1],[56,84,0.6667,0.37499,0.25691,0.14286,0.35714,0.57143,0.0,0.85714,3,0,0,3,0,9,0,0,4,0,0,6,0,0,3,0,0,5,0,0,2,0,0],[60,84,0.7143,0.24544,0.20279,0.14286,0.14288,0.32143,0.0,0.71429,5,0,0,5,0,14,0,0,5,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[64,84,0.7619,0.14286,0.11294,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,18,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,84,0.8095,0.125,0.14617,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,16,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[72,84,0.8571,0.1249,0.12236,0.0,0.14286,0.14286,0.0,0.571,11,0,0,11,0,16,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,84,0.9048,0.1383,0.145,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,13,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[80,84,0.9524,0.08482,0.12299,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,11,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[84,84,1.0,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.05804,"x":0.16071,"p":[[0,28,0.0,0.13393,0.25985,0.0,0.0,0.07143,0.0,1.0,24,1,24,24,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,1],[4,28,0.1429,0.14732,0.18029,0.0,0.14286,0.14286,0.0,1.0,9,1,1,9,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,28,0.2857,0.16071,0.17035,0.0,0.14286,0.17857,0.0,0.71429,11,0,0,11,0,13,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[12,28,0.4286,0.10268,0.1439,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,12,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,28,0.5714,0.08482,0.15916,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,10,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,28,0.7143,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f4adffbfe098b891","q":"Let  $n\\geq 2$  be an integer and  $a_1,a_2,\\cdots,a_n>1$  be real numbers. Prove that the inequality below holds.  $$ \\prod_{i=1}^n\\left(a_ia_{i+1}-\\frac{1}{a_ia_{i+1}}\\right)\\geq 2^n\\prod_{i=1}^n\\left(a_i-\\frac{1}{a_i}\\right) $$","t":[{"b":2,"e":0.0,"k":"flat","v":0.04464,"x":0.24103,"p":[[0,17,0.0,0.17409,0.2251,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,1,0,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[4,17,0.2353,0.24103,0.29756,0.0,0.0,0.4642,0.0,1.0,17,1,0,17,0,0,0,0,6,0,0,1,0,0,3,0,0,4,0,0,0,0,1],[8,17,0.4706,0.21875,0.22011,0.0,0.28571,0.32143,0.0,0.71429,14,0,0,14,0,0,0,0,10,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[12,17,0.7059,0.19195,0.26631,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,1,0,0,7,0,0,1,0,0,2,0,0,2,0,0,0,0,1],[16,17,0.9412,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[17,17,1.0,0.11161,0.27371,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,2]]},{"b":5,"e":0.57143,"k":"flat","v":0.24549,"x":0.41963,"p":[[0,60,0.0,0.29909,0.2865,0.0,0.28571,0.57111,0.0,1.0,13,1,0,13,0,0,0,0,5,0,0,5,0,0,5,0,0,3,0,0,0,0,1],[4,60,0.0667,0.38391,0.26591,0.28571,0.35714,0.46418,0.0,1.0,6,2,0,6,0,0,0,0,10,0,0,8,0,0,2,0,0,4,0,0,0,0,2],[8,60,0.1333,0.3214,0.22585,0.21429,0.28571,0.42858,0.0,0.71429,8,0,0,8,0,0,0,0,10,0,0,7,0,0,4,0,0,3,0,0,0,0,0],[12,60,0.2,0.24549,0.24279,0.0,0.2857,0.32143,0.0,0.85714,13,0,0,13,0,0,0,0,11,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[16,60,0.2667,0.26783,0.27137,0.0,0.2857,0.46418,0.0,1.0,13,1,0,13,0,0,0,0,10,0,0,1,0,0,5,0,0,2,0,0,0,0,1],[20,60,0.3333,0.31248,0.22425,0.10717,0.28571,0.4642,0.0,0.71429,8,0,0,8,0,1,0,0,10,0,0,5,0,0,6,0,0,2,0,0,0,0,0],[24,60,0.4,0.37051,0.25216,0.2857,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,0,0,0,10,0,0,4,0,0,4,0,0,7,0,0,0,0,0],[28,60,0.4667,0.32141,0.25504,0.10714,0.28571,0.4642,0.0,1.0,8,1,0,8,0,1,0,0,12,0,0,3,0,0,4,0,0,3,0,0,0,0,1],[32,60,0.5333,0.31248,0.273,0.0,0.2857,0.42857,0.0,1.0,9,2,0,9,0,1,0,0,11,0,0,5,0,0,2,0,0,2,0,0,0,0,2],[36,60,0.6,0.31694,0.24929,0.0,0.28571,0.4642,0.0,0.85714,9,0,0,9,0,0,0,0,11,0,0,4,0,0,4,0,0,3,0,0,1,0,0],[40,60,0.6667,0.39732,0.31488,0.21427,0.35714,0.57143,0.0,1.0,8,4,0,8,0,0,0,0,8,0,0,5,0,0,5,0,0,2,0,0,0,0,4],[44,60,0.7333,0.4107,0.33834,0.0,0.49979,0.71429,0.0,1.0,11,2,0,11,0,0,0,0,3,0,0,2,0,0,5,0,0,8,0,0,1,0,2],[48,60,0.8,0.29462,0.25235,0.0,0.28571,0.4642,0.0,0.71429,11,0,0,11,0,0,0,0,9,0,0,4,0,0,4,0,0,4,0,0,0,0,0],[52,60,0.8667,0.41963,0.2285,0.28571,0.42857,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,12,0,0,7,0,0,3,0,0,6,0,0,0,0,1],[56,60,0.9333,0.34821,0.27185,0.0,0.28571,0.57143,0.0,0.85714,9,0,0,9,0,0,0,0,9,0,0,4,0,0,3,0,0,6,0,0,1,0,0],[60,60,1.0,0.29017,0.25374,0.0,0.28571,0.4642,0.0,0.71429,11,0,0,11,0,1,0,0,8,0,0,4,0,0,4,0,0,4,0,0,0,0,0]]}]},{"i":"e4c565c18fae0742","q":"Find all functions  $f \\colon \\mathbb{R} \\to \\mathbb{R}$  such that\n\\[f(f(x) - y) = f(xy) + f(x)f(-y)\\]\nfor any two real numbers  $x, y$ .\n\n*Proposed by Pablo Valeriano*","t":[{"b":4,"e":0.57143,"k":"rising","v":0.16509,"x":0.58927,"p":[[0,275,0.0,0.16509,0.0883,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,19,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,275,0.0145,0.49107,0.26711,0.28571,0.42857,0.57143,0.0,1.0,1,5,1,1,0,2,0,0,7,0,0,11,0,0,5,0,0,0,0,0,1,0,5],[8,275,0.0291,0.54017,0.19799,0.42857,0.4286,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,15,0,0,8,0,0,1,0,0,2,0,3],[12,275,0.0436,0.53569,0.2988,0.28571,0.42857,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,7,0,0,9,0,0,4,0,0,0,0,0,2,0,7],[16,275,0.0582,0.46875,0.23483,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,1,0,0,7,0,0,14,0,0,5,0,0,0,0,0,0,0,4],[20,275,0.0727,0.4464,0.19478,0.28571,0.42857,0.571,0.14286,1.0,0,1,0,0,0,2,0,0,9,0,0,12,0,0,5,0,0,1,0,0,2,0,1],[24,275,0.0873,0.4375,0.20497,0.28571,0.42857,0.42857,0.0,1.0,1,2,0,1,0,1,0,0,8,0,0,15,0,0,4,0,0,0,0,0,1,0,2],[28,275,0.1018,0.47768,0.24119,0.28571,0.42857,0.46431,0.1429,1.0,0,3,0,0,0,1,0,0,11,0,0,12,0,0,1,0,0,1,0,0,3,0,3],[32,275,0.1164,0.41515,0.12552,0.28571,0.42857,0.571,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,10,0,0,8,0,0,1,0,0,0,0,0],[36,275,0.1309,0.41964,0.14698,0.28571,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,10,0,0,15,0,0,3,0,0,2,0,0,1,0,0],[40,275,0.1455,0.39286,0.10714,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,15,0,0,3,0,0,1,0,0,0,0,0],[44,275,0.16,0.49107,0.25238,0.39286,0.42857,0.46431,0.0,1.0,1,4,0,1,0,1,0,0,6,0,0,16,0,0,1,0,0,1,0,0,2,0,4],[48,275,0.1745,0.40625,0.18249,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,11,0,0,11,0,0,5,0,0,0,0,0,1,0,1],[52,275,0.1891,0.45087,0.18593,0.42857,0.42857,0.571,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,17,0,0,6,0,0,1,0,0,1,0,1],[56,275,0.2036,0.47768,0.21609,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,8,0,0,10,0,0,7,0,0,1,0,0,2,0,2],[60,275,0.2182,0.49104,0.24983,0.28571,0.42859,0.57143,0.0,1.0,1,3,0,1,0,2,0,0,8,0,0,6,0,0,9,0,0,1,0,0,2,0,3],[64,275,0.2327,0.50445,0.2575,0.28571,0.42857,0.57143,0.0,1.0,1,5,0,1,0,0,0,0,9,0,0,10,0,0,6,0,0,0,0,0,1,0,5],[68,275,0.2473,0.44195,0.1868,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,8,0,0,6,0,0,3,0,0,2,0,0],[72,275,0.2618,0.45982,0.15458,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,11,0,0,10,0,0,1,0,0,0,0,1],[76,275,0.2764,0.45089,0.25532,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,4,0,0,7,0,0,9,0,0,7,0,0,0,0,0,0,0,4],[80,275,0.2909,0.39731,0.20118,0.2857,0.35714,0.4286,0.14286,1.0,0,2,0,0,0,4,0,0,12,0,0,9,0,0,5,0,0,0,0,0,0,0,2],[84,275,0.3055,0.45084,0.17533,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,10,0,0,11,0,0,1,0,0,0,0,1],[88,275,0.32,0.39286,0.20825,0.28571,0.42857,0.42858,0.0,1.0,1,1,0,1,0,5,0,0,8,0,0,11,0,0,4,0,0,1,0,0,1,0,1],[92,275,0.3345,0.4107,0.1587,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,4,0,0,7,0,0,12,0,0,8,0,0,0,0,0,1,0,0],[96,275,0.3491,0.45981,0.24152,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,9,0,0,10,0,0,4,0,0,1,0,0,3,0,2],[100,275,0.3636,0.46427,0.22303,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,8,0,0,9,0,0,8,0,0,1,0,0,0,0,3],[104,275,0.3782,0.44195,0.22967,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,5,0,0,7,0,0,11,0,0,4,0,0,0,0,0,4,0,1],[108,275,0.3927,0.49999,0.26244,0.28571,0.42857,0.57143,0.0,1.0,2,3,0,2,0,1,0,0,7,0,0,7,0,0,8,0,0,1,0,0,3,0,3],[112,275,0.4073,0.4866,0.23652,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,6,0,0,13,0,0,3,0,0,1,0,0,4,0,2],[116,275,0.4218,0.48212,0.23076,0.28571,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,3,0,0,6,0,0,13,0,0,3,0,0,2,0,0,3,0,2],[120,275,0.4364,0.47767,0.22477,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,8,0,0,13,0,0,5,0,0,1,0,0,1,0,3],[124,275,0.4509,0.4732,0.22142,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,7,0,0,11,0,0,8,0,0,1,0,0,0,0,3],[128,275,0.4655,0.45536,0.23266,0.28571,0.42857,0.46431,0.0,1.0,1,3,0,1,0,1,0,0,9,0,0,13,0,0,3,0,0,1,0,0,1,0,3],[132,275,0.48,0.46874,0.19637,0.39286,0.42857,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,11,0,0,10,0,0,0,0,0,2,0,1],[136,275,0.4945,0.54015,0.21048,0.42857,0.42859,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,13,0,0,8,0,0,2,0,0,2,0,3],[140,275,0.5091,0.54908,0.28596,0.39286,0.42857,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,5,0,0,9,0,0,6,0,0,0,0,0,3,0,6],[144,275,0.5236,0.54015,0.23618,0.42857,0.42859,0.57143,0.14286,1.0,0,5,0,0,0,1,0,0,5,0,0,11,0,0,9,0,0,0,0,0,1,0,5],[148,275,0.5382,0.56695,0.24609,0.42857,0.4286,0.74996,0.1429,1.0,0,5,0,0,0,1,0,0,4,0,0,12,0,0,6,0,0,1,0,0,3,0,5],[152,275,0.5527,0.51784,0.18123,0.42857,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,11,0,0,10,0,0,3,0,0,2,0,1],[156,275,0.5673,0.54911,0.28596,0.39286,0.42859,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,6,0,0,12,0,0,1,0,0,2,0,0,3,0,6],[160,275,0.5818,0.58926,0.23891,0.42857,0.57121,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,10,0,0,8,0,0,2,0,0,3,0,5],[164,275,0.5964,0.48659,0.25965,0.28571,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,3,0,0,9,0,0,9,0,0,4,0,0,1,0,0,2,0,4],[168,275,0.6109,0.58927,0.24157,0.42857,0.57143,0.85711,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,9,0,0,9,0,0,1,0,0,5,0,4],[172,275,0.6255,0.5357,0.22868,0.42857,0.42859,0.6071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8,0.19043,0.28571,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,7,0,0,10,0,0,8,0,0,3,0,0,1,0,1],[248,260,0.9538,0.38393,0.25111,0.24999,0.35714,0.42857,0.0,1.0,3,1,0,3,0,5,0,0,8,0,0,9,0,0,1,0,0,3,0,0,2,0,1],[252,260,0.9692,0.4107,0.18122,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,11,0,0,8,0,0,7,0,0,2,0,0,1,0,0],[256,260,0.9846,0.49552,0.25501,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,5,0,0,4,0,0,10,0,0,5,0,0,4,0,0,0,0,4],[260,260,1.0,0.39283,0.22301,0.2857,0.42857,0.571,0.0,1.0,2,1,0,2,0,5,0,0,7,0,0,9,0,0,6,0,0,1,0,0,1,0,1]]}]},{"i":"f754d5e4e3234d9d","q":"Given a sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of positive integers in which every positive integer occurs exactly once. Prove that there exist integers $\\ell$ and $m$, $1<\\ell<m$, such that $a_{1}+a_{m}=2 a_{\\ell}$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.19197,"x":0.25446,"p":[[0,33,0.0,0.25446,0.24675,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,22,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,33,0.1212,0.23214,0.25191,0.14286,0.14286,0.14286,0.0,1.0,1,3,1,1,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,33,0.2424,0.19197,0.06785,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.19643,0.06916,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.24107,0.06621,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.21428,0.07143,0.14286,0.21428,0.28571,0.14286,0.28571,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.2366,0.06785,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.16062,"x":0.27678,"p":[[0,28,0.0,0.20982,0.21124,0.14286,0.14286,0.14286,0.0,1.0,1,2,1,1,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,28,0.1429,0.27678,0.31731,0.14286,0.14286,0.17857,0.0,1.0,3,5,3,3,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,28,0.2857,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.19643,0.15465,0.14286,0.14286,0.14289,0.14286,1.0,0,1,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,28,0.8571,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.17411,0.05906,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de29a59db54c587e","q":"Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \\leq N \\leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.","t":[{"b":1,"e":0.0,"k":"falling","v":0.23214,"x":0.6964,"p":[[0,18,0.0,0.6964,0.3973,0.53539,1.0,1.0,0.0,1.0,6,18,6,6,0,1,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,18],[4,18,0.2222,0.62054,0.46922,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,19],[8,18,0.4444,0.38393,0.45797,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,9],[12,18,0.6667,0.23214,0.36726,0.0,0.0,0.32144,0.0,1.0,20,3,0,20,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,3],[16,18,0.8889,0.24554,0.38504,0.0,0.0,0.42857,0.0,1.0,20,5,0,20,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,5],[18,18,1.0,0.24551,0.38996,0.0,0.0,0.571,0.0,1.0,22,5,0,22,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,5]]},{"b":5,"e":0.2857,"k":"falling","v":0.38392,"x":0.88839,"p":[[0,52,0.0,0.76786,0.40367,0.78571,1.0,1.0,0.0,1.0,4,24,3,4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[4,52,0.0769,0.88839,0.28288,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,26],[8,52,0.1538,0.72765,0.40464,0.46396,1.0,1.0,0.0,1.0,5,20,0,5,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,20],[12,52,0.2308,0.64284,0.43741,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,17],[16,52,0.3077,0.59375,0.44765,0.0,0.85714,1.0,0.0,1.0,9,15,0,9,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,15],[20,52,0.3846,0.66964,0.44239,0.14286,1.0,1.0,0.0,1.0,6,20,0,6,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,20],[24,52,0.4615,0.70089,0.40146,0.39286,1.0,1.0,0.0,1.0,5,19,0,5,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,19],[28,52,0.5385,0.74107,0.41255,0.53571,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,22],[32,52,0.6154,0.77677,0.35704,0.71429,1.0,1.0,0.0,1.0,5,19,0,5,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,19],[36,52,0.6923,0.875,0.26904,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,23],[40,52,0.7692,0.7232,0.40868,0.46396,1.0,1.0,0.0,1.0,6,19,0,6,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,19],[44,52,0.8462,0.61161,0.44641,0.14286,1.0,1.0,0.0,1.0,6,18,0,6,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[48,52,0.9231,0.58482,0.41705,0.14286,0.64286,1.0,0.0,1.0,3,15,0,3,0,9,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,15],[52,52,1.0,0.38392,0.36672,0.14286,0.14286,0.60714,0.0,1.0,4,7,0,4,0,15,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,7]]}]},{"i":"4ae2a53014a99d29","q":"Let $G$ be a finite undirected graph, $\\mathrm{n} \\in \\mathbb{N}^{*}, \\mathrm{~d} \\in \\mathbb{N}^{*}$ such that the degree of every vertex in $G$ is bounded by $d$, and $x$ a vertex of $G$. We denote $a(n, x)$ as the number of induced connected subgraphs of $G$ containing $n$ vertices including the vertex $x$ (an induced subgraph $U$ is a graph whose vertices are vertices of $G$ and if $x$ and $y$ are vertices of $U$, $x$ is connected by an edge to $y$ in $U$ if and only if $x$ is connected by an edge to $y$ in $G$).\n\n1. Show that $a(n, x) \\leqslant 2^{n d}$.\n2. Show that $a(n, x) \\leqslant\\left(\\frac{(d+1)^{d+1}}{d^{d}}\\right)^{n}$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.3125,"x":0.81696,"p":[[0,46,0.0,0.3125,0.34337,0.0,0.2857,0.46431,0.0,1.0,14,3,1,14,0,1,0,0,4,0,0,5,0,0,2,0,0,1,0,0,2,0,3],[4,46,0.087,0.77677,0.23403,0.57143,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,5,0,0,3,0,14],[8,46,0.1739,0.7031,0.30936,0.42859,0.85714,1.0,0.0,1.0,1,11,0,1,1,1,0,0,2,0,0,4,0,0,4,0,0,1,0,0,7,0,11],[12,46,0.2609,0.81696,0.20896,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,6,0,14],[16,46,0.3478,0.75445,0.26783,0.57143,0.85707,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,0,5,0,12],[20,46,0.4348,0.76786,0.2165,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,5,0,0,8,0,10],[24,46,0.5217,0.74102,0.22713,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,8,0,0,7,0,8],[28,46,0.6087,0.7723,0.19517,0.57143,0.857,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,8,0,9],[32,46,0.6957,0.78125,0.24996,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,2,0,0,2,0,0,8,0,0,6,0,12],[36,46,0.7826,0.74997,0.22018,0.57143,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,12,0,6],[40,46,0.8696,0.72761,0.22973,0.57132,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,0,8,0,7],[44,46,0.9565,0.63835,0.19227,0.571,0.71429,0.74996,0.143,0.85714,0,0,0,0,0,1,0,0,3,0,0,2,0,0,8,0,0,10,0,0,8,0,0],[46,46,1.0,0.64728,0.24481,0.571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,4,0,0,1,0,0,11,0,0,5,0,0,5,0,5]]},{"b":4,"e":1.0,"k":"rising","v":0.29467,"x":0.92857,"p":[[0,29,0.0,0.29467,0.32915,0.0,0.2143,0.57111,0.0,1.0,13,2,0,13,0,3,0,0,6,0,0,1,0,0,4,0,0,0,0,0,3,0,2],[4,29,0.1379,0.83259,0.20717,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,3,1,0,5,0,0,6,0,15],[8,29,0.2759,0.74998,0.26001,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,3,0,0,4,0,0,3,0,0,9,0,10],[12,29,0.4138,0.75444,0.23485,0.57143,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,8,0,9],[16,29,0.5517,0.875,0.17767,0.85711,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,8,0,17],[20,29,0.6897,0.92857,0.20825,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[24,29,0.8276,0.86605,0.17837,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,4,0,18],[28,29,0.9655,0.86158,0.19065,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,5,0,18],[29,29,1.0,0.81248,0.18365,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,9,0,11]]}]},{"i":"38e67fb80dfe2671","q":"Let $P_{1}, P_{2}, \\ldots, P_{2019}$ be non-constant polynomials with real coefficients such that\n\n$$\nP_{1}\\left(P_{2}(x)\\right)=P_{2}\\left(P_{3}(x)\\right)=\\ldots=P_{2019}\\left(P_{1}(x)\\right)\n$$\n\nfor all real $x$. Prove that $P_{1}=P_{2}=\\ldots=P_{2019}$.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.28125,"x":0.77232,"p":[[0,44,0.0,0.59819,0.24338,0.28571,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,8,0,0,1,0,0,7,0,0,6,0,0,7,0,2],[4,44,0.0909,0.75891,0.26831,0.57132,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,4,0,0,2,0,0,2,0,0,2,0,0,10,0,11],[8,44,0.1818,0.76339,0.28032,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,3,0,0,0,0,0,5,0,0,8,0,12],[12,44,0.2727,0.68303,0.3169,0.42857,0.78564,1.0,0.0,1.0,2,10,0,2,0,1,0,0,3,0,0,5,0,0,0,0,0,5,0,0,6,0,10],[16,44,0.3636,0.77232,0.19515,0.71429,0.85707,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,9,0,8],[20,44,0.4545,0.74554,0.29824,0.42857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,2,0,0,4,0,0,0,0,0,3,0,0,7,0,13],[24,44,0.5455,0.75445,0.24805,0.67857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,6,0,0,8,0,10],[28,44,0.6364,0.76339,0.26633,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,3,0,0,0,0,0,5,0,0,9,0,11],[32,44,0.7273,0.58034,0.27418,0.42857,0.4998,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,9,0,0,2,0,0,5,0,0,4,0,5],[36,44,0.8182,0.45981,0.19474,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,10,0,0,15,0,0,2,0,0,2,0,0,1,0,2],[40,44,0.9091,0.38838,0.14388,0.28571,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,12,0,0,14,0,0,2,0,0,1,0,0,1,0,0],[44,44,1.0,0.28125,0.42481,0.0,0.0,0.57143,0.0,1.0,20,8,0,20,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,8]]},{"b":2,"e":0.42857,"k":"flat","v":0.58482,"x":0.74554,"p":[[0,38,0.0,0.58482,0.27283,0.28571,0.57143,0.85704,0.0,1.0,1,3,0,1,0,2,0,0,6,0,0,2,0,0,6,0,0,6,0,0,6,0,3],[4,38,0.1053,0.67409,0.29066,0.42857,0.85714,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,5,0,0,3,0,0,2,0,0,3,0,0,11,0,6],[8,38,0.2105,0.70982,0.32436,0.39286,0.85714,1.0,0.0,1.0,1,11,0,1,0,3,0,0,4,0,0,1,0,0,0,0,0,4,0,0,8,0,11],[12,38,0.3158,0.72321,0.24984,0.71429,0.78571,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,3,0,0,3,0,0,0,0,0,9,0,0,10,0,6],[16,38,0.4211,0.74553,0.25187,0.53571,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,2,0,0,11,0,9],[20,38,0.5263,0.74554,0.22513,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,6,0,0,9,0,8],[24,38,0.6316,0.72324,0.30076,0.57143,0.85707,1.0,0.0,1.0,2,11,0,2,0,1,0,0,2,0,0,1,0,0,4,0,0,5,0,0,6,0,11],[28,38,0.7368,0.72765,0.2843,0.42857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,5,0,0,3,0,0,2,0,0,2,0,0,8,0,11],[32,38,0.8421,0.72766,0.26089,0.5354,0.78564,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,5,0,0,1,0,0,7,0,0,7,0,9],[36,38,0.9474,0.59821,0.30186,0.28571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,6,0,0,4,0,0,2,0,0,6,0,0,6,0,5],[38,38,1.0,0.60265,0.28288,0.28571,0.57121,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,8,0,0,5,0,0,5,0,0,2,0,0,4,0,7]]}]},{"i":"39996b45fe8f3c5a","q":"Let  $x,y$  be positive real numbers .Find the minimum of  $x+y+\\frac{|x-1|}{y}+\\frac{|y-1|}{x}$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.86157,"x":1.0,"p":[[0,96,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,96,0.0417,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,96,0.0833,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,96,0.125,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,96,0.1667,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,96,0.2083,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,96,0.25,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,96,0.2917,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,96,0.3333,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[36,96,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,96,0.4167,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,96,0.4583,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,96,0.5,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[52,96,0.5417,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,96,0.5833,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[60,96,0.625,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,96,0.6667,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[68,96,0.7083,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[72,96,0.75,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[76,96,0.7917,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[80,96,0.8333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,96,0.875,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[88,96,0.9167,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[92,96,0.9583,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[96,96,1.0,0.86157,0.1694,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,8,0,15]]},{"b":7,"e":0.71429,"k":"flat","v":0.933,"x":1.0,"p":[[0,104,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,104,0.0385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,104,0.0769,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,104,0.1154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,104,0.1538,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,104,0.1923,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[24,104,0.2308,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,104,0.2692,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,104,0.3077,0.98213,0.0692,1.0,1.0,1.0,0.714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,104,0.3462,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,104,0.3846,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[44,104,0.4231,0.94196,0.10631,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[48,104,0.4615,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[52,104,0.5,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,104,0.5385,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,104,0.5769,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,104,0.6154,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,104,0.6538,0.95534,0.12599,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[72,104,0.6923,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[76,104,0.7308,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,104,0.7692,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[84,104,0.8077,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[88,104,0.8462,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[92,104,0.8846,0.933,0.14728,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[96,104,0.9231,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[100,104,0.9615,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[104,104,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"bf425d2411d24e78","q":"Let $a$ and $b$ be positive integers bigger than 2. Prove that there exists a positive integer $k$ and a sequence $n_{1}, n_{2}, \\ldots, n_{k}$ consisting of positive integers, such that $n_{1}=a$, $n_{k}=b$, and $\\left(n_{i}+n_{i+1}\\right) \\mid n_{i} n_{i+1}$ for all $i=1,2, \\ldots, k-1$.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.11161,"x":0.2007,"p":[[0,43,0.0,0.15161,0.10064,0.14214,0.14286,0.1429,0.0,0.4286,6,0,1,6,0,19,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.11161,0.12234,0.0,0.14286,0.14287,0.0,0.42857,15,0,0,15,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.13822,0.13114,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,11,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.12054,0.16409,0.0,0.14286,0.14286,0.0,0.85714,13,0,0,13,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,43,0.3721,0.15624,0.13529,0.0,0.14286,0.2857,0.0,0.571,9,0,0,9,0,14,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,43,0.4651,0.19643,0.14617,0.10714,0.14286,0.28571,0.0,0.4286,8,0,0,8,0,9,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.14286,0.13363,0.0,0.14286,0.1786,0.0,0.4286,11,0,0,11,0,13,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.14286,0.13832,0.0,0.14286,0.14292,0.0,0.4286,11,0,0,11,0,14,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.17857,0.15972,0.0,0.14286,0.2857,0.0,0.57143,10,0,1,10,0,10,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[36,43,0.8372,0.2007,0.20475,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,10,0,0,6,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[40,43,0.9302,0.15616,0.12038,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,12,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.14286,0.13363,0.0,0.14286,0.1786,0.0,0.4286,11,0,0,11,0,13,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.05805,"x":0.15625,"p":[[0,60,0.0,0.14724,0.10403,0.14214,0.14286,0.1429,0.0,0.4286,7,0,2,7,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.15625,0.1488,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,9,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.09375,0.12169,0.0,0.0,0.14286,0.0,0.42857,17,0,3,17,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.57143,22,0,1,22,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,60,0.2667,0.0892,0.11148,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.10714,0.12372,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,60,0.4,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.14732,0.145,0.0,0.14286,0.28571,0.0,0.5714,12,0,0,12,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,60,0.5333,0.12938,0.11494,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,17,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.11161,0.11701,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,12,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.09375,0.10479,0.0,0.07143,0.14287,0.0,0.28571,16,0,0,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.08929,0.09943,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.05805,0.09356,0.0,0.0,0.14286,0.0,0.286,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,60,1.0,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9b9145421513171f","q":"Let $a, b, c$ be positive real numbers. Prove that\n\n$$\n\\left(\\left(3 a^{2}+1\\right)^{2}+2\\left(1+\\frac{3}{b}\\right)^{2}\\right)\\left(\\left(3 b^{2}+1\\right)^{2}+2\\left(1+\\frac{3}{c}\\right)^{2}\\right)\\left(\\left(3 c^{2}+1\\right)^{2}+2\\left(1+\\frac{3}{a}\\right)^{2}\\right) \\geq 48^{3}\n$$\n\nWhen does equality hold?","t":[{"b":0,"e":0.28571,"k":"flat","v":0.20089,"x":0.28571,"p":[[0,70,0.0,0.20089,0.11214,0.14286,0.2857,0.2857,0.0,0.4286,5,0,0,5,0,10,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,70,0.0571,0.28571,0.15567,0.14286,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,10,0,0,18,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[8,70,0.1143,0.24107,0.14032,0.14286,0.21428,0.28571,0.14286,0.71429,0,0,0,0,0,16,0,0,14,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[12,70,0.1714,0.25,0.12372,0.14286,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,14,0,0,15,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,70,0.2286,0.23214,0.06916,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.2009,0.07017,0.14286,0.14286,0.28571,0.14286,0.286,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.22768,0.07016,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.22321,0.08702,0.14286,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,13,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.21875,0.07129,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.22312,0.07945,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,12,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.22768,0.07873,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,11,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.20536,0.07936,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,16,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.20536,0.07936,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.20982,0.07129,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,17,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.22768,0.07016,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.20982,0.07973,0.14286,0.2143,0.28571,0.0,0.28571,1,0,0,1,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.20089,0.07873,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,17,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.23214,0.06916,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"rising","v":0.22768,"x":0.39284,"p":[[0,55,0.0,0.22768,0.18509,0.14286,0.14286,0.2857,0.0,1.0,2,1,0,2,0,18,0,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[4,55,0.0727,0.25,0.11294,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,10,0,0,19,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,55,0.1455,0.24107,0.12595,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,15,0,0,15,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,55,0.2182,0.27679,0.20806,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,17,0,0,10,0,0,0,0,0,2,0,0,2,0,0,0,0,1],[16,55,0.2909,0.27223,0.17268,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,13,0,0,13,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[20,55,0.3636,0.31696,0.21349,0.14286,0.28571,0.46429,0.14286,0.71429,0,0,0,0,0,15,0,0,8,0,0,1,0,0,3,0,0,5,0,0,0,0,0],[24,55,0.4364,0.26786,0.16656,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,13,0,0,13,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[28,55,0.5091,0.26786,0.17035,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,15,0,0,13,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[32,55,0.5818,0.29908,0.19015,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,13,0,0,13,0,0,0,0,0,2,0,0,4,0,0,0,0,0],[36,55,0.6545,0.32143,0.16752,0.24999,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,8,0,0,17,0,0,0,0,0,5,0,0,2,0,0,0,0,0],[40,55,0.7273,0.29464,0.17105,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,11,0,0,16,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[44,55,0.8,0.34366,0.20481,0.14286,0.28571,0.57143,0.14,0.71429,0,0,0,0,0,11,0,0,11,0,0,0,0,0,6,0,0,4,0,0,0,0,0],[48,55,0.8727,0.34821,0.2111,0.14286,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,12,0,0,9,0,0,0,0,0,7,0,0,4,0,0,0,0,0],[52,55,0.9455,0.36159,0.18551,0.2857,0.28571,0.57111,0.14286,0.71429,0,0,0,0,0,6,0,0,16,0,0,1,0,0,5,0,0,4,0,0,0,0,0],[55,55,1.0,0.39284,0.18897,0.28571,0.28571,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,16,0,0,1,0,0,6,0,0,5,0,0,0,0,0]]}]},{"i":"8f87fa3c87228b9f","q":"Let $A B C D E$ be a convex pentagon with the properties\n\n$$\n\\overline{A B}=\\overline{B C}=\\overline{C D}, \\angle B A E=\\angle D C B \\text { and } \\angle E D C=\\angle C B A .\n$$\n\nProve that the perpendicular from $E$ to $B C$ passes through the intersection of $A C$ and $B D$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.05357,"x":0.89732,"p":[[0,100,0.0,0.81249,0.25616,0.71429,0.92857,1.0,0.0,1.0,1,16,1,1,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,4,0,16],[4,100,0.04,0.89732,0.29284,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[8,100,0.08,0.8616,0.32239,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,26],[12,100,0.12,0.64732,0.4489,0.0,0.92857,1.0,0.0,1.0,10,16,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,16],[16,100,0.16,0.875,0.26426,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,24],[20,100,0.2,0.83482,0.35194,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,25],[24,100,0.24,0.83929,0.36202,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[28,100,0.28,0.80804,0.34183,0.71429,1.0,1.0,0.0,1.0,4,22,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,22],[32,100,0.32,0.81687,0.37172,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[36,100,0.36,0.80357,0.38755,0.96429,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,24],[40,100,0.4,0.6875,0.43805,0.0,1.0,1.0,0.0,1.0,9,19,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,19],[44,100,0.44,0.61607,0.47974,0.0,1.0,1.0,0.0,1.0,12,19,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,19],[48,100,0.48,0.66964,0.40945,0.25,0.92857,1.0,0.0,1.0,6,16,0,6,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,16],[52,100,0.52,0.29464,0.42248,0.0,0.0,0.64286,0.0,1.0,20,7,0,20,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,7],[56,100,0.56,0.31694,0.42517,0.0,0.0,0.67825,0.0,1.0,19,8,0,19,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,8],[60,100,0.6,0.21875,0.38793,0.0,0.0,0.28571,0.0,1.0,23,6,0,23,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[64,100,0.64,0.1696,0.32617,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3],[68,100,0.68,0.20089,0.37773,0.0,0.0,0.14286,0.0,1.0,23,5,0,23,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[72,100,0.72,0.05804,0.20159,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[76,100,0.76,0.17856,0.31742,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[80,100,0.8,0.33036,0.44239,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[84,100,0.84,0.16071,0.31084,0.0,0.0,0.17857,0.0,1.0,23,3,0,23,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[88,100,0.88,0.15624,0.3141,0.0,0.0,0.03575,0.0,1.0,24,3,0,24,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,3],[92,100,0.92,0.07589,0.20511,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[96,100,0.96,0.09813,0.16534,0.0,0.0,0.1786,0.0,0.57143,22,0,0,22,0,2,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[100,100,1.0,0.05357,0.12752,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.1875,"x":0.90178,"p":[[0,80,0.0,0.79018,0.22864,0.64286,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,4,0,0,7,0,13],[4,80,0.05,0.90178,0.26592,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,26],[8,80,0.1,0.875,0.28738,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[12,80,0.15,0.84375,0.3524,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[16,80,0.2,0.85268,0.33784,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,26],[20,80,0.25,0.62054,0.4412,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,17],[24,80,0.3,0.60714,0.43006,0.0,0.78571,1.0,0.0,1.0,9,14,0,9,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,14],[28,80,0.35,0.43748,0.40078,0.0,0.42857,0.85704,0.0,1.0,12,6,0,12,0,1,0,0,2,0,0,2,0,0,3,0,0,3,0,0,3,0,6],[32,80,0.4,0.31696,0.39404,0.0,0.0,0.71429,0.0,1.0,17,4,0,17,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,4],[36,80,0.45,0.60936,0.33311,0.28571,0.60721,1.0,0.0,1.0,3,9,0,3,0,1,0,0,5,0,0,3,0,0,4,1,0,3,0,0,3,0,9],[40,80,0.5,0.54463,0.35434,0.24999,0.64286,0.85714,0.0,1.0,5,6,0,5,0,3,0,0,3,0,0,3,0,0,2,0,0,6,0,0,4,0,6],[44,80,0.55,0.61605,0.38372,0.25,0.71429,1.0,0.0,1.0,6,10,0,6,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,0,5,0,10],[48,80,0.6,0.48214,0.39569,0.0,0.42857,0.89286,0.0,1.0,9,8,0,9,0,1,0,0,5,0,0,2,0,0,2,0,0,3,0,0,2,0,8],[52,80,0.65,0.4866,0.40384,0.0,0.5,0.85714,0.0,1.0,9,7,0,9,0,3,0,0,3,0,0,1,0,0,1,0,0,4,0,0,4,0,7],[56,80,0.7,0.61383,0.36372,0.28571,0.71429,0.89286,0.0,1.0,6,8,0,6,0,1,0,0,2,0,0,0,0,0,3,0,0,7,1,0,4,0,8],[60,80,0.75,0.61605,0.33964,0.28571,0.71429,0.85714,0.0,1.0,5,7,0,5,0,0,0,0,4,0,0,0,0,0,3,0,0,9,0,0,4,0,7],[64,80,0.8,0.44197,0.37177,0.0,0.42857,0.85714,0.0,1.0,10,3,0,10,0,1,0,0,4,0,0,3,0,0,1,0,0,4,0,0,6,0,3],[68,80,0.85,0.50444,0.39767,0.0,0.57121,0.85714,0.0,1.0,10,6,0,10,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,0,6,0,6],[72,80,0.9,0.40178,0.33963,0.10714,0.28571,0.71429,0.0,1.0,8,3,0,8,0,3,0,0,7,0,0,2,0,0,1,0,0,6,0,0,2,0,3],[76,80,0.95,0.37034,0.34611,0.0,0.35714,0.71429,0.0,1.0,11,2,0,11,0,3,0,0,2,0,0,5,0,0,1,0,0,5,0,0,3,0,2],[80,80,1.0,0.1875,0.32426,0.0,0.0,0.21429,0.0,1.0,22,2,0,22,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,2]]}]},{"i":"4b131c9165c1dfa8","q":"Let $A B C$ be an equilateral triangle and $P$ a point on its circumcircle. Set $D=$ $\\overline{P A} \\cap \\overline{B C}, E=\\overline{P B} \\cap \\overline{C A}, F=\\overline{P C} \\cap \\overline{A B}$. Prove that the area of triangle $D E F$ is twice the area of triangle $A B C$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.04018,"x":0.84375,"p":[[0,123,0.0,0.67857,0.40564,0.39285,0.92857,1.0,0.0,1.0,7,16,0,7,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,16],[4,123,0.0325,0.76339,0.37899,0.82143,1.0,1.0,0.0,1.0,6,18,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,18],[8,123,0.065,0.71875,0.38875,0.5,1.0,1.0,0.0,1.0,5,18,0,5,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0,3,0,18],[12,123,0.0976,0.74107,0.38206,0.75,0.92857,1.0,0.0,1.0,6,16,0,6,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,16],[16,123,0.1301,0.78571,0.32927,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,5,0,18],[20,123,0.1626,0.75446,0.38504,0.67857,1.0,1.0,0.0,1.0,6,19,0,6,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,19],[24,123,0.1951,0.75446,0.38504,0.74989,1.0,1.0,0.0,1.0,5,19,0,5,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,19],[28,123,0.2276,0.73213,0.36553,0.67846,0.85714,1.0,0.0,1.0,5,14,0,5,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,14],[32,123,0.2602,0.63839,0.42855,0.10714,0.85714,1.0,0.0,1.0,8,14,0,8,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,6,0,14],[36,123,0.2927,0.70536,0.39113,0.42857,0.85714,1.0,0.0,1.0,6,15,0,6,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,15],[40,123,0.3252,0.6875,0.40945,0.25,0.92857,1.0,0.0,1.0,6,16,0,6,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,16],[44,123,0.3577,0.84375,0.30797,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,22],[48,123,0.3902,0.78571,0.37115,0.85714,1.0,1.0,0.0,1.0,5,19,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,19],[52,123,0.4228,0.6875,0.39679,0.28571,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,4,0,16],[56,123,0.4553,0.54464,0.43366,0.0,0.71429,1.0,0.0,1.0,11,10,0,11,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,5,0,10],[60,123,0.4878,0.63393,0.43732,0.0,0.85714,1.0,0.0,1.0,10,14,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,14],[64,123,0.5203,0.51339,0.43428,0.0,0.57143,1.0,0.0,1.0,10,10,0,10,0,3,0,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,10],[68,123,0.5528,0.71875,0.38046,0.42857,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,5,0,16],[72,123,0.5854,0.61607,0.42624,0.10714,0.85714,1.0,0.0,1.0,8,12,0,8,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,7,0,12],[76,123,0.6179,0.45089,0.4317,0.0,0.35714,1.0,0.0,1.0,13,9,0,13,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,3,0,9],[80,123,0.6504,0.62945,0.45577,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,17],[84,123,0.6829,0.55804,0.44658,0.0,0.71429,1.0,0.0,1.0,11,13,0,11,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,13],[88,123,0.7154,0.42857,0.43006,0.0,0.2143,0.85714,0.0,1.0,13,7,0,13,0,3,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,7],[92,123,0.748,0.5,0.46702,0.0,0.57143,1.0,0.0,1.0,14,12,0,14,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,12],[96,123,0.7805,0.27679,0.42399,0.0,0.0,0.71429,0.0,1.0,22,7,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,7],[100,123,0.813,0.27902,0.43315,0.0,0.0,0.85714,0.0,1.0,21,7,0,21,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,7],[104,123,0.8455,0.375,0.45702,0.0,0.0,1.0,0.0,1.0,17,10,0,17,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,10],[108,123,0.878,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[112,123,0.9106,0.16071,0.35129,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[116,123,0.9431,0.21429,0.39609,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[120,123,0.9756,0.19643,0.37754,0.0,0.0,0.14286,0.0,1.0,23,5,0,23,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[123,123,1.0,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.89732,"p":[[0,66,0.0,0.54018,0.47074,0.0,0.78571,1.0,0.0,1.0,12,15,2,12,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,15],[4,66,0.0606,0.79464,0.37447,0.85714,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,21],[8,66,0.1212,0.77232,0.346,0.57143,1.0,1.0,0.0,1.0,2,20,0,2,0,2,0,0,3,0,0,0,0,0,2,0,0,1,0,0,2,0,20],[12,66,0.1818,0.89732,0.29066,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[16,66,0.2424,0.52679,0.48765,0.0,0.85714,1.0,0.0,1.0,14,15,0,14,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,15],[20,66,0.303,0.58036,0.45308,0.0,0.85714,1.0,0.0,1.0,11,12,0,11,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,12],[24,66,0.3636,0.73214,0.40838,0.71429,1.0,1.0,0.0,1.0,7,18,0,7,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,18],[28,66,0.4242,0.4375,0.46282,0.0,0.07143,1.0,0.0,1.0,16,9,0,16,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,9],[32,66,0.4848,0.62054,0.46649,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,17],[36,66,0.5455,0.47767,0.47057,0.0,0.42856,1.0,0.0,1.0,15,12,0,15,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,12],[40,66,0.6061,0.40179,0.4717,0.0,0.0,1.0,0.0,1.0,18,10,0,18,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,10],[44,66,0.6667,0.20982,0.39767,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[48,66,0.7273,0.21875,0.37794,0.0,0.0,0.32144,0.0,1.0,23,5,0,23,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,5],[52,66,0.7879,0.34821,0.42996,0.0,0.0,0.85714,0.0,1.0,18,5,0,18,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,5],[56,66,0.8485,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[60,66,0.9091,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"13055f7b685dfcff","q":"Let $a_{1}, a_{2}, \\ldots$ be an infinite sequence of positive integers such that, for all positive integers $m$ and $n$, we have that $a_{m+n}$ divides $a_{m} a_{n}-1$. Prove that there exists an integer $C$ such that, for all positive integers $k>C$, we have $a_{k}=1$.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.41514,"x":0.808,"p":[[0,88,0.0,0.41514,0.1571,0.2857,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,0,11,0,0,6,0,0,3,0,0,0,0,0],[4,88,0.0455,0.808,0.24903,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,14],[8,88,0.0909,0.77674,0.20809,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,10,0,9],[12,88,0.1364,0.75889,0.20341,0.67836,0.78564,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,8,0,0,9,0,7],[16,88,0.1818,0.76783,0.20749,0.67836,0.857,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,7,0,0,8,0,9],[20,88,0.2273,0.73656,0.22899,0.57132,0.857,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,5,0,0,12,0,6],[24,88,0.2727,0.72764,0.19021,0.67836,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,13,0,0,5,0,6],[28,88,0.3182,0.7589,0.2243,0.57132,0.78564,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,6,0,0,5,0,11],[32,88,0.3636,0.75445,0.20897,0.67857,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,11,0,0,3,0,10],[36,88,0.4091,0.68297,0.21649,0.57075,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,10,0,0,5,0,5],[40,88,0.4545,0.74995,0.20207,0.67857,0.78564,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,8,0,0,10,0,6],[44,88,0.5,0.65619,0.26453,0.53539,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,1,0,0,5,0,0,8,0,0,5,0,6],[48,88,0.5455,0.70088,0.24317,0.42859,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,7,0,0,7,0,7],[52,88,0.5909,0.70528,0.25491,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,6,0,0,6,0,8],[56,88,0.6364,0.63835,0.28119,0.42857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,4,0,0,5,0,0,4,0,0,8,0,5],[60,88,0.6818,0.74105,0.24599,0.57132,0.85707,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,6,0,0,8,0,9],[64,88,0.7273,0.62491,0.2442,0.5354,0.71429,0.85704,0.0,1.0,1,1,0,1,0,2,0,0,2,0,0,3,0,0,6,0,0,8,0,0,9,0,1],[68,88,0.7727,0.60711,0.30514,0.39285,0.57143,0.85714,0.0,1.0,1,6,0,1,0,4,0,0,3,0,0,3,0,0,6,0,0,3,0,0,6,0,6],[72,88,0.8182,0.56248,0.28332,0.28571,0.57143,0.857,0.0,1.0,1,2,0,1,0,4,0,0,5,0,0,2,0,0,5,0,0,6,0,0,7,0,2],[76,88,0.8636,0.47309,0.30404,0.14286,0.571,0.71429,0.0,1.0,2,2,0,2,0,9,0,0,2,0,0,2,0,0,5,0,0,7,0,0,3,0,2],[80,88,0.9091,0.50893,0.24982,0.28571,0.571,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,5,0,0,5,0,0,9,0,0,4,0,0,4,0,1],[84,88,0.9545,0.5938,0.31165,0.28571,0.57143,0.85786,0.0,1.0,1,7,0,1,0,3,0,0,6,0,0,2,0,0,7,0,0,1,0,0,5,0,7],[88,88,1.0,0.60266,0.32681,0.28571,0.64286,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,6,0,0,3,0,0,3,0,0,4,0,0,4,0,8]]},{"b":1,"e":0.571,"k":"rising","v":0.42403,"x":0.74105,"p":[[0,32,0.0,0.42403,0.17665,0.28571,0.42857,0.571,0.0,0.71429,1,0,0,1,0,3,0,0,7,0,0,9,0,0,9,0,0,3,0,0,0,0,0],[4,32,0.125,0.7053,0.20809,0.57132,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,9,0,0,3,0,7],[8,32,0.25,0.68747,0.32229,0.39286,0.78564,1.0,0.0,1.0,1,12,0,1,0,2,0,0,5,0,0,2,0,0,2,0,0,4,0,0,4,0,12],[12,32,0.375,0.65609,0.31932,0.42859,0.71429,1.0,0.0,1.0,2,10,0,2,0,2,0,0,3,0,0,2,0,0,5,0,0,5,0,0,3,0,10],[16,32,0.5,0.57585,0.25874,0.42857,0.57121,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,5,0,0,6,0,0,6,0,0,6,0,0,3,0,4],[20,32,0.625,0.65174,0.22287,0.571,0.71429,0.85704,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,1,0,0,6,0,0,11,0,0,7,0,2],[24,32,0.75,0.71873,0.23002,0.71421,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,13,0,0,5,0,7],[28,32,0.875,0.74105,0.16918,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,10,0,0,8,0,5],[32,32,1.0,0.62718,0.23059,0.571,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,5,0,0,1,0,0,7,0,0,11,0,0,4,1,2]]}]},{"i":"426a5eb94da9b66f","q":"Let $a, b$ be natural numbers with $a b>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisible by $a+b$. Prove that the quotient is at most $(a+b) / 4$. When is this quotient exactly equal to $(a+b) / 4$ ?","t":[{"b":0,"e":0.71429,"k":"flat","v":0.54464,"x":0.76786,"p":[[0,62,0.0,0.62947,0.20782,0.42859,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,5,0,0,4,0,0,8,0,0,10,0,0],[4,62,0.0645,0.60268,0.23347,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,2,0,0,9,0,0,3,0,0,11,0,0],[8,62,0.129,0.63839,0.22583,0.42857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,4,0,0,5,0,0,3,0,0,14,0,0],[12,62,0.1935,0.63393,0.20183,0.53572,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,0,2,0,0,12,0,0],[16,62,0.2581,0.56696,0.24086,0.39286,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,6,0,0,5,0,0,3,0,0,10,0,0],[20,62,0.3226,0.59375,0.22335,0.42857,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,7,0,0,8,0,0,2,0,0,9,0,1],[24,62,0.3871,0.625,0.23891,0.53571,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,2,0,0,7,0,0,5,0,0,12,0,0],[28,62,0.4516,0.65625,0.19515,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,7,0,0,11,0,0],[32,62,0.5161,0.57589,0.2435,0.42857,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,5,0,0,8,0,0,1,0,0,11,0,0],[36,62,0.5806,0.54464,0.22428,0.28571,0.57143,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,5,0,0,8,0,0,2,0,0,8,0,0],[40,62,0.6452,0.57143,0.24223,0.28571,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,4,0,0,4,0,0,6,0,0,9,0,0],[44,62,0.7097,0.74108,0.16143,0.57143,0.85714,0.85714,0.286,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,4,0,0,17,0,1],[48,62,0.7742,0.70535,0.17104,0.57143,0.78564,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,3,0,0,16,0,0],[52,62,0.8387,0.72767,0.19679,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,0,2,0,0,20,0,0],[56,62,0.9032,0.74552,0.16649,0.57143,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,1,0,0,19,0,1],[60,62,0.9677,0.75893,0.13092,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,16,0,1],[62,62,1.0,0.76786,0.13716,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,21,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.14286,"x":0.67855,"p":[[0,60,0.0,0.67855,0.21129,0.53539,0.78564,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,4,0,0,16,0,0],[4,60,0.0667,0.66518,0.19103,0.5357,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,5,0,0,13,0,0],[8,60,0.1333,0.60268,0.26422,0.28571,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,6,0,0,3,0,0,2,0,0,5,0,0,13,0,0],[12,60,0.2,0.65625,0.2392,0.53571,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,1,0,0,4,0,0,6,0,0,13,0,1],[16,60,0.2667,0.58929,0.21053,0.42857,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,6,0,0,6,0,0,7,0,0,4,0,0,9,0,0],[20,60,0.3333,0.58928,0.28291,0.28571,0.57143,0.85714,0.0,1.0,1,2,0,1,0,3,0,0,5,0,0,2,0,0,6,0,0,3,0,0,10,0,2],[24,60,0.4,0.54463,0.22428,0.28571,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,6,0,0,5,0,0,5,0,0,7,0,0],[28,60,0.4667,0.54464,0.24074,0.42857,0.50001,0.75,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,9,0,0,5,0,0,3,0,0,7,0,1],[32,60,0.5333,0.48661,0.24964,0.2857,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,5,0,0,7,0,0,6,0,0,5,0,0,2,0,0,7,0,0],[36,60,0.6,0.57589,0.23551,0.39286,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,8,0,0,7,0,0,4,0,0,3,0,0,9,0,1],[40,60,0.6667,0.34375,0.20473,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,10,0,0,10,0,0,7,0,0,1,0,0,2,0,0,2,0,0],[44,60,0.7333,0.36607,0.17474,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,6,0,0,11,0,0,10,0,0,2,0,0,2,0,0,1,0,0],[48,60,0.8,0.29464,0.10062,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,7,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.30804,0.17536,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,10,0,0,13,0,0,7,0,0,0,0,0,0,0,0,2,0,0],[56,60,0.9333,0.25,0.12372,0.14286,0.21428,0.28571,0.14286,0.57143,0,0,0,0,0,16,0,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[60,60,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"33bfa3e33dfa9528","q":"Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a, b, c$ are integers (not necessarily positive) satisfying the equations  $$ a^{n}+p b=b^{n}+p c=c^{n}+p a, $$  then $a=b=c$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.04464,"x":0.29006,"p":[[0,57,0.0,0.2275,0.15925,0.14286,0.14286,0.28571,0.14,1.0,0,1,0,0,0,19,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,57,0.0702,0.29006,0.23823,0.14286,0.28571,0.32143,0.0,1.0,4,1,0,4,0,11,0,0,9,0,0,1,0,0,5,0,0,0,0,0,1,0,1],[8,57,0.1404,0.14714,0.16165,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,18,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,57,0.2105,0.20088,0.25468,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,0,1,0,1],[16,57,0.2807,0.18304,0.1439,0.14286,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,14,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,57,0.3509,0.17848,0.17859,0.105,0.14286,0.2857,0.0,0.85714,8,0,0,8,0,15,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[24,57,0.4211,0.20071,0.18515,0.14,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,14,0,0,7,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[28,57,0.4912,0.13393,0.13803,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,17,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,57,0.5614,0.10715,0.14286,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,57,0.6316,0.08037,0.10064,0.0,0.0,0.14286,0.0,0.286,18,0,0,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.09375,0.11071,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.0625,0.08702,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.14277,"x":0.2857,"p":[[0,110,0.0,0.22321,0.08702,0.14286,0.2143,0.28571,0.14286,0.42857,0,0,0,0,0,16,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,110,0.0364,0.23652,0.15412,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,18,0,0,8,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[8,110,0.0727,0.23214,0.15047,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,19,0,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[12,110,0.1091,0.2857,0.23688,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,20,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[16,110,0.1455,0.25446,0.17029,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,19,0,0,7,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[20,110,0.1818,0.2009,0.09354,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,110,0.2182,0.2366,0.18764,0.14286,0.14286,0.2857,0.0,0.857,1,0,0,1,0,22,0,0,3,0,0,1,0,0,4,0,0,0,0,0,1,0,0],[28,110,0.2545,0.16964,0.07523,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,110,0.2909,0.18304,0.13474,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,19,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,110,0.3273,0.17393,0.10561,0.14286,0.14286,0.17857,0.0,0.57143,3,0,0,3,0,21,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,110,0.3636,0.19643,0.13243,0.14286,0.14286,0.1786,0.14286,0.85714,0,0,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,110,0.4,0.15178,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,110,0.4364,0.17857,0.13832,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,110,0.4727,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,110,0.5091,0.1875,0.10972,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,110,0.5455,0.16062,0.08566,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,23,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,110,0.5818,0.16509,0.05191,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,110,0.6182,0.2008,0.11773,0.14286,0.14286,0.2857,0.0,0.71429,1,0,0,1,0,20,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,110,0.6545,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,110,0.6909,0.16964,0.08328,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,110,0.7273,0.16062,0.05925,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,110,0.7636,0.16071,0.08564,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[88,110,0.8,0.16518,0.09522,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,25,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[92,110,0.8364,0.15625,0.08268,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,110,0.8727,0.15624,0.07449,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[100,110,0.9091,0.18294,0.08921,0.14286,0.14286,0.14286,0.14,0.5714,0,0,0,0,0,25,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[104,110,0.9455,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,110,0.9818,0.16965,0.06621,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[110,110,1.0,0.20527,0.11816,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,19,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"a130f5bf13648375","q":"Let $p>10$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n<p$ such that $p$ is a divisor of $5^{m} 7^{n}-1$.","t":[{"b":2,"e":0.71429,"k":"rising","v":0.27661,"x":0.72768,"p":[[0,58,0.0,0.27661,0.29228,0.14286,0.14286,0.2144,0.0,1.0,4,2,0,4,0,20,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,2],[4,58,0.069,0.5848,0.28203,0.39285,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,6,0,0,2,0,0,3,0,0,4,0,0,11,0,0,1,0,5],[8,58,0.1379,0.57142,0.23419,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,1,0,0,5,0,0,3,0,0,15,0,0,2,0,1],[12,58,0.2069,0.51786,0.2165,0.39286,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,6,0,0,3,0,0,15,0,0,0,0,0],[16,58,0.2759,0.58035,0.26948,0.42857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,6,0,0,1,0,0,5,0,0,2,0,0,12,0,0,3,0,3],[20,58,0.3448,0.47767,0.27107,0.14286,0.49979,0.71429,0.14286,1.0,0,1,0,0,0,11,0,0,0,0,0,5,0,0,2,0,0,12,0,0,1,0,1],[24,58,0.4138,0.53571,0.28793,0.14286,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,10,0,0,0,0,0,1,0,0,5,0,0,11,0,0,3,0,2],[28,58,0.4828,0.54911,0.23987,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,7,0,0,0,0,0,3,0,0,5,0,0,16,0,0,0,0,1],[32,58,0.5517,0.49998,0.27198,0.14286,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,10,0,0,0,0,0,5,0,0,3,0,0,9,0,0,5,0,0],[36,58,0.6207,0.54018,0.23887,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,2,0,0,5,0,0,5,0,0,13,0,0,1,0,1],[40,58,0.6897,0.56695,0.24086,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,0,0,0,3,0,0,8,0,0,12,0,0,1,0,2],[44,58,0.7586,0.58481,0.23517,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,0,0,0,5,0,0,4,0,0,15,0,0,1,0,2],[48,58,0.8276,0.51338,0.27399,0.1429,0.57141,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,1,0,0,4,0,0,3,0,0,12,0,0,1,0,2],[52,58,0.8966,0.62947,0.19186,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,1,0,0,5,0,0,19,0,0,3,0,0],[56,58,0.9655,0.71875,0.13115,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,23,0,0,5,0,1],[58,58,1.0,0.72768,0.06546,0.71429,0.71429,0.71429,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.31697,"x":0.78572,"p":[[0,42,0.0,0.31697,0.30037,0.14286,0.14286,0.57143,0.0,1.0,4,2,0,4,0,16,0,0,2,0,0,1,0,0,2,0,0,4,0,0,1,0,2],[4,42,0.0952,0.57142,0.25505,0.42857,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,1,0,0,3,0,0,6,0,0,11,0,0,3,0,2],[8,42,0.1905,0.58482,0.29093,0.25,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,1,0,0,1,0,0,1,0,0,16,0,0,1,0,4],[12,42,0.2857,0.46875,0.29502,0.14286,0.57143,0.71429,0.0,1.0,1,1,0,1,0,11,0,0,2,0,0,0,0,0,5,0,0,9,0,0,3,0,1],[16,42,0.381,0.45534,0.31019,0.14286,0.42857,0.71429,0.0,1.0,1,3,0,1,0,12,0,0,1,0,0,3,0,0,4,0,0,6,0,0,2,0,3],[20,42,0.4762,0.56695,0.28456,0.28571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,6,0,0,2,0,0,1,0,0,4,0,0,13,0,0,2,0,3],[24,42,0.5714,0.45536,0.2683,0.14286,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,12,0,0,1,0,0,4,0,0,0,0,0,14,0,0,1,0,0],[28,42,0.6667,0.5,0.26244,0.28571,0.5,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,4,0,0,5,0,0,3,0,0,10,0,0,1,0,2],[32,42,0.7619,0.45535,0.24598,0.14286,0.4998,0.71429,0.14286,1.0,0,1,0,0,0,10,0,0,1,0,0,5,0,0,7,0,0,8,0,0,0,0,1],[36,42,0.8571,0.50893,0.2922,0.25,0.50001,0.71429,0.0,1.0,2,2,0,2,0,6,0,0,2,0,0,6,0,0,1,0,0,10,0,0,3,0,2],[40,42,0.9524,0.64731,0.335,0.39286,0.71429,0.85714,0.0,1.0,3,7,0,3,0,3,0,0,2,0,0,1,0,0,2,0,0,6,0,0,8,0,7],[42,42,1.0,0.78572,0.21129,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,0,15,0,7]]}]},{"i":"a80810ff77554447","q":"Let $x, y, z$ be strictly positive real numbers such that $x y + y z + z x = 1$. Show that\n\n$$\n2\\left(x^{2}+y^{2}+z^{2}\\right)+\\frac{4}{3}\\left(\\frac{1}{x^{2}+1}+\\frac{1}{y^{2}+1}+\\frac{1}{z^{2}+1}\\right) \\geqslant 5\n$$","t":[{"b":3,"e":1.0,"k":"volatile","v":0.09375,"x":0.99107,"p":[[0,14,0.0,0.09375,0.25657,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,14,0.2857,0.56696,0.44246,0.14286,0.85714,1.0,0.0,1.0,6,14,0,6,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,14],[8,14,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,14,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[14,14,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":6,"e":0.0,"k":"flat","v":0.05804,"x":0.63839,"p":[[0,100,0.0,0.20089,0.38608,0.0,0.0,0.14286,0.0,1.0,23,6,0,23,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[4,100,0.04,0.55357,0.48148,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17],[8,100,0.08,0.63839,0.46426,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,19],[12,100,0.12,0.4241,0.46357,0.0,0.14286,1.0,0.0,1.0,14,11,0,14,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,11],[16,100,0.16,0.36607,0.44167,0.0,0.14286,1.0,0.0,1.0,15,10,0,15,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,10],[20,100,0.2,0.33482,0.44694,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,9],[24,100,0.24,0.29911,0.42312,0.0,0.0,0.57143,0.0,1.0,19,8,0,19,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,8],[28,100,0.28,0.12054,0.29474,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[32,100,0.32,0.11161,0.29609,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[36,100,0.36,0.18304,0.36111,0.0,0.0,0.14286,0.0,1.0,23,5,0,23,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[40,100,0.4,0.33482,0.4412,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,9],[44,100,0.44,0.25446,0.43114,0.0,0.0,0.35714,0.0,1.0,23,8,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[48,100,0.48,0.20089,0.391,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[52,100,0.52,0.17411,0.33832,0.0,0.0,0.14286,0.0,1.0,22,3,0,22,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[56,100,0.56,0.15625,0.33189,0.0,0.0,0.03571,0.0,1.0,24,4,0,24,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[60,100,0.6,0.10268,0.29066,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[64,100,0.64,0.28571,0.41802,0.0,0.0,0.60714,0.0,1.0,20,7,0,20,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,7],[68,100,0.68,0.36607,0.46282,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[72,100,0.72,0.18304,0.34853,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[76,100,0.76,0.16964,0.32818,0.0,0.0,0.14286,0.0,1.0,21,4,0,21,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[80,100,0.8,0.11161,0.29175,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[84,100,0.84,0.11607,0.28891,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[88,100,0.88,0.1875,0.3719,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[92,100,0.92,0.34375,0.45856,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,9],[96,100,0.96,0.32143,0.43886,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,9],[100,100,1.0,0.05804,0.19186,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"132f6aad383d6209","q":"Let $n$ be an integer such that $n>3$. Suppose that we choose three numbers from the set $\\{1,2, \\ldots, n\\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations.\n(a) Show that if we choose all three numbers greater than $n / 2$, then the values of these combinations are all distinct.\n(b) Let $p$ be a prime number such that $p \\leq \\sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p-1$.","t":[{"b":0,"e":0.571,"k":"falling","v":0.53571,"x":0.87955,"p":[[0,115,0.0,0.79463,0.22852,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,5,0,0,3,0,15],[4,115,0.0348,0.87955,0.13414,0.82143,0.85857,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,9,0,15],[8,115,0.0696,0.85254,0.21581,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,4,0,0,7,0,17],[12,115,0.1043,0.85267,0.19719,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,6,0,17],[16,115,0.1391,0.85714,0.16751,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,7,0,15],[20,115,0.1739,0.85714,0.15567,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,6,0,15],[24,115,0.2087,0.83035,0.1448,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,7,0,11],[28,115,0.2435,0.8168,0.1484,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,12,0,8],[32,115,0.2783,0.83482,0.18595,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,5,0,15],[36,115,0.313,0.79018,0.19227,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,6,0,11],[40,115,0.3478,0.75891,0.20342,0.67857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,10,0,0,5,0,9],[44,115,0.3826,0.81692,0.20282,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,5,0,14],[48,115,0.4174,0.79909,0.17447,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,12,0,0,4,0,11],[52,115,0.4522,0.84821,0.17474,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,9,0,14],[56,115,0.487,0.82142,0.19563,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,5,0,0,4,0,15],[60,115,0.5217,0.875,0.17768,0.82132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,6,0,18],[64,115,0.5565,0.81695,0.19962,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,2,0,15],[68,115,0.5913,0.77664,0.22003,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,6,0,0,5,0,12],[72,115,0.6261,0.83022,0.20033,0.71321,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,6,0,15],[76,115,0.6609,0.80355,0.22233,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,0,4,0,15],[80,115,0.6957,0.73199,0.2044,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,7,0,0,7,0,7],[84,115,0.7304,0.75451,0.22927,0.67857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,10,0,0,3,0,11],[88,115,0.7652,0.7632,0.19443,0.67536,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,9,0,0,7,0,8],[92,115,0.8,0.81687,0.17213,0.71429,0.85707,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,5,0,12],[96,115,0.8348,0.80353,0.21947,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,3,0,0,4,0,15],[100,115,0.8696,0.75893,0.22711,0.67857,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,0,13,0,7],[104,115,0.9043,0.79018,0.21424,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,6,0,0,10,0,10],[108,115,0.9391,0.65623,0.25967,0.4286,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,7,0,0,5,0,0,5,0,0,3,0,8],[112,115,0.9739,0.6205,0.22759,0.4286,0.64286,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,5,0,0,7,0,0,9,0,0,4,0,3],[115,115,1.0,0.53571,0.23146,0.39286,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,9,0,0,5,0,0,5,0,0,2,0,3]]},{"b":4,"e":1.0,"k":"flat","v":0.7857,"x":0.93304,"p":[[0,67,0.0,0.87946,0.13415,0.857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,11,0,14],[4,67,0.0597,0.85714,0.19885,0.85711,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,9,0,16],[8,67,0.1194,0.86606,0.15128,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,15],[12,67,0.1791,0.83928,0.1915,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,0,9,0,13],[16,67,0.2388,0.8348,0.17538,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,7,0,13],[20,67,0.2985,0.83928,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,8,0,13],[24,67,0.3582,0.80803,0.25657,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,7,0,15],[28,67,0.4179,0.81697,0.1897,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,10,0,11],[32,67,0.4776,0.83035,0.19377,0.82132,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,12,0,12],[36,67,0.5373,0.81694,0.24807,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,8,0,15],[40,67,0.597,0.7857,0.23147,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,6,0,12],[44,67,0.6567,0.93304,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[48,67,0.7164,0.88393,0.16536,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,16],[52,67,0.7761,0.90625,0.13175,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,9,0,18],[56,67,0.8358,0.84372,0.17629,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,6,0,15],[60,67,0.8955,0.89732,0.12993,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,11,0,16],[64,67,0.9552,0.91518,0.12807,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,19],[67,67,1.0,0.86606,0.17835,0.857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,9,0,16]]}]},{"i":"42406597af9ce0c0","q":"Let $N$ be the number of positive integers with 10 digits $\\overline{d_{9} d_{8} \\cdots d_{1} d_{0}}$ in base 10 (where $0 \\leq d_{i} \\leq 9$ for all $i$ and $d_{9}>0$ ) such that the polynomial\n\n$$\nd_{9} x^{9}+d_{8} x^{8}+\\cdots+d_{1} x+d_{0}\n$$\n\nis irreducible in $\\mathbb{Q}$. Prove that $N$ is even.\n(A polynomial is irreducible in $\\mathbb{Q}$ if it cannot be factored into two non-constant polynomials with rational coefficients.)","t":[{"b":0,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,17,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,17,0.2353,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,17,0.4706,0.95536,0.16536,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[12,17,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,17,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[17,17,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":1,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,29,0.0,0.91071,0.23891,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,27],[4,29,0.1379,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,29,0.2759,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,29,0.4138,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,29,0.5517,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,29,0.6897,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,29,0.8276,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"d8de504d6a2675c3","q":"Let real number  $\\lambda \\in (0,1)$ , complex sequence  $\\{z_n\\}$  s.t.  $z_1=1$ ,  $\\forall n, \\frac{z_{n+1}}{z_n}\\in \\{i,\\lambda \\}$ , find  $C_{\\text{min}}$  s.t.  $\\forall n,|z_1+z_2+\\cdots z_n|<C$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.15179,"x":0.86607,"p":[[0,37,0.0,0.67857,0.32927,0.28571,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,10,0,0,5,0,0,0,0,0,0,0,0,2,0,15],[4,37,0.1081,0.74107,0.33012,0.53571,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,1,0,0,4,0,0,3,0,0,3,0,0,1,0,17],[8,37,0.2162,0.69638,0.30673,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,5,0,0,4,0,0,5,0,0,2,0,0,1,0,14],[12,37,0.3243,0.71429,0.32143,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,6,0,0,4,0,0,2,0,0,2,0,0,1,0,16],[16,37,0.4324,0.86607,0.2257,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,22],[20,37,0.5405,0.73659,0.32165,0.42857,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,3,0,0,5,0,0,1,0,0,1,0,0,5,0,15],[24,37,0.6486,0.73661,0.31564,0.53571,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,2,0,0,3,0,0,1,0,0,5,0,0,4,0,14],[28,37,0.7568,0.75,0.30093,0.53571,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,5,0,0,2,0,0,1,0,0,6,0,0,1,0,16],[32,37,0.8649,0.60714,0.40877,0.21429,0.71429,1.0,0.0,1.0,8,13,0,8,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,1,0,13],[36,37,0.973,0.375,0.43558,0.0,0.07143,1.0,0.0,1.0,16,9,0,16,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,9],[37,37,1.0,0.15179,0.29001,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,2]]},{"b":4,"e":0.42857,"k":"falling","v":0.45982,"x":0.86607,"p":[[0,88,0.0,0.73214,0.31084,0.39286,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,8,0,0,3,0,0,1,0,0,1,0,0,3,0,16],[4,88,0.0455,0.75447,0.28845,0.42857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,18],[8,88,0.0909,0.78123,0.26961,0.4286,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,0,0,0,2,0,18],[12,88,0.1364,0.72767,0.31816,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,5,0,0,5,0,0,1,0,0,3,0,0,0,0,17],[16,88,0.1818,0.79911,0.31106,0.42857,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,4,0,0,4,0,0,0,0,0,0,0,0,2,0,21],[20,88,0.2273,0.74107,0.2976,0.42857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,7,0,0,3,0,0,0,0,0,5,0,0,1,0,16],[24,88,0.2727,0.85268,0.21572,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,4,0,19],[28,88,0.3182,0.83033,0.24859,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,4,0,0,1,0,20],[32,88,0.3636,0.75891,0.30814,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,2,0,0,5,0,0,1,0,0,4,0,0,0,0,18],[36,88,0.4091,0.85714,0.22868,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,1,0,22],[40,88,0.4545,0.84372,0.22123,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,19],[44,88,0.5,0.74106,0.27535,0.42857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,7,0,0,3,0,0,2,0,0,2,0,15],[48,88,0.5455,0.82141,0.24224,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,5,0,0,4,0,17],[52,88,0.5909,0.82588,0.23074,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,0,4,0,17],[56,88,0.6364,0.86607,0.23128,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,5,0,21],[60,88,0.6818,0.84821,0.2111,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,5,0,18],[64,88,0.7273,0.80357,0.23623,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,5,0,15],[68,88,0.7727,0.82142,0.23419,0.67857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,2,0,0,5,0,17],[72,88,0.8182,0.78125,0.2696,0.42857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,3,0,0,6,0,0,1,0,0,1,0,0,5,0,16],[76,88,0.8636,0.71428,0.26963,0.42857,0.78564,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,11,0,0,0,0,0,3,0,0,4,0,12],[80,88,0.9091,0.81696,0.2237,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,6,0,0,0,0,18],[84,88,0.9545,0.67848,0.27681,0.42857,0.71429,1.0,0.14,1.0,0,11,0,0,0,1,0,0,3,0,0,8,0,0,2,0,0,6,0,0,1,0,11],[88,88,1.0,0.45982,0.17399,0.42857,0.42857,0.4286,0.2857,1.0,0,2,0,0,0,0,0,0,7,0,0,19,0,0,2,0,0,2,0,0,0,0,2]]}]},{"i":"51787ea80937541a","q":"On a polynomial of degree three it is allowed to perform the following two operations arbitrarily many times:\n\n(i) reverse the order of its coefficients including zeroes (for instance, from the polynomial $x^{3}-2 x^{2}-3$ we can obtain $-3 x^{3}-2 x+1$;\n\n(ii) change polynomial $P(x)$ to the polynomial $P(x+1)$.\n\nIs it possible to obtain the polynomial $x^{3}-3 x^{2}+3 x-3$ from the polynomial $x^{3}-2$ ?\n\nThe answer is no.","t":[{"b":4,"e":1.0,"k":"rising","v":0.34378,"x":0.75888,"p":[[0,45,0.0,0.46429,0.26726,0.1429,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,9,0,0,2,0,0,7,0,0,7,0,0,3,0,0,1,0,3],[4,45,0.0889,0.39732,0.23072,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,1,0,0,19,0,0,5,0,0,0,0,0,3,0,0,1,0,2],[8,45,0.1778,0.34378,0.1551,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,19,0,0,7,0,0,2,0,0,0,0,0,0,0,1],[12,45,0.2667,0.40176,0.2065,0.2857,0.28571,0.46418,0.14286,1.0,0,1,0,0,0,1,0,0,21,0,0,2,0,0,2,0,0,4,0,0,1,0,1],[16,45,0.3556,0.45981,0.23887,0.28571,0.35714,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,15,0,0,5,0,0,2,0,0,6,0,0,1,0,2],[20,45,0.4444,0.41071,0.26426,0.2857,0.28571,0.60714,0.0,1.0,3,1,0,3,0,2,0,0,14,0,0,3,0,0,2,0,0,4,0,0,3,0,1],[24,45,0.5333,0.45086,0.2627,0.2857,0.28571,0.60682,0.0,1.0,1,3,0,1,0,1,0,0,16,0,0,4,0,0,2,0,0,3,0,0,2,0,3],[28,45,0.6222,0.41072,0.19479,0.28571,0.28571,0.60714,0.0,0.71429,1,0,0,1,0,0,0,0,18,0,0,4,0,0,1,0,0,8,0,0,0,0,0],[32,45,0.7111,0.50446,0.25501,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,13,0,0,4,0,0,2,0,0,7,0,0,3,0,2],[36,45,0.8,0.48213,0.28514,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,10,0,0,3,0,0,4,0,0,5,0,0,1,0,4],[40,45,0.8889,0.52677,0.2911,0.28571,0.4998,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,12,0,0,2,0,0,3,0,0,6,0,0,3,0,4],[44,45,0.9778,0.73214,0.22231,0.71429,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,3,0,0,1,0,0,13,0,0,7,0,6],[45,45,1.0,0.75888,0.20655,0.71429,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,8,0,7]]},{"b":5,"e":0.0,"k":"flat","v":0.37946,"x":0.74553,"p":[[0,68,0.0,0.52232,0.17354,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,6,0,0,16,0,0,4,0,0,0,0,1],[4,68,0.0588,0.44643,0.29179,0.2857,0.28571,0.71429,0.0,1.0,3,3,0,3,0,1,0,0,15,0,0,1,0,0,2,0,0,5,0,0,2,0,3],[8,68,0.1176,0.4866,0.25219,0.2857,0.42857,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,14,0,0,4,0,0,3,0,0,5,0,0,3,0,2],[12,68,0.1765,0.60714,0.28347,0.28571,0.57143,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,8,0,0,5,0,0,3,0,0,6,0,0,1,0,8],[16,68,0.2353,0.67857,0.27894,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,5,0,0,3,0,0,3,0,0,8,0,0,3,0,9],[20,68,0.2941,0.64731,0.28791,0.42857,0.64286,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,6,0,0,4,0,0,5,0,0,3,0,0,5,0,8],[24,68,0.3529,0.62052,0.28927,0.28571,0.71429,0.85704,0.0,1.0,1,5,0,1,0,1,0,0,8,0,0,1,0,0,2,0,0,8,0,0,6,0,5],[28,68,0.4118,0.60266,0.28061,0.28571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,9,0,0,4,0,0,3,0,0,3,0,0,7,0,5],[32,68,0.4706,0.66963,0.26592,0.4286,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,4,0,0,4,0,0,7,0,0,5,0,7],[36,68,0.5294,0.60267,0.27137,0.28571,0.71429,0.75,0.0,1.0,1,4,0,1,0,1,0,0,7,0,0,2,0,0,3,0,0,10,0,0,4,0,4],[40,68,0.5882,0.74553,0.26663,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,4,0,0,1,0,0,4,0,0,5,0,0,5,0,12],[44,68,0.6471,0.4955,0.28342,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,7,0,0,5,0,0,5,0,0,4,0,0,3,0,3],[48,68,0.7059,0.43304,0.27545,0.28571,0.28571,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,12,0,0,2,0,0,4,0,0,5,0,0,3,0,1],[52,68,0.7647,0.37946,0.30432,0.14286,0.28571,0.60714,0.0,1.0,7,2,0,7,0,2,0,0,10,0,0,3,0,0,2,0,0,4,0,0,2,0,2],[56,68,0.8235,0.5,0.33312,0.28571,0.42857,0.71429,0.0,1.0,6,5,1,6,0,0,0,0,5,0,0,7,0,0,1,0,0,6,0,0,2,0,5],[60,68,0.8824,0.55357,0.34022,0.2857,0.57143,0.85714,0.0,1.0,4,6,0,4,0,3,0,0,3,0,0,4,0,0,3,0,0,5,0,0,4,0,6],[64,68,0.9412,0.55355,0.31893,0.28571,0.57143,0.85704,0.0,1.0,3,4,0,3,0,2,0,0,7,0,0,1,0,0,4,0,0,5,0,0,6,0,4],[68,68,1.0,0.64283,0.28347,0.42859,0.64271,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,5,0,0,2,0,0,7,0,0,4,0,0,5,0,7]]}]},{"i":"f10c3db3af5d90ec","q":"Let $\\mathrm{Zij} ABC$ be an acute triangle. Let $\\mathrm{Zij} H$ be the foot of the altitude from $C$ to $AB$. Suppose that $|AH|=3|BH|$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively. Let $P$ be a point such that $|NP|=|NC|$ and $|CP|=|CB|$, and such that $B$ and $P$ lie on opposite sides of the line $AC$.\nProve that $\\angle APM = \\angle PBA$.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.54911,"x":0.57589,"p":[[0,12,0.0,0.54911,0.2372,0.28571,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,7,0,0,1,0,0,15,0,0,2,0,0,1,0,4],[4,12,0.3333,0.56249,0.04971,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0],[8,12,0.6667,0.5714,0.03571,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0],[12,12,1.0,0.57589,0.02486,0.57143,0.57143,0.57143,0.5714,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.49552,"x":0.57142,"p":[[0,45,0.0,0.49552,0.23415,0.2857,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,5,0,0,5,0,0,2,0,0,16,0,0,1,0,0,0,0,3],[4,45,0.0889,0.56249,0.26229,0.28571,0.57143,0.57143,0.2857,1.0,0,7,0,0,0,0,0,0,11,0,0,1,0,0,13,0,0,0,0,0,0,0,7],[8,45,0.1778,0.49554,0.16745,0.42857,0.57143,0.57143,0.0,0.85714,1,0,1,1,0,1,0,0,4,0,0,6,0,0,17,0,0,2,0,0,1,0,0],[12,45,0.2667,0.50891,0.2111,0.28571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,7,0,0,5,0,0,12,0,0,3,0,0,1,0,2],[16,45,0.3556,0.57142,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[20,45,0.4444,0.52229,0.13174,0.57143,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,3,0,0,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0],[24,45,0.5333,0.57141,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[28,45,0.6222,0.5714,0.0001,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[32,45,0.7111,0.56249,0.04971,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0],[36,45,0.8,0.5714,0.0001,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0],[40,45,0.8889,0.55357,0.06916,0.57143,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0],[44,45,0.9778,0.54018,0.11701,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,0,0,0,28,0,0,1,0,0,0,0,0],[45,45,1.0,0.57142,7e-05,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0]]}]},{"i":"6583f8ea03fdd6d5","q":"Let $ABC$ be a triangle with acute angles. We denote $D$ as the foot of the altitude from $A$ in triangle $ABC$, then $E$ as the midpoint of the segment $[AD]$, and $\\omega$ as the circle with diameter $[AD]$. Next, let $X$ be the intersection point between $\\omega$ and the line $(BE)$, such that $B$ and $X$ are on opposite sides of the line $(AD)$. Similarly, let $Y$ be the intersection point between $\\omega$ and the line $(CE)$, such that $C$ and $Y$ are on opposite sides of the line $(AD)$. Finally, suppose there exists a point $Z$, other than $D$, lying on the line $(AD)$ and on both circumcircles of $BDX$ and $CDY$. Prove that $AB = AC$.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.15607,"x":0.375,"p":[[0,26,0.0,0.36599,0.31332,0.14286,0.14295,0.57143,0.14,1.0,0,5,0,0,0,17,0,0,5,0,0,1,0,0,3,0,0,1,0,0,0,0,5],[4,26,0.1538,0.3392,0.29402,0.14286,0.2857,0.32143,0.0,1.0,1,4,0,1,0,14,0,0,9,0,0,2,0,0,1,0,0,0,0,0,1,0,4],[8,26,0.3077,0.375,0.31491,0.14286,0.28571,0.42857,0.0,1.0,1,6,0,1,0,11,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[12,26,0.4615,0.20537,0.15542,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,26,0.6154,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.16072,0.05922,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.16964,0.06621,0.14286,0.14286,0.1429,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.14732,"x":0.33474,"p":[[0,41,0.0,0.32134,0.2767,0.14286,0.2143,0.28571,0.14,1.0,0,4,0,0,0,16,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,4],[4,41,0.0976,0.33474,0.30854,0.14286,0.1429,0.42857,0.0,1.0,1,5,0,1,0,17,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,5],[8,41,0.1951,0.23214,0.21651,0.14286,0.14286,0.28571,0.0,1.0,2,2,0,2,0,19,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,41,0.2927,0.21866,0.1597,0.14286,0.14286,0.2857,0.14,1.0,0,1,0,0,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,41,0.3902,0.2408,0.20971,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,18,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,41,0.4878,0.20535,0.15542,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,41,0.5854,0.18304,0.06423,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.1875,0.07523,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,20,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.14733,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.17402,0.06905,0.14286,0.14286,0.1786,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.14732,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.17402,0.07774,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,21,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8c09ae61f470c1c8","q":"Let $n \\geq 3$ be a positive integer. A triangulation of a convex $n$-gon is a set of $n-3$ of its diagonals which do not intersect in the interior of the polygon. Along with the $n$ sides, these diagonals separate the polygon into $n-2$ disjoint triangles. Any triangulation can be viewed as a graph: the vertices of the graph are the corners of the polygon, and the $n$ sides and $n-3$ diagonals are the edges.\nFor a fixed $n$-gon, different triangulations correspond to different graphs. Prove that all of these graphs have the same chromatic number.","t":[{"b":4,"e":1.0,"k":"flat","v":0.89732,"x":0.98214,"p":[[0,62,0.0,0.91071,0.19805,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,25],[4,62,0.0645,0.92857,0.17496,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,27],[8,62,0.129,0.91964,0.16342,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,0,0,25],[12,62,0.1935,0.89732,0.1684,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,1,0,22],[16,62,0.2581,0.90625,0.20079,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,26],[20,62,0.3226,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[24,62,0.3871,0.94643,0.15046,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,28],[28,62,0.4516,0.95089,0.1365,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[32,62,0.5161,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[36,62,0.5806,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[40,62,0.6452,0.90625,0.16602,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,23],[44,62,0.7097,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[48,62,0.7742,0.95522,0.12621,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[52,62,0.8387,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[56,62,0.9032,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,26],[60,62,0.9677,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[62,62,1.0,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26]]},{"b":6,"e":1.0,"k":"flat","v":0.74107,"x":0.90625,"p":[[0,24,0.0,0.80804,0.22477,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,8,0,0,0,0,17],[4,24,0.1667,0.90625,0.18766,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,0,0,25],[8,24,0.3333,0.7945,0.21416,0.71321,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,10,0,0,0,0,15],[12,24,0.5,0.82589,0.21646,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,8,0,0,0,0,18],[16,24,0.6667,0.74107,0.24074,0.42857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,9,0,0,0,0,13],[20,24,0.8333,0.75892,0.22712,0.57132,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,8,0,0,1,0,13],[24,24,1.0,0.86161,0.19393,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,8,0,0,0,0,20]]}]},{"i":"7e03dcb29b8b00a7","q":"Set  $A$  consists of  $2016$  positive integers. All prime divisors of these numbers are smaller than  $30.$  Prove that there are four distinct numbers  $a, b, c$  and  $d$  in  $A$  such that  $abcd$  is a perfect square.","t":[{"b":1,"e":0.2857,"k":"falling","v":0.41518,"x":0.88393,"p":[[0,22,0.0,0.88393,0.22428,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,25],[4,22,0.1818,0.82143,0.26486,0.42857,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,22],[8,22,0.3636,0.73213,0.27375,0.42857,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,1,0,0,0,0,16],[12,22,0.5455,0.67411,0.26301,0.42857,0.64286,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,14,0,0,1,0,0,4,0,0,1,0,11],[16,22,0.7273,0.61607,0.26592,0.42857,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,19,0,0,1,0,0,1,0,0,0,0,10],[20,22,0.9091,0.54911,0.24772,0.42857,0.42857,0.50002,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,21,0,0,0,0,0,1,0,0,0,0,7],[22,22,1.0,0.41518,0.11495,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,20,0,0,0,0,0,3,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"falling","v":0.52232,"x":0.80804,"p":[[0,28,0.0,0.80804,0.25407,0.53571,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,1,0,0,0,0,20],[4,28,0.1429,0.75893,0.2683,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,17],[8,28,0.2857,0.67857,0.28347,0.42857,0.42859,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,14],[12,28,0.4286,0.79464,0.26949,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,20],[16,28,0.5714,0.73661,0.2699,0.42857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,1,0,0,2,0,15],[20,28,0.7143,0.62054,0.24643,0.42857,0.42857,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,6,0,0,0,0,8],[24,28,0.8571,0.60267,0.24932,0.42857,0.42857,0.89275,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,2,0,0,1,0,8],[28,28,1.0,0.52232,0.20079,0.42857,0.42857,0.42857,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,24,0,0,0,0,0,3,0,0,0,0,4]]}]},{"i":"08935b8d133d77a7","q":"Show that, for every positive integer $n$, there exists a monic polynomial of degree $n$ with integer coefficients such that the coefficients are decreasing and the roots of the polynomial are all integers.","t":[{"b":1,"e":0.42857,"k":"falling","v":0.2366,"x":0.68302,"p":[[0,37,0.0,0.68302,0.28956,0.42857,0.64286,1.0,0.0,1.0,1,11,1,1,0,1,0,0,2,0,0,5,0,0,7,0,0,2,0,0,3,0,11],[4,37,0.1081,0.40624,0.29256,0.14286,0.28571,0.60714,0.14286,1.0,0,3,0,0,0,12,0,0,7,0,0,3,0,0,2,0,0,3,0,0,2,0,3],[8,37,0.2162,0.46866,0.30781,0.24999,0.28571,0.71429,0.0,1.0,1,4,0,1,0,7,0,0,9,0,0,1,0,0,3,0,0,5,0,0,2,0,4],[12,37,0.3243,0.34821,0.25985,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,13,0,0,9,0,0,4,0,0,1,0,0,1,0,0,2,0,2],[16,37,0.4324,0.38839,0.30563,0.14286,0.14288,0.60714,0.14286,1.0,0,3,0,0,0,17,0,0,1,0,0,4,0,0,2,0,0,3,0,0,2,0,3],[20,37,0.5405,0.32143,0.30304,0.14286,0.14286,0.32143,0.0,1.0,2,3,0,2,0,17,0,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,3],[24,37,0.6486,0.32134,0.31344,0.14286,0.14286,0.28571,0.0,1.0,1,4,0,1,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,0,2,0,4],[28,37,0.7568,0.2366,0.14107,0.14286,0.14288,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[32,37,0.8649,0.30356,0.20122,0.14286,0.2857,0.42857,0.0,0.85714,1,0,0,1,0,13,0,0,8,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[36,37,0.973,0.49103,0.1954,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,12,0,0,7,0,0,3,0,0,2,0,1],[37,37,1.0,0.46427,0.14724,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,14,0,0,6,0,0,5,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.37498,"x":0.8616,"p":[[0,38,0.0,0.60711,0.33881,0.25001,0.71429,0.89286,0.0,1.0,1,8,0,1,0,7,0,0,1,0,0,3,0,0,3,0,0,4,0,0,5,0,8],[4,38,0.1053,0.8616,0.18379,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,6,0,17],[8,38,0.2105,0.71875,0.32436,0.42859,0.85714,1.0,0.0,1.0,1,14,0,1,0,4,0,0,0,0,0,4,0,0,1,0,0,5,0,0,3,0,14],[12,38,0.3158,0.65176,0.28108,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,5,0,0,5,0,0,3,0,0,5,0,8],[16,38,0.4211,0.70536,0.2549,0.53571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,5,0,0,7,0,8],[20,38,0.5263,0.44196,0.25595,0.24999,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,7,0,0,2,0,0,5,0,0,3,0,1],[24,38,0.6316,0.41072,0.28514,0.25001,0.28571,0.57143,0.0,1.0,2,3,0,2,0,6,0,0,11,0,0,2,0,0,5,0,0,1,0,0,2,0,3],[28,38,0.7368,0.45534,0.26349,0.24999,0.42857,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,6,0,0,3,0,0,5,0,0,7,0,0,2,0,1],[32,38,0.8421,0.38382,0.22997,0.14286,0.35714,0.4642,0.0,0.85714,1,0,0,1,0,8,0,0,7,0,0,8,0,0,3,0,0,2,0,0,3,0,0],[36,38,0.9474,0.37498,0.21052,0.2857,0.28571,0.46418,0.0,1.0,1,1,0,1,0,5,0,0,13,0,0,5,0,0,4,0,0,3,0,0,0,0,1],[38,38,1.0,0.37944,0.18762,0.28571,0.28571,0.42857,0.14286,0.857,0,0,0,0,0,4,0,0,15,0,0,7,0,0,2,0,0,2,0,0,2,0,0]]}]},{"i":"e8e2b96ff46d4057","q":"Point $D$ belongs to the side $[A C]$ of the equilateral triangle $A B C$. We denote $F$ as the orthogonal projection of $D$ on ($B C$), then $K$ as the orthogonal projection of $F$ on ($A B$), and finally $E$ as the orthogonal projection of $K$ on ($C A$). Let $L$ be the midpoint of [BC], and $P$ the intersection of the lines ($K E$) and (FD).\n\nProve that the line (BP) passes through the midpoint of [AL].","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0758,"p":[[0,81,0.0,0.0758,0.22009,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,81,0.0494,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,81,0.0988,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,81,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,81,0.1975,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,81,0.2469,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,81,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,81,0.3457,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,81,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,81,0.4938,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,81,0.5432,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.03563,0.17491,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,81,0.642,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,81,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,81,0.7901,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,50,0.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,50,0.4,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.02232,0.10171,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,50,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.05357,0.18471,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"42e384856109faad","q":"Suppose P is a point in the interior of a triangle ABC, that x; y; z are\nthe distances from P to A; B; C, respectively, and that p; q; r are the per-\npendicular distances from P to the sides BC; CA; AB, respectively. Prove\nthat $xyz \u0015\\geq 8pqr$ ;\nwith equality implying that the triangle ABC is equilateral.","t":[{"b":2,"e":0.0,"k":"falling","v":0.14286,"x":0.53572,"p":[[0,52,0.0,0.53572,0.31743,0.28571,0.57143,0.71429,0.0,1.0,3,6,0,3,0,3,0,0,5,0,0,3,0,0,4,0,0,8,0,0,0,0,6],[4,52,0.0769,0.39282,0.24741,0.25,0.35714,0.571,0.0,1.0,2,2,0,2,0,6,0,0,8,0,0,7,0,0,4,0,0,3,0,0,0,0,2],[8,52,0.1538,0.29464,0.18536,0.14286,0.2857,0.32143,0.0,0.71429,1,0,0,1,0,12,0,0,11,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[12,52,0.2308,0.27661,0.20509,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,11,0,0,8,0,0,4,0,0,2,0,0,3,0,0,0,0,0],[16,52,0.3077,0.29911,0.20316,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,10,0,0,8,0,0,7,0,0,0,0,0,4,0,0,0,0,0],[20,52,0.3846,0.24554,0.17582,0.14286,0.2857,0.32143,0.0,0.71429,6,0,0,6,0,8,0,0,10,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[24,52,0.4615,0.28125,0.21275,0.14286,0.2857,0.32143,0.0,1.0,4,1,0,4,0,9,0,0,11,0,0,4,0,0,2,0,0,1,0,0,0,0,1],[28,52,0.5385,0.24108,0.21558,0.14286,0.2143,0.28571,0.0,1.0,7,1,0,7,0,9,0,0,9,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[32,52,0.6154,0.27679,0.19212,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,11,0,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,1],[36,52,0.6923,0.25,0.16366,0.14286,0.2143,0.28571,0.0,0.71429,2,0,0,2,0,14,0,0,10,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[40,52,0.7692,0.25668,0.20031,0.14286,0.24999,0.42857,0.0,0.71429,6,0,0,6,0,9,0,1,7,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[44,52,0.8462,0.23205,0.19154,0.14214,0.14288,0.32143,0.0,0.71429,7,0,0,7,0,10,0,0,7,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[48,52,0.9231,0.17856,0.17854,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,9,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[52,52,1.0,0.14286,0.15152,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,17,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.16518,"x":0.60714,"p":[[0,95,0.0,0.50446,0.27661,0.28571,0.5,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,6,0,0,5,0,0,4,0,0,8,0,0,1,0,3],[4,95,0.0421,0.52676,0.2911,0.28571,0.57143,0.71429,0.0,1.0,2,5,0,2,0,3,0,0,6,0,0,2,0,0,8,0,0,6,0,0,0,0,5],[8,95,0.0842,0.38393,0.22711,0.2857,0.35714,0.42858,0.0,1.0,1,2,0,1,0,6,0,0,9,0,0,10,0,0,2,0,0,2,0,0,0,0,2],[12,95,0.1263,0.47544,0.22133,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,5,0,1,3,0,0,10,0,0,3,0,0,9,0,0,0,0,1],[16,95,0.1684,0.60714,0.25505,0.42857,0.64286,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,5,0,0,8,0,0,2,0,0,9,0,0,1,0,6],[20,95,0.2105,0.49553,0.28567,0.25,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,3,0,0,7,0,0,3,0,0,6,0,0,1,0,4],[24,95,0.2526,0.54013,0.28734,0.28571,0.571,0.71429,0.14286,1.0,0,6,0,0,0,6,0,0,3,0,0,6,0,0,6,0,0,5,0,0,0,0,6],[28,95,0.2947,0.50446,0.24218,0.42857,0.57143,0.57143,0.0,1.0,2,3,0,2,0,2,0,0,3,0,0,7,0,0,12,0,0,3,0,0,0,0,3],[32,95,0.3368,0.43304,0.30406,0.14286,0.42857,0.71429,0.0,1.0,2,3,0,2,0,10,0,0,3,0,0,4,0,0,3,0,0,6,0,0,1,0,3],[36,95,0.3789,0.43302,0.28004,0.14286,0.42857,0.71429,0.0,1.0,2,2,0,2,0,8,0,0,4,0,0,5,0,0,4,0,0,6,0,0,1,0,2],[40,95,0.4211,0.36607,0.29652,0.14286,0.28571,0.4286,0.0,1.0,5,3,0,5,0,7,0,0,5,0,0,8,0,0,1,0,0,2,0,0,1,0,3],[44,95,0.4632,0.33482,0.25407,0.14286,0.2857,0.46431,0.0,1.0,3,1,0,3,0,11,0,0,6,0,0,4,0,0,2,0,0,5,0,0,0,0,1],[48,95,0.5053,0.39284,0.2812,0.14289,0.28571,0.57143,0.0,1.0,4,2,0,4,0,5,0,0,8,0,0,5,0,0,4,0,0,2,0,0,2,0,2],[52,95,0.5474,0.28571,0.22868,0.14286,0.2857,0.42857,0.0,1.0,3,2,0,3,0,12,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,2],[56,95,0.5895,0.375,0.26666,0.14289,0.28571,0.46431,0.0,1.0,3,2,0,3,0,6,0,0,10,0,0,5,0,0,2,0,0,3,0,0,1,0,2],[60,95,0.6316,0.35719,0.26246,0.14286,0.28571,0.42858,0.0,1.0,2,3,0,2,0,9,0,0,8,0,0,7,0,0,2,0,0,1,0,0,0,0,3],[64,95,0.6737,0.31241,0.24862,0.14286,0.28571,0.42857,0.0,1.0,4,2,0,4,0,9,0,0,8,0,0,6,0,0,2,0,0,1,0,0,0,0,2],[68,95,0.7158,0.37499,0.24678,0.14286,0.35714,0.57111,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,7,0,0,3,0,0,5,0,0,0,0,1],[72,95,0.7579,0.25446,0.21049,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,12,0,0,4,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[76,95,0.8,0.36161,0.2696,0.14286,0.2857,0.46429,0.0,1.0,2,2,0,2,0,11,0,0,6,0,0,5,0,0,1,0,0,5,0,0,0,0,2],[80,95,0.8421,0.25445,0.19797,0.14286,0.14286,0.42857,0.0,0.71429,5,0,0,5,0,12,0,0,6,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[84,95,0.8842,0.22321,0.15542,0.14286,0.1429,0.28571,0.0,0.71429,4,0,0,4,0,13,0,0,11,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[88,95,0.9263,0.18304,0.14827,0.14286,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,15,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[92,95,0.9684,0.19197,0.14555,0.10714,0.14286,0.28571,0.0,0.4286,8,0,0,8,0,10,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[95,95,1.0,0.16518,0.12428,0.10714,0.14286,0.2857,0.0,0.4286,8,0,0,8,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8a0358b141f498ee","q":"Show that\n\n$$\n(x+y+z)\\left(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}\\right) \\geq 4\\left(\\frac{x}{x y+1}+\\frac{y}{y z+1}+\\frac{z}{z x+1}\\right)^{2}\n$$\n\nfor any real positive numbers $x, y$ and $z$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.14732,"x":0.95089,"p":[[0,84,0.0,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[4,84,0.0476,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,84,0.0952,0.79464,0.39438,1.0,1.0,1.0,0.0,1.0,6,25,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,25],[12,84,0.1429,0.79464,0.38949,0.96425,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,24],[16,84,0.1905,0.75893,0.4141,0.75,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,23],[20,84,0.2381,0.79463,0.37448,0.89275,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,24],[24,84,0.2857,0.83929,0.36202,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[28,84,0.3333,0.63839,0.47243,0.0,1.0,1.0,0.0,1.0,11,20,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20],[32,84,0.381,0.74554,0.41762,0.42857,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,23],[36,84,0.4286,0.80804,0.38896,1.0,1.0,1.0,0.0,1.0,6,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[40,84,0.4762,0.73214,0.41611,0.42857,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,22],[44,84,0.5238,0.77679,0.3976,0.85714,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[48,84,0.5714,0.81696,0.32778,0.78571,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,23],[52,84,0.619,0.75,0.43301,0.75,1.0,1.0,0.0,1.0,8,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[56,84,0.6667,0.78571,0.39123,0.85714,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,23],[60,84,0.7143,0.76339,0.4159,0.85714,1.0,1.0,0.0,1.0,7,24,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24],[64,84,0.7619,0.76339,0.4159,0.85714,1.0,1.0,0.0,1.0,7,24,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24],[68,84,0.8095,0.83482,0.33902,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,24],[72,84,0.8571,0.72321,0.42996,0.32143,1.0,1.0,0.0,1.0,8,21,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,21],[76,84,0.9048,0.84375,0.31412,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,24],[80,84,0.9524,0.38839,0.47947,0.0,0.0,1.0,0.0,1.0,19,12,0,19,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12],[84,84,1.0,0.14732,0.34531,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4]]},{"b":6,"e":1.0,"k":"flat","v":0.71875,"x":0.95089,"p":[[0,32,0.0,0.87054,0.31209,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,27],[4,32,0.125,0.83036,0.36498,1.0,1.0,1.0,0.0,1.0,5,26,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26],[8,32,0.25,0.90179,0.27067,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[12,32,0.375,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[16,32,0.5,0.71875,0.44961,0.0,1.0,1.0,0.0,1.0,9,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[20,32,0.625,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[24,32,0.75,0.87946,0.21162,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,0,2,0,23],[28,32,0.875,0.90179,0.19704,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,3,0,24],[32,32,1.0,0.79464,0.25738,0.42859,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,2,0,0,0,0,19]]}]},{"i":"4e745901870ffaa7","q":"Prove that for any positive integers $p, q$ such that $\\sqrt{11}>\\frac{p}{q}$, the following inequality holds:\n\n$$\n\\sqrt{11}-\\frac{p}{q}>\\frac{1}{2 p q}\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,123,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,123,0.0325,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,123,0.065,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,123,0.0976,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,123,0.1301,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,123,0.1626,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,123,0.1951,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[28,123,0.2276,0.93304,0.21424,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[32,123,0.2602,0.94195,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[36,123,0.2927,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,123,0.3252,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[44,123,0.3577,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[48,123,0.3902,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[52,123,0.4228,0.9375,0.16342,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[56,123,0.4553,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,123,0.4878,0.9241,0.18552,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[64,123,0.5203,0.97098,0.05607,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,1,25],[68,123,0.5528,0.94639,0.12764,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[72,123,0.5854,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[76,123,0.6179,0.94642,0.15872,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[80,123,0.6504,0.89732,0.22654,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,21],[84,123,0.6829,0.93746,0.14265,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,25],[88,123,0.7154,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[92,123,0.748,0.90625,0.25155,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[96,123,0.7805,0.93303,0.17852,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,25],[100,123,0.813,0.93302,0.1923,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[104,123,0.8455,0.9375,0.19541,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[108,123,0.878,0.97321,0.10375,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[112,123,0.9106,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[116,123,0.9431,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[120,123,0.9756,0.94642,0.13247,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[123,123,1.0,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24]]},{"b":2,"e":0.28571,"k":"falling","v":0.27232,"x":1.0,"p":[[0,88,0.0,0.96873,0.06906,1.0,1.0,1.0,0.714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[4,88,0.0455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,88,0.0909,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,88,0.1364,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,88,0.1818,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,88,0.2273,0.9375,0.19541,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[24,88,0.2727,0.96428,0.12878,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[28,88,0.3182,0.9375,0.18877,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[32,88,0.3636,0.94196,0.21087,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[36,88,0.4091,0.95534,0.10976,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[40,88,0.4545,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,88,0.5,0.94196,0.19516,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[48,88,0.5455,0.98213,0.05924,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,88,0.5909,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,88,0.6364,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[60,88,0.6818,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,88,0.7273,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,88,0.7727,0.96428,0.10715,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[72,88,0.8182,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[76,88,0.8636,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[80,88,0.9091,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[84,88,0.9545,0.69196,0.35554,0.28571,0.85714,1.0,0.0,1.0,1,15,0,1,0,3,0,0,6,0,0,2,0,0,0,0,0,1,0,0,4,0,15],[88,88,1.0,0.27232,0.19351,0.14286,0.2857,0.28571,0.0,1.0,1,1,0,1,0,12,0,0,16,0,0,0,0,0,0,0,0,2,0,0,0,0,1]]}]},{"i":"d36e4a13ceacc768","q":"The angle  $POQ$  is given ( $OP$  and  $OQ$  are rays). Let  $M$  and  $N$  be points inside the angle  $POQ$  such that  $\\angle POM = \\angle QON$  and  $\\angle POM < \\angle PON$ . Consider two circles: one touches the rays  $OP$  and  $ON$ , the other touches the rays  $OM$  and  $OQ$ . Denote by  $B$  and  $C$  the points of their intersection. Prove that  $\\angle POC = \\angle QOB$ .","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.70982,"p":[[0,52,0.0,0.43291,0.34352,0.105,0.42836,0.71429,0.0,1.0,8,4,1,8,0,2,0,0,6,0,0,0,0,0,6,0,0,5,0,0,1,0,4],[4,52,0.0769,0.53125,0.45208,0.0,0.71429,1.0,0.0,1.0,11,13,0,11,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,13],[8,52,0.1538,0.70982,0.38711,0.39286,1.0,1.0,0.0,1.0,4,18,0,4,0,3,0,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,18],[12,52,0.2308,0.48656,0.42536,0.0,0.571,1.0,0.0,1.0,12,9,0,12,0,1,0,0,1,0,0,0,0,0,3,0,0,5,0,0,1,0,9],[16,52,0.3077,0.52231,0.46237,0.0,0.64286,1.0,0.0,1.0,13,13,0,13,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,13],[20,52,0.3846,0.60714,0.44892,0.0,0.92857,1.0,0.0,1.0,9,16,0,9,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,16],[24,52,0.4615,0.41964,0.42996,0.0,0.28571,1.0,0.0,1.0,13,9,0,13,0,2,0,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,9],[28,52,0.5385,0.60711,0.38795,0.28571,0.71429,1.0,0.0,1.0,7,10,0,7,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,5,0,10],[32,52,0.6154,0.64732,0.40874,0.25,0.85714,1.0,0.0,1.0,7,14,0,7,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,0,3,0,14],[36,52,0.6923,0.6116,0.42743,0.0,0.78564,1.0,0.0,1.0,9,14,0,9,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,14],[40,52,0.7692,0.52679,0.44954,0.0,0.64286,1.0,0.0,1.0,12,12,0,12,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,12],[44,52,0.8462,0.54464,0.46488,0.0,0.78564,1.0,0.0,1.0,12,14,0,12,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,14],[48,52,0.9231,0.33479,0.42797,0.0,0.0,0.85704,0.0,1.0,19,6,0,19,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,6],[52,52,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.10043,"x":0.7232,"p":[[0,36,0.0,0.46871,0.3077,0.2857,0.571,0.60714,0.0,1.0,6,3,0,6,0,1,0,0,5,0,0,2,0,0,10,0,0,3,0,0,2,0,3],[4,36,0.1111,0.54911,0.4603,0.0,0.71429,1.0,0.0,1.0,12,14,0,12,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,14],[8,36,0.2222,0.71427,0.3993,0.49967,1.0,1.0,0.0,1.0,6,18,0,6,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,18],[12,36,0.3333,0.67855,0.40248,0.39287,0.92857,1.0,0.0,1.0,7,16,0,7,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,16],[16,36,0.4444,0.58926,0.42069,0.0,0.71429,1.0,0.0,1.0,9,13,0,9,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,1,0,13],[20,36,0.5556,0.52665,0.4336,0.0,0.71429,1.0,0.0,1.0,12,10,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,1,0,10],[24,36,0.6667,0.7232,0.36933,0.57132,1.0,1.0,0.0,1.0,5,17,0,5,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,1,0,17],[28,36,0.7778,0.56695,0.39687,0.24999,0.71429,1.0,0.0,1.0,7,11,0,7,0,1,0,0,4,0,0,2,0,0,1,0,0,5,0,0,1,0,11],[32,36,0.8889,0.19196,0.37561,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[36,36,1.0,0.10043,0.27012,0.0,0.0,0.0,0.0,1.0,27,2,0,27,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2]]}]},{"i":"b05ae434dc86e17d","q":"Prove that for all non-negative numbers  $x,y,z$  satisfying  $x+y+z=1$ , one has\n\\[1 \\le \\frac{x}{1-yz}+\\frac{y}{1-zx}+\\frac{z}{1-xy} \\le \\frac{9}{8}.\\]","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14286,"x":0.33036,"p":[[0,27,0.0,0.26339,0.25281,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,23,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[4,27,0.1481,0.25893,0.26592,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,26,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[8,27,0.2963,0.25446,0.22227,0.14286,0.14286,0.32143,0.14286,1.0,0,2,0,0,0,23,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[12,27,0.4444,0.33036,0.3223,0.14286,0.14286,0.42857,0.14286,1.0,0,5,0,0,0,22,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,5],[16,27,0.5926,0.24545,0.19643,0.14286,0.14286,0.32143,0.14,1.0,0,1,0,0,0,23,0,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[20,27,0.7407,0.16518,0.12428,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,27,0.8889,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.4286,"k":"flat","v":0.19643,"x":0.39732,"p":[[0,87,0.0,0.20536,0.13333,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,25,0,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[4,87,0.046,0.24107,0.22142,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[8,87,0.092,0.20536,0.2111,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,87,0.1379,0.19643,0.17035,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,26,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[16,87,0.1839,0.27679,0.28557,0.14286,0.14286,0.14286,0.14286,1.0,0,4,0,0,0,25,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4],[20,87,0.2299,0.38839,0.33165,0.14286,0.14286,0.42857,0.14286,1.0,0,6,0,0,0,17,0,0,2,0,0,6,0,0,0,0,0,0,0,0,1,0,6],[24,87,0.2759,0.26786,0.28065,0.14286,0.14286,0.1786,0.14286,1.0,0,4,0,0,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[28,87,0.3218,0.30357,0.28516,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,22,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,4],[32,87,0.3678,0.35268,0.28118,0.14286,0.28571,0.42857,0.0,1.0,1,4,0,1,0,14,0,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,4],[36,87,0.4138,0.28125,0.25626,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,22,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[40,87,0.4598,0.23661,0.19103,0.14286,0.14286,0.42857,0.0,1.0,2,1,0,2,0,20,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[44,87,0.5057,0.35714,0.32927,0.14286,0.14286,0.42857,0.14286,1.0,0,6,0,0,0,20,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,6],[48,87,0.5517,0.32589,0.27947,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,18,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,4],[52,87,0.5977,0.29911,0.24317,0.14286,0.14286,0.42857,0.0,1.0,1,2,0,1,0,18,0,0,1,0,0,8,0,0,1,0,0,1,0,0,0,0,2],[56,87,0.6437,0.35714,0.30514,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,19,0,0,0,0,0,6,0,0,1,0,0,1,0,0,1,0,4],[60,87,0.6897,0.33929,0.30671,0.14286,0.14286,0.42857,0.14286,1.0,0,5,0,0,0,20,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,5],[64,87,0.7356,0.34374,0.28315,0.14286,0.14286,0.42857,0.14286,1.0,0,4,0,0,0,18,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,4],[68,87,0.7816,0.24107,0.22142,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[72,87,0.8276,0.28125,0.22156,0.14286,0.14286,0.42857,0.14286,1.0,0,2,0,0,0,19,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,2],[76,87,0.8736,0.33036,0.23266,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,13,0,0,3,0,0,12,0,0,0,0,0,1,0,0,0,0,2],[80,87,0.9195,0.39732,0.23072,0.14286,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,9,0,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,3],[84,87,0.9655,0.35714,0.24485,0.14286,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,13,0,0,2,0,0,14,0,0,0,0,0,0,0,0,0,0,3],[87,87,1.0,0.35268,0.17491,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"971bc8bd0d414777","q":"Prove that for all positive real numbers  $a, b, c$  with  $\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c} =1$  the following inequality holds:   $$ 3(ab+bc+ca)+\\frac{9}{a+b+c} \\le \\frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1 $$","t":[{"b":4,"e":0.0,"k":"falling","v":0.14723,"x":0.39286,"p":[[0,103,0.0,0.375,0.14174,0.39286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,2,0,0,22,0,0,1,0,0,1,0,0,0,0,0],[4,103,0.0388,0.39286,0.19562,0.39286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,2,0,0,3,0,0,21,0,0,0,0,0,2,0,0,0,0,1],[8,103,0.0777,0.375,0.15872,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,4,0,0,21,0,0,0,0,0,0,0,0,0,0,1],[12,103,0.1165,0.36606,0.10675,0.28571,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,4,0,0,7,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[16,103,0.1553,0.36159,0.19555,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,4,0,0,12,0,0,2,0,0,4,0,0,0,0,0],[20,103,0.1942,0.37053,0.12805,0.42857,0.42857,0.42857,0.0,0.571,1,0,0,1,0,5,0,0,1,0,0,24,0,0,1,0,0,0,0,0,0,0,0],[24,103,0.233,0.37947,0.12169,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,23,0,0,0,0,0,1,0,0,0,0,0],[28,103,0.2718,0.36607,0.14258,0.2857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,4,0,0,18,0,0,2,0,0,1,0,0,0,0,0],[32,103,0.3107,0.32143,0.15567,0.14286,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,8,0,0,4,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[36,103,0.3495,0.35714,0.19562,0.2857,0.42857,0.42857,0.0,1.0,3,1,0,3,0,4,0,0,5,0,0,18,0,0,0,0,0,1,0,0,0,0,1],[40,103,0.3883,0.34821,0.12846,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,2,0,0,9,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[44,103,0.4272,0.36161,0.14279,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,2,0,0,22,0,0,0,0,0,1,0,0,0,0,0],[48,103,0.466,0.36165,0.14721,0.2857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,5,0,0,18,0,0,0,0,0,2,0,0,0,0,0],[52,103,0.5049,0.35267,0.14277,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,6,0,0,5,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[56,103,0.5437,0.37053,0.17807,0.28571,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,4,0,0,2,0,0,19,0,0,2,0,0,2,0,0,0,0,0],[60,103,0.5825,0.37946,0.20395,0.2857,0.42857,0.42857,0.0,1.0,2,1,0,2,0,4,0,0,7,0,0,15,0,0,0,0,0,3,0,0,0,0,1],[64,103,0.6214,0.38839,0.21498,0.25,0.42857,0.42857,0.0,1.0,2,1,0,2,0,6,0,0,3,0,0,15,0,0,2,0,0,3,0,0,0,0,1],[68,103,0.6602,0.36161,0.17122,0.28571,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,3,0,0,5,0,0,18,0,0,1,0,0,2,0,0,0,0,0],[72,103,0.699,0.31696,0.18466,0.14286,0.35714,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,4,0,0,13,0,0,2,0,0,0,0,0,1,0,0],[76,103,0.7379,0.36607,0.14698,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,3,0,0,5,0,0,20,0,0,1,0,0,1,0,0,0,0,0],[80,103,0.7767,0.30804,0.1992,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,11,0,0,5,0,0,11,0,0,2,0,0,0,0,0,0,0,1],[84,103,0.8155,0.27232,0.20316,0.14286,0.21428,0.42857,0.0,0.71429,6,0,0,6,0,10,0,0,1,0,0,13,0,0,0,0,0,2,0,0,0,0,0],[88,103,0.8544,0.29464,0.15947,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,10,0,0,6,0,0,13,0,0,0,0,0,1,0,0,0,0,0],[92,103,0.8932,0.30803,0.16409,0.24999,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,12,0,0,9,0,0,2,0,0,1,0,0,0,0,0],[96,103,0.932,0.32589,0.25812,0.14286,0.28571,0.42857,0.0,1.0,4,3,0,4,0,7,0,0,9,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[100,103,0.9709,0.14723,0.14054,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,10,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[103,103,1.0,0.16518,0.13882,0.10714,0.14286,0.17857,0.0,0.42857,8,0,0,8,0,16,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.24107,"x":0.41071,"p":[[0,161,0.0,0.3884,0.12492,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,4,0,0,5,0,0,20,0,0,2,0,0,1,0,0,0,0,0],[4,161,0.0248,0.37946,0.18074,0.2857,0.42857,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,5,0,0,18,0,0,1,0,0,1,0,0,0,0,1],[8,161,0.0497,0.39284,0.18897,0.28571,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,2,0,0,5,0,0,16,0,0,2,0,0,4,0,0,0,0,0],[12,161,0.0745,0.41071,0.17768,0.2857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,5,0,0,12,0,0,5,0,0,4,0,0,0,0,0],[16,161,0.0994,0.36143,0.16384,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,8,0,0,15,0,0,0,0,0,3,0,0,0,0,0],[20,161,0.1242,0.36607,0.14258,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,6,0,0,18,0,0,0,0,0,2,0,0,0,0,0],[24,161,0.1491,0.36161,0.14719,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,3,0,0,6,0,0,19,0,0,1,0,0,1,0,0,0,0,0],[28,161,0.1739,0.37054,0.20782,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,2,0,0,15,0,0,2,0,0,2,0,0,0,0,1],[32,161,0.1988,0.38838,0.15249,0.39286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,4,0,0,21,0,0,1,0,0,2,0,0,0,0,0],[36,161,0.2236,0.38839,0.17215,0.28571,0.42857,0.42857,0.0,1.0,1,1,0,1,0,1,0,0,12,0,0,15,0,0,0,0,0,2,0,0,0,0,1],[40,161,0.2484,0.33928,0.14174,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,13,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[44,161,0.2733,0.35714,0.21129,0.14286,0.42857,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,4,0,0,14,0,0,3,0,0,1,0,0,0,0,1],[48,161,0.2981,0.3482,0.1634,0.24999,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,5,0,0,2,0,0,19,0,0,3,0,0,0,0,0,0,0,0],[52,161,0.323,0.33473,0.15826,0.14286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,7,0,0,3,0,0,19,0,0,0,0,0,1,0,0,0,0,0],[56,161,0.3478,0.34821,0.17835,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,4,0,0,13,0,0,3,0,0,2,0,0,0,0,0],[60,161,0.3727,0.3125,0.15335,0.14286,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,6,0,0,6,0,0,16,0,0,1,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Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from  $ 1$  to  $ 15$  in clockwise order. Committee rules state that a Martian must occupy chair  $ 1$  and an Earthling must occupy chair  $ 15$ . Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is  $ N\\cdot (5!)^3$ . Find  $ N$ .","t":[{"b":5,"e":0.14286,"k":"falling","v":0.06232,"x":0.98661,"p":[[0,47,0.0,0.92411,0.21124,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,28],[4,47,0.0851,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[8,47,0.1702,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,47,0.2553,0.80357,0.31084,0.57143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,22],[16,47,0.3404,0.77454,0.33174,0.53539,1.0,1.0,0.0,1.0,1,21,0,1,1,1,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,21],[20,47,0.4255,0.74107,0.35434,0.39286,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,20],[24,47,0.5106,0.68304,0.32485,0.42857,0.57143,1.0,0.0,1.0,1,15,0,1,0,2,0,0,3,0,0,4,0,0,7,0,0,0,0,0,0,0,15],[28,47,0.5957,0.60268,0.33452,0.42857,0.57143,1.0,0.0,1.0,3,11,0,3,0,2,0,0,2,0,0,4,0,0,10,0,0,0,0,0,0,0,11],[32,47,0.6809,0.63838,0.34439,0.42859,0.57143,1.0,0.0,1.0,3,13,0,3,0,2,0,0,2,0,0,2,0,0,10,0,0,0,0,0,0,0,13],[36,47,0.766,0.77008,0.33917,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,3,0,0,2,0,1,0,0,0,4,0,0,0,0,0,0,0,21],[40,47,0.8511,0.83036,0.3163,0.82143,1.0,1.0,0.0,1.0,2,23,0,2,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,23],[44,47,0.9362,0.26331,0.36268,0.0,0.14286,0.32143,0.0,1.0,14,5,0,14,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[47,47,1.0,0.06232,0.07067,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.09152,"x":1.0,"p":[[0,103,0.0,0.875,0.28065,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,26],[4,103,0.0388,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,103,0.0777,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,103,0.1165,0.81696,0.31183,0.57143,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,23],[16,103,0.1553,0.84822,0.25238,0.57143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,0,23],[20,103,0.1942,0.76116,0.28661,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,2,0,1,7,0,0,0,0,0,0,0,18],[24,103,0.233,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,103,0.2718,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,103,0.3107,0.92857,0.23419,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[36,103,0.3495,0.92411,0.17852,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,27],[40,103,0.3883,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[44,103,0.4272,0.89286,0.28571,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[48,103,0.466,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[52,103,0.5049,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[56,103,0.5437,0.98219,0.09918,1.0,1.0,1.0,0.43,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[60,103,0.5825,0.91518,0.26453,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[64,103,0.6214,0.94197,0.1885,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[68,103,0.6602,0.9375,0.19541,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[72,103,0.699,0.9375,0.18189,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[76,103,0.7379,0.92857,0.23419,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[80,103,0.7767,0.87946,0.29038,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,27],[84,103,0.8155,0.89286,0.26486,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,27],[88,103,0.8544,0.84821,0.31932,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[92,103,0.8932,0.83482,0.3483,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[96,103,0.932,0.81696,0.33165,0.89286,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,24],[100,103,0.9709,0.48438,0.45762,0.125,0.14286,1.0,0.0,1.0,7,14,0,7,1,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[103,103,1.0,0.09152,0.11605,0.0,0.10714,0.14286,0.0,0.57143,15,0,0,15,1,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"5f3e6940eabad9a5","q":"The numbers  $1,2,...,2n-1,2n$  are divided into two disjoint sets,  $a_1 < a_2 < ... < a_n$  and  $b_1 > b_2 > ... > b_n$ . Prove that   $$ |a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2. $$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.04018,"x":0.66964,"p":[[0,51,0.0,0.5,0.45316,0.0,0.42857,1.0,0.0,1.0,10,11,0,10,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,11],[4,51,0.0784,0.65179,0.44741,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,17],[8,51,0.1569,0.66964,0.44095,0.14286,1.0,1.0,0.0,1.0,6,19,0,6,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,19],[12,51,0.2353,0.54464,0.46625,0.0,0.85714,1.0,0.0,1.0,12,12,0,12,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12],[16,51,0.3137,0.62947,0.39747,0.14289,0.85714,1.0,0.0,1.0,7,10,0,7,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,10],[20,51,0.3922,0.59375,0.46306,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,15],[24,51,0.4706,0.54911,0.47125,0.0,0.85714,1.0,0.0,1.0,12,14,0,12,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,14],[28,51,0.549,0.54911,0.44623,0.0,0.85714,1.0,0.0,1.0,9,12,0,9,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,12],[32,51,0.6275,0.53125,0.44925,0.0,0.78571,1.0,0.0,1.0,11,10,0,11,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,10],[36,51,0.7059,0.5,0.46015,0.0,0.5,1.0,0.0,1.0,11,11,0,11,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,11],[40,51,0.7843,0.51786,0.46531,0.0,0.78571,1.0,0.0,1.0,12,12,0,12,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,12],[44,51,0.8627,0.44643,0.45841,0.0,0.14286,1.0,0.0,1.0,12,11,0,12,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,11],[48,51,0.9412,0.16964,0.33204,0.0,0.0,0.14286,0.0,1.0,21,4,0,21,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[51,51,1.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.01786,"x":0.70089,"p":[[0,84,0.0,0.47768,0.46922,0.0,0.14286,1.0,0.0,1.0,12,13,0,12,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,13],[4,84,0.0476,0.42848,0.46708,0.0,0.14286,1.0,0.0,1.0,13,12,0,13,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,12],[8,84,0.0952,0.49107,0.45588,0.0,0.42857,1.0,0.0,1.0,11,12,0,11,0,5,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,12],[12,84,0.1429,0.55804,0.44515,0.14286,0.78571,1.0,0.0,1.0,7,14,0,7,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,14],[16,84,0.1905,0.48214,0.47749,0.0,0.42857,1.0,0.0,1.0,15,12,0,15,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,12],[20,84,0.2381,0.46428,0.40876,0.10714,0.35714,0.85714,0.0,1.0,8,6,0,8,0,8,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,6],[24,84,0.2857,0.5,0.43741,0.0,0.64286,0.89286,0.0,1.0,11,8,0,11,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,8],[28,84,0.3333,0.29911,0.42312,0.0,0.0,0.85714,0.0,1.0,17,7,0,17,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,7],[32,84,0.381,0.37947,0.44408,0.0,0.14286,1.0,0.0,1.0,14,9,0,14,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,9],[36,84,0.4286,0.50893,0.45166,0.0,0.71429,1.0,0.0,1.0,11,11,0,11,0,4,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,11],[40,84,0.4762,0.47331,0.4482,0.0,0.1429,1.0,0.0,1.0,10,10,0,10,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,10],[44,84,0.5238,0.63839,0.41952,0.10714,0.85714,1.0,0.0,1.0,8,13,0,8,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,6,0,13],[48,84,0.5714,0.61148,0.43479,0.0,0.85714,1.0,0.0,1.0,9,13,0,9,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,13],[52,84,0.619,0.70089,0.39018,0.46429,0.85714,1.0,0.0,1.0,5,15,0,5,0,3,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,15],[56,84,0.6667,0.53125,0.45908,0.0,0.85707,1.0,0.0,1.0,10,12,0,10,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,12],[60,84,0.7143,0.5,0.46015,0.0,0.71429,1.0,0.0,1.0,13,11,0,13,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,11],[64,84,0.7619,0.62054,0.4512,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,17],[68,84,0.8095,0.45982,0.45699,0.0,0.14288,1.0,0.0,1.0,11,12,0,11,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,12],[72,84,0.8571,0.54464,0.44239,0.10714,0.78571,1.0,0.0,1.0,8,12,0,8,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,12],[76,84,0.9048,0.17411,0.30875,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,3,0,1],[80,84,0.9524,0.07143,0.18898,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[84,84,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4369175ba9cb8c05","q":"The set  $ S$  consists of  $ n > 2$  points in the plane. The set  $ P$  consists of  $ m$  lines in the plane such that every line in  $ P$  is an axis of symmetry for  $ S$ . Prove that  $ m\\leq n$ , and determine when equality holds.","t":[{"b":0,"e":1.0,"k":"flat","v":0.5892,"x":0.79908,"p":[[0,22,0.0,0.69194,0.22049,0.5354,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,4,0,0,9,0,5],[4,22,0.1818,0.79908,0.21978,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,3,0,15],[8,22,0.3636,0.69637,0.16659,0.57143,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,12,0,0,2,0,5],[12,22,0.5455,0.6696,0.18365,0.571,0.71429,0.85704,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,8,0,0,6,0,3],[16,22,0.7273,0.5892,0.17406,0.5354,0.57141,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,6,0,0,15,0,0,4,0,0,4,0,1],[20,22,0.9091,0.63386,0.17106,0.571,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,8,0,0,4,0,2],[22,22,1.0,0.62048,0.16983,0.4286,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,6,0,0,4,0,2]]},{"b":7,"e":0.85714,"k":"rising","v":0.57142,"x":0.93076,"p":[[0,33,0.0,0.67407,0.25813,0.42859,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,7,0,0,2,0,0,4,0,0,5,0,0,8,0,6],[4,33,0.1212,0.6964,0.23078,0.4286,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,6,0,0,4,0,0,5,0,8],[8,33,0.2424,0.72319,0.23129,0.4286,0.78564,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,2,0,0,7,0,9],[12,33,0.3636,0.66063,0.20751,0.42859,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,5,0,0,3,0,6],[16,33,0.4848,0.57142,0.21129,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,9,0,0,7,0,0,6,0,0,2,0,3],[20,33,0.6061,0.61602,0.19046,0.5354,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,6,0,0,13,0,0,6,0,0,1,0,4],[24,33,0.7273,0.76335,0.2541,0.571,0.85707,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,4,0,0,3,0,14],[28,33,0.8485,0.93076,0.1468,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,1,25],[32,33,0.9697,0.87053,0.21535,0.82132,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,21],[33,33,1.0,0.91518,0.1885,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,25]]}]},{"i":"a85399e0e54189bf","q":"There are  $2021$  people at a meeting. It is known that one person at the meeting doesn't have any friends there and another person has only one friend there. In addition, it is true that, given any  $4$  people, at least  $2$  of them are friends. Show that there are  $2018$  people at the meeting that are all friends with each other.\n*Note.*If  $A$  is friend of  $B$  then  $B$  is a friend of  $A$ .","t":[{"b":5,"e":1.0,"k":"rising","v":0.55357,"x":1.0,"p":[[0,30,0.0,0.55357,0.40049,0.24999,0.28571,1.0,0.0,1.0,2,14,0,2,0,6,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[4,30,0.1333,0.75447,0.36637,0.28571,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,22],[8,30,0.2667,0.66071,0.37244,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0,2,0,16],[12,30,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,30,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,30,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.50893,"x":1.0,"p":[[0,20,0.0,0.50893,0.38785,0.14286,0.28571,1.0,0.0,1.0,1,11,1,1,0,10,0,0,8,0,0,0,0,0,0,0,0,1,0,0,1,0,11],[4,20,0.2,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,20,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,20,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"1cc72665eefaca87","q":"The polynomial of seven variables  $$ Q(x_1,x_2,\\ldots,x_7)=(x_1+x_2+\\ldots+x_7)^2+2(x_1^2+x_2^2+\\ldots+x_7^2) $$  is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:  $$ Q(x_1,\\ldots,x_7)=P_1(x_1,\\ldots,x_7)^2+P_2(x_1,\\ldots,x_7)^2+\\ldots+P_7(x_1,\\ldots,x_7)^2. $$  Find all possible values of  $P_1(1,1,\\ldots,1)$ .\n\n*(A. Yuran)*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.88393,"x":1.0,"p":[[0,111,0.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,111,0.036,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,111,0.0721,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,111,0.1081,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[16,111,0.1441,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,111,0.1802,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,111,0.2162,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,111,0.2523,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,111,0.2883,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,111,0.3243,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,111,0.3604,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,111,0.3964,0.92411,0.20198,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,28],[48,111,0.4324,0.94196,0.15916,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,26],[52,111,0.4685,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[56,111,0.5045,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[60,111,0.5405,0.92411,0.1988,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,26],[64,111,0.5766,0.92856,0.17499,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[68,111,0.6126,0.88393,0.25111,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[72,111,0.6486,0.91071,0.21354,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,26],[76,111,0.6847,0.88839,0.22794,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,24],[80,111,0.7207,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[84,111,0.7568,0.90625,0.22759,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,24],[88,111,0.7928,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[92,111,0.8288,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,111,0.8649,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[100,111,0.9009,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,111,0.9369,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[108,111,0.973,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[111,111,1.0,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18]]},{"b":1,"e":0.71429,"k":"flat","v":0.88392,"x":1.0,"p":[[0,56,0.0,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[4,56,0.0714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,56,0.1429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,56,0.2143,0.97991,0.04803,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,27],[16,56,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,56,0.3571,0.94196,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[24,56,0.4286,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,56,0.5,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[32,56,0.5714,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,56,0.6429,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,56,0.7143,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[44,56,0.7857,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,56,0.8571,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,56,0.9286,0.92409,0.11289,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,11,0,19],[56,56,1.0,0.88392,0.13092,0.857,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,15]]}]},{"i":"e1731877c2de840d","q":"Show that for all strictly positive real numbers $\\mathrm{a}, \\mathrm{b}, \\mathrm{c}$:\n\n$$\n\\frac{b c}{a^{2}+2 b c}+\\frac{c a}{b^{2}+2 c a}+\\frac{a b}{c^{2}+2 a b} \\leqslant 1 \\leqslant \\frac{a^{2}}{a^{2}+2 b c}+\\frac{b^{2}}{b^{2}+2 c a}+\\frac{c^{2}}{c^{2}+2 a b}\n$$","t":[{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,8,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,8,0.5,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,33,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,33,0.1212,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,33,0.2424,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,33,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,33,0.4848,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,33,0.8485,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,33,0.9697,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[33,33,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"987bc8cf6b13b9ff","q":"Three non-collinear lattice points  $A,B,C$  lie on the plane  $1+3x+5y+7z=0$ . The minimal possible area of triangle  $ABC$  can be expressed as  $\\frac{\\sqrt{m}}{n}$  where  $m,n$  are positive integers such that there does not exists a prime  $p$  dividing  $n$  with  $p^2$  dividing  $m$ . Compute  $100m+n$ .\n\n*Proposed by Yannick Yao*","t":[{"b":2,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,57,0.0,0.91964,0.19212,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,27],[4,57,0.0702,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,57,0.1404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,57,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,57,0.2807,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,57,0.3509,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,57,0.4211,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,57,0.4912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,57,0.5614,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,57,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,57,0.7018,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,57,0.7719,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,57,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,57,0.9123,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,57,0.9825,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[57,57,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,34,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,34,0.1176,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,34,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,34,0.3529,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,34,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,34,0.5882,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,34,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,34,0.9412,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[34,34,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"133b3de6df737bc6","q":"Solve the equation in postive integers   $$ x^2+y^2+1998=1997x-1999y. $$","t":[{"b":3,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,134,0.0,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,134,0.0299,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,134,0.0597,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,134,0.0896,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,134,0.1194,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,134,0.1493,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[24,134,0.1791,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,134,0.209,0.91964,0.24206,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[32,134,0.2388,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,134,0.2687,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,134,0.2985,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,134,0.3284,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,134,0.3582,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,134,0.3881,0.91518,0.24448,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[56,134,0.4179,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[60,134,0.4478,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,134,0.4776,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[68,134,0.5075,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[72,134,0.5373,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[76,134,0.5672,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,134,0.597,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[84,134,0.6269,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[88,134,0.6567,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[92,134,0.6866,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[96,134,0.7164,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[100,134,0.7463,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,134,0.7761,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[108,134,0.806,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,134,0.8358,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[116,134,0.8657,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[120,134,0.8955,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[124,134,0.9254,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[128,134,0.9552,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[132,134,0.9851,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[134,134,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":4,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,110,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,110,0.0364,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,110,0.0727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,110,0.1091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,110,0.1455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,110,0.1818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,110,0.2182,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,110,0.2545,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,110,0.2909,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,110,0.3273,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,110,0.3636,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,110,0.4,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,110,0.4364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,110,0.4727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,110,0.5091,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,110,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,110,0.5818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,110,0.6182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,110,0.6545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,110,0.6909,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,110,0.7273,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[84,110,0.7636,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,110,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,110,0.8364,0.99553,0.02486,1.0,1.0,1.0,0.8571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,110,0.8727,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,110,0.9091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[104,110,0.9455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,110,0.9818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[110,110,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"98e0fdd44c8c544a","q":"Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.","t":[{"b":0,"e":1.0,"k":"rising","v":0.62054,"x":0.89286,"p":[[0,12,0.0,0.62054,0.23313,0.42857,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,8,0,0,8,0,0,6,0,0,1,0,6],[4,12,0.3333,0.71429,0.25505,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,4,0,0,8,0,0,5,0,0,2,0,11],[8,12,0.6667,0.89286,0.13832,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,19],[12,12,1.0,0.85268,0.15765,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,5,0,15]]},{"b":5,"e":0.71429,"k":"flat","v":0.63839,"x":0.83929,"p":[[0,31,0.0,0.63839,0.22299,0.53571,0.57143,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,6,0,0,12,0,0,4,0,0,2,0,6],[4,31,0.129,0.64284,0.20825,0.5354,0.57143,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,7,0,0,13,0,0,3,0,0,2,0,6],[8,31,0.2581,0.79463,0.16344,0.71429,0.71429,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,0,2,0,11],[12,31,0.3871,0.83929,0.13243,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,7,0,11],[16,31,0.5161,0.75892,0.12078,0.71429,0.71429,0.74996,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,0,3,0,5],[20,31,0.6452,0.79464,0.12846,0.71429,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,6,0,7],[24,31,0.7742,0.80357,0.15047,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,1,0,11],[28,31,0.9032,0.79464,0.15126,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,2,0,10],[31,31,1.0,0.76338,0.15408,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,15,0,0,2,0,8]]}]},{"i":"07b4299c31691873","q":"A nondegenerate triangle with perimeter  $1$  has side lengths  $a, b,$  and  $c$ . Prove that \\[\\left|\\frac{a - b}{c + ab}\\right| + \\left|\\frac{b - c}{a + bc}\\right| + \\left|\\frac{c - a}{b + ac}\\right| < 2.\\]\n*Proposed by Andrew Wen*","t":[{"b":1,"e":0.57143,"k":"falling","v":0.12501,"x":0.58473,"p":[[0,86,0.0,0.58473,0.31627,0.39286,0.57143,0.85714,0.0,1.0,2,7,0,2,0,4,0,0,2,0,0,4,0,0,5,0,0,6,0,0,2,0,7],[4,86,0.0465,0.20536,0.23128,0.0,0.14286,0.32143,0.0,0.85714,11,0,0,11,0,11,0,0,2,0,0,5,0,0,0,0,0,2,0,0,1,0,0],[8,86,0.093,0.17409,0.16647,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,12,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,86,0.1395,0.20089,0.21683,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,12,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[16,86,0.186,0.24098,0.28672,0.105,0.14286,0.28571,0.0,1.0,8,3,0,8,0,14,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[20,86,0.2326,0.25,0.24223,0.14286,0.14286,0.32143,0.0,1.0,6,1,0,6,0,15,0,0,3,0,0,1,0,0,5,0,0,1,0,0,0,0,1],[24,86,0.2791,0.20089,0.21683,0.10714,0.14286,0.2857,0.0,1.0,8,1,0,8,0,15,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[28,86,0.3256,0.20982,0.2448,0.10714,0.14286,0.28571,0.0,1.0,8,2,0,8,0,15,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[32,86,0.3721,0.23214,0.24678,0.10714,0.14286,0.28571,0.0,0.857,8,0,0,8,0,14,0,0,3,0,0,2,0,0,0,0,0,4,0,0,1,0,0],[36,86,0.4186,0.16964,0.17655,0.0,0.14286,0.17857,0.0,0.71429,10,0,0,10,0,14,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[40,86,0.4651,0.17857,0.19562,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,17,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[44,86,0.5116,0.18304,0.15663,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,17,0,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[48,86,0.5581,0.21875,0.2696,0.0,0.14286,0.32143,0.0,0.85714,13,0,0,13,0,9,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0],[52,86,0.6047,0.24106,0.24854,0.0,0.14286,0.32143,0.0,1.0,9,1,0,9,0,10,0,0,5,0,0,3,0,0,2,0,0,2,0,0,0,0,1],[56,86,0.6512,0.12501,0.09279,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,86,0.6977,0.20982,0.20511,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,13,0,0,6,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[64,86,0.7442,0.21875,0.21124,0.14286,0.14286,0.28571,0.0,0.85714,6,0,0,6,0,16,0,0,4,0,0,3,0,0,0,0,0,2,0,0,1,0,0],[68,86,0.7907,0.21429,0.19562,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,14,0,0,4,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[72,86,0.8372,0.30356,0.25189,0.14286,0.14288,0.57111,0.0,0.71429,6,0,0,6,0,11,0,0,2,0,0,4,0,0,4,0,0,5,0,0,0,0,0],[76,86,0.8837,0.23213,0.23889,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,13,0,0,4,0,0,2,0,0,1,0,0,3,0,0,1,0,0],[80,86,0.9302,0.20982,0.18552,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,16,0,0,4,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[84,86,0.9767,0.33479,0.29794,0.14286,0.28571,0.57111,0.0,1.0,7,2,0,7,0,8,0,0,4,0,0,3,0,0,5,0,0,2,0,0,1,0,2],[86,86,1.0,0.27232,0.26812,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,14,0,0,3,0,0,3,0,0,1,0,0,3,0,0,1,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.01339,"x":0.59375,"p":[[0,216,0.0,0.59375,0.34646,0.25,0.71429,0.85714,0.0,1.0,3,7,0,3,0,5,0,0,1,0,0,3,0,0,2,0,0,6,0,0,5,0,7],[4,216,0.0185,0.22759,0.1919,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,17,0,0,5,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[8,216,0.037,0.22321,0.2878,0.0,0.14286,0.32143,0.0,1.0,13,2,0,13,0,9,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,2],[12,216,0.0556,0.25,0.29451,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,13,0,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[16,216,0.0741,0.12946,0.20935,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,9,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,216,0.0926,0.14732,0.19719,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,17,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[24,216,0.1111,0.12054,0.17896,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,6,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[28,216,0.1296,0.13839,0.24868,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,8,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[32,216,0.1481,0.13393,0.2257,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,10,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[36,216,0.1667,0.13837,0.20035,0.0,0.0,0.2857,0.0,0.857,18,0,0,18,0,4,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[40,216,0.1852,0.11607,0.18707,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,8,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[44,216,0.2037,0.11159,0.24673,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[48,216,0.2222,0.09821,0.20652,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[52,216,0.2407,0.19196,0.25904,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,14,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[56,216,0.2593,0.21429,0.31135,0.0,0.14286,0.14286,0.0,1.0,13,3,0,13,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[60,216,0.2778,0.23214,0.2714,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,10,0,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[64,216,0.2963,0.31249,0.29544,0.14286,0.14286,0.57143,0.0,1.0,7,1,0,7,0,11,0,0,3,0,0,1,0,0,3,0,0,5,0,0,1,0,1],[68,216,0.3148,0.28124,0.27544,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,5,0,0,2,0,0,3,0,0,1,0,0,2,0,1],[72,216,0.3333,0.22767,0.31512,0.0,0.14286,0.17857,0.0,1.0,11,4,0,11,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[76,216,0.3519,0.28571,0.26,0.14286,0.14286,0.46429,0.0,1.0,5,1,0,5,0,14,0,0,3,0,0,2,0,0,5,0,0,1,0,0,1,0,1],[80,216,0.3704,0.25,0.28571,0.10714,0.14286,0.28571,0.0,1.0,8,3,0,8,0,12,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,3],[84,216,0.3889,0.3125,0.3223,0.10714,0.14286,0.57143,0.0,1.0,8,3,0,8,0,11,0,0,2,0,0,2,0,0,2,0,0,4,0,0,0,0,3],[88,216,0.4074,0.23661,0.19103,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,10,0,0,6,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[92,216,0.4259,0.30357,0.27837,0.14286,0.21428,0.42857,0.0,1.0,6,2,0,6,0,10,0,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,2],[96,216,0.4444,0.18304,0.18638,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,14,0,0,4,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[100,216,0.463,0.25,0.27199,0.0,0.14286,0.32143,0.0,1.0,11,1,0,11,0,7,0,0,6,0,0,2,0,0,1,0,0,4,0,0,0,0,1],[104,216,0.4815,0.20536,0.19212,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,12,0,0,6,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[108,216,0.5,0.29911,0.26332,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,17,0,0,4,0,0,3,0,0,1,0,0,3,0,0,0,0,2],[112,216,0.5185,0.22322,0.25985,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,13,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,2],[116,216,0.537,0.23214,0.17405,0.14286,0.21428,0.28571,0.0,0.71429,6,0,0,6,0,10,0,0,9,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[120,216,0.5556,0.25446,0.24932,0.14286,0.14286,0.42857,0.0,0.85714,7,0,0,7,0,14,0,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,0],[124,216,0.5741,0.19643,0.26184,0.0,0.14286,0.28571,0.0,1.0,12,2,0,12,0,11,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[128,216,0.5926,0.20982,0.24479,0.0,0.14286,0.2857,0.0,1.0,9,1,0,9,0,14,0,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,1],[132,216,0.6111,0.16518,0.18249,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,10,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[136,216,0.6296,0.28569,0.23416,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,7,0,0,5,0,0,9,0,0,2,0,0,1,0,0,0,0,1],[140,216,0.6481,0.27232,0.29528,0.0,0.14286,0.42857,0.0,1.0,10,2,0,10,0,8,0,0,5,0,0,3,0,0,0,0,0,4,0,0,0,0,2],[144,216,0.6667,0.28571,0.28347,0.14286,0.14286,0.42857,0.0,1.0,6,2,0,6,0,13,0,0,3,0,0,5,0,0,0,0,0,2,0,0,1,0,2],[148,216,0.6852,0.16518,0.26027,0.0,0.07143,0.14286,0.0,1.0,16,2,0,16,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[152,216,0.7037,0.25891,0.23537,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,5,0,0,5,0,0,1,0,0,2,0,0,0,0,1],[156,216,0.7222,0.1942,0.16774,0.125,0.14286,0.28571,0.0,0.57143,7,0,0,7,1,14,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[160,216,0.7407,0.23661,0.21609,0.10714,0.14286,0.42857,0.0,0.71429,8,0,0,8,0,11,0,0,3,0,0,7,0,0,0,0,0,3,0,0,0,0,0],[164,216,0.7593,0.22768,0.21387,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,9,0,0,10,0,0,2,0,0,0,0,0,2,0,0,1,0,0],[168,216,0.7778,0.25447,0.24932,0.14286,0.14286,0.32143,0.0,0.85714,6,0,0,6,0,14,0,0,4,0,0,4,0,0,0,0,0,1,0,0,3,0,0],[172,216,0.7963,0.16964,0.25614,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,11,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,1],[176,216,0.8148,0.09821,0.20341,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[180,216,0.8333,0.11607,0.25111,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[184,216,0.8519,0.09822,0.20025,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[188,216,0.8704,0.12054,0.18596,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,9,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[192,216,0.8889,0.15625,0.2212,0.0,0.07143,0.2857,0.0,1.0,16,1,0,16,0,6,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[196,216,0.9074,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[200,216,0.9259,0.12499,0.17764,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,7,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[204,216,0.9444,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[208,216,0.963,0.0625,0.18877,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[212,216,0.9815,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[216,216,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d6a2cf4f227fbf46","q":"Solve in  $ \\mathbb{Z}^2 $  the equation:  $ x^2\\left( 1+x^2 \\right) =-1+21^y. $ *Lucian Petrescu*","t":[{"b":4,"e":1.0,"k":"rising","v":0.38391,"x":1.0,"p":[[0,58,0.0,0.54018,0.33643,0.28571,0.28571,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,11],[4,58,0.069,0.5,0.32143,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[8,58,0.1379,0.47768,0.31055,0.28571,0.28571,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[12,58,0.2069,0.38391,0.23807,0.28571,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[16,58,0.2759,0.70089,0.36485,0.28571,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[20,58,0.3448,0.87947,0.26025,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[24,58,0.4138,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,58,0.4828,0.93304,0.2082,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[32,58,0.5517,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[36,58,0.6207,0.92411,0.21124,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[40,58,0.6897,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,58,0.7586,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[48,58,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,58,0.8966,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[56,58,0.9655,0.92857,0.20825,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[58,58,1.0,0.9375,0.18189,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,28]]},{"b":6,"e":1.0,"k":"rising","v":0.41964,"x":1.0,"p":[[0,66,0.0,0.55357,0.3458,0.28571,0.28571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[4,66,0.0606,0.41964,0.2788,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[8,66,0.1212,0.57589,0.35081,0.28571,0.28571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[12,66,0.1818,0.41964,0.2788,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,66,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,66,0.303,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,66,0.3636,0.93304,0.2082,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[28,66,0.4242,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,66,0.4848,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,66,0.5455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,66,0.6061,0.93304,0.2082,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[44,66,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,66,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,66,0.7879,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,66,0.8485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,66,0.9091,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[64,66,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[66,66,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"64e307e689e3f288","q":"Positive integers  $a, b, c$  satisfy  $\\frac{ab}{a - b} = c .$  what is the largest possible value of  $a+ b+ c$  not exceeding  $99$ ?","t":[{"b":3,"e":1.0,"k":"flat","v":0.92857,"x":0.95982,"p":[[0,5,0.0,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[4,5,0.8,0.95982,0.06423,0.85714,1.0,1.0,0.8571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[5,5,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.8571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]},{"b":4,"e":1.0,"k":"flat","v":0.88392,"x":0.9375,"p":[[0,6,0.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[4,6,0.6667,0.93312,0.07965,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[6,6,1.0,0.88392,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,9]]}]},{"i":"229ec1252f999d6b","q":"Prove that the number of 5 -tuples of positive integers $(a, b, c, d, e)$ satisfying the equation\n\n$$\na b c d e=5(b c d e+a c d e+a b d e+a b c e+a b c d)\n$$\n\nis an odd integer.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.95089,"x":0.98214,"p":[[0,35,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,35,0.1143,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,35,0.2286,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,35,0.3429,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,35,0.4571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,35,0.5714,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[24,35,0.6857,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[28,35,0.8,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[32,35,0.9143,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[35,35,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]},{"b":7,"e":1.0,"k":"rising","v":0.74554,"x":1.0,"p":[[0,62,0.0,0.74554,0.40679,0.57143,1.0,1.0,0.0,1.0,5,22,1,5,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,22],[4,62,0.0645,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[8,62,0.129,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,62,0.1935,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,62,0.2581,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[20,62,0.3226,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,62,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,62,0.4516,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,62,0.5161,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,62,0.5806,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,62,0.6452,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[44,62,0.7097,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[48,62,0.7742,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,62,0.8387,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,62,0.9032,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,62,0.9677,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[62,62,1.0,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28]]}]},{"i":"acd6a5f4a4cd42c9","q":"A point  $ P$  in the interior of triangle  $ ABC$  satisfies\r\n\r\n\\[ \\angle BPC \\minus{} \\angle BAC \\equal{} \\angle CPA \\minus{} \\angle CBA \\equal{} \\angle APB \\minus{} \\angle ACB.\\]\r\n\r\nProve that \\[ \\bar{PA} \\cdot \\bar{BC} \\equal{} \\bar{PB} \\cdot \\bar{AC} \\equal{} \\bar{PC} \\cdot \\bar{AB}.\\]","t":[{"b":5,"e":0.2857,"k":"flat","v":0.33473,"x":0.47767,"p":[[0,100,0.0,0.38839,0.25564,0.14286,0.35714,0.71429,0.14286,0.71429,0,0,0,0,0,15,0,0,1,0,0,5,0,0,0,0,0,11,0,0,0,0,0],[4,100,0.04,0.41515,0.17983,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,5,0,0,6,0,0,11,0,0,9,0,0,0,0,0,0,0,1],[8,100,0.08,0.44194,0.19677,0.39286,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,1,0,0,9,0,0,11,0,0,4,0,0,0,0,0],[12,100,0.12,0.4508,0.19934,0.39286,0.42857,0.57143,0.14,1.0,0,1,0,0,0,6,0,0,2,0,0,11,0,0,9,0,0,3,0,0,0,0,1],[16,100,0.16,0.45982,0.15458,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,2,0,0,11,0,0,13,0,0,2,0,0,0,0,0],[20,100,0.2,0.45536,0.1448,0.42857,0.5,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,2,0,0,10,0,0,16,0,0,0,0,0,0,0,0],[24,100,0.24,0.42412,0.18721,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,9,0,0,9,0,0,2,0,0,1,0,0],[28,100,0.28,0.47767,0.15814,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,4,0,0,7,0,0,15,0,0,3,0,0,0,0,0],[32,100,0.32,0.45533,0.17653,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,3,0,0,9,0,0,11,0,0,4,0,0,0,0,0],[36,100,0.36,0.45534,0.14479,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,3,0,0,13,0,0,11,0,0,2,0,0,0,0,0],[40,100,0.4,0.41515,0.17983,0.24999,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,2,0,0,8,0,0,13,0,0,1,0,0,0,0,0],[44,100,0.44,0.38392,0.15334,0.28571,0.42857,0.4642,0.14286,0.57143,0,0,0,0,0,7,0,0,4,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[48,100,0.48,0.39727,0.16644,0.2857,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,7,0,0,4,0,0,11,0,0,9,0,0,1,0,0,0,0,0],[52,100,0.52,0.4598,0.15864,0.42857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,15,0,0,11,0,0,0,0,0,0,0,1],[56,100,0.56,0.4375,0.16342,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,1,0,0,11,0,0,13,0,0,1,0,0,0,0,0],[60,100,0.6,0.41517,0.12035,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,4,0,0,18,0,0,7,0,0,0,0,0,0,0,0],[64,100,0.64,0.37948,0.15405,0.2857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,7,0,0,4,0,0,15,0,0,5,0,0,1,0,0,0,0,0],[68,100,0.68,0.41068,0.14169,0.39286,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,5,0,0,3,0,0,15,0,0,9,0,0,0,0,0,0,0,0],[72,100,0.72,0.37503,0.17035,0.24999,0.42857,0.42895,0.0,0.71429,1,0,0,1,0,7,0,0,3,0,0,14,0,0,6,0,0,1,0,0,0,0,0],[76,100,0.76,0.38837,0.17938,0.2857,0.42857,0.571,0.0,0.71429,1,0,0,1,0,6,0,0,5,0,0,11,0,0,7,0,0,2,0,0,0,0,0],[80,100,0.8,0.38393,0.16917,0.14286,0.42857,0.46431,0.14286,0.71429,0,0,0,0,0,9,0,0,1,0,0,14,0,0,7,0,0,1,0,0,0,0,0],[84,100,0.84,0.37054,0.1551,0.24999,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,8,0,0,3,0,0,16,0,0,4,0,0,1,0,0,0,0,0],[88,100,0.88,0.37054,0.13296,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,5,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[92,100,0.92,0.39283,0.14282,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,3,0,0,18,0,0,6,0,0,0,0,0,0,0,0],[96,100,0.96,0.33473,0.16612,0.14286,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,12,0,0,2,0,0,14,0,0,3,0,0,1,0,0,0,0,0],[100,100,1.0,0.36607,0.14698,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,2,0,0,21,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.25893,"x":0.47318,"p":[[0,110,0.0,0.39731,0.25934,0.14286,0.28571,0.71429,0.0,0.71429,1,0,0,1,0,12,0,0,4,0,0,2,0,0,2,0,0,11,0,0,0,0,0],[4,110,0.0364,0.43303,0.1838,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,5,0,0,8,0,0,11,0,0,3,0,0,0,0,0],[8,110,0.0727,0.45538,0.15333,0.39286,0.4293,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,5,0,0,9,0,0,13,0,0,2,0,0,0,0,0],[12,110,0.1091,0.42857,0.14286,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,3,0,0,15,0,0,9,0,0,1,0,0,0,0,0],[16,110,0.1455,0.47318,0.14029,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,10,0,0,14,0,0,2,0,0,0,0,0],[20,110,0.1818,0.42856,0.18557,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,7,0,0,12,0,0,1,0,0,1,0,0],[24,110,0.2182,0.41962,0.18875,0.14286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,0,0,0,9,0,0,12,0,0,2,0,0,0,0,0],[28,110,0.2545,0.43747,0.1554,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,4,0,0,12,0,0,10,0,0,2,0,0,0,0,0],[32,110,0.2909,0.40622,0.16406,0.28571,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,6,0,0,4,0,0,13,0,0,7,0,0,2,0,0,0,0,0],[36,110,0.3273,0.37946,0.18074,0.24999,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,3,0,0,15,0,0,3,0,0,3,0,0,0,0,0],[40,110,0.3636,0.38839,0.15251,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,5,0,0,3,0,0,16,0,0,7,0,0,0,0,0,0,0,0],[44,110,0.4,0.37935,0.16223,0.28571,0.42857,0.4642,0.0,0.57143,1,0,0,1,0,6,0,0,4,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[48,110,0.4364,0.33926,0.16265,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,1,0,0,14,0,0,5,0,0,0,0,0,0,0,0],[52,110,0.4727,0.35258,0.175,0.14286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,7,0,0,3,0,0,15,0,0,4,0,0,1,0,0,0,0,0],[56,110,0.5091,0.35268,0.16745,0.14289,0.42857,0.42858,0.0,0.57143,1,0,0,1,0,8,0,0,5,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[60,110,0.5455,0.38393,0.15746,0.2857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,6,0,0,5,0,0,16,0,0,4,0,0,0,0,0,1,0,0],[64,110,0.5818,0.33482,0.15407,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,5,0,0,14,0,0,2,0,0,1,0,0,0,0,0],[68,110,0.6182,0.30804,0.17896,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,13,0,0,4,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[72,110,0.6545,0.32594,0.1483,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,1,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[76,110,0.6909,0.30357,0.15047,0.14286,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,2,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[80,110,0.7273,0.33482,0.14987,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,10,0,0,5,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[84,110,0.7636,0.34375,0.16698,0.14286,0.42857,0.42858,0.0,0.71429,1,0,0,1,0,9,0,0,3,0,0,15,0,0,3,0,0,1,0,0,0,0,0],[88,110,0.8,0.2633,0.15619,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,15,0,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[92,110,0.8364,0.27205,0.14022,0.14286,0.21428,0.42857,0.14,0.57143,0,0,0,0,0,16,0,0,4,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[96,110,0.8727,0.29464,0.13803,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,13,0,0,5,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[100,110,0.9091,0.35714,0.12372,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,6,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[104,110,0.9455,0.33929,0.13717,0.14286,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,4,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[108,110,0.9818,0.25893,0.13092,0.14286,0.14286,0.42857,0.14286,0.42857,0,0,0,0,0,17,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[110,110,1.0,0.32588,0.1299,0.14286,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,9,0,0,6,0,0,16,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"8880ca26204afc9f","q":"The intersection of two squares with perimeter  $8$  is a rectangle with diagonal length  $1$ . Given that the distance between the centers of the two squares is  $2$ , the  perimeter of the rectangle can be expressed as  $P$ . Find  $10P$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.875,"x":0.98661,"p":[[0,25,0.0,0.875,0.21354,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,20],[4,25,0.16,0.89732,0.23483,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,26],[8,25,0.32,0.88839,0.22227,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,24],[12,25,0.48,0.93304,0.13825,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[16,25,0.64,0.91964,0.17835,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,26],[20,25,0.8,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,25,0.96,0.92857,0.17857,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[25,25,1.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]},{"b":5,"e":1.0,"k":"flat","v":0.85267,"x":0.91071,"p":[[0,8,0.0,0.85267,0.2612,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,0,3,0,22],[4,8,0.5,0.90625,0.21011,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,25],[8,8,1.0,0.91071,0.20124,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,24]]}]},{"i":"925bd3ee6f25278a","q":"The point  $D$  on the altitude  $AA_1$  of an acute triangle  $ABC$  is such that  $\\angle BDC=90^\\circ$ ;  $H$  is the orthocentre of  $ABC$ . A circle \nwith diameter  $AH$  is constructed. Prove that the tangent drawn from  $B$  \nto this circle is equal to  $BD$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.44643,"x":0.84816,"p":[[0,22,0.0,0.52679,0.29974,0.28571,0.50001,0.71429,0.0,1.0,2,7,0,2,0,0,0,0,11,0,0,3,0,0,7,0,0,2,0,0,0,0,7],[4,22,0.1818,0.44643,0.32093,0.28571,0.28571,0.57143,0.0,1.0,3,7,0,3,0,1,0,0,16,0,0,2,0,0,3,0,0,0,0,0,0,0,7],[8,22,0.3636,0.82584,0.22234,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,19],[12,22,0.5455,0.80356,0.26668,0.57143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,4,0,0,0,0,0,8,0,0,0,0,0,0,0,20],[16,22,0.7273,0.81695,0.26059,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,0,0,0,8,0,0,0,0,0,1,0,20],[20,22,0.9091,0.84816,0.21713,0.57143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,1,0,0,0,0,21],[22,22,1.0,0.82141,0.25507,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,0,0,0,8,0,0,1,0,0,2,0,19]]},{"b":4,"e":0.2857,"k":"flat","v":0.36155,"x":0.49102,"p":[[0,19,0.0,0.44196,0.3338,0.2857,0.28571,0.71429,0.0,1.0,5,6,0,5,0,0,0,0,15,0,0,1,0,0,2,0,0,2,0,0,1,0,6],[4,19,0.2105,0.48213,0.37415,0.2857,0.28571,1.0,0.0,1.0,5,9,0,5,0,2,0,0,12,0,0,0,0,0,2,0,0,1,0,0,1,0,9],[8,19,0.4211,0.49102,0.30915,0.2857,0.49979,0.71429,0.0,1.0,4,4,0,4,0,0,0,0,11,0,0,1,0,0,6,0,0,3,0,0,3,0,4],[12,19,0.6316,0.36155,0.22578,0.2857,0.28571,0.571,0.0,1.0,4,1,0,4,0,1,0,0,16,0,0,1,0,0,7,0,0,2,0,0,0,0,1],[16,19,0.8421,0.3839,0.21556,0.28571,0.28571,0.57111,0.0,0.85714,3,0,0,3,0,0,0,0,17,0,0,2,0,0,6,0,0,2,0,0,2,0,0],[19,19,1.0,0.44637,0.23349,0.28571,0.28571,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,16,0,0,2,0,0,9,0,0,1,0,0,0,0,3]]}]},{"i":"70b2295f684d2686","q":"Is it possible to find a set  $A$  of eleven positive integers such that no six elements of  $A$  have a sum which is divisible by  $6$ ?","t":[{"b":5,"e":0.14286,"k":"flat","v":0.13813,"x":0.4106,"p":[[0,19,0.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.4106,0.32494,0.14286,0.28571,0.71429,0.0,1.0,5,4,0,5,0,6,0,0,7,0,0,2,0,0,2,0,0,6,0,0,0,0,4],[8,19,0.4211,0.34822,0.28107,0.14286,0.2857,0.50002,0.0,1.0,3,1,0,3,0,12,0,0,5,0,0,4,0,0,0,0,0,5,0,0,2,0,1],[12,19,0.6316,0.24536,0.22092,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,7,0,0,3,0,0,1,0,0,0,0,0,1,0,1],[16,19,0.8421,0.16036,0.09286,0.14214,0.14286,0.1429,0.0,0.4286,4,0,0,4,0,21,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.13813,0.04349,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.12938,"x":0.2857,"p":[[0,33,0.0,0.16947,0.14917,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,33,0.1212,0.24991,0.27668,0.105,0.14286,0.28571,0.0,1.0,8,2,0,8,0,12,0,0,5,0,0,3,0,0,0,0,0,1,0,0,1,0,2],[8,33,0.2424,0.22322,0.19541,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,13,0,0,3,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[12,33,0.3636,0.2857,0.22586,0.14286,0.14288,0.42858,0.0,1.0,3,1,0,3,0,14,0,0,4,0,0,7,0,0,1,0,0,2,0,0,0,0,1],[16,33,0.4848,0.26775,0.17769,0.14286,0.1429,0.42858,0.0,0.71429,2,0,0,2,0,15,0,0,5,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[20,33,0.6061,0.21419,0.18899,0.14286,0.14286,0.14292,0.0,1.0,2,1,0,2,0,23,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[24,33,0.7273,0.18295,0.0961,0.14286,0.14286,0.1429,0.0,0.4286,1,0,0,1,0,24,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.14733,0.05629,0.14286,0.14286,0.1429,0.0,0.2857,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.13384,0.0612,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.12938,0.05484,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f090c5dad97730e7","q":"In an acute triangle  $ABC$  the line  $OI$  is parallel to side  $BC$ . Prove that the center of the nine-point circle of triangle  $ABC$  lies on the line  $MI$ , where  $M$  is the midpoint of  $BC$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.77674,"x":0.88837,"p":[[0,105,0.0,0.77674,0.19216,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,6,0,0,12,0,7],[4,105,0.0381,0.84375,0.16115,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,15,0,10],[8,105,0.0762,0.86605,0.09411,0.85714,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,21,0,7],[12,105,0.1143,0.84372,0.10333,0.85714,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,21,0,5],[16,105,0.1524,0.81693,0.10859,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,17,0,4],[20,105,0.1905,0.85936,0.09194,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,1,0,21,0,6],[24,105,0.2286,0.83703,0.10603,0.83918,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,1,0,19,0,5],[28,105,0.2667,0.87053,0.09689,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,8],[32,105,0.3048,0.84373,0.16117,0.85714,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,8],[36,105,0.3429,0.85264,0.11006,0.85711,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,19,0,7],[40,105,0.381,0.85266,0.11002,0.82132,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,16,0,8],[44,105,0.419,0.85713,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[48,105,0.4571,0.84374,0.13996,0.857,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,17,0,8],[52,105,0.4952,0.84375,0.11495,0.85714,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,22,0,5],[56,105,0.5333,0.84374,0.10329,0.82143,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,18,0,6],[60,105,0.5714,0.85267,0.16164,0.82132,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,12,0,12],[64,105,0.6095,0.86602,0.10075,0.85714,0.85714,0.89286,0.57,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,8],[68,105,0.6476,0.85711,0.1288,0.857,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,17,0,9],[72,105,0.6857,0.84821,0.15126,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,17,0,9],[76,105,0.7238,0.8616,0.11564,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,17,0,9],[80,105,0.7619,0.84373,0.1204,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,20,0,6],[84,105,0.8,0.87053,0.09006,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,19,0,8],[88,105,0.8381,0.82142,0.16366,0.82132,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,6,0,0,18,0,6],[92,105,0.8762,0.86159,0.16164,0.857,0.85714,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,11],[96,105,0.9143,0.875,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,9],[100,105,0.9524,0.85267,0.10402,0.85714,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,23,0,5],[104,105,0.9905,0.88837,0.10559,0.85714,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,19,0,11],[105,105,1.0,0.86607,0.09407,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,8]]},{"b":6,"e":0.85714,"k":"flat","v":0.77229,"x":0.89732,"p":[[0,185,0.0,0.79909,0.12298,0.71429,0.857,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,14,0,4],[4,185,0.0216,0.89732,0.08918,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12],[8,185,0.0432,0.84821,0.07936,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,22,0,4],[12,185,0.0649,0.83479,0.12434,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,21,0,5],[16,185,0.0865,0.84821,0.13803,0.85711,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,17,0,8],[20,185,0.1081,0.83482,0.20237,0.85714,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,18,0,9],[24,185,0.1297,0.83929,0.13716,0.82143,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,16,0,8],[28,185,0.1514,0.83926,0.12758,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,12,0,9],[32,185,0.173,0.83704,0.15491,0.85714,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,4,0,0,17,0,8],[36,185,0.1946,0.83936,0.09284,0.85711,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,21,0,4],[40,185,0.2162,0.80803,0.11071,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,20,0,2],[44,185,0.2378,0.83929,0.15465,0.85714,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,7],[48,185,0.2595,0.83926,0.13246,0.857,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,18,0,7],[52,185,0.2811,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,18,0,11],[56,185,0.3027,0.87946,0.14334,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,13,0,14],[60,185,0.3243,0.85266,0.09099,0.85714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 $ ABC $  be an isosceles triangle with  $ AC=BC $  . Take points  $ D $  on side  $AC$  and  $E$  on side  $BC$  and  $ F $  the intersection of bisectors of angles  $ DEB $  and  $ADE$  such that  $ F$  lies on side  $AB$ . Prove that  $F$  is the midpoint of  $AB$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.21428,"p":[[0,59,0.0,0.21428,0.27894,0.0,0.0,0.57143,0.0,1.0,18,1,2,18,0,1,0,0,4,0,0,0,0,0,8,0,0,0,0,0,0,0,1],[4,59,0.0678,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,59,0.1356,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,59,0.2034,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,59,0.2712,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,59,0.339,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,59,0.4068,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,59,0.6102,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,59,0.8136,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[59,59,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.2232,"p":[[0,97,0.0,0.2232,0.27183,0.0,0.0,0.57111,0.0,0.71429,18,0,0,18,0,1,0,0,1,0,0,3,0,0,7,0,0,2,0,0,0,0,0],[4,97,0.0412,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,97,0.0825,0.04464,0.1448,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,97,0.1237,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,97,0.1649,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,97,0.2062,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,97,0.2474,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,97,0.2887,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,97,0.3299,0.08482,0.23107,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[36,97,0.3711,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,97,0.4124,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,97,0.4536,0.07143,0.17496,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[48,97,0.4948,0.07588,0.22008,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[52,97,0.5361,0.05357,0.16656,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[56,97,0.5773,0.06695,0.20508,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,97,0.6186,0.02231,0.10163,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,97,0.6598,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,97,0.701,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,97,0.7423,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,97,0.7835,0.03571,0.11294,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,97,0.8247,0.07589,0.18893,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[84,97,0.866,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,97,0.9072,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,97,0.9485,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,97,0.9897,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[97,97,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"14ea1b1a5ea662b8","q":"Let  $ a\\in \\mathbb{R} $  and  $ f_1(x),f_2(x),\\ldots,f_n(x): \\mathbb{R} \\rightarrow \\mathbb{R} $  are the additive functions such that for every  $ x\\in \\mathbb{R} $  we have  $ f_1(x)f_2(x) \\cdots f_n(x) =ax^n $ . Show that there exists  $ b\\in \\mathbb {R} $  and  $ i\\in {\\{1,2,\\ldots,n}\\} $  such that for every  $ x\\in \\mathbb{R} $  we have  $ f_i(x)=bx $ .","t":[{"b":4,"e":1.0,"k":"rising","v":0.52232,"x":0.94643,"p":[[0,36,0.0,0.52232,0.29148,0.28571,0.42859,0.85714,0.0,1.0,2,3,0,2,0,2,0,0,7,0,0,6,0,0,4,0,0,2,0,0,6,0,3],[4,36,0.1111,0.94643,0.14174,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[8,36,0.2222,0.86606,0.22287,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,5,0,20],[12,36,0.3333,0.88839,0.20119,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,5,0,21],[16,36,0.4444,0.89286,0.2369,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,24],[20,36,0.5556,0.89732,0.19638,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,23],[24,36,0.6667,0.88393,0.24856,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,3,0,24],[28,36,0.7778,0.91071,0.18814,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,23],[32,36,0.8889,0.89286,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[36,36,1.0,0.84821,0.15126,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,18,0,9]]},{"b":7,"e":0.42857,"k":"flat","v":0.58926,"x":0.94643,"p":[[0,40,0.0,0.58926,0.30462,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,10,0,0,6,0,0,2,0,0,2,0,0,2,0,9],[4,40,0.1,0.89286,0.20825,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,22],[8,40,0.2,0.84821,0.17474,0.85714,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,16,0,11],[12,40,0.3,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[16,40,0.4,0.87946,0.14773,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,12,0,14],[20,40,0.5,0.85714,0.18211,0.82143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,14],[24,40,0.6,0.77679,0.26471,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,2,0,0,0,0,0,2,0,0,0,0,0,6,0,0,11,0,10],[28,40,0.7,0.85714,0.20516,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,11,0,15],[32,40,0.8,0.73212,0.25444,0.57132,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,5,0,0,11,0,7],[36,40,0.9,0.80804,0.2412,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,4,0,0,6,0,15],[40,40,1.0,0.59819,0.2299,0.53539,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,2,0,0,12,0,0,5,0,0,4,0,3]]}]},{"i":"16651200f4d1212d","q":"In an acute triangle  $ABC$ , let  $D$  be a point in  $BC$  such that  $AD$  is the angle bisector of  $\\angle{BAC}$ . Let  $E \\neq B$  be the point of intersection of the circumcircle of triangle  $ABD$  with the line perpendicular to  $AD$  drawn through  $B$ . Let  $O$  be the circumcenter of triangle  $ABC$ . Prove that  $E$ ,  $O$ , and  $A$  are collinear.","t":[{"b":4,"e":0.57143,"k":"flat","v":0.32588,"x":0.71873,"p":[[0,90,0.0,0.5491,0.35911,0.2857,0.42859,1.0,0.0,1.0,4,9,1,4,0,0,0,0,10,0,0,4,0,0,0,0,0,2,0,0,3,0,9],[4,90,0.0444,0.44196,0.33381,0.2857,0.28571,0.75,0.0,1.0,6,4,0,6,0,1,0,0,10,0,0,3,0,0,3,0,0,1,0,0,4,0,4],[8,90,0.0889,0.32588,0.24283,0.2857,0.28571,0.28571,0.0,1.0,6,1,0,6,0,0,0,0,19,0,0,0,0,0,3,0,0,2,0,0,1,0,1],[12,90,0.1333,0.40178,0.32229,0.24999,0.28571,0.60714,0.0,1.0,7,2,0,7,0,1,0,0,12,0,0,0,0,0,4,0,0,1,0,0,5,0,2],[16,90,0.1778,0.43303,0.36331,0.10714,0.28571,0.75,0.0,1.0,8,5,0,8,0,1,0,0,10,0,0,0,0,0,2,0,0,3,0,0,3,0,5],[20,90,0.2222,0.45089,0.35734,0.2857,0.28571,0.85714,0.0,1.0,7,5,0,7,0,0,0,0,12,0,0,0,0,0,2,0,0,2,0,0,4,0,5],[24,90,0.2667,0.57588,0.40796,0.25,0.64286,1.0,0.0,1.0,6,13,0,6,0,2,0,0,6,0,0,0,0,0,2,0,0,2,0,0,1,0,13],[28,90,0.3111,0.51338,0.38937,0.2857,0.28571,1.0,0.0,1.0,5,11,0,5,0,2,0,0,11,0,0,0,0,0,1,0,0,2,0,0,0,0,11],[32,90,0.3556,0.50893,0.3443,0.28571,0.28571,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,13,0,0,1,0,0,1,0,0,4,0,0,2,0,7],[36,90,0.4,0.55357,0.40994,0.2857,0.35716,1.0,0.0,1.0,6,13,0,6,0,1,0,0,9,0,0,1,0,0,0,0,0,1,0,0,1,0,13],[40,90,0.4444,0.52232,0.36876,0.28571,0.35714,1.0,0.0,1.0,5,9,0,5,0,0,0,0,11,0,0,2,0,0,1,0,0,2,0,0,2,0,9],[44,90,0.4889,0.54018,0.3792,0.2857,0.35716,1.0,0.0,1.0,5,10,0,5,0,0,0,0,11,0,0,1,0,0,2,0,0,0,0,0,3,0,10],[48,90,0.5333,0.40625,0.35555,0.24999,0.28571,0.75,0.0,1.0,7,6,0,7,0,1,0,0,14,0,0,1,0,0,0,0,0,1,0,0,2,0,6],[52,90,0.5778,0.5,0.40406,0.21427,0.28571,1.0,0.0,1.0,8,10,0,8,0,0,0,0,9,0,0,1,0,0,1,0,0,1,0,0,2,0,10],[56,90,0.6222,0.54464,0.41563,0.2857,0.28571,1.0,0.0,1.0,7,12,0,7,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,3,0,12],[60,90,0.6667,0.4241,0.32827,0.2857,0.28571,0.53568,0.0,1.0,4,6,0,4,0,0,0,0,19,0,0,1,0,0,0,0,0,0,0,0,2,0,6],[64,90,0.7111,0.43303,0.37027,0.24999,0.28571,0.85714,0.0,1.0,6,7,0,6,0,2,0,0,14,0,0,0,0,0,0,0,0,0,0,0,3,0,7],[68,90,0.7556,0.49999,0.34441,0.28571,0.28571,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,16,0,0,1,0,0,1,0,0,1,0,0,2,0,8],[72,90,0.8,0.36607,0.35703,0.0,0.28571,0.60714,0.0,1.0,10,5,0,10,0,0,0,0,13,0,0,0,0,0,1,0,0,1,0,0,2,0,5],[76,90,0.8444,0.41071,0.34947,0.2857,0.28571,0.53571,0.0,1.0,6,7,0,6,0,0,0,0,17,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[80,90,0.8889,0.45535,0.34706,0.2857,0.28571,0.89275,0.0,1.0,4,8,0,4,0,0,0,0,17,0,0,2,0,0,0,0,0,0,0,0,1,0,8],[84,90,0.9333,0.35259,0.33315,0.14214,0.28571,0.28571,0.0,1.0,7,6,0,7,0,2,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[88,90,0.9778,0.71873,0.25874,0.53539,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,6,0,0,5,0,10],[90,90,1.0,0.6562,0.24448,0.53539,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,3,0,0,1,0,0,4,0,0,4,0,0,10,0,0,6,0,4]]},{"b":6,"e":0.42857,"k":"falling","v":0.04018,"x":0.75891,"p":[[0,165,0.0,0.59374,0.33524,0.28571,0.4998,1.0,0.0,1.0,3,10,0,3,0,0,0,0,6,0,0,7,0,0,1,0,0,4,0,0,1,0,10],[4,165,0.0242,0.52678,0.3489,0.28571,0.42857,0.85714,0.0,1.0,4,6,0,4,0,1,0,0,11,0,0,0,0,0,2,0,0,3,0,0,5,0,6],[8,165,0.0485,0.32587,0.26055,0.24999,0.28571,0.42857,0.0,1.0,7,1,0,7,0,1,0,0,15,0,0,2,0,0,3,0,0,1,0,0,2,0,1],[12,165,0.0727,0.36161,0.3174,0.24999,0.28571,0.42857,0.0,1.0,7,4,0,7,0,1,0,0,15,0,0,2,0,0,1,0,0,0,0,0,2,0,4],[16,165,0.097,0.58929,0.36201,0.28571,0.71429,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,10,0,0,0,0,0,1,0,0,2,0,0,7,0,8],[20,165,0.1212,0.47768,0.35644,0.28571,0.28571,0.85714,0.0,1.0,6,6,0,6,0,1,0,0,10,0,0,0,0,0,4,0,0,2,0,0,3,0,6],[24,165,0.1455,0.49999,0.39609,0.24999,0.28571,0.89286,0.0,1.0,7,8,0,7,0,1,0,0,10,0,0,0,0,0,0,0,0,1,0,0,5,0,8],[28,165,0.1697,0.50446,0.3214,0.2857,0.35714,0.75,0.0,1.0,2,7,0,2,0,2,0,0,12,0,0,2,0,0,4,0,0,2,0,0,1,0,7],[32,165,0.1939,0.42411,0.34531,0.2857,0.28571,0.60714,0.0,1.0,7,6,0,7,0,0,0,0,12,0,0,2,0,0,3,0,0,1,0,0,1,0,6],[36,165,0.2182,0.52678,0.36672,0.28571,0.35714,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,11,0,0,1,0,0,2,0,0,1,0,0,4,0,8],[40,165,0.2424,0.40624,0.32164,0.10714,0.35716,0.57143,0.0,1.0,8,2,0,8,0,2,0,0,6,0,0,1,0,0,9,0,0,0,0,0,4,0,2],[44,165,0.2667,0.54464,0.37191,0.28571,0.42857,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,10,0,0,2,0,0,1,0,0,0,0,0,6,0,8],[48,165,0.2909,0.75891,0.34152,0.49968,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,0,0,20],[52,165,0.3152,0.56249,0.36235,0.28571,0.4998,1.0,0.0,1.0,4,9,0,4,0,1,0,0,8,0,0,3,0,0,2,0,0,1,0,0,4,0,9],[56,165,0.3394,0.66071,0.36202,0.28571,0.78571,1.0,0.0,1.0,3,14,0,3,0,0,0,0,8,0,0,1,0,0,1,0,0,3,0,0,2,0,14],[60,165,0.3636,0.5,0.36422,0.28571,0.28571,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,11,0,0,1,0,0,2,0,0,2,0,0,2,0,8],[64,165,0.3879,0.48214,0.3859,0.24999,0.28571,0.85714,0.0,1.0,7,7,0,7,0,1,0,0,10,0,0,1,0,0,0,0,0,1,0,0,5,0,7],[68,165,0.4121,0.57589,0.37199,0.28571,0.64286,1.0,0.0,1.0,5,11,0,5,0,0,0,0,8,0,0,1,0,0,2,0,0,5,0,0,0,0,11],[72,165,0.4364,0.57143,0.36596,0.28571,0.64286,1.0,0.0,1.0,5,9,0,5,0,0,0,0,8,0,0,1,0,0,2,0,0,4,0,0,3,0,9],[76,165,0.4606,0.51784,0.31894,0.28571,0.35714,0.75,0.0,1.0,2,7,0,2,0,1,0,0,13,0,0,1,0,0,4,0,0,3,0,0,1,0,7],[80,165,0.4848,0.4598,0.35307,0.2857,0.28571,0.85714,0.0,1.0,6,6,0,6,0,0,0,0,13,0,0,0,0,0,3,0,0,1,0,0,3,0,6],[84,165,0.5091,0.54462,0.33775,0.2857,0.42857,1.0,0.0,1.0,2,9,0,2,0,1,0,0,12,0,0,2,0,0,4,0,0,0,0,0,2,0,9],[88,165,0.5333,0.48214,0.36727,0.24999,0.28571,0.85714,0.0,1.0,6,7,0,6,0,2,0,0,9,0,0,0,0,0,3,0,0,3,0,0,2,0,7],[92,165,0.5576,0.47766,0.33618,0.2857,0.35714,0.75,0.0,1.0,5,5,0,5,0,1,0,0,10,0,0,2,0,0,3,0,0,3,0,0,3,0,5],[96,165,0.5818,0.41516,0.31411,0.2857,0.28571,0.60714,0.0,1.0,7,2,0,7,0,0,0,0,11,0,0,1,0,0,5,0,0,2,0,0,4,0,2],[100,165,0.6061,0.5,0.35535,0.28571,0.28571,0.89286,0.0,1.0,4,8,0,4,0,2,0,0,11,0,0,1,0,0,2,0,0,3,0,0,1,0,8],[104,165,0.6303,0.41071,0.3567,0.14286,0.28571,0.71429,0.0,1.0,7,6,0,7,0,2,0,0,12,0,0,1,0,0,0,0,0,3,0,0,1,0,6],[108,165,0.6545,0.49991,0.34079,0.28571,0.42857,0.85714,0.0,1.0,5,4,0,5,0,1,0,0,10,0,0,0,0,0,2,0,0,5,0,0,5,0,4],[112,165,0.6788,0.40178,0.31224,0.14286,0.28571,0.71429,0.0,1.0,7,2,0,7,0,2,0,0,9,0,0,1,0,0,4,0,0,5,0,0,2,0,2],[116,165,0.703,0.45088,0.34276,0.24999,0.35714,0.60714,0.0,1.0,6,6,0,6,0,2,0,0,8,0,0,2,0,0,6,0,0,1,0,0,1,0,6],[120,165,0.7273,0.40625,0.25533,0.28571,0.28571,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,18,0,0,2,0,0,4,0,0,0,0,0,3,0,2],[124,165,0.7515,0.49999,0.37115,0.25,0.42836,0.85714,0.0,1.0,6,6,0,6,0,2,0,0,8,0,0,0,0,0,4,0,0,0,0,0,6,0,6],[128,165,0.7758,0.67855,0.34256,0.39286,0.857,1.0,0.0,1.0,3,11,0,3,0,1,0,0,4,0,0,2,0,0,1,0,0,4,0,0,6,0,11],[132,165,0.8,0.43752,0.26228,0.2857,0.42857,0.57143,0.0,1.0,3,2,0,3,0,3,0,0,8,0,0,4,0,0,8,0,0,3,0,0,1,0,2],[136,165,0.8242,0.51783,0.34021,0.24999,0.57143,0.85704,0.0,1.0,5,4,0,5,0,3,0,0,4,0,0,1,0,0,7,0,0,2,0,0,6,0,4],[140,165,0.8485,0.55356,0.36899,0.28571,0.57143,1.0,0.0,1.0,5,9,0,5,0,1,0,0,7,0,0,2,0,0,3,0,0,2,0,0,3,0,9],[144,165,0.8727,0.56696,0.35443,0.2857,0.71429,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,6,0,0,3,0,0,0,0,0,5,0,0,5,0,7],[148,165,0.897,0.62497,0.31492,0.5354,0.64286,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,2,0,0,2,0,0,8,0,0,6,0,0,2,0,8],[152,165,0.9212,0.5223,0.31054,0.28571,0.49979,0.85704,0.0,1.0,2,4,0,2,0,2,0,0,11,0,0,1,0,0,3,0,0,4,0,0,5,0,4],[156,165,0.9455,0.46427,0.34809,0.0,0.57143,0.71429,0.0,1.0,9,2,0,9,0,2,0,0,1,0,0,0,0,0,9,0,0,4,0,0,5,0,2],[160,165,0.9697,0.10714,0.23958,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[164,165,0.9939,0.11161,0.24415,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[165,165,1.0,0.04018,0.10853,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b3ebec82843b7739","q":"Let  $(x_n)$  be a sequence of positive integers defined as follows:  $x_1$  is a fixed six-digit number and for any  $n \\geq 1$ ,  $x_{n+1}$  is a prime divisor of  $x_n + 1$ . Find  $x_{19} + x_{20}$ .","t":[{"b":3,"e":0.571,"k":"flat","v":0.68296,"x":0.91071,"p":[[0,53,0.0,0.81249,0.17657,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,8,0,11],[4,53,0.0755,0.90177,0.13095,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,13,0,16],[8,53,0.1509,0.86606,0.12343,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,19,0,9],[12,53,0.2264,0.84373,0.14448,0.857,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,17,0,8],[16,53,0.3019,0.85713,0.14725,0.85711,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,15,0,11],[20,53,0.3774,0.82138,0.16372,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,14,0,9],[24,53,0.4528,0.91071,0.1171,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,14,0,16],[28,53,0.5283,0.79906,0.16703,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,13,0,7],[32,53,0.6038,0.8348,0.2086,0.82132,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,12,0,12],[36,53,0.6792,0.79911,0.20159,0.67857,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,9,0,11],[40,53,0.7547,0.76786,0.1915,0.71429,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,10,0,7],[44,53,0.8302,0.75891,0.24338,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,3,0,0,11,0,9],[48,53,0.9057,0.74997,0.21726,0.57132,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,0,6,0,10],[52,53,0.9811,0.68296,0.24677,0.571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,3,0,0,9,0,0,5,0,0,5,0,7],[53,53,1.0,0.70532,0.26712,0.571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,3,0,0,6,0,0,5,0,0,4,0,10]]},{"b":6,"e":0.4286,"k":"flat","v":0.75,"x":0.83035,"p":[[0,39,0.0,0.7723,0.21388,0.71429,0.85714,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,12,0,7],[4,39,0.1026,0.75,0.20825,0.67857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,9,0,0,7,0,8],[8,39,0.2051,0.79911,0.17807,0.67857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,11,0,9],[12,39,0.3077,0.78122,0.18895,0.57143,0.85707,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,9,0,9],[16,39,0.4103,0.83034,0.15337,0.71429,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,14,0,9],[20,39,0.5128,0.79017,0.2082,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,12,0,9],[24,39,0.6154,0.79908,0.12304,0.71429,0.85714,0.85714,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,20,0,2],[28,39,0.7179,0.83035,0.18013,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,0,11,0,11],[32,39,0.8205,0.77229,0.28764,0.57143,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,0,7,0,14],[36,39,0.9231,0.78125,0.18893,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,15,0,6],[39,39,1.0,0.76783,0.20749,0.71429,0.857,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,11,0,6]]}]},{"i":"0b9b29f2b3eb6243","q":"The triangle $A B C$ is right-angled at $A$. Let $M$ be the midpoint of the segment $B C$. The point $D$ lies on the side $A C$ and satisfies $\\overline{A D}=\\overline{A M}$. The intersection point of the circumcircles of triangles $A M C$ and $B D C$, different from $C$, is called $P$. Prove that $C P$ bisects the angle at $C$ of triangle $A B C$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.08928,"x":0.16964,"p":[[0,32,0.0,0.14732,0.15355,0.0,0.07143,0.2857,0.0,0.42857,16,0,0,16,0,1,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.09821,0.1357,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.16517,0.15198,0.0,0.2857,0.2857,0.0,0.42857,14,0,0,14,0,1,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.125,0.15872,0.0,0.0,0.28571,0.0,0.4286,19,0,0,19,0,1,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.13839,0.14934,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,3,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.16964,0.1448,0.0,0.2857,0.28571,0.0,0.4286,13,0,0,13,0,1,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.16518,0.15612,0.0,0.21428,0.28571,0.0,0.42857,14,0,0,14,0,2,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.15625,0.1488,0.0,0.2857,0.28571,0.0,0.42857,15,0,0,15,0,0,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.08928,0.13243,0.0,0.0,0.17857,0.0,0.42857,21,0,0,21,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.10268,"x":0.19196,"p":[[0,29,0.0,0.125,0.15047,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,2,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.14277,0.15152,0.0,0.07,0.28571,0.0,0.4286,16,0,0,16,0,2,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.19196,0.1411,0.0,0.2857,0.28571,0.0,0.4286,9,0,0,9,0,6,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.14723,0.14501,0.0,0.14143,0.28571,0.0,0.42857,15,0,0,15,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.19196,0.15815,0.0,0.2857,0.28571,0.0,0.4286,11,0,0,11,0,4,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.16965,0.16918,0.0,0.14286,0.28571,0.0,0.4286,14,0,0,14,0,4,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.14732,0.16554,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,1,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.10268,0.1394,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,2,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.11161,0.14166,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,2,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3214ee2565145308","q":"Let  $ABC$  a triangle. Let  $D$  be a point on the circumcircle of this triangle and let  $E , F$  be the feet of the perpendiculars from  $A$  on  $DB, DC$ , respectively. Finally, let  $N$  be the midpoint of  $EF$ . Let  $M \\ne N$  be the midpoint of  the side  $BC$  . Prove that the lines  $NA$  and  $NM$  are perpendicular.","t":[{"b":3,"e":0.0,"k":"falling","v":0.28125,"x":0.88838,"p":[[0,88,0.0,0.71874,0.29121,0.5354,0.71429,1.0,0.0,1.0,2,13,1,2,0,0,0,0,1,0,0,5,0,0,2,0,0,9,0,0,0,0,13],[4,88,0.0455,0.88838,0.24675,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,22],[8,88,0.0909,0.83035,0.32427,0.85708,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,20],[12,88,0.1364,0.84821,0.28107,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,21],[16,88,0.1818,0.74106,0.35613,0.67857,0.85714,1.0,0.0,1.0,4,14,0,4,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,9,0,14],[20,88,0.2273,0.61606,0.39194,0.25,0.85714,1.0,0.0,1.0,6,10,0,6,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,0,7,0,10],[24,88,0.2727,0.59375,0.43757,0.0,0.78564,1.0,0.0,1.0,10,14,0,10,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,14],[28,88,0.3182,0.52679,0.38206,0.0,0.64286,0.78571,0.0,1.0,9,8,0,9,0,0,0,0,1,0,0,4,0,0,2,0,0,8,0,0,0,0,8],[32,88,0.3636,0.54688,0.44557,0.0,0.78571,1.0,0.0,1.0,11,11,0,11,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,1,11],[36,88,0.4091,0.49553,0.44029,0.0,0.71429,0.85714,0.0,1.0,13,7,0,13,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,8,0,7],[40,88,0.4545,0.52679,0.45096,0.0,0.57144,1.0,0.0,1.0,12,13,0,12,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,13],[44,88,0.5,0.65625,0.40699,0.32143,0.85714,1.0,0.0,1.0,8,13,0,8,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,5,0,13],[48,88,0.5455,0.66294,0.42266,0.0,0.85714,1.0,0.0,1.0,9,13,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,7,0,13],[52,88,0.5909,0.62053,0.43243,0.0,0.85707,1.0,0.0,1.0,10,13,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,13],[56,88,0.6364,0.61607,0.45378,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,15],[60,88,0.6818,0.85266,0.2934,0.85711,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,21],[64,88,0.7273,0.73213,0.36202,0.71429,0.85714,1.0,0.0,1.0,5,13,0,5,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,13],[68,88,0.7727,0.62946,0.40384,0.21427,0.78564,1.0,0.0,1.0,8,11,0,8,0,0,0,0,2,0,0,0,0,0,0,0,0,6,0,0,5,0,11],[72,88,0.8182,0.58481,0.43501,0.0,0.85714,1.0,0.0,1.0,10,10,0,10,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,10],[76,88,0.8636,0.71429,0.37287,0.67857,0.85714,1.0,0.0,1.0,6,12,0,6,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,10,0,12],[80,88,0.9091,0.35714,0.43595,0.0,0.0,0.85714,0.0,1.0,19,4,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,4],[84,88,0.9545,0.28125,0.38213,0.0,0.0,0.71429,0.0,1.0,19,2,0,19,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,2],[88,88,1.0,0.38838,0.42443,0.0,0.0,0.74996,0.0,1.0,17,5,0,17,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,5]]},{"b":4,"e":0.0,"k":"falling","v":0.0625,"x":0.94642,"p":[[0,107,0.0,0.69195,0.31966,0.5354,0.71429,1.0,0.0,1.0,3,12,0,3,0,1,0,0,1,0,0,3,0,0,1,0,0,11,0,0,0,0,12],[4,107,0.0374,0.91518,0.18851,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,22],[8,107,0.0748,0.83482,0.26027,0.82132,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,0,7,0,17],[12,107,0.1121,0.88838,0.21941,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,22],[16,107,0.1495,0.94642,0.09943,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[20,107,0.1869,0.86605,0.26473,0.857,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,22],[24,107,0.2243,0.79463,0.34615,0.85711,1.0,1.0,0.0,1.0,4,18,0,4,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,8,0,18],[28,107,0.2617,0.71428,0.40406,0.42857,1.0,1.0,0.0,1.0,7,17,0,7,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,17],[32,107,0.2991,0.82142,0.31135,0.85711,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,7,0,19],[36,107,0.3364,0.73213,0.369,0.71421,0.85714,1.0,0.0,1.0,6,14,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,14],[40,107,0.3738,0.82142,0.30304,0.857,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,18],[44,107,0.4112,0.66963,0.42773,0.0,0.85714,1.0,0.0,1.0,9,15,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,15],[48,107,0.4486,0.67856,0.41032,0.32143,0.85714,1.0,0.0,1.0,8,14,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,14],[52,107,0.486,0.49998,0.45033,0.0,0.64264,1.0,0.0,1.0,13,10,0,13,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,10],[56,107,0.5234,0.58482,0.43938,0.0,0.85707,1.0,0.0,1.0,11,11,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,11],[60,107,0.5607,0.79017,0.35711,0.85714,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,18],[64,107,0.5981,0.6116,0.44925,0.0,0.85714,1.0,0.0,1.0,11,13,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,13],[68,107,0.6355,0.57142,0.41342,0.10714,0.71429,1.0,0.0,1.0,8,10,0,8,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,0,5,0,10],[72,107,0.6729,0.55357,0.43995,0.0,0.85714,1.0,0.0,1.0,11,9,0,11,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,9],[76,107,0.7103,0.68303,0.38587,0.60714,0.85714,1.0,0.0,1.0,7,10,0,7,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,11,0,10],[80,107,0.7477,0.55355,0.4161,0.0,0.71429,0.89286,0.0,1.0,10,8,0,10,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,7,0,8],[84,107,0.785,0.52678,0.42021,0.0,0.85707,0.85714,0.0,1.0,10,5,0,10,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,12,0,5],[88,107,0.8224,0.66071,0.42068,0.0,0.85714,1.0,0.0,1.0,9,12,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,12],[92,107,0.8598,0.49552,0.44317,0.0,0.71429,0.85714,0.0,1.0,14,6,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,6],[96,107,0.8972,0.70534,0.38785,0.67835,0.85714,1.0,0.0,1.0,7,13,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,13],[100,107,0.9346,0.72767,0.3542,0.71429,0.85714,1.0,0.0,1.0,4,14,0,4,0,2,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,14],[104,107,0.972,0.78571,0.33312,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,17],[107,107,1.0,0.0625,0.2141,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"7abe0233681d354b","q":"It is said that a sequence $\\left(u_{n}\\right)_{n \\geqslant 0}$ of non-zero natural numbers is Sicilian if $^{-10}$\n\n$$\nu_{n+1} \\in\\left\\{u_{n} / 2, u_{n} / 3,2 u_{n}+1,3 u_{n}+1\\right\\}\n$$\n\nfor all $n \\geqslant 0$. Prove that, for every integer $k \\geqslant 1$, there exists a Sicilian sequence $\\left(u_{n}\\right)_{n \\geqslant 0}$ and an integer $\\ell \\geqslant 0$ such that $u_{0}=k$ and $u_{\\ell}=1$.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.20982,"x":0.30357,"p":[[0,71,0.0,0.20982,0.10092,0.14286,0.2857,0.28571,0.0,0.28571,4,0,3,4,0,9,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.26785,0.07784,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,71,0.1127,0.27678,0.04971,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,71,0.169,0.30357,0.08564,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,26,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,71,0.2254,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,4,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,71,0.2817,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,71,0.338,0.29017,0.05629,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.25893,0.07523,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,5,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,71,0.4507,0.25893,0.06621,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,7,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.28571,0.08748,0.2857,0.28571,0.28571,0.0,0.57143,1,0,1,1,0,2,0,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,71,0.5634,0.27232,0.07457,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,71,0.6197,0.28125,0.06667,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,4,0,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.25,0.07143,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.28571,0.08748,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[56,71,0.7887,0.27678,0.06121,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,4,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.25446,0.06901,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.25446,0.09932,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,8,0,0,21,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[68,71,0.9577,0.24553,0.1197,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,7,0,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[71,71,1.0,0.25446,0.08552,0.24999,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,7,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.21875,"x":0.2991,"p":[[0,60,0.0,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.25893,0.06621,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.28125,0.08364,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,5,0,0,24,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,60,0.3333,0.2991,0.05486,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,60,0.4,0.26339,0.06298,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.26786,0.06916,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,6,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.28571,0.07143,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.2633,0.10789,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,6,0,0,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,60,0.7333,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.27678,0.06121,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.28572,0.03571,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.27223,0.04192,0.28571,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.29017,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b8ad7d41ae72c0ab","q":"Let  $ABC$  be a right-angled triangle with the right angle at  $B$  and circumcircle  $c$ . Denote by  $D$  the midpoint of the shorter arc  $AB$  of  $c$ . Let  $P$  be the point on the side  $AB$  such that  $CP=CD$  and let  $X$  and  $Y$  be two distinct points on  $c$  satisfying  $AX=AY=PD$ . Prove that  $X, Y$  and  $P$  are collinear.\n\n*Proposed by Dominik Burek, Poland*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.84818,"x":0.97768,"p":[[0,57,0.0,0.87946,0.20858,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,20],[4,57,0.0702,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,57,0.1404,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,57,0.2105,0.95535,0.11539,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[16,57,0.2807,0.91515,0.14671,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,5,0,22],[20,57,0.3509,0.93304,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[24,57,0.4211,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[28,57,0.4912,0.94196,0.10013,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[32,57,0.5614,0.94196,0.12807,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,24],[36,57,0.6316,0.92857,0.15152,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,23],[40,57,0.7018,0.90179,0.15746,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,6,0,20],[44,57,0.7719,0.94641,0.12246,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[48,57,0.8421,0.92411,0.15561,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[52,57,0.9123,0.94197,0.13295,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26],[56,57,0.9825,0.90179,0.22141,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[57,57,1.0,0.84818,0.24209,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,2,0,20]]},{"b":6,"e":1.0,"k":"flat","v":0.84817,"x":0.98659,"p":[[0,82,0.0,0.91516,0.14226,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[4,82,0.0488,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,82,0.0976,0.94195,0.17809,1.0,1.0,1.0,0.1429,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[12,82,0.1463,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,82,0.1951,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[20,82,0.2439,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[24,82,0.2927,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[28,82,0.3415,0.93749,0.15545,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[32,82,0.3902,0.95535,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[36,82,0.439,0.92857,0.18898,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,27],[40,82,0.4878,0.94196,0.12807,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[44,82,0.5366,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[48,82,0.5854,0.97321,0.06625,1.0,1.0,1.0,0.714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[52,82,0.6341,0.94196,0.19186,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[56,82,0.6829,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,21],[60,82,0.7317,0.89728,0.15259,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,20],[64,82,0.7805,0.875,0.15046,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,7,0,16],[68,82,0.8293,0.84819,0.19543,0.82132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,8,0,16],[72,82,0.878,0.8973,0.13943,0.85711,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,18],[76,82,0.9268,0.84817,0.20502,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,5,0,17],[80,82,0.9756,0.91071,0.14617,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,21],[82,82,1.0,0.89286,0.18898,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,19]]}]},{"i":"ac334953c2d2cabf","q":"Let  $ABCD$  be a trapezoid, where  $AB$  and  $CD$  are parallel. Let  $P$  be a point on the  side  $BC$ . Show that the parallels to  $AP$  and  $PD$  intersect through  $C$  and  $B$  to  $DA$ , respectively.","t":[{"b":2,"e":0.2857,"k":"flat","v":0.3124,"x":0.4375,"p":[[0,12,0.0,0.3124,0.22433,0.14286,0.1429,0.42857,0.0,0.85714,2,0,0,2,0,15,0,0,0,0,0,10,0,0,1,0,0,3,0,0,1,0,0],[4,12,0.3333,0.33036,0.18707,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,1,0,0,15,0,0,0,0,0,3,0,0,0,0,0],[8,12,0.6667,0.31687,0.15466,0.14286,0.35714,0.42857,0.14,0.71429,0,0,0,0,0,12,0,0,4,0,0,14,0,0,1,0,0,1,0,0,0,0,0],[12,12,1.0,0.4375,0.20183,0.42857,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,4,0,0,3,0,0,21,0,0,0,0,0,1,0,0,1,0,2]]},{"b":6,"e":0.42857,"k":"flat","v":0.30796,"x":0.38382,"p":[[0,17,0.0,0.35269,0.23685,0.14286,0.42857,0.42857,0.0,1.0,2,1,0,2,0,11,0,0,1,0,0,13,0,0,1,0,0,2,0,0,1,0,1],[4,17,0.2353,0.32143,0.15152,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,12,0,0,2,0,0,17,0,0,0,0,0,1,0,0,0,0,0],[8,17,0.4706,0.37938,0.1977,0.24999,0.42857,0.4286,0.14,1.0,0,1,0,0,0,8,0,0,5,0,0,15,0,0,0,0,0,3,0,0,0,0,1],[12,17,0.7059,0.34374,0.18849,0.14286,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,6,0,0,13,0,0,1,0,0,1,0,0,0,0,1],[16,17,0.9412,0.30796,0.18603,0.14286,0.2857,0.4286,0.0,0.71429,1,0,0,1,0,14,0,0,2,0,0,11,0,0,2,0,0,2,0,0,0,0,0],[17,17,1.0,0.38382,0.22149,0.14286,0.42857,0.4286,0.14,1.0,0,1,0,0,0,10,0,0,4,0,0,11,0,0,3,0,0,2,0,0,1,0,1]]}]},{"i":"457bef0d636653ae","q":"In an acute-angled triangle $A B C$, a point $D$ lies on the segment $B C$. Let $O_{1}, O_{2}$ denote the circumcentres of triangles $A B D$ and $A C D$, respectively. Prove that the line joining the circumcentre of triangle $A B C$ and the orthocentre of triangle $\\mathrm{O}_{1} \\mathrm{O}_{2} \\mathrm{D}$ is parallel to $B C$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.12501,"x":0.21875,"p":[[0,79,0.0,0.19188,0.11075,0.14286,0.1429,0.28571,0.0,0.42857,5,0,0,5,0,12,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,79,0.0506,0.21875,0.08736,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,11,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,79,0.1013,0.16062,0.10566,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,79,0.1519,0.16071,0.1171,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,10,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,79,0.2025,0.1384,0.10404,0.0,0.14286,0.17857,0.0,0.286,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,79,0.2532,0.12938,0.09005,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,79,0.3038,0.14278,0.09451,0.14214,0.14286,0.14286,0.0,0.286,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,79,0.3544,0.15625,0.09688,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,79,0.4051,0.16964,0.09061,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,18,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,79,0.4557,0.15616,0.10327,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,79,0.5063,0.1383,0.10998,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,79,0.557,0.12501,0.09943,0.0,0.14286,0.1429,0.0,0.28571,10,0,0,10,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,79,0.6076,0.15179,0.08702,0.14286,0.14286,0.1429,0.0,0.28571,5,0,0,5,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,79,0.6582,0.1517,0.09407,0.14286,0.14286,0.1786,0.0,0.28571,6,0,0,6,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,79,0.7089,0.14733,0.11,0.0,0.14286,0.2857,0.0,0.286,9,0,0,9,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,79,0.7595,0.14286,0.09449,0.14286,0.14286,0.1429,0.0,0.28571,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,79,0.8101,0.13839,0.0977,0.10714,0.14286,0.1429,0.0,0.28571,8,0,0,8,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,79,0.8608,0.14286,0.09449,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,79,0.9114,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,79,0.962,0.12937,0.10325,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[79,79,1.0,0.1384,0.10404,0.0,0.14286,0.1786,0.0,0.286,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.11161,"x":0.20982,"p":[[0,105,0.0,0.16062,0.10566,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,105,0.0381,0.18303,0.10248,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,13,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,105,0.0762,0.17411,0.09932,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,15,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,105,0.1143,0.15625,0.11495,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,11,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,105,0.1524,0.13384,0.10676,0.0,0.14286,0.1786,0.0,0.28571,10,0,0,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,105,0.1905,0.15625,0.09688,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,105,0.2286,0.18304,0.08171,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,105,0.2667,0.1383,0.08363,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,105,0.3048,0.15178,0.11811,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,10,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,105,0.3429,0.16071,0.09942,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,105,0.381,0.16518,0.08072,0.14286,0.14286,0.1786,0.0,0.28571,3,0,0,3,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,105,0.419,0.17857,0.09449,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,105,0.4571,0.15625,0.09006,0.14286,0.14286,0.1786,0.0,0.28571,5,0,0,5,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,105,0.4952,0.13393,0.10062,0.0,0.14286,0.14287,0.0,0.28571,9,0,0,9,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,105,0.5333,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,105,0.5714,0.13831,0.06666,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,105,0.6095,0.12054,0.08073,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,105,0.6476,0.13393,0.10677,0.0,0.14286,0.17857,0.0,0.28571,10,0,0,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,105,0.6857,0.20518,0.10688,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,11,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,105,0.7238,0.20982,0.08736,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,13,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,105,0.7619,0.16518,0.1017,0.14286,0.14286,0.2857,0.0,0.28571,6,0,0,6,0,15,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,105,0.8,0.1383,0.10998,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,105,0.8381,0.14286,0.07986,0.14286,0.14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 $ABC$  be a triangle such that in its interior there exists a point  $D$  with  $\\angle DAC = \\angle DCA = 30^o$  and  $ \\angle DBA = 60^o$ . Denote  $E$  the midpoint of the segment  $BC$ , and take  $F$  on the segment  $AC$  so that  $AF = 2FC$ . Prove that  $DE \\perp EF$ 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 $E$  be an interior point of the convex quadrilateral  $ABCD$ . Construct triangles  $\\triangle ABF,\\triangle BCG,\\triangle CDH$  and  $\\triangle DAI$  on the outside of the quadrilateral such that the similarities  $\\triangle ABF\\sim\\triangle DCE,\\triangle BCG\\sim \\triangle ADE,\\triangle CDH\\sim\\triangle BAE$  and  $ \\triangle DAI\\sim\\triangle CBE$  hold. Let  $P,Q,R$  and  $S$  be the projections of  $E$  on the lines  $AB,BC,CD$  and  $DA$ , respectively. Prove that if the quadrilateral  $PQRS$  is cyclic, then\n\\[EF\\cdot CD=EG\\cdot DA=EH\\cdot AB=EI\\cdot BC.\\]","t":[{"b":0,"e":0.14286,"k":"flat","v":0.0424,"x":0.22759,"p":[[0,82,0.0,0.17857,0.11845,0.10714,0.21428,0.28571,0.0,0.28571,8,0,0,8,0,8,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,82,0.0488,0.19643,0.1171,0.14286,0.2857,0.28571,0.0,0.4286,6,0,0,6,0,9,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,82,0.0976,0.19643,0.09942,0.14286,0.21428,0.28571,0.0,0.28571,4,0,0,4,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,82,0.1463,0.17857,0.11845,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,11,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,82,0.1951,0.21652,0.09696,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,1,11,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,82,0.2439,0.20536,0.09406,0.14286,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,82,0.2927,0.19196,0.09852,0.14286,0.14288,0.28571,0.0,0.28571,4,0,0,4,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,82,0.3415,0.22759,0.12305,0.14286,0.2857,0.28571,0.0,0.4286,5,0,0,5,0,6,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,82,0.3902,0.16518,0.09523,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,82,0.439,0.15392,0.10731,0.105,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,13,0,1,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,82,0.4878,0.14732,0.10999,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,82,0.5366,0.1875,0.11538,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,16,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,82,0.5854,0.15178,0.11258,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,82,0.6341,0.16955,0.1154,0.14214,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,13,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,82,0.6829,0.15393,0.098,0.14214,0.14286,0.2857,0.0,0.28571,6,0,0,6,1,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,82,0.7317,0.11152,0.1114,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,82,0.7805,0.09822,0.09063,0.0,0.14286,0.14286,0.0,0.286,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,82,0.8293,0.10929,0.1056,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,1,15,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,82,0.878,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.08018,0.09395,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.10268,0.10853,0.0,0.14286,0.14287,0.0,0.28571,15,0,0,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.0424,0.07332,0.0,0.0,0.08896,0.0,0.28571,23,0,0,23,1,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.08482,"x":0.1875,"p":[[0,55,0.0,0.1741,0.15865,0.0,0.14286,0.28571,0.0,0.71429,11,0,1,11,0,6,0,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,55,0.0727,0.15839,0.10676,0.14214,0.14286,0.2857,0.0,0.42857,6,0,0,6,1,16,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.15178,0.11258,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.14286,0.10714,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.1875,0.10374,0.14286,0.1429,0.28571,0.0,0.28571,5,0,0,5,0,12,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.08929,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.10705,0.08745,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.14277,0.11294,0.0,0.14286,0.17857,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.11607,0.09062,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.10268,0.07563,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,1,19,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.14733,0.0977,0.14286,0.14286,0.1786,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.09143,0.09814,0.0,0.10571,0.14286,0.0,0.28571,15,0,0,15,1,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.08483,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"21478e44cee66382","q":"Let  $I$  be the incenter of triangle  $ABC$ . The circle of centre  $A$  and radius  $AI$  intersects the circumcircle of triangle  $ABC$  in  $M$  and  $N$ . Prove that the line  $MN$  is tangent to the incircle of triangle  $ABC$","t":[{"b":3,"e":0.14286,"k":"falling","v":0.16517,"x":0.83926,"p":[[0,94,0.0,0.8214,0.1637,0.71429,0.85707,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,5,0,12],[4,94,0.0426,0.75891,0.20959,0.67857,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,7,0,0,11,0,0,4,0,9],[8,94,0.0851,0.7857,0.23421,0.71429,0.78564,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,0,0,0,5,0,0,9,0,0,3,0,13],[12,94,0.1277,0.77679,0.29867,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,1,0,16],[16,94,0.1702,0.81695,0.22655,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,2,0,16],[20,94,0.2128,0.75889,0.22432,0.67857,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,0,0,0,6,0,0,9,0,0,6,0,9],[24,94,0.2553,0.67411,0.31589,0.57143,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,1,0,0,1,0,0,5,0,0,10,0,0,1,0,10],[28,94,0.2979,0.81696,0.23753,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,14],[32,94,0.3404,0.73212,0.26905,0.67857,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,2,0,0,0,0,0,4,0,0,9,0,0,6,0,9],[36,94,0.383,0.69196,0.30116,0.57143,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,0,0,0,2,0,0,4,0,0,9,0,0,4,0,9],[40,94,0.4255,0.83926,0.27143,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,0,2,0,21],[44,94,0.4681,0.7232,0.33682,0.57143,0.85714,1.0,0.0,1.0,4,13,0,4,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,5,0,13],[48,94,0.5106,0.65625,0.34043,0.53572,0.71429,1.0,0.0,1.0,5,10,0,5,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,0,2,0,10],[52,94,0.5532,0.70535,0.33491,0.57143,0.71429,1.0,0.0,1.0,4,13,0,4,0,0,0,0,2,0,0,0,0,0,4,0,0,7,0,0,2,0,13],[56,94,0.5957,0.61606,0.3984,0.25,0.71429,1.0,0.0,1.0,7,12,0,7,0,1,0,0,2,0,0,0,0,0,4,0,0,3,0,0,3,0,12],[60,94,0.6383,0.68302,0.38088,0.28571,0.85714,1.0,0.0,1.0,5,14,0,5,0,1,0,0,3,0,0,0,0,0,1,0,0,4,0,0,4,0,14],[64,94,0.6809,0.64284,0.40089,0.28571,0.78564,1.0,0.0,1.0,7,15,0,7,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,0,1,0,15],[68,94,0.7234,0.65625,0.36919,0.5,0.71429,1.0,0.0,1.0,6,11,0,6,0,0,0,0,2,0,0,0,0,0,3,0,0,6,0,0,4,0,11],[72,94,0.766,0.53124,0.41069,0.14286,0.57121,1.0,0.0,1.0,7,11,0,7,0,3,0,0,5,0,0,0,0,0,3,0,0,1,0,0,2,0,11],[76,94,0.8085,0.58928,0.3989,0.24999,0.71429,1.0,0.0,1.0,7,12,0,7,0,1,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,12],[80,94,0.8511,0.6875,0.37362,0.57143,0.85714,1.0,0.0,1.0,6,13,0,6,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,5,0,13],[84,94,0.8936,0.57587,0.34531,0.28571,0.57143,0.85714,0.0,1.0,5,7,0,5,0,0,0,0,6,0,0,0,0,0,6,0,0,4,0,0,4,0,7],[88,94,0.9362,0.41071,0.35129,0.0,0.42859,0.71429,0.0,1.0,10,3,0,10,0,2,0,0,3,0,0,2,0,0,6,0,0,3,0,0,3,0,3],[92,94,0.9787,0.16517,0.22896,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,4,0,0,3,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[94,94,1.0,0.23658,0.21308,0.0,0.2857,0.42857,0.0,0.57143,11,0,0,11,0,4,0,0,8,0,0,3,0,0,6,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"falling","v":0.45084,"x":0.87945,"p":[[0,76,0.0,0.7991,0.15093,0.71429,0.78564,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,14,0,0,9,0,7],[4,76,0.0526,0.80356,0.24937,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,1,0,17],[8,76,0.1053,0.76338,0.2826,0.71429,0.78571,1.0,0.0,1.0,3,12,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,4,0,12],[12,76,0.1579,0.73213,0.24419,0.71429,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,4,0,8],[16,76,0.2105,0.7366,0.29257,0.57143,0.71429,1.0,0.0,1.0,3,12,0,3,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,3,0,12],[20,76,0.2632,0.79464,0.25238,0.71429,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,5,0,13],[24,76,0.3158,0.72321,0.32525,0.57143,0.85714,1.0,0.0,1.0,4,12,0,4,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,5,0,12],[28,76,0.3684,0.6875,0.30606,0.57143,0.71429,1.0,0.0,1.0,3,10,0,3,0,0,0,0,2,0,0,1,0,0,7,0,0,5,0,0,4,0,10],[32,76,0.4211,0.69643,0.25692,0.71429,0.71429,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,5,0,5],[36,76,0.4737,0.66962,0.3102,0.57143,0.71429,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,1,0,0,0,0,0,8,0,0,6,0,0,5,0,8],[40,76,0.5263,0.72098,0.28255,0.57143,0.71429,1.0,0.0,1.0,2,11,0,2,0,0,0,1,1,0,0,0,0,0,7,0,0,7,0,0,3,0,11],[44,76,0.5789,0.78125,0.26722,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,2,0,15],[48,76,0.6316,0.72766,0.23788,0.67857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,5,0,7],[52,76,0.6842,0.76337,0.21314,0.67857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,10,0,0,4,0,10],[56,76,0.7368,0.80355,0.16658,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,4,0,11],[60,76,0.7895,0.73659,0.27919,0.57143,0.85707,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,2,0,0,6,0,0,4,0,0,6,0,11],[64,76,0.8421,0.69195,0.21163,0.57132,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,7,0,0,7,0,5],[68,76,0.8947,0.58034,0.26949,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,3,0,0,2,0,0,5,0,0,9,0,0,5,0,0,2,0,5],[72,76,0.9474,0.87945,0.19922,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,4,0,21],[76,76,1.0,0.45084,0.31155,0.14286,0.4998,0.71429,0.0,1.0,5,3,0,5,0,4,0,0,5,0,0,2,0,0,6,0,0,6,0,0,1,0,3]]}]},{"i":"c202f67054b0da19","q":"Let  $P^{*}$  be the set of primes less than  $10000$ . Find all possible primes  $p\\in P^{*}$  such that for each subset  $S=\\{p_{1},p_{2},...,p_{k}\\}$  of  $P^{*}$  with  $k\\geq 2$  and each  $p\\not\\in S$ , there is a  $q\\in P^{*}-S$  such that  $q+1$  divides  $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.53568,"x":0.81247,"p":[[0,30,0.0,0.61606,0.31428,0.57143,0.64286,0.85714,0.0,1.0,5,5,2,5,0,0,0,0,1,0,0,0,0,0,10,0,0,5,0,0,6,0,5],[4,30,0.1333,0.63393,0.43877,0.0,0.92857,1.0,0.0,1.0,9,16,0,9,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,16],[8,30,0.2667,0.60268,0.44426,0.0,0.85714,1.0,0.0,1.0,10,14,0,10,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,14],[12,30,0.4,0.53568,0.4374,0.0,0.71429,1.0,0.0,1.0,12,10,0,12,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,10],[16,30,0.5333,0.70087,0.37006,0.5713,0.78571,1.0,0.0,1.0,6,15,0,6,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,1,0,15],[20,30,0.6667,0.81247,0.20961,0.71429,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,1,0,14],[24,30,0.8,0.66962,0.3102,0.42857,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,3,0,0,6,0,0,4,0,0,4,0,0,1,0,12],[28,30,0.9333,0.73667,0.27462,0.5354,0.857,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,5,0,0,3,0,0,4,0,0,5,0,12],[30,30,1.0,0.59371,0.25282,0.42857,0.42859,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,14,0,0,4,0,0,3,0,0,1,0,7]]},{"b":2,"e":0.0,"k":"falling","v":0.00447,"x":0.78119,"p":[[0,40,0.0,0.43304,0.3416,0.0,0.57143,0.71429,0.0,1.0,10,2,2,10,0,0,0,0,4,0,0,1,0,0,6,0,0,6,0,0,3,0,2],[4,40,0.1,0.78119,0.29015,0.571,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,4,0,16],[8,40,0.2,0.5625,0.44022,0.0,0.71429,1.0,0.0,1.0,11,12,0,11,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,12],[12,40,0.3,0.55355,0.4267,0.0,0.71429,1.0,0.0,1.0,11,10,1,11,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,10],[16,40,0.4,0.34373,0.39424,0.0,0.0,0.71429,0.0,1.0,17,4,0,17,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,2,0,4],[20,40,0.5,0.45534,0.44669,0.0,0.42835,1.0,0.0,1.0,14,10,0,14,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,10],[24,40,0.6,0.4598,0.41914,0.0,0.4286,1.0,0.0,1.0,12,9,0,12,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,0,1,0,9],[28,40,0.7,0.05357,0.1915,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[32,40,0.8,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4d671535aff02e9c","q":"Let  $X_1,X_2,..$  be independent random variables with the same distribution, and let  $S_n=X_1+X_2+...+X_n, n=1,2,...$ . For what real numbers  $c$  is the following statement true:  $$ P\\left(\\left| \\frac{S_{2n}}{2n}- c \\right| \\leqslant \\left| \\frac{S_n}{n}-c\\right| \\right)\\geqslant \\frac{1}{2} $$","t":[{"b":0,"e":1.0,"k":"rising","v":0.66516,"x":1.0,"p":[[0,42,0.0,0.66516,0.27108,0.57142,0.57143,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,5,0,0,0,0,0,12,0,0,1,0,0,5,0,8],[4,42,0.0952,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[8,42,0.1905,0.9375,0.12846,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,24],[12,42,0.2857,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[16,42,0.381,0.92411,0.14279,0.85714,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,23],[20,42,0.4762,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[24,42,0.5714,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[28,42,0.6667,0.87054,0.20628,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,9,0,18],[32,42,0.7619,0.89284,0.1786,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,7,0,20],[36,42,0.8571,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,42,0.9524,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":2,"e":0.85714,"k":"rising","v":0.58481,"x":0.97321,"p":[[0,16,0.0,0.58481,0.28428,0.28571,0.57143,0.75,0.0,1.0,1,7,1,1,0,2,0,0,6,0,0,0,0,0,13,0,0,2,0,0,1,0,7],[4,16,0.25,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[8,16,0.5,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,16,0.75,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,16,1.0,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26]]}]},{"i":"88f59c495e363baa","q":"A sequence  $a_1,a_2,...,a_{2007}$  where  $a_i \\in\\{2,3\\}$  for  $i = 1,2,...,2007$  and an integer sequence  $x_1,x_2,...,x_{2007}$  satis\ffies the following:  $a_ix_i + x_{i+2 }\\equiv 0$  ( $mod 5$ ) , where the indices are taken modulo  $2007$ . Prove that  $x_1,x_2,...,x_{2007}$  are all multiples of  $5$ .","t":[{"b":1,"e":0.6,"k":"flat","v":0.91517,"x":0.99554,"p":[[0,56,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,56,0.0714,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,56,0.1429,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[12,56,0.2143,0.92411,0.13825,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,23],[16,56,0.2857,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[20,56,0.3571,0.91517,0.13767,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,20],[24,56,0.4286,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[28,56,0.5,0.9375,0.11812,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[32,56,0.5714,0.94286,0.12556,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,6,0,24],[36,56,0.6429,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[40,56,0.7143,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[44,56,0.7857,0.93304,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[48,56,0.8571,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[52,56,0.9286,0.95981,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[56,56,1.0,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23]]},{"b":6,"e":0.57143,"k":"volatile","v":0.61601,"x":0.97767,"p":[[0,8,0.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,8,0.5,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[8,8,1.0,0.61601,0.16536,0.571,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,19,0,0,1,0,0,3,0,3]]}]},{"i":"fcd677ae8133332a","q":"Let  $X$  be a variable point on the side  $BC$  of a triangle  $ABC$ . Let  $B'$  and  $C'$  be points on the rays  $[XB$  and  $[XC$ , respectively, satisfying  $B'X=BC=C'X$ . The line passing through  $X$  and parallel to  $AB'$  cuts the line  $AC$  at  $Y$  and the line passing through  $X$  and parallel to  $AC'$  cuts the line  $AB$  at  $Z$ . Prove that all lines  $YZ$  pass through a fixed point as  $X$  varies on the line segment  $BC$ 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 $ABCD$  be a cyclic quadrilateral with circumcircle  $\\omega$  and let  $AC$  and  $BD$  intersect at  $X$ . Let the line through  $A$  parallel to  $BD$  intersect line  $CD$  at  $E$  and  $\\omega$   at  $Y \\ne A$ . If  $AB = 10, AD = 24, XA = 17$ , and  $XB = 21$ , then the area of  $\\vartriangle DEY$  can be written in simplest form as  $\\frac{m}{n}$  . Find  $m + n$ 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 $\\Gamma_{1}$  be a circle.  $AB$  is a diameter,  $\\ell$  is the tangent at  $B$ , and  $M$  is a point on  $\\Gamma_{1}$  other than  $A$ .  $\\Gamma_{2}$  is a circle tangent to  $\\ell$ , and also to  $\\Gamma_{1}$  at  $M$ .\r\n\r\na) Determine the point of tangency  $P$  of  $\\ell$  and  $\\Gamma_{2}$  and find the locus of the center of  $\\Gamma_{2}$  as  $M$  varies.\r\n\r\nb) Show that there exists a circle that is always orthogonal to  $\\Gamma_{2}$ , regardless of the position of  $M$ .","t":[{"b":2,"e":0.571,"k":"flat","v":0.57585,"x":0.69196,"p":[[0,44,0.0,0.69196,0.14334,0.71429,0.71429,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,4,0,0,23,0,0,4,0,0],[4,44,0.0909,0.63838,0.13826,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,11,0,0,2,0,1],[8,44,0.1818,0.63389,0.10065,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,14,0,0,1,0,0],[12,44,0.2727,0.61161,0.07349,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,10,0,0,0,0,0],[16,44,0.3636,0.60709,0.13833,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,21,0,0,8,0,0,2,0,0],[20,44,0.4545,0.63833,0.09443,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,0,3,0,0],[24,44,0.5455,0.62496,0.08567,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,8,0,0,2,0,0],[28,44,0.6364,0.59812,0.07527,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,2,0,0,2,0,0],[32,44,0.7273,0.58029,0.04972,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0],[36,44,0.8182,0.57585,0.0563,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0],[40,44,0.9091,0.58478,0.06537,0.5713,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0,0,0,0],[44,44,1.0,0.58922,0.05925,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"flat","v":0.56695,"x":0.67854,"p":[[0,101,0.0,0.67854,0.14288,0.71429,0.71429,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,6,0,0,22,0,0,3,0,0],[4,101,0.0396,0.65624,0.10013,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,11,0,0,4,0,0],[8,101,0.0792,0.62051,0.09183,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,8,0,0,2,0,0],[12,101,0.1188,0.60713,0.06187,0.57143,0.57143,0.60714,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0],[16,101,0.1584,0.60258,0.05911,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0],[20,101,0.198,0.61603,0.07526,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0],[24,101,0.2376,0.60268,0.06902,0.57143,0.57143,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,8,0,0,0,0,0],[28,101,0.2772,0.62942,0.07876,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,0,1,0,0],[32,101,0.3168,0.56695,0.11564,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,1,0,0,26,0,0,4,0,0,0,0,0],[36,101,0.3564,0.58029,0.04972,0.57142,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0],[40,101,0.396,0.6071,0.06188,0.57143,0.57143,0.60714,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0],[44,101,0.4356,0.58028,0.0346,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[48,101,0.4752,0.59372,0.06299,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0],[52,101,0.5149,0.5714,0.10713,0.57143,0.57143,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,2,0,0,2,0,0,23,0,0,4,0,0,1,0,0],[56,101,0.5545,0.57582,0.11564,0.5714,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,26,0,0,5,0,0,0,0,0],[60,101,0.5941,0.58922,0.04727,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0],[64,101,0.6337,0.59367,0.06301,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,6,0,0,0,0,0],[68,101,0.6733,0.6071,0.07143,0.57143,0.57143,0.57143,0.571,0.857,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,6,0,0,1,0,0],[72,101,0.7129,0.5892,0.06918,0.57142,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,3,0,0,1,0,0],[76,101,0.7525,0.5714,0.03571,0.57143,0.57143,0.57143,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0],[80,101,0.7921,0.5803,0.0346,0.57143,0.57143,0.57143,0.57,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[84,101,0.8317,0.59365,0.06301,0.5714,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0],[88,101,0.8713,0.58475,0.04166,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0],[92,101,0.9109,0.58922,0.05925,0.57143,0.57143,0.57143,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,2,0,0,1,0,0],[96,101,0.9505,0.58028,0.0346,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[100,101,0.9901,0.58459,0.04126,0.57142,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0],[101,101,1.0,0.5983,0.06657,0.57143,0.57143,0.57143,0.5714,0.86,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,4,0,0,1,0,0]]}]},{"i":"5907370db35e4358","q":"Let  $a, \\ b \\in \\mathbb{Z_{+}}$ . Denote  $f(a, b)$  the number sequences  $s_1, \\ s_2, \\ ..., \\ s_a$ ,  $s_i \\in \\mathbb{Z}$  such that  $|s_1|+|s_2|+...+|s_a| \\le b$ . Show that  $f(a, b)=f(b, a)$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.98661,"x":0.99554,"p":[[0,5,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,5,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[5,5,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,32,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,32,0.125,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,32,0.25,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,32,0.375,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"574d6c545268049e","q":"Let  $a$  and  $b$  be positive integers,  $b<a$ , such that  $a^3+b^3+ab$  is divisible by  $ab(a-b)$ . Prove that  $ab$  is a perfect cube.","t":[{"b":5,"e":0.0,"k":"falling","v":0.11161,"x":0.81695,"p":[[0,131,0.0,0.66518,0.27574,0.57143,0.71429,0.78571,0.0,1.0,1,8,0,1,0,3,0,0,1,0,0,1,0,0,5,0,0,13,0,0,0,0,8],[4,131,0.0305,0.81695,0.22372,0.71429,0.85714,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,0,0,16],[8,131,0.0611,0.70982,0.28456,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,2,0,0,0,0,0,1,0,0,1,0,0,16,0,0,0,0,10],[12,131,0.0916,0.67634,0.30722,0.57143,0.71429,1.0,0.0,1.0,2,10,0,2,1,2,0,0,0,0,0,1,0,0,4,0,0,12,0,0,0,0,10],[16,131,0.1221,0.59366,0.38161,0.14286,0.71429,1.0,0.0,1.0,6,10,0,6,0,3,0,0,1,0,0,0,0,0,3,0,0,8,0,0,1,0,10],[20,131,0.1527,0.63839,0.39283,0.35714,0.71429,1.0,0.0,1.0,6,14,0,6,0,2,0,0,0,0,0,3,0,0,1,0,0,6,0,0,0,0,14],[24,131,0.1832,0.66071,0.38755,0.46429,0.71429,1.0,0.0,1.0,6,14,0,6,0,2,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,14],[28,131,0.2137,0.54911,0.36789,0.14286,0.71429,0.75,0.0,1.0,7,7,0,7,0,2,0,0,1,0,0,2,0,0,2,0,0,10,0,0,1,0,7],[32,131,0.2443,0.45518,0.35603,0.0,0.64071,0.71429,0.0,1.0,9,3,0,9,0,4,0,0,0,0,0,0,0,0,3,0,0,13,0,0,0,0,3],[36,131,0.2748,0.47321,0.40789,0.0,0.57143,0.85714,0.0,1.0,10,7,0,10,0,4,0,0,0,0,0,2,0,0,0,0,0,7,0,0,2,0,7],[40,131,0.3053,0.33482,0.39867,0.0,0.14286,0.71429,0.0,1.0,14,6,0,14,0,6,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,6],[44,131,0.3359,0.50892,0.35703,0.10714,0.57143,0.71429,0.0,1.0,8,6,0,8,0,1,0,0,0,0,0,5,0,0,5,0,0,6,0,0,1,0,6],[48,131,0.3664,0.35259,0.36072,0.0,0.14286,0.71429,0.0,1.0,11,3,0,11,0,7,0,0,1,0,0,0,0,0,1,0,0,9,0,0,0,0,3],[52,131,0.3969,0.36161,0.39445,0.0,0.14286,0.71429,0.0,1.0,14,5,0,14,0,4,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,5],[56,131,0.4275,0.39732,0.33069,0.0,0.5,0.71429,0.0,1.0,10,1,0,10,0,4,0,0,0,0,0,2,0,0,3,0,0,12,0,0,0,0,1],[60,131,0.458,0.39284,0.35535,0.0,0.21429,0.71429,0.0,1.0,9,3,0,9,0,7,0,0,1,0,0,0,0,0,2,0,0,10,0,0,0,0,3],[64,131,0.4885,0.34375,0.36919,0.0,0.14286,0.71429,0.0,1.0,13,4,0,13,0,5,0,0,0,0,0,1,0,0,4,0,0,5,0,0,0,0,4],[68,131,0.5191,0.27676,0.31929,0.0,0.0,0.57143,0.0,0.71429,18,0,0,18,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,0,0,0],[72,131,0.5496,0.36607,0.39113,0.0,0.14286,0.71429,0.0,1.0,14,4,0,14,0,4,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,4],[76,131,0.5802,0.29911,0.35778,0.0,0.07143,0.60714,0.0,1.0,16,3,0,16,0,3,0,0,0,0,0,2,0,0,3,0,0,5,0,0,0,0,3],[80,131,0.6107,0.50447,0.4134,0.0,0.64286,1.0,0.0,1.0,10,9,0,10,0,3,0,0,0,0,0,0,0,0,3,0,0,7,0,0,0,0,9],[84,131,0.6412,0.36161,0.38957,0.0,0.14286,0.71429,0.0,1.0,14,4,0,14,0,4,0,0,0,0,0,0,0,0,2,0,0,7,0,0,1,0,4],[88,131,0.6718,0.42856,0.35355,0.0,0.42857,0.71429,0.0,1.0,9,4,0,9,0,4,0,0,0,0,0,4,0,0,3,0,0,8,0,0,0,0,4],[92,131,0.7023,0.34375,0.329,0.0,0.14286,0.71429,0.0,1.0,10,1,0,10,0,7,0,0,2,0,0,0,0,0,1,0,0,11,0,0,0,0,1],[96,131,0.7328,0.40165,0.41859,0.0,0.14286,0.71429,0.0,1.0,14,7,0,14,0,3,0,0,0,0,0,0,0,0,2,0,0,6,0,0,0,0,7],[100,131,0.7634,0.41964,0.4164,0.0,0.14286,0.71429,0.0,1.0,12,7,0,12,0,5,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,7],[104,131,0.7939,0.46429,0.3993,0.0,0.71429,0.71429,0.0,1.0,10,6,0,10,0,4,0,0,1,0,0,0,0,0,0,0,0,10,0,0,1,0,6],[108,131,0.8244,0.4375,0.37786,0.0,0.57143,0.71429,0.0,1.0,12,4,0,12,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,0,0,0,4],[112,131,0.855,0.4866,0.37772,0.14286,0.57143,0.74996,0.0,1.0,7,6,0,7,0,5,0,0,2,0,0,1,0,0,2,0,0,7,0,0,2,0,6],[116,131,0.8855,0.42857,0.39609,0.0,0.42857,0.71429,0.0,1.0,11,6,0,11,0,4,0,0,0,0,0,2,0,0,2,0,0,6,0,0,1,0,6],[120,131,0.916,0.4375,0.37954,0.0,0.5,0.71429,0.0,1.0,9,5,0,9,0,6,0,0,0,0,0,1,0,0,1,0,0,10,0,0,0,0,5],[124,131,0.9466,0.36607,0.41023,0.0,0.14286,0.71429,0.0,1.0,13,7,0,13,0,6,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,7],[128,131,0.9771,0.27232,0.35956,0.0,0.0,0.71429,0.0,1.0,17,3,0,17,0,4,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0,0,3],[131,131,1.0,0.11161,0.15865,0.0,0.0,0.14287,0.0,0.57143,18,0,0,18,0,8,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.62054,"x":0.88839,"p":[[0,91,0.0,0.62054,0.22477,0.53571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,0,0,0,5,0,0,4,0,0,17,0,0,0,0,3],[4,91,0.044,0.7366,0.22899,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,18,0,0,0,0,9],[8,91,0.0879,0.80357,0.17405,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,0,2,0,12],[12,91,0.1319,0.67408,0.33165,0.67825,0.71429,1.0,0.0,1.0,5,10,0,5,0,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,1,0,10],[16,91,0.1758,0.72768,0.26812,0.67857,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,0,0,0,2,0,0,3,0,0,12,0,0,1,0,11],[20,91,0.2198,0.75893,0.19377,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,20,0,0,0,0,9],[24,91,0.2637,0.70089,0.27283,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,0,0,0,2,0,0,1,0,0,14,0,0,3,0,8],[28,91,0.3077,0.70982,0.28231,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,3,0,0,1,0,0,0,0,0,2,0,0,14,0,0,1,0,10],[32,91,0.3516,0.69195,0.28596,0.57143,0.71429,1.0,0.0,1.0,3,9,0,3,0,0,0,0,1,0,0,1,0,0,4,0,0,13,0,0,1,0,9],[36,91,0.3956,0.76786,0.1915,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,0,0,9],[40,91,0.4396,0.73659,0.25533,0.57143,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,0,0,0,3,0,0,4,0,0,10,0,0,2,0,11],[44,91,0.4835,0.66509,0.28276,0.57143,0.71429,1.0,0.14,1.0,0,9,0,0,0,5,0,0,0,0,0,2,0,0,5,0,0,11,0,0,0,0,9],[48,91,0.5275,0.63839,0.28568,0.42857,0.71429,0.74996,0.0,1.0,2,5,0,2,0,3,0,0,0,0,0,4,0,0,0,0,0,15,0,0,3,0,5],[52,91,0.5714,0.70089,0.31614,0.71429,0.71429,1.0,0.0,1.0,2,11,0,2,0,3,0,0,1,0,0,1,0,0,0,0,0,12,0,0,2,0,11],[56,91,0.6154,0.70536,0.28557,0.67857,0.71429,1.0,0.0,1.0,3,9,0,3,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,3,0,9],[60,91,0.6593,0.70088,0.30169,0.53571,0.71429,1.0,0.0,1.0,2,10,0,2,0,2,0,0,0,0,0,4,0,0,1,0,0,9,0,0,4,0,10],[64,91,0.7033,0.69194,0.25782,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,0,0,0,1,0,0,5,0,0,14,0,0,2,0,7],[68,91,0.7473,0.62054,0.28928,0.57143,0.71429,0.71429,0.0,1.0,2,5,0,2,0,4,0,0,0,0,0,1,0,0,5,0,0,13,0,0,2,0,5],[72,91,0.7912,0.73214,0.31693,0.71429,0.78564,1.0,0.0,1.0,2,13,0,2,0,3,0,0,0,0,0,1,0,0,1,0,0,9,0,0,3,0,13],[76,91,0.8352,0.88839,0.13236,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,6,0,17],[80,91,0.8791,0.74553,0.25688,0.67857,0.78564,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,1,0,0,3,0,0,8,0,0,6,0,10],[84,91,0.9231,0.75893,0.24856,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,10,0,0,4,0,11],[88,91,0.967,0.83033,0.20655,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,9,0,14],[91,91,1.0,0.66069,0.27375,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,10,0,0,3,0,0,2,0,0,6,0,8]]}]},{"i":"3cdf03d80dece61a","q":"Let  $D$  be the foot of the internal angle bisector of the angle  $A$  of a triangle  $ABC$ . Let  $E$  be a point on side  $[AC]$  such that  $|CE|= |CD|$  and  $|AE|=6\\sqrt 5$ ; let  $F$  be a point on the ray  $[AB$  such that  $|DB|=|BF|$  and  $|AB|<|AF| = 8\\sqrt 5$ . What is  $|AD|$ ? $ \n\\textbf{(A)}\\ 10\\sqrt 5\n\\qquad\\textbf{(B)}\\ 8\n\\qquad\\textbf{(C)}\\ 4\\sqrt{15}\n\\qquad\\textbf{(D)}\\ 7\\sqrt 5\n\\qquad\\textbf{(E)}\\ \\text{None of above}\n$","t":[{"b":0,"e":0.42857,"k":"flat","v":0.17857,"x":0.46874,"p":[[0,51,0.0,0.39286,0.32537,0.14286,0.14286,0.42858,0.14286,1.0,0,6,0,0,0,17,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,6],[4,51,0.0784,0.30357,0.29179,0.14286,0.14286,0.42857,0.0,1.0,2,4,1,2,0,19,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,4],[8,51,0.1569,0.46874,0.3179,0.14286,0.42857,0.60682,0.14286,1.0,0,7,0,0,0,11,0,0,0,0,0,12,0,0,1,0,0,1,0,0,0,0,7],[12,51,0.2353,0.25893,0.22428,0.14286,0.14286,0.42857,0.14286,1.0,0,2,0,0,0,23,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,2],[16,51,0.3137,0.37054,0.27632,0.14286,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,15,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,4],[20,51,0.3922,0.26786,0.19805,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,20,0,0,2,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[24,51,0.4706,0.24107,0.19704,0.14286,0.14286,0.21432,0.14286,1.0,0,1,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[28,51,0.549,0.27679,0.17105,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,18,0,0,2,0,0,10,0,0,0,0,0,2,0,0,0,0,0],[32,51,0.6275,0.25,0.23419,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,25,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,2],[36,51,0.7059,0.22321,0.17835,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,25,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[40,51,0.7843,0.17857,0.15567,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,51,0.8627,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.42857,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.14277,"x":0.43304,"p":[[0,17,0.0,0.37491,0.34586,0.14286,0.14286,0.42858,0.14,1.0,0,7,0,0,0,20,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,7],[4,17,0.2353,0.41071,0.34947,0.14286,0.14286,0.71429,0.14286,1.0,0,7,0,0,0,18,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,7],[8,17,0.4706,0.43304,0.36155,0.14286,0.14286,0.89286,0.14286,1.0,0,8,0,0,0,17,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,8],[12,17,0.7059,0.1875,0.10374,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,27,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"871b5567abaac79b","q":"Let  $I \\subset \\mathbb{R}$  be an open interval and  $f:I \\to \\mathbb{R}$  a twice differentiable function such that  $f(x)f''(x)=0,$  for any  $x \\in I.$  Prove that  $f''(x)=0,$  for any  $x \\in I.$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.68749,"x":0.70536,"p":[[0,4,0.0,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[4,4,1.0,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.66071,"x":0.72321,"p":[[0,44,0.0,0.66071,0.10564,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,16,0,0,1,0,1],[4,44,0.0909,0.66518,0.09182,0.67857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,24,0,0,0,0,0],[8,44,0.1818,0.70089,0.08268,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,0,0,1],[12,44,0.2727,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[16,44,0.3636,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[20,44,0.4545,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[24,44,0.5455,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[28,44,0.6364,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[32,44,0.7273,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.7143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,44,0.8182,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[40,44,0.9091,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[44,44,1.0,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0]]}]},{"i":"cb4af47a17725b64","q":"Let  $a_1, a_2... $  be an infinite sequence of real numbers such that  $a_{n+1}=\\sqrt{{a_n}^2+a_n-1}$ . Prove that  $a_1  \\notin (-2,1)$ *Proposed by Oleg Mushkarov and Nikolai Nikolov*","t":[{"b":3,"e":0.4286,"k":"falling","v":0.64729,"x":0.86607,"p":[[0,20,0.0,0.85713,0.16366,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,16,0,11],[4,20,0.2,0.86606,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[8,20,0.4,0.86607,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[12,20,0.6,0.82138,0.13834,0.857,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,26,0,2],[16,20,0.8,0.83033,0.14913,0.85713,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0],[20,20,1.0,0.64729,0.26483,0.5713,0.78564,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,1,0,0,2,0,0,9,0,0,1,0,0,16,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.86605,"x":0.89729,"p":[[0,32,0.0,0.87498,0.07784,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,6],[4,32,0.125,0.86605,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[8,32,0.25,0.86607,0.07087,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,4],[12,32,0.375,0.87052,0.08273,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,6],[16,32,0.5,0.87053,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5],[20,32,0.625,0.88838,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[24,32,0.75,0.89729,0.06425,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[28,32,0.875,0.87946,0.08073,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,8],[32,32,1.0,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9]]}]},{"i":"344a8fe758e0b466","q":"Let  $n$  be a positive integer. A board with a format  $n*n$  is divided in  $n*n$  equal squares.Determine all integers  $n$ \u22653 such that the board can be covered in  $2*1$  (or  $1*2$ ) pieces so that there is exactly one empty square in each row and each column.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.66517,"x":0.94196,"p":[[0,24,0.0,0.68301,0.24676,0.57143,0.57143,1.0,0.0,1.0,1,9,1,1,0,1,0,0,0,0,0,0,0,0,18,0,0,1,0,0,2,0,9],[4,24,0.1667,0.66517,0.36528,0.28571,0.78571,1.0,0.0,1.0,1,15,0,1,0,6,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,15],[8,24,0.3333,0.94196,0.12807,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,25],[12,24,0.5,0.91518,0.18509,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[16,24,0.6667,0.85265,0.2409,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,5,0,0,1,0,21],[20,24,0.8333,0.83929,0.1948,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,10,0,0,3,0,16],[24,24,1.0,0.78572,0.19233,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,13,0,0,3,0,11]]},{"b":4,"e":0.57143,"k":"falling","v":0.30357,"x":0.625,"p":[[0,45,0.0,0.625,0.21053,0.57143,0.57143,0.60714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,18,0,0,1,0,0,1,0,6],[4,45,0.0889,0.54463,0.30605,0.25,0.57143,0.85714,0.0,1.0,1,5,0,1,0,7,0,0,2,0,0,0,0,0,13,0,0,0,0,0,4,0,5],[8,45,0.1778,0.5089,0.30501,0.28571,0.4998,0.75,0.0,1.0,1,5,0,1,0,6,0,0,5,0,0,4,0,0,7,0,0,1,0,0,3,0,5],[12,45,0.2667,0.47311,0.33972,0.14286,0.49979,0.75,0.0,1.0,1,6,0,1,0,12,0,0,2,0,0,1,0,0,7,0,0,1,0,0,2,0,6],[16,45,0.3556,0.58477,0.22969,0.53539,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,4,0,0,2,0,0,15,0,0,3,0,0,2,0,4],[20,45,0.4444,0.53569,0.26486,0.28571,0.571,0.60714,0.14286,1.0,0,5,0,0,0,2,0,0,9,0,0,4,0,0,9,0,0,1,0,0,2,0,5],[24,45,0.5333,0.54015,0.2737,0.28571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,5,0,0,0,0,0,13,0,0,3,0,0,2,0,4],[28,45,0.6222,0.51785,0.26666,0.28571,0.57143,0.60714,0.0,1.0,1,4,0,1,0,2,0,0,8,0,0,4,0,0,9,0,0,2,0,0,2,0,4],[32,45,0.7111,0.44196,0.27049,0.2857,0.35714,0.57143,0.0,1.0,2,3,0,2,0,4,0,0,10,0,0,1,0,0,10,0,0,1,0,0,1,0,3],[36,45,0.8,0.55801,0.28651,0.28571,0.571,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,9,0,0,4,0,0,8,0,0,0,0,0,2,0,7],[40,45,0.8889,0.34373,0.20157,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,5,0,0,20,0,0,2,0,0,3,0,0,0,0,0,0,0,2],[44,45,0.9778,0.32142,0.11842,0.28571,0.28571,0.32164,0.14286,0.57143,0,0,0,0,0,4,0,0,20,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[45,45,1.0,0.30357,0.15465,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,4,0,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"ea8fb79dbef84459","q":"Let  $f$  be a polynomial with real coefficients of degree  $n$ . Suppose that  $\\displaystyle \\frac{f(x)-f(y)}{x-y}$  is an integer for all  $0 \\leq x<y \\leq n$ . Prove that  $a-b | f(a)-f(b)$  for all distinct integers  $a,b$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.09375,"x":0.29016,"p":[[0,102,0.0,0.09821,0.1448,0.0,0.0,0.2857,0.0,0.4286,21,0,1,21,0,2,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,102,0.0392,0.23653,0.31058,0.0,0.14143,0.32164,0.0,1.0,15,3,0,15,0,4,0,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[8,102,0.0784,0.25,0.22016,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,6,0,0,7,0,0,7,0,0,1,0,0,1,0,0,1,0,0],[12,102,0.1176,0.2232,0.19862,0.0,0.2143,0.32143,0.0,0.71429,10,0,0,10,0,6,0,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[16,102,0.1569,0.25,0.19233,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,7,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[20,102,0.1961,0.19643,0.23623,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,4,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[24,102,0.2353,0.17857,0.2369,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,5,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[28,102,0.2745,0.1875,0.23266,0.0,0.14286,0.32143,0.0,1.0,15,1,0,15,0,5,0,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[32,102,0.3137,0.19196,0.23585,0.0,0.07143,0.32143,0.0,0.71429,16,0,0,16,0,4,0,0,4,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[36,102,0.3529,0.29016,0.29769,0.0,0.28571,0.5711,0.0,1.0,13,1,0,13,0,2,0,0,5,0,0,3,0,0,3,0,0,5,0,0,0,0,1],[40,102,0.3922,0.20982,0.1988,0.0,0.21428,0.42857,0.0,0.57143,13,0,0,13,0,3,0,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[44,102,0.4314,0.2232,0.21704,0.0,0.2143,0.32143,0.0,0.71429,12,0,0,12,0,4,0,0,8,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[48,102,0.4706,0.27232,0.22689,0.0,0.28571,0.42857,0.0,0.71429,10,0,0,10,0,2,0,0,8,0,0,8,0,0,1,0,0,3,0,0,0,0,0],[52,102,0.5098,0.22767,0.19837,0.0,0.2143,0.42857,0.0,0.71429,10,0,0,10,0,6,0,0,6,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[56,102,0.549,0.17857,0.21724,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,5,0,0,6,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[60,102,0.5882,0.21875,0.22011,0.0,0.21428,0.28571,0.0,0.85714,12,0,0,12,0,4,0,0,9,0,0,3,0,0,3,0,0,0,0,0,1,0,0],[64,102,0.6275,0.23214,0.22232,0.0,0.21428,0.42857,0.0,0.85714,11,0,0,11,0,5,0,0,7,0,0,5,0,0,3,0,0,0,0,0,1,0,0],[68,102,0.6667,0.1875,0.23538,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,6,0,0,4,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[72,102,0.7059,0.15625,0.17985,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,6,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[76,102,0.7451,0.15625,0.18336,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,6,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[80,102,0.7843,0.15625,0.20316,0.0,0.14286,0.1786,0.0,0.71429,15,0,0,15,0,9,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[84,102,0.8235,0.20536,0.22851,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,4,0,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[88,102,0.8627,0.13393,0.2111,0.0,0.0,0.1786,0.0,1.0,18,1,0,18,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[92,102,0.902,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[96,102,0.9412,0.10714,0.14726,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,7,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[100,102,0.9804,0.14286,0.20516,0.0,0.0,0.17857,0.0,0.71429,18,0,0,18,0,6,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[102,102,1.0,0.15625,0.21535,0.0,0.14286,0.1786,0.0,1.0,15,1,0,15,0,9,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,1]]},{"b":4,"e":0.0,"k":"rising","v":0.09374,"x":0.26337,"p":[[0,19,0.0,0.09374,0.15811,0.0,0.0,0.14286,0.0,0.571,22,0,1,22,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,19,0.2105,0.25447,0.27833,0.0,0.21429,0.42857,0.0,1.0,14,1,1,14,0,2,0,0,5,0,0,5,0,0,2,0,0,3,0,0,0,0,1],[8,19,0.4211,0.18747,0.21256,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,3,0,0,3,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[12,19,0.6316,0.23652,0.23857,0.0,0.14288,0.42857,0.0,0.85714,12,0,0,12,0,5,0,0,4,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[16,19,0.8421,0.26337,0.23446,0.0,0.28571,0.42857,0.0,0.71429,11,0,0,11,0,4,0,0,3,0,0,9,0,0,3,0,0,2,0,0,0,0,0],[19,19,1.0,0.24551,0.2556,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,5,0,0,2,0,0,5,0,0,4,0,0,3,0,0,0,0,0]]}]},{"i":"f80ff7cc9f306487","q":"Let  $n \\ge 3$  be an integer. A frog is to jump along the real axis, starting at the point  $0$  and making  $n$  jumps: one of length  $1$ , one of length  $2$ ,  $\\dots$  , one of length  $n$ . It may perform these  $n$  jumps in any order. If at some point the frog is sitting on a number  $a \\le 0$ , its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number  $a > 0$ , its next jump must be to the left (towards the negative numbers). Find the largest positive integer  $k$  for which the frog can perform its jumps in such an order that it never lands on any of the numbers  $1, 2, \\dots , k$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.57589,"x":0.85264,"p":[[0,113,0.0,0.57589,0.19061,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,2,0,0,10,0,0,14,0,0,1,0,0],[4,113,0.0354,0.72766,0.23788,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,1,0,0,0,0,0,6,0,0,8,0,0,9,0,6],[8,113,0.0708,0.80356,0.228,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,0,11,0,11],[12,113,0.1062,0.70536,0.24206,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,8,0,0,8,0,6],[16,113,0.1416,0.74553,0.19474,0.57143,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,6,0,0,12,0,5],[20,113,0.177,0.7723,0.17808,0.67857,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,15,0,5],[24,113,0.2124,0.78123,0.21426,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,11,0,8],[28,113,0.2478,0.74998,0.22869,0.67857,0.85707,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,0,0,0,5,0,0,6,0,0,12,0,6],[32,113,0.2832,0.82142,0.17496,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,9,0,11],[36,113,0.3186,0.77676,0.20808,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,7,0,0,9,0,9],[40,113,0.354,0.81696,0.19637,0.82132,0.85714,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,15,0,9],[44,113,0.3894,0.77229,0.18164,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,12,0,6],[48,113,0.4248,0.83034,0.16148,0.82143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,15,0,9],[52,113,0.4602,0.79458,0.20189,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,0,16,0,7],[56,113,0.4956,0.7857,0.19886,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,10,0,10],[60,113,0.531,0.78124,0.20822,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,9,0,0,7,0,10],[64,113,0.5664,0.81249,0.19047,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,7,0,12],[68,113,0.6018,0.79462,0.1673,0.71429,0.85707,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,10,0,0,9,0,8],[72,113,0.6372,0.76784,0.20749,0.71429,0.85714,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,14,0,5],[76,113,0.6726,0.80356,0.20125,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,0,8,0,11],[80,113,0.708,0.72308,0.2549,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,0,0,0,4,0,0,8,0,0,7,0,8],[84,113,0.7434,0.85264,0.13598,0.857,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,15,0,10],[88,113,0.7788,0.79012,0.19886,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,0,8,0,11],[92,113,0.8142,0.78123,0.19558,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,5,0,0,13,0,7],[96,113,0.8496,0.76783,0.21945,0.57143,0.85707,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,6,0,0,7,0,10],[100,113,0.885,0.77676,0.1888,0.67857,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,9,0,0,5,0,10],[104,113,0.9204,0.74105,0.21559,0.57143,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,0,8,0,8],[108,113,0.9558,0.6964,0.24937,0.5354,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,5,0,0,9,0,6],[112,113,0.9912,0.74105,0.1692,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,10,0,0,8,0,5],[113,113,1.0,0.72319,0.21411,0.57142,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,9,0,0,6,0,7]]},{"b":5,"e":0.571,"k":"rising","v":0.53123,"x":0.81249,"p":[[0,72,0.0,0.53123,0.19959,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,3,0,0,16,0,0,4,0,0,3,0,0],[4,72,0.0556,0.69641,0.31693,0.57132,0.857,1.0,0.0,1.0,2,9,0,2,0,3,0,0,1,0,0,1,0,0,3,0,0,5,0,0,8,0,9],[8,72,0.1111,0.73661,0.23449,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,5,0,0,10,0,7],[12,72,0.1667,0.7098,0.25377,0.57132,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,1,0,0,2,0,0,4,0,0,5,0,0,12,0,5],[16,72,0.2222,0.71429,0.27433,0.57143,0.78571,0.85714,0.0,1.0,3,6,0,3,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,10,0,6],[20,72,0.2778,0.76784,0.20125,0.71429,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,6,0,0,13,0,6],[24,72,0.3333,0.79462,0.20499,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,4,0,0,7,0,12],[28,72,0.3889,0.74554,0.19475,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,8,0,0,9,0,6],[32,72,0.4444,0.74997,0.23148,0.57143,0.78571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,7,0,9],[36,72,0.5,0.81249,0.17657,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,10,0,10],[40,72,0.5556,0.70089,0.25344,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,0,0,0,3,0,0,3,0,0,9,0,0,9,0,5],[44,72,0.6111,0.76784,0.2468,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,7,0,0,5,0,12],[48,72,0.6667,0.72768,0.19351,0.67857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,11,0,0,9,0,4],[52,72,0.7222,0.68301,0.2194,0.4286,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,7,0,0,7,0,5],[56,72,0.7778,0.70978,0.19722,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,9,0,0,5,0,0,10,0,4],[60,72,0.8333,0.71425,0.20826,0.67856,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,10,0,0,11,0,3],[64,72,0.8889,0.78571,0.18898,0.71429,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,12,0,7],[68,72,0.9444,0.77232,0.20472,0.71429,0.85707,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,11,0,0,10,0,7],[72,72,1.0,0.75892,0.16145,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,12,0,0,12,0,3]]}]},{"i":"47d44924999756e8","q":"Consider the base 27 number \n\\[\n  n = ABCDEFGHIJKLMNOPQRSTUVWXYZ ,\n\\]\t\nwhere each letter has the value of its position in the alphabet.  What remainder do you get when you divide  $n$  by 100?  (The remainder is an integer between 0 and 99, inclusive.)","t":[{"b":6,"e":0.71429,"k":"flat","v":0.78125,"x":0.97321,"p":[[0,41,0.0,0.80804,0.20705,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,0,1,0,16],[4,41,0.0976,0.91071,0.17035,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,25],[8,41,0.1951,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[12,41,0.2927,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,41,0.3902,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[20,41,0.4878,0.94643,0.1171,1.0,1.0,1.0,0.5714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[24,41,0.5854,0.91964,0.14258,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[28,41,0.6829,0.84375,0.18681,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,4,0,16],[32,41,0.7805,0.87054,0.2,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,20],[36,41,0.878,0.8482,0.21707,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,1,0,20],[40,41,0.9756,0.87051,0.18685,0.71429,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,0,0,21],[41,41,1.0,0.78125,0.18552,0.67857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,10,0,0,3,0,11]]},{"b":7,"e":0.57143,"k":"falling","v":0.62052,"x":0.92411,"p":[[0,59,0.0,0.87946,0.19597,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,1,0,22],[4,59,0.0678,0.92411,0.13825,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,24],[8,59,0.1356,0.75892,0.20025,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,7,0,0,2,0,11],[12,59,0.2034,0.81696,0.18638,0.57143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,15],[16,59,0.2712,0.78557,0.16374,0.71321,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,12,0,0,3,0,10],[20,59,0.339,0.81249,0.18709,0.71429,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,10,0,0,0,0,15],[24,59,0.4068,0.67409,0.18293,0.57143,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,7,0,0,1,0,6],[28,59,0.4746,0.74107,0.18363,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,8,0,0,2,0,9],[32,59,0.5424,0.77229,0.17448,0.57143,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,9,0,0,3,0,10],[36,59,0.6102,0.69197,0.17169,0.57143,0.64286,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,9,0,0,1,0,6],[40,59,0.678,0.75446,0.17941,0.57143,0.71429,1.0,0.5714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,9,0,0,1,0,10],[44,59,0.7458,0.73661,0.18595,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,10,0,0,1,0,9],[48,59,0.8136,0.65177,0.16728,0.57143,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,16,0,0,10,0,0,0,0,4],[52,59,0.8814,0.66963,0.14914,0.57143,0.64286,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,12,0,0,0,0,4],[56,59,0.9492,0.67408,0.12995,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,15,0,0,0,0,3],[59,59,1.0,0.62052,0.11634,0.57143,0.64286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,16,0,0,0,0,0]]}]},{"i":"1edd4aa69a9ee397","q":"Anton and Britta play a game with the set  $M=\\left \\{ 1,2,\\dots,n-1 \\right \\}$  where  $n \\geq 5$  is an odd integer. In each step Anton removes a number from  $M$  and puts it in his set  $A$ , and Britta removes a number from  $M$  and puts it in her set  $B$  (both  $A$  and  $B$  are empty to begin with). When  $M$  is empty, Anton picks two distinct numbers  $x_1, x_2$  from  $A$  and shows them to Britta. Britta then picks two distinct numbers  $y_1, y_2$  from  $B$ . Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\\frac{n-1}{2}}\\equiv 1\\mod n$ otherwise Anton wins. Find all  $n$  for which Britta has a winning strategy.","t":[{"b":1,"e":1.0,"k":"rising","v":0.21428,"x":0.99554,"p":[[0,76,0.0,0.21428,0.12877,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,1,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,76,0.0526,0.56246,0.36235,0.2857,0.571,1.0,0.0,1.0,4,10,0,4,0,3,0,0,4,0,0,3,0,0,5,0,0,2,0,0,1,0,10],[8,76,0.1053,0.47759,0.33054,0.2857,0.42857,0.71429,0.0,1.0,5,5,0,5,0,2,0,0,6,0,0,6,0,0,2,0,0,4,0,0,2,0,5],[12,76,0.1579,0.52679,0.34337,0.28571,0.50001,0.78571,0.0,1.0,4,8,0,4,0,3,0,0,4,0,0,5,0,0,4,0,0,4,0,0,0,0,8],[16,76,0.2105,0.54459,0.27534,0.39286,0.4998,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,5,0,0,8,0,0,6,0,0,4,0,0,0,0,6],[20,76,0.2632,0.50004,0.26725,0.42857,0.42857,0.60714,0.0,1.0,2,5,0,2,0,1,0,0,4,0,0,15,0,0,2,0,0,3,0,0,0,0,5],[24,76,0.3158,0.45094,0.27689,0.28571,0.42857,0.57143,0.0,1.0,4,4,0,4,0,1,0,0,6,0,0,9,0,0,7,0,0,1,0,0,0,0,4],[28,76,0.3684,0.51343,0.31714,0.39286,0.4286,0.71429,0.0,1.0,5,6,0,5,0,0,0,0,3,0,0,10,0,0,4,0,0,3,0,0,1,0,6],[32,76,0.4211,0.51339,0.30691,0.28571,0.5005,0.71429,0.0,1.0,2,6,0,2,0,4,0,0,5,0,0,5,0,0,7,0,0,2,0,0,1,0,6],[36,76,0.4737,0.4375,0.21705,0.28571,0.42857,0.4286,0.0,1.0,1,2,0,1,0,1,0,0,9,0,0,15,0,0,2,0,0,0,0,0,2,0,2],[40,76,0.5263,0.40625,0.27919,0.14289,0.42857,0.57143,0.0,1.0,4,3,0,4,0,5,0,0,4,0,0,9,0,0,6,0,0,0,0,0,1,0,3],[44,76,0.5789,0.45066,0.31963,0.14286,0.42857,0.71429,0.0,1.0,3,5,0,3,0,7,0,0,4,0,0,6,0,0,2,0,0,5,0,0,0,0,5],[48,76,0.6316,0.40624,0.32753,0.14286,0.28571,0.57143,0.0,1.0,7,4,0,7,0,2,0,0,8,0,0,4,0,0,4,0,0,1,0,0,2,0,4],[52,76,0.6842,0.58482,0.33761,0.28571,0.71429,0.89286,0.0,1.0,3,8,0,3,0,2,0,0,6,0,0,2,0,0,2,0,0,7,0,0,2,0,8],[56,76,0.7368,0.89286,0.18211,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,21],[60,76,0.7895,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[64,76,0.8421,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[68,76,0.8947,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,76,0.9474,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,76,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":0.14286,"k":"flat","v":0.13392,"x":0.42848,"p":[[0,48,0.0,0.17857,0.14286,0.0,0.28571,0.28571,0.0,0.42857,12,0,0,12,0,1,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.42848,0.41194,0.0,0.35714,1.0,0.0,1.0,9,10,0,9,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,10],[8,48,0.1667,0.37499,0.32878,0.0,0.42857,0.4642,0.0,1.0,9,4,0,9,0,3,0,0,2,0,0,10,0,0,2,0,0,1,0,0,1,0,4],[12,48,0.25,0.32143,0.29233,0.0,0.35714,0.42857,0.0,1.0,9,3,0,9,0,4,0,0,3,0,0,12,0,0,0,0,0,1,0,0,0,0,3],[16,48,0.3333,0.31695,0.24931,0.14286,0.28571,0.4286,0.0,1.0,7,1,0,7,0,4,0,0,8,0,0,7,0,0,2,0,0,3,0,0,0,0,1],[20,48,0.4167,0.22318,0.18181,0.10714,0.1429,0.32143,0.0,0.571,8,0,0,8,0,9,0,0,7,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[24,48,0.5,0.21418,0.17497,0.105,0.14286,0.42857,0.0,0.571,8,0,0,8,0,11,0,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[28,48,0.5833,0.22321,0.23673,0.0,0.14286,0.32143,0.0,1.0,11,1,0,11,0,7,0,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[32,48,0.6667,0.28561,0.21433,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,8,0,0,8,0,0,7,0,0,3,0,0,0,0,0,0,0,1],[36,48,0.75,0.20983,0.14279,0.14286,0.14288,0.28571,0.0,0.4286,5,0,0,5,0,14,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.23204,0.17037,0.14286,0.1429,0.28571,0.0,0.71429,5,0,0,5,0,12,0,0,8,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[44,48,0.9167,0.2008,0.18164,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,10,0,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[48,48,1.0,0.13392,0.15122,0.0,0.14286,0.1786,0.0,0.571,14,0,0,14,0,10,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"0691b870fe07558d","q":"Define the sequence of integers  $a_1, a_2, a_3, \\ldots$  by  $a_1 = 1$ , and\n\\[ a_{n+1} = \\left(n+1-\\gcd(a_n,n) \\right) \\times a_n \\]\nfor all integers  $n \\ge 1$ .\nProve that  $\\frac{a_{n+1}}{a_n}=n$  if and only if  $n$  is prime or  $n=1$ .\n*Here  $\\gcd(s,t)$  denotes the greatest common divisor of  $s$  and  $t$ .*","t":[{"b":3,"e":0.85714,"k":"flat","v":0.80803,"x":0.96429,"p":[[0,62,0.0,0.80803,0.16982,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,11,0,9],[4,62,0.0645,0.91964,0.13803,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[8,62,0.129,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,62,0.1935,0.92856,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[16,62,0.2581,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[20,62,0.3226,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[24,62,0.3871,0.91517,0.15096,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,23],[28,62,0.4516,0.90179,0.16146,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,21],[32,62,0.5161,0.94643,0.14617,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[36,62,0.5806,0.92857,0.14286,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,25],[40,62,0.6452,0.9375,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[44,62,0.7097,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[48,62,0.7742,0.91964,0.14698,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,22],[52,62,0.8387,0.83928,0.15872,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,11,0,11],[56,62,0.9032,0.91517,0.10013,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[60,62,0.9677,0.86606,0.10677,0.82132,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,14,0,10],[62,62,1.0,0.82142,0.10105,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,17,0,4]]},{"b":6,"e":1.0,"k":"flat","v":0.83481,"x":0.97768,"p":[[0,37,0.0,0.89285,0.14286,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,6,0,18],[4,37,0.1081,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[8,37,0.2162,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,37,0.3243,0.87498,0.16271,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,18],[16,37,0.4324,0.88393,0.15335,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,8,0,17],[20,37,0.5405,0.83481,0.18251,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,4,0,15],[24,37,0.6486,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[28,37,0.7568,0.87946,0.11904,0.85711,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,12,0,13],[32,37,0.8649,0.89285,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[36,37,0.973,0.9107,0.12756,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,19],[37,37,1.0,0.86606,0.15129,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,8,0,15]]}]},{"i":"0d941d881fcf2618","q":"Consider  $4n$  points in the plane, with no three points collinear. Using these points as vertices, we form  $\\binom{4n}{3}$  triangles. Show that there exists a point  $X$  of the plane that belongs to the interior of at least  $2n^3$  of these triangles.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.0,"x":0.25888,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.21875,0.22583,0.0,0.2857,0.28571,0.0,1.0,11,1,0,11,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[8,26,0.3077,0.18304,0.18977,0.0,0.2143,0.28571,0.0,0.85714,13,0,0,13,0,3,0,0,13,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[12,26,0.4615,0.20089,0.23107,0.0,0.14285,0.32143,0.0,0.85714,16,0,0,16,0,0,0,0,8,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[16,26,0.6154,0.20982,0.22011,0.0,0.2857,0.28571,0.0,0.85714,14,0,0,14,0,0,0,0,12,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[20,26,0.7692,0.23213,0.24934,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,2,0,0,10,0,0,2,0,0,3,0,0,1,0,0,0,0,1],[24,26,0.9231,0.21429,0.17128,0.0,0.28571,0.32143,0.0,0.4286,11,0,0,11,0,2,0,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.25888,0.16528,0.14286,0.2857,0.42857,0.0,0.57143,6,0,0,6,0,5,0,0,12,0,0,7,0,0,2,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.01339,"x":0.28571,"p":[[0,55,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,6,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.28571,0.27433,0.0,0.28571,0.42857,0.0,1.0,11,2,1,11,0,0,0,0,11,0,0,6,0,0,0,0,0,2,0,0,0,0,2],[8,55,0.1455,0.26339,0.26027,0.0,0.2857,0.32143,0.0,1.0,11,1,0,11,0,1,0,0,12,0,0,5,0,0,0,0,0,0,0,0,2,0,1],[12,55,0.2182,0.22768,0.24964,0.0,0.2857,0.28571,0.0,1.0,12,2,0,12,0,2,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[16,55,0.2909,0.28571,0.26726,0.0,0.28571,0.32143,0.0,1.0,9,2,0,9,0,2,0,0,13,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[20,55,0.3636,0.21429,0.16366,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,2,0,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,55,0.4364,0.20982,0.22299,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,2,0,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[28,55,0.5091,0.14732,0.18723,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,1,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[32,55,0.5818,0.09375,0.17717,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,55,0.6545,0.11607,0.20959,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,55,0.7273,0.13838,0.16931,0.0,0.0,0.2857,0.0,0.571,18,0,0,18,0,1,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,55,0.8,0.15178,0.15542,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,3,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.12054,0.14334,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.12938,0.1488,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,3,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.14731,0.1616,0.0,0.14286,0.2857,0.0,0.571,15,0,0,15,0,5,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6c738107700b28b5","q":"Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \\neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. (Croatia)","t":[{"b":4,"e":0.4286,"k":"flat","v":0.40406,"x":0.6071,"p":[[0,49,0.0,0.40406,0.15833,0.28571,0.28571,0.51786,0.14286,0.71429,0,0,0,0,0,1,0,0,16,0,0,6,0,1,4,0,0,4,0,0,0,0,0],[4,49,0.0816,0.60265,0.174,0.571,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,3,0,0,11,0,0,12,0,0,0,0,2],[8,49,0.1633,0.6071,0.14287,0.57143,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,1,0,0,15,0,0,12,0,0,0,0,1],[12,49,0.2449,0.49995,0.15969,0.42857,0.57141,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,3,0,0,5,0,0,17,0,0,4,0,0,0,0,0],[16,49,0.3265,0.46423,0.17493,0.28571,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,7,0,0,5,0,0,14,0,0,2,0,0,1,0,0],[20,49,0.4082,0.4732,0.20958,0.28571,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,6,0,0,2,0,0,14,0,0,4,0,0,0,0,1],[24,49,0.4898,0.48212,0.19479,0.39286,0.571,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,4,0,0,7,0,0,12,0,0,4,0,0,0,0,1],[28,49,0.5714,0.41514,0.21826,0.14289,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,8,0,0,4,0,0,4,0,0,10,0,0,5,0,0,0,0,0],[32,49,0.6531,0.41069,0.20436,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,6,0,0,6,0,0,10,0,0,2,0,0,1,0,0],[36,49,0.7347,0.45087,0.20236,0.28571,0.57121,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,4,0,0,3,0,0,13,0,0,5,0,0,0,0,0],[40,49,0.8163,0.51334,0.13765,0.4286,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,3,0,0,20,0,0,3,0,0,0,0,0],[44,49,0.898,0.50445,0.14278,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,6,0,0,16,0,0,4,0,0,0,0,0],[48,49,0.9796,0.46427,0.15152,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,6,0,0,11,0,0,4,0,0,0,0,0],[49,49,1.0,0.48214,0.16656,0.28571,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,1,0,0,15,0,0,3,0,0,1,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.39285,"x":0.56688,"p":[[0,64,0.0,0.41737,0.17135,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,1,0,0,13,0,1,10,0,0,3,0,0,2,0,0,2,0,0],[4,64,0.0625,0.56247,0.12846,0.57132,0.57143,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,2,0,0,18,0,0,8,0,0,0,0,0],[8,64,0.125,0.56688,0.09094,0.571,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,22,0,0,5,0,0,0,0,0],[12,64,0.1875,0.50888,0.16339,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,4,0,0,15,0,0,6,0,0,0,0,0],[16,64,0.25,0.50445,0.1556,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,6,0,0,15,0,0,5,0,0,0,0,0],[20,64,0.3125,0.49103,0.1554,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,4,0,0,3,0,0,20,0,0,2,0,0,0,0,0],[24,64,0.375,0.49107,0.20183,0.28571,0.57143,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,5,0,0,0,0,0,20,0,0,2,0,0,0,0,1],[28,64,0.4375,0.54452,0.19393,0.4286,0.57143,0.71429,0.14,1.0,0,1,0,0,0,3,0,0,2,0,0,5,0,0,13,0,0,7,0,0,1,0,1],[32,64,0.5,0.51783,0.20123,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,3,0,0,3,0,0,16,0,0,4,0,0,1,0,1],[36,64,0.5625,0.42854,0.21127,0.2857,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,6,0,0,3,0,0,11,0,0,5,0,0,0,0,0],[40,64,0.625,0.54461,0.20339,0.39286,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,1,0,0,11,0,0,10,0,0,2,0,0],[44,64,0.6875,0.433,0.2157,0.2857,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,7,0,0,5,0,0,8,0,0,5,0,0,1,0,0],[48,64,0.75,0.39285,0.20515,0.25,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,7,0,0,5,0,0,8,0,0,4,0,0,0,0,0],[52,64,0.8125,0.51329,0.23121,0.28571,0.57143,0.71429,0.14,1.0,0,1,0,0,0,5,0,0,5,0,0,2,0,0,9,0,0,9,0,0,1,0,1],[56,64,0.875,0.46868,0.21789,0.28571,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,7,0,0,4,0,0,3,0,0,9,0,0,9,0,0,0,0,0],[60,64,0.9375,0.42409,0.22723,0.14286,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,9,0,0,5,0,0,4,0,0,7,0,0,6,0,0,1,0,0],[64,64,1.0,0.39732,0.24932,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,11,0,0,7,0,0,1,0,0,7,0,0,4,0,0,1,0,1]]}]},{"i":"395a4165931f4dcc","q":"Denote $\\{1,2, \\ldots, n\\}$ by $[n]$, and let $S$ be the set of all permutations of [ $n$ ]. Call a subset $T$ of $S$ good if every permutation $\\sigma$ in $S$ may be written as $t_{1} t_{2}$ for elements $t_{1}$ and $t_{2}$ of $T$, where the product of two permutations is defined to be their composition. Call a subset of $U$ of $S$ extremely good if every permutation $\\sigma$ in $S$ may be written as $s^{-1} u s$ for elements $s$ of $S$ and $u$ of $U$. Let $\\tau$ be the smallest value of $|T| /|S|$ for all good subsets $T$, and let $v$ be the smallest value of $|U| /|S|$ for all extremely good subsets $U$. Prove that $\\sqrt{v} \\geq \\tau$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13839,"x":0.30357,"p":[[0,53,0.0,0.27678,0.04971,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.30357,0.20748,0.14289,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,6,0,0,17,0,0,1,0,0,2,0,0,1,0,0,2,0,0],[8,53,0.1509,0.28125,0.18029,0.14286,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,7,0,0,17,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[12,53,0.2264,0.25893,0.1448,0.14286,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,8,0,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[16,53,0.3019,0.25,0.16751,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,12,0,0,15,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[20,53,0.3774,0.22321,0.13333,0.14286,0.14288,0.28571,0.0,0.71429,2,0,0,2,0,15,0,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[24,53,0.4528,0.24107,0.18363,0.14286,0.2857,0.28571,0.0,0.85714,4,0,0,4,0,11,0,0,14,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[28,53,0.5283,0.19196,0.10479,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,17,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,53,0.6038,0.18303,0.08917,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,20,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,53,0.6792,0.17857,0.07986,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,53,0.7547,0.17857,0.09449,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,53,0.8302,0.21875,0.10092,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,16,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,53,0.9057,0.17857,0.09449,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,53,0.9811,0.17411,0.05905,0.14286,0.14286,0.1429,0.14286,0.28571,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.08036,"x":0.28125,"p":[[0,55,0.0,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.24107,0.21558,0.14286,0.28571,0.28571,0.0,1.0,6,1,0,6,0,9,0,0,13,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[8,55,0.1455,0.26786,0.16269,0.14286,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,7,0,0,18,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[12,55,0.2182,0.26339,0.23987,0.14286,0.28571,0.28571,0.0,1.0,6,2,0,6,0,8,0,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[16,55,0.2909,0.21875,0.13825,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,16,0,0,12,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,55,0.3636,0.22321,0.12339,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,13,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,55,0.4364,0.25,0.16751,0.14286,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,9,0,0,13,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[28,55,0.5091,0.19643,0.12242,0.14286,0.21428,0.28571,0.0,0.42857,6,0,0,6,0,10,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.18304,0.13474,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,10,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.17848,0.1557,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,10,0,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[40,55,0.7273,0.15178,0.12846,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,12,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.12947,0.10326,0.0,0.14286,0.1429,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.0892,0.09274,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e24bf764a057abc6","q":"Does there exist a set  $S$  of  $100$  points in a plane such that the center of mass of any  $10$  points in  $S$  is also a point in  $S$ ?","t":[{"b":3,"e":0.0,"k":"falling","v":0.04455,"x":0.28125,"p":[[0,47,0.0,0.2633,0.2318,0.105,0.21428,0.42858,0.0,0.71429,8,0,0,8,0,8,0,0,7,0,0,2,0,0,4,0,0,3,0,0,0,0,0],[4,47,0.0851,0.22319,0.33105,0.0,0.0,0.32142,0.0,1.0,17,3,0,17,0,6,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,3],[8,47,0.1702,0.20955,0.30724,0.0,0.14143,0.14286,0.0,1.0,13,3,0,13,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,3],[12,47,0.2553,0.28125,0.33021,0.0,0.14286,0.42858,0.0,1.0,12,3,0,12,0,7,0,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,3],[16,47,0.3404,0.04455,0.10964,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,47,0.4255,0.19643,0.25939,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,9,0,0,3,0,0,1,0,0,0,0,0,4,0,0,1,0,0],[24,47,0.5106,0.13839,0.24086,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[28,47,0.5957,0.1783,0.26248,0.0,0.0,0.2857,0.0,1.0,17,1,0,17,0,6,0,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[32,47,0.6809,0.1963,0.28267,0.0,0.0,0.32143,0.0,0.85714,18,0,0,18,0,4,0,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,0],[36,47,0.766,0.13839,0.2461,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[40,47,0.8511,0.22765,0.25215,0.0,0.14286,0.28571,0.0,1.0,11,1,0,11,0,8,0,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[44,47,0.9362,0.12929,0.18335,0.0,0.0,0.14287,0.0,0.71429,17,0,0,17,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[47,47,1.0,0.10687,0.19557,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.0,"k":"falling","v":0.05777,"x":0.37052,"p":[[0,54,0.0,0.37052,0.20157,0.2857,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,2,0,0,9,0,0,7,0,0,8,0,0,2,0,0,0,0,0],[4,54,0.0741,0.19643,0.26426,0.0,0.14286,0.21429,0.0,1.0,13,2,0,13,0,11,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,2],[8,54,0.1481,0.13393,0.26471,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[12,54,0.2222,0.09822,0.24338,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,54,0.2963,0.13375,0.23941,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[20,54,0.3704,0.12491,0.23076,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[24,54,0.4444,0.11161,0.25439,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[28,54,0.5185,0.18295,0.27488,0.0,0.14286,0.14286,0.0,1.0,13,2,0,13,0,14,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[32,54,0.5926,0.16509,0.27459,0.0,0.07,0.14286,0.0,1.0,16,2,0,16,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[36,54,0.6667,0.12054,0.19597,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,54,0.7407,0.23652,0.25158,0.14214,0.14286,0.21432,0.0,0.85714,7,0,0,7,0,17,0,0,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0],[44,54,0.8148,0.16509,0.27226,0.0,0.14143,0.14286,0.0,1.0,15,2,0,15,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[48,54,0.8889,0.10268,0.15663,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,54,0.963,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,12,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[54,54,1.0,0.05777,0.06984,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42b90d4395152224","q":"Let  $n$  be a positive integer, and let  $1=d_1<d_2<\\dots < d_k=n$  denote all positive divisors of  $n$ , If the following conditions are satisfied:  $$  2d_2+d_4+d_5=d_7 $$    $$  d_3 d_6 d_7=n $$    $$  (d_6+d_7)^2=n+1 $$  find all possible values of  $n$ .","t":[{"b":5,"e":0.0,"k":"falling","v":0.01339,"x":0.9308,"p":[[0,300,0.0,0.34819,0.29218,0.0,0.42857,0.57143,0.0,0.85714,11,0,6,11,0,1,0,0,3,0,0,5,0,0,5,0,0,6,0,0,1,0,0],[4,300,0.0133,0.83033,0.25616,0.71429,0.92857,1.0,0.0,1.0,2,16,1,2,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,16],[8,300,0.0267,0.92857,0.12877,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,23],[12,300,0.04,0.87054,0.19999,0.85714,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,18],[16,300,0.0533,0.88838,0.20121,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,20],[20,300,0.0667,0.875,0.24936,0.85714,1.0,1.0,0.0,1.0,2,21,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,21],[24,300,0.08,0.89286,0.20203,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,21],[28,300,0.0933,0.70982,0.36506,0.57143,0.85714,1.0,0.0,1.0,6,12,4,6,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,12],[32,300,0.1067,0.80803,0.26149,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,14],[36,300,0.12,0.84821,0.14698,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,9,0,12],[40,300,0.1333,0.87054,0.21829,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,5,0,20],[44,300,0.1467,0.80801,0.27807,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,15],[48,300,0.16,0.76339,0.30432,0.67857,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,3,0,15],[52,300,0.1733,0.69196,0.36615,0.57143,0.78564,1.0,0.0,1.0,6,13,1,6,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,3,0,13],[56,300,0.1867,0.79908,0.26454,0.71429,0.85714,1.0,0.0,1.0,2,15,1,2,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,4,0,15],[60,300,0.2,0.79017,0.2575,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,10,0,11],[64,300,0.2133,0.82143,0.26,0.71429,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,16],[68,300,0.2267,0.86829,0.21139,0.82132,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,4,0,0,5,0,19],[72,300,0.24,0.79018,0.35353,0.71429,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,21],[76,300,0.2533,0.82143,0.27433,0.71429,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,6,0,17],[80,300,0.2667,0.77232,0.35689,0.71429,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,18],[84,300,0.28,0.84374,0.25596,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,3,0,20],[88,300,0.2933,0.78572,0.30514,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,16],[92,300,0.3067,0.79461,0.26474,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,3,0,15],[96,300,0.32,0.88391,0.14917,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,7,0,17],[100,300,0.3333,0.86161,0.23551,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,19],[104,300,0.3467,0.85714,0.24223,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,20],[108,300,0.36,0.76338,0.28708,0.67857,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,0,3,0,14],[112,300,0.3733,0.83929,0.28291,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,21],[116,300,0.3867,0.88829,0.178,0.85711,1.0,1.0,0.14,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,18],[120,300,0.4,0.85722,0.2143,0.85714,0.93,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,16],[124,300,0.4133,0.83927,0.23078,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,7,0,16],[128,300,0.4267,0.86607,0.16342,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,4,0,17],[132,300,0.44,0.89732,0.1394,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,18],[136,300,0.4533,0.76338,0.28034,0.71429,0.85714,1.0,0.0,1.0,3,9,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,11,0,9],[140,300,0.4667,0.80356,0.27606,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,6,0,0,8,0,14],[144,300,0.48,0.82143,0.27433,0.82143,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,8,0,16],[148,300,0.4933,0.87498,0.21055,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,18],[152,300,0.5067,0.86606,0.24728,0.85711,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,19],[156,300,0.52,0.9308,0.09015,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,10,0,19],[160,300,0.5333,0.87052,0.2063,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,18],[164,300,0.5467,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[168,300,0.56,0.43304,0.41571,0.0,0.35714,1.0,0.0,1.0,12,9,0,12,0,1,0,0,3,0,0,4,0,0,0,0,0,3,0,0,0,0,9],[172,300,0.5733,0.22768,0.36045,0.0,0.0,0.32143,0.0,1.0,20,5,0,20,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[176,300,0.5867,0.2991,0.39505,0.0,0.0,0.57143,0.0,1.0,17,5,1,17,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,5],[180,300,0.6,0.49106,0.37954,0.0,0.5712,0.85714,0.0,1.0,10,4,0,10,0,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,6,0,4],[184,300,0.6133,0.30357,0.41149,0.0,0.0,0.75,0.0,1.0,18,6,1,1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,0,0,0,0,0,7,0,0,2,0,0,5,0,0,5,1,9],[256,319,0.8025,0.70087,0.31413,0.5713,0.71429,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,3,0,0,4,0,0,6,0,0,4,0,11],[260,319,0.815,0.69195,0.3483,0.42857,0.85714,1.0,0.0,1.0,4,14,0,4,0,0,0,0,1,0,0,5,0,0,3,0,0,2,0,0,3,0,14],[264,319,0.8276,0.70086,0.26089,0.53539,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,5,0,0,4,0,0,7,0,0,4,0,9],[268,319,0.8401,0.68749,0.37702,0.28571,0.92857,1.0,0.0,1.0,4,16,0,4,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,0,2,0,16],[272,319,0.8527,0.64731,0.31539,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,3,0,0,7,0,0,1,0,0,4,0,0,5,0,9],[276,319,0.8652,0.73213,0.25693,0.42857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,6,0,0,1,0,0,4,0,0,9,0,9],[280,319,0.8777,0.70089,0.31816,0.42859,0.71429,1.0,0.0,1.0,3,12,0,3,0,0,0,0,2,0,0,4,0,0,0,0,0,9,0,0,2,0,12],[284,319,0.8903,0.64732,0.35711,0.42857,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,2,0,0,5,0,0,2,0,0,3,0,0,3,0,12],[288,319,0.9028,0.69196,0.35194,0.42857,0.85714,1.0,0.0,1.0,3,14,0,3,0,1,0,0,3,0,0,4,0,0,1,0,0,2,0,0,4,0,14],[292,319,0.9154,0.45981,0.28734,0.28571,0.42857,0.60714,0.0,1.0,4,4,0,4,0,0,0,0,8,0,0,10,0,0,2,0,0,3,0,0,1,0,4],[296,319,0.9279,0.558,0.34136,0.39286,0.4998,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,3,0,0,8,0,0,3,0,0,3,0,0,2,0,8],[300,319,0.9404,0.43304,0.26119,0.28571,0.42857,0.60714,0.0,1.0,3,1,0,3,0,3,0,0,7,0,0,9,0,0,2,0,0,4,0,0,3,0,1],[304,319,0.953,0.32142,0.29013,0.0,0.28571,0.42857,0.0,1.0,10,3,0,10,0,1,0,0,6,0,0,10,0,0,2,0,0,0,0,0,0,0,3],[308,319,0.9655,0.41518,0.3542,0.0,0.42857,0.71429,0.0,1.0,9,5,0,9,0,2,0,0,3,0,0,8,0,0,0,0,0,4,0,0,1,0,5],[312,319,0.9781,0.36606,0.32914,0.0,0.42857,0.4642,0.0,1.0,10,3,0,10,0,2,0,0,3,0,0,9,0,0,1,0,0,2,0,0,2,0,3],[316,319,0.9906,0.34812,0.30297,0.105,0.28571,0.42857,0.0,1.0,8,3,0,8,0,2,0,0,8,0,0,9,0,0,0,0,0,0,0,0,2,0,3],[319,319,1.0,0.06697,0.11837,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d8ac2410da4d410a","q":"Let  $d(n)$  denote the number of positive divisors of the natural number  $n$ . Find all the natural numbers  $n$  such that   $$ d(n) = \\frac{n}{5} $$  .","t":[{"b":2,"e":1.0,"k":"flat","v":0.95535,"x":0.97321,"p":[[0,6,0.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,6,0.6667,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[6,6,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":7,"e":1.0,"k":"flat","v":0.89286,"x":0.99554,"p":[[0,31,0.0,0.91964,0.16728,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,23],[4,31,0.129,0.89286,0.19562,0.85714,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,19],[8,31,0.2581,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[12,31,0.3871,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[16,31,0.5161,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,31,0.6452,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,31,0.7742,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,31,0.9032,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"103e5b101cf34078","q":"Determine the smallest positive integer  $q$  with the following property: for  every  integer  $m$  with   $1\\leqslant m\\leqslant 1006$ , there  exists  an  integer  $n$  such that  $$ \\dfrac{m}{1007}q<n<\\dfrac{m+1}{1008}q $$  .","t":[{"b":0,"e":0.57143,"k":"rising","v":0.52679,"x":0.79909,"p":[[0,148,0.0,0.52679,0.30606,0.28571,0.57143,0.75,0.0,1.0,3,3,1,3,0,3,0,0,5,0,0,3,0,0,5,0,0,5,0,0,5,0,3],[4,148,0.027,0.79909,0.24965,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,7,0,0,5,0,14],[8,148,0.0541,0.76339,0.249,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,0,13,0,7],[12,148,0.0811,0.70982,0.25123,0.67857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,1,0,0,3,0,0,12,0,0,5,0,7],[16,148,0.1081,0.79463,0.22285,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,0,9,0,11],[20,148,0.1351,0.76339,0.22477,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,14,0,6],[24,148,0.1622,0.63381,0.30725,0.28571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,5,0,0,1,0,0,3,0,0,7,0,0,5,0,7],[28,148,0.1892,0.63829,0.32351,0.39286,0.71429,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,2,0,0,1,0,0,7,0,0,8,0,6],[32,148,0.2162,0.625,0.31894,0.2857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,5,0,0,2,0,0,0,0,0,5,0,0,10,0,5],[36,148,0.2432,0.65178,0.25739,0.53571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,3,0,0,2,0,0,3,0,0,12,0,0,4,0,5],[40,148,0.2703,0.73661,0.30952,0.67857,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,5,0,0,0,0,0,1,0,0,5,0,0,7,0,12],[44,148,0.2973,0.74107,0.28446,0.71429,0.78571,1.0,0.0,1.0,2,11,0,2,0,0,0,0,3,0,0,0,0,0,2,0,0,9,0,0,5,0,11],[48,148,0.3243,0.66964,0.31019,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,1,0,0,0,0,0,3,0,0,9,0,0,5,0,8],[52,148,0.3514,0.69196,0.26752,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,0,0,0,6,0,0,11,0,0,4,0,7],[56,148,0.3784,0.71426,0.2287,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,1,0,0,8,0,0,7,0,0,9,0,5],[60,148,0.4054,0.68749,0.18364,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,0,0,0,1,0,0,10,0,0,11,0,0,7,0,2],[64,148,0.4324,0.66071,0.22517,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,0,0,0,9,0,0,9,0,0,8,0,2],[68,148,0.4595,0.71427,0.17129,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,14,0,0,5,0,4],[72,148,0.4865,0.68303,0.21048,0.57143,0.71429,0.85704,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,14,0,0,6,0,3],[76,148,0.5135,0.66518,0.22192,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,2,0,0,7,0,0,10,0,0,8,0,2],[80,148,0.5405,0.69195,0.15199,0.71429,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,19,0,0,6,0,0],[84,148,0.5676,0.66962,0.19379,0.57143,0.71429,0.75,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,0,0,0,11,0,0,10,0,0,5,0,3],[88,148,0.5946,0.69196,0.23176,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,3,0,0,4,0,0,11,0,0,6,0,5],[92,148,0.6216,0.60267,0.23072,0.57143,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,2,0,0,1,0,0,9,0,0,11,0,0,6,0,0],[96,148,0.6486,0.67857,0.19885,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,11,0,0,8,0,2],[100,148,0.6757,0.71873,0.23279,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,1,0,0,5,0,0,7,0,0,9,0,6],[104,148,0.7027,0.66071,0.23623,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,3,0,0,0,0,0,7,0,0,9,0,0,9,0,2],[108,148,0.7297,0.66963,0.23266,0.57143,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,2,0,0,0,0,0,6,0,0,11,0,0,10,0,1],[112,148,0.7568,0.64732,0.23686,0.57143,0.71429,0.75,0.0,1.0,1,3,0,1,0,2,0,0,0,0,0,4,0,0,5,0,0,12,0,0,5,0,3],[116,148,0.7838,0.74107,0.21558,0.71429,0.78564,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,10,0,0,12,0,4],[120,148,0.8108,0.62943,0.24186,0.5354,0.71429,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,4,0,0,7,0,0,11,0,0,1,0,5],[124,148,0.8378,0.65177,0.22571,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,7,0,0,8,0,3],[128,148,0.8649,0.63836,0.223,0.571,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,2,0,0,6,0,0,10,0,0,8,0,1],[132,148,0.8919,0.62498,0.28515,0.39288,0.71429,0.85714,0.0,1.0,1,4,0,1,0,3,0,0,4,0,0,1,0,0,4,0,0,8,0,0,7,0,4],[136,148,0.9189,0.73212,0.19481,0.67857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,10,0,0,10,0,4],[140,148,0.9459,0.67409,0.20277,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,11,0,0,8,0,2],[144,148,0.973,0.73213,0.19804,0.67857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,12,0,0,7,0,5],[148,148,1.0,0.68749,0.18364,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,12,0,0,6,0,3]]},{"b":7,"e":0.0,"k":"falling","v":0.13384,"x":0.83036,"p":[[0,139,0.0,0.50892,0.32328,0.24999,0.64286,0.71429,0.0,1.0,4,3,1,4,0,4,0,0,5,0,0,1,0,0,2,0,0,10,0,0,3,0,3],[4,139,0.0288,0.83036,0.16146,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,10,0,11],[8,139,0.0576,0.76339,0.22192,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,0,0,0,5,0,0,10,0,0,6,0,9],[12,139,0.0863,0.76777,0.26687,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,10],[16,139,0.1151,0.73661,0.30117,0.71429,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,6,0,11],[20,139,0.1439,0.75892,0.20652,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,14,0,0,7,0,7],[24,139,0.1727,0.76339,0.22192,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,14,0,0,6,0,8],[28,139,0.2014,0.78123,0.22012,0.71429,0.85707,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,0,9,0,8],[32,139,0.2302,0.7857,0.2143,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,7,0,0,5,0,12],[36,139,0.259,0.75891,0.25615,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,1,0,0,2,0,0,7,0,0,8,0,10],[40,139,0.2878,0.80357,0.20748,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,8,0,0,10,0,10],[44,139,0.3165,0.62053,0.29797,0.42857,0.71429,0.85704,0.0,1.0,3,5,0,3,0,1,0,0,3,0,0,2,0,0,3,0,0,11,0,0,4,0,5],[48,139,0.3453,0.75445,0.2059,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,14,0,0,6,0,7],[52,139,0.3741,0.71875,0.28456,0.67857,0.71429,1.0,0.0,1.0,1,10,0,1,0,3,0,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,10],[56,139,0.4029,0.63393,0.333,0.39286,0.71429,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,3,0,0,1,0,0,3,0,0,7,0,0,6,0,7],[60,139,0.4317,0.74107,0.25111,0.67857,0.85707,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,10,0,7],[64,139,0.4604,0.60714,0.26245,0.53571,0.57143,0.75,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,1,0,0,9,0,0,7,0,0,4,0,4],[68,139,0.4892,0.59373,0.33141,0.2857,0.71429,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,5,0,0,1,0,0,2,0,0,6,0,0,6,0,6],[72,139,0.518,0.65178,0.29437,0.53569,0.71429,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,2,0,0,2,0,0,4,0,0,8,0,0,6,0,6],[76,139,0.5468,0.59374,0.37134,0.2857,0.71429,0.89286,0.0,1.0,7,8,0,7,0,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,5,0,8],[80,139,0.5755,0.62053,0.36528,0.2857,0.71429,1.0,0.0,1.0,5,10,0,5,0,2,0,0,2,0,0,1,0,0,2,0,0,8,0,0,2,0,10],[84,139,0.6043,0.62499,0.34023,0.39286,0.71429,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,2,0,0,2,0,0,2,0,0,7,0,0,6,0,7],[88,139,0.6331,0.49107,0.34058,0.14286,0.57143,0.71429,0.0,1.0,7,3,0,7,0,2,0,0,3,0,0,2,0,0,3,0,0,9,0,0,3,0,3],[92,139,0.6619,0.61161,0.36111,0.35715,0.71429,1.0,0.0,1.0,5,9,0,5,0,3,0,0,0,0,0,1,0,0,5,0,0,6,0,0,3,0,9],[96,139,0.6906,0.4732,0.37361,0.0,0.64286,0.74996,0.0,1.0,9,3,0,9,0,3,0,0,2,0,0,0,0,0,2,0,0,8,0,0,5,0,3],[100,139,0.7194,0.54463,0.34337,0.39286,0.64286,0.75,0.0,1.0,7,5,0,7,0,0,0,0,1,0,0,5,0,0,3,0,0,8,0,0,3,0,5],[104,139,0.7482,0.61161,0.38004,0.14286,0.71429,1.0,0.0,1.0,4,9,0,4,0,6,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,9],[108,139,0.777,0.41069,0.36725,0.0,0.35714,0.75,0.0,1.0,9,3,0,9,0,6,0,0,1,0,0,1,0,0,5,0,0,2,0,0,5,0,3],[112,139,0.8058,0.35268,0.3589,0.10714,0.14286,0.71429,0.0,1.0,8,3,0,8,0,11,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,3],[116,139,0.8345,0.42411,0.4218,0.0,0.35716,0.85714,0.0,1.0,14,6,0,14,0,2,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,6],[120,139,0.8633,0.39723,0.40527,0.0,0.21428,0.85714,0.0,1.0,13,5,0,13,0,3,0,0,1,0,0,2,0,0,1,0,0,3,0,0,4,0,5],[124,139,0.8921,0.30803,0.38977,0.0,0.07143,0.71429,0.0,1.0,16,4,0,16,0,4,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,4],[128,139,0.9209,0.46875,0.37327,0.14286,0.57143,0.75,0.0,1.0,7,4,0,7,0,7,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,4],[132,139,0.9496,0.41963,0.37785,0.0,0.35693,0.71429,0.0,1.0,9,4,0,9,0,7,0,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,4],[136,139,0.9784,0.48214,0.38258,0.0,0.5,0.85714,0.0,1.0,9,5,0,9,0,2,0,0,2,0,0,3,0,0,2,0,0,4,0,0,5,0,5],[139,139,1.0,0.13384,0.2111,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"5431bc2e3e798d13","q":"Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set  $S$  of  $n>1$  collinear points. Then, for every unordered pair of points  $\\{X,Y\\}$  in  $S$ , Elmo draws the circle with diameter  $XY$  so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of  $n$ , what is the minimum number of colors Elmo could have used?\n\n*Michael Ren*","t":[{"b":4,"e":0.4286,"k":"rising","v":0.1875,"x":0.69197,"p":[[0,71,0.0,0.1875,0.13092,0.0,0.2857,0.28571,0.0,0.28571,10,0,10,10,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.63838,0.25501,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,4,0,0,4,0,0,3,0,0,10,0,0,4,0,5],[8,71,0.1127,0.62052,0.22478,0.53539,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,11,0,0,3,0,3],[12,71,0.169,0.64284,0.24485,0.5354,0.71429,0.75,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,0,3,0,0,4,0,0,12,0,0,4,0,4],[16,71,0.2254,0.58929,0.26184,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,5,0,0,6,0,0,7,0,0,2,0,5],[20,71,0.2817,0.69197,0.23176,0.42859,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,10,0,0,4,0,7],[24,71,0.338,0.61164,0.26055,0.42857,0.64286,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,2,0,0,7,0,0,4,0,0,9,0,0,1,0,6],[28,71,0.3944,0.62947,0.24185,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,5,0,0,3,0,0,8,0,0,7,0,3],[32,71,0.4507,0.6607,0.2442,0.4286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,9,0,0,2,0,7],[36,71,0.507,0.59374,0.18935,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,6,0,0,6,0,0,13,0,0,2,0,1],[40,71,0.5634,0.5713,0.17486,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,12,0,0,7,0,0,7,0,0,3,0,1],[44,71,0.6197,0.56697,0.23279,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,5,0,0,8,0,0,7,0,0,2,0,3],[48,71,0.6761,0.53572,0.21724,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,8,0,0,8,0,0,7,0,0,1,0,2],[52,71,0.7324,0.57589,0.19719,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,13,0,0,4,0,0,9,0,0,2,0,2],[56,71,0.7887,0.52679,0.26351,0.42857,0.42857,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,12,0,0,4,0,0,2,0,0,3,0,4],[60,71,0.8451,0.57128,0.18547,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,12,0,0,6,0,0,8,0,0,3,0,1],[64,71,0.9014,0.55802,0.19018,0.42857,0.57121,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,10,0,0,4,0,0,10,0,0,2,0,1],[68,71,0.9577,0.4464,0.10561,0.42857,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,17,0,0,10,0,0,0,0,0,0,0,0],[71,71,1.0,0.41076,0.10565,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,20,0,0,3,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.18304,"x":0.64286,"p":[[0,52,0.0,0.18304,0.13475,0.0,0.28571,0.28571,0.0,0.286,11,0,11,11,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.64286,0.22016,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,2,0,0,5,0,0,16,0,0,2,0,3],[8,52,0.1538,0.60713,0.24485,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,2,0,0,8,0,0,8,0,0,3,0,4],[12,52,0.2308,0.48659,0.20472,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,7,0,0,7,0,0,6,0,0,7,0,0,2,0,0],[16,52,0.3077,0.58479,0.22688,0.42857,0.64286,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,7,0,0,4,0,0,10,0,0,5,0,1],[20,52,0.3846,0.56249,0.28107,0.28571,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,3,0,0,1,0,0,4,0,0,10,0,0,5,0,2],[24,52,0.4615,0.52678,0.22428,0.42857,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,11,0,0,5,0,0,6,0,0,2,0,2],[28,52,0.5385,0.54463,0.19377,0.42857,0.42859,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,12,0,0,3,0,0,10,0,0,0,0,2],[32,52,0.6154,0.50445,0.19556,0.42857,0.4286,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,10,0,0,4,0,0,10,0,0,1,0,0],[36,52,0.6923,0.51783,0.25441,0.28571,0.4998,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,6,0,0,6,0,0,5,0,0,2,0,3],[40,52,0.7692,0.5982,0.26108,0.42857,0.57143,0.71429,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,9,0,0,5,0,0,6,0,0,0,0,7],[44,52,0.8462,0.47767,0.14986,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,17,0,0,5,0,0,6,0,0,0,0,0],[48,52,0.9231,0.49552,0.10704,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,18,0,0,11,0,0,1,0,0,1,0,0],[52,52,1.0,0.45982,0.11701,0.42857,0.42857,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,20,0,0,6,0,0,1,0,0,1,0,0]]}]},{"i":"cffdb52d9300c764","q":"Consider a regular octahedron  $ABCDEF$  with lower vertex  $E$ , upper vertex  $F$ , middle cross-section  $ABCD$ , midpoint  $M$  and circumscribed sphere  $k$ . Further, let  $X$  be an arbitrary point inside the face  $ABF$ . Let the line  $EX$  intersect  $k$  in  $E$  and  $Z$ , and the plane  $ABCD$  in  $Y$ .\nShow that  $\\sphericalangle{EMZ}=\\sphericalangle{EYF}$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.03572,"x":0.16964,"p":[[0,44,0.0,0.16964,0.21852,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,9,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[4,44,0.0909,0.13393,0.29001,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[8,44,0.1818,0.15625,0.23787,0.0,0.0,0.2857,0.0,0.85714,18,0,0,18,0,5,0,0,5,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[12,44,0.2727,0.04911,0.14987,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,44,0.3636,0.09375,0.13175,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.07588,0.13351,0.0,0.0,0.14286,0.0,0.571,22,0,0,22,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,44,0.5455,0.06251,0.1126,0.0,0.0,0.03571,0.0,0.286,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.07589,0.14279,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.03572,0.08751,0.0,0.0,0.0,0.0,0.286,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.04018,0.1197,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"rising","v":0.125,"x":0.28571,"p":[[0,29,0.0,0.13391,0.22567,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,6,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[4,29,0.1379,0.23661,0.24643,0.0,0.14286,0.42857,0.0,0.85714,12,0,0,12,0,6,0,0,3,0,0,7,0,0,1,0,0,2,0,0,1,0,0],[8,29,0.2759,0.125,0.20124,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,29,0.4138,0.18079,0.23279,0.0,0.07143,0.30352,0.0,1.0,16,1,0,16,0,4,0,0,4,0,1,5,0,0,1,0,0,0,0,0,0,0,1],[16,29,0.5517,0.18749,0.21258,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,3,0,0,8,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[20,29,0.6897,0.27677,0.22284,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,5,0,0,11,0,0,4,0,0,4,0,0,0,0,0,0,0,1],[24,29,0.8276,0.1964,0.22227,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,2,0,0,9,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[28,29,0.9655,0.13829,0.15761,0.0,0.07,0.2857,0.0,0.571,16,0,0,16,0,4,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[29,29,1.0,0.28571,0.22588,0.14286,0.28571,0.42857,0.0,0.85714,7,0,0,7,0,5,0,0,10,0,0,5,0,0,2,0,0,2,0,0,1,0,0]]}]},{"i":"7f0e22a75ae3685a","q":"Given  $n$  real numbers  $a_1 \\leq a_2 \\leq \\cdots \\leq a_n$ , define\n\\[M_1=\\frac 1n \\sum_{i=1}^{n} a_i , \\quad M_2=\\frac{2}{n(n-1)} \\sum_{1 \\leq i<j \\leq n} a_ia_j, \\quad Q=\\sqrt{M_1^2-M_2}\\]\nProve that\n\\[a_1 \\leq  M_1 - Q \\leq  M_1 + Q \\leq a_n\\]\nand that equality holds if and only if  $a_1 = a_2 = \\cdots = a_n.$","t":[{"b":3,"e":1.0,"k":"flat","v":0.85267,"x":0.96428,"p":[[0,84,0.0,0.89284,0.13366,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,15],[4,84,0.0476,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[8,84,0.0952,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[12,84,0.1429,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[16,84,0.1905,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[20,84,0.2381,0.91071,0.15465,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,17],[24,84,0.2857,0.9241,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[28,84,0.3333,0.93302,0.1071,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[32,84,0.381,0.89732,0.15663,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,10,0,18],[36,84,0.4286,0.90622,0.11076,0.85711,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[40,84,0.4762,0.93736,0.12364,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[44,84,0.5238,0.9375,0.11811,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,22],[48,84,0.5714,0.95089,0.15815,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[52,84,0.619,0.91071,0.12753,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,18],[56,84,0.6667,0.89732,0.17941,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[60,84,0.7143,0.89286,0.22016,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,23],[64,84,0.7619,0.87053,0.19678,0.82132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,5,0,19],[68,84,0.8095,0.86604,0.2313,0.85711,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,17],[72,84,0.8571,0.88838,0.21349,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,21],[76,84,0.9048,0.93749,0.15545,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[80,84,0.9524,0.85267,0.26119,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,6,0,20],[84,84,1.0,0.90177,0.17658,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,22]]},{"b":5,"e":1.0,"k":"falling","v":0.71428,"x":0.98213,"p":[[0,21,0.0,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,20],[4,21,0.1905,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[8,21,0.381,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,21,0.5714,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[16,21,0.7619,0.91517,0.15096,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,9,0,20],[20,21,0.9524,0.7455,0.29394,0.5354,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,4,0,0,3,0,0,1,0,0,7,0,13],[21,21,1.0,0.71428,0.2988,0.42859,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,5,0,0,3,0,0,0,0,0,4,0,0,7,0,11]]}]},{"i":"e84aaa640bedd948","q":"Given a family  $C$  of circles of the same radius  $R$ , which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to  $R\\sqrt3$  .","t":[{"b":1,"e":0.0,"k":"falling","v":0.02232,"x":0.23214,"p":[[0,36,0.0,0.23214,0.29179,0.0,0.0,0.35714,0.0,1.0,17,1,0,17,0,0,0,0,7,0,0,0,0,0,5,0,0,1,0,0,1,0,1],[4,36,0.1111,0.17408,0.25931,0.0,0.0,0.571,0.0,0.57143,22,0,0,22,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[8,36,0.2222,0.14286,0.26245,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[12,36,0.3333,0.10268,0.24804,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[16,36,0.4444,0.17402,0.28289,0.0,0.0,0.25,0.0,1.0,21,1,0,21,0,3,0,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,1],[20,36,0.5556,0.11161,0.22513,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[24,36,0.6667,0.0625,0.15947,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,36,0.7778,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[32,36,0.8889,0.07143,0.21129,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[36,36,1.0,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.08929,"x":0.29467,"p":[[0,29,0.0,0.29467,0.32329,0.0,0.07143,0.57143,0.0,1.0,16,2,0,16,0,1,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,2],[4,29,0.1379,0.20536,0.31931,0.0,0.0,0.46429,0.0,1.0,20,2,0,20,0,3,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,2],[8,29,0.2759,0.17857,0.28571,0.0,0.0,0.46429,0.0,0.85714,22,0,0,22,0,1,0,0,0,0,0,1,0,0,6,0,0,0,0,0,2,0,0],[12,29,0.4138,0.08929,0.21651,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[16,29,0.5517,0.12946,0.25344,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,1],[20,29,0.6897,0.15624,0.25592,0.0,0.0,0.35702,0.0,0.71429,23,0,0,23,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,0],[24,29,0.8276,0.13392,0.25983,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,0,1],[28,29,0.9655,0.19196,0.33237,0.0,0.0,0.35713,0.0,1.0,23,3,0,23,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,0,3],[29,29,1.0,0.14732,0.28901,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,1]]}]},{"i":"8f0e565a48284d03","q":"Let  $ABC$  be an acute triangle with the perimeter of  $2s$ . We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers  $A, B$ , and  $C$ , respectively. Prove that there exists a circle with the radius of  $s$  which contains all the three circles. \n\n*Proposed by Josef Tkadlec, Czechia*","t":[{"b":3,"e":0.14286,"k":"falling","v":0.00893,"x":0.5223,"p":[[0,24,0.0,0.37054,0.33666,0.14286,0.14286,0.46429,0.0,1.0,1,6,0,1,0,18,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,6],[4,24,0.1667,0.5223,0.28707,0.24999,0.571,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,2,0,0,5,0,0,4,0,0,8,0,0,1,0,4],[8,24,0.3333,0.49552,0.2696,0.24999,0.4998,0.71429,0.14286,1.0,0,3,0,0,0,8,0,0,2,0,0,6,0,0,6,0,0,6,0,0,1,0,3],[12,24,0.5,0.24999,0.24483,0.14286,0.14286,0.32143,0.0,1.0,6,1,0,6,0,15,0,0,3,0,0,2,0,0,3,0,0,2,0,0,0,0,1],[16,24,0.6667,0.05804,0.16311,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,24,0.8333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.35715,"x":0.90623,"p":[[0,50,0.0,0.35715,0.33503,0.14286,0.14286,0.42858,0.0,1.0,1,6,0,1,0,18,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,6],[4,50,0.08,0.69187,0.30761,0.53572,0.71429,1.0,0.0,1.0,2,11,0,2,0,1,0,0,2,0,0,3,0,0,5,0,0,4,0,0,4,0,11],[8,50,0.16,0.87723,0.18319,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,1,0,2,0,20],[12,50,0.24,0.70533,0.3233,0.53539,0.71429,1.0,0.0,1.0,2,14,0,2,0,2,0,0,1,0,0,3,0,0,4,0,0,5,0,0,1,0,14],[16,50,0.32,0.73658,0.26754,0.42857,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,7,0,0,2,0,0,3,0,0,4,0,13],[20,50,0.4,0.81249,0.21262,0.67857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,4,0,15],[24,50,0.48,0.79014,0.25505,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,3,0,0,3,0,16],[28,50,0.56,0.78569,0.23148,0.57132,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,5,0,0,4,0,14],[32,50,0.64,0.83476,0.19604,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,7,0,15],[36,50,0.72,0.70979,0.22157,0.5354,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,5,0,0,7,0,7],[40,50,0.8,0.80356,0.20749,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,4,0,14],[44,50,0.88,0.81246,0.20027,0.71429,0.85707,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,6,0,13],[48,50,0.96,0.81249,0.20653,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,3,0,15],[50,50,1.0,0.90623,0.1499,0.85711,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,21]]}]},{"i":"d35d861da8d0acba","q":"Define the infinite products \\[ A = \\prod\\limits_{i=2}^{\\infty} \\left(1-\\frac{1}{n^3}\\right) \\text{ and }  B = \\prod\\limits_{i=1}^{\\infty}\\left(1+\\frac{1}{n(n+1)}\\right). \\] If  $\\tfrac{A}{B} = \\tfrac{m}{n}$  where  $m,n$  are relatively prime positive integers, determine  $100m+n$ .\n\n*Proposed by Lewis Chen*","t":[{"b":0,"e":0.71429,"k":"falling","v":0.625,"x":0.88393,"p":[[0,41,0.0,0.80804,0.20079,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,8,0,0,3,0,14],[4,41,0.0976,0.88393,0.16917,0.71429,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,1,0,21],[8,41,0.1951,0.72768,0.16506,0.57143,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,0,2,0,7],[12,41,0.2927,0.67411,0.15251,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,8,0,0,2,0,4],[16,41,0.3902,0.73214,0.17768,0.57143,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,6,0,0,3,0,8],[20,41,0.4878,0.69643,0.15465,0.57143,0.64286,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,9,0,0,2,0,5],[24,41,0.5854,0.69196,0.16016,0.57143,0.57143,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,9,0,0,0,0,6],[28,41,0.6829,0.70536,0.17835,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,8,0,0,3,0,6],[32,41,0.7805,0.62946,0.07873,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,14,0,0,0,0,0],[36,41,0.878,0.64286,0.10714,0.57143,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,15,0,0,0,0,1],[40,41,0.9756,0.625,0.09279,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,9,0,0,0,0,1],[41,41,1.0,0.625,0.12242,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,8,0,0,0,0,2]]},{"b":2,"e":1.0,"k":"rising","v":0.74554,"x":0.99554,"p":[[0,5,0.0,0.74554,0.23619,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,5,0,0,3,0,12],[4,5,0.8,0.85268,0.22442,0.67857,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,2,0,0,1,0,21],[5,5,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"e3eb32b7fac16785","q":"Let  $p=4k+3$  be a prime number. Find the number of different residues mod p of  $(x^{2}+y^{2})^{2}$  where  $(x,p)=(y,p)=1.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,49,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,49,0.0816,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,49,0.1633,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,49,0.2449,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,49,0.3265,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,49,0.4082,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,49,0.4898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,49,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,49,0.6531,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,49,0.7347,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,49,0.8163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,49,0.898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,49,0.9796,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[49,49,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,27,0.0,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,27,0.1481,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,27,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,27,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[27,27,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"e58280592a6187d6","q":"Let  $w$  be a circle and  $AB$  a line not intersecting  $w$ . Given a point  $P_{0}$  on  $w$ , define the sequence  $P_{0},P_{1},\\ldots $  as follows:  $P_{n\\plus{}1}$  is the second intersection with  $w$  of the line passing through  $B$  and the second intersection of the line  $AP_{n}$  with  $w$ . Prove that for a positive integer  $k$ , if  $P_{0}\\equal{}P_{k}$  for some choice of  $P_{0}$ , then  $P_{0}\\equal{}P_{k}$  for any choice of  $P_{0}$ .\n\n*Gheorge Eckstein*","t":[{"b":0,"e":1.0,"k":"rising","v":0.75,"x":1.0,"p":[[0,18,0.0,0.75,0.3896,0.67857,1.0,1.0,0.0,1.0,6,19,6,6,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,19],[4,18,0.2222,0.95982,0.1439,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[8,18,0.4444,0.88393,0.22142,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,24],[12,18,0.6667,0.96426,0.10109,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[16,18,0.8889,0.95089,0.18073,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":0.57143,"k":"falling","v":0.48658,"x":1.0,"p":[[0,42,0.0,0.87054,0.26088,0.85714,1.0,1.0,0.0,1.0,2,22,2,2,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,22],[4,42,0.0952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,42,0.1905,0.8482,0.25239,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,0,0,22],[12,42,0.2857,0.91963,0.17476,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,25],[16,42,0.381,0.88838,0.17031,0.82143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,3,0,21],[20,42,0.4762,0.8482,0.22852,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,0,3,0,20],[24,42,0.5714,0.79907,0.19191,0.57143,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,5,0,13],[28,42,0.6667,0.66962,0.22712,0.57143,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,1,0,0,17,0,0,2,0,0,0,0,9],[32,42,0.7619,0.61157,0.2402,0.53539,0.57143,0.74996,0.2857,1.0,0,6,0,0,0,0,0,0,7,0,0,1,0,0,14,0,0,2,0,0,2,0,6],[36,42,0.8571,0.5223,0.15814,0.42859,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,2,0,0,21,0,0,0,0,0,1,0,1],[40,42,0.9524,0.58923,0.16269,0.571,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,23,0,0,0,0,0,1,0,3],[42,42,1.0,0.48658,0.12297,0.39286,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,3,0,0,21,0,0,0,0,0,0,0,0]]}]},{"i":"4b0cee1efddf80fe","q":"Let  $ABC$  be an isosceles triangle with  $AB = BC$ . Point  $E$  lies on the side  $AB$ , and  $ED$  is the perpendicular from  $E$  to  $BC$ . It is known that  $AE = DE$ . Find  $\\angle DAC$ .","t":[{"b":3,"e":0.4286,"k":"falling","v":0.32143,"x":0.68749,"p":[[0,67,0.0,0.68749,0.3223,0.28571,0.85714,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,5,0,0,1,0,0,2,0,0,1,0,0,9,0,10],[4,67,0.0597,0.41964,0.27879,0.14286,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,11,0,0,3,0,0,9,0,0,3,0,0,2,0,0,0,0,4],[8,67,0.1194,0.40179,0.27301,0.14286,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,10,0,0,8,0,0,4,0,0,6,0,0,0,0,0,0,0,4],[12,67,0.1791,0.38839,0.27947,0.14286,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,12,0,0,7,0,0,4,0,0,4,0,0,0,0,0,2,0,3],[16,67,0.2388,0.41964,0.27879,0.24999,0.35714,0.42858,0.14286,1.0,0,5,0,0,0,8,0,0,8,0,0,9,0,0,2,0,0,0,0,0,0,0,5],[20,67,0.2985,0.37499,0.25691,0.14286,0.28571,0.42857,0.0,1.0,1,3,0,1,0,9,0,0,8,0,0,7,0,0,3,0,0,1,0,0,0,0,3],[24,67,0.3582,0.3839,0.19701,0.25,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,10,0,0,6,0,0,1,0,0,0,0,1],[28,67,0.4179,0.43302,0.2823,0.14286,0.42857,0.57111,0.14286,1.0,0,5,0,0,0,9,0,0,5,0,0,9,0,0,4,0,0,0,0,0,0,0,5],[32,67,0.4776,0.46875,0.30771,0.25,0.42857,0.60714,0.14286,1.0,0,6,0,0,0,8,0,0,7,0,0,6,0,0,3,0,0,1,0,0,1,0,6],[36,67,0.5373,0.38838,0.25059,0.14289,0.28571,0.57111,0.14286,1.0,0,3,0,0,0,10,0,0,7,0,0,6,0,0,6,0,0,0,0,0,0,0,3],[40,67,0.597,0.44195,0.23243,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,7,0,0,4,0,0,10,0,0,6,0,0,1,0,0,3,0,1],[44,67,0.6567,0.38391,0.24855,0.14286,0.28571,0.46418,0.14286,1.0,0,3,0,0,0,10,0,0,7,0,0,7,0,0,5,0,0,0,0,0,0,0,3],[48,67,0.7164,0.41964,0.28779,0.24999,0.28571,0.42858,0.14286,1.0,0,5,0,0,0,8,0,0,9,0,0,9,0,0,0,0,0,0,0,0,1,0,5],[52,67,0.7761,0.34817,0.26707,0.14286,0.2143,0.46393,0.14286,1.0,0,2,0,0,0,16,0,0,4,0,0,4,0,0,4,0,0,0,0,0,2,0,2],[56,67,0.8358,0.37947,0.23585,0.14286,0.42857,0.4286,0.0,1.0,1,2,0,1,0,10,0,0,2,0,0,12,0,0,4,0,0,1,0,0,0,0,2],[60,67,0.8955,0.32143,0.17857,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,8,0,0,8,0,0,3,0,0,2,0,0,0,0,0],[64,67,0.9552,0.33481,0.18072,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,11,0,0,7,0,0,8,0,0,5,0,0,0,0,0,1,0,0],[67,67,1.0,0.44642,0.16655,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,11,0,0,11,0,0,1,0,0,1,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.27678,"x":0.81694,"p":[[0,100,0.0,0.81694,0.23486,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,3,0,0,5,0,16],[4,100,0.04,0.37052,0.25469,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,10,0,0,6,0,0,9,0,0,2,0,0,0,0,0,2,0,2],[8,100,0.08,0.5357,0.3677,0.14286,0.4286,1.0,0.14286,1.0,0,11,0,0,0,11,0,0,3,0,0,3,0,0,3,0,0,1,0,0,0,0,11],[12,100,0.12,0.37051,0.21384,0.24999,0.28571,0.46418,0.14286,1.0,0,1,0,0,0,8,0,0,11,0,0,5,0,0,5,0,0,1,0,0,1,0,1],[16,100,0.16,0.41518,0.27747,0.14286,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,11,0,0,4,0,0,8,0,0,3,0,0,1,0,0,2,0,3],[20,100,0.2,0.44643,0.33645,0.14286,0.35714,0.71429,0.0,1.0,1,7,0,1,0,11,0,0,4,0,0,6,0,0,1,0,0,2,0,0,0,0,7],[24,100,0.24,0.48215,0.33834,0.1429,0.35714,0.89286,0.14286,1.0,0,8,0,0,0,9,0,0,7,0,0,6,0,0,0,0,0,1,0,0,1,0,8],[28,100,0.28,0.37044,0.22556,0.14289,0.35714,0.42857,0.14,1.0,0,2,0,0,0,9,0,0,7,0,0,12,0,0,1,0,0,0,0,0,1,0,2],[32,100,0.32,0.41514,0.27046,0.14289,0.35714,0.571,0.14286,1.0,0,4,0,0,0,9,0,0,7,0,0,7,0,0,4,0,0,1,0,0,0,0,4],[36,100,0.36,0.44643,0.32093,0.14286,0.42857,0.57143,0.0,1.0,1,6,0,1,0,10,0,0,4,0,0,6,0,0,4,0,0,0,0,0,1,0,6],[40,100,0.4,0.37054,0.26452,0.14286,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,11,0,0,9,0,0,6,0,0,1,0,0,1,0,0,1,0,3],[44,100,0.44,0.375,0.23891,0.24999,0.35714,0.42857,0.0,1.0,1,3,0,1,0,7,0,0,8,0,0,12,0,0,1,0,0,0,0,0,0,0,3],[48,100,0.48,0.39732,0.25187,0.14286,0.42857,0.46431,0.14286,1.0,0,3,0,0,0,10,0,0,5,0,0,9,0,0,4,0,0,1,0,0,0,0,3],[52,100,0.52,0.36607,0.23941,0.14286,0.28571,0.46431,0.14286,1.0,0,2,0,0,0,11,0,0,8,0,0,5,0,0,4,0,0,2,0,0,0,0,2],[56,100,0.56,0.41517,0.28427,0.14286,0.35714,0.57111,0.14286,1.0,0,3,0,0,0,11,0,0,5,0,0,7,0,0,3,0,0,0,0,0,3,0,3],[60,100,0.6,0.33482,0.16982,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,12,0,0,3,0,0,12,0,0,4,0,0,1,0,0,0,0,0],[64,100,0.64,0.39284,0.26963,0.14286,0.28571,0.4286,0.14286,1.0,0,4,0,0,0,9,0,0,10,0,0,6,0,0,2,0,0,1,0,0,0,0,4],[68,100,0.68,0.36607,0.22851,0.14286,0.35714,0.42857,0.14286,1.0,0,2,0,0,0,11,0,0,5,0,0,10,0,0,3,0,0,1,0,0,0,0,2],[72,100,0.72,0.54007,0.31901,0.28571,0.42859,0.85714,0.14,1.0,0,6,0,0,0,6,0,0,7,0,0,4,0,0,3,0,0,1,0,0,5,0,6],[76,100,0.76,0.3482,0.1954,0.14286,0.42857,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,5,0,0,12,0,0,4,0,0,0,0,0,0,0,1],[80,100,0.8,0.42411,0.29121,0.14286,0.42857,0.57143,0.14286,1.0,0,5,0,0,0,11,0,0,4,0,0,7,0,0,5,0,0,0,0,0,0,0,5],[84,100,0.84,0.38393,0.27067,0.14286,0.35714,0.42857,0.14286,1.0,0,4,0,0,0,12,0,0,4,0,0,10,0,0,2,0,0,0,0,0,0,0,4],[88,100,0.88,0.36589,0.22588,0.1429,0.35714,0.42857,0.0,1.0,1,2,0,1,0,8,0,0,7,0,0,11,0,0,2,0,0,1,0,0,0,0,2],[92,100,0.92,0.43302,0.2823,0.14289,0.42857,0.57111,0.14286,1.0,0,5,0,0,0,9,0,0,5,0,0,9,0,0,4,0,0,0,0,0,0,0,5],[96,100,0.96,0.31695,0.15038,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,7,0,0,10,0,0,4,0,0,0,0,0,0,0,0],[100,100,1.0,0.27678,0.138,0.14286,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,14,0,0,8,0,0,8,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"dcca5ad9899d17f0","q":"Let $A=A(x, y)$ and $B=B(x, y)$ be two-variable polynomials with real coefficients. Suppose that $A(x, y) / B(x, y)$ is a polynomial in $x$ for infinitely many values of $y$, and a polynomial in $y$ for infinitely many values of $x$. Prove that $B$ divides $A$, meaning there exists a third polynomial $C$ with real coefficients such that $A=B \\cdot C$.","t":[{"b":4,"e":0.42857,"k":"rising","v":0.20981,"x":0.58472,"p":[[0,24,0.0,0.20981,0.26721,0.0,0.0,0.42857,0.0,1.0,17,1,0,17,0,1,0,0,4,0,0,7,0,0,1,0,0,0,0,0,1,0,1],[4,24,0.1667,0.39732,0.38088,0.0,0.35714,0.64286,0.0,1.0,11,5,0,11,0,4,0,0,1,0,0,2,0,0,6,0,0,0,0,0,3,0,5],[8,24,0.3333,0.38839,0.36811,0.0,0.42857,0.60714,0.0,1.0,13,4,0,13,0,0,0,0,0,0,0,8,0,0,3,0,0,1,0,0,3,0,4],[12,24,0.5,0.58472,0.31828,0.42857,0.57143,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,0,0,0,8,0,0,6,0,0,3,0,0,3,0,7],[16,24,0.6667,0.43749,0.28557,0.1429,0.42859,0.60714,0.0,1.0,5,2,0,5,0,4,0,0,1,0,0,9,0,0,5,0,0,5,0,0,1,0,2],[20,24,0.8333,0.40624,0.33523,0.0,0.42857,0.57143,0.0,1.0,11,3,0,11,0,0,0,0,1,0,0,5,0,0,8,0,0,3,0,0,1,0,3],[24,24,1.0,0.43302,0.25123,0.28571,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,0,0,0,5,0,0,11,0,0,5,0,0,2,0,0,4,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.22768,"x":0.57146,"p":[[0,30,0.0,0.22768,0.274,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,2,0,0,4,0,0,3,0,0,5,0,0,1,0,0,0,0,1],[4,30,0.1333,0.34822,0.31122,0.0,0.35714,0.57143,0.0,1.0,10,2,0,10,0,4,0,0,2,0,0,4,0,0,6,0,0,4,0,0,0,0,2],[8,30,0.2667,0.41075,0.369,0.0,0.42929,0.71429,0.0,1.0,11,4,0,11,0,3,0,0,0,0,0,3,0,0,5,0,0,4,0,0,2,0,4],[12,30,0.4,0.44195,0.37857,0.10714,0.35714,0.75,0.0,1.0,8,7,0,8,0,4,0,0,4,0,0,2,0,0,4,0,0,2,0,0,1,0,7],[16,30,0.5333,0.39732,0.31891,0.10714,0.42857,0.60714,0.0,1.0,8,3,0,8,0,3,0,0,4,0,0,4,0,0,5,0,0,5,0,0,0,0,3],[20,30,0.6667,0.57146,0.28345,0.42857,0.57143,0.71429,0.0,1.0,3,6,0,3,0,1,0,0,0,0,0,8,0,0,10,0,0,3,0,0,1,0,6],[24,30,0.8,0.55355,0.24678,0.42857,0.42857,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,1,0,0,14,0,0,5,0,0,3,0,0,4,0,3],[28,30,0.9333,0.53125,0.08917,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,20,0,0,0,0,0,1,0,0],[30,30,1.0,0.47768,0.08459,0.42857,0.4286,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,17,0,0,13,0,0,0,0,0,0,0,0]]}]},{"i":"e1f438e0b43a87ea","q":"Determine the integers  $0 \\le a \\le b \\le c \\le d$  such that:   $$ 2^n= a^2 + b^2 + c^2 + d^2. $$","t":[{"b":1,"e":0.57143,"k":"flat","v":0.67411,"x":0.83036,"p":[[0,27,0.0,0.83036,0.19377,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,5,0,15],[4,27,0.1481,0.76339,0.14987,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,9,0,5],[8,27,0.2963,0.73214,0.15465,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,11,0,0,9,0,3],[12,27,0.4444,0.77677,0.16343,0.67857,0.78564,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,9,0,7],[16,27,0.5926,0.67411,0.18638,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,7,0,0,7,0,3],[20,27,0.7407,0.69196,0.13882,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,16,0,0,4,0,2],[24,27,0.8889,0.6875,0.14032,0.57143,0.71429,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,12,0,0,7,0,1],[27,27,1.0,0.70088,0.15715,0.57143,0.71429,0.75,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,14,0,0,5,0,3]]},{"b":4,"e":0.71429,"k":"flat","v":0.68303,"x":0.82143,"p":[[0,31,0.0,0.81696,0.18977,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,10,0,11],[4,31,0.129,0.78125,0.24219,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,4,0,0,5,0,0,4,0,14],[8,31,0.2581,0.68303,0.20119,0.57143,0.71429,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,11,0,0,3,0,5],[12,31,0.3871,0.82143,0.21129,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,2,0,17],[16,31,0.5161,0.76339,0.17353,0.67857,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,10,0,0,7,0,7],[20,31,0.6452,0.74554,0.17029,0.57143,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,6,0,0,6,0,7],[24,31,0.7742,0.79911,0.17076,0.67857,0.78571,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,5,0,11],[28,31,0.9032,0.71875,0.28901,0.57143,0.71429,1.0,0.0,1.0,1,11,1,1,0,3,0,0,0,0,0,1,0,0,6,0,0,6,0,0,4,0,11],[31,31,1.0,0.74107,0.20025,0.67857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,13,0,0,3,0,8]]}]},{"i":"f901c1c35d719ecb","q":"Let  $ABC$  be an acute triangle. Let  $H$  denote its orthocenter and  $D, E$  and  $F$  the feet of its altitudes from  $A, B$  and  $C$ , respectively. Let the common point of  $DF$  and the altitude through  $B$  be  $P$ . The line perpendicular to  $BC$  through  $P$  intersects  $AB$  in  $Q$ . Furthermore,  $EQ$  intersects the altitude through  $A$  in  $N$ . Prove that  $N$  is the midpoint of  $AH$ .\n\nProposed by Karl 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0.14286,0.857,0,0,0,0,0,12,0,0,9,0,0,7,0,0,2,0,0,1,0,0,1,0,0],[176,195,0.9026,0.21876,0.1287,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,20,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[180,195,0.9231,0.25884,0.15342,0.14286,0.1429,0.42857,0.14,0.71429,0,0,0,0,0,18,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[184,195,0.9436,0.21865,0.11287,0.14286,0.14286,0.28571,0.14,0.571,0,0,0,0,0,20,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[188,195,0.9641,0.24553,0.14387,0.14286,0.14288,0.32164,0.0,0.57143,1,0,0,1,0,17,0,0,6,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[192,195,0.9846,0.25438,0.15465,0.14286,0.14288,0.32143,0.0,0.71429,1,0,0,1,0,16,0,0,7,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[195,195,1.0,0.20294,0.12002,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,19,0,0,7,0,1,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.22322,"x":0.55351,"p":[[0,126,0.0,0.3281,0.28054,0.14286,0.1429,0.571,0.0,1.0,4,2,2,4,0,13,0,0,3,0,0,2,0,1,4,0,0,3,0,0,0,0,2],[4,126,0.0317,0.42175,0.23857,0.14286,0.42857,0.58932,0.14,1.0,0,1,0,0,0,9,0,0,6,0,0,4,0,0,5,1,0,6,0,0,0,0,1],[8,126,0.0635,0.37497,0.23889,0.14286,0.28571,0.571,0.0,1.0,1,1,0,1,0,10,0,0,6,0,0,6,0,0,3,0,0,5,0,0,0,0,1],[12,126,0.0952,0.4866,0.24707,0.28571,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,5,0,0,6,0,0,4,0,0,9,0,0,0,0,2],[16,126,0.127,0.55351,0.28291,0.28571,0.571,0.71429,0.14286,1.0,0,6,0,0,0,5,0,0,4,0,0,5,0,0,6,0,0,6,0,0,0,0,6],[20,126,0.1587,0.37496,0.2194,0.25,0.28571,0.571,0.0,1.0,1,1,0,1,0,7,0,0,10,0,0,5,0,0,5,0,0,3,0,0,0,0,1],[24,126,0.1905,0.37496,0.24676,0.14286,0.42857,0.571,0.0,1.0,2,2,0,2,0,9,0,0,4,0,0,8,0,0,6,0,0,1,0,0,0,0,2],[28,126,0.2222,0.41518,0.24836,0.25,0.42857,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,6,0,0,9,0,0,2,0,0,5,0,0,0,0,2],[32,126,0.254,0.40624,0.23448,0.14289,0.42857,0.57111,0.0,1.0,1,1,0,1,0,8,0,0,4,0,0,10,0,0,2,0,0,6,0,0,0,0,1],[36,126,0.2857,0.40622,0.2425,0.14286,0.42857,0.571,0.0,1.0,1,2,0,1,0,8,0,0,4,0,0,10,0,0,4,0,0,3,0,0,0,0,2],[40,126,0.3175,0.44641,0.24934,0.24999,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,6,0,0,5,0,0,3,0,0,8,0,0,1,0,1],[44,126,0.3492,0.41963,0.2736,0.14286,0.42857,0.66082,0.0,1.0,1,2,0,1,1,9,0,0,4,0,0,4,0,0,4,1,0,6,0,0,0,0,2],[48,126,0.381,0.41516,0.22405,0.28571,0.42857,0.5711,0.14286,1.0,0,2,0,0,0,7,0,0,6,0,0,10,0,0,5,0,0,2,0,0,0,0,2],[52,126,0.4127,0.43738,0.30508,0.14286,0.42857,0.60714,0.0,1.0,2,4,0,2,0,9,0,0,3,0,0,6,0,0,4,0,0,3,0,0,1,0,4],[56,126,0.4444,0.3482,0.2671,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,10,0,0,6,0,0,6,0,0,1,0,0,4,0,0,0,0,2],[60,126,0.4762,0.40614,0.2554,0.14286,0.42857,0.57143,0.0,1.0,1,2,0,1,0,11,0,0,0,0,0,9,0,0,6,0,0,3,0,0,0,0,2],[64,126,0.5079,0.41515,0.24315,0.14289,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,9,0,0,5,0,0,8,0,0,4,0,0,4,0,0,0,0,2],[68,126,0.5397,0.43301,0.25872,0.14289,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,9,0,0,4,0,0,8,0,0,5,0,0,3,0,0,0,0,3],[72,126,0.5714,0.34375,0.21683,0.14286,0.35714,0.42857,0.0,1.0,2,1,0,2,0,9,0,0,5,0,0,11,0,0,2,0,0,2,0,0,0,0,1],[76,126,0.6032,0.37944,0.19099,0.25002,0.35714,0.571,0.14286,0.71429,0,0,0,0,0,8,0,0,8,0,0,7,0,0,5,0,0,4,0,0,0,0,0],[80,126,0.6349,0.36605,0.24465,0.14286,0.28571,0.571,0.14286,1.0,0,1,0,0,0,13,0,0,6,0,0,4,0,0,2,0,0,6,0,0,0,0,1],[84,126,0.6667,0.40841,0.21366,0.25,0.42857,0.571,0.0,1.0,1,1,0,1,0,7,0,0,3,0,1,9,0,0,8,0,0,2,0,0,0,0,1],[88,126,0.6984,0.33923,0.18807,0.14286,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,10,0,0,6,0,0,8,0,0,5,0,0,2,0,0,0,0,0],[92,126,0.7302,0.31916,0.22369,0.14286,0.2857,0.42858,0.0,1.0,2,1,0,2,0,10,0,1,9,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[96,126,0.7619,0.30352,0.1776,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,13,0,0,5,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[100,126,0.7937,0.2632,0.18259,0.14286,0.14288,0.32143,0.0,0.71429,2,0,0,2,0,15,0,0,7,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[104,126,0.8254,0.25669,0.17483,0.14286,0.2857,0.37489,0.0,0.71429,4,0,0,4,0,11,0,0,8,0,1,5,0,0,2,0,0,1,0,0,0,0,0],[108,126,0.8571,0.22759,0.20476,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,17,0,0,3,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[112,126,0.8889,0.29451,0.17836,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,14,0,0,5,0,0,7,0,0,4,0,0,1,0,0,0,0,0],[116,126,0.9206,0.32574,0.20277,0.14286,0.28571,0.4286,0.14,1.0,0,1,0,0,0,13,0,0,7,0,0,5,0,0,6,0,0,0,0,0,0,0,1],[120,126,0.9524,0.25,0.16366,0.14286,0.14288,0.42857,0.0,0.71429,2,0,0,2,0,16,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[124,126,0.9841,0.22758,0.15918,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,17,0,0,5,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[126,126,1.0,0.22322,0.14258,0.14286,0.1429,0.28571,0.0,0.57143,4,0,0,4,0,13,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"351fd1a422a5dd7a","q":"Let $a_{1}, a_{2}, a_{3}, \\ldots, a_{n}$ be a sequence of non-negative integers, where $n$ is a positive integer.\n\nLet\n\n$$\nA_{n}=\\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}\n$$\n\nProve that\n\n$$\na_{1}!a_{2}!\\ldots a_{n}!\\geq\\left(\\left\\lfloor A_{n}\\right\\rfloor!\\right)^{n}\n$$\n\nwhere $\\left\\lfloor A_{n}\\right\\rfloor$ is the greatest integer less than or equal to $A_{n}$, and $a!=1 \\times 2 \\times \\cdots \\times a$ for $a \\geq 1$ (and $0!=1$ ). When does equality hold?","t":[{"b":1,"e":0.0,"k":"flat","v":0.36607,"x":0.54911,"p":[[0,15,0.0,0.50443,0.22862,0.28571,0.42857,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,9,0,0,4,0,0,2,0,0,4,0,2],[4,15,0.2667,0.54911,0.28146,0.28571,0.42857,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,11,0,0,5,0,0,2,0,0,4,0,0,5,0,4],[8,15,0.5333,0.54464,0.3489,0.28571,0.42857,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,11,0,0,2,0,0,2,0,0,1,0,0,4,0,8],[12,15,0.8,0.54017,0.36199,0.28571,0.57121,0.85714,0.0,1.0,7,7,0,7,0,0,0,0,2,0,0,6,0,0,4,0,0,2,0,0,4,0,7],[15,15,1.0,0.36607,0.33681,0.10714,0.28571,0.57143,0.0,1.0,8,4,0,8,0,4,0,0,8,0,0,3,0,0,2,0,0,1,0,0,2,0,4]]},{"b":5,"e":0.14286,"k":"flat","v":0.42411,"x":0.63393,"p":[[0,16,0.0,0.42411,0.21275,0.28571,0.28571,0.46431,0.2857,1.0,0,2,0,0,0,0,0,0,19,0,0,5,0,0,3,0,0,2,0,0,1,0,2],[4,16,0.25,0.58034,0.3008,0.28571,0.57143,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,9,0,0,1,0,0,7,0,0,3,0,0,4,0,6],[8,16,0.5,0.63393,0.23128,0.42857,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,3,0,0,5,0,0,10,0,2],[12,16,0.75,0.58482,0.28428,0.28571,0.57143,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,10,0,0,3,0,0,3,0,0,4,0,0,7,0,4],[16,16,1.0,0.46428,0.26245,0.28571,0.35714,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,14,0,0,7,0,0,2,0,0,1,0,0,2,0,4]]}]},{"i":"21cde0f83f281bca","q":"Let $\\mathrm{n} \\geqslant 1$ be an integer and $\\mathrm{x}_{1}, \\ldots, \\mathrm{x}_{\\mathrm{n}}$ be positive real numbers. Show that\n\n$$\n\\left(\\frac{x_{1}}{1}+\\frac{x_{2}}{2}+\\ldots+\\frac{x_{n}}{n}\\right)\\left(1 \\cdot x_{1}+2 \\cdot x_{2}+\\ldots+n \\cdot x_{n}\\right) \\leqslant \\frac{(n+1)^{2}}{4 n}\\left(x_{1}+x_{2}+\\ldots+x_{n}\\right)^{2}\n$$\n\nWhat are the cases of equality?","t":[{"b":0,"e":0.42857,"k":"flat","v":0.44643,"x":0.53125,"p":[[0,22,0.0,0.50893,0.18189,0.42857,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,4,0,0,14,0,0,1,0,0,12,0,0,0,0,0],[4,22,0.1818,0.52679,0.15335,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,19,0,0,0,0,0,12,0,0,0,0,0],[8,22,0.3636,0.51339,0.12807,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,0,9,0,0,0,0,0],[12,22,0.5455,0.53125,0.1439,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,0,12,0,0,0,0,0],[16,22,0.7273,0.49107,0.11811,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0],[20,22,0.9091,0.50446,0.12364,0.42857,0.42857,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,1,0,0,8,0,0,0,0,0],[22,22,1.0,0.44643,0.06916,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"flat","v":0.50893,"x":0.56696,"p":[[0,18,0.0,0.50893,0.15947,0.42857,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,17,0,0,4,0,0,9,0,0,0,0,0],[4,18,0.2222,0.56696,0.17307,0.42857,0.5,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,15,0,0,1,0,0,14,0,0,0,0,1],[8,18,0.4444,0.56696,0.16554,0.42857,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,14,0,0,0,0,1],[12,18,0.6667,0.53125,0.1439,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,0,12,0,0,0,0,0],[16,18,0.8889,0.55357,0.14174,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0],[18,18,1.0,0.55357,0.14174,0.42857,0.42859,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0]]}]},{"i":"885058a570e1dba6","q":"Let $\\omega=e^{2 \\pi i / 5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k>1$ such that\n\n$$\n\\left(a+b \\omega+c \\omega^{2}+d \\omega^{3}\\right)^{k}=1+\\omega\n$$","t":[{"b":1,"e":0.0,"k":"falling","v":0.19643,"x":0.72321,"p":[[0,50,0.0,0.61161,0.3867,0.14286,0.64286,1.0,0.0,1.0,4,13,1,4,0,5,0,0,1,0,0,1,0,0,5,0,0,2,0,0,1,0,13],[4,50,0.08,0.72321,0.37617,0.42857,1.0,1.0,0.0,1.0,4,17,0,4,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,17],[8,50,0.16,0.68748,0.36147,0.42857,0.85714,1.0,0.0,1.0,3,14,0,3,0,3,0,0,1,0,0,3,0,0,2,0,0,2,0,0,4,0,14],[12,50,0.24,0.60713,0.43006,0.10714,0.85714,1.0,0.0,1.0,8,15,0,8,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0,2,0,15],[16,50,0.32,0.50893,0.43144,0.0,0.57144,1.0,0.0,1.0,10,10,0,10,0,3,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,10],[20,50,0.4,0.49098,0.4134,0.0,0.42857,1.0,0.0,1.0,9,10,0,9,0,2,0,0,3,0,0,5,0,0,0,0,0,1,0,0,2,0,10],[24,50,0.48,0.5089,0.4071,0.10714,0.49979,1.0,0.0,1.0,8,9,0,8,0,4,0,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,9],[28,50,0.56,0.38393,0.40159,0.0,0.35714,0.71429,0.0,1.0,14,7,0,14,0,1,0,0,1,0,0,5,0,0,1,0,0,3,0,0,0,0,7],[32,50,0.64,0.43749,0.41793,0.0,0.42857,0.85714,0.0,1.0,13,7,0,13,0,1,0,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,7],[36,50,0.72,0.38839,0.40913,0.0,0.28571,0.85714,0.0,1.0,14,6,0,14,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,0,4,0,6],[40,50,0.8,0.5491,0.38813,0.25,0.64286,0.89286,0.0,1.0,6,8,0,6,0,2,0,0,6,0,0,1,0,0,1,0,0,2,0,0,6,0,8],[44,50,0.88,0.41964,0.3976,0.0,0.35714,0.85714,0.0,1.0,11,6,0,11,0,3,0,0,2,0,0,3,0,0,3,0,0,0,0,0,4,0,6],[48,50,0.96,0.19643,0.29397,0.0,0.0,0.2857,0.0,1.0,17,2,0,17,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,2],[50,50,1.0,0.21875,0.3214,0.0,0.0,0.42858,0.0,1.0,19,2,0,19,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,0,1,0,2]]},{"b":2,"e":0.0,"k":"falling","v":0.17857,"x":0.52679,"p":[[0,22,0.0,0.52679,0.41563,0.14286,0.35714,1.0,0.0,1.0,4,12,0,4,0,9,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,12],[4,22,0.1818,0.47768,0.43096,0.0,0.42857,1.0,0.0,1.0,11,9,0,11,0,3,0,0,0,0,0,4,0,0,0,0,0,1,0,0,4,0,9],[8,22,0.3636,0.4017,0.38212,0.0,0.28571,0.75,0.0,1.0,10,6,0,10,0,3,0,0,5,0,0,3,0,0,1,0,0,2,0,0,2,0,6],[12,22,0.5455,0.37945,0.3285,0.14286,0.28571,0.71429,0.0,1.0,6,2,0,6,0,8,0,0,5,0,0,2,0,0,2,0,0,3,0,0,4,0,2],[16,22,0.7273,0.33034,0.3415,0.0,0.14286,0.60714,0.0,1.0,11,1,0,11,0,6,0,0,3,0,0,1,0,0,3,0,0,2,0,0,5,0,1],[20,22,0.9091,0.31249,0.29544,0.14286,0.14286,0.4642,0.0,1.0,7,2,0,7,0,11,0,0,1,0,0,5,0,0,2,0,0,4,0,0,0,0,2],[22,22,1.0,0.17857,0.25754,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,15,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,2]]}]},{"i":"2655faf18b91e27b","q":"Let $ABC$ be a triangle and $O$ the center of its circumcircle. Let $d$ be the line parallel to $(BC)$ passing through $O$. Let $A'$ be the symmetric point of $A$ with respect to $(BC)$. The line parallel to $(A'B)$ passing through $C$ intersects $d$ at $C_1$, and the lines $(A'C)$ and $(BC_1)$ intersect at $C_2$. The line parallel to $(A'C)$ passing through $B$ intersects $d$ at $B_1$, and the lines $(A'B)$ and $(CB_1)$ intersect at $B_2$. Show that the points $A, A', B_2, C_2$ are concyclic.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.10679,"x":0.16956,"p":[[0,121,0.0,0.10679,0.06166,0.105,0.14286,0.14286,0.0,0.143,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,121,0.0331,0.16956,0.14915,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,121,0.0661,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,121,0.0992,0.14259,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,121,0.1322,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,121,0.1653,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,121,0.1983,0.14278,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,121,0.2314,0.13822,0.0435,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,121,0.2645,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,121,0.2975,0.14697,0.02494,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,121,0.3306,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,121,0.3636,0.16483,0.08083,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,121,0.3967,0.15161,0.04975,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,121,0.4298,0.15161,0.06124,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,121,0.4628,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,121,0.4959,0.15161,0.04975,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,121,0.5289,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,121,0.562,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,121,0.595,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,121,0.6281,0.14715,0.04354,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,121,0.6612,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,121,0.6942,0.14268,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,121,0.7273,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,121,0.7603,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,121,0.7934,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,121,0.8264,0.14724,0.04352,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,121,0.8595,0.14715,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,121,0.8926,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,121,0.9256,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,121,0.9587,0.13813,0.02482,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,121,0.9917,0.13376,0.03454,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[121,121,1.0,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.09795,"x":0.15179,"p":[[0,59,0.0,0.09795,0.06604,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.15179,0.04974,0.14286,0.14286,0.14286,0.0,0.286,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.14259,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.14277,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,59,0.4068,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.13831,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.1383,0.07563,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.12501,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.13366,0.03452,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"342cccc38e29717a","q":"Let $a, b, c$ be three real numbers such that $1 \\geq a \\geq b \\geq c \\geq 0$. Prove that if $\\lambda$ is a root of the cubic equation $x^{3}+a x^{2}+b x+c=0$ (real or complex), then $|\\lambda| \\leq 1$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.17411,"x":0.29911,"p":[[0,18,0.0,0.27232,0.39506,0.0,0.0,0.57143,0.0,1.0,19,6,0,19,0,2,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,6],[4,18,0.2222,0.29911,0.32608,0.0,0.21431,0.57143,0.0,1.0,14,3,0,14,0,2,0,0,3,0,0,2,0,0,8,0,0,0,0,0,0,0,3],[8,18,0.4444,0.24554,0.35756,0.0,0.0,0.57143,0.0,1.0,19,4,0,19,0,3,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,4],[12,18,0.6667,0.17411,0.18808,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,3,0,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[16,18,0.8889,0.22767,0.18507,0.0,0.21429,0.42857,0.0,0.57143,9,0,0,9,0,7,0,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[18,18,1.0,0.19643,0.1915,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,11,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"volatile","v":0.19196,"x":0.66518,"p":[[0,31,0.0,0.19196,0.37391,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[4,31,0.129,0.54015,0.36897,0.24999,0.57143,1.0,0.0,1.0,6,10,0,6,0,2,0,0,2,0,0,3,0,0,9,0,0,0,0,0,0,0,10],[8,31,0.2581,0.42858,0.39769,0.0,0.5,0.67857,0.0,1.0,11,8,0,11,0,3,0,0,1,0,0,1,0,0,8,0,0,0,0,0,0,0,8],[12,31,0.3871,0.54018,0.42668,0.0,0.57143,1.0,0.0,1.0,10,13,0,10,0,0,0,0,2,0,0,2,0,0,5,0,0,0,0,0,0,0,13],[16,31,0.5161,0.58929,0.41304,0.10714,0.57143,1.0,0.0,1.0,8,14,0,8,0,1,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,14],[20,31,0.6452,0.36161,0.3813,0.0,0.28571,0.57143,0.0,1.0,14,6,0,14,0,1,0,0,2,0,0,2,0,0,7,0,0,0,0,0,0,0,6],[24,31,0.7742,0.66518,0.37899,0.5,0.78571,1.0,0.0,1.0,5,14,0,5,0,2,0,0,1,0,0,0,0,0,5,0,0,3,0,0,2,0,14],[28,31,0.9032,0.54464,0.4,0.10714,0.57143,1.0,0.0,1.0,8,11,0,8,0,1,0,0,3,0,0,0,0,0,8,0,0,0,0,0,1,0,11],[31,31,1.0,0.65623,0.36397,0.42859,0.71421,1.0,0.0,1.0,5,13,0,5,0,0,0,0,2,0,0,2,0,0,7,0,0,0,0,0,3,0,13]]}]},{"i":"e5c77d3ae4f68923","q":"Let $a, b$ and $c$ be positive integers satisfying the equation\n\n$$\n(a, b)+[a, b]=2021^{c} .\n$$\n\nIf $|a-b|$ is a prime number, prove that the number $(a+b)^{2}+4$ is composite.\nHere, $(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $[a, b]$ denotes the least common multiple of $a$ and $b$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.20089,"x":0.56244,"p":[[0,81,0.0,0.36606,0.1634,0.28571,0.42857,0.4642,0.0,0.57143,2,0,1,2,0,3,0,0,10,0,0,9,0,0,8,0,0,0,0,0,0,0,0],[4,81,0.0494,0.45086,0.2528,0.2857,0.42859,0.57143,0.0,1.0,3,2,3,3,0,2,0,0,7,0,0,5,0,0,10,0,0,2,0,0,1,0,2],[8,81,0.0988,0.4687,0.24282,0.2857,0.4998,0.57143,0.0,1.0,1,1,1,1,0,6,0,0,3,0,0,6,0,0,9,0,0,4,0,0,2,0,1],[12,81,0.1481,0.48656,0.24705,0.28571,0.571,0.57143,0.0,1.0,2,1,1,2,0,2,0,0,7,0,0,3,0,0,12,0,0,1,0,0,4,0,1],[16,81,0.1975,0.56244,0.24983,0.42857,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,4,0,0,2,0,0,5,0,0,13,0,0,1,0,0,3,0,4],[20,81,0.2469,0.35268,0.24739,0.14286,0.28571,0.42857,0.0,1.0,2,2,0,2,0,9,0,0,8,0,0,6,0,0,3,0,0,2,0,0,0,0,2],[24,81,0.2963,0.43971,0.29992,0.14289,0.35714,0.57143,0.0,1.0,1,5,0,1,1,7,0,0,7,0,0,3,0,0,7,0,0,1,0,0,0,0,5],[28,81,0.3457,0.36161,0.26483,0.14286,0.35714,0.57143,0.0,1.0,4,1,0,4,0,9,0,0,3,0,0,5,0,0,8,0,0,0,0,0,2,0,1],[32,81,0.3951,0.43748,0.3173,0.14289,0.42857,0.60714,0.0,1.0,4,3,0,4,0,6,0,0,5,0,0,4,0,0,5,0,0,1,0,0,4,0,3],[36,81,0.4444,0.36605,0.23671,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,9,0,0,2,0,0,7,0,0,9,0,0,1,0,0,0,0,1],[40,81,0.4938,0.41963,0.25489,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,6,0,0,3,0,0,7,0,0,7,0,0,4,0,0,1,0,1],[44,81,0.5432,0.35711,0.2743,0.14286,0.35714,0.57111,0.0,1.0,4,2,0,4,0,10,0,0,2,0,0,6,0,0,7,0,0,0,0,0,1,0,2],[48,81,0.5926,0.41515,0.21235,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,9,0,0,5,0,0,10,0,0,1,0,0,2,0,0],[52,81,0.642,0.38383,0.19055,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,8,0,0,7,0,0,7,0,0,3,0,0,0,0,0],[56,81,0.6914,0.37499,0.26183,0.14286,0.42857,0.57143,0.0,1.0,4,2,0,4,0,7,0,0,4,0,0,6,0,0,8,0,0,1,0,0,0,0,2],[60,81,0.7407,0.42855,0.25999,0.28571,0.42857,0.57143,0.0,1.0,4,2,0,4,0,3,0,0,4,0,0,8,0,0,9,0,0,1,0,0,1,0,2],[64,81,0.7901,0.49552,0.2624,0.28571,0.49979,0.57143,0.0,1.0,2,3,0,2,0,2,0,0,6,0,0,6,0,0,9,0,0,2,0,0,2,0,3],[68,81,0.8395,0.32586,0.2531,0.14286,0.28571,0.46418,0.0,1.0,5,1,0,5,0,8,0,0,6,0,0,5,0,0,5,0,0,1,0,0,1,0,1],[72,81,0.8889,0.33925,0.27833,0.14286,0.28571,0.571,0.0,1.0,4,1,0,4,0,11,0,0,5,0,0,2,0,0,6,0,0,0,0,0,3,0,1],[76,81,0.9383,0.24554,0.15251,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,18,0,0,4,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[80,81,0.9877,0.20089,0.14664,0.14286,0.14286,0.1786,0.0,0.71429,2,0,0,2,0,22,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[81,81,1.0,0.20981,0.19225,0.14286,0.14286,0.2857,0.0,0.85714,6,0,0,6,0,16,0,0,4,0,0,3,0,0,2,0,0,0,0,0,1,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.13831,"x":0.46425,"p":[[0,76,0.0,0.37052,0.1816,0.2857,0.28571,0.57143,0.0,0.71429,1,0,1,1,0,6,0,0,10,0,0,4,0,0,10,0,0,1,0,0,0,0,0],[4,76,0.0526,0.46425,0.26243,0.28571,0.571,0.57143,0.0,1.0,3,2,0,3,0,3,0,0,5,0,0,4,0,0,12,0,0,1,0,0,2,0,2],[8,76,0.1053,0.42405,0.27773,0.24999,0.42857,0.57143,0.0,1.0,5,2,0,5,0,3,0,0,5,0,0,4,0,0,10,0,0,2,0,0,1,0,2],[12,76,0.1579,0.40625,0.27458,0.25,0.42857,0.57143,0.0,1.0,3,3,0,3,0,5,0,0,7,0,0,8,0,0,4,0,0,1,0,0,1,0,3],[16,76,0.2105,0.33482,0.16213,0.2857,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,8,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[20,76,0.2632,0.41069,0.24934,0.14286,0.42857,0.57143,0.0,1.0,4,1,0,4,0,5,0,0,2,0,0,8,0,0,8,0,0,4,0,0,0,0,1],[24,76,0.3158,0.32583,0.22076,0.14286,0.28571,0.571,0.0,0.85714,4,0,0,4,0,9,0,0,4,0,0,6,0,0,8,0,0,0,0,0,1,0,0],[28,76,0.3684,0.32579,0.27259,0.14286,0.2857,0.46418,0.0,0.85714,5,0,0,5,0,10,0,0,5,0,0,4,0,0,3,0,0,1,0,0,4,0,0],[32,76,0.4211,0.32588,0.21791,0.14286,0.28571,0.57143,0.0,0.71429,5,0,0,5,0,6,0,0,8,0,0,2,0,0,10,0,0,1,0,0,0,0,0],[36,76,0.4737,0.37944,0.2438,0.14286,0.35714,0.57143,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,4,0,0,8,0,0,3,0,0,0,0,1],[40,76,0.5263,0.44639,0.28513,0.25,0.42857,0.57143,0.0,1.0,2,2,0,2,0,6,0,0,7,0,0,3,0,0,7,0,0,1,0,0,4,0,2],[44,76,0.5789,0.34375,0.28763,0.14286,0.28571,0.4286,0.0,1.0,6,3,0,6,0,7,0,0,4,0,0,8,0,0,3,0,0,1,0,0,0,0,3],[48,76,0.6316,0.35245,0.18566,0.14289,0.42857,0.4642,0.0,0.57143,3,0,0,3,0,6,0,0,4,0,0,11,0,0,8,0,0,0,0,0,0,0,0],[52,76,0.6842,0.24551,0.2265,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,15,0,0,5,0,0,2,0,0,3,0,0,1,0,0,0,0,1],[56,76,0.7368,0.25445,0.22226,0.14286,0.14286,0.42857,0.0,1.0,5,1,0,5,0,14,0,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,1],[60,76,0.7895,0.24988,0.17858,0.14286,0.21428,0.42857,0.0,0.57143,5,0,0,5,0,11,0,0,7,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[64,76,0.8421,0.14733,0.17307,0.0,0.14286,0.14287,0.0,0.71429,12,0,0,12,0,14,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[68,76,0.8947,0.1384,0.10403,0.14286,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,21,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.23204,0.15874,0.14286,0.1429,0.28571,0.0,0.57143,4,0,0,4,0,13,0,0,9,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[76,76,1.0,0.13831,0.11564,0.105,0.14286,0.1429,0.0,0.57143,8,0,0,8,0,19,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"7febe9a7df8e009f","q":"Let $k$ be an integer coprime with $n$ satisfying $1 \\leqslant k < n$. Augustin colors the integers of $\\{1,2, \\ldots, n-1\\}$ with as many colors as he wishes. However, if $\\mathbf{j}$ is an integer satisfying $1 \\leqslant j \\leqslant n-1$, the integers $j$ and $n-j$ are of the same color. Additionally, if $i$ is an integer satisfying $1 \\leqslant i \\leqslant n$ and $i \\neq k$, the integers $i$ and $|i-k|$ are of the same color.\nProve that Augustin has colored all the integers the same color.","t":[{"b":0,"e":0.857,"k":"flat","v":0.62945,"x":0.76785,"p":[[0,27,0.0,0.71429,0.29233,0.42857,0.85714,1.0,0.0,1.0,1,9,0,1,0,2,0,0,1,0,0,5,0,0,2,0,0,2,0,0,10,0,9],[4,27,0.1481,0.74547,0.22516,0.571,0.85707,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,3,0,0,10,0,8],[8,27,0.2963,0.76785,0.24936,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,5,0,0,8,0,11],[12,27,0.4444,0.72767,0.26332,0.42859,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,3,0,0,0,0,0,4,0,0,11,0,8],[16,27,0.5926,0.73213,0.20125,0.57143,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,5,0,0,15,0,3],[20,27,0.7407,0.66069,0.23351,0.42857,0.857,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,12,0,0,1,0,0,0,0,0,15,0,2],[24,27,0.8889,0.62945,0.22548,0.42857,0.42859,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,1,0,0,10,0,3],[27,27,1.0,0.63392,0.20182,0.42857,0.57143,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,3,0,0,11,0,1]]},{"b":5,"e":0.28571,"k":"flat","v":0.59375,"x":0.80803,"p":[[0,33,0.0,0.71425,0.23959,0.57132,0.85707,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,4,0,0,12,0,5],[4,33,0.1212,0.77678,0.22286,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,5,0,0,13,0,8],[8,33,0.2424,0.80803,0.20705,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,0,10,0,11],[12,33,0.3636,0.78569,0.22306,0.67857,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,0,9,0,11],[16,33,0.4848,0.78123,0.223,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,10,0,10],[20,33,0.6061,0.78568,0.18214,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,8,0,0,10,0,8],[24,33,0.7273,0.67411,0.27019,0.39286,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,1,0,0,0,0,0,8,0,0,11,0,4],[28,33,0.8485,0.76336,0.21904,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,7,0,0,10,0,8],[32,33,0.9697,0.59375,0.2372,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,9,0,0,2,0,0,8,0,0,3,0,4],[33,33,1.0,0.5982,0.27533,0.28571,0.71429,0.857,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,1,0,0,1,0,0,11,0,0,7,0,2]]}]},{"i":"b5a3035d1dceb6c9","q":"Let $n$ be a natural number and $X=\\{1,2, \\ldots, n\\}$. For subsets $A$ and $B$ of $X$ we define $A \\Delta B$ to be the set of all those elements of $X$ which belong to exactly one of $A$ and $B$. Let $\\mathcal{F}$ be a collection of subsets of $X$ such that for any two distinct elements $A$ and $B$ in $\\mathcal{F}$ the set $A \\Delta B$ has at least two elements. Show that $\\mathcal{F}$ has at most $2^{n-1}$ elements. Find all such collections $\\mathcal{F}$ with $2^{n-1}$ elements.","t":[{"b":1,"e":0.571,"k":"falling","v":0.74996,"x":0.96875,"p":[[0,9,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,9,0.4444,0.91518,0.17807,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[8,9,0.8889,0.80354,0.24939,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,0,0,0,6,0,16],[9,9,1.0,0.74996,0.23422,0.57143,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,2,0,0,4,0,12]]},{"b":4,"e":0.42857,"k":"flat","v":0.8482,"x":0.96427,"p":[[0,18,0.0,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[4,18,0.2222,0.92857,0.15152,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,24],[8,18,0.4444,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[12,18,0.6667,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[16,18,0.8889,0.86606,0.20808,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,6,0,19],[18,18,1.0,0.8482,0.21412,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,5,0,18]]}]},{"i":"af47305c345934a3","q":"Let  $p$  be an odd prime number. How many  $p$ -element subsets  $A$  of  $\\{1,2,\\ldots \\ 2p\\}$  are there, the sum of whose elements is divisible by  $p$ ?","t":[{"b":1,"e":0.14286,"k":"flat","v":0.1383,"x":0.14286,"p":[[0,36,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.13839,"x":0.14286,"p":[[0,16,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"27788904e22f4fe2","q":"Let $p$ be an odd prime, and put $N=\\frac{1}{4}\\left(p^{3}-p\\right)-1$. The numbers $1,2, \\ldots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \\leqslant N$, denote by $r(n)$ the fraction of integers in $\\{1,2, \\ldots, n\\}$ that are red.  Prove that there exists a positive integer $a \\in\\{1,2, \\ldots, p-1\\}$ such that $r(n) \\neq a / p$ for all $n=1,2, \\ldots, N$. (Netherlands)","t":[{"b":1,"e":0.14286,"k":"falling","v":0.11384,"x":0.46427,"p":[[0,51,0.0,0.46427,0.1821,0.42857,0.42857,0.42858,0.0,1.0,1,1,1,1,0,1,0,0,2,0,0,21,0,0,2,0,0,3,0,0,1,0,1],[4,51,0.0784,0.31249,0.19702,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,7,0,0,14,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[8,51,0.1569,0.33481,0.17715,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,8,0,0,9,0,0,10,0,0,1,0,0,3,0,0,0,0,0],[12,51,0.2353,0.39274,0.19894,0.28571,0.35714,0.4642,0.14,0.85714,0,0,0,0,0,6,0,0,10,0,0,8,0,0,4,0,0,2,0,0,2,0,0],[16,51,0.3137,0.29465,0.13803,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,10,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[20,51,0.3922,0.33033,0.1836,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,7,0,0,8,0,0,4,0,0,2,0,0,0,0,0],[24,51,0.4706,0.375,0.16656,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,9,0,0,14,0,0,3,0,0,0,0,0,0,0,1],[28,51,0.549,0.31697,0.19799,0.24999,0.28571,0.28579,0.0,1.0,1,1,0,1,0,7,0,0,17,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[32,51,0.6275,0.31696,0.19475,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,10,0,0,9,0,0,9,0,0,1,0,0,1,0,0,0,0,1],[36,51,0.7059,0.22321,0.12846,0.14286,0.21428,0.28571,0.0,0.57143,3,0,0,3,0,13,0,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,51,0.7843,0.26786,0.14174,0.14286,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,12,0,0,11,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[44,51,0.8627,0.30356,0.1587,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,11,0,0,7,0,0,1,0,0,2,0,0,0,0,0],[48,51,0.9412,0.15179,0.11258,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,21,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.11384,0.08349,0.05357,0.14286,0.14286,0.0,0.42857,8,0,0,8,1,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"falling","v":0.15616,"x":0.45981,"p":[[0,71,0.0,0.45981,0.15865,0.42857,0.42857,0.4286,0.0,0.85714,1,0,1,1,0,1,0,0,1,0,0,22,0,0,2,0,0,4,0,0,1,0,0],[4,71,0.0563,0.32589,0.16457,0.24999,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,8,0,0,14,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[8,71,0.1127,0.41069,0.21941,0.28571,0.42857,0.571,0.14286,1.0,0,1,0,0,0,6,0,0,9,0,0,8,0,0,5,0,0,1,0,0,2,0,1],[12,71,0.169,0.36607,0.2111,0.2857,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,7,0,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,2],[16,71,0.2254,0.35268,0.18205,0.14286,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,9,0,0,9,0,0,7,0,0,4,0,0,3,0,0,0,0,0],[20,71,0.2817,0.36606,0.23401,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,10,0,0,8,0,0,8,0,0,3,0,0,0,0,0,1,0,2],[24,71,0.338,0.32143,0.13832,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,14,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[28,71,0.3944,0.36161,0.21719,0.14286,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,11,0,0,5,0,0,11,0,0,0,0,0,4,0,0,0,0,1],[32,71,0.4507,0.33482,0.20705,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,11,0,0,7,0,0,1,0,0,1,0,0,1,0,1],[36,71,0.507,0.32143,0.24484,0.14286,0.28571,0.42857,0.0,1.0,2,2,0,2,0,12,0,0,6,0,0,8,0,0,0,0,0,2,0,0,0,0,2],[40,71,0.5634,0.2767,0.11822,0.14286,0.28571,0.42857,0.14,0.4286,0,0,0,0,0,12,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.29017,0.15352,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,6,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[48,71,0.6761,0.375,0.19805,0.28571,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,9,0,0,11,0,0,1,0,0,3,0,0,0,0,1],[52,71,0.7324,0.32147,0.21431,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,11,0,0,5,0,0,9,0,0,3,0,0,1,0,0,0,0,1],[56,71,0.7887,0.30803,0.16793,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,9,0,0,9,0,0,1,0,0,2,0,0,0,0,0],[60,71,0.8451,0.31696,0.17762,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,11,0,0,8,0,0,2,0,0,0,0,0,0,0,1],[64,71,0.9014,0.24553,0.15663,0.14286,0.14286,0.42857,0.0,0.71429,2,0,0,2,0,16,0,0,5,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[68,71,0.9577,0.20534,0.17102,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,26,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[71,71,1.0,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2e91315dfe4928a6","q":"Let be a continuous function  $ f:\\mathbb{R}\\longrightarrow\\mathbb{R} $  that has the property that   $$  xf(x)\\ge \\int_0^x f(t)dt ,  $$  for all real numbers  $ x. $  Prove that**a)** the mapping  $ x\\mapsto \\frac{1}{x}\\int_0^x  f(t) dt $  is nondecreasing on the restrictions  $ \\mathbb{R}_{<0 } $  and   $ \\mathbb{R}_{>0 } . $  **b)** if  $ \\int_x^{x+1} f(t)dt=\\int_{x-1}^x f(t)dt , $  for any real number  $ x, $  then  $ f $  is constant.\n\n\n*Mihai Piticari*","t":[{"b":1,"e":0.85714,"k":"flat","v":0.81695,"x":0.95982,"p":[[0,31,0.0,0.81695,0.24805,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,1,0,0,2,0,19],[4,31,0.129,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[8,31,0.2581,0.88838,0.21941,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,24],[12,31,0.3871,0.94196,0.17076,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[16,31,0.5161,0.9375,0.17474,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[20,31,0.6452,0.95982,0.12993,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[24,31,0.7742,0.89729,0.19315,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,7,0,21],[28,31,0.9032,0.91071,0.15047,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,10,0,19],[31,31,1.0,0.88838,0.08557,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,9]]},{"b":7,"e":0.85714,"k":"flat","v":0.74554,"x":0.96429,"p":[[0,16,0.0,0.81695,0.25314,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,0,0,0,1,0,20],[4,16,0.25,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[8,16,0.5,0.93304,0.15966,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[12,16,0.75,0.84821,0.22851,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,7,0,18],[16,16,1.0,0.74554,0.23887,0.57143,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,2,0,0,13,0,7]]}]},{"i":"3d5ac26544047227","q":"Let be a finite group  $ G $  that has an element  $ a\\neq 1 $  for which exists a prime number  $ p $  such that  $ x^{1+p}=a^{-1}xa, $  for all  $ x\\in G. $ **a)** Prove that the order of  $ G $  is a power of  $ p. $ **b)** Show that  $ H:=\\{x\\in G|\\text{ord} (x)=p\\}\\le G $  and  $ \\text{ord}^2(H)>\\text{ord}(G). $","t":[{"b":2,"e":0.42857,"k":"rising","v":0.13393,"x":0.54015,"p":[[0,38,0.0,0.13393,0.10677,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.3392,0.19159,0.14286,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,9,0,0,11,0,0,8,0,0,1,0,0,1,0,0,2,0,0],[8,38,0.2105,0.3482,0.18187,0.2857,0.28571,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,12,0,0,11,0,0,1,0,0,1,0,0,0,0,1],[12,38,0.3158,0.30803,0.15198,0.2857,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,6,0,0,17,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[16,38,0.4211,0.30808,0.11361,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,14,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[20,38,0.5263,0.30357,0.16269,0.14286,0.2857,0.28571,0.14286,0.85714,0,0,0,0,0,9,0,0,16,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[24,38,0.6316,0.37049,0.23103,0.14286,0.28571,0.46418,0.0,1.0,1,1,0,1,0,8,0,0,9,0,0,6,0,0,5,0,0,0,0,0,2,0,1],[28,38,0.7368,0.54015,0.25687,0.28571,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,6,0,0,4,0,0,6,0,0,7,0,0,5,0,1],[32,38,0.8421,0.42856,0.20515,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,5,0,0,8,0,0,5,0,0,7,0,0,0,0,0],[36,38,0.9474,0.3571,0.24219,0.14286,0.42857,0.571,0.0,0.85714,4,0,0,4,0,8,0,0,3,0,0,7,0,0,7,0,0,1,0,0,2,0,0],[38,38,1.0,0.29018,0.22156,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,5,0,0,4,0,0,10,0,0,3,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.11152,"x":0.60265,"p":[[0,49,0.0,0.11152,0.1114,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.32143,0.15568,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,16,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[8,49,0.1633,0.2857,0.14283,0.14286,0.2857,0.28571,0.14286,0.85714,0,0,0,0,0,9,0,0,18,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[12,49,0.2449,0.37051,0.14219,0.28571,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,1,0,0,20,0,0,4,0,0,5,0,0,2,0,0,0,0,0],[16,49,0.3265,0.34374,0.19184,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,22,0,0,1,0,0,2,0,0,1,0,0,1,0,1],[20,49,0.4082,0.3125,0.11539,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,21,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[24,49,0.4898,0.29464,0.12846,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,7,0,0,20,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[28,49,0.5714,0.3214,0.14281,0.2857,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,6,0,0,18,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[32,49,0.6531,0.48659,0.274,0.28571,0.42857,0.60714,0.14286,1.0,0,4,0,0,0,4,0,0,11,0,0,3,0,0,6,0,0,2,0,0,2,0,4],[36,49,0.7347,0.33918,0.18822,0.24999,0.28571,0.42857,0.14,1.0,0,1,0,0,0,8,0,0,13,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[40,49,0.8163,0.42408,0.25624,0.25002,0.28571,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,9,0,0,1,0,0,7,0,0,4,0,0,2,0,1],[44,49,0.898,0.58467,0.22706,0.42857,0.57143,0.71429,0.14,1.0,0,3,0,0,0,2,0,0,3,0,0,5,0,0,12,0,0,3,0,0,4,0,3],[48,49,0.9796,0.60265,0.19145,0.5354,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,5,0,0,12,0,0,7,0,0,3,0,2],[49,49,1.0,0.54014,0.16262,0.4286,0.57143,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,6,0,0,14,0,0,7,0,0,1,0,0]]}]},{"i":"9037ddeab82491eb","q":"Let n be the integer $4 \\times 201420142014 \\ldots 2014$ (where 2014 is written 117819 times). Show that $2014^{3}$ divides n.","t":[{"b":1,"e":0.71429,"k":"falling","v":0.66964,"x":0.88839,"p":[[0,35,0.0,0.85268,0.14054,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,7,0,13],[4,35,0.1143,0.88839,0.13709,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,18],[8,35,0.2286,0.875,0.16269,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,2,0,19],[12,35,0.3429,0.88392,0.14483,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,4,0,18],[16,35,0.4571,0.86161,0.16554,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,3,0,17],[20,35,0.5714,0.75893,0.15335,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,0,7,0,6],[24,35,0.6857,0.66964,0.18013,0.57143,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,8,0,0,2,0,5],[28,35,0.8,0.71873,0.17308,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,9,0,0,6,0,5],[32,35,0.9143,0.69195,0.16016,0.57143,0.71429,0.75,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,9,0,0,4,0,4],[35,35,1.0,0.67409,0.1439,0.57143,0.64286,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,9,0,0,5,0,2]]},{"b":3,"e":0.57143,"k":"falling","v":0.69641,"x":0.94196,"p":[[0,67,0.0,0.84821,0.18877,0.82143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,11,0,13],[4,67,0.0597,0.84821,0.18536,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,10,0,0,2,0,17],[8,67,0.1194,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26],[12,67,0.1791,0.85714,0.18211,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,5,0,17],[16,67,0.2388,0.80357,0.14617,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,3,0,10],[20,67,0.2985,0.79911,0.17075,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,0,3,0,11],[24,67,0.3582,0.76786,0.15465,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,16,0,0,6,0,6],[28,67,0.4179,0.80804,0.13651,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,5,0,9],[32,67,0.4776,0.79464,0.17835,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,7,0,10],[36,67,0.5373,0.78125,0.15561,0.71429,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,13,0,0,9,0,6],[40,67,0.597,0.75893,0.19045,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,13,0,0,5,0,8],[44,67,0.6567,0.75893,0.16145,0.67857,0.71429,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,5,0,7],[48,67,0.7164,0.75,0.14725,0.57143,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,7,0,5],[52,67,0.7761,0.73214,0.17405,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,10,0,0,7,0,5],[56,67,0.8358,0.74554,0.18118,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,12,0,0,3,0,8],[60,67,0.8955,0.7589,0.16539,0.67857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,9,0,0,10,0,5],[64,67,0.9552,0.74553,0.18808,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,7,0,0,7,0,7],[67,67,1.0,0.69641,0.18473,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,7,0,0,5,0,5]]}]},{"i":"743e327f590ed553","q":"Let $a, b, c, x, y, z$ be positive real numbers such that $a+b+c=x+y+z$ and abc $=x y z$. Further, suppose that $a \\leq x<y<z \\leq c$ and $a<b<c$. Prove that $a=x, b=y$ and $c=z$.","t":[{"b":3,"e":1.0,"k":"rising","v":0.7991,"x":0.97768,"p":[[0,15,0.0,0.7991,0.34967,0.71429,1.0,1.0,0.0,1.0,4,22,0,4,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,22],[4,15,0.2667,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[8,15,0.5333,0.90625,0.18073,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,24],[12,15,0.8,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[15,15,1.0,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":4,"e":0.28571,"k":"volatile","v":0.32143,"x":0.66518,"p":[[0,7,0.0,0.66518,0.4265,0.21429,1.0,1.0,0.0,1.0,8,18,0,8,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,18],[4,7,0.5714,0.36161,0.3273,0.0,0.42857,0.42857,0.0,1.0,10,5,0,10,0,0,0,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,5],[7,7,1.0,0.32143,0.19885,0.28571,0.35714,0.42857,0.0,1.0,6,1,0,6,0,0,0,0,10,0,0,15,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"09a732ab167ae432","q":"12. 3b.(GBR 4) A sequence of polynomials $P_{m}(x, y, z), m=0,1,2, \\ldots$, in $x, y$, and $z$ is defined by $P_{0}(x, y, z)=1$ and by  $$ P_{m}(x, y, z)=(x+z)(y+z) P_{m-1}(x, y, z+1)-z^{2} P_{m-1}(x, y, z) $$  for $m>0$. Prove that each $P_{m}(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z$.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.15179,"x":0.47767,"p":[[0,84,0.0,0.36607,0.27879,0.14286,0.2143,0.46429,0.14286,1.0,0,3,0,0,0,16,0,0,2,0,0,6,0,0,2,0,0,3,0,0,0,0,3],[4,84,0.0476,0.30357,0.32291,0.14286,0.14286,0.25,0.0,1.0,3,4,0,3,0,21,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,4],[8,84,0.0952,0.26786,0.28291,0.14286,0.14286,0.1786,0.0,1.0,2,3,0,2,0,22,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[12,84,0.1429,0.2545,0.2194,0.14286,0.14286,0.14287,0.14286,0.85714,0,0,0,0,0,25,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0],[16,84,0.1905,0.32124,0.30939,0.14286,0.14286,0.46418,0.0,1.0,2,3,0,2,0,20,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,3],[20,84,0.2381,0.41963,0.36759,0.14286,0.14286,0.85714,0.0,1.0,2,6,0,2,0,16,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,6],[24,84,0.2857,0.22765,0.23377,0.14286,0.14286,0.1786,0.0,1.0,5,1,0,5,0,19,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,1],[28,84,0.3333,0.26339,0.27918,0.14286,0.14286,0.2857,0.0,1.0,5,2,0,5,0,17,0,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,2],[32,84,0.381,0.39731,0.34019,0.14286,0.14286,0.71429,0.14286,1.0,0,5,0,0,0,19,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,5],[36,84,0.4286,0.38391,0.30184,0.14286,0.14286,0.71429,0.0,1.0,1,2,0,1,0,16,0,0,1,0,0,3,0,0,2,0,0,5,0,0,2,0,2],[40,84,0.4762,0.39282,0.3499,0.14286,0.14286,0.71429,0.0,1.0,2,5,0,2,0,17,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,5],[44,84,0.5238,0.45536,0.38372,0.14286,0.21428,1.0,0.0,1.0,2,9,0,2,0,14,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,9],[48,84,0.5714,0.39732,0.34019,0.14286,0.14286,0.60714,0.14286,1.0,0,6,0,0,0,18,0,0,1,0,0,4,0,0,1,0,0,1,0,0,1,0,6],[52,84,0.619,0.46875,0.38504,0.14286,0.21435,0.89286,0.0,1.0,1,8,0,1,0,15,0,0,2,0,0,2,0,0,0,0,0,0,0,0,4,0,8],[56,84,0.6667,0.41498,0.35432,0.14286,0.14286,0.857,0.0,1.0,1,4,0,1,0,17,0,0,1,0,0,2,0,0,1,0,0,0,0,0,6,0,4],[60,84,0.7143,0.46872,0.36982,0.14286,0.28571,0.85704,0.0,1.0,2,5,0,2,0,14,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,5],[64,84,0.7619,0.47767,0.36353,0.14286,0.28574,0.78571,0.14286,1.0,0,8,0,0,0,16,0,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,8],[68,84,0.8095,0.34374,0.31714,0.14286,0.14286,0.42858,0.14286,1.0,0,5,0,0,0,21,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,5],[72,84,0.8571,0.27232,0.26573,0.14286,0.14286,0.32143,0.0,1.0,3,2,0,3,0,19,0,0,2,0,0,3,0,0,0,0,0,3,0,0,0,0,2],[76,84,0.9048,0.31697,0.27137,0.14286,0.14286,0.42858,0.0,1.0,2,2,0,2,0,17,0,0,1,0,0,6,0,0,0,0,0,4,0,0,0,0,2],[80,84,0.9524,0.21875,0.18891,0.14286,0.14286,0.1429,0.0,0.85714,1,0,0,1,0,24,0,0,2,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[84,84,1.0,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.12938,"x":0.34375,"p":[[0,53,0.0,0.26339,0.16793,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,20,0,0,1,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[4,53,0.0755,0.26786,0.26904,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,24,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,2],[8,53,0.1509,0.2767,0.26953,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,25,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[12,53,0.2264,0.2767,0.27884,0.14286,0.14286,0.17857,0.0,1.0,1,3,0,1,0,23,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,3],[16,53,0.3019,0.33034,0.30184,0.14286,0.14286,0.46418,0.14286,1.0,0,3,0,0,0,22,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,3],[20,53,0.3774,0.34375,0.33854,0.14286,0.14286,0.5,0.0,1.0,1,5,0,1,0,21,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,5],[24,53,0.4528,0.28569,0.30091,0.14286,0.14286,0.21432,0.0,1.0,3,3,0,3,0,21,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,3],[28,53,0.5283,0.33927,0.34208,0.14286,0.14286,0.57111,0.0,1.0,5,5,0,5,0,15,0,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,5],[32,53,0.6038,0.29462,0.30289,0.14286,0.14286,0.21429,0.0,1.0,2,3,0,2,0,22,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,3],[36,53,0.6792,0.26784,0.34021,0.10714,0.14286,0.14286,0.0,1.0,8,5,0,8,0,17,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[40,53,0.7547,0.16071,0.22517,0.10714,0.14286,0.14286,0.0,1.0,8,2,0,8,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,53,0.8302,0.13839,0.17672,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[48,53,0.9057,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,53,0.9811,0.15179,0.10677,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,29,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[53,53,1.0,0.15625,0.15303,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,25,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0]]}]},{"i":"cfbd7db94f79f045","q":"We have $\\mathrm{a}+\\mathrm{b}$ bowls aligned in a row. The first $a$ bowls contain an apple, while the last $b$ bowls contain a pear.\nAn operation consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, where $i$ and $j$ are integers such that $i-j$ is even (A bowl can contain multiple fruits). We want to reach the final situation where the first $b$ bowls contain a pear and the last $a$ bowls contain an apple. Show that this is possible if and only if $ab$ is even.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.50891,"x":0.80789,"p":[[0,48,0.0,0.50891,0.17472,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,19,0,0,4,0,0,2,0,0,3,0,1],[4,48,0.0833,0.80789,0.16991,0.71321,0.78571,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,4,0,12],[8,48,0.1667,0.74102,0.20655,0.67857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,12,0,0,5,0,7],[12,48,0.25,0.80357,0.18813,0.71429,0.78571,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,3,0,13],[16,48,0.3333,0.79018,0.17122,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,2,0,11],[20,48,0.4167,0.74548,0.18813,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,11,0,0,0,0,10],[24,48,0.5,0.71868,0.16168,0.57143,0.71429,0.857,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,12,0,0,4,0,5],[28,48,0.5833,0.78122,0.16749,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,16,0,0,3,0,9],[32,48,0.6667,0.79906,0.18166,0.57143,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,0,2,0,13],[36,48,0.75,0.72762,0.15717,0.57143,0.71429,0.857,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,11,0,0,7,0,4],[40,48,0.8333,0.77675,0.14702,0.71429,0.71429,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,0,6,0,7],[44,48,0.9167,0.70981,0.18724,0.57143,0.71429,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,13,0,0,1,0,7],[48,48,1.0,0.61601,0.11539,0.5714,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,8,0,0,1,0,1]]},{"b":7,"e":0.85714,"k":"flat","v":0.53124,"x":0.75888,"p":[[0,36,0.0,0.53124,0.22083,0.42857,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,16,0,0,4,0,0,0,0,0,4,0,3],[4,36,0.1111,0.75888,0.16921,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,14,0,0,3,0,8],[8,36,0.2222,0.73209,0.1547,0.57143,0.71429,0.71429,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,15,0,0,0,0,7],[12,36,0.3333,0.69192,0.12935,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,20,0,0,0,0,3],[16,36,0.4444,0.68746,0.17292,0.57142,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,11,0,0,2,0,5],[20,36,0.5556,0.68747,0.14035,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,15,0,0,2,0,3],[24,36,0.6667,0.72765,0.13537,0.71429,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,21,0,0,0,0,5],[28,36,0.7778,0.66956,0.12085,0.57132,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,18,0,0,0,0,2],[32,36,0.8889,0.67853,0.15974,0.57143,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,13,0,0,3,0,3],[36,36,1.0,0.62494,0.18472,0.571,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,13,0,0,6,0,0,3,0,3]]}]},{"i":"3cec22c32143305c","q":"a)Let $a,b,c\\in\\mathbb{R}$  and  $a^2+b^2+c^2=1$ .Prove that: $|a-b|+|b-c|+|c-a|\\le2\\sqrt{2}$ b) Let  $a_1,a_2,..a_{2019}\\in\\mathbb{R}$  and  $\\sum_{i=1}^{2019}a_i^2=1$ .Find the maximum of: $S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$","t":[{"b":2,"e":0.71429,"k":"rising","v":0.52235,"x":0.75149,"p":[[0,23,0.0,0.52235,0.16209,0.42857,0.4286,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,17,0,0,7,0,0,3,0,0,2,0,1],[4,23,0.1739,0.69939,0.24584,0.64286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,3,0,0,2,0,0,2,0,1,14,0,0,1,0,8],[8,23,0.3478,0.71875,0.21866,0.71429,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,13,0,0,6,0,6],[12,23,0.5217,0.65179,0.19541,0.4286,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,12,0,0,4,0,3],[16,23,0.6957,0.6875,0.22428,0.57143,0.71429,0.85704,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,2,0,0,5,0,0,13,0,0,4,0,5],[20,23,0.8696,0.75149,0.19406,0.70238,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,3,0,2,11,0,0,6,0,7],[23,23,1.0,0.67857,0.23958,0.42857,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,11,0,0,3,0,7]]},{"b":6,"e":1.0,"k":"rising","v":0.57588,"x":0.80357,"p":[[0,18,0.0,0.57588,0.28679,0.42857,0.4998,0.85714,0.0,1.0,3,6,3,3,0,0,0,0,0,0,0,13,0,0,5,0,0,2,0,0,3,0,6],[4,18,0.2222,0.72321,0.22851,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,13,0,0,0,0,10],[8,18,0.4444,0.7589,0.19048,0.67857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,11,0,0,4,0,9],[12,18,0.6667,0.71578,0.17874,0.71429,0.71429,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,2,1,0,1,0,0,1,0,0,19,0,0,4,0,4],[16,18,0.8889,0.80357,0.14174,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,4,0,9],[18,18,1.0,0.73214,0.20438,0.71429,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,16,0,0,5,0,6]]}]},{"i":"ec0d6d38b6963b0d","q":"$ABC$  is a triangle, and  $E$  and  $F$  are points on the segments  $BC$  and  $CA$  respectively, such that  $\\frac{CE}{CB}+\\frac{CF}{CA}=1$  and  $\\angle CEF=\\angle CAB$ . Suppose that  $M$  is the midpoint of  $EF$  and  $G$  is the point of intersection between  $CM$  and  $AB$ . Prove that triangle  $FEG$  is similar to triangle  $ABC$ .","t":[{"b":5,"e":0.28571,"k":"flat","v":0.18303,"x":0.32589,"p":[[0,70,0.0,0.22759,0.16316,0.14286,0.1429,0.28571,0.0,0.71429,4,0,0,4,0,14,0,0,8,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[4,70,0.0571,0.26338,0.17534,0.14286,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,9,0,0,12,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[8,70,0.1143,0.29464,0.13803,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,6,0,0,19,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[12,70,0.1714,0.21875,0.15561,0.14286,0.21428,0.28571,0.0,0.57143,6,0,0,6,0,10,0,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,70,0.2286,0.25893,0.15746,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,12,0,0,12,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[20,70,0.2857,0.23214,0.20748,0.14286,0.14288,0.28571,0.0,1.0,5,1,0,5,0,13,0,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[24,70,0.3429,0.25892,0.16143,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,10,0,0,8,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[28,70,0.4,0.2366,0.1411,0.14286,0.21428,0.28571,0.0,0.57143,2,0,0,2,0,14,0,0,12,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[32,70,0.4571,0.25893,0.20341,0.14286,0.21428,0.28571,0.0,0.85714,4,0,0,4,0,12,0,0,10,0,0,1,0,0,3,0,0,1,0,0,1,0,0],[36,70,0.5143,0.26785,0.17035,0.14286,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,8,0,0,16,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[40,70,0.5714,0.32589,0.22084,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,9,0,0,13,0,0,5,0,0,1,0,0,0,0,0,2,0,1],[44,70,0.6286,0.31696,0.18808,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,13,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[48,70,0.6857,0.25446,0.17762,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,13,0,0,13,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[52,70,0.7429,0.21875,0.15561,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,20,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,70,0.8,0.24107,0.11538,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,10,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[60,70,0.8571,0.2366,0.12169,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,13,0,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[64,70,0.9143,0.30134,0.21409,0.14286,0.28571,0.28571,0.0,1.0,2,1,1,2,0,9,0,0,15,0,0,1,0,0,2,0,0,1,1,0,0,0,1],[68,70,0.9714,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.18303,0.09606,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,15,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.20536,"x":0.49107,"p":[[0,65,0.0,0.20536,0.15126,0.14286,0.14286,0.28571,0.0,0.57143,6,0,0,6,0,12,0,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,65,0.0615,0.23214,0.12242,0.14286,0.21428,0.28571,0.0,0.57143,1,0,0,1,0,15,0,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,65,0.1231,0.30357,0.20124,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,14,0,0,9,0,0,5,0,0,1,0,0,1,0,0,2,0,0],[12,65,0.1846,0.25,0.17857,0.14286,0.2857,0.28571,0.0,0.85714,5,0,0,5,0,7,0,0,16,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[16,65,0.2462,0.23213,0.17402,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,13,0,0,7,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[20,65,0.3077,0.28125,0.22724,0.14286,0.2857,0.28571,0.0,1.0,5,1,0,5,0,8,0,0,12,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[24,65,0.3692,0.25893,0.20652,0.14286,0.28571,0.28571,0.0,1.0,4,1,0,4,0,11,0,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[28,65,0.4308,0.35268,0.2525,0.14286,0.28571,0.4643,0.0,1.0,2,1,0,2,0,9,0,0,10,0,0,3,0,0,3,0,0,2,0,0,2,0,1],[32,65,0.4923,0.24098,0.14486,0.14286,0.21428,0.28571,0.0,0.57143,2,0,0,2,0,14,0,0,11,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[36,65,0.5538,0.24999,0.14722,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,14,0,0,13,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[40,65,0.6154,0.29007,0.19724,0.14286,0.28571,0.32143,0.0,0.85714,3,0,0,3,0,9,0,0,12,0,0,3,0,0,3,0,0,1,0,0,1,0,0],[44,65,0.6769,0.39286,0.20203,0.28571,0.35714,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,10,0,0,6,0,0,6,0,0,3,0,0,1,0,0],[48,65,0.7385,0.35714,0.23145,0.14289,0.28571,0.46429,0.14286,1.0,0,2,0,0,0,10,0,0,11,0,0,3,0,0,5,0,0,1,0,0,0,0,2],[52,65,0.8,0.41512,0.27045,0.2857,0.35714,0.57143,0.0,1.0,3,2,0,3,0,4,0,0,9,0,0,5,0,0,5,0,0,2,0,0,2,0,2],[56,65,0.8615,0.40625,0.24251,0.24999,0.42857,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,7,0,0,7,0,0,5,0,0,3,0,0,0,0,2],[60,65,0.9231,0.49105,0.28106,0.24999,0.57143,0.71429,0.0,1.0,2,3,0,2,0,6,0,0,2,0,0,4,0,0,9,0,0,5,0,0,1,0,3],[64,65,0.9846,0.36604,0.21703,0.2857,0.28571,0.571,0.0,0.85714,2,0,0,2,0,4,0,0,15,0,0,2,0,0,5,0,0,2,0,0,2,0,0],[65,65,1.0,0.49107,0.19864,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,3,0,0,6,0,0,14,0,0,3,0,0,2,0,0]]}]},{"i":"1e8be379feb6eaee","q":"$10$  real numbers are given  $a_1,a_2,\\ldots ,a_{10} $ , and the  $45$  sums of two of these numbers are formed  $a_i+a_j $ ,  $1\\leq i&lt;j\\leq 10$  . It is known that not all these sums are integers. Determine the minimum value of  $k$  such that it is possible that among the  $45$  sums there are  $k$  that are not integers and  $45-k$  that are integers.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.79018,"x":0.92409,"p":[[0,68,0.0,0.82589,0.18808,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,8,0,13],[4,68,0.0588,0.86607,0.14698,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,7,0,15],[8,68,0.1176,0.85714,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,5,0,14],[12,68,0.1765,0.87053,0.15714,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,4,0,17],[16,68,0.2353,0.82142,0.16368,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,5,0,12],[20,68,0.2941,0.86607,0.14258,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,6,0,15],[24,68,0.3529,0.83928,0.13716,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,8,0,11],[28,68,0.4118,0.86161,0.14054,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,4,0,15],[32,68,0.4706,0.84375,0.17627,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,7,0,14],[36,68,0.5294,0.87051,0.14886,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,6,0,16],[40,68,0.5882,0.92409,0.12873,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[44,68,0.6471,0.87507,0.17036,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,4,0,18],[48,68,0.7059,0.86159,0.15357,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,4,0,16],[52,68,0.7647,0.81695,0.20898,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,7,0,0,3,0,15],[56,68,0.8235,0.87499,0.15467,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,3,0,18],[60,68,0.8824,0.87946,0.13882,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,16],[64,68,0.9412,0.875,0.15872,0.82132,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,7,0,17],[68,68,1.0,0.79018,0.24996,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,6,0,0,5,0,14]]},{"b":3,"e":0.14286,"k":"falling","v":0.25445,"x":0.875,"p":[[0,37,0.0,0.77231,0.25471,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,13],[4,37,0.1081,0.875,0.15047,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,5,0,17],[8,37,0.2162,0.80357,0.19805,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,0,7,0,12],[12,37,0.3243,0.80802,0.15409,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,8,0,9],[16,37,0.4324,0.83482,0.15612,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,10,0,11],[20,37,0.5405,0.76337,0.2161,0.71429,0.78564,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,12,0,0,10,0,6],[24,37,0.6486,0.75891,0.24075,0.71429,0.78564,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,10,0,0,7,0,9],[28,37,0.7568,0.72768,0.27283,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,0,0,0,1,0,0,4,0,0,8,0,0,5,0,10],[32,37,0.8649,0.63384,0.31745,0.42857,0.71429,0.85714,0.0,1.0,2,4,0,2,0,5,0,0,0,0,0,2,0,0,3,0,0,5,0,0,11,0,4],[36,37,0.973,0.25445,0.32089,0.0,0.14286,0.46418,0.0,1.0,13,1,0,13,0,9,0,0,1,0,0,1,0,0,3,0,0,0,0,0,4,0,1],[37,37,1.0,0.30803,0.29038,0.14286,0.14286,0.57143,0.0,0.85714,7,0,0,7,0,10,0,0,5,0,0,1,0,0,3,0,0,2,0,0,4,0,0]]}]},{"i":"7ff00770c066a60e","q":"Let $x$ and $y$ be positive integers and assume that $z=4 x y /(x+y)$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4 n-1$ where $n$ is a positive integer.","t":[{"b":4,"e":0.0,"k":"volatile","v":0.00447,"x":0.65624,"p":[[0,8,0.0,0.65624,0.43133,0.14286,1.0,1.0,0.0,1.0,7,17,0,7,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,17],[4,8,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.74107,"x":0.88393,"p":[[0,28,0.0,0.74107,0.37701,0.71429,1.0,1.0,0.0,1.0,6,18,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,18],[4,28,0.1429,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[8,28,0.2857,0.86161,0.14054,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,16],[12,28,0.4286,0.87946,0.13882,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,1,0,18],[16,28,0.5714,0.83482,0.14773,0.71429,0.71429,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,14],[20,28,0.7143,0.86161,0.14934,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,17],[24,28,0.8571,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19],[28,28,1.0,0.82143,0.13832,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,12]]}]},{"i":"a9cc6a1ac2d21814","q":"A group of 6 students decided to make *study groups* and *service activity groups* according to the following principle:\n\nEach group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.\n\nSupposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.45085,"x":0.89731,"p":[[0,88,0.0,0.56695,0.22723,0.4286,0.57143,0.71429,0.0,1.0,2,1,1,2,0,1,0,0,1,0,0,5,0,0,12,0,0,6,0,0,4,0,1],[4,88,0.0455,0.82589,0.21048,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,0,7,0,14],[8,88,0.0909,0.78125,0.24218,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,1,0,0,6,0,0,5,0,0,5,0,13],[12,88,0.1364,0.80355,0.19806,0.57143,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,2,0,0,10,0,11],[16,88,0.1818,0.7857,0.2342,0.714,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,6,0,0,7,0,12],[20,88,0.2273,0.89731,0.12993,0.85711,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,11,0,16],[24,88,0.2727,0.79462,0.19213,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,0,11,0,9],[28,88,0.3182,0.74553,0.21646,0.57143,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,5,0,0,8,0,8],[32,88,0.3636,0.75446,0.20277,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,4,0,0,5,0,10],[36,88,0.4091,0.82589,0.18466,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,5,0,14],[40,88,0.4545,0.77677,0.23942,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,2,0,0,9,0,11],[44,88,0.5,0.79016,0.18894,0.71429,0.85707,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,8,0,0,7,0,10],[48,88,0.5455,0.75892,0.18012,0.57143,0.857,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,5,0,0,11,0,6],[52,88,0.5909,0.73214,0.27837,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,2,0,0,9,0,10],[56,88,0.6364,0.74997,0.21131,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,6,0,0,6,0,9],[60,88,0.6818,0.74553,0.22227,0.57143,0.85707,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,3,0,0,9,0,8],[64,88,0.7273,0.73659,0.24512,0.57143,0.857,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,4,0,0,5,0,0,3,0,0,10,0,8],[68,88,0.7727,0.72766,0.17986,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,8,0,0,6,0,6],[72,88,0.8182,0.63392,0.24984,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,5,0,0,5,0,0,8,0,0,4,0,5],[76,88,0.8636,0.74107,0.19373,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,0,7,0,7],[80,88,0.9091,0.69642,0.21053,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,6,0,0,7,0,5],[84,88,0.9545,0.45085,0.22897,0.2857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,10,0,0,5,0,0,8,0,0,2,0,0,1,0,2],[88,88,1.0,0.45526,0.27774,0.2857,0.28571,0.60714,0.0,1.0,1,3,0,1,0,3,0,0,14,0,0,2,0,0,4,0,0,2,0,0,3,0,3]]},{"b":3,"e":0.57143,"k":"flat","v":0.52229,"x":0.8125,"p":[[0,119,0.0,0.52229,0.23312,0.42857,0.57143,0.57143,0.0,1.0,2,1,2,2,0,1,0,0,4,0,0,5,0,0,13,0,0,2,0,0,4,0,1],[4,119,0.0336,0.79016,0.21125,0.57143,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,3,0,0,9,0,11],[8,119,0.0672,0.79909,0.21088,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,4,0,0,4,0,14],[12,119,0.1008,0.63386,0.29002,0.571,0.57143,0.85714,0.0,1.0,1,6,0,1,0,4,0,0,1,0,0,1,0,0,10,0,0,3,0,0,6,0,6],[16,119,0.1345,0.67856,0.30305,0.53539,0.78571,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,3,0,0,2,0,0,5,0,0,3,0,0,8,0,8],[20,119,0.1681,0.67851,0.26966,0.571,0.64286,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,0,0,0,3,0,0,10,0,0,3,0,0,5,0,8],[24,119,0.2017,0.59361,0.28387,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,1,0,0,5,0,0,8,0,0,2,0,0,7,0,4],[28,119,0.2353,0.66962,0.31832,0.42859,0.71429,1.0,0.0,1.0,2,9,0,2,0,2,0,0,3,0,0,2,0,0,3,0,0,5,0,0,6,0,9],[32,119,0.2689,0.54017,0.30667,0.28571,0.5,0.85704,0.0,1.0,2,5,0,2,0,3,0,0,5,0,0,6,0,0,4,0,0,3,0,0,4,0,5],[36,119,0.3025,0.57585,0.35443,0.14286,0.57143,0.89286,0.0,1.0,3,8,0,3,0,6,0,0,1,0,0,1,0,0,7,0,0,2,0,0,4,0,8],[40,119,0.3361,0.7366,0.31157,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,2,0,0,1,0,0,0,0,0,6,0,0,1,0,0,8,0,12],[44,119,0.3697,0.56249,0.32131,0.24999,0.57143,0.85714,0.0,1.0,1,5,0,1,0,7,0,0,3,0,0,1,0,0,5,0,0,5,0,0,5,0,5],[48,119,0.4034,0.69194,0.24773,0.57132,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,2,0,0,7,0,0,4,0,0,10,0,5],[52,119,0.437,0.7232,0.23403,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,8,0,0,9,0,6],[56,119,0.4706,0.72765,0.20002,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,4,0,0,9,0,6],[60,119,0.5042,0.66517,0.27107,0.57142,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,1,0,0,2,0,0,8,0,0,2,0,0,10,0,5],[64,119,0.5378,0.68748,0.28669,0.42859,0.78564,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,4,0,0,2,0,0,5,0,0,9,0,7],[68,119,0.5714,0.73657,0.26514,0.57143,0.857,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,2,0,0,6,0,0,4,0,0,7,0,10],[72,119,0.605,0.6607,0.31084,0.57132,0.78571,0.85714,0.0,1.0,2,6,0,2,0,3,0,0,2,0,0,0,0,0,6,0,0,3,0,0,10,0,6],[76,119,0.6387,0.58924,0.2714,0.39286,0.57143,0.857,0.0,1.0,1,3,0,1,0,3,0,0,4,0,0,1,0,0,9,0,0,5,0,0,6,0,3],[80,119,0.6723,0.62944,0.30694,0.42859,0.57143,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,3,0,0,2,0,0,8,0,0,2,0,0,6,0,7],[84,119,0.7059,0.67409,0.29717,0.5354,0.71429,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,0,0,0,4,0,0,5,0,0,5,0,0,6,0,8],[88,119,0.7395,0.65619,0.24708,0.571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,1,0,0,12,0,0,4,0,0,6,0,5],[92,119,0.7731,0.8125,0.14913,0.71429,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,17,0,6],[96,119,0.8067,0.80351,0.16274,0.71429,0.85707,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,12,0,8],[100,119,0.8403,0.71875,0.22724,0.57143,0.57143,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,1,0,0,4,0,10],[104,119,0.8739,0.57588,0.09095,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,1,0,0,2,0,0],[108,119,0.9076,0.60265,0.15459,0.57143,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,23,0,0,1,0,0,1,0,3],[112,119,0.9412,0.63611,0.17256,0.57132,0.57143,0.57143,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,1,21,0,0,1,0,0,1,0,5],[116,119,0.9748,0.56249,0.03457,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0],[119,119,1.0,0.59819,0.16918,0.5354,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,18,0,0,2,0,0,0,0,4]]}]},{"i":"23f93ce8f5c20f11","q":"For integer  $a$ ,  $a \\neq 0$ ,  $v_2(a)$  is greatest nonnegative integer  $k$  such that  $2^k | a$ . For given  $n \\in \\mathbb{N}$  determine highest possible cardinality of subset  $A$  of set  $ \\{1,2,3,...,2^n \\} $  with following property:\nFor all  $x, y \\in A$ ,  $x \\neq y$ , number  $v_2(x-y)$  is even.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.03571,"x":0.98214,"p":[[0,18,0.0,0.74551,0.24417,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,1,0,0,6,0,0,9,0,0,4,0,10],[4,18,0.2222,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,18,0.4444,0.89732,0.29284,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[12,18,0.6667,0.1875,0.33964,0.0,0.0,0.17857,0.0,1.0,21,4,0,21,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[16,18,0.8889,0.19196,0.37048,0.0,0.0,0.14286,0.0,1.0,23,5,0,23,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[18,18,1.0,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"flat","v":0.7455,"x":0.99554,"p":[[0,46,0.0,0.7455,0.24934,0.57143,0.71429,1.0,0.0,1.0,1,11,1,1,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,0,4,0,11],[4,46,0.087,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,46,0.1739,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,46,0.2609,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,46,0.3478,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[20,46,0.4348,0.95089,0.16213,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[24,46,0.5217,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,46,0.6087,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,46,0.6957,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,46,0.7826,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[40,46,0.8696,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[44,46,0.9565,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[46,46,1.0,0.88392,0.20341,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,5,0,21]]}]},{"i":"a408566701a556ea","q":"For positive integers  $k$  and  $n$ , we know  $k \\geq n!$ . Prove that  $ \\phi (k) \\geq (n-1)!$","t":[{"b":0,"e":1.0,"k":"rising","v":0.2499,"x":0.54908,"p":[[0,29,0.0,0.2499,0.20206,0.105,0.2857,0.32143,0.0,0.71429,8,0,0,8,0,6,0,0,10,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[4,29,0.1379,0.46419,0.31952,0.14286,0.4286,0.74996,0.0,1.0,4,1,0,4,0,6,0,0,4,0,0,3,0,0,3,0,0,4,0,0,7,0,1],[8,29,0.2759,0.54908,0.28596,0.28571,0.71429,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,3,0,0,4,0,0,2,0,0,11,0,0,4,0,2],[12,29,0.4138,0.39732,0.30667,0.14286,0.28571,0.57143,0.0,1.0,3,3,0,3,0,9,0,0,6,0,0,3,0,0,5,0,0,0,0,0,3,0,3],[16,29,0.5517,0.41516,0.35598,0.14286,0.28571,0.85704,0.0,1.0,6,3,0,6,0,8,0,0,4,0,0,1,0,0,3,0,0,1,0,0,6,0,3],[20,29,0.6897,0.35239,0.33324,0.105,0.2143,0.71429,0.0,1.0,8,1,0,8,0,8,0,0,4,0,0,1,0,0,2,0,0,3,0,0,5,0,1],[24,29,0.8276,0.30357,0.27837,0.14286,0.14286,0.46429,0.0,0.85714,7,0,0,7,0,10,0,0,4,0,0,3,0,0,1,0,0,5,0,0,2,0,0],[28,29,0.9655,0.40177,0.2271,0.24999,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,7,0,0,6,0,0,9,0,0,4,0,0,2,0,0,3,0,0],[29,29,1.0,0.46426,0.29879,0.14286,0.42857,0.71429,0.0,1.0,1,3,0,1,0,8,0,0,5,0,0,5,0,0,4,0,0,2,0,0,4,0,3]]},{"b":2,"e":0.28571,"k":"rising","v":0.23661,"x":0.69193,"p":[[0,51,0.0,0.25893,0.22711,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,15,0,0,6,0,0,2,0,0,2,0,0,1,0,0,2,0,0],[4,51,0.0784,0.23661,0.249,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,9,0,0,5,0,0,3,0,0,2,0,0,1,0,0,2,0,0],[8,51,0.1569,0.2857,0.27892,0.14286,0.14286,0.32143,0.0,1.0,6,2,0,6,0,12,0,0,6,0,0,1,0,0,2,0,0,3,0,0,0,0,2],[12,51,0.2353,0.36606,0.25488,0.14286,0.2857,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,6,0,0,1,0,0,6,0,0,4,0,0,2,0,0],[16,51,0.3137,0.39729,0.31078,0.14286,0.28571,0.71429,0.0,1.0,5,2,0,5,0,8,0,0,4,0,0,2,0,0,3,0,0,7,0,0,1,0,2],[20,51,0.3922,0.36159,0.26481,0.14286,0.28571,0.57111,0.0,0.85714,4,0,0,4,0,8,0,0,5,0,0,6,0,0,4,0,0,1,0,0,4,0,0],[24,51,0.4706,0.40622,0.2837,0.14286,0.35714,0.71429,0.0,1.0,5,1,0,5,0,5,0,0,6,0,0,2,0,0,4,0,0,9,0,0,0,0,1],[28,51,0.549,0.43747,0.305,0.14286,0.49979,0.71429,0.0,0.85714,5,0,0,5,0,6,0,0,3,0,0,2,0,0,5,0,0,6,0,0,5,0,0],[32,51,0.6275,0.33926,0.29394,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,5,0,0,3,0,0,4,0,0,4,0,0,6,0,0,0,0,1],[36,51,0.7059,0.37051,0.30272,0.14286,0.35714,0.57143,0.0,1.0,7,1,0,7,0,6,0,0,3,0,0,5,0,0,4,0,0,3,0,0,3,0,1],[40,51,0.7843,0.44641,0.32683,0.14286,0.49979,0.71429,0.0,1.0,4,2,0,4,0,9,0,0,1,0,0,2,0,0,6,0,0,3,0,0,5,0,2],[44,51,0.8627,0.57141,0.26725,0.28571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,1,0,0,6,0,0,8,0,0,4,0,3],[48,51,0.9412,0.62944,0.2547,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,4,0,0,4,0,0,6,0,0,7,0,4],[51,51,1.0,0.69193,0.22049,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,3,0,0,7,0,0,9,0,0,6,0,5]]}]},{"i":"34a759751252d7cd","q":"Let $x$ and $y$ be two real numbers. We define\n\n$$\nM=\\max \\{x y+1, x y-x-y+3,-2 x y+x+y+2\\} .\n$$\n\nProve that $M \\geqslant 2$, and determine the cases of equality.","t":[{"b":1,"e":1.0,"k":"rising","v":0.67409,"x":1.0,"p":[[0,64,0.0,0.75,0.22868,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,16,0,0,1,0,10],[4,64,0.0625,0.75445,0.19311,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,15,0,0,2,0,9],[8,64,0.125,0.77677,0.21111,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,12,0,0,2,0,12],[12,64,0.1875,0.80356,0.21356,0.71429,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,12,0,0,0,0,15],[16,64,0.25,0.72761,0.19357,0.57132,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,11,0,0,2,0,8],[20,64,0.3125,0.88838,0.19477,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,1,0,23],[24,64,0.375,0.81697,0.1931,0.71429,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,13,0,0,2,0,14],[28,64,0.4375,0.85714,0.19233,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,0,0,0,19],[32,64,0.5,0.78571,0.17496,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,17,0,0,2,0,10],[36,64,0.5625,0.76783,0.20441,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,14,0,0,1,0,11],[40,64,0.625,0.75892,0.22428,0.67836,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,11,0,0,1,0,12],[44,64,0.6875,0.77675,0.19869,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,14,0,0,0,0,12],[48,64,0.75,0.67409,0.18293,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,18,0,0,2,0,3],[52,64,0.8125,0.8125,0.14032,0.71429,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,9,0,8],[56,64,0.875,0.87946,0.12428,0.82132,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,10,0,14],[60,64,0.9375,0.875,0.10565,0.85711,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,14,0,11],[64,64,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":0.42857,"k":"falling","v":0.49996,"x":0.80357,"p":[[0,27,0.0,0.80357,0.22517,0.71429,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,14,0,0,1,0,14],[4,27,0.1481,0.56249,0.20805,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,9,0,0,4,0,0,13,0,0,1,0,1],[8,27,0.2963,0.63835,0.17492,0.57132,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,1,0,0,5,0,0,20,0,0,1,0,1],[12,27,0.4444,0.59819,0.20959,0.42857,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,5,0,0,6,0,0,13,0,0,0,0,3],[16,27,0.5926,0.60267,0.15459,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,7,0,0,4,0,0,19,0,0,0,0,0],[20,27,0.7407,0.49996,0.23689,0.28571,0.571,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,6,0,0,5,0,0,7,0,0,10,0,0,0,0,1],[24,27,0.8889,0.58925,0.22517,0.4286,0.64286,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,2,0,0,5,0,0,7,0,0,13,0,0,1,0,2],[27,27,1.0,0.54015,0.21048,0.42857,0.4998,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,11,0,0,4,0,0,10,0,0,0,0,2]]}]},{"i":"70b64de4d3b14618","q":"Given that  $40!=\\overline{abcdef283247897734345611269596115894272pqrstuvwx}$ , find  $a,b,c,d,e,f,p,q,r,s,t,u,v,w,x$ .","t":[{"b":4,"e":0.42857,"k":"flat","v":0.41518,"x":0.4375,"p":[[0,15,0.0,0.41518,0.04164,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.4375,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[8,15,0.5333,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.41964,0.03458,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.38839,"x":0.42411,"p":[[0,11,0.0,0.38839,0.08917,0.42857,0.42857,0.42857,0.0,0.42857,1,0,1,1,0,0,0,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bcd7455c9a9f06fc","q":"From the point  $P$  outside a circle  $\\omega$  with center  $O$  draw the tangents  $PA$  and  $PB$  where  $A$  and  $B$  belong to  $\\omega$ .In a random point  $M$  in the chord  $AB$  we draw the perpendicular to  $OM$ , which intersects  $PA$  and  $PB$  in  $C$  and  $D$ . Prove that  $M$  is the midpoint  $CD$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.86159,"x":0.99107,"p":[[0,62,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,62,0.0645,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[8,62,0.129,0.9732,0.09068,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[12,62,0.1935,0.94195,0.19189,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[16,62,0.2581,0.96429,0.11845,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[20,62,0.3226,0.86159,0.31439,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,26],[24,62,0.3871,0.91072,0.25939,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[28,62,0.4516,0.91518,0.24186,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[32,62,0.5161,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,62,0.5806,0.87947,0.29903,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[40,62,0.6452,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,62,0.7097,0.94642,0.19805,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[48,62,0.7742,0.95982,0.12992,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[52,62,0.8387,0.95089,0.18074,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[56,62,0.9032,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[60,62,0.9677,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[62,62,1.0,0.96429,0.11845,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29]]},{"b":2,"e":1.0,"k":"flat","v":0.81696,"x":0.9866,"p":[[0,105,0.0,0.94197,0.20158,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[4,105,0.0381,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,105,0.0762,0.95089,0.14555,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[12,105,0.1143,0.96875,0.08553,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,105,0.1524,0.91071,0.24936,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,27],[20,105,0.1905,0.91517,0.2522,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[24,105,0.2286,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,105,0.2667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,105,0.3048,0.95982,0.1394,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[36,105,0.3429,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[40,105,0.381,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[44,105,0.419,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[48,105,0.4571,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,105,0.4952,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[56,105,0.5333,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[60,105,0.5714,0.93304,0.20511,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,28],[64,105,0.6095,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[68,105,0.6476,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[72,105,0.6857,0.81696,0.343,0.71429,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,23],[76,105,0.7238,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[80,105,0.7619,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[84,105,0.8,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[88,105,0.8381,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[92,105,0.8762,0.91964,0.24727,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[96,105,0.9143,0.94642,0.14617,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[100,105,0.9524,0.90179,0.20652,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,0,0,25],[104,105,0.9905,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[105,105,1.0,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30]]}]},{"i":"3adffb87a580cbb2","q":"We define  $ f: \\mathbb{N} \\rightarrow \\mathbb{N}$ ,  $ f(n) \\equal{} \\sum_{k \\equal{} 1}^{n}(k,n)$ . \r\n\r\na) Show that if  $ \\gcd(m,n)\\equal{}1$  then we have  $ f(mn)\\equal{}f(m)\\cdot f(n)$ ;\r\n\r\nb) Show that  $ \\sum_{d|n}f(d) \\equal{} nd(n)$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.83929,"x":0.87946,"p":[[0,26,0.0,0.8616,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3],[4,26,0.1538,0.83929,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,24,0,2],[8,26,0.3077,0.86606,0.03458,0.85714,0.85714,0.85714,0.857,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[12,26,0.4615,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[16,26,0.6154,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[20,26,0.7692,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[24,26,0.9231,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[26,26,1.0,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4]]},{"b":6,"e":0.85714,"k":"flat","v":0.8616,"x":0.87946,"p":[[0,18,0.0,0.8616,0.06667,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,3],[4,18,0.2222,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[8,18,0.4444,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[12,18,0.6667,0.87052,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3],[16,18,0.8889,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[18,18,1.0,0.87053,0.04164,0.85714,0.85714,0.85714,0.857,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3]]}]},{"i":"b8dc8ccf45f6437b","q":"$2019$  coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from  $ 1$  to  $99$ , the second player in one move can take an even number of coins from  $2$  to  $100$ . The player who can not make a move is lost. Who has the winning strategy in this game?","t":[{"b":3,"e":1.0,"k":"rising","v":0.0625,"x":0.87498,"p":[[0,53,0.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.42857,20,0,5,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.36152,0.34072,0.14286,0.14286,0.46431,0.0,1.0,2,5,0,2,0,18,0,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,5],[8,53,0.1509,0.3883,0.29073,0.14286,0.42857,0.57143,0.0,1.0,5,1,0,5,0,8,0,0,0,0,0,8,0,0,6,0,0,0,0,0,4,0,1],[12,53,0.2264,0.37036,0.34612,0.14286,0.14286,0.53571,0.0,1.0,4,4,0,4,0,14,0,0,1,0,0,5,0,0,0,0,0,0,0,0,4,0,4],[16,53,0.3019,0.37946,0.30433,0.14286,0.35714,0.4286,0.0,1.0,5,4,0,5,0,6,0,0,5,0,0,9,0,0,2,0,0,0,0,0,1,0,4],[20,53,0.3774,0.41063,0.35676,0.14286,0.28571,0.75,0.0,1.0,5,5,0,5,0,10,0,0,2,0,0,4,0,0,2,0,0,1,0,0,3,0,5],[24,53,0.4528,0.39277,0.31751,0.14286,0.28571,0.71429,0.0,1.0,3,3,0,3,0,12,0,0,2,0,0,5,0,0,1,0,0,4,0,0,2,0,3],[28,53,0.5283,0.42401,0.33979,0.14286,0.42857,0.64286,0.0,1.0,3,5,0,3,0,11,0,0,1,0,0,7,0,0,2,0,0,0,0,0,3,0,5],[32,53,0.6038,0.55356,0.36553,0.14286,0.57143,0.89286,0.0,1.0,4,8,0,4,0,5,0,0,2,0,0,3,0,0,4,0,0,2,0,0,4,0,8],[36,53,0.6792,0.64272,0.36243,0.25,0.85714,1.0,0.0,1.0,3,9,0,3,0,5,0,0,1,0,0,1,0,0,2,0,0,3,0,0,8,0,9],[40,53,0.7547,0.78571,0.27432,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,0,11,0,12],[44,53,0.8302,0.81695,0.28847,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,20],[48,53,0.9057,0.77679,0.32525,0.71429,0.85714,1.0,0.0,1.0,3,14,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,14],[52,53,0.9811,0.87498,0.15049,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,11,0,15],[53,53,1.0,0.82142,0.22303,0.82132,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,12,0,12]]},{"b":7,"e":0.57143,"k":"rising","v":0.05348,"x":0.74554,"p":[[0,99,0.0,0.05348,0.11145,0.0,0.0,0.14071,0.0,0.57143,23,0,9,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,99,0.0404,0.42411,0.37878,0.14286,0.21429,0.85714,0.0,1.0,4,7,0,4,0,12,0,0,2,0,0,4,0,0,0,0,0,0,0,0,3,0,7],[8,99,0.0808,0.42402,0.31446,0.14286,0.42857,0.57143,0.0,1.0,3,4,0,3,0,10,0,0,0,0,0,7,0,0,6,0,0,0,0,0,2,0,4],[12,99,0.1212,0.37937,0.32662,0.14286,0.14286,0.60714,0.0,1.0,4,2,0,4,0,13,0,0,0,0,0,5,0,0,2,0,0,1,0,0,5,0,2],[16,99,0.1616,0.31678,0.30678,0.14286,0.14286,0.46428,0.0,1.0,3,4,0,3,0,18,0,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,4],[20,99,0.202,0.37505,0.33646,0.14286,0.35714,0.57143,0.0,1.0,7,3,0,7,0,8,0,0,1,0,0,7,0,0,2,0,0,0,0,0,4,0,3],[24,99,0.2424,0.41063,0.37421,0.14286,0.21429,0.85714,0.0,1.0,6,6,0,6,0,10,0,0,1,0,0,4,0,0,2,0,0,0,0,0,3,0,6],[28,99,0.2828,0.58036,0.38122,0.14286,0.57143,1.0,0.0,1.0,2,12,0,2,0,8,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,12],[32,99,0.3232,0.66516,0.34369,0.42857,0.85707,1.0,0.0,1.0,2,9,0,2,0,5,0,0,0,0,0,3,0,0,2,0,0,2,0,0,9,0,9],[36,99,0.3636,0.50893,0.36411,0.14286,0.42857,1.0,0.0,1.0,3,9,0,3,0,7,0,0,3,0,0,5,0,0,3,0,0,1,0,0,1,0,9],[40,99,0.404,0.49107,0.34614,0.14286,0.42857,0.85714,0.0,1.0,3,6,0,3,0,7,0,0,3,0,0,5,0,0,4,0,0,0,0,0,4,0,6],[44,99,0.4444,0.49999,0.32927,0.14286,0.42857,0.85714,0.0,1.0,2,4,0,2,0,8,0,0,1,0,0,8,0,0,1,0,0,2,0,0,6,0,4],[48,99,0.4848,0.58464,0.34346,0.25001,0.57143,1.0,0.0,1.0,1,9,0,1,0,7,0,0,1,0,0,6,0,0,2,0,0,3,0,0,3,0,9],[52,99,0.5253,0.53112,0.34106,0.14286,0.57143,0.85714,0.0,1.0,1,7,0,1,0,9,0,0,2,0,0,3,0,0,5,0,0,2,0,0,3,0,7],[56,99,0.5657,0.58471,0.3561,0.14286,0.64286,0.89286,0.0,1.0,1,8,0,1,0,9,0,0,0,0,0,4,0,0,2,0,0,2,0,0,6,0,8],[60,99,0.6061,0.67856,0.30305,0.42859,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,1,0,0,5,0,0,6,0,0,2,0,0,5,0,10],[64,99,0.6465,0.65625,0.37263,0.42857,0.85714,1.0,0.0,1.0,4,13,0,4,0,3,0,0,0,0,0,4,0,0,3,0,0,1,0,0,4,0,13],[68,99,0.6869,0.58482,0.32411,0.42857,0.57143,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,0,0,0,9,0,0,5,0,0,3,0,0,2,0,8],[72,99,0.7273,0.54009,0.33653,0.14286,0.57143,0.85714,0.0,1.0,2,7,0,2,0,7,0,0,1,0,0,4,0,0,7,0,0,1,0,0,3,0,7],[76,99,0.7677,0.64285,0.33312,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,5,0,0,1,0,0,5,0,0,3,0,0,2,0,0,5,0,10],[80,99,0.8081,0.52232,0.2917,0.39286,0.42857,0.85714,0.0,1.0,2,3,0,2,0,3,0,0,3,0,0,11,0,0,3,0,0,0,0,0,7,0,3],[84,99,0.8485,0.6741,0.33357,0.42857,0.85707,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,1,0,0,8,0,0,0,0,0,1,0,0,4,0,13],[88,99,0.8889,0.55803,0.3017,0.39285,0.42857,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,4,0,0,10,0,0,2,0,0,3,0,0,2,0,7],[92,99,0.9293,0.74554,0.27137,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,7,0,0,4,0,0,1,0,0,4,0,14],[96,99,0.9697,0.69193,0.28819,0.42857,0.85707,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,3,0,0,3,0,0,3,0,0,3,0,0,9,0,8],[99,99,1.0,0.63393,0.3213,0.42857,0.64286,1.0,0.0,1.0,1,9,0,1,0,4,0,0,0,0,0,10,0,0,1,0,0,1,0,0,6,0,9]]}]},{"i":"6b1cd6564bcb585c","q":"If  $x$ ,  $y$ ,  $z$  are positive numbers satisfying \n\\[x+\\frac{y}{z}=y+\\frac{z}{x}=z+\\frac{x}{y}=2.\\]\nFind all the possible values of  $x+y+z$ .","t":[{"b":0,"e":0.2857,"k":"flat","v":0.13839,"x":0.37053,"p":[[0,142,0.0,0.22321,0.25238,0.0,0.14288,0.2857,0.0,1.0,10,2,0,10,0,7,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,142,0.0282,0.25446,0.23072,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,6,0,0,15,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[8,142,0.0563,0.32141,0.21427,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,13,0,0,5,0,0,2,0,0,2,0,0,0,0,1],[12,142,0.0845,0.37053,0.29851,0.14286,0.28571,0.71429,0.0,1.0,6,2,0,6,0,4,0,0,11,0,0,1,0,0,0,0,0,8,0,0,0,0,2],[16,142,0.1127,0.20533,0.18872,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,4,0,0,13,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[20,142,0.1408,0.23214,0.23351,0.0,0.2857,0.28571,0.0,1.0,10,1,0,10,0,5,0,0,12,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[24,142,0.169,0.20089,0.21975,0.0,0.14286,0.28571,0.0,1.0,11,1,0,11,0,7,0,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[28,142,0.1972,0.25893,0.2126,0.14286,0.28571,0.28571,0.0,1.0,6,1,0,6,0,5,0,0,18,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[32,142,0.2254,0.23214,0.21943,0.0,0.2857,0.28571,0.0,1.0,10,1,0,10,0,3,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[36,142,0.2535,0.18303,0.1684,0.0,0.14288,0.28571,0.0,0.71429,11,0,0,11,0,6,0,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[40,142,0.2817,0.23661,0.16602,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,11,0,0,13,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[44,142,0.3099,0.18304,0.21199,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,4,0,0,11,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[48,142,0.338,0.19643,0.17035,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,8,0,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[52,142,0.3662,0.17411,0.1461,0.0,0.21428,0.28571,0.0,0.57143,11,0,0,11,0,5,0,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,142,0.3944,0.1875,0.1357,0.0,0.2857,0.28571,0.0,0.42857,9,0,0,9,0,6,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,142,0.4225,0.19196,0.14987,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,4,0,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,142,0.4507,0.25446,0.09268,0.14286,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,8,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[68,142,0.4789,0.22768,0.16311,0.14286,0.21428,0.28571,0.0,0.71429,4,0,0,4,0,12,0,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[72,142,0.507,0.13839,0.14054,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,7,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,142,0.5352,0.1875,0.16917,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,8,0,0,10,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[80,142,0.5634,0.20535,0.12339,0.14286,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,5,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,142,0.5915,0.26786,0.23623,0.14286,0.2143,0.28571,0.0,1.0,5,1,0,5,0,11,0,0,10,0,0,1,0,0,1,0,0,3,0,0,0,0,1],[88,142,0.6197,0.21429,0.19233,0.10714,0.21428,0.28571,0.0,0.85714,8,0,0,8,0,8,0,0,13,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[92,142,0.6479,0.15625,0.13533,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,6,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,142,0.6761,0.20982,0.19556,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,6,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[100,142,0.7042,0.22321,0.15947,0.14286,0.2857,0.28571,0.0,0.71429,7,0,0,7,0,6,0,0,15,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[104,142,0.7324,0.20982,0.18205,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,6,0,0,14,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[108,142,0.7606,0.18303,0.11425,0.14286,0.21428,0.28571,0.0,0.28571,7,0,0,7,0,9,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,142,0.7887,0.18303,0.16457,0.0,0.21428,0.28571,0.0,0.71429,11,0,0,11,0,5,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[116,142,0.8169,0.21875,0.17122,0.14286,0.2143,0.28571,0.0,0.71429,6,0,0,6,0,10,0,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[120,142,0.8451,0.28571,0.18898,0.24999,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,4,0,0,19,0,0,2,0,0,0,0,0,2,0,0,1,0,0],[124,142,0.8732,0.17411,0.20119,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,8,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[128,142,0.9014,0.17857,0.11845,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,11,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[132,142,0.9296,0.20982,0.15561,0.14286,0.2857,0.28571,0.0,0.71429,7,0,0,7,0,8,0,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[136,142,0.9577,0.23661,0.16982,0.14286,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,9,0,0,14,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[140,142,0.9859,0.19194,0.16207,0.0,0.21428,0.28571,0.0,0.571,10,0,0,10,0,6,0,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[142,142,1.0,0.16964,0.11538,0.10714,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,10,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.15178,"x":0.3214,"p":[[0,155,0.0,0.18304,0.15663,0.0,0.14288,0.28571,0.0,0.71429,9,0,0,9,0,9,0,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,155,0.0258,0.25437,0.21651,0.14214,0.2857,0.28571,0.0,1.0,7,1,0,7,0,6,0,0,13,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[8,155,0.0516,0.3214,0.22584,0.14286,0.28571,0.42857,0.0,0.857,4,0,0,4,0,7,0,0,10,0,0,4,0,0,3,0,0,3,0,0,1,0,0],[12,155,0.0774,0.2857,0.19883,0.14286,0.28571,0.28571,0.0,1.0,4,1,0,4,0,6,0,0,15,0,0,3,0,0,3,0,0,0,0,0,0,0,1],[16,155,0.1032,0.25,0.22588,0.14286,0.21428,0.28571,0.0,1.0,7,1,0,7,0,9,0,0,9,0,0,3,0,0,2,0,0,1,0,0,0,0,1],[20,155,0.129,0.25446,0.24675,0.10714,0.28571,0.28571,0.0,1.0,8,2,0,8,0,5,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[24,155,0.1548,0.20536,0.19541,0.0,0.21428,0.28571,0.0,0.71429,11,0,0,11,0,5,0,0,11,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[28,155,0.1806,0.26786,0.23077,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,6,0,0,12,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[32,155,0.2065,0.27679,0.2141,0.14286,0.28571,0.28571,0.0,1.0,3,1,0,3,0,9,0,0,16,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[36,155,0.2323,0.20089,0.21086,0.0,0.2857,0.28571,0.0,1.0,12,1,0,12,0,3,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[40,155,0.2581,0.19643,0.14174,0.10714,0.28571,0.28571,0.0,0.57143,8,0,0,8,0,7,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,155,0.2839,0.25,0.17857,0.14286,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,9,0,0,16,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[48,155,0.3097,0.26786,0.17405,0.14286,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,4,0,0,18,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[52,155,0.3355,0.1875,0.13092,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,155,0.3613,0.24554,0.27254,0.10714,0.14286,0.28571,0.0,1.0,8,3,0,8,0,10,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[60,155,0.3871,0.22768,0.11214,0.14286,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,7,0,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[64,155,0.4129,0.20089,0.11769,0.14286,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,5,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,155,0.4387,0.20089,0.17445,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,2,0,0,17,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[72,155,0.4645,0.23205,0.20442,0.14214,0.28571,0.28571,0.0,1.0,7,1,0,7,0,7,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[76,155,0.4903,0.20089,0.18161,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,8,0,0,12,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[80,155,0.5161,0.26786,0.17035,0.14289,0.28571,0.28571,0.0,1.0,3,1,0,3,0,6,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[84,155,0.5419,0.22768,0.22263,0.10714,0.21428,0.28571,0.0,1.0,8,1,0,8,0,8,0,0,12,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[88,155,0.5677,0.20535,0.12846,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,3,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,155,0.5935,0.27678,0.2111,0.14286,0.28571,0.28571,0.0,1.0,5,1,0,5,0,5,0,0,17,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[96,155,0.6194,0.20089,0.12299,0.14286,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,6,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,155,0.6452,0.20089,0.17075,0.0,0.2857,0.28571,0.0,0.71429,11,0,0,11,0,2,0,0,16,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[104,155,0.671,0.19197,0.12682,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,6,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,155,0.6968,0.21428,0.15152,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,6,0,0,17,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[112,155,0.7226,0.26339,0.21162,0.14286,0.2857,0.28571,0.0,1.0,5,1,0,5,0,7,0,0,16,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[116,155,0.7484,0.19641,0.16653,0.0,0.21428,0.28571,0.0,0.57143,10,0,0,10,0,6,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[120,155,0.7742,0.22768,0.17445,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,6,0,0,16,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[124,155,0.8,0.20535,0.19212,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,6,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[128,155,0.8258,0.22322,0.17835,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,8,0,0,13,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[132,155,0.8516,0.15625,0.14445,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,5,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[136,155,0.8774,0.27232,0.22406,0.14286,0.28571,0.32143,0.0,1.0,6,1,0,6,0,7,0,0,11,0,0,5,0,0,0,0,0,2,0,0,0,0,1],[140,155,0.9032,0.22321,0.13333,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,12,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[144,155,0.929,0.20089,0.16311,0.0,0.2857,0.28571,0.0,0.71429,10,0,0,10,0,3,0,0,17,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[148,155,0.9548,0.21875,0.23954,0.0,0.28571,0.28571,0.0,1.0,11,2,0,11,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[152,155,0.9806,0.15178,0.13333,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,4,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[155,155,1.0,0.18303,0.1197,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d4e16484ccf740a8","q":"In a country there are  $n>100$  cities and initially no roads. The government randomly determined the cost of building a two-way road between any two cities, using all amounts from  $1$  to  $\\frac{n(n-1)}{2}$  thalers once (all options are equally likely). The mayor of each city chooses the cheapest of the  $n-1$  roads emanating from that city and it is built (this may be the mutual desired of the mayors of both cities being connected, or only one of the two).\nAfter the construction of these roads, the cities are divided into  $M$  connected components (between cities of the same connected component, you can get along the constructed roads, possibly via other cities, but this is not possible for cities of different components). Find the expected value of the random variable  $M$ .\n*Proposed by F. Petrov*","t":[{"b":2,"e":1.0,"k":"flat","v":0.90179,"x":1.0,"p":[[0,24,0.0,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[4,24,0.1667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,24,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,24,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,24,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,24,0.8333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,24,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,52,0.0,0.92411,0.24996,1.0,1.0,1.0,0.0,1.0,2,29,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[4,52,0.0769,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,52,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,52,0.2308,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,52,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,52,0.3846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,52,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,52,0.5385,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,52,0.6154,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,52,0.6923,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8236d816e3de566d","q":"In a room, there are 2005 fruit crates, each containing one or more types of fruit, with each type of fruit being present in whole numbers.\na) Show that it is always possible to select 669 fruit crates that together contain at least one third of all the apples and at least one third of all the pears.\nb) Can the crates in part a) always be chosen such that they also contain at least one third of all the peaches?","t":[{"b":0,"e":1.0,"k":"flat","v":0.27233,"x":0.39732,"p":[[0,29,0.0,0.30803,0.0724,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,29,0.1379,0.39732,0.31689,0.28571,0.28571,0.57143,0.0,1.0,5,4,0,5,0,1,0,0,17,0,0,0,0,0,2,0,0,0,0,0,3,0,4],[8,29,0.2759,0.27233,0.13533,0.2857,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,29,0.4138,0.36607,0.30917,0.2857,0.28571,0.32143,0.0,1.0,6,5,0,6,0,0,0,0,18,0,0,2,0,0,0,0,0,1,0,0,0,0,5],[16,29,0.5517,0.33036,0.23538,0.2857,0.28571,0.32143,0.0,1.0,5,1,0,5,0,0,0,0,19,0,0,4,0,0,0,0,0,1,0,0,2,0,1],[20,29,0.6897,0.34821,0.26229,0.2857,0.28571,0.28571,0.0,1.0,4,3,0,4,0,1,0,0,20,0,0,2,0,0,1,0,0,0,0,0,1,0,3],[24,29,0.8276,0.38393,0.28669,0.28571,0.28571,0.28571,0.0,1.0,3,4,0,3,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[28,29,0.9655,0.28572,0.13832,0.2857,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,2,0,0,26,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[29,29,1.0,0.34373,0.16696,0.2857,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,2,0,0,0,0,0,1,0,1]]},{"b":6,"e":0.28571,"k":"flat","v":0.25894,"x":0.38839,"p":[[0,28,0.0,0.25894,0.10972,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,28,0.1429,0.29018,0.25874,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,0,0,0,17,0,0,1,0,0,0,0,0,3,0,0,1,0,1],[8,28,0.2857,0.30357,0.25938,0.10714,0.28571,0.28571,0.0,0.85714,8,0,0,8,0,2,0,0,15,0,0,0,0,0,1,0,0,4,0,0,2,0,0],[12,28,0.4286,0.27677,0.24726,0.0,0.2857,0.28571,0.0,1.0,9,1,0,9,0,1,0,0,16,0,0,1,0,0,2,0,0,1,0,0,1,0,1],[16,28,0.5714,0.30357,0.31693,0.0,0.28571,0.28571,0.0,1.0,10,3,0,10,0,2,0,0,14,0,0,0,0,0,0,0,0,1,0,0,2,0,3],[20,28,0.7143,0.36605,0.31929,0.21427,0.28571,0.571,0.0,1.0,8,4,0,8,0,0,0,0,14,0,0,1,0,0,3,0,0,1,0,0,1,0,4],[24,28,0.8571,0.36607,0.3008,0.2857,0.28571,0.32144,0.0,1.0,5,4,0,5,0,2,0,0,17,0,0,1,0,0,1,0,0,1,0,0,1,0,4],[28,28,1.0,0.38839,0.22371,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0,1,0,0,2,0,2]]}]},{"i":"6ac266cf6b08c799","q":"Let the circumcircle of a triangle  $ABC$  be  $\\Gamma$ . The tangents to  $\\Gamma$  at  $B,C$  meet at point  $E$ . For a point  $F$  on line  $BC$  which is not on the segment  $BC$ , let the midpoint of  $EF$  be  $G$ . Lines  $GB,GC$  meet  $\\Gamma$  again at points  $I,H$  respectively. Let  $M$  be the midpoint of  $BC$ . Prove that the points  $F,I,H,M$  lie on a circle.\n\nProposed by *Mehmet Can Ba\u015ftemir*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.21868,"x":0.43746,"p":[[0,19,0.0,0.34374,0.30274,0.0,0.35714,0.57143,0.0,1.0,10,1,0,10,0,3,0,0,3,0,0,6,0,0,4,0,0,3,0,0,2,0,1],[4,19,0.2105,0.21868,0.23676,0.0,0.14286,0.42858,0.0,0.71429,13,0,0,13,0,7,0,0,2,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[8,19,0.4211,0.43746,0.24725,0.28571,0.4998,0.57143,0.0,0.85714,5,0,0,5,0,1,0,0,5,0,0,5,0,0,9,0,0,6,0,0,1,0,0],[12,19,0.6316,0.35704,0.31546,0.0,0.28571,0.57143,0.0,1.0,9,2,0,9,0,4,0,0,4,0,0,4,0,0,5,0,0,2,0,0,2,0,2],[16,19,0.8421,0.42853,0.33118,0.14286,0.42857,0.71429,0.0,1.0,7,2,0,7,0,4,0,0,3,0,0,4,0,0,5,0,0,2,0,0,5,0,2],[19,19,1.0,0.38835,0.26779,0.24999,0.42857,0.57143,0.0,1.0,6,2,0,6,0,2,0,0,7,0,0,4,0,0,10,0,0,1,0,0,0,0,2]]},{"b":4,"e":0.42857,"k":"rising","v":0.25883,"x":0.48646,"p":[[0,18,0.0,0.25883,0.28448,0.0,0.14288,0.42858,0.0,1.0,12,2,0,12,0,5,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,2],[4,18,0.2222,0.36608,0.21995,0.25,0.42857,0.46525,0.0,0.71429,5,0,0,5,0,3,0,0,5,0,0,11,0,0,4,0,0,4,0,0,0,0,0],[8,18,0.4444,0.41511,0.2153,0.28571,0.42857,0.57111,0.0,0.71429,4,0,0,4,0,2,0,0,4,0,0,9,0,0,9,0,0,4,0,0,0,0,0],[12,18,0.6667,0.47314,0.22706,0.2857,0.571,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,6,0,0,5,0,0,12,0,0,1,0,0,4,0,0],[16,18,0.8889,0.48646,0.23117,0.28571,0.571,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,3,0,0,4,0,0,12,0,0,5,0,0,1,0,1],[18,18,1.0,0.47316,0.28219,0.2857,0.571,0.57143,0.0,1.0,5,3,0,5,0,1,0,0,4,0,0,4,0,0,11,0,0,4,0,0,0,0,3]]}]},{"i":"edcdb5c9b643f499","q":"Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and \neach cevian starting from that vertex, is possible to construct a triangle.","t":[{"b":1,"e":0.571,"k":"flat","v":0.20516,"x":0.39279,"p":[[0,77,0.0,0.31667,0.24169,0.14214,0.28571,0.4286,0.0,0.85714,5,0,1,5,0,8,0,0,7,0,0,5,0,0,2,0,0,4,0,0,1,0,0],[4,77,0.0519,0.39279,0.27428,0.14286,0.49979,0.60714,0.0,0.71429,6,0,0,6,0,6,0,0,2,0,0,2,0,0,8,0,0,8,0,0,0,0,0],[8,77,0.1039,0.38841,0.32971,0.0,0.42857,0.71429,0.0,1.0,9,1,0,9,0,5,0,0,0,0,0,5,0,0,4,0,0,4,0,0,4,0,1],[12,77,0.1558,0.30356,0.2389,0.0,0.28571,0.4642,0.0,0.71429,10,0,0,10,0,1,0,0,6,0,0,7,0,0,6,0,0,2,0,0,0,0,0],[16,77,0.2078,0.3839,0.27532,0.14286,0.42857,0.57143,0.0,1.0,5,2,0,5,0,6,0,0,3,0,0,7,0,0,6,0,0,3,0,0,0,0,2],[20,77,0.2597,0.27679,0.25738,0.0,0.2857,0.42857,0.0,1.0,9,1,0,9,0,5,0,0,9,0,0,3,0,0,2,0,0,3,0,0,0,0,1],[24,77,0.3117,0.22768,0.1984,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,14,0,0,5,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[28,77,0.3636,0.25891,0.25611,0.0,0.2143,0.4286,0.0,1.0,11,1,0,11,0,5,0,0,5,0,0,5,0,0,4,0,0,1,0,0,0,0,1],[32,77,0.4156,0.29017,0.24609,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,3,0,0,9,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[36,77,0.4675,0.37498,0.29611,0.10714,0.42857,0.57143,0.0,1.0,8,1,0,8,0,4,0,0,3,0,0,4,0,0,6,0,0,5,0,0,1,0,1],[40,77,0.5195,0.31249,0.27533,0.10714,0.2143,0.57111,0.0,1.0,8,1,0,8,0,8,0,0,1,0,0,6,0,0,6,0,0,1,0,0,1,0,1],[44,77,0.5714,0.33472,0.25161,0.14214,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,5,0,0,5,0,0,5,0,0,5,0,0,5,0,0,0,0,0],[48,77,0.6234,0.28569,0.29664,0.0,0.14288,0.4642,0.0,1.0,11,1,0,11,0,6,0,0,4,0,0,3,0,0,2,0,0,4,0,0,1,0,1],[52,77,0.6753,0.20516,0.20186,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,8,0,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[56,77,0.7273,0.28118,0.21563,0.10714,0.28571,0.4286,0.0,0.71429,8,0,0,8,0,5,0,0,6,0,0,7,0,0,5,0,0,1,0,0,0,0,0],[60,77,0.7792,0.34809,0.23406,0.14286,0.35714,0.571,0.0,0.71429,5,0,0,5,0,6,0,0,5,0,0,6,0,0,6,0,0,4,0,0,0,0,0],[64,77,0.8312,0.35265,0.23951,0.14286,0.35714,0.57143,0.0,0.71429,6,0,0,6,0,4,0,0,6,0,0,5,0,0,7,0,0,4,0,0,0,0,0],[68,77,0.8831,0.37491,0.28746,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,6,0,0,2,0,0,9,0,0,3,0,0,2,0,0,3,0,1],[72,77,0.9351,0.31243,0.2706,0.0,0.28571,0.571,0.0,0.71429,10,0,0,10,0,4,0,0,4,0,0,3,0,0,6,0,0,5,0,0,0,0,0],[76,77,0.987,0.29462,0.16723,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,9,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[77,77,1.0,0.25873,0.18371,0.14,0.28571,0.42857,0.0,0.571,7,0,0,7,0,7,0,0,5,0,0,11,0,0,2,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14,"k":"flat","v":0.20508,"x":0.38838,"p":[[0,51,0.0,0.27679,0.28333,0.10714,0.1429,0.32143,0.0,1.0,8,2,4,8,0,9,0,0,7,0,0,2,0,0,2,0,0,1,0,0,1,0,2],[4,51,0.0784,0.38383,0.32629,0.14214,0.35714,0.71429,0.0,1.0,7,2,0,7,0,7,0,0,2,0,0,5,0,0,2,0,0,4,0,0,3,0,2],[8,51,0.1569,0.38838,0.32777,0.10714,0.42857,0.71429,0.0,1.0,8,1,0,8,0,6,0,0,1,0,0,5,0,0,2,0,0,5,0,0,4,0,1],[12,51,0.2353,0.28571,0.30723,0.0,0.14286,0.57143,0.0,1.0,14,1,0,14,0,3,0,0,2,0,0,3,0,0,4,0,0,5,0,0,0,0,1],[16,51,0.3137,0.23212,0.23347,0.0,0.14286,0.42858,0.0,0.71429,11,0,0,11,0,8,0,0,3,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[20,51,0.3922,0.32591,0.29929,0.0,0.28571,0.571,0.0,0.85714,11,0,0,11,0,2,0,0,5,0,0,5,0,0,2,0,0,4,0,0,3,0,0],[24,51,0.4706,0.2991,0.29527,0.0,0.21431,0.57111,0.0,1.0,10,2,0,10,0,6,0,0,4,0,0,3,0,0,5,0,0,2,0,0,0,0,2],[28,51,0.549,0.37054,0.2942,0.14286,0.35714,0.60714,0.0,0.85714,7,0,0,7,0,6,0,0,3,0,0,4,0,0,4,0,0,5,0,0,3,0,0],[32,51,0.6275,0.26775,0.29615,0.0,0.14286,0.57111,0.0,1.0,12,1,0,12,0,7,0,0,3,0,0,1,0,0,3,0,0,5,0,0,0,0,1],[36,51,0.7059,0.30356,0.20746,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,11,0,0,6,0,0,6,0,0,3,0,0,3,0,0,0,0,0],[40,51,0.7843,0.27229,0.2657,0.0,0.14286,0.4642,0.0,0.85714,10,0,0,10,0,8,0,0,2,0,0,4,0,0,4,0,0,3,0,0,1,0,0],[44,51,0.8627,0.31695,0.2806,0.10714,0.2143,0.57143,0.0,0.85714,8,0,0,8,0,8,0,0,3,0,0,3,0,0,3,0,0,6,0,0,1,0,0],[48,51,0.9412,0.25437,0.20436,0.14214,0.21435,0.42857,0.0,0.71429,7,0,0,7,0,9,0,0,5,0,0,8,0,0,1,0,0,2,0,0,0,0,0],[51,51,1.0,0.20508,0.1555,0.14214,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"908aca6fb8350945","q":"Prove that  $$ \\operatorname{Re}\\left(\\operatorname{Li}_2\\left(\\frac{1-i\\sqrt3}2\\right)+\\operatorname{Li}_2\\left(\\frac{\\sqrt3-i}{2\\sqrt3}\\right)\\right)=\\frac{7\\pi^2}{72}-\\frac{\\ln^23}8 $$  where as usual  $$ \\operatorname{Li}_2(z)=-\\int^z_0\\frac{\\ln(1-t)}tdt,z\\in\\mathbb C\\setminus[1,\\infty) $$  *Proposed by Paolo Perfetti*","t":[{"b":6,"e":0.71429,"k":"flat","v":0.65179,"x":0.77679,"p":[[0,32,0.0,0.69196,0.22335,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,8,0,0,4,0,0,4,0,0,11,0,4],[4,32,0.125,0.65179,0.26471,0.42857,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,4,0,0,1,0,0,5,0,0,3,0,0,2,0,0,16,0,1],[8,32,0.25,0.70535,0.20805,0.57143,0.78564,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,6,0,0,15,0,1],[12,32,0.375,0.75893,0.18707,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,20,0,1],[16,32,0.5,0.77679,0.13333,0.67857,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,23,0,0],[20,32,0.625,0.76785,0.15465,0.71429,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,20,0,1],[24,32,0.75,0.70982,0.16164,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,5,0,0,13,0,1],[28,32,0.875,0.77232,0.1063,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,18,0,0],[32,32,1.0,0.75,0.13832,0.57143,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,19,0,0]]},{"b":7,"e":0.571,"k":"flat","v":0.62497,"x":0.74554,"p":[[0,78,0.0,0.64286,0.23958,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,10,0,0,2,0,0,4,0,0,10,0,3],[4,78,0.0513,0.64284,0.21429,0.57143,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,3,0,0,12,0,0,2,0,0,11,0,1],[8,78,0.1026,0.6875,0.2299,0.5357,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,6,0,0,5,0,0,2,0,0,15,0,2],[12,78,0.1538,0.72765,0.19021,0.57143,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,5,0,0,17,0,1],[16,78,0.2051,0.67411,0.20896,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,5,0,0,8,0,0,2,0,0,14,0,1],[20,78,0.2564,0.69197,0.17536,0.57143,0.78571,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,1,0,0,16,0,0],[24,78,0.3077,0.74554,0.1665,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,2,0,0,20,0,0],[28,78,0.359,0.70982,0.18723,0.57143,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,4,0,0,17,0,0],[32,78,0.4103,0.70534,0.20498,0.5713,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,6,0,0,5,0,0,2,0,0,17,0,1],[36,78,0.4615,0.62497,0.22232,0.5713,0.57143,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,1,0,0,2,0,0,13,0,0,1,0,0,12,0,0],[40,78,0.5128,0.73214,0.22517,0.57143,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,0,19,0,2],[44,78,0.5641,0.69643,0.20124,0.57143,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,3,0,0,17,0,0],[48,78,0.6154,0.71427,0.17858,0.57143,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,3,0,0,17,0,0],[52,78,0.6667,0.67856,0.20204,0.57132,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,6,0,0,8,0,0,2,0,0,14,0,1],[56,78,0.7179,0.63391,0.22851,0.42857,0.64286,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,4,0,0,11,0,1],[60,78,0.7692,0.74554,0.1665,0.57143,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,5,0,0,17,0,1],[64,78,0.8205,0.70089,0.19999,0.57143,0.85714,0.85714,0.1429,0.85714,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,0,2,0,0,18,0,0],[68,78,0.8718,0.71875,0.19061,0.57143,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,2,0,0,17,0,1],[72,78,0.9231,0.64732,0.23415,0.53571,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,7,0,0,13,0,0],[76,78,0.9744,0.64284,0.22869,0.57132,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,2,0,0,8,0,0,5,0,0,11,0,1],[78,78,1.0,0.64728,0.21425,0.5354,0.71429,0.85714,0.1429,0.85714,0,0,0,0,0,1,0,0,3,0,0,4,0,0,7,0,0,4,0,0,13,0,0]]}]},{"i":"8587ea07f8c4f5f7","q":"Suppose that  $S$  is a finite set of points in the plane such that the area of triangle  $\\triangle ABC$  is at most  $1$  whenever  $A,B,$  and  $C$  are in  $S.$  Show that there exists a triangle of area  $4$  that (together with its interior) covers the set  $S.$","t":[{"b":0,"e":1.0,"k":"volatile","v":0.28116,"x":0.96875,"p":[[0,19,0.0,0.32143,0.41032,0.0,0.14286,0.74996,0.0,1.0,15,6,1,15,0,6,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,6],[4,19,0.2105,0.39732,0.44996,0.0,0.14286,1.0,0.0,1.0,15,10,0,15,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,10],[8,19,0.4211,0.40161,0.3805,0.105,0.28571,0.71429,0.0,1.0,8,7,0,8,0,7,0,0,3,0,0,3,0,0,1,0,0,3,0,0,0,0,7],[12,19,0.6316,0.28116,0.36158,0.0,0.14286,0.32143,0.0,1.0,13,5,0,13,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[16,19,0.8421,0.85714,0.27664,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,22],[19,19,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]},{"b":3,"e":0.57143,"k":"rising","v":0.22313,"x":0.93079,"p":[[0,22,0.0,0.22313,0.32527,0.0,0.14143,0.2857,0.0,1.0,15,3,1,15,0,8,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[4,22,0.1818,0.40177,0.39356,0.14286,0.14288,0.89286,0.0,1.0,7,8,0,7,0,10,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,8],[8,22,0.3636,0.62945,0.40699,0.14286,0.92857,1.0,0.0,1.0,4,16,0,4,0,5,0,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,16],[12,22,0.5455,0.87053,0.28203,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,23],[16,22,0.7273,0.9241,0.20198,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[20,22,0.9091,0.93079,0.11225,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,1,21],[22,22,1.0,0.9241,0.12364,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20]]}]},{"i":"9ce295beb2919271","q":"Suppose function  $f:[0,\\infty)\\to[0,\\infty)$  satisfies \n(1) $\\forall x,y \\geq 0,$  we have  $f(x)f(y)\\leq y^2f(\\frac{x}{2})+x^2f(\\frac{y}{2})$ ;\n(2) $\\forall 0 \\leq x \\leq 1, f(x) \\leq 2016$ .\nProve that  $f(x)\\leq x^2$  for all  $x\\geq 0$ .","t":[{"b":0,"e":0.57143,"k":"falling","v":0.47319,"x":0.91964,"p":[[0,60,0.0,0.91964,0.18877,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,26],[4,60,0.0667,0.87946,0.23449,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[8,60,0.1333,0.89286,0.15152,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,18],[12,60,0.2,0.81696,0.29931,0.75,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,5,0,0,0,0,0,0,0,0,3,0,21],[16,60,0.2667,0.81696,0.27254,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,4,0,0,3,0,0,1,0,0,2,0,20],[20,60,0.3333,0.87947,0.23717,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,22],[24,60,0.4,0.7366,0.35012,0.53571,0.92857,1.0,0.0,1.0,2,16,0,2,0,4,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,16],[28,60,0.4667,0.77679,0.21997,0.57143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,4,0,0,6,0,12],[32,60,0.5333,0.77677,0.27185,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,2,0,0,10,0,12],[36,60,0.6,0.78123,0.28569,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,0,2,0,17],[40,60,0.6667,0.67856,0.26487,0.42857,0.64286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,3,0,0,3,0,10],[44,60,0.7333,0.70089,0.25595,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,4,0,0,8,0,0,4,0,0,6,0,8],[48,60,0.8,0.60713,0.2369,0.42857,0.57143,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,8,0,0,8,0,0,4,0,0,3,0,5],[52,60,0.8667,0.55357,0.16656,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,8,0,0,9,0,0,12,0,0,0,0,0],[56,60,0.9333,0.48214,0.1171,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,15,0,0,10,0,0,3,0,0,0,0,0],[60,60,1.0,0.47319,0.12593,0.39286,0.571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,7,0,0,16,0,0,1,0,0,0,0,0]]},{"b":1,"e":1.0,"k":"flat","v":0.91071,"x":0.98661,"p":[[0,116,0.0,0.91518,0.17807,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[4,116,0.0345,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[8,116,0.069,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[12,116,0.1034,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[16,116,0.1379,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[20,116,0.1724,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[24,116,0.2069,0.91071,0.20438,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,26],[28,116,0.2414,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[32,116,0.2759,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,116,0.3103,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,116,0.3448,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,116,0.3793,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[48,116,0.4138,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[52,116,0.4483,0.95981,0.1143,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[56,116,0.4828,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[60,116,0.5172,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[64,116,0.5517,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[68,116,0.5862,0.95089,0.16982,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[72,116,0.6207,0.92857,0.19233,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,28],[76,116,0.6552,0.93304,0.15966,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,27],[80,116,0.6897,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[84,116,0.7241,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[88,116,0.7586,0.94196,0.13767,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[92,116,0.7931,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[96,116,0.8276,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[100,116,0.8621,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[104,116,0.8966,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[108,116,0.931,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[112,116,0.9655,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[116,116,1.0,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23]]}]},{"i":"8cc47e53b150aacf","q":"RUS   Let $n$ be a positive integer. Given a sequence $\\varepsilon_{1}, \\ldots, \\varepsilon_{n-1}$ with $\\varepsilon_{i}=0$ or $\\varepsilon_{i}=1$ for each $i=1, \\ldots, n-1$, the sequences $a_{0}, \\ldots, a_{n}$ and $b_{0}, \\ldots, b_{n}$ are constructed by the following rules:  $$ \\begin{gathered} a_{0}=b_{0}=1, \\quad a_{1}=b_{1}=7, \\\\ a_{i+1}=\\left\\{\\begin{array}{ll} 2 a_{i-1}+3 a_{i}, & \\text { if } \\varepsilon_{i}=0, \\\\ 3 a_{i-1}+a_{i}, & \\text { if } \\varepsilon_{i}=1, \\end{array} \\text { for each } i=1, \\ldots, n-1,\\right. \\\\ b_{i+1}=\\left\\{\\begin{array}{ll} 2 b_{i-1}+3 b_{i}, & \\text { if } \\varepsilon_{n-i}=0, \\\\ 3 b_{i-1}+b_{i}, & \\text { if } \\varepsilon_{n-i}=1, \\end{array} \\text { for each } i=1, \\ldots, n-1 .\\right. \\end{gathered} $$  Prove that $a_{n}=b_{n}$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.44194,"x":0.93304,"p":[[0,48,0.0,0.44194,0.21534,0.28571,0.42857,0.571,0.0,1.0,1,2,1,1,0,1,0,0,10,0,0,11,0,0,5,0,0,1,0,0,1,0,2],[4,48,0.0833,0.79013,0.20203,0.57143,0.78564,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,6,0,0,3,0,13],[8,48,0.1667,0.74107,0.22428,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,10,0,0,1,0,11],[12,48,0.25,0.78125,0.22011,0.57143,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,7,0,0,3,0,13],[16,48,0.3333,0.75446,0.24284,0.57143,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,6,0,0,2,0,13],[20,48,0.4167,0.79018,0.23141,0.57143,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,6,0,0,0,0,16],[24,48,0.5,0.83034,0.2156,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0,0,0,19],[28,48,0.5833,0.86161,0.19719,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,1,0,20],[32,48,0.6667,0.93303,0.14279,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[36,48,0.75,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[40,48,0.8333,0.91963,0.13807,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,23],[44,48,0.9167,0.88837,0.18471,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,3,0,22],[48,48,1.0,0.72097,0.22331,0.57143,0.64286,1.0,0.357,1.0,0,11,0,0,0,0,0,0,0,0,1,4,0,0,11,0,0,4,0,0,1,0,11]]},{"b":7,"e":0.42857,"k":"flat","v":0.53124,"x":0.79462,"p":[[0,39,0.0,0.56473,0.2848,0.28571,0.4286,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,10,0,0,6,0,1,2,0,0,2,0,0,4,0,6],[4,39,0.1026,0.71428,0.22016,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,12,0,0,4,0,7],[8,39,0.2051,0.70979,0.26363,0.57143,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,1,0,0,1,0,0,6,0,0,12,0,0,0,0,10],[12,39,0.3077,0.79462,0.25241,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,8,0,0,0,0,17],[16,39,0.4103,0.6607,0.21355,0.57132,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,4,0,0,8,0,0,10,0,0,1,0,6],[20,39,0.5128,0.72764,0.2352,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,1,0,0,7,0,0,9,0,0,2,0,10],[24,39,0.6154,0.74552,0.2194,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,5,0,0,1,0,12],[28,39,0.7179,0.70982,0.25376,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,7,0,0,1,0,11],[32,39,0.8205,0.64058,0.26215,0.41068,0.71429,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,6,0,1,1,0,0,6,0,0,8,0,0,2,0,7],[36,39,0.9231,0.69194,0.21163,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,9,0,0,2,0,7],[39,39,1.0,0.53124,0.21498,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,8,0,0,5,0,0,8,0,0,7,0,0,1,0,2]]}]},{"i":"b6e7cb6186c82fe4","q":"Prove that:\n(a) if  $y<\\frac12$  and  $n\\ge3$  is a natural number then  $(y+1)^n\\ge y^n+(1+2y)^\\frac n2$ ;\n(b) if  $x,y,z$  and  $n\\ge3$  are natural numbers for which  $x^2-1\\le2y$  then  $x^n+y^n\\ne z^n$ 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in real numbers the system of equations: \\begin{align*}\n\\frac{1}{xy}&=\\frac{x}{z}+1 \n\\frac{1}{yz}&=\\frac{y}{x}+1 \n\\frac{1}{zx}&=\\frac{z}{y}+1 \n\\end{align*}","t":[{"b":0,"e":0.57143,"k":"flat","v":0.5357,"x":0.76784,"p":[[0,165,0.0,0.6161,0.22707,0.42857,0.71429,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,4,0,0,5,0,0,9,0,0,6,0,2],[4,165,0.0242,0.55804,0.22968,0.28571,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,9,0,0,4,0,0,9,0,0,4,0,0,3,0,3],[8,165,0.0485,0.625,0.21053,0.42857,0.71429,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,7,0,0,4,0,0,9,0,0,6,0,2],[12,165,0.0727,0.63839,0.23686,0.42857,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,5,0,0,4,0,0,9,0,0,5,0,0,3,0,6],[16,165,0.097,0.57143,0.25254,0.39286,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,6,0,0,7,0,0,4,0,0,2,0,5],[20,165,0.1212,0.5357,0.23958,0.28571,0.4998,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,9,0,0,6,0,0,6,0,0,3,0,0,5,0,2],[24,165,0.1455,0.71428,0.23419,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,5,0,0,6,0,8],[28,165,0.1697,0.68303,0.22227,0.42857,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,7,0,0,9,0,4],[32,165,0.1939,0.60714,0.20825,0.42857,0.64286,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,7,0,0,4,0,0,7,0,0,9,0,0],[36,165,0.2182,0.58929,0.23351,0.39286,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,4,0,0,6,0,0,7,0,0,4,0,3],[40,165,0.2424,0.69196,0.25281,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,8,0,0,3,0,0,3,0,0,7,0,8],[44,165,0.2667,0.60268,0.26422,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,5,0,0,6,0,0,5,0,0,4,0,5],[48,165,0.2909,0.76784,0.19806,0.67857,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,7,0,0,9,0,8],[52,165,0.3152,0.625,0.23891,0.42857,0.64286,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,6,0,0,4,0,0,5,0,0,8,0,3],[56,165,0.3394,0.66964,0.23808,0.42857,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,3,0,0,7,0,6],[60,165,0.3636,0.59821,0.23538,0.42857,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,7,0,0,5,0,0,7,0,0,4,0,0,6,0,3],[64,165,0.3879,0.65177,0.2549,0.42857,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,6,0,0,5,0,0,4,0,0,5,0,0,6,0,6],[68,165,0.4121,0.64286,0.22588,0.42857,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,4,0,0,6,0,0,6,0,0,6,0,0,6,0,4],[72,165,0.4364,0.68302,0.21049,0.57143,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,3,0,0,9,0,4],[76,165,0.4606,0.73213,0.23624,0.57132,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,5,0,0,5,0,10],[80,165,0.4848,0.63839,0.23686,0.42857,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,4,0,0,7,0,0,6,0,0,6,0,0,3,0,6],[84,165,0.5091,0.67856,0.21129,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,5,0,0,10,0,3],[88,165,0.5333,0.65179,0.22851,0.42859,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,7,0,0,7,0,0,4,0,0,6,0,5],[92,165,0.5576,0.58929,0.21053,0.42857,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,9,0,0,4,0,0,5,0,0,9,0,0],[96,165,0.5818,0.6875,0.22428,0.53571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,7,0,0,4,0,7],[100,165,0.6061,0.625,0.23623,0.42857,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,6,0,0,8,0,0,2,0,0,7,0,4],[104,165,0.6303,0.64732,0.2448,0.42857,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,7,0,0,6,0,0,5,0,0,3,0,7],[108,165,0.6545,0.69643,0.23891,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,4,0,0,7,0,7],[112,165,0.6788,0.71428,0.19885,0.57143,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,2,0,0,8,0,6],[116,165,0.703,0.71429,0.21724,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,6,0,0,10,0,5],[120,165,0.7273,0.66518,0.20705,0.53571,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,9,0,0,7,0,3],[124,165,0.7515,0.70536,0.21706,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,9,0,0,6,0,6],[128,165,0.7758,0.66518,0.21011,0.53572,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,5,0,0,7,0,4],[132,165,0.8,0.68304,0.21349,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,6,0,0,10,0,3],[136,165,0.8242,0.66071,0.24157,0.42857,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,5,0,0,4,0,0,6,0,0,6,0,0,5,0,6],[140,165,0.8485,0.69643,0.21053,0.53572,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,6,0,0,8,0,5],[144,165,0.8727,0.70536,0.19865,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,8,0,0,7,0,5],[148,165,0.897,0.66964,0.24598,0.42857,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,10,0,0,5,0,0,1,0,0,7,0,7],[152,165,0.9212,0.75,0.18898,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,7,0,0,8,0,7],[156,165,0.9455,0.70982,0.18724,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,6,0,0,11,0,3],[160,165,0.9697,0.68303,0.19475,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,6,0,0,7,0,4],[164,165,0.9939,0.58927,0.17035,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,6,0,0,12,0,0,7,0,0,3,0,1],[165,165,1.0,0.55804,0.17985,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,11,0,0,9,0,0,6,0,0,1,0,2]]},{"b":2,"e":0.2857,"k":"flat","v":0.48214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function  $ f : \\mathbb{N} \\to \\mathbb{Z}$  is defined by  $ f(0) \\equal{} 2$ ,  $ f(1) \\equal{} 503$  and  $ f(n \\plus{} 2) \\equal{} 503f(n \\plus{} 1) \\minus{} 1996f(n)$  for all  $ n \\in\\mathbb{N}$ . Let  $ s_1$ ,  $ s_2$ ,  $ \\ldots$ ,  $ s_k$  be arbitrary integers not smaller than  $ k$ , and let  $ p(s_i)$  be an arbitrary prime divisor of  $ f\\left(2^{s_i}\\right)$ , ( $ i \\equal{} 1, 2, \\ldots, k$ ). Prove that, for any positive integer  $ t$  ( $ t\\le k$ ), we have  $ 2^t \\Big | \\sum_{i \\equal{} 1}^kp(s_i)$  if and only if  $ 2^t | k$ .","t":[{"b":4,"e":1.0,"k":"flat","v":0.9866,"x":1.0,"p":[[0,23,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,23,0.1739,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,23,0.3478,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,23,0.5217,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,28,0.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[24,28,0.8571,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"49abfe495c6ac298","q":"Suppose that  $a$ ,  $b$ ,  $c$ , and  $d$  are real numbers such that  $a+b+c+d=8$ . Compute the minimum possible value of \\[20(a^2+b^2+c^2+d^2)-\\sum_{\\text{sym}}a^3b,\\] where the sum is over all  $12$  symmetric terms.\n\n*Derek Liu*","t":[{"b":1,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,62,0.0,0.91518,0.21975,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,27],[4,62,0.0645,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,62,0.129,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,62,0.1935,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,62,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,62,0.3226,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,62,0.3871,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,62,0.4516,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,62,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,62,0.5806,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[40,62,0.6452,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,62,0.7097,0.98864,0.06327,1.0,1.0,1.0,0.63636,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,31],[48,62,0.7742,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,62,0.8387,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,62,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,62,0.9677,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[62,62,1.0,0.97524,0.07371,1.0,1.0,1.0,0.63636,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,28]]},{"b":2,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,147,0.0,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[4,147,0.0272,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,147,0.0544,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,147,0.0816,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,147,0.1088,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,147,0.1361,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,147,0.1633,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,147,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,147,0.2177,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,147,0.2449,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,147,0.2721,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,147,0.2993,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,147,0.3265,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,147,0.3537,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,147,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,147,0.4082,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,147,0.4354,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,147,0.4626,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,147,0.4898,0.98864,0.06327,1.0,1.0,1.0,0.63636,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,31],[76,147,0.517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,147,0.5442,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,147,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,147,0.5986,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,147,0.6259,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,147,0.6531,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,147,0.6803,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,147,0.7075,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,147,0.7347,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,147,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,147,0.7891,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,147,0.8163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,147,0.8435,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[128,147,0.8707,0.98864,0.06327,1.0,1.0,1.0,0.63636,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,31],[132,147,0.898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[136,147,0.9252,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[140,147,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[144,147,0.9796,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[147,147,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"e12489630fa829fc","q":"The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.","t":[{"b":4,"e":0.85714,"k":"flat","v":0.60265,"x":0.75888,"p":[[0,45,0.0,0.66069,0.23352,0.57143,0.71429,0.71429,0.0,1.0,2,4,1,2,0,0,0,0,1,0,0,1,0,0,8,0,0,13,0,0,3,0,4],[4,45,0.0889,0.73657,0.2205,0.67825,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,9,0,0,8,0,7],[8,45,0.1778,0.63833,0.24482,0.42857,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,6,0,0,6,0,0,5,0,0,7,0,4],[12,45,0.2667,0.72321,0.20183,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,10,0,0,8,0,5],[16,45,0.3556,0.72762,0.20002,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,6,0,0,10,0,5],[20,45,0.4444,0.7187,0.19064,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,6,0,0,8,0,5],[24,45,0.5333,0.60265,0.24416,0.42857,0.71429,0.75,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,3,0,0,4,0,0,10,0,0,7,0,1],[28,45,0.6222,0.68747,0.2683,0.57132,0.78564,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,5,0,0,1,0,0,6,0,0,3,0,0,10,0,6],[32,45,0.7111,0.6339,0.26472,0.39286,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,2,0,0,6,0,0,4,0,0,7,0,5],[36,45,0.8,0.74105,0.20653,0.57143,0.78564,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,7,0,0,10,0,6],[40,45,0.8889,0.68747,0.1801,0.571,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,9,0,0,7,0,3],[44,45,0.9778,0.70978,0.19722,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,7,0,0,9,0,4],[45,45,1.0,0.75888,0.1654,0.57143,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,0,17,0,3]]},{"b":7,"e":0.42857,"k":"rising","v":0.59816,"x":0.86159,"p":[[0,85,0.0,0.60713,0.18558,0.53539,0.64286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,3,0,0,8,0,0,12,0,0,3,0,1],[4,85,0.0471,0.74996,0.15974,0.71429,0.71429,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,16,0,0,4,0,6],[8,85,0.0941,0.59816,0.24074,0.53539,0.64286,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,4,0,0,2,0,0,8,0,0,9,0,0,6,0,1],[12,85,0.1412,0.62497,0.24936,0.42857,0.71429,0.74996,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,3,0,0,5,0,0,9,0,0,3,0,5],[16,85,0.1882,0.67409,0.28624,0.53539,0.78564,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,3,0,0,2,0,0,5,0,0,3,0,0,10,0,6],[20,85,0.2353,0.62943,0.26211,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,4,0,0,6,0,0,7,0,0,2,0,0,7,0,5],[24,85,0.2824,0.71424,0.26002,0.571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,3,0,0,6,0,0,4,0,0,5,0,10],[28,85,0.3294,0.68748,0.27534,0.42859,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,2,0,0,3,0,0,5,0,0,7,0,8],[32,85,0.3765,0.72312,0.30309,0.60714,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,6,0,0,0,0,0,0,0,0,5,0,0,9,0,10],[36,85,0.4235,0.69863,0.23129,0.571,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,6,1,0,7,0,6],[40,85,0.4706,0.70979,0.26363,0.571,0.78564,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,1,0,0,6,0,10],[44,85,0.5176,0.75445,0.24546,0.67846,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,0,8,0,10],[48,85,0.5647,0.7232,0.22851,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,7,0,0,6,0,8],[52,85,0.6118,0.78571,0.22303,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,6,0,0,8,0,11],[56,85,0.6588,0.83034,0.2156,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,5,0,16],[60,85,0.7059,0.75442,0.22657,0.57132,0.85707,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,4,0,0,7,0,10],[64,85,0.7529,0.78569,0.22589,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,4,0,0,11,0,10],[68,85,0.8,0.84818,0.1854,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,5,0,16],[72,85,0.8471,0.83034,0.19379,0.82132,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,12,0,12],[76,85,0.8941,0.81246,0.20028,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,6,0,13],[80,85,0.9412,0.86159,0.17309,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,7,0,16],[84,85,0.9882,0.8214,0.18561,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,6,0,13],[85,85,1.0,0.77229,0.16701,0.71429,0.857,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,12,0,5]]}]},{"i":"a64434ad26b40353","q":"The real numbers $a, b, c, d$ satisfy simultaneously the equations\n\n$$\na b c-d=1, b c d-a=2, c d a-b=3, d a b-c=-6\n$$\n\nProve that $a+b+c+d \\neq 0$.","t":[{"b":2,"e":0.571,"k":"rising","v":0.21866,"x":0.567,"p":[[0,107,0.0,0.21866,0.20202,0.14214,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,15,0,0,1,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[4,107,0.0374,0.39286,0.27894,0.14286,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,4,0,0,1,0,0,7,0,0,6,0,0,5,0,0,2,0,0],[8,107,0.0748,0.37945,0.23584,0.14286,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,4,0,0,4,0,0,8,0,0,6,0,0,5,0,0,0,0,0],[12,107,0.1121,0.55802,0.16888,0.57143,0.57143,0.60714,0.0,0.71429,2,0,0,2,0,0,0,0,1,0,0,1,0,0,20,0,0,8,0,0,0,0,0],[16,107,0.1495,0.567,0.15351,0.57143,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,3,0,0,18,0,0,9,0,0,0,0,0],[20,107,0.1869,0.55355,0.17034,0.57132,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,1,0,0,14,0,0,11,0,0,0,0,0],[24,107,0.2243,0.43295,0.22736,0.28571,0.50001,0.57143,0.0,0.71429,4,0,0,4,0,3,0,0,2,0,0,7,0,0,11,0,0,5,0,0,0,0,0],[28,107,0.2617,0.47767,0.2101,0.42857,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,1,0,0,7,0,0,13,0,0,6,0,0,0,0,0],[32,107,0.2991,0.5491,0.15612,0.57132,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,5,0,0,19,0,0,5,0,0,1,0,0],[36,107,0.3364,0.52679,0.20025,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,1,0,0,8,0,0,10,0,0,9,0,0,1,0,0],[40,107,0.3738,0.50893,0.19541,0.4286,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,0,0,0,2,0,0,4,0,0,17,0,0,6,0,0,0,0,0],[44,107,0.4112,0.51344,0.19837,0.4286,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,0,0,0,5,0,0,15,0,0,6,0,0,1,0,0],[48,107,0.4486,0.49107,0.20183,0.42857,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,1,0,0,1,0,0,7,0,0,14,0,0,6,0,0,0,0,0],[52,107,0.486,0.54017,0.15458,0.57132,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,4,0,0,19,0,0,6,0,0,0,0,0],[56,107,0.5234,0.54464,0.18013,0.57143,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,2,0,0,3,0,0,16,0,0,9,0,0,0,0,0],[60,107,0.5607,0.52229,0.14986,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,9,0,0,14,0,0,6,0,0,0,0,0],[64,107,0.5981,0.55357,0.15872,0.57143,0.57143,0.60714,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,3,0,0,18,0,0,8,0,0,0,0,0],[68,107,0.6355,0.53567,0.19884,0.571,0.57143,0.57143,0.0,0.85714,3,0,0,3,0,0,0,0,1,0,0,2,0,0,19,0,0,6,0,0,1,0,0],[72,107,0.6729,0.49553,0.19876,0.42857,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,1,0,0,1,0,0,5,0,0,17,0,0,5,0,0,0,0,0],[76,107,0.7103,0.52676,0.16145,0.571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,0,2,0,0,20,0,0,5,0,0,0,0,0],[80,107,0.7477,0.46872,0.19636,0.39286,0.57143,0.57143,0.0,0.857,2,0,0,2,0,2,0,0,4,0,0,5,0,0,16,0,0,2,0,0,1,0,0],[84,107,0.785,0.50444,0.2051,0.39288,0.57143,0.60714,0.0,0.71429,1,0,0,1,0,4,0,0,3,0,0,1,0,0,15,0,0,8,0,0,0,0,0],[88,107,0.8224,0.48212,0.19803,0.42857,0.57141,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,3,0,0,6,0,0,13,0,0,6,0,0,0,0,0],[92,107,0.8598,0.51774,0.15892,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,6,0,0,17,0,0,5,0,0,0,0,0],[96,107,0.8972,0.55802,0.13533,0.57132,0.57143,0.60714,0.1429,0.71429,0,0,0,0,0,1,0,0,2,0,0,4,0,0,17,0,0,8,0,0,0,0,0],[100,107,0.9346,0.46872,0.20276,0.28571,0.5712,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,5,0,0,5,0,0,12,0,0,6,0,0,0,0,0],[104,107,0.972,0.50668,0.19012,0.39286,0.57143,0.58932,0.0,0.71429,1,0,0,1,0,2,0,0,5,0,0,2,0,0,14,1,0,7,0,0,0,0,0],[107,107,1.0,0.5223,0.16981,0.42857,0.5712,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,9,0,0,8,0,0,10,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"rising","v":0.24106,"x":0.66517,"p":[[0,90,0.0,0.24106,0.25613,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,5,0,0,3,0,0,5,0,0,2,0,0,4,0,0,0,0,0],[4,90,0.0444,0.48659,0.27861,0.14286,0.57143,0.71429,0.0,0.85714,5,0,0,5,0,4,0,0,0,0,0,1,0,0,9,0,0,12,0,0,1,0,0],[8,90,0.0889,0.35268,0.28344,0.14286,0.28571,0.57143,0.0,1.0,7,1,0,7,0,6,0,0,4,0,0,4,0,0,4,0,0,6,0,0,0,0,1],[12,90,0.1333,0.34374,0.26209,0.14286,0.28571,0.57143,0.0,0.71429,7,0,0,7,0,5,0,0,6,0,0,2,0,0,6,0,0,6,0,0,0,0,0],[16,90,0.1778,0.39284,0.29014,0.10714,0.42859,0.71429,0.0,0.85714,8,0,0,8,0,3,0,0,3,0,0,3,0,0,6,0,0,8,0,0,1,0,0],[20,90,0.2222,0.35713,0.25253,0.14286,0.35714,0.57143,0.0,0.71429,6,0,0,6,0,5,0,0,5,0,0,5,0,0,5,0,0,6,0,0,0,0,0],[24,90,0.2667,0.36829,0.24426,0.14286,0.42857,0.57143,0.0,0.71429,6,0,0,6,0,3,0,0,6,0,0,6,0,0,5,1,0,5,0,0,0,0,0],[28,90,0.3111,0.32589,0.26301,0.0,0.42857,0.57143,0.0,0.857,9,0,0,9,0,4,0,0,2,0,0,8,0,0,5,0,0,3,0,0,1,0,0],[32,90,0.3556,0.33036,0.28221,0.0,0.28571,0.57143,0.0,0.85714,10,0,0,10,0,3,0,0,4,0,0,4,0,0,5,0,0,5,0,0,1,0,0],[36,90,0.4,0.36161,0.26722,0.10714,0.42857,0.57143,0.0,0.71429,8,0,0,8,0,3,0,0,3,0,0,7,0,0,4,0,0,7,0,0,0,0,0],[40,90,0.4444,0.38839,0.26782,0.25,0.28571,0.71429,0.0,0.85714,6,0,0,6,0,2,0,0,9,0,0,3,0,0,3,0,0,8,0,0,1,0,0],[44,90,0.4889,0.35713,0.24483,0.14286,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,4,0,0,5,0,0,7,0,0,6,0,0,3,0,0,1,0,0],[48,90,0.5333,0.29016,0.29338,0.0,0.28571,0.42857,0.0,1.0,12,2,0,12,0,2,0,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,2],[52,90,0.5778,0.26553,0.25586,0.0,0.21428,0.46428,0.0,0.71429,11,0,0,11,0,5,0,0,5,0,1,2,0,0,4,0,0,4,0,0,0,0,0],[56,90,0.6222,0.36161,0.2575,0.14286,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,5,0,0,4,0,0,6,0,0,7,0,0,2,0,0,2,0,0],[60,90,0.6667,0.55801,0.15714,0.5713,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,2,0,0,17,0,0,9,0,0,0,0,0],[64,90,0.7111,0.55356,0.15872,0.53539,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,4,0,0,14,0,0,10,0,0,0,0,0],[68,90,0.7556,0.5714,0.14725,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,2,0,0,17,0,0,10,0,0,0,0,0],[72,90,0.8,0.58481,0.16506,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,3,0,0,15,0,0,11,0,0,1,0,0],[76,90,0.8444,0.5491,0.13415,0.5354,0.57143,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,2,0,0,5,0,0,17,0,0,7,0,0,0,0,0],[80,90,0.8889,0.58482,0.16115,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,5,0,0,12,0,0,12,0,0,1,0,0],[84,90,0.9333,0.65177,0.14258,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,21,0,0,1,0,0],[88,90,0.9778,0.66517,0.11634,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,24,0,0,1,0,0],[90,90,1.0,0.61603,0.09742,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,14,0,0,0,0,0]]}]},{"i":"26227bbea64f81df","q":"There are  $11$  empty boxes. In one move, a player can put one coin in each of some  $10$  boxes. Two people play, taking turns. The winner is  the player after whose move in one of the boxes there will be  $21$  coins. Who has a winning strategy?","t":[{"b":0,"e":0.0,"k":"falling","v":0.00893,"x":0.16964,"p":[[0,55,0.0,0.16516,0.21459,0.0,0.0,0.28571,0.0,0.71429,18,0,2,18,0,0,0,0,10,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[4,55,0.0727,0.07589,0.15966,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,55,0.1455,0.13393,0.23673,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,2,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[12,55,0.2182,0.16964,0.18363,0.0,0.14285,0.28571,0.0,0.57143,16,0,0,16,0,0,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,55,0.2909,0.12053,0.13882,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.10714,0.16751,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,1,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,55,0.4364,0.10266,0.16837,0.0,0.0,0.2857,0.0,0.57143,22,0,0,22,0,1,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,55,0.5091,0.10714,0.22303,0.0,0.0,0.07143,0.0,0.85714,24,0,0,24,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[32,55,0.5818,0.10713,0.21426,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,55,0.6545,0.0625,0.18877,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,55,0.7273,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,55,0.8727,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.03571,"x":0.21428,"p":[[0,54,0.0,0.06696,0.13825,0.0,0.0,0.0,0.0,0.57143,25,0,6,25,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,54,0.0741,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.14731,0.21863,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[20,54,0.3704,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.08928,0.16268,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,54,0.5185,0.08034,0.14694,0.0,0.0,0.07143,0.0,0.571,24,0,0,24,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,54,0.5926,0.09821,0.1729,0.0,0.0,0.2857,0.0,0.71429,23,0,0,23,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,54,0.6667,0.11159,0.16259,0.0,0.0,0.28571,0.0,0.571,21,0,0,21,0,0,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,54,0.7407,0.12054,0.22619,0.0,0.0,0.17857,0.0,0.85714,23,0,0,23,0,1,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[44,54,0.8148,0.17411,0.22227,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[48,54,0.8889,0.21428,0.25999,0.0,0.2857,0.28571,0.0,1.0,15,1,0,15,0,0,0,0,13,0,0,0,0,0,1,0,0,1,0,0,1,0,1],[52,54,0.963,0.19641,0.16653,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,0,0,0,18,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[54,54,1.0,0.20535,0.14698,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"a8eb304747f5ff4e","q":"A prime number  $p$  is a **moderate** number if for every  $2$  positive integers  $k > 1$  and  $m$ , there exists k positive integers  $n_1, n_2, ..., n_k $  such that \\[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \\]\nIf  $q$  is the smallest **moderate** number, then determine the smallest prime  $r$  which is not moderate and  $q < r$ .","t":[{"b":6,"e":0.71429,"k":"flat","v":0.44637,"x":0.73661,"p":[[0,69,0.0,0.44637,0.15868,0.28571,0.571,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,6,0,0,6,0,0,17,0,0,0,0,0,0,0,0],[4,69,0.058,0.70535,0.2141,0.57142,0.71429,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,11,0,0,3,0,7],[8,69,0.1159,0.62498,0.21353,0.42857,0.64286,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,9,0,0,4,0,0,8,0,0,5,0,3],[12,69,0.1739,0.67406,0.19961,0.571,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,9,0,0,5,0,0,7,0,4],[16,69,0.2319,0.62943,0.22264,0.42857,0.71429,0.85704,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,8,0,0,3,0,0,8,0,0,6,0,3],[20,69,0.2899,0.68747,0.22142,0.5354,0.71429,0.857,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,5,0,0,9,0,0,3,0,7],[24,69,0.3478,0.73661,0.20237,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,11,0,0,4,0,8],[28,69,0.4058,0.69192,0.2086,0.57143,0.71429,0.85704,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,10,0,0,3,0,6],[32,69,0.4638,0.6116,0.22083,0.42857,0.71429,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,8,0,0,2,0,0,9,0,0,6,0,2],[36,69,0.5217,0.63386,0.18879,0.4286,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,10,0,0,1,0,4],[40,69,0.5797,0.56234,0.2312,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,6,0,0,4,0,0,10,0,0,4,0,1],[44,69,0.6377,0.63834,0.22585,0.5354,0.57143,0.85714,0.1429,1.0,0,4,0,0,0,1,0,0,3,0,0,4,0,0,9,0,0,6,0,0,5,0,4],[48,69,0.6957,0.55801,0.20934,0.42857,0.571,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,10,0,0,5,0,0,8,0,0,2,0,2],[52,69,0.7536,0.62497,0.1948,0.4286,0.64286,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,6,0,0,7,0,0,11,0,0,2,0,3],[56,69,0.8116,0.60708,0.23146,0.42859,0.5712,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,7,0,0,9,0,0,5,0,0,2,0,5],[60,69,0.8696,0.55353,0.18813,0.42857,0.4998,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,13,0,0,7,0,0,5,0,0,2,0,2],[64,69,0.9275,0.51333,0.15507,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,8,0,0,13,0,0,3,0,0,2,0,0],[68,69,0.9855,0.5714,0.24744,0.42857,0.571,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,8,0,0,5,0,0,6,0,0,1,0,5],[69,69,1.0,0.53122,0.21198,0.42857,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,11,0,0,4,0,0,7,0,0,3,0,1]]},{"b":7,"e":1.0,"k":"rising","v":0.49105,"x":0.77227,"p":[[0,61,0.0,0.49105,0.1006,0.42857,0.57121,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,10,0,0,18,0,0,0,0,0,0,0,0],[4,61,0.0656,0.59817,0.27301,0.42857,0.57121,0.74996,0.14286,1.0,0,6,0,0,0,3,0,0,4,0,0,6,0,0,4,0,0,7,0,0,2,0,6],[8,61,0.1311,0.66068,0.21055,0.5354,0.64286,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,7,0,0,4,0,5],[12,61,0.1967,0.68302,0.18467,0.57132,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,13,0,0,2,0,5],[16,61,0.2623,0.65624,0.23921,0.4286,0.71429,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,8,0,0,4,0,0,9,0,0,0,0,8],[20,61,0.3279,0.69638,0.18817,0.571,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,9,0,0,3,0,6],[24,61,0.3934,0.67853,0.21726,0.571,0.71429,0.85704,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,9,0,0,3,0,6],[28,61,0.459,0.69195,0.22619,0.42857,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,5,0,0,11,0,4],[32,61,0.5246,0.64274,0.1786,0.571,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,15,0,0,6,0,0,2,0,4],[36,61,0.5902,0.65174,0.17105,0.571,0.64286,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,10,0,0,3,0,3],[40,61,0.6557,0.73658,0.19271,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,9,0,0,6,0,7],[44,61,0.7213,0.64728,0.19881,0.5354,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,11,0,0,4,0,3],[48,61,0.7869,0.6339,0.17105,0.5354,0.64286,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,11,0,0,3,0,2],[52,61,0.8525,0.62494,0.17036,0.571,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,13,0,0,6,0,0,4,0,2],[56,61,0.918,0.64276,0.17499,0.571,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,15,0,0,5,0,0,4,0,3],[60,61,0.9836,0.77227,0.20476,0.57143,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,0,5,0,11],[61,61,1.0,0.76782,0.17772,0.57143,0.78571,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,5,0,0,8,0,8]]}]},{"i":"0b11a147df7b9c8c","q":"Through the endpoints  $A$  and  $B$  of a diameter  $AB$  of a given circle, the tangents  $\\ell$   and  $m$  have been drawn. Let  $C\\ne A$  be a point on  $\\ell$   and let  $q_1,q_2$  be two rays from  $C$ . Ray  $q_i$  cuts the circle in  $D_i$  and  $E_i$  with  $D_i$  between  $C$  and  $E_i, i = 1,2$ . Rays  $AD_1,AD_2,AE_1,AE_2$  meet  $m$  in the respective points  $M_1,M_2,N_1,N_2$ . Prove that  $M_1M_2 = N_1N_2$ .","t":[{"b":2,"e":0.85714,"k":"falling","v":0.60235,"x":0.93304,"p":[[0,80,0.0,0.9241,0.13356,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,19],[4,80,0.05,0.8839,0.15751,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,9,0,17],[8,80,0.1,0.92409,0.10096,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[12,80,0.15,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[16,80,0.2,0.89732,0.17941,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,16],[20,80,0.25,0.88838,0.14167,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,15,0,14],[24,80,0.3,0.91964,0.13333,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,18],[28,80,0.35,0.89732,0.12993,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,0,13],[32,80,0.4,0.83036,0.25364,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,10,0,16],[36,80,0.45,0.87945,0.16411,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,15,0,14],[40,80,0.5,0.88393,0.14032,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,14,0,14],[44,80,0.55,0.83928,0.20748,0.8214,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,10,0,14],[48,80,0.6,0.8258,0.28534,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,9,0,17],[52,80,0.65,0.82141,0.26246,0.82132,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,0,7,0,17],[56,80,0.7,0.89727,0.12503,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,14,0,15],[60,80,0.75,0.83928,0.22798,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,13,0,14],[64,80,0.8,0.70531,0.32329,0.571,0.85714,1.0,0.0,1.0,2,10,0,2,0,3,0,0,1,0,0,1,0,0,4,0,0,2,0,0,9,0,10],[68,80,0.85,0.71429,0.26486,0.42857,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,6,0,0,0,0,0,1,0,0,16,0,5],[72,80,0.9,0.6875,0.28221,0.42857,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,7,0,0,1,0,0,3,0,0,10,0,7],[76,80,0.95,0.76339,0.27342,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,0,14,0,9],[80,80,1.0,0.60235,0.25712,0.4286,0.71214,0.857,0.0,1.0,1,1,0,1,0,3,0,0,1,0,0,6,0,0,4,0,0,7,0,0,9,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.65629,"x":0.93303,"p":[[0,106,0.0,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[4,106,0.0377,0.84823,0.2228,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,0,8,0,17],[8,106,0.0755,0.93303,0.11837,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,21],[12,106,0.1132,0.87945,0.19597,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,10,0,18],[16,106,0.1509,0.67411,0.32972,0.42857,0.85714,1.0,0.0,1.0,2,11,0,2,0,2,0,0,1,0,0,8,0,0,0,0,0,2,0,0,6,0,11],[20,106,0.1887,0.66068,0.28291,0.42857,0.857,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,7,0,0,2,0,0,1,0,0,12,0,5],[24,106,0.2264,0.80359,0.22227,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,13,0,10],[28,106,0.2642,0.82139,0.21132,0.857,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,17,0,9],[32,106,0.3019,0.81251,0.22989,0.71429,0.85714,1.0,0.143,1.0,0,13,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,3,0,0,10,0,13],[36,106,0.3396,0.85267,0.14054,0.85714,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,20,0,8],[40,106,0.3774,0.74997,0.30095,0.71429,0.85714,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,15,0,8],[44,106,0.4151,0.65629,0.34226,0.42857,0.85714,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,1,0,0,6,0,0,0,0,0,1,0,0,12,0,7],[48,106,0.4528,0.76336,0.25409,0.57132,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,0,13,0,9],[52,106,0.4906,0.69194,0.30328,0.67846,0.857,0.85714,0.0,1.0,4,5,0,4,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,12,0,5],[56,106,0.5283,0.70982,0.29339,0.53572,0.85714,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,0,18,0,4],[60,106,0.566,0.73665,0.29033,0.42965,0.85714,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,1,0,0,5,0,0,0,0,0,0,0,0,15,0,8],[64,106,0.6038,0.70536,0.28779,0.4286,0.85714,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,16,0,5],[68,106,0.6415,0.6607,0.31894,0.42857,0.85714,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,6,0,0,1,0,0,1,0,0,11,0,7],[72,106,0.6792,0.67855,0.32537,0.53539,0.85714,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,1,0,0,2,0,0,1,0,0,4,0,0,14,0,5],[76,106,0.717,0.67854,0.24223,0.4286,0.64271,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,10,0,0,5,0,0,1,0,0,9,0,6],[80,106,0.7547,0.78121,0.24484,0.57132,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,4,0,0,5,0,0,0,0,0,11,0,11],[84,106,0.7925,0.8125,0.19377,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,13,0,10],[88,106,0.8302,0.79013,0.20204,0.67857,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,1,0,0,16,0,7],[92,106,0.8679,0.78569,0.16752,0.67857,0.85714,0.85714,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,0,19,0,4],[96,106,0.9057,0.83034,0.11541,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,21,0,4],[100,106,0.9434,0.83482,0.16793,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,14,0,10],[104,106,0.9811,0.78571,0.14725,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,20,0,2],[106,106,1.0,0.74552,0.15867,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,9,0,0,16,0,1]]}]},{"i":"cfe7bd2ff567f354","q":"A positive integer  $n$   is *acceptable* if the sum of the squares of its proper divisors is equal to  $2n+4$   (a divisor of  $n$   is *proper* if it is different from  $1$   and of  $n$  ). Find all acceptable numbers less than  $10000$ ,","t":[{"b":2,"e":0.71429,"k":"flat","v":0.81694,"x":0.94643,"p":[[0,61,0.0,0.82143,0.16366,0.71429,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,0,2,0,13],[4,61,0.0656,0.89732,0.16841,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,20],[8,61,0.1311,0.93747,0.12854,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[12,61,0.1967,0.88393,0.15335,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,19],[16,61,0.2623,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[20,61,0.3279,0.91517,0.12807,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,19],[24,61,0.3934,0.89732,0.12492,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,16],[28,61,0.459,0.87945,0.15201,0.82143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,17],[32,61,0.5246,0.86604,0.14704,0.82143,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,14],[36,61,0.5902,0.92411,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[40,61,0.6557,0.85267,0.16164,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,6,0,15],[44,61,0.7213,0.85714,0.14286,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,6,0,14],[48,61,0.7869,0.88838,0.13237,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,6,0,17],[52,61,0.8525,0.89732,0.12993,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,18],[56,61,0.918,0.86159,0.18725,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,3,0,19],[60,61,0.9836,0.81694,0.19641,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,6,0,0,1,0,16],[61,61,1.0,0.84372,0.1651,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,4,0,15]]},{"b":6,"e":0.71429,"k":"flat","v":0.80803,"x":0.95982,"p":[[0,69,0.0,0.80803,0.16213,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,1,0,12],[4,69,0.058,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[8,69,0.1159,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,20],[12,69,0.1739,0.93304,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[16,69,0.2319,0.90178,0.1729,0.85714,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,19],[20,69,0.2899,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[24,69,0.3478,0.88392,0.19045,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,18],[28,69,0.4058,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[32,69,0.4638,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[36,69,0.5217,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[40,69,0.5797,0.91071,0.14174,0.85714,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,21],[44,69,0.6377,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[48,69,0.6957,0.92856,0.12376,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,23],[52,69,0.7536,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[56,69,0.8116,0.90624,0.11633,0.857,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,18],[60,69,0.8696,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[64,69,0.9275,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[68,69,0.9855,0.90179,0.14913,0.71429,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,2,0,21],[69,69,1.0,0.89719,0.13493,0.71429,1.0,1.0,0.71,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20]]}]},{"i":"c3bda1400fc15345","q":"A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\\frac{p}{2}$ such that $\\frac{a b-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.","t":[{"b":5,"e":1.0,"k":"rising","v":0.47765,"x":0.87051,"p":[[0,94,0.0,0.47765,0.19101,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,6,0,0,13,0,0,8,0,0,1,0,0,2,0,1],[4,94,0.0426,0.87051,0.22123,0.82132,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,20],[8,94,0.0851,0.79016,0.22865,0.71429,0.85714,1.0,0.0,1.0,1,11,1,1,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,0,8,0,11],[12,94,0.1277,0.76784,0.2896,0.57143,0.9285,1.0,0.0,1.0,2,16,2,2,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,1,0,16],[16,94,0.1702,0.82587,0.19801,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,0,7,0,14],[20,94,0.2128,0.83481,0.18251,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,4,0,15],[24,94,0.2553,0.78569,0.24224,0.71421,0.85714,1.0,0.0,1.0,1,13,1,1,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,5,0,13],[28,94,0.2979,0.7098,0.24869,0.4286,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,5,0,0,2,0,11],[32,94,0.3404,0.808,0.21015,0.67856,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,3,0,15],[36,94,0.383,0.83036,0.22428,0.71429,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,4,0,16],[40,94,0.4255,0.82586,0.17766,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,9,0,12],[44,94,0.4681,0.82587,0.19146,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,6,0,14],[48,94,0.5106,0.84819,0.22,0.71429,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,0,0,19],[52,94,0.5532,0.82581,0.18127,0.71429,0.857,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,2,0,15],[56,94,0.5957,0.81699,0.20582,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,8,0,13],[60,94,0.6383,0.80357,0.20124,0.67857,0.85707,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,3,0,14],[64,94,0.6809,0.81689,0.22378,0.71429,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,15],[68,94,0.7234,0.85709,0.17135,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,4,0,17],[72,94,0.766,0.81694,0.18296,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,5,0,13],[76,94,0.8085,0.7455,0.21352,0.571,0.78571,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,0,7,0,9],[80,94,0.8511,0.79905,0.19521,0.67857,0.85707,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,6,0,12],[84,94,0.8936,0.84372,0.1834,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,6,0,15],[88,94,0.9362,0.76334,0.19108,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,7,0,0,6,0,9],[92,94,0.9787,0.83925,0.18477,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,4,0,16],[94,94,1.0,0.80352,0.20128,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,6,0,13]]},{"b":6,"e":0.2857,"k":"rising","v":0.41969,"x":0.79907,"p":[[0,58,0.0,0.41969,0.10678,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,17,0,0,5,0,0,1,0,0,0,0,0],[4,58,0.069,0.79461,0.2524,0.57143,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,0,2,0,16],[8,58,0.1379,0.74998,0.23691,0.57143,0.85707,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,5,0,0,7,0,10],[12,58,0.2069,0.79907,0.24966,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,0,5,0,15],[16,58,0.2759,0.78125,0.22011,0.67857,0.85707,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,7,0,0,5,0,12],[20,58,0.3448,0.75445,0.25813,0.57132,0.85707,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,5,0,0,4,0,13],[24,58,0.4138,0.7321,0.22801,0.571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,6,0,0,2,0,11],[28,58,0.4828,0.73659,0.25282,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,7,0,0,2,0,12],[32,58,0.5517,0.76783,0.25193,0.57132,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,0,5,0,13],[36,58,0.6207,0.74104,0.24858,0.571,0.85707,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,3,0,0,6,0,11],[40,58,0.6897,0.72317,0.27652,0.5354,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,4,0,0,4,0,0,2,0,0,8,0,10],[44,58,0.7586,0.76782,0.2308,0.57143,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,7,0,0,4,0,12],[48,58,0.8276,0.7187,0.2187,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,5,0,0,5,0,8],[52,58,0.8966,0.64285,0.24999,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,9,0,0,4,0,0,3,0,0,6,0,6],[56,58,0.9655,0.66962,0.22711,0.53539,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,4,0,0,9,0,4],[58,58,1.0,0.71427,0.24485,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,8,0,0,4,0,9]]}]},{"i":"88cbe1662196177d","q":"In the triangle  $ABC$  let  $B'$  and  $C'$  be the midpoints of the sides  $AC$  and  $AB$  respectively and  $H$  the foot of the altitude passing through the vertex  $A$ . Prove that the circumcircles of the triangles  $AB'C'$ , $BC'H$ , and  $B'CH$  have a common point  $I$  and that the line  $HI$  passes through the midpoint of the segment  $B'C'.$","t":[{"b":1,"e":0.0,"k":"falling","v":0.03125,"x":0.23213,"p":[[0,90,0.0,0.18748,0.22137,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,4,0,0,3,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[4,90,0.0444,0.23213,0.21941,0.0,0.21428,0.42857,0.0,0.57143,13,0,0,13,0,3,0,0,3,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[8,90,0.0889,0.11606,0.20338,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[12,90,0.1333,0.19639,0.23345,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,1,0,0,0,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[16,90,0.1778,0.15625,0.2,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,4,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[20,90,0.2222,0.18304,0.20896,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,5,0,0,3,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[24,90,0.2667,0.08036,0.15947,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,90,0.3111,0.20533,0.23124,0.0,0.07143,0.42857,0.0,0.57143,16,0,0,16,0,3,0,0,1,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[32,90,0.3556,0.14731,0.20353,0.0,0.0,0.21429,0.0,0.57143,18,0,0,18,0,6,0,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[36,90,0.4,0.07588,0.16742,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[40,90,0.4444,0.15179,0.21998,0.0,0.0,0.42857,0.0,0.57143,21,0,0,21,0,1,0,0,0,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[44,90,0.4889,0.15624,0.22404,0.0,0.0,0.32143,0.0,0.57143,20,0,0,20,0,2,0,0,2,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[48,90,0.5333,0.11606,0.19374,0.0,0.0,0.14292,0.0,0.57143,22,0,0,22,0,3,0,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[52,90,0.5778,0.11606,0.19374,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[56,90,0.6222,0.15177,0.21408,0.0,0.0,0.32143,0.0,0.57143,19,0,0,19,0,4,0,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[60,90,0.6667,0.18289,0.22926,0.0,0.0,0.32143,0.0,0.57143,18,0,0,18,0,1,0,0,5,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[64,90,0.7111,0.12947,0.21237,0.0,0.0,0.1786,0.0,0.57143,22,0,0,22,0,2,0,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[68,90,0.7556,0.13837,0.22436,0.0,0.0,0.21429,0.0,0.57143,22,0,0,22,0,2,0,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[72,90,0.8,0.1741,0.21347,0.0,0.0,0.42857,0.0,0.57143,17,0,0,17,0,4,0,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[76,90,0.8444,0.10713,0.17493,0.0,0.0,0.1786,0.0,0.571,22,0,0,22,0,2,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[80,90,0.8889,0.21424,0.22579,0.0,0.1429,0.42857,0.0,0.57143,14,0,0,14,0,4,0,0,4,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[84,90,0.9333,0.08036,0.15127,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[88,90,0.9778,0.0759,0.15561,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[90,90,1.0,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.16515,"x":0.36145,"p":[[0,138,0.0,0.22319,0.22282,0.0,0.14286,0.42857,0.0,0.57143,14,0,0,14,0,3,0,0,2,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[4,138,0.029,0.20977,0.24988,0.0,0.0,0.4642,0.0,0.57143,17,0,0,17,0,3,0,0,0,0,0,4,0,0,8,0,0,0,0,0,0,0,0],[8,138,0.058,0.17857,0.20825,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,7,0,0,0,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[12,138,0.087,0.17407,0.2222,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,3,0,0,1,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[16,138,0.1159,0.17853,0.22581,0.0,0.0,0.42857,0.0,0.57143,17,0,0,17,0,5,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[20,138,0.1449,0.21426,0.21424,0.0,0.14288,0.42858,0.0,0.57143,13,0,0,13,0,5,0,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[24,138,0.1739,0.2678,0.23616,0.0,0.28571,0.42858,0.0,0.57143,13,0,0,13,0,0,0,0,4,0,0,8,0,0,7,0,0,0,0,0,0,0,0],[28,138,0.2029,0.16517,0.22332,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,5,0,0,0,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[32,138,0.2319,0.25887,0.24849,0.0,0.14286,0.571,0.0,0.57143,13,0,0,13,0,4,0,0,0,0,0,6,0,0,9,0,0,0,0,0,0,0,0],[36,138,0.2609,0.29017,0.23819,0.0,0.28574,0.57143,0.0,0.57143,9,0,0,9,0,7,0,0,0,0,0,6,0,0,10,0,0,0,0,0,0,0,0],[40,138,0.2899,0.32586,0.21788,0.14286,0.42857,0.57111,0.0,0.57143,4,0,0,4,0,11,0,0,0,0,0,6,0,0,11,0,0,0,0,0,0,0,0],[44,138,0.3188,0.36145,0.22586,0.14286,0.42859,0.57143,0.0,0.57143,5,0,0,5,0,7,0,0,0,0,0,6,0,0,14,0,0,0,0,0,0,0,0],[48,138,0.3478,0.29453,0.25741,0.0,0.28574,0.57143,0.0,0.57143,11,0,0,11,0,5,0,0,0,0,0,3,0,0,13,0,0,0,0,0,0,0,0],[52,138,0.3768,0.21871,0.22858,0.0,0.14286,0.4642,0.0,0.57143,11,0,0,11,0,11,0,0,0,0,0,2,0,0,8,0,0,0,0,0,0,0,0],[56,138,0.4058,0.30798,0.22893,0.14286,0.2857,0.5711,0.0,0.57143,6,0,0,6,0,9,0,0,3,0,0,2,0,0,12,0,0,0,0,0,0,0,0],[60,138,0.4348,0.27658,0.22293,0.14,0.14288,0.571,0.0,0.57143,7,0,0,7,0,10,0,0,2,0,0,4,0,0,9,0,0,0,0,0,0,0,0],[64,138,0.4638,0.22313,0.22573,0.0,0.14286,0.42857,0.0,0.57143,11,0,0,11,0,10,0,0,0,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[68,138,0.4928,0.20972,0.21125,0.0,0.14286,0.32143,0.0,0.57143,9,0,0,9,0,14,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[72,138,0.5217,0.16515,0.19592,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,14,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[76,138,0.5507,0.29459,0.21991,0.14286,0.14286,0.571,0.0,0.57143,5,0,0,5,0,12,0,0,1,0,0,4,0,0,10,0,0,0,0,0,0,0,0],[80,138,0.5797,0.23658,0.21898,0.10714,0.14286,0.4642,0.0,0.57143,8,0,0,8,0,13,0,0,1,0,0,2,0,0,8,0,0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triangle  $ABC$ , the point  $H$  is the orthocenter. A circle centered at point  $H$  and  with radius  $AH$  intersects the lines  $AB$  and  $AC$  at points  $E$  and  $D$ , respectively. The point  $X$  is the symmetric of the point  $A$  with respect to the line  $BC$  . Prove that  $XH$  is the bisector of the angle  $DXE$ .\n\n(Matthew of Kursk)","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.10714,"p":[[0,77,0.0,0.10714,0.15152,0.0,0.0,0.2857,0.0,0.42857,20,0,3,20,0,3,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,77,0.1039,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,77,0.1558,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,77,0.2078,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,77,0.3117,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,77,0.3636,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,77,0.4156,0.06697,0.18893,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,77,0.4675,0.05357,0.10565,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,77,0.5195,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,77,0.5714,0.04018,0.15251,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,77,0.6234,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.03125,"x":0.10714,"p":[[0,52,0.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.06697,0.18552,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,52,0.1538,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,52,0.2308,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,52,0.3846,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.08027,0.17832,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,52,0.5385,0.08473,0.10625,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.08018,0.07922,0.0,0.14,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.09812,0.06616,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c37149789bdcebb9","q":"Lateral sidelines  $AB$  and  $CD$  of a trapezoid  $ABCD$  ( $AD >BC$ ) meet at point  $P$ . Let  $Q$  be a point of segment  $AD$  such that  $BQ = CQ$ . Prove that the line passing through the circumcenters of triangles  $AQC$  and  $BQD$  is perpendicular to  $PQ$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0134,"x":0.07589,"p":[[0,65,0.0,0.04911,0.09851,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.05804,0.17807,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,65,0.1846,0.07143,0.18211,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,65,0.2462,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.07589,0.18552,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,65,0.5538,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.06696,0.18552,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,65,0.6769,0.04018,0.10853,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,65,0.7385,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.07581,0.10088,0.0,0.0,0.14286,0.0,0.286,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.05794,0.10004,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.08482,"p":[[0,113,0.0,0.07588,0.14274,0.0,0.0,0.03575,0.0,0.571,24,0,0,24,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,113,0.0354,0.05358,0.11152,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,113,0.0708,0.08482,0.24186,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,113,0.1062,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,113,0.1416,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,113,0.177,0.05804,0.18161,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,113,0.2124,0.0625,0.17835,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,113,0.2478,0.04911,0.09181,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,113,0.2832,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,113,0.3186,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,113,0.354,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,113,0.3894,0.04018,0.08171,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,113,0.4248,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,113,0.4602,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,113,0.4956,0.07134,0.18554,0.0,0.0,0.035,0.0,1.0,24,1,0,24,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,113,0.531,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,113,0.5664,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,113,0.6018,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,113,0.6372,0.05357,0.15465,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[76,113,0.6726,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,113,0.708,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,113,0.7434,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[88,113,0.7788,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,113,0.8142,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[96,113,0.8496,0.06696,0.19227,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[100,113,0.885,0.03572,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[104,113,0.9204,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,113,0.9558,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,113,0.9912,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[113,113,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,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 $ ABC$  be an acute triangle and  $ CL$  be the angle bisector of  $ \\angle ACB$ . The point  $ P$  lies on  the segment  $CL$  such that  $ \\angle APB\\equal{}\\pi\\minus{}\\frac{_1}{^2}\\angle ACB$ . Let  $ k_1$  and  $ k_2$  be the circumcircles of the triangles  $ APC$  and  $ BPC$ .  $ BP\\cap k_1\\equal{}Q, AP\\cap k_2\\equal{}R$ . The tangents to  $ k_1$  at  $ Q$  and  $ k_2$  at  $ B$  intersect at  $ S$  and the tangents to  $ k_1$  at  $ A$  and  $ k_2$  at  $ R$  intersect at  $ T$ . Prove that  $ AS\\equal{}BT.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.02455,"x":0.09822,"p":[[0,77,0.0,0.02455,0.06341,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.06027,0.12256,0.0,0.0,0.01786,0.0,0.42857,24,0,0,24,1,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,77,0.1039,0.08027,0.19539,0.0,0.0,0.035,0.0,1.0,24,1,0,24,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,77,0.1558,0.06018,0.09953,0.0,0.0,0.14071,0.0,0.28571,22,0,0,22,1,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,77,0.2078,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,77,0.3117,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,77,0.3636,0.07367,0.10638,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,1,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,77,0.4156,0.06241,0.10055,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,77,0.4675,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,77,0.5195,0.08927,0.15042,0.0,0.0,0.14287,0.0,0.571,22,0,0,22,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,77,0.5714,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,77,0.6234,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.06688,0.10699,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.08036,0.12339,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.06688,0.10699,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.09822,0.19704,0.0,0.0,0.14287,0.0,1.0,22,1,0,22,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[72,77,0.9351,0.06696,0.11837,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02231,"x":0.09821,"p":[[0,60,0.0,0.04455,0.09731,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.05799,0.09181,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,2,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.0625,0.11812,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.09821,0.13092,0.0,0.0,0.17857,0.0,0.42857,19,0,0,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.04687,0.10352,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,1,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.06696,0.10705,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.04902,0.09843,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.05581,0.10524,0.0,0.0,0.08929,0.0,0.42857,23,0,0,23,1,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.03116,0.07758,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.02231,0.10163,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,60,1.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bd9b35e9c1f5aa98","q":"The Xantians are the inhabitants, potentially in infinite number, of the planet Xanta. Regarding themselves and their peers, the Xantians are capable of feeling two types of emotions, which they call love and respect. It has been observed that:\n\n- Each Xantian loves one and only one Xantian, and respects one and only one Xantian.\n- If $A$ loves $B$, then any Xantian who respects $A$ also loves $B$.\n- If $A$ respects $B$, then any Xantian who loves $A$ also respects $B$.\n- Each Xantian is loved by at least one Xantian.\n\nIs it true that each Xantian respects the Xantian they love?","t":[{"b":6,"e":1.0,"k":"rising","v":0.58927,"x":1.0,"p":[[0,42,0.0,0.80804,0.23313,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,5,0,0,0,0,18],[4,42,0.0952,0.64283,0.21725,0.42857,0.57143,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,7,0,0,0,0,7],[8,42,0.1905,0.66963,0.23266,0.42857,0.64286,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,7,0,0,0,0,9],[12,42,0.2857,0.64284,0.20517,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,11,0,0,6,0,0,9,0,0,0,0,6],[16,42,0.381,0.58927,0.23891,0.42857,0.4998,0.71429,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,15,0,0,5,0,0,5,0,0,0,0,6],[20,42,0.4762,0.85712,0.20828,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,0,0,21],[24,42,0.5714,0.76339,0.25407,0.42857,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,5,0,0,0,0,16],[28,42,0.6667,0.77232,0.2547,0.42857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,3,0,0,2,0,16],[32,42,0.7619,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[36,42,0.8571,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,42,0.9524,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.73661,"x":1.0,"p":[[0,19,0.0,0.77676,0.21411,0.57143,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,8,0,0,2,0,13],[4,19,0.2105,0.79018,0.22011,0.71429,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,0,0,15],[8,19,0.4211,0.79464,0.23673,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,6,0,0,0,0,17],[12,19,0.6316,0.73661,0.25028,0.42857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,5,0,0,0,0,14],[16,19,0.8421,0.83929,0.21053,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,0,0,19],[19,19,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"437c729a61388252","q":"Let $b, n>1$ be integers. Suppose that for each $k>1$ there exists an integer $a_{k}$ such that $b-a_{k}^{n}$ is divisible by $k$. Prove that $b=A^{n}$ for some integer $A$.  (Canada)","t":[{"b":4,"e":0.14286,"k":"falling","v":0.45089,"x":0.95534,"p":[[0,17,0.0,0.95534,0.1871,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,17,0.2353,0.95089,0.16602,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[8,17,0.4706,0.9107,0.22519,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,26],[12,17,0.7059,0.87945,0.26754,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,26],[16,17,0.9412,0.56696,0.43373,0.14286,0.71429,1.0,0.0,1.0,3,15,0,3,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,15],[17,17,1.0,0.45089,0.39947,0.14286,0.21429,1.0,0.0,1.0,5,9,0,5,0,11,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,9]]},{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,30,0.0,0.9375,0.20806,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[4,30,0.1333,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,30,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,30,0.4,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,30,0.5333,0.95982,0.1525,1.0,1.0,1.0,0.1429,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[20,30,0.6667,0.95088,0.1541,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[24,30,0.8,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,30,0.9333,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[30,30,1.0,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]}]},{"i":"682a850dbbb1dcae","q":"An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:\n\n(a) $x *(x * y)=y$ for all $x, y \\in \\mathbb{Z}$;\n\n(b) $(x * y) * y=x$ for all $x, y \\in \\mathbb{Z}$.\n\nThe professor claims in his book that\n\n(C1) the operation $*$ is commutative: $x * y=y * x$ for all $x, y \\in \\mathbb{Z}$.\n\n(C2) the operation $*$ is associative: $(x * y) * z=x *(y * z)$ for all $x, y, z \\in \\mathbb{Z}$.\n\nWhich of these claims follow from the stated axioms?","t":[{"b":1,"e":1.0,"k":"flat","v":0.66072,"x":1.0,"p":[[0,114,0.0,0.66072,0.3458,0.42857,0.71429,1.0,0.0,1.0,3,14,2,3,0,1,0,0,0,0,0,11,0,0,0,0,0,2,0,0,1,0,14],[4,114,0.0351,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,114,0.0702,0.93304,0.17852,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,28],[12,114,0.1053,0.94643,0.19805,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[16,114,0.1404,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[20,114,0.1754,0.9375,0.19865,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[24,114,0.2105,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[28,114,0.2456,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,114,0.2807,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[36,114,0.3158,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[40,114,0.3509,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,114,0.386,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,114,0.4211,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,114,0.4561,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,114,0.4912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,114,0.5263,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[64,114,0.5614,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[68,114,0.5965,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[72,114,0.6316,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[76,114,0.6667,0.9375,0.13333,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,26],[80,114,0.7018,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[84,114,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,114,0.7719,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[92,114,0.807,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[96,114,0.8421,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[100,114,0.8772,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,26],[104,114,0.9123,0.94642,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[108,114,0.9474,0.91518,0.15093,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,24],[112,114,0.9825,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[114,114,1.0,0.8079,0.16989,0.71429,0.71429,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,1,0,13]]},{"b":5,"e":0.71429,"k":"flat","v":0.85266,"x":0.99554,"p":[[0,81,0.0,0.91518,0.18509,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,26],[4,81,0.0494,0.92857,0.20203,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,28],[8,81,0.0988,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,81,0.1481,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,81,0.1975,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,81,0.2469,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,81,0.2963,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,81,0.3457,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,81,0.3951,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,81,0.4444,0.98661,0.07457,1.0,1.0,1.0,0.5714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,81,0.4938,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,81,0.5432,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,81,0.5926,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[52,81,0.642,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,81,0.6914,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[60,81,0.7407,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,81,0.7901,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[68,81,0.8395,0.96429,0.11293,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[72,81,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,81,0.9383,0.91951,0.12362,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,21],[80,81,0.9877,0.9107,0.1372,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,21],[81,81,1.0,0.85266,0.16167,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,16]]}]},{"i":"8d2744c3a07b11fe","q":"Consider a row of cells numbered $0,1, \\ldots, k$ from left to right where, for each $i \\geqslant 1$, cell number $i$ contains $x_{i}$ tokens. Initially, there are no tokens on cell number 0. Alice and Bob then take turns playing according to the following rules:\n\n- Bob chooses a set $S$ of tokens, not necessarily all on the same cell.\n- Alice can then either eliminate all tokens that are not in $S$ but then move each token in $S$ from the cell it occupies to the neighboring cell to its left (such a token thus moves from cell number $i$ to cell number $i-1$), or eliminate all tokens that are in $S$ but then move each token that is not in $S$ from the cell it occupies to the neighboring cell to its left.\nBob wins the game if he manages to bring a token to cell number 0, and Alice wins if she manages to eliminate all the tokens.\n\n1) Prove that Alice has a winning strategy if $\\sum_{i=1}^{k} 2^{-i} x_{i}<1$.\n2) Is it true that if $\\sum_{i=1}^{k} 2^{-i} x_{i} \\geqslant 1$ then Bob has a winning strategy?","t":[{"b":4,"e":0.42857,"k":"flat","v":0.48672,"x":0.82589,"p":[[0,46,0.0,0.48672,0.13274,0.42857,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,10,0,0,17,0,0,1,0,0,0,0,0],[4,46,0.087,0.82589,0.13235,0.71429,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,17,0,6],[8,46,0.1739,0.79464,0.14698,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,17,0,4],[12,46,0.2609,0.73661,0.20238,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,5,0,0,8,0,7],[16,46,0.3478,0.81695,0.16842,0.85713,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,18,0,7],[20,46,0.4348,0.73201,0.18473,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,0,9,0,5],[24,46,0.5217,0.71871,0.20043,0.571,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,5,0,0,8,0,6],[28,46,0.6087,0.70076,0.19018,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,8,0,0,6,0,5],[32,46,0.6957,0.6829,0.15863,0.57143,0.71214,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,6,0,0,10,0,1],[36,46,0.7826,0.75892,0.16919,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,13,0,4],[40,46,0.8696,0.70522,0.16341,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,8,0,0,12,0,1],[44,46,0.9565,0.65634,0.1886,0.42857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,7,0,0,8,0,2],[46,46,1.0,0.55357,0.19149,0.42857,0.42857,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,3,0,0,6,0,1]]},{"b":6,"e":0.85714,"k":"rising","v":0.49554,"x":0.73213,"p":[[0,34,0.0,0.49554,0.15561,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,0,0,0,12,0,0,15,0,0,3,0,0,0,0,0],[4,34,0.1176,0.73213,0.21054,0.5354,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,4,0,0,11,0,6],[8,34,0.2353,0.70535,0.17473,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,8,0,0,7,0,4],[12,34,0.3529,0.66071,0.17405,0.53571,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,7,0,0,9,0,1],[16,34,0.4706,0.60267,0.198,0.42857,0.57143,0.75,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,8,0,0,2,0,0,5,0,3],[20,34,0.5882,0.60272,0.18113,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,5,0,0,6,0,0,6,0,1],[24,34,0.7059,0.60712,0.20825,0.42857,0.4998,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,16,0,0,4,0,0,3,0,0,6,0,3],[28,34,0.8235,0.65625,0.18851,0.42857,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,0,4,0,0,13,0,0],[32,34,0.9412,0.69641,0.14617,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,7,0,0,10,0,1],[34,34,1.0,0.72768,0.13997,0.57143,0.78571,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,4,0,0,16,0,0]]}]},{"i":"f6c9e34b66ea161d","q":"Consider the set  $F$  of all polynomials whose coefficients are in the set of  $\\{0,1\\}$ . Let  $q(x) = x^3 + x +1$ . The number of polynomials  $p(x)$  in  $F$  of degree  $14$  such that the product  $p(x)q(x)$  is also in  $F$  is:","t":[{"b":3,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,31,0.0,0.94196,0.17076,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[4,31,0.129,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,31,0.2581,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,31,0.3871,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,31,0.6452,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,31,0.7742,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,31,0.9032,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[31,31,1.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]},{"b":4,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,33,0.0,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,33,0.1212,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,33,0.2424,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,33,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,33,0.4848,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,33,0.6061,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,33,0.8485,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,33,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d32b1f172e01e9a1","q":"Let  $n$  be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer  $l \\geq 0$ , there are exactly  $n$  cards that have the number  $l$  written on them. A move consists of picking  $100$  cards from the infinite set of cards and discarding them. Find the least possible value of  $n$  for which there is an infinitely long series of moves such that for each positive integer  $k$ , the sum of the numbers written on the  $100$  chosen cards during the  $k$ -th move is equal to  $k$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.04464,"x":0.75436,"p":[[0,74,0.0,0.12946,0.19351,0.0,0.0,0.42857,0.0,0.42857,22,0,5,22,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[4,74,0.0541,0.65625,0.391,0.35714,0.85714,1.0,0.0,1.0,5,14,0,5,0,3,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,14],[8,74,0.1081,0.45076,0.43309,0.0,0.2857,1.0,0.0,1.0,11,10,0,11,0,4,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,10],[12,74,0.1622,0.54464,0.42773,0.14286,0.57143,1.0,0.0,1.0,7,13,0,7,0,4,0,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,13],[16,74,0.2162,0.65625,0.41166,0.25,0.85714,1.0,0.0,1.0,7,15,0,7,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,15],[20,74,0.2703,0.56696,0.43811,0.0,0.78571,1.0,0.0,1.0,9,13,0,9,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,13],[24,74,0.3243,0.47321,0.41869,0.10714,0.28571,1.0,0.0,1.0,8,10,0,8,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,0,2,0,10],[28,74,0.3784,0.5892,0.38268,0.25001,0.57143,1.0,0.0,1.0,4,12,0,4,0,4,0,0,2,0,0,5,0,0,2,0,0,1,0,0,2,0,12],[32,74,0.4324,0.75436,0.34686,0.67846,1.0,1.0,0.0,1.0,2,18,0,2,0,3,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,18],[36,74,0.4865,0.70534,0.38289,0.42857,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,17],[40,74,0.5405,0.59822,0.38038,0.28571,0.64286,1.0,0.0,1.0,4,12,0,4,0,3,0,0,4,0,0,3,0,0,2,0,0,2,0,0,2,0,12],[44,74,0.5946,0.61607,0.38867,0.24999,0.71429,1.0,0.0,1.0,3,14,0,3,0,5,0,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,14],[48,74,0.6486,0.75433,0.30565,0.71321,0.85714,1.0,0.0,1.0,2,15,0,2,0,1,0,0,1,0,0,3,0,0,0,0,0,8,0,0,2,0,15],[52,74,0.7027,0.67858,0.41187,0.25002,1.0,1.0,0.0,1.0,5,18,0,5,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,18],[56,74,0.7568,0.67857,0.37627,0.39286,0.85714,1.0,0.0,1.0,4,14,0,4,0,3,0,0,1,0,0,2,0,0,0,0,0,5,0,0,3,0,14],[60,74,0.8108,0.58035,0.38455,0.25,0.64264,1.0,0.0,1.0,5,12,0,5,0,3,0,0,2,0,0,5,0,0,1,0,0,4,0,0,0,0,12],[64,74,0.8649,0.62944,0.37942,0.42857,0.71429,1.0,0.0,1.0,7,11,0,7,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,3,0,11],[68,74,0.9189,0.28563,0.34996,0.0,0.14286,0.5,0.0,1.0,12,3,0,12,0,9,0,0,2,0,0,1,0,0,0,0,0,3,0,0,2,0,3],[72,74,0.973,0.08036,0.18877,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[74,74,1.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.2679,"x":0.70536,"p":[[0,57,0.0,0.2679,0.27839,0.0,0.42857,0.42857,0.0,1.0,14,2,6,14,0,1,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,2],[4,57,0.0702,0.62937,0.42246,0.14286,1.0,1.0,0.0,1.0,4,17,0,4,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,17],[8,57,0.1404,0.59372,0.38649,0.14286,0.71429,1.0,0.0,1.0,4,12,0,4,0,5,0,0,2,0,0,2,0,0,2,0,0,4,0,0,1,0,12],[12,57,0.2105,0.57579,0.38223,0.25,0.57143,1.0,0.0,1.0,5,11,0,5,0,3,0,0,3,0,0,2,0,0,5,0,0,1,0,0,2,0,11],[16,57,0.2807,0.29009,0.3244,0.0,0.14286,0.4286,0.0,1.0,9,4,0,9,0,10,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,4],[20,57,0.3509,0.36607,0.37617,0.0,0.28571,0.71429,0.0,1.0,12,5,0,12,0,2,0,0,5,0,0,3,0,0,1,0,0,2,0,0,2,0,5],[24,57,0.4211,0.4241,0.35442,0.14286,0.2857,0.71429,0.0,1.0,4,6,0,4,0,10,0,0,3,0,0,4,0,0,1,0,0,3,0,0,1,0,6],[28,57,0.4912,0.37496,0.36199,0.0,0.28571,0.60714,0.0,1.0,11,4,0,11,0,2,0,0,5,0,0,3,0,0,3,0,0,1,0,0,3,0,4],[32,57,0.5614,0.36606,0.36058,0.0,0.28571,0.71429,0.0,1.0,10,4,0,10,0,5,0,0,4,0,0,2,0,0,2,0,0,3,0,0,2,0,4],[36,57,0.6316,0.57156,0.35525,0.28571,0.57143,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,4,0,0,1,0,0,6,0,0,2,0,0,4,0,8],[40,57,0.7018,0.49552,0.39927,0.10714,0.4998,1.0,0.0,1.0,8,9,1,8,0,4,0,0,1,0,0,3,0,0,3,0,0,3,0,0,1,0,9],[44,57,0.7719,0.6339,0.2922,0.571,0.64286,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,1,0,0,9,0,0,6,0,0,3,0,7],[48,57,0.8421,0.64731,0.28791,0.57143,0.71429,0.85714,0.0,1.0,3,5,0,3,0,2,0,0,0,0,0,0,0,0,6,0,0,12,0,0,4,0,5],[52,57,0.9123,0.65179,0.23673,0.57143,0.71429,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,0,0,0,1,0,0,8,0,0,13,0,0,4,0,3],[56,57,0.9825,0.70533,0.14701,0.57143,0.71429,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,10,0,0,6,0,3],[57,57,1.0,0.70536,0.15947,0.57143,0.71429,0.71429,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,13,0,0,2,0,5]]}]},{"i":"482ad1ddef2d29b5","q":"Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile.  The game ends if a player cannot move, in which case the other player wins.\n\nDetermine all pairs  $(a,b)$  of positive integers such that if initially the two piles have  $a$   and  $b$  coins respectively, then Bob has a winning strategy.\n\nProposed by Dimitris Christophides, Cyprus","t":[{"b":1,"e":0.71429,"k":"rising","v":0.21857,"x":0.83034,"p":[[0,91,0.0,0.27222,0.29746,0.0,0.14286,0.42857,0.0,1.0,11,2,10,11,0,7,0,0,4,0,0,3,0,0,2,0,0,3,0,0,0,0,2],[4,91,0.044,0.21857,0.26488,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,13,0,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[8,91,0.0879,0.33036,0.34151,0.0,0.14286,0.60714,0.0,1.0,9,3,0,9,0,9,0,0,2,0,0,3,0,0,1,0,0,3,0,0,2,0,3],[12,91,0.1319,0.25427,0.27143,0.0,0.14286,0.42857,0.0,1.0,9,2,1,9,0,10,0,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,2],[16,91,0.1758,0.3124,0.28225,0.14286,0.21428,0.4642,0.0,1.0,5,2,0,5,0,11,0,0,7,0,0,1,0,0,2,0,0,4,0,0,0,0,2],[20,91,0.2198,0.25444,0.26177,0.0,0.14286,0.32143,0.0,1.0,10,1,0,10,0,7,0,0,7,0,0,1,0,0,5,0,0,0,0,0,1,0,1],[24,91,0.2637,0.31249,0.27764,0.14286,0.21428,0.4642,0.0,1.0,6,1,0,6,0,10,0,0,5,0,0,3,0,0,2,0,0,4,0,0,1,0,1],[28,91,0.3077,0.21876,0.24481,0.0,0.14286,0.28571,0.0,1.0,10,1,0,10,0,10,0,0,6,0,0,2,0,0,2,0,0,0,0,0,1,0,1],[32,91,0.3516,0.24552,0.20587,0.14286,0.14288,0.42857,0.0,0.71429,7,0,0,7,0,10,0,0,6,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[36,91,0.3956,0.32587,0.30141,0.10714,0.21429,0.571,0.0,1.0,8,3,0,8,0,8,0,0,1,0,0,6,0,0,6,0,0,0,0,0,0,0,3],[40,91,0.4396,0.24999,0.26962,0.0,0.14286,0.32143,0.0,1.0,9,2,0,9,0,10,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,2],[44,91,0.4835,0.33927,0.28737,0.14286,0.2143,0.46418,0.0,1.0,4,2,0,4,0,12,0,0,3,0,0,5,0,0,2,0,0,3,0,0,1,0,2],[48,91,0.5275,0.45088,0.29904,0.24999,0.42857,0.57143,0.0,1.0,4,4,0,4,0,4,0,0,3,0,0,9,0,0,5,0,0,2,0,0,1,0,4],[52,91,0.5714,0.41518,0.22119,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,5,0,0,10,0,0,6,0,0,2,0,0,1,0,1],[56,91,0.6154,0.39284,0.21723,0.2857,0.35714,0.57143,0.0,0.85714,3,0,0,3,0,2,0,0,11,0,0,6,0,0,5,0,0,4,0,0,1,0,0],[60,91,0.6593,0.46874,0.24803,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,4,0,0,6,0,0,5,0,0,8,0,0,1,0,1],[64,91,0.7033,0.55357,0.28516,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,3,0,0,5,0,0,6,0,0,4,0,0,6,0,0,2,0,5],[68,91,0.7473,0.61151,0.27962,0.28571,0.71429,0.85704,0.14,1.0,0,5,0,0,0,3,0,0,6,0,0,3,0,0,1,0,0,10,0,0,4,0,5],[72,91,0.7912,0.65177,0.25985,0.42859,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,5,0,0,5,0,0,4,0,0,8,0,5],[76,91,0.8352,0.60713,0.3093,0.42857,0.57143,1.0,0.0,1.0,2,9,0,2,0,1,0,0,4,0,0,6,0,0,5,0,0,4,0,0,1,0,9],[80,91,0.8791,0.65622,0.26693,0.571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,1,0,0,2,0,0,8,0,0,4,0,0,8,0,5],[84,91,0.9231,0.5982,0.24856,0.42857,0.57143,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,6,0,0,4,0,0,7,0,0,4,0,4],[88,91,0.967,0.64732,0.28118,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,4,0,0,4,0,8],[91,91,1.0,0.83034,0.15338,0.71429,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,9,0,11]]},{"b":6,"e":0.14286,"k":"falling","v":0.12491,"x":0.35714,"p":[[0,69,0.0,0.35714,0.29233,0.10714,0.42857,0.57141,0.0,1.0,8,1,5,8,0,4,0,0,3,0,0,8,0,0,4,0,0,1,0,0,3,0,1],[4,69,0.058,0.30356,0.26181,0.14286,0.14286,0.42858,0.0,0.857,6,0,0,6,0,12,0,0,0,0,0,7,0,0,1,0,0,5,0,0,1,0,0],[8,69,0.1159,0.2231,0.24725,0.0,0.14286,0.42857,0.0,0.857,12,0,0,12,0,8,0,0,3,0,0,4,0,0,2,0,0,2,0,0,1,0,0],[12,69,0.1739,0.17402,0.16264,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,13,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,69,0.2319,0.17391,0.17399,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,11,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[20,69,0.2899,0.24554,0.26058,0.10714,0.14286,0.42857,0.0,1.0,8,1,0,8,0,14,0,0,0,0,0,5,0,0,2,0,0,1,0,0,1,0,1],[24,69,0.3478,0.23214,0.21651,0.14286,0.14286,0.32143,0.0,0.85714,7,0,0,7,0,13,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[28,69,0.4058,0.25883,0.22993,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,5,0,0,3,0,0,5,0,0,0,0,0,0,0,1],[32,69,0.4638,0.24991,0.26248,0.105,0.14286,0.28571,0.0,1.0,8,1,0,8,0,11,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,1],[36,69,0.5217,0.24534,0.2294,0.14214,0.14286,0.28571,0.0,0.85714,6,0,0,6,0,13,0,0,7,0,0,1,0,0,1,0,0,3,0,0,1,0,0],[40,69,0.5797,0.13838,0.14496,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,15,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,69,0.6377,0.15616,0.10927,0.14286,0.14286,0.14287,0.0,0.42857,6,0,0,6,0,19,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.13831,0.09771,0.14214,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,20,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.21874,0.21122,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,15,0,0,3,0,0,2,0,0,4,0,0,0,0,0,1,0,0],[56,69,0.8116,0.16062,0.17406,0.0,0.14286,0.1786,0.0,0.71429,11,0,0,11,0,13,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[60,69,0.8696,0.12491,0.13242,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,69,0.9275,0.14277,0.13832,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,16,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[68,69,0.9855,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.12929,0.05482,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"929a1405d3c81e68","q":"Each of the points  $G$  and  $H$  lying from different sides of the plane of hexagon  $ABCDEF$  is connected with all vertices of the hexagon. \nIs it possible to mark 18 segments thus formed by the numbers  $1, 2, 3, \\ldots, 18$  and arrange some real numbers at points  $A, B, C, D, E, F, G, H$  so that each segment is marked with the difference of the numbers at its ends?\n\n*Proposed by A. Golovanov*","t":[{"b":2,"e":1.0,"k":"flat","v":0.63393,"x":0.81694,"p":[[0,78,0.0,0.70098,0.38694,0.5,0.85714,1.0,0.0,1.0,6,12,1,6,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,12],[4,78,0.0513,0.7723,0.30276,0.85714,0.85714,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,7],[8,78,0.1026,0.77232,0.31916,0.85714,0.85714,1.0,0.0,1.0,3,11,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,11],[12,78,0.1538,0.75892,0.36673,0.82143,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,16],[16,78,0.2051,0.6875,0.3984,0.25,0.85714,1.0,0.0,1.0,6,12,0,6,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,0,12],[20,78,0.2564,0.64732,0.42104,0.1429,0.85714,1.0,0.0,1.0,7,13,0,7,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,8,0,13],[24,78,0.3077,0.70535,0.40554,0.57143,0.85714,1.0,0.0,1.0,7,15,0,7,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,15],[28,78,0.359,0.64286,0.43154,0.0,0.85714,1.0,0.0,1.0,9,13,0,9,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,13],[32,78,0.4103,0.70982,0.41571,0.53571,0.92857,1.0,0.0,1.0,8,16,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,16],[36,78,0.4615,0.69196,0.37983,0.57144,0.85714,1.0,0.0,1.0,5,11,0,5,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,11],[40,78,0.5128,0.70533,0.37447,0.67846,0.85714,1.0,0.0,1.0,6,10,0,6,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,10],[44,78,0.5641,0.6875,0.41101,0.24999,0.85714,1.0,0.0,1.0,7,15,0,7,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,15],[48,78,0.6154,0.77232,0.34043,0.82143,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,9,0,15],[52,78,0.6667,0.73222,0.30675,0.67846,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,0,10,0,10],[56,78,0.7179,0.76339,0.35644,0.71429,0.92857,1.0,0.0,1.0,4,16,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,16],[60,78,0.7692,0.67409,0.36811,0.28571,0.85714,1.0,0.0,1.0,4,12,0,4,0,2,0,0,3,0,0,0,0,0,2,0,0,3,0,0,6,0,12],[64,78,0.8205,0.70981,0.36681,0.28571,0.92857,1.0,0.0,1.0,2,16,0,2,0,4,0,0,3,0,0,0,0,0,2,0,0,1,0,0,4,0,16],[68,78,0.8718,0.81694,0.2772,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,4,0,18],[72,78,0.9231,0.63393,0.38122,0.28571,0.85714,1.0,0.0,1.0,3,14,0,3,0,2,0,0,7,0,0,2,0,0,1,0,0,0,0,0,3,0,14],[76,78,0.9744,0.63829,0.39457,0.28571,0.85714,1.0,0.0,1.0,4,14,0,4,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0,4,0,14],[78,78,1.0,0.78123,0.32338,0.67846,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,0,5,0,18]]},{"b":5,"e":1.0,"k":"rising","v":0.52232,"x":0.99554,"p":[[0,151,0.0,0.80357,0.34209,0.85714,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,18],[4,151,0.0265,0.81696,0.28175,0.82143,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,14],[8,151,0.053,0.65625,0.40541,0.25001,0.85714,1.0,0.0,1.0,7,11,0,7,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,10,0,11],[12,151,0.0795,0.76786,0.29397,0.71429,0.85714,1.0,0.0,1.0,3,9,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,9],[16,151,0.106,0.69196,0.38317,0.39288,0.85714,1.0,0.0,1.0,6,12,0,6,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,10,0,12],[20,151,0.1325,0.64732,0.4103,0.0,0.85714,0.89286,0.0,1.0,9,8,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,8],[24,151,0.1589,0.73214,0.32684,0.82132,0.85714,0.85714,0.0,1.0,3,7,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,7],[28,151,0.1854,0.81696,0.25313,0.85714,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,16,0,11],[32,151,0.2119,0.68303,0.33642,0.2857,0.85714,0.85714,0.0,1.0,1,6,0,1,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,0,16,0,6],[36,151,0.2384,0.83481,0.20858,0.85714,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,7],[40,151,0.2649,0.8125,0.24338,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,8],[44,151,0.2914,0.80803,0.249,0.85708,0.85714,1.0,0.0,1.0,2,9,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,16,0,9],[48,151,0.3179,0.71429,0.35714,0.71429,0.85714,1.0,0.0,1.0,4,10,0,4,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,10],[52,151,0.3444,0.7366,0.33141,0.85711,0.85714,0.89286,0.0,1.0,3,8,0,3,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,17,0,8],[56,151,0.3709,0.69642,0.36377,0.82132,0.85714,0.85714,0.0,1.0,6,6,0,6,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,6],[60,151,0.3974,0.76338,0.30642,0.85714,0.85714,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,8],[64,151,0.4238,0.75893,0.32818,0.85714,0.85714,1.0,0.0,1.0,3,10,0,3,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,10],[68,151,0.4503,0.75,0.32341,0.85711,0.85714,1.0,0.0,1.0,2,9,0,2,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,9],[72,151,0.4768,0.74553,0.3108,0.85711,0.85714,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,0,7],[76,151,0.5033,0.70972,0.33229,0.78571,0.85714,0.85714,0.0,1.0,4,5,0,4,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,19,0,5],[80,151,0.5298,0.71874,0.31639,0.857,0.85714,0.85714,0.0,1.0,3,3,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,3],[84,151,0.5563,0.70982,0.34346,0.64286,0.85714,1.0,0.0,1.0,2,10,0,2,0,5,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,10],[88,151,0.5828,0.75008,0.33315,0.82132,0.85714,1.0,0.0,1.0,5,9,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,9],[92,151,0.6093,0.67402,0.34499,0.57143,0.85714,0.85714,0.0,1.0,3,4,0,3,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,4],[96,151,0.6358,0.76339,0.26633,0.82143,0.85714,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,19,0,5],[100,151,0.6623,0.60714,0.35535,0.14286,0.85714,0.85714,0.0,1.0,2,3,0,2,0,8,0,0,1,0,0,0,0,0,1,0,0,1,0,0,16,0,3],[104,151,0.6887,0.65625,0.35867,0.14289,0.85714,0.85714,0.0,1.0,3,5,0,3,0,6,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,5],[108,151,0.7152,0.69195,0.35195,0.46396,0.85714,1.0,0.0,1.0,2,9,0,2,0,6,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,9],[112,151,0.7417,0.65625,0.36045,0.14286,0.85714,0.85714,0.0,1.0,1,6,0,1,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,6],[116,151,0.7682,0.66517,0.37048,0.24999,0.85714,0.89286,0.0,1.0,4,8,0,4,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,0,13,0,8],[120,151,0.7947,0.52232,0.43829,0.0,0.85707,0.89286,0.0,1.0,10,8,0,10,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,8],[124,151,0.8212,0.69196,0.39303,0.24999,0.85714,1.0,0.0,1.0,5,13,0,5,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,13],[128,151,0.8477,0.61607,0.41563,0.14286,0.85714,1.0,0.0,1.0,6,11,0,6,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,11],[132,151,0.8742,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[136,151,0.9007,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[140,151,0.9272,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[144,151,0.9536,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[148,151,0.9801,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[151,151,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"af18fbdd5b226177","q":"It is said that a pair of integers ( $\\mathbf{a}, \\mathrm{b}$ ) is Cypriot if $\\mathrm{a} \\geqslant \\mathrm{b} \\geqslant 2$, if a and b are coprime, and if $a+b$ divides $a^{b}+b^{a}$.\nProve that there are infinitely many distinct Cypriot pairs.","t":[{"b":5,"e":1.0,"k":"flat","v":0.9241,"x":0.99107,"p":[[0,59,0.0,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,59,0.0678,0.9241,0.20198,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[8,59,0.1356,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,59,0.2034,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,59,0.2712,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,59,0.339,0.95981,0.08173,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[24,59,0.4068,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,59,0.4746,0.95087,0.12177,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[32,59,0.5424,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,59,0.6102,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[40,59,0.678,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,59,0.7458,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[48,59,0.8136,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,59,0.8814,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,59,0.9492,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[59,59,1.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,40,0.0,0.98213,0.04728,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,40,0.1,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,40,0.2,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,40,0.3,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,40,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,40,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,40,0.7,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,40,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,40,0.9,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"10a4beed00d64905","q":"Find all pairs  $(a,b)$  of positive integers such that  $a^{2017}+b$  is a multiple of  $ab$ .","t":[{"b":0,"e":0.14286,"k":"falling","v":0.37504,"x":0.56249,"p":[[0,37,0.0,0.56249,0.25738,0.42857,0.57121,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,8,0,0,7,0,0,4,0,0,0,0,6],[4,37,0.1081,0.50445,0.24996,0.28571,0.42857,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,14,0,0,6,0,0,1,0,0,7,0,0,0,0,4],[8,37,0.2162,0.5,0.23146,0.28571,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,13,0,0,6,0,0,3,0,0,7,0,0,0,0,3],[12,37,0.3243,0.44643,0.23351,0.28571,0.42857,0.46429,0.14286,1.0,0,3,0,0,0,2,0,0,13,0,0,9,0,0,1,0,0,4,0,0,0,0,3],[16,37,0.4324,0.46429,0.17857,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,12,0,0,2,0,0,7,0,0,1,0,0],[20,37,0.5405,0.45981,0.19144,0.28571,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,9,0,0,2,0,0,7,0,0,0,0,1],[24,37,0.6486,0.41518,0.16115,0.28571,0.42857,0.46431,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,9,0,0,6,0,0,1,0,0,0,0,1],[28,37,0.7568,0.41964,0.18189,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,10,0,0,2,0,0,4,0,0,0,0,1],[32,37,0.8649,0.46875,0.17581,0.39286,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,17,0,0,0,0,0,4,0,0,3,0,0],[36,37,0.973,0.39285,0.13362,0.28571,0.42857,0.42857,0.2857,0.857,0,0,0,0,0,0,0,0,15,0,0,13,0,0,2,0,0,1,0,0,1,0,0],[37,37,1.0,0.37504,0.15466,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,17,0,0,10,0,0,0,0,0,4,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.11161,"x":0.70982,"p":[[0,48,0.0,0.49999,0.29014,0.2857,0.49979,0.71429,0.14286,1.0,0,4,0,0,0,7,0,0,7,0,0,2,0,0,4,0,0,7,0,0,1,0,4],[4,48,0.0833,0.67857,0.26726,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,7,0,0,2,0,0,3,0,0,10,0,0,0,0,10],[8,48,0.1667,0.61161,0.27487,0.28571,0.71429,0.71429,0.14286,1.0,0,7,0,0,0,1,0,0,9,0,0,2,0,0,2,0,0,11,0,0,0,0,7],[12,48,0.25,0.70982,0.29984,0.42857,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,7,0,0,4,0,0,2,0,0,4,0,0,0,0,15],[16,48,0.3333,0.63839,0.32534,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,10,0,0,1,0,0,1,0,0,6,0,0,0,0,12],[20,48,0.4167,0.625,0.22798,0.42857,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,6,0,0,4,0,0,4,0,0,13,0,0,0,0,5],[24,48,0.5,0.49997,0.23957,0.28571,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,15,0,0,3,0,0,3,0,0,8,0,0,0,0,3],[28,48,0.5833,0.51784,0.2714,0.28571,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,11,0,0,9,0,0,1,0,0,4,0,0,0,0,6],[32,48,0.6667,0.43749,0.17472,0.28571,0.35714,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,5,0,0,4,0,0,7,0,0,0,0,0],[36,48,0.75,0.46875,0.21793,0.28571,0.42857,0.60714,0.0,1.0,1,2,0,1,0,0,0,0,10,0,0,11,0,0,2,0,0,6,0,0,0,0,2],[40,48,0.8333,0.42411,0.28679,0.2857,0.42857,0.60714,0.0,1.0,5,3,0,5,0,1,0,0,9,0,0,6,0,0,3,0,0,5,0,0,0,0,3],[44,48,0.9167,0.37054,0.22829,0.28571,0.28571,0.42858,0.0,1.0,3,1,0,3,0,4,0,0,10,0,0,8,0,0,2,0,0,4,0,0,0,0,1],[48,48,1.0,0.11161,0.1504,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,13,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"e8ced34479be46dc","q":"Ana and Banana play a game. First, Ana picks a real number  $p$  with  $0 \\le p \\le 1$ . Then, Banana picks an integer  $h$  greater than  $1$  and creates a spaceship with  $h$  hit points. Now every minute, Ana decreases the spaceship's hit points by  $2$  with probability  $1-p$ , and by  $3$  with probability  $p$ . Ana wins if and only if the number of hit points is reduced to exactly  $0$  at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select  $p$  and  $h$  optimally, compute the integer closest to  $1000p$ .\n\n*Proposed by Lewis Chen*","t":[{"b":2,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,90,0.0,0.86161,0.10403,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,23,0,6],[4,90,0.0444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,90,0.0889,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[12,90,0.1333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,90,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,90,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,90,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,90,0.3111,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,90,0.3556,0.97991,0.04803,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,27],[36,90,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,90,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,90,0.4889,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,90,0.5333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,90,0.5778,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[56,90,0.6222,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,90,0.6667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[64,90,0.7111,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,90,0.7556,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,90,0.8,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[76,90,0.8444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,90,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,90,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,90,0.9778,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[90,90,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"rising","v":0.81027,"x":1.0,"p":[[0,195,0.0,0.81027,0.19853,0.85714,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,0,0,0,0,1,0,1,0,0,22,0,5],[4,195,0.0205,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,195,0.041,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,195,0.0615,0.99777,0.01242,1.0,1.0,1.0,0.9286,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[16,195,0.0821,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[20,195,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,195,0.1231,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,195,0.1436,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,195,0.1641,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,195,0.1846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,195,0.2051,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,195,0.2256,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,195,0.2462,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,195,0.2667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,195,0.2872,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[60,195,0.3077,0.97545,0.0524,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,1,26],[64,195,0.3282,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,195,0.3487,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[72,195,0.3692,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[76,195,0.3897,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[80,195,0.4103,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[84,195,0.4308,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[88,195,0.4513,0.97991,0.04803,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,27],[92,195,0.4718,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[96,195,0.4923,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[100,195,0.5128,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[104,195,0.5333,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[108,195,0.5538,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[112,195,0.5744,0.98884,0.0362,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,29],[116,195,0.5949,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[120,195,0.6154,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[124,195,0.6359,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[128,195,0.6564,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[132,195,0.6769,0.9375,0.14698,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,24],[136,195,0.6974,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[140,195,0.7179,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[144,195,0.7385,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[148,195,0.759,0.91518,0.20473,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,25],[152,195,0.7795,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[156,195,0.8,0.90179,0.16146,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,20],[160,195,0.8205,0.91964,0.16728,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,24],[164,195,0.841,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[168,195,0.8615,0.9442,0.14144,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,1,25],[172,195,0.8821,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[176,195,0.9026,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[180,195,0.9231,0.95313,0.09037,0.91074,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,1,23],[184,195,0.9436,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[188,195,0.9641,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[192,195,0.9846,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[195,195,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"dab71556a27ece7f","q":"Determine the locus of points  $M$  in the plane of a given rhombus  $ABCD$  such that  $MA\\cdot MC+MB\\cdot MD=AB^2$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,76,0.0,0.92857,0.19233,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,26],[4,76,0.0526,0.96429,0.11845,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[8,76,0.1053,0.95089,0.14987,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[12,76,0.1579,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,76,0.2105,0.95535,0.13571,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[20,76,0.2632,0.95088,0.17718,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[24,76,0.3158,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,76,0.3684,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[32,76,0.4211,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[36,76,0.4737,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,76,0.5263,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,76,0.5789,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,76,0.6316,0.85714,0.27432,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,0,3,0,22],[52,76,0.6842,0.95982,0.12993,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[56,76,0.7368,0.92411,0.19228,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,27],[60,76,0.7895,0.86607,0.26711,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,4,0,23],[64,76,0.8421,0.94642,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[68,76,0.8947,0.91503,0.19856,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[72,76,0.9474,0.92857,0.15972,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,25],[76,76,1.0,0.90624,0.2008,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,23]]},{"b":5,"e":0.14286,"k":"falling","v":0.27667,"x":0.9866,"p":[[0,39,0.0,0.89731,0.22935,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,25],[4,39,0.1026,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,39,0.2051,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,39,0.3077,0.89272,0.29242,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[16,39,0.4103,0.95535,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[20,39,0.5128,0.91518,0.22548,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[24,39,0.6154,0.94187,0.16741,1.0,1.0,1.0,0.14,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[28,39,0.7179,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[32,39,0.8205,0.89732,0.22084,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,25],[36,39,0.9231,0.76329,0.33062,0.4286,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,5,0,0,2,0,0,1,0,0,2,0,0,0,0,20],[39,39,1.0,0.27667,0.18538,0.14286,0.2857,0.32143,0.0,0.71429,3,0,0,3,0,11,0,0,10,0,0,2,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"b5f8ab414c482b95","q":"Find all pairs  $(p,q)$  of prime numbers such that  $$  p(p^2 - p - 1) = q(2q + 3) . $$","t":[{"b":2,"e":1.0,"k":"flat","v":0.82143,"x":0.99554,"p":[[0,143,0.0,0.87946,0.24772,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,22],[4,143,0.028,0.87723,0.23092,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,1,0,0,0,1,0,2,0,0,5,0,21],[8,143,0.0559,0.83036,0.25364,0.82143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,17],[12,143,0.0839,0.89286,0.19885,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,22],[16,143,0.1119,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[20,143,0.1399,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,143,0.1678,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[28,143,0.1958,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[32,143,0.2238,0.82143,0.28348,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,3,0,20],[36,143,0.2517,0.91517,0.17448,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,24],[40,143,0.2797,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[44,143,0.3077,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[48,143,0.3357,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[52,143,0.3636,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,143,0.3916,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,143,0.4196,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[64,143,0.4476,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[68,143,0.4755,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[72,143,0.5035,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[76,143,0.5315,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[80,143,0.5594,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[84,143,0.5874,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,143,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,143,0.6434,0.96428,0.06186,0.96429,1.0,1.0,0.8571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[96,143,0.6713,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[100,143,0.6993,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[104,143,0.7273,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[108,143,0.7552,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[112,143,0.7832,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[116,143,0.8112,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[120,143,0.8392,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[124,143,0.8671,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[128,143,0.8951,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[132,143,0.9231,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[136,143,0.951,0.95534,0.09746,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[140,143,0.979,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[143,143,1.0,0.94642,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22]]},{"b":3,"e":0.14286,"k":"falling","v":0.16964,"x":0.92857,"p":[[0,78,0.0,0.92857,0.21129,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[4,78,0.0513,0.87054,0.25595,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,22],[8,78,0.1026,0.88839,0.24932,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,24],[12,78,0.1538,0.91071,0.15872,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,18],[16,78,0.2051,0.69196,0.35555,0.28571,0.85714,1.0,0.0,1.0,1,13,0,1,0,6,0,0,2,0,0,0,0,0,2,0,0,2,0,0,6,0,13],[20,78,0.2564,0.83482,0.2969,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,0,3,0,22],[24,78,0.3077,0.81696,0.29066,0.82132,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,0,5,0,19],[28,78,0.359,0.83482,0.28146,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,21],[32,78,0.4103,0.82589,0.28512,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,19],[36,78,0.4615,0.88839,0.19475,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,20],[40,78,0.5128,0.80356,0.28291,0.857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,10,0,15],[44,78,0.5641,0.80804,0.28259,0.75001,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,5,0,0,0,0,0,0,0,0,6,0,18],[48,78,0.6154,0.79018,0.31539,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,18],[52,78,0.6667,0.8125,0.26591,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,0,3,0,18],[56,78,0.7179,0.66518,0.34183,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,4,0,0,4,0,0,2,0,0,1,0,0,6,0,0,1,0,13],[60,78,0.7692,0.53116,0.3917,0.14286,0.57143,0.89286,0.0,1.0,2,8,0,2,0,12,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,8],[64,78,0.8205,0.6383,0.38141,0.14286,0.85714,1.0,0.0,1.0,2,12,0,2,0,7,0,0,2,0,0,1,0,0,0,0,0,3,0,0,5,0,12],[68,78,0.8718,0.8125,0.27994,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,3,0,0,3,0,19],[72,78,0.9231,0.72767,0.33949,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,8,0,14],[76,78,0.9744,0.50893,0.36759,0.14286,0.35714,1.0,0.14286,1.0,0,9,0,0,0,13,0,0,3,0,0,1,0,0,2,0,0,3,0,0,1,0,9],[78,78,1.0,0.16964,0.06621,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6edc1ce5822cc183","q":"Find all quadruples of real numbers  $(a,b,c,d)$  satisfying the system of equations\n\\[\\begin{cases}(b+c+d)^{2010}=3a (a+c+d)^{2010}=3b (a+b+d)^{2010}=3c (a+b+c)^{2010}=3d\\end{cases}\\]","t":[{"b":1,"e":1.0,"k":"rising","v":0.5982,"x":1.0,"p":[[0,26,0.0,0.5982,0.27067,0.42857,0.57121,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,3,0,0,11,0,0,7,0,0,1,0,0,1,0,8],[4,26,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,26,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"rising","v":0.61607,"x":1.0,"p":[[0,32,0.0,0.61607,0.25614,0.42857,0.57143,0.85704,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,9,0,0,9,0,0,2,0,0,2,0,7],[4,32,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,32,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,32,0.375,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,32,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"90cfd43967ccb4cb","q":"Every of  $n$  guests invited to a dinner has got an invitation denoted by a number from  $1$  to  $n$ . The guests will be sitting around a round table with  $n$  seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.\nFind all  $n$  for which he can select the guests in such an order to serve all the guests.","t":[{"b":2,"e":0.0,"k":"falling","v":0.08464,"x":0.93304,"p":[[0,41,0.0,0.83928,0.14617,0.85714,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,4],[4,41,0.0976,0.93304,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[8,41,0.1951,0.84821,0.14698,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,12,0,11],[12,41,0.2927,0.86607,0.21998,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,10,0,17],[16,41,0.3902,0.625,0.37923,0.28571,0.85707,1.0,0.0,1.0,5,11,0,5,0,1,0,0,3,0,0,5,0,0,0,0,0,1,0,0,6,0,11],[20,41,0.4878,0.60271,0.34575,0.2857,0.78564,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,7,0,0,2,0,0,1,0,0,1,0,0,9,0,7],[24,41,0.5854,0.63839,0.3369,0.28571,0.71429,0.85714,0.0,1.0,4,6,0,4,0,1,0,0,4,0,0,0,0,0,2,0,0,6,0,0,9,0,6],[28,41,0.6829,0.70526,0.29454,0.42859,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,5,0,0,2,0,0,1,0,0,2,0,0,13,0,7],[32,41,0.7805,0.57141,0.32927,0.28571,0.57143,0.85714,0.0,1.0,4,4,0,4,0,1,0,0,5,0,0,3,0,0,4,0,0,2,0,0,9,0,4],[36,41,0.878,0.5625,0.26471,0.42857,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,4,0,0,7,0,0,4,0,0,5,0,0,8,0,1],[40,41,0.9756,0.125,0.20124,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,4,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[41,41,1.0,0.08464,0.16692,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":4,"e":1.0,"k":"flat","v":0.36159,"x":0.85266,"p":[[0,46,0.0,0.82142,0.14728,0.85714,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,26,0,2],[4,46,0.087,0.83482,0.26271,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0,10,0,16],[8,46,0.1739,0.75445,0.27255,0.67857,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,2,0,0,2,0,0,2,0,0,4,0,0,10,0,10],[12,46,0.2609,0.75,0.29881,0.71429,0.85714,1.0,0.0,1.0,2,11,0,2,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,0,9,0,11],[16,46,0.3478,0.7142,0.28365,0.57143,0.85714,1.0,0.0,1.0,1,9,0,1,0,2,0,0,1,0,0,3,0,0,4,0,0,4,0,0,8,0,9],[20,46,0.4348,0.40616,0.36968,0.14214,0.21431,0.85714,0.0,1.0,7,2,0,7,0,9,0,0,3,0,0,0,0,0,1,0,0,2,0,0,8,0,2],[24,46,0.5217,0.51775,0.30051,0.2857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,6,0,0,2,0,0,7,0,0,4,0,0,3,0,4],[28,46,0.6087,0.53123,0.29716,0.2857,0.4286,0.74996,0.0,1.0,1,4,0,1,0,5,0,0,4,0,0,7,0,0,2,0,0,5,0,0,4,0,4],[32,46,0.6957,0.41964,0.24984,0.14286,0.42857,0.60714,0.0,0.85714,2,0,0,2,0,8,0,0,2,0,0,8,0,0,4,0,0,6,0,0,2,0,0],[36,46,0.7826,0.36159,0.2901,0.14286,0.2857,0.57111,0.0,1.0,4,1,0,4,0,8,0,0,9,0,0,2,0,0,2,0,0,2,0,0,4,0,1],[40,46,0.8696,0.70981,0.2382,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,8,0,0,6,0,0,2,0,0,6,0,9],[44,46,0.9565,0.817,0.22646,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,3,0,0,7,0,15],[46,46,1.0,0.85266,0.16937,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,9,0,14]]}]},{"i":"710d954c6754e1c9","q":"Find all the polynomials with real coefficients which satisfy  $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$  for all  $x\\in \\mathbb{R}$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.73213,"x":1.0,"p":[[0,52,0.0,0.73213,0.22233,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,10,0,0,3,0,9],[4,52,0.0769,0.89732,0.17582,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,1,0,23],[8,52,0.1538,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[12,52,0.2308,0.92857,0.14725,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,25],[16,52,0.3077,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[20,52,0.3846,0.94196,0.13767,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[24,52,0.4615,0.89732,0.14827,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,3,0,20],[28,52,0.5385,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,52,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,52,0.6923,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,52,0.7692,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,52,0.8462,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,52,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,52,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.71429,"k":"flat","v":0.67855,"x":0.89286,"p":[[0,56,0.0,0.73212,0.26667,0.5354,0.78564,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,5,0,0,4,0,12],[4,56,0.0714,0.84372,0.22123,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,19],[8,56,0.1429,0.88839,0.18466,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,1,0,22],[12,56,0.2143,0.8125,0.22428,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,3,0,16],[16,56,0.2857,0.89286,0.16366,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,3,0,20],[20,56,0.3571,0.82142,0.15974,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,2,0,13],[24,56,0.4286,0.75,0.15568,0.57143,0.71429,0.85714,0.5714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,13,0,0,3,0,7],[28,56,0.5,0.72767,0.16116,0.57143,0.71429,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,4,0,6],[32,56,0.5714,0.72321,0.17835,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,14,0,0,3,0,6],[36,56,0.6429,0.75444,0.17584,0.57143,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,11,0,0,0,0,10],[40,56,0.7143,0.71875,0.15765,0.57143,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,16,0,0,2,0,5],[44,56,0.7857,0.72767,0.13055,0.67857,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,17,0,0,3,0,4],[48,56,0.8571,0.75891,0.15747,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,0,0,4,0,7],[52,56,0.9286,0.70089,0.15303,0.57143,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,10,0,0,2,0,5],[56,56,1.0,0.67855,0.14727,0.57143,0.64286,0.71429,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,11,0,0,1,0,4]]}]},{"i":"d8a55c4bf82684a9","q":"Find the natural numbers  $ n\\ge 2 $  which have the property that the ring of integers modulo  $ n $  has exactly an element that is not a sum of two squares.","t":[{"b":2,"e":1.0,"k":"rising","v":0.77677,"x":1.0,"p":[[0,51,0.0,0.77677,0.2141,0.71421,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,6,0,0,10,0,9],[4,51,0.0784,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,51,0.1569,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,51,0.2353,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,51,0.3137,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,51,0.3922,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,51,0.4706,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,51,0.549,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[32,51,0.6275,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,51,0.7059,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[48,51,0.9412,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[51,51,1.0,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30]]},{"b":5,"e":0.85714,"k":"flat","v":0.78571,"x":0.96875,"p":[[0,40,0.0,0.78571,0.1821,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,16,0,5],[4,40,0.1,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,40,0.2,0.9375,0.1234,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25],[12,40,0.3,0.96427,0.08754,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,40,0.4,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[20,40,0.5,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[24,40,0.6,0.9375,0.12846,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,23],[28,40,0.7,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[32,40,0.8,0.92411,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,22],[36,40,0.9,0.94642,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[40,40,1.0,0.87486,0.13732,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,8,0,15]]}]},{"i":"a3625c4f70c8a80a","q":"Fix a triangle  $ABC$ . Let  $\\Gamma_1$  the circle through  $B$ , tangent  to edge in  $A$ . Let   $\\Gamma_2$  the circle through C tangent to edge  $AB$  in  $A$ . The second intersection of  $\\Gamma_1$  and  $\\Gamma_2$  is denoted by  $D$ . The line  $AD$  has second intersection  $E$  with the circumcircle of   $\\vartriangle ABC$ . Show that  $D$  is the midpoint of the segment  $AE$ .","t":[{"b":2,"e":0.57143,"k":"rising","v":0.29464,"x":0.5,"p":[[0,77,0.0,0.29464,0.14258,0.2857,0.28571,0.28571,0.0,0.71429,3,0,1,3,0,0,0,0,25,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[4,77,0.0519,0.38392,0.19377,0.28571,0.28571,0.35714,0.2857,1.0,0,2,0,0,0,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,0,0,2],[8,77,0.1039,0.2991,0.11495,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,0,0,0,26,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,77,0.1558,0.4732,0.14032,0.28571,0.57143,0.57143,0.1429,0.57143,0,0,0,0,0,1,0,0,9,0,0,1,0,0,21,0,0,0,0,0,0,0,0],[16,77,0.2078,0.5,0.12372,0.5,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,8,0,0,0,0,0,24,0,0,0,0,0,0,0,0],[20,77,0.2597,0.4241,0.17307,0.28571,0.57141,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,12,0,0,1,0,0,17,0,0,0,0,0,0,0,0],[24,77,0.3117,0.39285,0.18558,0.28571,0.28571,0.57143,0.0,0.57143,3,0,0,3,0,0,0,0,14,0,0,0,0,0,15,0,0,0,0,0,0,0,0],[28,77,0.3636,0.4732,0.16535,0.28571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,0,0,0,21,0,0,1,0,0,0,0,0],[32,77,0.4156,0.45534,0.19703,0.28571,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,2,0,0,4,0,0,0,0,0,23,0,0,0,0,0,0,0,0],[36,77,0.4675,0.46426,0.20823,0.28571,0.57143,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,7,0,0,1,0,0,20,0,0,0,0,0,0,0,1],[40,77,0.5195,0.48659,0.12807,0.28571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,1,0,0,22,0,0,0,0,0,0,0,0],[44,77,0.5714,0.45086,0.18593,0.28571,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,0,0,0,7,0,0,1,0,0,21,0,0,0,0,0,0,0,0],[48,77,0.6234,0.45085,0.19918,0.28571,0.57143,0.57143,0.0,0.57143,4,0,0,4,0,0,0,0,5,0,0,1,0,0,22,0,0,0,0,0,0,0,0],[52,77,0.6753,0.48658,0.1631,0.53539,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,5,0,0,1,0,0,24,0,0,0,0,0,0,0,0],[56,77,0.7273,0.47321,0.18363,0.5,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,0,0,0,5,0,0,0,0,0,24,0,0,0,0,0,0,0,0],[60,77,0.7792,0.49553,0.13355,0.42857,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,4,0,0,3,0,0,23,0,0,0,0,0,0,0,0],[64,77,0.8312,0.32588,0.24019,0.0,0.35714,0.57143,0.0,0.57143,9,0,0,9,0,2,0,0,5,0,0,3,0,0,13,0,0,0,0,0,0,0,0],[68,77,0.8831,0.41518,0.18336,0.28571,0.50001,0.57143,0.0,0.57143,3,0,0,3,0,0,0,0,10,0,0,3,0,0,16,0,0,0,0,0,0,0,0],[72,77,0.9351,0.36161,0.21424,0.24999,0.35714,0.57143,0.0,0.57143,5,0,0,5,0,3,0,0,8,0,0,2,0,0,14,0,0,0,0,0,0,0,0],[76,77,0.987,0.42409,0.20665,0.28571,0.57143,0.57143,0.0,0.57143,5,0,0,5,0,0,0,0,4,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[77,77,1.0,0.45982,0.16263,0.2857,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,6,0,0,3,0,0,20,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.14722,"x":0.32142,"p":[[0,77,0.0,0.32142,0.16366,0.2857,0.28571,0.28571,0.0,1.0,2,1,1,2,0,0,0,0,24,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[4,77,0.0519,0.29017,0.13592,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,0,0,0,26,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[8,77,0.1039,0.2723,0.13994,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,2,0,0,22,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,77,0.1558,0.27678,0.10677,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,0,0,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,77,0.2078,0.28125,0.10999,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,2,0,0,25,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,77,0.2597,0.26339,0.11355,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,0,0,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,77,0.3117,0.27678,0.09407,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,1,0,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,77,0.3636,0.2991,0.12037,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,2,0,0,26,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[32,77,0.4156,0.26339,0.10779,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,77,0.4675,0.28566,0.11311,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,2,0,0,24,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[40,77,0.5195,0.25446,0.09268,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,77,0.5714,0.30357,0.12753,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,0,0,0,26,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[48,77,0.6234,0.2991,0.09689,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,26,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[52,77,0.6753,0.2723,0.11491,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,1,0,0,26,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[56,77,0.7273,0.299,0.13062,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,2,0,0,23,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[60,77,0.7792,0.25892,0.10374,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,77,0.8312,0.24105,0.11534,0.24999,0.2857,0.28571,0.0,0.571,4,0,0,4,0,4,0,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,77,0.8831,0.26785,0.07784,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.2366,0.09851,0.2857,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.14722,0.14497,0.0,0.14286,0.2857,0.0,0.571,13,0,0,13,0,7,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[77,77,1.0,0.16071,0.15464,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,3,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"32914364f0d1e54a","q":"Consider  $\\triangle ABC$  and a point  $M$  in its interior so that  $\\angle MAB = 10^\\circ$ ,  $\\angle MBA = 20^\\circ$ ,  $\\angle MCA = 30^\\circ$  and  $\\angle MAC = 40^\\circ$ . What is  $\\angle MBC$ ?","t":[{"b":6,"e":1.0,"k":"rising","v":0.80804,"x":0.97766,"p":[[0,11,0.0,0.82587,0.24154,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,17],[4,11,0.3636,0.80804,0.25657,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,3,0,17],[8,11,0.7273,0.8348,0.18597,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,6,0,14],[11,11,1.0,0.97766,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.77679,"x":0.8616,"p":[[0,13,0.0,0.8125,0.24337,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,3,0,15],[4,13,0.3077,0.77679,0.18536,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,14,0,0,1,0,11],[8,13,0.6154,0.8616,0.14054,0.71429,0.9285,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,16],[12,13,0.9231,0.83926,0.18127,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,0,1,0,16],[13,13,1.0,0.83929,0.14174,0.71429,0.71429,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,14]]}]},{"i":"333a671ac547d8a3","q":"Consider an equilateral triangle  $\\triangle ABC$ . The points  $K$  and  $L$  divide the leg  $BC$  into three equal parts, the point  $M$  divides the leg  $AC$  in the ratio  $1:2$ , counting from the vertex  $A$ . Prove that  $\\angle AKM+\\angle ALM=30^{\\circ}$ . \n\nProposed by V. Proizvolov","t":[{"b":5,"e":0.14286,"k":"flat","v":0.13831,"x":0.20089,"p":[[0,35,0.0,0.20089,0.12807,0.14286,0.14286,0.17857,0.0,0.4286,2,0,0,2,0,22,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.14724,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.16956,0.09064,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.16956,0.09064,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.11161,"x":0.19634,"p":[[0,32,0.0,0.1875,0.12595,0.14286,0.14286,0.14287,0.0,0.4286,3,0,0,3,0,22,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.16965,0.10374,0.14286,0.14286,0.14287,0.0,0.42857,2,0,0,2,0,26,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.19634,0.12756,0.14286,0.14286,0.1429,0.0,0.42857,2,0,0,2,0,23,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.1384,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.15625,0.13054,0.14286,0.14286,0.14287,0.0,0.4286,7,0,0,7,0,20,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.12054,0.11904,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,18,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.16956,0.12598,0.14286,0.14286,0.14286,0.0,0.4286,5,0,0,5,0,21,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.13393,0.14258,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,19,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"45a010ad37927fdc","q":"For an integer  $n$  let  $M (n) = \\{n, n + 1, n + 2, n + 3, n + 4\\}$ . Furthermore, be  $S (n)$  sum of squares and  $P (n)$  the product of the squares of the elements of  $M (n)$ . For which integers  $n$  is  $S (n)$  a divisor of  $P (n)$  ?","t":[{"b":1,"e":0.57143,"k":"falling","v":0.65175,"x":0.91071,"p":[[0,129,0.0,0.91071,0.12753,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,19],[4,129,0.031,0.79017,0.11285,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,11,0,4],[8,129,0.062,0.82143,0.11845,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,14,0,6],[12,129,0.093,0.82589,0.1461,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,12,0,8],[16,129,0.124,0.7589,0.12079,0.71429,0.78564,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,0,16,0,0],[20,129,0.155,0.7232,0.15127,0.67857,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,13,0,0,9,0,2],[24,129,0.186,0.75445,0.11973,0.71429,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,0,8,0,3],[28,129,0.2171,0.73213,0.09281,0.71429,0.71429,0.75,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,0,8,0,0],[32,129,0.2481,0.72766,0.13997,0.67857,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,12,0,0,11,0,1],[36,129,0.2791,0.65625,0.17075,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,1,0,0,9,0,0,14,0,0,6,0,0],[40,129,0.3101,0.70088,0.10928,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,21,0,0,5,0,0],[44,129,0.3411,0.72321,0.17473,0.71429,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,15,0,0,13,0,0],[48,129,0.3721,0.70981,0.13116,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,20,0,0,7,0,0],[52,129,0.4031,0.72319,0.11262,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,15,0,0,10,0,0],[56,129,0.4341,0.72319,0.11262,0.67857,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,15,0,0,8,0,1],[60,129,0.4651,0.71875,0.08364,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,0,3,0,1],[64,129,0.4961,0.68301,0.11704,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,15,0,0,6,0,0],[68,129,0.5271,0.74553,0.10554,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,16,0,0,12,0,0],[72,129,0.5581,0.70535,0.12341,0.71429,0.71429,0.71429,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,20,0,0,3,0,2],[76,129,0.5891,0.65179,0.20806,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,1,0,0,4,0,0,16,0,0,7,0,0],[80,129,0.6202,0.70087,0.13997,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,19,0,0,7,0,0],[84,129,0.6512,0.69195,0.15613,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,19,0,0,5,0,1],[88,129,0.6822,0.67856,0.14285,0.71429,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,20,0,0,5,0,0],[92,129,0.7132,0.6964,0.11155,0.57143,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,15,0,0,7,0,0],[96,129,0.7442,0.71872,0.10406,0.71429,0.71429,0.74996,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,18,0,0,8,0,0],[100,129,0.7752,0.69195,0.13416,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,17,0,0,5,0,1],[104,129,0.8062,0.68749,0.14479,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,19,0,0,6,0,0],[108,129,0.8372,0.7455,0.11705,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,0,11,0,1],[112,129,0.8682,0.6875,0.14914,0.57143,0.71429,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,15,0,0,8,0,0],[116,129,0.8992,0.70979,0.13118,0.67857,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,14,0,0,10,0,0],[120,129,0.9302,0.67409,0.11971,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,18,0,0,4,0,0],[124,129,0.9612,0.68299,0.15461,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,15,0,0,6,0,1],[128,129,0.9922,0.6607,0.14617,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,0,15,0,0,5,0,0],[129,129,1.0,0.65175,0.17835,0.571,0.71429,0.74996,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,12,0,0,8,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.74107,"x":0.84821,"p":[[0,101,0.0,0.84375,0.18336,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,6,0,15],[4,101,0.0396,0.84821,0.14258,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,9,0,12],[8,101,0.0792,0.77229,0.11219,0.71429,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,13,0,2],[12,101,0.1188,0.83927,0.10564,0.857,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,20,0,5],[16,101,0.1584,0.80789,0.10491,0.71429,0.857,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,14,0,4],[20,101,0.198,0.84374,0.09688,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,17,0,6],[24,101,0.2376,0.76336,0.09857,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,12,0,1],[28,101,0.2772,0.7857,0.101,0.71429,0.857,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,19,0,0],[32,101,0.3168,0.77678,0.13333,0.71429,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,0,0,17,0,1],[36,101,0.3564,0.79017,0.07972,0.71429,0.78564,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,15,0,1],[40,101,0.396,0.76339,0.12681,0.71429,0.71429,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,0,10,0,3],[44,101,0.4356,0.79911,0.09354,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,16,0,2],[48,101,0.4752,0.79463,0.12342,0.71429,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,20,0,1],[52,101,0.5149,0.77677,0.10676,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,17,0,0,12,0,2],[56,101,0.5545,0.75,0.13363,0.71429,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,16,0,0,12,0,1],[60,101,0.5941,0.75893,0.11538,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,17,0,0,12,0,1],[64,101,0.6337,0.79018,0.10091,0.71429,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,20,0,0],[68,101,0.6733,0.80355,0.12244,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,18,0,3],[72,101,0.7129,0.7455,0.13709,0.71429,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,15,0,0],[76,101,0.7525,0.74107,0.12078,0.71429,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,19,0,0,11,0,0],[80,101,0.7921,0.79909,0.07876,0.71429,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,20,0,0],[84,101,0.8317,0.76339,0.09852,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,9,0,2],[88,101,0.8713,0.76339,0.09852,0.71429,0.71429,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,14,0,0],[92,101,0.9109,0.78125,0.09438,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,12,0,2],[96,101,0.9505,0.75893,0.0974,0.71429,0.71429,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,13,0,0],[100,101,0.9901,0.77678,0.11258,0.71429,0.85707,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,0,0,17,0,0],[101,101,1.0,0.77676,0.08704,0.71429,0.71429,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,13,0,1]]}]},{"i":"9bfb760d22c928d9","q":"Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers satisfying $a_{0}=1$ and $a_{n}=a_{\\lfloor 7 n / 9\\rfloor}+a_{\\lfloor n / 9\\rfloor}$ for $n=1,2, \\ldots$ Prove that there exists a positive integer $k$ with $a_{k}<\\frac{k}{2001 !}$.\n\n(Here $\\lfloor x\\rfloor$ denotes the largest integer not greater than $x$.)","t":[{"b":2,"e":1.0,"k":"flat","v":0.91518,"x":0.98661,"p":[[0,24,0.0,0.91964,0.21706,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,27],[4,24,0.1667,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,24,0.3333,0.91518,0.23107,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[12,24,0.5,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,24,0.6667,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[20,24,0.8333,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,24,1.0,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.91517,"x":0.99554,"p":[[0,22,0.0,0.91517,0.20161,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,26],[4,22,0.1818,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,22,0.3636,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,22,0.9091,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[22,22,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"f6235c6c14dbfa69","q":"Let $ABC$ be an acute triangle with $AB \\neq AC$. We denote $\\Gamma$ as its circumcircle, $H$ as its orthocenter, and $O$ as the center of $\\Gamma$. Let $M$ be the midpoint of $[BC]$. The line $(AM)$ intersects $\\Gamma$ again at $N$, and the circle with diameter $[AM]$ intersects $\\Gamma$ again at $P$.\n\nProve that the lines $(AP)$, $(BC)$, and $(OH)$ are concurrent if and only if $AH = HN$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.16518,"p":[[0,79,0.0,0.10714,0.15972,0.0,0.0,0.2857,0.0,0.4286,21,0,2,21,0,2,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,79,0.0506,0.16518,0.18249,0.0,0.07143,0.32143,0.0,0.42857,16,0,0,16,0,3,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,79,0.1013,0.11161,0.1665,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,79,0.1519,0.11598,0.16535,0.0,0.0,0.14287,0.0,0.4286,19,0,0,19,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,79,0.2025,0.10268,0.14826,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,1,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,79,0.2532,0.125,0.17767,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,0,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,79,0.3038,0.09822,0.14914,0.0,0.0,0.1786,0.0,0.42857,21,0,0,21,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,79,0.3544,0.07589,0.14279,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,79,0.4051,0.10714,0.17857,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,3,0,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[36,79,0.4557,0.08928,0.12752,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,79,0.5063,0.09375,0.15815,0.0,0.0,0.17857,0.0,0.42857,23,0,0,23,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[44,79,0.557,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,79,0.6076,0.08036,0.14258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,79,0.6582,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,79,0.7089,0.07143,0.14726,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,79,0.7595,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,79,0.8101,0.09375,0.1411,0.0,0.0,0.17857,0.0,0.42857,21,0,0,21,0,3,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,79,0.8608,0.12946,0.16115,0.0,0.0,0.2857,0.0,0.42857,18,0,0,18,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[72,79,0.9114,0.08036,0.14258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,79,0.962,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[79,79,1.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.05357,"x":0.29464,"p":[[0,109,0.0,0.0759,0.12869,0.0,0.0,0.1429,0.0,0.42857,23,0,0,23,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,109,0.0367,0.17857,0.18898,0.0,0.07143,0.42857,0.0,0.4286,16,0,0,16,0,1,0,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,109,0.0734,0.12946,0.17261,0.0,0.0,0.28571,0.0,0.42857,19,0,0,19,0,3,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,109,0.1101,0.05357,0.12242,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,109,0.1468,0.19643,0.17768,0.0,0.2857,0.32143,0.0,0.4286,13,0,0,13,0,2,0,0,9,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[20,109,0.1835,0.05804,0.12807,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,109,0.2202,0.11161,0.15866,0.0,0.0,0.14287,0.0,0.42857,19,0,0,19,0,6,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,109,0.2569,0.09822,0.15746,0.0,0.0,0.1786,0.0,0.4286,22,0,0,22,0,2,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,109,0.2936,0.12947,0.17262,0.0,0.0,0.28571,0.0,0.4286,19,0,0,19,0,3,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[36,109,0.3303,0.12946,0.16506,0.0,0.0,0.28571,0.0,0.42857,19,0,0,19,0,1,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,109,0.367,0.17857,0.17128,0.0,0.21428,0.28571,0.0,0.42857,14,0,0,14,0,2,0,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[44,109,0.4037,0.11161,0.15458,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,2,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,109,0.4404,0.11161,0.1504,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,1,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,109,0.4771,0.16518,0.17169,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,3,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[56,109,0.5138,0.14732,0.16935,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,2,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[60,109,0.5505,0.19196,0.17717,0.0,0.21428,0.32143,0.0,0.4286,13,0,0,13,0,3,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[64,109,0.5872,0.17857,0.17496,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,3,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[68,109,0.6239,0.12054,0.16409,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,1,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[72,109,0.6606,0.17857,0.17496,0.0,0.14286,0.32143,0.0,0.4286,13,0,0,13,0,6,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[76,109,0.6972,0.19643,0.17768,0.0,0.2857,0.32143,0.0,0.4286,13,0,0,13,0,2,0,0,9,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[80,109,0.7339,0.16518,0.16793,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,2,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[84,109,0.7706,0.12054,0.16016,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,3,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[88,109,0.8073,0.14732,0.18029,0.0,0.0,0.28571,0.0,0.4286,18,0,0,18,0,2,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[92,109,0.844,0.19643,0.16269,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,4,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[96,109,0.8807,0.18303,0.16457,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,2,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[100,109,0.9174,0.25894,0.14914,0.14289,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,3,0,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[104,109,0.9541,0.25447,0.17029,0.0,0.28571,0.42857,0.0,0.4286,9,0,0,9,0,0,0,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[108,109,0.9908,0.26786,0.11152,0.24999,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,6,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[109,109,1.0,0.29464,0.1234,0.28571,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,2,0,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cca7f20cd07e07fd","q":"Let $a, b, c$ be integers satisfying $0<a<c-1$ and $1<b<c$. For each $k$, $0 \\leq k \\leq a$, let $r_{k}, 0 \\leq r_{k}<c$, be the remainder of $k b$ when divided by $c$. Prove that the two sets $\\left\\{r_{0}, r_{1}, r_{2}, \\ldots, r_{a}\\right\\}$ and $\\{0,1,2, \\ldots, a\\}$ are different.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.07143,"p":[[0,31,0.0,0.03116,0.07758,0.0,0.0,0.0,0.0,0.2857,27,0,2,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.06696,0.15146,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,5,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,31,0.3871,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.07143,0.17128,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,31,0.7742,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.12491,"p":[[0,44,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,3,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.08482,0.17445,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,44,0.1818,0.12491,0.20438,0.0,0.0,0.1786,0.0,0.71429,20,0,0,20,0,4,0,0,5,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[12,44,0.2727,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.04018,0.08918,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.02233,0.05188,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ffdac79545f023af","q":"Let $S$ be a non-empty set of strictly positive integers satisfying the following property: For all integers $a, b \\in S$, the integer $a b + 1$ also belongs to $S$.\nShow that the set of prime numbers that do not divide any of the elements of $S$ is finite.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.12491,"x":0.37947,"p":[[0,52,0.0,0.27232,0.16115,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,7,0,0,6,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.33473,0.2132,0.14286,0.35714,0.46429,0.0,0.85714,4,0,0,4,0,7,0,0,5,0,0,8,0,0,7,0,0,0,0,0,1,0,0],[8,52,0.1538,0.29469,0.19868,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,5,0,0,8,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[12,52,0.2308,0.37947,0.23313,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,6,0,0,5,0,0,8,0,0,8,0,0,0,0,0,1,0,1],[16,52,0.3077,0.28569,0.17492,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,5,0,0,11,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[20,52,0.3846,0.27236,0.16892,0.14286,0.2857,0.42857,0.0,0.57143,3,0,0,3,0,11,0,0,8,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[24,52,0.4615,0.24557,0.1646,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,10,0,0,8,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[28,52,0.5385,0.28562,0.16374,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,9,0,0,8,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[32,52,0.6154,0.23214,0.17405,0.14286,0.14286,0.42857,0.0,0.57143,6,0,0,6,0,13,0,0,1,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[36,52,0.6923,0.16518,0.08827,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,22,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.12491,0.08562,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.14277,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.12929,0.09004,0.14,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.11589,"x":0.49554,"p":[[0,15,0.0,0.28558,0.13855,0.14286,0.28571,0.42857,0.0,0.43,1,0,0,1,0,12,0,0,5,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.49554,0.20198,0.42857,0.42857,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,3,0,0,13,0,0,10,0,0,1,0,0,1,0,2],[8,15,0.5333,0.26786,0.21651,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,11,0,0,3,0,0,7,0,0,4,0,0,0,0,0,1,0,0],[12,15,0.8,0.12045,0.10169,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c8996890d6af7141","q":"Let $S=\\{1, \\ldots, n\\}$, with $n \\geqslant 3$ an integer, and let $k$ be a strictly positive integer. We denote $S^{k}$ as the set of $k$-tuples of elements of $S$. Let $f: S^{k} \\rightarrow S$ be such that, if $x=\\left(x_{1}, \\ldots, x_{k}\\right) \\in S^{k}$ and $y=\\left(y_{1}, \\ldots, y_{k}\\right) \\in S^{k}$ with $x_{i} \\neq y_{i}$ for all $1 \\leqslant i \\leqslant k$, then $f(x) \\neq f(y)$.\nShow that there exists $\\ell$ with $1 \\leqslant \\ell \\leqslant k$ and a function $g: S \\rightarrow S$ satisfying, for all $x_{1}, \\ldots, x_{k} \\in S$, $f\\left(x_{1}, \\ldots, x_{k}\\right)=g\\left(x_{\\ell}\\right)$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.01339,"x":0.11607,"p":[[0,52,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.04018,0.10853,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,52,0.1538,0.0357,0.10708,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,52,0.2308,0.08481,0.20155,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,52,0.3077,0.05358,0.14617,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,52,0.3846,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.11607,0.18013,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,7,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[32,52,0.6154,0.09821,0.21852,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[36,52,0.6923,0.0625,0.18877,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,52,0.7692,0.06687,0.11279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.08927,0.16265,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,9,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[48,52,0.9231,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.10713,0.13828,0.0,0.14286,0.14286,0.0,0.571,15,0,0,15,0,14,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.07589,"p":[[0,40,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.07589,0.21124,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,40,0.5,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.04455,0.09052,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.05348,0.18119,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,40,0.8,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.07589,0.18552,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,40,1.0,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"51f09fb4adffb216","q":"Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a concave function and let $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ be continuous. Given that\n\n$$\nf(x+y)+f(x-y)-2 f(x)=g(x) y^{2}\n$$\n\nfor all $x, y \\in \\mathbb{R}$, prove that $f$ is a quadratic function.\n(Bulgaria)","t":[{"b":0,"e":1.0,"k":"rising","v":0.67855,"x":1.0,"p":[[0,111,0.0,0.67855,0.30723,0.42857,0.57121,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,2,0,0,10,0,0,3,0,0,0,0,0,1,0,14],[4,111,0.036,0.9375,0.14699,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,26],[8,111,0.0721,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,25],[12,111,0.1081,0.92411,0.17122,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,23],[16,111,0.1441,0.93304,0.15966,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,24],[20,111,0.1802,0.92409,0.12873,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[24,111,0.2162,0.94643,0.15464,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,28],[28,111,0.2523,0.85712,0.20518,0.82132,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,6,0,18],[32,111,0.2883,0.88393,0.20652,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,4,0,22],[36,111,0.3243,0.88839,0.18466,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,20],[40,111,0.3604,0.92411,0.14718,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,7,0,22],[44,111,0.3964,0.91964,0.16727,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,24],[48,111,0.4324,0.88838,0.18811,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,22],[52,111,0.4685,0.85268,0.22441,0.67857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,1,0,0,2,0,21],[56,111,0.5045,0.85714,0.23419,0.78571,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,0,0,0,2,0,22],[60,111,0.5405,0.88391,0.19048,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,4,0,21],[64,111,0.5766,0.87052,0.1968,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,4,0,20],[68,111,0.6126,0.93304,0.14718,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,24],[72,111,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,111,0.6847,0.89731,0.1646,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,21],[80,111,0.7207,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[84,111,0.7568,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,111,0.7928,0.94196,0.14223,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,27],[92,111,0.8288,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,111,0.8649,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[100,111,0.9009,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[104,111,0.9369,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[108,111,0.973,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[111,111,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"rising","v":0.57589,"x":1.0,"p":[[0,124,0.0,0.57589,0.35622,0.28571,0.42857,1.0,0.0,1.0,3,11,0,3,0,3,0,0,3,0,0,8,0,0,2,0,0,1,0,0,1,0,11],[4,124,0.0323,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,124,0.0645,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,124,0.0968,0.91964,0.19212,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,26],[16,124,0.129,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[20,124,0.1613,0.86607,0.22851,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,0,4,0,21],[24,124,0.1935,0.88839,0.20433,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,22],[28,124,0.2258,0.92857,0.16366,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[32,124,0.2581,0.91963,0.1595,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,23],[36,124,0.2903,0.91071,0.20124,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,23],[40,124,0.3226,0.9375,0.16342,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[44,124,0.3548,0.91517,0.17445,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,9,0,21],[48,124,0.3871,0.91516,0.1781,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,4,0,24],[52,124,0.4194,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[56,124,0.4516,0.9375,0.14698,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,25],[60,124,0.4839,0.93304,0.15966,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,26],[64,124,0.5161,0.90177,0.16539,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,6,0,21],[68,124,0.5484,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[72,124,0.5806,0.92409,0.16749,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,3,0,25],[76,124,0.6129,0.87053,0.20935,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,22],[80,124,0.6452,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,27],[84,124,0.6774,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[88,124,0.7097,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[92,124,0.7419,0.95981,0.12496,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[96,124,0.7742,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[100,124,0.8065,0.90179,0.20025,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,25],[104,124,0.8387,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[108,124,0.871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,124,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,124,0.9355,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[120,124,0.9677,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,124,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"14783a59756c22cc","q":"Find all polynomials with integer coefficients such that for all positive integers  $n$  satisfies  $P(n!)=|P(n)|!$","t":[{"b":3,"e":1.0,"k":"flat","v":0.72308,"x":0.81237,"p":[[0,21,0.0,0.73659,0.0883,0.71429,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,3],[4,21,0.1905,0.77232,0.12299,0.71429,0.71429,0.71429,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,0,0,7],[8,21,0.381,0.72308,0.04974,0.71429,0.71429,0.71429,0.71,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[12,21,0.5714,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[16,21,0.7619,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[20,21,0.9524,0.76339,0.11633,0.71429,0.71429,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,6],[21,21,1.0,0.81237,0.1358,0.71429,0.71429,1.0,0.71,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,11]]},{"b":5,"e":0.71429,"k":"flat","v":0.73214,"x":0.80357,"p":[[0,77,0.0,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[4,77,0.0519,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[8,77,0.1039,0.76339,0.11633,0.71429,0.71429,0.71429,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,6],[12,77,0.1558,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[16,77,0.2078,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[20,77,0.2597,0.75446,0.10853,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[24,77,0.3117,0.73214,0.09279,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,0,0,3],[28,77,0.3636,0.75445,0.10855,0.71429,0.71429,0.71429,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[32,77,0.4156,0.75446,0.10853,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[36,77,0.4675,0.76786,0.11152,0.71429,0.71429,0.71429,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,6],[40,77,0.5195,0.73661,0.08828,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,3],[44,77,0.5714,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[48,77,0.6234,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[52,77,0.6753,0.7723,0.12302,0.71429,0.71429,0.71429,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,0,0,7],[56,77,0.7273,0.75446,0.10853,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[60,77,0.7792,0.75433,0.10858,0.71429,0.71429,0.71429,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,5],[64,77,0.8312,0.80357,0.13243,0.71429,0.71429,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,10],[68,77,0.8831,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[72,77,0.9351,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[76,77,0.987,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[77,77,1.0,0.74987,0.09454,0.71429,0.71429,0.71429,0.71,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4]]}]},{"i":"b4fc08dc65c80a79","q":"Let $p$ be a prime number. Let $\\mathbb{F}_{p}$ denote the integers modulo $p$, and let $\\mathbb{F}_{p}[x]$ be the set of polynomials with coefficients in $\\mathbb{F}_{p}$. Define $\\Psi: \\mathbb{F}_{p}[x] \\rightarrow \\mathbb{F}_{p}[x]$ by  $$ \\Psi\\left(\\sum_{i=0}^{n} a_{i} x^{i}\\right)=\\sum_{i=0}^{n} a_{i} x^{x^{i}} . $$  Prove that for nonzero polynomials $F, G \\in \\mathbb{F}_{p}[x]$,  $$ \\Psi(\\operatorname{gcd}(F, G))=\\operatorname{gcd}(\\Psi(F), \\Psi(G)) $$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,22,0.0,0.04009,0.11418,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.05357,0.17035,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,22,0.3636,0.08036,0.19865,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[12,22,0.5455,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,22,0.7273,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[22,22,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"rising","v":0.04464,"x":0.30802,"p":[[0,19,0.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.25447,0.32875,0.0,0.07145,0.32143,0.0,1.0,16,3,0,16,0,1,0,0,7,0,0,2,0,0,1,0,0,1,0,0,1,0,3],[8,19,0.4211,0.12946,0.29312,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[12,19,0.6316,0.06696,0.20818,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[16,19,0.8421,0.29462,0.33867,0.0,0.2857,0.46418,0.0,1.0,14,4,0,14,0,1,0,0,7,0,0,2,0,0,3,0,0,1,0,0,0,0,4],[19,19,1.0,0.30802,0.3136,0.0,0.28571,0.57143,0.0,1.0,12,2,0,12,0,2,0,0,7,0,0,1,0,0,5,0,0,2,0,0,1,0,2]]}]},{"i":"c7a314ab52c785be","q":"Let $n \\geqslant 2$ and let $x_{1}, x_{2}, \\ldots, x_{n}$ be real numbers such that $x_{1}+x_{2}+\\cdots+x_{n}=0$ and $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{n}^{2}=1$.\nShow that there exists $i$ such that $x_{i} \\geqslant \\frac{1}{\\sqrt{n(n-1)}}$.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.3125,"x":0.58481,"p":[[0,28,0.0,0.3125,0.26107,0.0,0.42857,0.42857,0.0,1.0,10,1,0,10,0,2,0,0,3,0,0,11,0,0,3,0,0,2,0,0,0,0,1],[4,28,0.1429,0.49554,0.30301,0.28571,0.57143,0.60714,0.0,1.0,4,5,0,4,0,3,0,0,2,0,0,6,0,0,9,0,0,3,0,0,0,0,5],[8,28,0.2857,0.53125,0.28175,0.42857,0.57143,0.71429,0.0,1.0,4,3,0,4,0,1,0,0,1,0,0,7,0,0,9,0,0,4,0,0,3,0,3],[12,28,0.4286,0.44196,0.31816,0.28571,0.42857,0.46429,0.0,1.0,5,6,0,5,0,2,0,0,5,0,0,12,0,0,1,0,0,1,0,0,0,0,6],[16,28,0.5714,0.58039,0.32326,0.42857,0.57143,1.0,0.0,1.0,4,9,0,4,0,1,0,0,1,0,0,5,0,0,11,0,0,1,0,0,0,0,9],[20,28,0.7143,0.45089,0.30117,0.24999,0.42857,0.57143,0.0,1.0,5,3,0,5,0,3,0,0,4,0,0,5,0,0,8,0,0,2,0,0,2,0,3],[24,28,0.8571,0.58481,0.22968,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,1,0,0,6,0,0,12,0,0,6,0,0,2,0,3],[28,28,1.0,0.57143,0.13832,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,5,0,0,19,0,0,4,0,0,1,0,1]]},{"b":1,"e":0.71429,"k":"rising","v":0.22322,"x":0.59813,"p":[[0,31,0.0,0.22322,0.2878,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,4,0,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,2],[4,31,0.129,0.375,0.22232,0.14286,0.42857,0.46429,0.0,0.71429,5,0,0,5,0,4,0,0,1,0,0,14,0,0,4,0,0,4,0,0,0,0,0],[8,31,0.2581,0.54464,0.28446,0.42857,0.57143,0.60714,0.0,1.0,3,6,0,3,0,1,0,0,2,0,0,7,0,0,11,0,0,2,0,0,0,0,6],[12,31,0.3871,0.35713,0.23144,0.14286,0.42857,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,6,0,0,7,0,0,9,0,0,0,0,0,0,0,1],[16,31,0.5161,0.51786,0.29827,0.42857,0.42857,0.60714,0.0,1.0,4,6,0,4,0,1,0,0,1,0,0,11,0,0,7,0,0,2,0,0,0,0,6],[20,31,0.6452,0.43304,0.2977,0.25,0.42857,0.57143,0.0,1.0,6,3,0,6,0,2,0,0,3,0,0,9,0,0,5,0,0,3,0,0,1,0,3],[24,31,0.7742,0.52679,0.39679,0.14286,0.57143,1.0,0.0,1.0,7,9,0,7,0,4,0,0,2,0,0,1,0,0,3,0,0,4,0,0,2,0,9],[28,31,0.9032,0.47768,0.31463,0.14286,0.57143,0.71429,0.0,1.0,3,3,0,3,0,6,0,0,5,0,0,1,0,0,7,0,0,3,0,0,4,0,3],[31,31,1.0,0.59813,0.32242,0.28571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,4,0,0,2,0,0,3,0,0,6,0,0,4,0,7]]}]},{"i":"cf9f58c976434811","q":"Find all primes  $p$  such that there exist positive integers  $q$  and  $r$  such that  $p \\nmid q$ ,  $3 \\nmid q$ ,  $p^3 = r^3 - q^2$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.87499,"x":1.0,"p":[[0,25,0.0,0.91518,0.16698,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,5,0,23],[4,25,0.16,0.87499,0.19152,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,6,0,19],[8,25,0.32,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,25,0.48,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[25,25,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,29,0.0,0.9465,0.13241,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[4,29,0.1379,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,27],[8,29,0.2759,0.89286,0.19562,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,22],[12,29,0.4138,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,29,0.5517,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,29,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,29,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6daecd681fba829f","q":"Two circles are tangent externaly at a point  $B$ . A line tangent to one of the circles at a point  $A$  intersects the other circle at points  $C$  and  $D$ . Show that  $A$  is equidistant to the lines  $BC$  and  $BD$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.03571,"x":0.18304,"p":[[0,37,0.0,0.18304,0.212,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,18,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[4,37,0.1081,0.13393,0.11811,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,19,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.09822,0.07524,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.12054,0.10171,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.125,0.09942,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.11161,0.08553,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.10268,0.08917,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04464,"x":0.14286,"p":[[0,36,0.0,0.13384,0.14257,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,14,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,36,0.1111,0.14286,0.13832,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,17,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.08482,0.11214,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.08929,0.11152,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0759,0.11837,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.08483,0.11214,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"66ce9799fb6f47c9","q":"a) Evaluate \n\n\\[\\lim_{n\\to \\infty} \\underbrace{\\sqrt{a+\\sqrt{a+\\ldots+\\sqrt{a+\\sqrt{b}}}}}_{n\\ \\text{square roots}}\\]\n\nwith  $a,b>0$ .\n\nb)Let  $(a_n)_{n\\ge 1}$  and  $(x_n)_{n\\ge 1}$  such that  $a_n>0$  and\n\n\\[x_n=\\sqrt{a_n+\\sqrt{a_{n-1}+\\ldots+\\sqrt{a_2+\\sqrt{a_1}}}},\\ \\forall n\\in \\mathbb{N}^*\\]\n\nProve that:\n\n1)  $(x_n)_{n\\ge 1}$  is bounded if and only if  $(a_n)_{n\\ge 1}$  is bounded.\n2)  $(x_n)_{n\\ge 1}$  is convergent if and only if  $(a_n)_{n\\ge 1}$  is convergent.\n\n*Valentin Matrosenco*","t":[{"b":0,"e":1.0,"k":"flat","v":0.94643,"x":0.99554,"p":[[0,22,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,22,0.1818,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,22,0.3636,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,22,0.5455,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[16,22,0.7273,0.95089,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[20,22,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[22,22,1.0,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]},{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,29,0.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,29,0.1379,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,29,0.2759,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,29,0.4138,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,29,0.5517,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,29,0.6897,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,29,0.8276,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,29,0.9655,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[29,29,1.0,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19]]}]},{"i":"ae31609aa84d19b3","q":"Let n > 3 be a given integer. Find the largest integer d (in terms of n) such that for\nany set S of n integers, there are four distinct (but not necessarily disjoint) nonempty\nsubsets, the sum of the elements of each of which is divisible by d.","t":[{"b":6,"e":0.0,"k":"flat","v":0.02679,"x":0.19643,"p":[[0,73,0.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,6,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,73,0.0548,0.19643,0.16656,0.14286,0.14286,0.1429,0.0,0.71429,3,0,0,3,0,23,0,0,2,0,0,0,0,0,3,0,0,1,0,0,0,0,0],[8,73,0.1096,0.15625,0.16115,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,73,0.1644,0.1159,0.09057,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,21,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.12947,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,73,0.3288,0.12947,0.07457,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,73,0.3836,0.16071,0.18123,0.0,0.14286,0.1786,0.0,0.71429,11,0,0,11,0,13,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[32,73,0.4384,0.17411,0.10555,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,19,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.11598,0.08325,0.105,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,73,0.5479,0.16071,0.13243,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,15,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,73,0.6027,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,73,0.6575,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,73,0.7123,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,73,0.7671,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,73,0.9315,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,73,0.9863,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.11607,"x":0.19187,"p":[[0,64,0.0,0.12946,0.09688,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.16964,0.09062,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,24,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.17857,0.15152,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[12,64,0.1875,0.14714,0.06668,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.19187,0.11075,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,25,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[20,64,0.3125,0.12947,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,64,0.5,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.16518,0.08828,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,64,0.8125,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b640cef7c10adcc","q":"Let f be a function from  $\\{1, 2, . . . , 22\\}$  to the positive integers such that  $mn | f(m) + f(n)$  for all  $m, n \\in \\{1, 2, . . . , 22\\}$ . If  $d$  is the number of positive divisors of  $f(20)$ , compute the minimum possible value of  $d$ .","t":[{"b":4,"e":0.14286,"k":"falling","v":0.07143,"x":0.62945,"p":[[0,81,0.0,0.45087,0.23718,0.28571,0.49979,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,6,0,0,4,0,0,9,0,0,6,0,0,0,0,1],[4,81,0.0494,0.62945,0.32705,0.39285,0.71429,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,0,9,0,6],[8,81,0.0988,0.49999,0.37796,0.14286,0.35716,0.85714,0.0,1.0,4,7,0,4,0,7,0,0,5,0,0,1,0,0,2,0,0,1,0,0,5,0,7],[12,81,0.1481,0.42402,0.3264,0.14286,0.28571,0.71429,0.0,1.0,2,3,0,2,0,12,0,0,3,0,0,3,0,0,2,0,0,3,0,0,4,0,3],[16,81,0.1975,0.37947,0.31866,0.14286,0.14288,0.57143,0.0,1.0,1,4,0,1,0,16,0,0,2,0,0,4,0,0,2,0,0,1,0,0,2,0,4],[20,81,0.2469,0.43303,0.34159,0.14286,0.28571,0.75,0.0,1.0,2,5,0,2,0,11,0,0,5,0,0,3,0,0,1,0,0,2,0,0,3,0,5],[24,81,0.2963,0.36161,0.30719,0.14286,0.2857,0.57143,0.0,1.0,4,4,0,4,0,11,0,0,3,0,0,5,0,0,4,0,0,1,0,0,0,0,4],[28,81,0.3457,0.35713,0.33502,0.14286,0.14295,0.60714,0.0,1.0,7,2,0,7,0,10,0,0,2,0,0,2,0,0,3,0,0,2,0,0,4,0,2],[32,81,0.3951,0.50445,0.35171,0.14289,0.35714,0.85714,0.0,1.0,3,4,0,3,0,6,0,0,7,0,0,1,0,0,1,0,0,2,0,0,8,0,4],[36,81,0.4444,0.31696,0.31487,0.10714,0.21428,0.46429,0.0,1.0,8,2,0,8,0,8,0,0,6,0,0,2,0,0,2,0,0,1,0,0,3,0,2],[40,81,0.4938,0.44642,0.31893,0.24999,0.42857,0.60714,0.0,1.0,5,4,0,5,0,3,0,0,5,0,0,8,0,0,3,0,0,1,0,0,3,0,4],[44,81,0.5432,0.3125,0.2911,0.14286,0.14288,0.42858,0.0,1.0,6,1,0,6,0,11,0,0,4,0,0,4,0,0,1,0,0,2,0,0,3,0,1],[48,81,0.5926,0.30356,0.28737,0.14286,0.14286,0.4642,0.0,1.0,7,1,0,7,0,11,0,0,2,0,0,4,0,0,3,0,0,2,0,0,2,0,1],[52,81,0.642,0.2589,0.28888,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,11,0,0,5,0,0,0,0,0,3,0,0,0,0,0,3,0,1],[56,81,0.6914,0.25447,0.28288,0.0,0.14286,0.32143,0.0,1.0,10,2,0,10,0,9,0,0,5,0,0,2,0,0,2,0,0,2,0,0,0,0,2],[60,81,0.7407,0.32142,0.33119,0.0,0.2143,0.57143,0.0,1.0,10,2,0,10,0,6,0,0,5,0,0,2,0,0,3,0,0,0,0,0,4,0,2],[64,81,0.7901,0.35267,0.3174,0.14286,0.21428,0.57143,0.0,1.0,5,3,0,5,0,11,0,0,4,0,0,2,0,0,4,0,0,1,0,0,2,0,3],[68,81,0.8395,0.22322,0.25738,0.0,0.14286,0.2857,0.0,1.0,9,1,0,9,0,14,0,0,2,0,0,2,0,0,2,0,0,1,0,0,1,0,1],[72,81,0.8889,0.25446,0.28733,0.0,0.14286,0.4286,0.0,1.0,10,1,0,10,0,11,0,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,1],[76,81,0.9383,0.25893,0.27302,0.0,0.14286,0.4286,0.0,1.0,12,1,0,12,0,6,0,0,2,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[80,81,0.9877,0.19187,0.19761,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,8,0,0,5,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[81,81,1.0,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.14732,"x":0.7321,"p":[[0,63,0.0,0.4598,0.28734,0.25,0.4286,0.60714,0.0,1.0,3,3,0,3,0,5,0,0,4,0,0,5,0,0,7,0,0,4,0,0,1,0,3],[4,63,0.0635,0.7232,0.26949,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,1,0,0,4,0,0,3,0,0,12,0,7],[8,63,0.127,0.69194,0.28818,0.53539,0.85707,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,3,0,0,5,0,0,1,0,0,11,0,7],[12,63,0.1905,0.7321,0.26428,0.571,0.85707,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,2,0,0,5,0,0,4,0,0,7,0,10],[16,63,0.254,0.66516,0.3126,0.42857,0.85714,0.85714,0.0,1.0,3,6,1,3,0,1,0,0,1,0,0,4,0,0,5,0,0,0,0,0,12,0,6],[20,63,0.3175,0.65622,0.29637,0.42857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,1,0,0,5,0,0,5,0,0,2,0,0,8,0,7],[24,63,0.381,0.62498,0.29395,0.42857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,1,0,0,5,0,0,4,0,0,3,0,0,10,0,4],[28,63,0.4444,0.62945,0.30063,0.42857,0.71429,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,4,0,0,4,0,0,4,0,0,4,0,0,7,0,6],[32,63,0.5079,0.65176,0.29654,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,3,0,0,3,0,0,4,0,0,5,0,0,5,0,8],[36,63,0.5714,0.54009,0.32692,0.25,0.57143,0.85714,0.0,1.0,3,5,0,3,0,5,0,0,1,0,0,5,0,0,6,0,0,2,0,0,5,0,5],[40,63,0.6349,0.47319,0.22141,0.39286,0.4286,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,9,0,0,10,0,0,2,0,0,2,0,1],[44,63,0.6984,0.5982,0.24856,0.42857,0.57143,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,5,0,0,6,0,0,7,0,0,2,0,0,9,0,2],[48,63,0.7619,0.46874,0.26542,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,6,0,0,5,0,0,8,0,0,1,0,0,5,0,1],[52,63,0.8254,0.4463,0.26675,0.14286,0.42857,0.57143,0.14,1.0,0,3,0,0,0,9,0,0,3,0,0,8,0,0,7,0,0,0,0,0,2,0,3],[56,63,0.8889,0.34375,0.24707,0.14286,0.21428,0.57143,0.0,0.85714,1,0,0,1,0,15,0,0,3,0,0,3,0,0,5,0,0,3,0,0,2,0,0],[60,63,0.9524,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.19195,0.13172,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,26,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"a5acb623b0343388","q":"What is the number of ways in which one can color the squares of a  $4\\times 4$  chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?","t":[{"b":0,"e":0.14286,"k":"falling","v":0.21876,"x":0.51786,"p":[[0,38,0.0,0.51786,0.42521,0.14286,0.14288,1.0,0.14286,1.0,0,14,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[4,38,0.1053,0.29018,0.32239,0.14286,0.14286,0.14286,0.0,1.0,1,5,0,1,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[8,38,0.2105,0.37054,0.37773,0.14286,0.14286,0.78571,0.0,1.0,1,8,0,1,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[12,38,0.3158,0.29902,0.33765,0.14286,0.14286,0.1429,0.0,1.0,1,6,0,1,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[16,38,0.4211,0.2366,0.25655,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[20,38,0.5263,0.30348,0.32492,0.14286,0.14286,0.14287,0.14,1.0,0,5,0,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[24,38,0.6316,0.23206,0.24939,0.14286,0.14286,0.1429,0.14,1.0,0,3,0,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[28,38,0.7368,0.24545,0.28627,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[32,38,0.8421,0.22768,0.24964,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[36,38,0.9474,0.21876,0.2525,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[38,38,1.0,0.24107,0.28891,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]},{"b":2,"e":1.0,"k":"falling","v":0.18751,"x":0.57143,"p":[[0,47,0.0,0.57143,0.43006,0.14286,0.64286,1.0,0.0,1.0,1,16,0,1,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,47,0.0851,0.37045,0.36751,0.14286,0.14286,0.57143,0.14,1.0,0,8,0,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[8,47,0.1702,0.25429,0.28519,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[12,47,0.2553,0.27233,0.31614,0.14286,0.14286,0.1429,0.0,1.0,2,5,0,2,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,47,0.3404,0.29464,0.31122,0.14286,0.14286,0.17857,0.0,1.0,1,4,0,1,0,23,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[20,47,0.4255,0.24107,0.2911,0.14286,0.14286,0.14286,0.0,1.0,3,4,0,3,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,47,0.5106,0.18751,0.2126,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,47,0.5957,0.26786,0.31693,0.14286,0.14286,0.14286,0.0,1.0,2,5,0,2,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[32,47,0.6809,0.23652,0.29151,0.14286,0.14286,0.14286,0.0,1.0,3,4,0,3,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[36,47,0.766,0.29465,0.34057,0.14286,0.14286,0.14287,0.0,1.0,2,6,0,2,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[40,47,0.8511,0.2409,0.28897,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[44,47,0.9362,0.21875,0.23686,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[47,47,1.0,0.24554,0.28624,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]}]},{"i":"613efd2ac6a41348","q":"A function \\[\\text{f:(0,}\\infty \\text{) }\\to \\text{(0,}\\infty \\text{)}\\] is called contract if,  for every numbers  $x,y\\in \\text{(0,}\\infty \\text{)}$  we have,  $\\underset{n\\to \\infty }{\\mathop{\\lim }}\\,\\left( {{f}^{n}}\\left( x \\right)-{{f}^{n}}\\left( y \\right) \\right)=0$  where  ${{f}^{n}}=\\underbrace{f\\circ f\\circ ...\\circ f}_{n\\ f\\text{'s}}$ a) Consider \\[f:\\text{(0,}\\infty \\text{) }\\to \\text{(0,}\\infty \\text{)}\\] a function contract, continue with the property that has a fixed point, that existing  ${{x}_{0}}\\in \\text{(0,}\\infty \\text{) }$  there so that  $f\\left( {{x}_{0}} \\right)={{x}_{0}}.$  Show that  $f\\left( x \\right)>x,$  for every  $x\\in \\text{(0,}{{x}_{0}}\\text{)}\\,$  and  $f\\left( x \\right)<x$ , for every  $x\\in \\text{(}{{x}_{0}}\\text{,}\\infty \\text{)}\\,$ .\nb) Show that the given function \\[f\\text{:(0,}\\infty \\text{) }\\to \\text{(0,}\\infty \\text{)}\\] given by  $f\\left( x \\right)=x+\\frac{1}{x}$  is contracted but has no fix number.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.56472,"x":0.76783,"p":[[0,30,0.0,0.65845,0.18877,0.571,0.71429,0.85704,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,1,3,0,1,7,1,0,8,0,0,7,0,2],[4,30,0.1333,0.76783,0.17405,0.71429,0.85707,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,6,0,0,17,0,3],[8,30,0.2667,0.56472,0.20544,0.42859,0.57143,0.71429,0.07143,0.85714,0,0,0,0,1,2,0,0,2,0,0,4,0,0,11,0,0,8,0,0,4,0,0],[12,30,0.4,0.61161,0.23753,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,7,0,0,3,0,0,8,0,0,7,0,2],[16,30,0.5333,0.66292,0.19582,0.571,0.71429,0.80357,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,6,0,0,4,0,0,12,1,0,7,0,1],[20,30,0.6667,0.63836,0.18893,0.5354,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,13,0,0,7,0,0],[24,30,0.8,0.62945,0.18162,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,16,0,0,2,0,1],[28,30,0.9333,0.70311,0.14298,0.69646,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,1,0,14,0,0,10,0,0],[30,30,1.0,0.64283,0.12373,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,15,0,0,3,0,0]]},{"b":4,"e":0.71429,"k":"falling","v":0.50444,"x":0.67187,"p":[[0,15,0.0,0.67187,0.1863,0.57143,0.71429,0.85704,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,10,0,0,8,1,1],[4,15,0.2667,0.66294,0.2316,0.55355,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,0,0,0,4,0,1,4,0,0,8,0,0,10,0,2],[8,15,0.5333,0.587,0.25439,0.42857,0.60714,0.80375,0.0,1.0,2,1,0,2,0,2,0,0,0,0,0,7,0,0,5,2,0,5,1,0,7,0,1],[12,15,0.8,0.59371,0.16793,0.4286,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,8,0,0,11,0,0,5,0,0,6,0,0],[15,15,1.0,0.50444,0.16456,0.42857,0.57141,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,1,3,0,0,7,0,0,12,1,0,6,0,0,0,0,0]]}]},{"i":"e56b8df696ddcb7e","q":"For a fixed positive integer  $k$ , prove that there exist infinitely many primes  $p$  such that there is an integer  $w$ , where  $w^2-1$  is not divisible by  $p$ , and the order of  $w$  in modulus  $p$  is the same as the order of  $w$  in modulus  $p^k$ .\n\n*Author: James Rickards*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.67411,"x":0.70982,"p":[[0,45,0.0,0.70535,0.14697,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,23,0,0,1,0,3],[4,45,0.0889,0.70088,0.12038,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,23,0,0,1,0,2],[8,45,0.1778,0.6875,0.09062,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,24,0,0,0,0,1],[12,45,0.2667,0.70088,0.11497,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,25,0,0,0,0,2],[16,45,0.3556,0.70982,0.07563,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,0,0,0,1],[20,45,0.4444,0.68302,0.10556,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0],[24,45,0.5333,0.67411,0.1394,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0],[28,45,0.6222,0.68275,0.08545,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,27,0,0,0,0,0],[32,45,0.7111,0.70534,0.03463,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[36,45,0.8,0.70088,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[40,45,0.8889,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[44,45,0.9778,0.69196,0.08073,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,29,0,0,0,0,0],[45,45,1.0,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.55803,"x":0.75892,"p":[[0,34,0.0,0.75892,0.13092,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,23,0,0,1,0,6],[4,34,0.1176,0.61161,0.18293,0.57143,0.71429,0.71429,0.0,0.71429,1,0,1,1,0,0,0,0,4,0,0,1,0,0,4,0,0,22,0,0,0,0,0],[8,34,0.2353,0.58478,0.17262,0.53465,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,1,0,0,6,0,0,18,0,0,0,0,0],[12,34,0.3529,0.60713,0.16367,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,20,0,0,0,0,0],[16,34,0.4706,0.61606,0.15336,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,0,0,0,7,0,0,20,0,0,0,0,0],[20,34,0.5882,0.62053,0.16602,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,0,0,0,3,0,0,23,0,0,0,0,0],[24,34,0.7059,0.58481,0.16888,0.57132,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,0,0,0,8,0,0,17,0,0,0,0,0],[28,34,0.8235,0.58034,0.17835,0.57132,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,0,0,0,8,0,0,17,0,0,0,0,0],[32,34,0.9412,0.56235,0.17093,0.49968,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,0,0,0,10,0,0,14,0,0,0,0,0],[34,34,1.0,0.55803,0.17985,0.28571,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,0,0,0,8,0,0,15,0,0,0,0,0]]}]},{"i":"3682f338dd70841e","q":"$A,B,C$  are playing backgammon tournament. At first,  $A$  plays with  $B$ . Then the winner plays with  $C$ . As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that  $C$  wins the tournament? $ \\textbf{(A)}\\ \\frac27\n\\qquad\\textbf{(B)}\\ \\frac13\n\\qquad\\textbf{(C)}\\ \\frac3{14}\n\\qquad\\textbf{(D)}\\ \\frac 17\n\\qquad\\textbf{(E)}\\ \\text{None}\n$","t":[{"b":0,"e":1.0,"k":"flat","v":0.80357,"x":1.0,"p":[[0,49,0.0,0.87054,0.19352,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,0,0,21],[4,49,0.0816,0.875,0.22798,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,0,0,24],[8,49,0.1633,0.82143,0.27664,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,19],[12,49,0.2449,0.91071,0.15872,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,3,0,22],[16,49,0.3265,0.89286,0.22588,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,24],[20,49,0.4082,0.80357,0.26666,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,0,0,0,19],[24,49,0.4898,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[28,49,0.5714,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[32,49,0.6531,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[36,49,0.7347,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[40,49,0.8163,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,49,0.898,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,49,0.9796,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[49,49,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.79464,"x":0.91518,"p":[[0,39,0.0,0.86607,0.2141,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,1,0,21],[4,39,0.1026,0.85268,0.26841,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,0,0,22],[8,39,0.2051,0.86161,0.20973,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,1,0,21],[12,39,0.3077,0.79464,0.30917,0.71429,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,0,0,19],[16,39,0.4103,0.88839,0.14167,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,2,0,19],[20,39,0.5128,0.91518,0.17807,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[24,39,0.6154,0.86161,0.20973,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,0,0,21],[28,39,0.7179,0.84821,0.2257,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,0,1,0,20],[32,39,0.8205,0.85268,0.2435,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,5,0,0,0,0,22],[36,39,0.9231,0.84821,0.20806,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,0,2,0,19],[39,39,1.0,0.87499,0.1777,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,1,0,20]]}]},{"i":"876150922577bbc2","q":"17. C5 (USS 5) The right triangles $A B C$ and $A B_{1} C_{1}$ are similar and have opposite orientation. The right angles are at $C$ and $C_{1}$, and we also have $\\measuredangle C A B=\\measuredangle C_{1} A B_{1}$. Let $M$ be the point of intersection of the lines $B C_{1}$ and $B_{1} C$. Prove that if the lines $A M$ and $C C_{1}$ exist, they are perpendicular.","t":[{"b":0,"e":0.2857,"k":"falling","v":0.22525,"x":0.91964,"p":[[0,125,0.0,0.80356,0.15467,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,18,0,0,1,0,11],[4,125,0.032,0.88838,0.22228,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,22],[8,125,0.064,0.90625,0.22477,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,25],[12,125,0.096,0.91964,0.16342,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,24],[16,125,0.128,0.82588,0.28289,0.67857,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,21],[20,125,0.16,0.89508,0.19397,0.857,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,1,0,5,0,20],[24,125,0.192,0.83258,0.31674,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,0,2,0,23],[28,125,0.224,0.76782,0.26428,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,0,0,0,4,0,0,4,0,0,6,0,13],[32,125,0.256,0.72766,0.31209,0.57142,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,3,0,0,0,0,0,4,0,0,5,0,0,4,0,13],[36,125,0.288,0.84373,0.24319,0.71421,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,0,0,1,0,21],[40,125,0.32,0.79908,0.24188,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,7,0,0,2,0,16],[44,125,0.352,0.80356,0.26906,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,18],[48,125,0.384,0.875,0.23623,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,23],[52,125,0.416,0.77678,0.2549,0.71429,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,9,0,0,1,0,15],[56,125,0.448,0.74103,0.32033,0.53539,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,5,0,0,1,0,0,3,0,0,3,0,0,1,0,17],[60,125,0.48,0.70979,0.28681,0.53539,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,6,0,0,1,0,0,4,0,0,5,0,0,3,0,12],[64,125,0.512,0.59372,0.33524,0.2857,0.64286,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,6,0,0,2,0,0,3,0,0,4,0,0,4,0,8],[68,125,0.544,0.7232,0.35164,0.42859,0.92857,1.0,0.0,1.0,3,16,0,3,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0,3,0,16],[72,125,0.576,0.64283,0.33693,0.28571,0.71429,1.0,0.0,1.0,3,11,0,3,0,0,0,0,6,0,0,2,0,0,3,0,0,5,0,0,2,0,11],[76,125,0.608,0.63392,0.36235,0.28571,0.71429,1.0,0.0,1.0,3,12,0,3,0,2,0,0,5,0,0,3,0,0,1,0,0,3,0,0,3,0,12],[80,125,0.64,0.72765,0.3395,0.42859,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,17],[84,125,0.672,0.63839,0.39283,0.28571,0.71429,1.0,0.0,1.0,7,13,0,7,0,0,0,0,2,0,0,0,0,0,4,0,0,4,0,0,2,0,13],[88,125,0.704,0.60711,0.3522,0.28571,0.71429,1.0,0.0,1.0,4,9,0,4,0,1,0,0,5,0,0,2,0,0,3,0,0,4,1,0,2,1,9],[92,125,0.736,0.57592,0.3263,0.28571,0.57121,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,7,0,0,4,0,0,2,0,0,5,0,0,2,0,8],[96,125,0.768,0.6964,0.32879,0.53539,0.78564,1.0,0.0,1.0,3,13,0,3,0,0,0,0,3,0,0,2,0,0,5,0,0,3,0,0,3,0,13],[100,125,0.8,0.57137,0.30723,0.28571,0.571,0.78571,0.0,1.0,2,8,0,2,0,1,0,0,6,0,0,6,0,0,4,0,0,5,0,0,0,0,8],[104,125,0.832,0.56247,0.36235,0.2857,0.57121,1.0,0.0,1.0,5,9,0,5,0,0,0,0,8,0,0,1,0,0,3,0,0,4,0,0,2,0,9],[108,125,0.864,0.7165,0.32069,0.53539,0.85714,1.0,0.0,1.0,2,13,0,2,0,2,0,0,1,0,0,3,0,0,2,1,0,4,0,0,4,0,13],[112,125,0.896,0.76339,0.28259,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,6,0,0,3,0,15],[116,125,0.928,0.67187,0.38667,0.28571,0.85714,1.0,0.0,1.0,4,15,0,4,0,2,0,0,4,0,0,1,0,0,1,1,0,0,0,0,4,0,15],[120,125,0.96,0.5714,0.37627,0.25,0.64286,1.0,0.0,1.0,4,10,0,4,0,4,0,0,5,0,0,0,0,0,3,0,0,4,0,0,2,0,10],[124,125,0.992,0.22525,0.24429,0.0,0.1429,0.32143,0.0,0.857,12,0,0,12,0,6,0,0,6,0,0,4,0,0,1,0,0,1,1,0,1,0,0],[125,125,1.0,0.22768,0.24448,0.0,0.1429,0.28571,0.0,0.85714,12,0,0,12,0,5,0,0,8,0,0,3,0,0,0,0,0,3,0,0,1,0,0]]},{"b":1,"e":0.14,"k":"falling","v":0.24552,"x":0.96875,"p":[[0,65,0.0,0.79462,0.147,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,6,0,8],[4,65,0.0615,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[8,65,0.1231,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[12,65,0.1846,0.84375,0.30589,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,24],[16,65,0.2462,0.86158,0.25629,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,0,0,23],[20,65,0.3077,0.86158,0.23006,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,21],[24,65,0.3692,0.87052,0.2458,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,23],[28,65,0.4308,0.81472,0.30246,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,1,1,20],[32,65,0.4923,0.88393,0.24856,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,25],[36,65,0.5538,0.76337,0.31667,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,2,0,16],[40,65,0.6154,0.83482,0.24513,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,0,3,0,19],[44,65,0.6769,0.79461,0.30918,0.67857,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,19],[48,65,0.7385,0.70536,0.3808,0.39285,1.0,1.0,0.0,1.0,3,18,0,3,0,3,0,1,1,0,0,2,0,0,2,0,0,0,1,0,1,0,18],[52,65,0.8,0.71875,0.32485,0.42857,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,3,0,0,5,0,0,0,0,0,2,1,0,3,1,14],[56,65,0.8615,0.62498,0.40366,0.25,0.78564,1.0,0.0,1.0,5,15,0,5,0,3,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,15],[60,65,0.9231,0.74107,0.33586,0.39286,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,5,0,0,1,0,0,0,0,0,5,0,0,0,0,18],[64,65,0.9846,0.279,0.24892,0.10714,0.2857,0.42857,0.0,1.0,8,1,0,8,0,6,0,1,7,0,0,4,0,0,3,0,0,2,0,0,0,0,1],[65,65,1.0,0.24552,0.19636,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,7,0,0,7,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"56c098ede0d7a70a","q":"Let $a, b, c$ be positive real numbers such that $a+b+c=4 \\sqrt[3]{a b c}$. Prove that  $$ 2(a b+b c+c a)+4 \\min \\left(a^{2}, b^{2}, c^{2}\\right) \\geq a^{2}+b^{2}+c^{2} $$","t":[{"b":5,"e":0.57143,"k":"falling","v":0.64732,"x":0.98214,"p":[[0,20,0.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,20,0.2,0.92409,0.17494,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,25],[8,20,0.4,0.92857,0.18898,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[12,20,0.6,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,20,0.8,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[20,20,1.0,0.64732,0.16745,0.57143,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,15,0,0,7,0,0,3,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.93749,"x":1.0,"p":[[0,35,0.0,0.94643,0.12753,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[4,35,0.1143,0.93749,0.15545,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,27],[8,35,0.2286,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,35,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"32c74c90fa78323f","q":"14. (IND) Let $b, m, n$ be positive integers such that $b>1$ and $m \\neq n$. Prove that if $b^{m}-1$ and $b^{n}-1$ have the same prime divisors, then $b+1$ is a power of 2 .","t":[{"b":3,"e":1.0,"k":"rising","v":0.29909,"x":0.81695,"p":[[0,29,0.0,0.29909,0.33188,0.0,0.14286,0.46418,0.0,1.0,9,4,0,9,0,10,0,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,4],[4,29,0.1379,0.5357,0.32733,0.2857,0.42857,0.85714,0.0,1.0,1,7,0,1,0,6,0,0,5,0,0,5,0,0,2,0,0,4,0,0,2,0,7],[8,29,0.2759,0.5,0.32927,0.25,0.42857,0.71429,0.0,1.0,4,6,0,4,0,4,0,0,3,0,0,6,0,0,3,0,0,6,0,0,0,0,6],[12,29,0.4138,0.58481,0.32996,0.28571,0.71429,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,2,0,0,4,0,0,2,0,0,7,0,0,3,0,7],[16,29,0.5517,0.65175,0.2922,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,4,0,0,2,0,0,8,0,0,2,0,0,4,0,9],[20,29,0.6897,0.74105,0.28221,0.57132,0.78564,1.0,0.0,1.0,1,14,0,1,0,0,0,0,3,0,0,3,0,0,4,0,0,5,0,0,2,0,14],[24,29,0.8276,0.81695,0.19962,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,3,0,15],[28,29,0.9655,0.70981,0.21867,0.5354,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,7,0,0,6,0,7],[29,29,1.0,0.79462,0.18192,0.71429,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,5,0,11]]},{"b":6,"e":0.57143,"k":"rising","v":0.21875,"x":0.79462,"p":[[0,19,0.0,0.21875,0.27196,0.0,0.14286,0.28571,0.0,1.0,12,2,0,12,0,9,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[4,19,0.2105,0.71428,0.25505,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,6,0,0,2,0,0,6,0,0,6,0,9],[8,19,0.4211,0.64731,0.28791,0.42857,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,2,0,0,8,0,0,2,0,0,7,0,0,3,0,8],[12,19,0.6316,0.7723,0.23921,0.57132,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,5,0,0,3,0,14],[16,19,0.8421,0.76785,0.21354,0.67857,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,8,0,0,6,0,10],[19,19,1.0,0.79462,0.17475,0.67857,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,7,0,10]]}]},{"i":"e5854f1177b7023b","q":"8. C3 (MCD) Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. (a) Prove that Peter can always transfer all his money into two accounts. (b) Can Peter always transfer all his money into one account?","t":[{"b":2,"e":0.2857,"k":"flat","v":0.37054,"x":0.57143,"p":[[0,94,0.0,0.37054,0.12299,0.28571,0.35714,0.42858,0.0,0.57143,1,0,0,1,0,0,0,0,15,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[4,94,0.0426,0.54015,0.09931,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,18,0,0,4,0,0,0,0,0],[8,94,0.0851,0.52677,0.10971,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,10,0,0,16,0,0,4,0,0,0,0,0],[12,94,0.1277,0.49549,0.12865,0.42857,0.5712,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,8,0,0,17,0,0,2,0,0,0,0,0],[16,94,0.1702,0.53571,0.10101,0.4286,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,7,0,0,20,0,0,3,0,0,0,0,0],[20,94,0.2128,0.52677,0.09061,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,19,0,0,2,0,0,0,0,0],[24,94,0.2553,0.49106,0.12338,0.42857,0.5714,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,8,0,0,16,0,0,2,0,0,0,0,0],[28,94,0.2979,0.52677,0.09739,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,20,0,0,2,0,0,0,0,0],[32,94,0.3404,0.53561,0.10687,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,11,0,0,15,0,0,5,0,0,0,0,0],[36,94,0.383,0.51784,0.13716,0.42857,0.57141,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,11,0,0,12,0,0,6,0,0,0,0,0],[40,94,0.4255,0.53571,0.10714,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,11,0,0,15,0,0,5,0,0,0,0,0],[44,94,0.4681,0.51785,0.11151,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,9,0,0,17,0,0,3,0,0,0,0,0],[48,94,0.5106,0.50893,0.1234,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,7,0,0,17,0,0,3,0,0,0,0,0],[52,94,0.5532,0.53122,0.12491,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,9,0,0,14,0,0,6,0,0,0,0,0],[56,94,0.5957,0.54016,0.11143,0.53539,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,5,0,0,20,0,0,4,0,0,0,0,0],[60,94,0.6383,0.51338,0.11768,0.42857,0.57141,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,11,0,0,14,0,0,4,0,0,0,0,0],[64,94,0.6809,0.54908,0.12428,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,6,0,0,16,0,0,7,0,0,0,0,0],[68,94,0.7234,0.54024,0.12736,0.42859,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,8,0,0,14,0,0,7,0,0,0,0,0],[72,94,0.766,0.56248,0.12338,0.42859,0.57143,0.71407,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,7,0,0,14,0,0,9,0,0,0,0,0],[76,94,0.8085,0.54017,0.17399,0.42859,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,3,0,0,13,0,0,10,0,0,0,0,0],[80,94,0.8511,0.57143,0.14725,0.4286,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,5,0,0,10,0,0,13,0,0,0,0,0],[84,94,0.8936,0.51335,0.15091,0.42857,0.57141,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,8,0,0,15,0,0,5,0,0,0,0,0],[88,94,0.9362,0.52677,0.12078,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,12,0,0,12,0,0,6,0,0,0,0,0],[92,94,0.9787,0.51783,0.12241,0.42857,0.5712,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,11,0,0,13,0,0,5,0,0,0,0,0],[94,94,1.0,0.46875,0.1085,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,14,0,0,12,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.37052,"x":0.58034,"p":[[0,43,0.0,0.37052,0.14663,0.2857,0.28571,0.46418,0.0,0.71429,1,0,0,1,0,0,0,0,19,0,0,4,0,0,7,0,0,1,0,0,0,0,0],[4,43,0.093,0.53124,0.08917,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,20,0,0,2,0,0,0,0,0],[8,43,0.186,0.54461,0.11538,0.5713,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,2,0,0,22,0,0,4,0,0,0,0,0],[12,43,0.2791,0.54459,0.14479,0.571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,4,0,0,19,0,0,6,0,0,0,0,0],[16,43,0.3721,0.52674,0.1097,0.42857,0.5714,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,10,0,0,16,0,0,4,0,0,0,0,0],[20,43,0.4651,0.50892,0.12845,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,8,0,0,15,0,0,4,0,0,0,0,0],[24,43,0.5581,0.50895,0.12842,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,8,0,0,15,0,0,4,0,0,0,0,0],[28,43,0.6512,0.54909,0.11904,0.57132,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,5,0,0,20,0,0,5,0,0,0,0,0],[32,43,0.7442,0.53128,0.11421,0.42965,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,7,0,0,20,0,0,3,0,0,0,0,0],[36,43,0.8372,0.55802,0.13997,0.57132,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,6,0,0,19,0,0,5,0,0,1,0,0],[40,43,0.9302,0.58034,0.07936,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,0,6,0,0,0,0,0],[43,43,1.0,0.57588,0.10999,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,21,0,0,4,0,0,0,0,1]]}]},{"i":"1ac116f51c959cdf","q":"$N$  is a positive integer. Call all positive divisors of  $N$  which are different from  $1$  and  $N$  *beautiful divisors*.We call  $N$  a special number when it has at least  $2$  *beautiful divisors* and difference of any  $2$  *beautiful divisors* divides  $N$  as well. Find all special numbers.","t":[{"b":4,"e":0.2857,"k":"flat","v":0.37052,"x":0.50891,"p":[[0,67,0.0,0.37052,0.2078,0.28571,0.35714,0.42857,0.0,0.85714,3,0,0,3,0,3,0,0,10,0,0,9,0,0,3,0,0,3,0,0,1,0,0],[4,67,0.0597,0.5,0.16364,0.42857,0.4286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,12,0,0,5,0,0,9,0,0,0,0,0],[8,67,0.1194,0.46875,0.15663,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,11,0,0,9,0,0,3,0,0,1,0,0],[12,67,0.1791,0.43317,0.24078,0.28571,0.42857,0.57143,0.0,1.0,3,2,0,3,0,1,0,0,7,0,0,12,0,0,4,0,0,2,0,0,1,0,2],[16,67,0.2388,0.44643,0.20124,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,5,0,0,13,0,0,4,0,0,4,0,0,2,0,0],[20,67,0.2985,0.47319,0.18706,0.42857,0.42859,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,4,0,0,15,0,0,5,0,0,5,0,0,0,0,1],[24,67,0.3582,0.44195,0.20315,0.28571,0.42857,0.57111,0.0,1.0,1,1,0,1,0,1,0,0,9,0,0,12,0,0,5,0,0,1,0,0,2,0,1],[28,67,0.4179,0.49107,0.19541,0.39286,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,11,0,0,8,0,0,1,0,0,3,0,1],[32,67,0.4776,0.50891,0.2257,0.39286,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,12,0,0,5,0,0,2,0,0,2,0,3],[36,67,0.5373,0.41962,0.18875,0.2857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,12,0,0,7,0,0,6,0,0,2,0,0,2,0,0],[40,67,0.597,0.47767,0.17716,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,12,0,0,4,0,0,6,0,0,0,0,1],[44,67,0.6567,0.48657,0.16309,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,11,0,0,8,0,0,5,0,0,1,0,0],[48,67,0.7164,0.45089,0.19597,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,10,0,0,10,0,0,3,0,0,6,0,0,0,0,1],[52,67,0.7761,0.45087,0.13413,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,12,0,0,10,0,0,2,0,0,0,0,0],[56,67,0.8358,0.45981,0.15457,0.42857,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,16,0,0,4,0,0,4,0,0,1,0,0],[60,67,0.8955,0.41072,0.15465,0.28571,0.42857,0.46431,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,0,12,0,0,5,0,0,3,0,0,0,0,0],[64,67,0.9552,0.48215,0.14171,0.42857,0.42857,0.57141,0.28571,0.857,0,0,0,0,0,0,0,0,4,0,0,19,0,0,3,0,0,5,0,0,1,0,0],[67,67,1.0,0.4153,0.08245,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,21,0,0,4,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.375,"x":0.50444,"p":[[0,98,0.0,0.38838,0.18637,0.28571,0.42857,0.4642,0.0,0.85714,1,0,0,1,0,4,0,0,10,0,0,9,0,0,5,0,0,2,0,0,1,0,0],[4,98,0.0408,0.47758,0.19774,0.28571,0.42857,0.57143,0.14,1.0,0,1,0,0,0,1,0,0,8,0,0,13,0,0,3,0,0,4,0,0,2,0,1],[8,98,0.0816,0.44194,0.17624,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,13,0,0,6,0,0,0,0,0,3,0,0],[12,98,0.1224,0.42857,0.18211,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,6,0,0,10,0,0,6,0,0,5,0,0,0,0,0],[16,98,0.1633,0.46862,0.15679,0.42857,0.42857,0.57111,0.14,1.0,0,1,0,0,0,1,0,0,5,0,0,15,0,0,8,0,0,2,0,0,0,0,1],[20,98,0.2041,0.50444,0.20511,0.39286,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,8,0,0,12,0,0,5,0,0,3,0,0,2,0,2],[24,98,0.2449,0.46432,0.17856,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,15,0,0,6,0,0,2,0,0,1,0,1],[28,98,0.2857,0.46871,0.18975,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,1,0,0,9,0,0,11,0,0,5,0,0,4,0,0,1,0,1],[32,98,0.3265,0.47317,0.1836,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,13,0,0,2,0,0,6,0,0,2,0,0],[36,98,0.3673,0.46427,0.17856,0.39286,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,5,0,0,12,0,0,6,0,0,5,0,0,1,0,0],[40,98,0.4082,0.4598,0.1776,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,8,0,0,12,0,0,5,0,0,5,0,0,1,0,0],[44,98,0.449,0.49105,0.21409,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,9,0,0,4,0,0,6,0,0,2,0,1],[48,98,0.4898,0.44194,0.22118,0.28571,0.42857,0.4642,0.0,1.0,1,2,0,1,0,2,0,0,8,0,0,13,0,0,3,0,0,2,0,0,1,0,2],[52,98,0.5306,0.44194,0.17983,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,11,0,0,4,0,0,4,0,0,0,0,1],[56,98,0.5714,0.45985,0.19798,0.28571,0.42857,0.46525,0.14286,1.0,0,1,0,0,0,1,0,0,9,0,0,14,0,0,3,0,0,1,0,0,3,0,1],[60,98,0.6122,0.42855,0.20822,0.28571,0.42857,0.571,0.0,1.0,2,1,0,2,0,0,0,0,11,0,0,10,0,0,3,0,0,5,0,0,0,0,1],[64,98,0.6531,0.4598,0.17027,0.42857,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,3,0,0,2,0,0,19,0,0,3,0,0,3,0,0,2,0,0],[68,98,0.6939,0.4598,0.16261,0.39286,0.42857,0.4642,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,16,0,0,4,0,0,1,0,0,3,0,0],[72,98,0.7347,0.41527,0.16507,0.28571,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,19,0,0,1,0,0,2,0,0,0,0,1],[76,98,0.7755,0.44197,0.16506,0.28571,0.42857,0.4643,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,13,0,0,4,0,0,3,0,0,0,0,1],[80,98,0.8163,0.42856,0.15971,0.28571,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,10,0,0,15,0,0,1,0,0,4,0,0,1,0,0],[84,98,0.8571,0.39287,0.11293,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,18,0,0,1,0,0,0,0,0,1,0,0],[88,98,0.898,0.41516,0.10924,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,19,0,0,2,0,0,2,0,0,0,0,0],[92,98,0.9388,0.41968,0.14697,0.28571,0.42857,0.42895,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,15,0,0,5,0,0,1,0,0,1,0,0],[96,98,0.9796,0.41071,0.1171,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,16,0,0,5,0,0,1,0,0,0,0,0],[98,98,1.0,0.375,0.13716,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,12,0,0,15,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"1ed4124caf1e2fb7","q":"Having lost a game of checkers and my temper, I dash all the pieces to the ground but one. This last checker, which is perfectly circular in shape, remains completely on the board, and happens to cover equal areas of red and black squares. Prove that the center of this piece must lie on a boundary between two squares (or at a junction of four).","t":[{"b":4,"e":0.28571,"k":"rising","v":0.2634,"x":0.86161,"p":[[0,20,0.0,0.2634,0.14334,0.14286,0.28571,0.28571,0.0,0.71429,2,0,1,2,0,10,0,0,13,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[4,20,0.2,0.7098,0.34161,0.5354,0.85714,1.0,0.0,1.0,3,13,0,3,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,0,5,0,13],[8,20,0.4,0.74107,0.27993,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,4,0,0,6,0,12],[12,20,0.6,0.69643,0.29826,0.53571,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,4,0,0,1,0,0,1,0,0,8,0,0,6,0,9],[16,20,0.8,0.86161,0.19719,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,6,0,17],[20,20,1.0,0.76786,0.22799,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,8,0,9]]},{"b":6,"e":1.0,"k":"rising","v":0.2679,"x":0.88393,"p":[[0,36,0.0,0.2679,0.1547,0.14286,0.28571,0.42857,0.0,0.71429,3,0,2,3,0,9,0,0,11,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[4,36,0.1111,0.58927,0.33834,0.28571,0.57121,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,3,0,0,6,0,0,2,0,0,2,0,0,5,0,8],[8,36,0.2222,0.75451,0.27014,0.53607,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,6,0,0,1,0,15],[12,36,0.3333,0.74105,0.31429,0.42857,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,2,0,0,4,0,0,2,0,0,0,0,0,7,0,14],[16,36,0.4444,0.82589,0.29175,0.82143,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,5,0,19],[20,36,0.5556,0.86607,0.23673,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,6,0,20],[24,36,0.6667,0.88393,0.21558,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,4,0,22],[28,36,0.7778,0.78125,0.26483,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,6,0,0,4,0,15],[32,36,0.8889,0.86604,0.22854,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,19],[36,36,1.0,0.80802,0.18423,0.82132,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,18,0,6]]}]},{"i":"d4c305ffd88d1872","q":"When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out:\n(1) If two teams play against each other, there are no more team playing against both of them.\n(2) If two teams do not play against each other, then there is always exactly two other teams playing against them both.\nProve that all teams play the same number of games.","t":[{"b":2,"e":0.71429,"k":"rising","v":0.19196,"x":0.40179,"p":[[0,28,0.0,0.21426,0.2923,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,15,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,2],[4,28,0.1429,0.19196,0.14987,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,19,0,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,28,0.2857,0.24106,0.29109,0.14286,0.14286,0.14286,0.0,1.0,5,3,0,5,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,3],[12,28,0.4286,0.30357,0.33264,0.10714,0.14286,0.60714,0.0,1.0,8,2,0,8,0,13,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,2],[16,28,0.5714,0.23214,0.26184,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,19,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,1],[20,28,0.7143,0.27231,0.25593,0.14286,0.14286,0.32144,0.0,1.0,3,1,0,3,0,19,0,0,2,0,0,1,0,0,3,0,0,2,0,0,1,0,1],[24,28,0.8571,0.25,0.22304,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,19,0,0,3,0,0,1,0,0,4,0,0,0,0,0,2,0,0],[28,28,1.0,0.40179,0.26351,0.14286,0.42859,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,1,0,0,4,0,0,8,0,0,3,0,0,3,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.08929,"x":0.25446,"p":[[0,10,0.0,0.25446,0.28061,0.14286,0.14286,0.2857,0.0,1.0,6,2,0,6,0,17,0,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,2],[4,10,0.4,0.08929,0.11152,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,10,0.8,0.25,0.30929,0.10714,0.14286,0.17857,0.0,1.0,8,3,0,8,0,16,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[10,10,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dea199dd6d466e56","q":"Fredek runs a private hotel. He claims that whenever  $ n \\ge 3$  guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of  $ n$  is Fredek right? (Acquaintance is a symmetric relation.)","t":[{"b":1,"e":0.2857,"k":"rising","v":0.52676,"x":0.78124,"p":[[0,54,0.0,0.52676,0.17654,0.42859,0.57143,0.57143,0.0,1.0,2,1,2,2,0,0,0,0,0,0,0,7,0,0,21,0,0,0,0,0,1,0,1],[4,54,0.0741,0.7098,0.17308,0.57143,0.85707,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,12,0,0,1,0,0,17,0,0],[8,54,0.1481,0.63841,0.25992,0.42859,0.71429,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,6,0,0,5,0,0,3,0,0,12,0,2],[12,54,0.2222,0.66069,0.20748,0.57143,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,3,0,0,10,0,2],[16,54,0.2963,0.65174,0.19214,0.571,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,13,0,0,2,0,0,9,0,2],[20,54,0.3704,0.67859,0.19557,0.57132,0.64286,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,4,0,0,10,0,0,1,0,0,14,0,1],[24,54,0.4444,0.69193,0.21462,0.57132,0.78564,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,1,0,0,13,0,3],[28,54,0.5185,0.76785,0.20124,0.57143,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,2,0,0,13,0,7],[32,54,0.5926,0.76784,0.15466,0.57143,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,17,0,3],[36,54,0.6667,0.64266,0.23418,0.571,0.57143,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,0,0,0,1,0,0,15,0,0,1,0,0,10,0,2],[40,54,0.7407,0.66521,0.17712,0.5714,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,2,0,0,11,0,1],[44,54,0.8148,0.68745,0.20026,0.57132,0.64286,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,11,0,0,2,0,0,11,0,3],[48,54,0.8889,0.65175,0.2141,0.571,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,3,0,0,10,0,0,4,0,0,8,0,3],[52,54,0.963,0.78124,0.17853,0.67857,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,4,0,0,14,0,6],[54,54,1.0,0.71426,0.189,0.57143,0.64286,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,3,0,0,7,0,6]]},{"b":4,"e":0.42857,"k":"flat","v":0.33928,"x":0.68304,"p":[[0,29,0.0,0.47765,0.24899,0.42857,0.57143,0.57143,0.0,1.0,4,2,4,4,0,2,0,0,0,0,0,7,0,0,16,0,0,0,0,0,1,0,2],[4,29,0.1379,0.6607,0.20439,0.57143,0.57143,0.85714,0.0,1.0,1,1,1,1,0,0,0,0,1,0,0,1,0,0,16,0,0,0,0,0,12,0,1],[8,29,0.2759,0.6652,0.207,0.53575,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,0,9,0,0,1,0,0,12,0,2],[12,29,0.4138,0.65625,0.21975,0.42859,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,4,0,0,13,0,1],[16,29,0.5517,0.68304,0.20434,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,2,0,0,13,0,2],[20,29,0.6897,0.63387,0.19542,0.42859,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,0,4,0,0,11,0,0],[24,29,0.8276,0.49995,0.19231,0.42857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,14,0,0,8,0,0,1,0,0,3,0,1],[28,29,0.9655,0.41068,0.14169,0.28571,0.42857,0.4642,0.1429,0.85714,0,0,0,0,0,1,0,0,12,0,0,11,0,0,7,0,0,0,0,0,1,0,0],[29,29,1.0,0.33928,0.06916,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b1bd90b4f965af1c","q":"In a 2025 by 2025 grid, every cell initially contains a `1'. Every minute, we simultaneously replace the number in each cell with the sum of numbers in the cells that share an edge with it. (For example, after the first minute, the number 2 is written in each of the four\ncorner cells.)\nAfter 2025 minutes, we colour the board in checkerboard fashion, such that the top left corner is black. Find the difference between the sum of numbers in black cells and the sum of numbers in white cells.\n\n*Proposed by chorn*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.73661,"x":0.96875,"p":[[0,78,0.0,0.73661,0.32165,0.53571,0.85714,1.0,0.0,1.0,1,15,1,1,0,3,0,0,2,0,0,2,0,0,1,0,0,5,0,0,3,0,15],[4,78,0.0513,0.91071,0.23351,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,27],[8,78,0.1026,0.91518,0.17076,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,25],[12,78,0.1538,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[16,78,0.2051,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[20,78,0.2564,0.95089,0.1048,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[24,78,0.3077,0.93304,0.17852,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,27],[28,78,0.359,0.93304,0.17122,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,27],[32,78,0.4103,0.89286,0.15152,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,2,0,20],[36,78,0.4615,0.92411,0.17491,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,26],[40,78,0.5128,0.89286,0.17128,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,0,0,22],[44,78,0.5641,0.91071,0.16656,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0,0,24],[48,78,0.6154,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[52,78,0.6667,0.86161,0.27078,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,23],[56,78,0.7179,0.83929,0.27837,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,1,0,21],[60,78,0.7692,0.83482,0.25281,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,5,0,0,0,0,21],[64,78,0.8205,0.77232,0.29636,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,5,0,0,0,0,18],[68,78,0.8718,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[72,78,0.9231,0.80357,0.17768,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,0,0,13],[76,78,0.9744,0.82143,0.17496,0.71429,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,0,0,15],[78,78,1.0,0.88393,0.1729,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,21]]},{"b":6,"e":1.0,"k":"rising","v":0.76786,"x":1.0,"p":[[0,102,0.0,0.76786,0.30252,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,3,0,0,1,0,0,6,0,0,1,0,17],[4,102,0.0392,0.90625,0.20079,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,24],[8,102,0.0784,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[12,102,0.1176,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[16,102,0.1569,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[20,102,0.1961,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[24,102,0.2353,0.91518,0.13767,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,23],[28,102,0.2745,0.91518,0.14664,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,1,0,23],[32,102,0.3137,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[36,102,0.3529,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[40,102,0.3922,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[44,102,0.4314,0.92411,0.18205,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,26],[48,102,0.4706,0.91071,0.14617,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,22],[52,102,0.5098,0.90625,0.15815,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,23],[56,102,0.549,0.87054,0.23244,0.71429,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,23],[60,102,0.5882,0.91518,0.17807,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,25],[64,102,0.6275,0.79911,0.23381,0.71429,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,10,0,0,0,0,16],[68,102,0.6667,0.8125,0.20958,0.71429,0.85714,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,10,0,0,0,0,16],[72,102,0.7059,0.91964,0.19212,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,26],[76,102,0.7451,0.84375,0.19678,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,10,0,0,2,0,17],[80,102,0.7843,0.80804,0.23313,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,11,0,0,2,0,15],[84,102,0.8235,0.78122,0.23688,0.67857,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,9,0,0,0,0,15],[88,102,0.8627,0.89286,0.18211,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,1,0,22],[92,102,0.902,0.79018,0.26483,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0,8,0,0,2,0,16],[96,102,0.9412,0.85268,0.23551,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,5,0,0,1,0,21],[100,102,0.9804,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[102,102,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"c0432e8397a1ca62","q":"On the sides $AB$, $AC$, and $BC$ of triangle $ABC$, points $M$, $X$, and $Y$ are given such that $AX = MX$ and $BY = MY$. Let $K$ and $L$ be the midpoints of segments $AY$ and $BX$, respectively, and let $O$ be the center of the circumcircle of triangle $ABC$. If $O_{1}$ and $O_{2}$ are the points symmetric to point $O$ with respect to $K$ and $L$, respectively, prove that points $X$, $Y$, $O_{1}$, and $O_{2}$ lie on the same circle.\n\n(Marco 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 $P(x)$  with integer coefficients is given. For some positive integer  $n$  numbers  $P(0),P(1),\\dots,P(2^n+1)$  are all divisible by  $2^{2^n}$ . Prove that values of  $P(x)$  in all integer points are divisible by  $2^{2^n}$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.20527,"x":0.87946,"p":[[0,30,0.0,0.87946,0.18595,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,4,0,20],[4,30,0.1333,0.58034,0.19541,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,12,0,0,8,0,0,5,0,0,2,0,3],[8,30,0.2667,0.63393,0.16728,0.57143,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,15,0,0,7,0,0,2,0,3],[12,30,0.4,0.66072,0.24419,0.42859,0.57143,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,8,0,0,8,0,0,6,0,0,1,0,8],[16,30,0.5333,0.65178,0.25985,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,7,0,0,6,0,0,4,0,0,3,0,8],[20,30,0.6667,0.51787,0.23889,0.53575,0.57143,0.57143,0.0,1.0,3,2,0,3,0,2,0,0,1,0,0,2,0,0,18,0,0,4,0,0,0,0,2],[24,30,0.8,0.49552,0.2575,0.42857,0.57143,0.71429,0.0,1.0,5,1,0,5,0,0,0,0,1,0,0,8,0,0,8,0,0,8,0,0,1,0,1],[28,30,0.9333,0.37054,0.28316,0.10714,0.42857,0.57143,0.0,1.0,8,1,0,8,0,2,0,0,5,0,0,5,0,0,8,0,0,1,0,0,2,0,1],[30,30,1.0,0.20527,0.26713,0.0,0.0,0.32143,0.0,0.85714,17,0,0,17,0,3,0,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,0]]},{"b":4,"e":0.57143,"k":"falling","v":0.53125,"x":0.91518,"p":[[0,36,0.0,0.91518,0.17445,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,25],[4,36,0.1111,0.69197,0.22899,0.57143,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,5,0,0,10,0,0,6,0,0,1,0,9],[8,36,0.2222,0.74107,0.20341,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,5,0,0,1,0,11],[12,36,0.3333,0.6875,0.22428,0.57143,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,4,0,0,13,0,0,3,0,0,3,0,8],[16,36,0.4444,0.73661,0.20858,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,5,0,0,5,0,9],[20,36,0.5556,0.79464,0.2111,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,3,0,0,2,0,15],[24,36,0.6667,0.6607,0.1948,0.57143,0.57143,0.74996,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,6,0,0,3,0,5],[28,36,0.7778,0.66964,0.23538,0.57143,0.64286,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,4,0,0,10,0,0,7,0,0,2,0,7],[32,36,0.8889,0.66518,0.19103,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,4,0,0,10,0,0,10,0,0,3,0,4],[36,36,1.0,0.53125,0.11971,0.53571,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,22,0,0,1,0,0,1,0,0]]}]},{"i":"c30a2b6c50d14421","q":"Let  $(a_n)_{n \\geq 0}$  be the sequence of integers defined recursively by  $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$  for  $n \\geq 0.$  Find the common divisors of  $a_{1986}$  and  $a_{6891}.$","t":[{"b":1,"e":0.71429,"k":"rising","v":0.66518,"x":0.95982,"p":[[0,19,0.0,0.66518,0.32657,0.28571,0.57143,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,11,0,0,2,0,0,4,0,0,0,0,0,0,0,15],[4,19,0.2105,0.94196,0.16312,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,28],[8,19,0.4211,0.90622,0.16987,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,24],[12,19,0.6316,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[16,19,0.8421,0.90179,0.14914,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,22],[19,19,1.0,0.82588,0.16652,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,0,1,0,14]]},{"b":7,"e":0.57143,"k":"volatile","v":0.57143,"x":0.97321,"p":[[0,14,0.0,0.57143,0.30514,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,7,0,0,2,0,0,1,0,0,0,0,10],[4,14,0.2857,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[8,14,0.5714,0.90625,0.15815,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,23],[12,14,0.8571,0.83927,0.18816,0.67857,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,6,0,0,0,0,18],[14,14,1.0,0.7232,0.15127,0.57143,0.71429,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,14,0,0,3,0,5]]}]},{"i":"92f4e87a7703dc6c","q":"A four-digit number  $n=\\overline{a b c d}$ , where  $a, b, c$  and  $d$  are digits, with  $a \\neq 0$ , is said to be *guanaco* if the product  $\\overline{a b} \\times \\overline{c d}$  is a positive divisor of  $n$ . Find all guanaco numbers.","t":[{"b":3,"e":0.2857,"k":"falling","v":0.22321,"x":0.56696,"p":[[0,21,0.0,0.56696,0.34899,0.28571,0.42857,1.0,0.0,1.0,2,11,0,2,0,3,0,0,6,0,0,7,0,0,1,0,0,2,0,0,0,0,11],[4,21,0.1905,0.46875,0.30563,0.2857,0.42857,0.60714,0.0,1.0,3,5,0,3,0,4,0,0,3,0,0,13,0,0,1,0,0,1,0,0,2,0,5],[8,21,0.381,0.35713,0.23689,0.14286,0.42857,0.42858,0.0,1.0,4,2,0,4,0,5,0,0,5,0,0,13,0,0,3,0,0,0,0,0,0,0,2],[12,21,0.5714,0.22321,0.23403,0.0,0.21428,0.42857,0.0,1.0,13,1,0,13,0,3,0,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[16,21,0.7619,0.33473,0.17363,0.14286,0.35714,0.42857,0.0,0.71429,2,0,0,2,0,7,0,0,7,0,0,11,0,0,4,0,0,1,0,0,0,0,0],[20,21,0.9524,0.28569,0.18894,0.14286,0.28571,0.42858,0.0,0.57143,5,0,0,5,0,7,0,0,9,0,0,5,0,0,6,0,0,0,0,0,0,0,0],[21,21,1.0,0.2991,0.18334,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,6,0,0,6,0,0,11,0,0,4,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.12947,"x":0.40624,"p":[[0,102,0.0,0.40624,0.35374,0.14286,0.35714,0.60714,0.0,1.0,7,6,0,7,0,6,0,0,3,0,0,5,0,0,3,0,0,2,0,0,0,0,6],[4,102,0.0392,0.38839,0.30977,0.14286,0.28571,0.42857,0.0,1.0,4,3,0,4,0,6,0,0,9,0,0,6,0,0,0,0,0,0,0,0,4,0,3],[8,102,0.0784,0.31697,0.34392,0.0,0.14293,0.46429,0.0,1.0,9,5,0,9,0,8,0,0,6,0,0,1,0,0,2,0,0,1,0,0,0,0,5],[12,102,0.1176,0.29446,0.3051,0.14214,0.14288,0.28571,0.0,1.0,6,4,0,6,0,11,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,4],[16,102,0.1569,0.23652,0.22763,0.105,0.14288,0.28571,0.0,1.0,8,1,0,8,0,9,0,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[20,102,0.1961,0.299,0.27287,0.14286,0.28571,0.42857,0.0,1.0,6,3,0,6,0,8,0,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,3],[24,102,0.2353,0.29469,0.2473,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,4,0,0,8,0,0,7,0,0,3,0,0,0,0,0,1,0,1],[28,102,0.2745,0.21875,0.22299,0.0,0.14286,0.32143,0.0,1.0,11,1,0,11,0,6,0,0,7,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[32,102,0.3137,0.18749,0.20338,0.0,0.14286,0.32143,0.0,0.57143,15,0,0,15,0,3,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[36,102,0.3529,0.19195,0.22189,0.0,0.14286,0.2857,0.0,1.0,12,1,0,12,0,9,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[40,102,0.3922,0.15625,0.16888,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,8,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[44,102,0.4314,0.24999,0.23144,0.0,0.14288,0.42857,0.0,1.0,9,1,0,9,0,8,0,0,3,0,0,9,0,0,2,0,0,0,0,0,0,0,1],[48,102,0.4706,0.29016,0.32238,0.0,0.14286,0.42857,0.0,1.0,11,4,0,11,0,6,0,0,4,0,0,5,0,0,2,0,0,0,0,0,0,0,4],[52,102,0.5098,0.26786,0.22517,0.14286,0.2143,0.42857,0.0,1.0,6,1,0,6,0,10,0,0,5,0,0,8,0,0,1,0,0,1,0,0,0,0,1],[56,102,0.549,0.16955,0.17657,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,5,0,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[60,102,0.5882,0.17411,0.1461,0.0,0.14286,0.2857,0.0,0.57143,9,0,0,9,0,11,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,102,0.6275,0.19643,0.21053,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,5,0,0,7,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[68,102,0.6667,0.13384,0.18877,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,5,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[72,102,0.7059,0.16072,0.15872,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,10,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[76,102,0.7451,0.19643,0.24419,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,6,0,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,1],[80,102,0.7843,0.20533,0.1986,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,6,0,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[84,102,0.8235,0.20982,0.17122,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,5,0,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[88,102,0.8627,0.19187,0.1772,0.0,0.14286,0.32143,0.0,0.57143,11,0,0,11,0,8,0,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[92,102,0.902,0.27677,0.21108,0.14286,0.2857,0.42857,0.0,1.0,5,1,0,5,0,9,0,0,7,0,0,8,0,0,2,0,0,0,0,0,0,0,1],[96,102,0.9412,0.22322,0.22286,0.0,0.14286,0.32143,0.0,1.0,9,1,0,9,0,10,0,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,1],[100,102,0.9804,0.12947,0.17627,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,8,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[102,102,1.0,0.23661,0.17717,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,7,0,0,6,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"049b4e0e13e1c46a","q":"Let  $ A_1A_2 \\ldots A_n$  be a convex polygon,  $ n \\geq 4.$  Prove that  $ A_1A_2 \\ldots A_n$  is cyclic if and only if to each vertex  $ A_j$  one can assign a pair  $ (b_j, c_j)$  of real numbers,  $ j = 1, 2, \\ldots, n,$  so that  $ A_iA_j = b_jc_i - b_ic_j$  for all  $ i, j$  with  $ 1 \\leq i < j \\leq n.$","t":[{"b":5,"e":0.2857,"k":"falling","v":0.22769,"x":0.8348,"p":[[0,28,0.0,0.8348,0.09526,0.85714,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,26,0,2],[4,28,0.1429,0.74998,0.18559,0.57143,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,16,0,3],[8,28,0.2857,0.70971,0.19745,0.57143,0.71429,0.85714,0.14,1.0,0,1,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,7,0,0,14,0,1],[12,28,0.4286,0.56249,0.26472,0.28571,0.57143,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,6,0,0,1,0,0,10,0,0,3,0,0,8,0,1],[16,28,0.5714,0.43079,0.28596,0.1429,0.4285,0.57143,0.0,1.0,2,1,0,2,0,8,0,1,5,0,0,0,0,0,10,0,0,0,0,0,5,0,1],[20,28,0.7143,0.22769,0.19839,0.14286,0.1429,0.28571,0.0,0.85714,6,0,0,6,0,13,0,0,7,0,0,2,0,0,3,0,0,0,0,0,1,0,0],[24,28,0.8571,0.32575,0.22059,0.14286,0.28571,0.57143,0.0,0.85714,3,0,0,3,0,9,0,0,9,0,0,1,0,0,8,0,0,1,0,0,1,0,0],[28,28,1.0,0.26085,0.16442,0.14286,0.24785,0.28571,0.0,0.57143,2,0,0,2,0,13,0,1,9,0,0,2,0,0,5,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.64505,"x":0.81247,"p":[[0,37,0.0,0.79911,0.1063,0.82143,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,24,0,0],[4,37,0.1081,0.81247,0.11537,0.857,0.85714,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,25,0,1],[8,37,0.2162,0.72318,0.17107,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,6,0,0,13,0,2],[12,37,0.3243,0.71873,0.16165,0.57143,0.78564,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,5,0,0,16,0,0],[16,37,0.4324,0.7723,0.14224,0.71429,0.85714,0.85714,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,17,0,2],[20,37,0.5405,0.72766,0.16116,0.57143,0.78564,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,7,0,0,16,0,0],[24,37,0.6486,0.75446,0.12992,0.71429,0.78564,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,9,0,0,15,0,1],[28,37,0.7568,0.64505,0.19023,0.5713,0.57143,0.85714,0.21429,0.85714,0,0,0,0,0,0,0,1,2,0,0,3,0,0,11,0,0,4,0,0,11,0,0],[32,37,0.8649,0.78129,0.15135,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,20,0,2],[36,37,0.973,0.78569,0.12876,0.71429,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,22,0,0],[37,37,1.0,0.74997,0.14289,0.57143,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,19,0,0]]}]},{"i":"f8d48da9a6ce6a44","q":"Positive integers  $a,b,c,x,y,z$  satisfy:\r\n\r $a^2+b^2=c^2$ ,  $x^2+y^2=z^2$ \r\n\r\nand \r\n\r $|x-a| \\leq 1$  ,  $|y-b| \\leq 1$ .\r\n\r\nProve that sets  $\\{a,b\\}$  and  $\\{x,y\\}$  are equal.","t":[{"b":0,"e":0.14,"k":"flat","v":0.14733,"x":0.68301,"p":[[0,93,0.0,0.26321,0.30957,0.0,0.14288,0.32143,0.0,1.0,11,3,0,11,0,8,0,0,5,0,0,2,0,0,1,0,0,2,0,0,0,0,3],[4,93,0.043,0.68301,0.35127,0.42857,0.78571,1.0,0.0,1.0,2,15,0,2,0,3,0,0,2,0,0,4,0,0,2,0,0,3,0,0,1,0,15],[8,93,0.086,0.61159,0.40443,0.25,0.78564,1.0,0.0,1.0,5,14,0,5,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,0,2,0,14],[12,93,0.129,0.57589,0.40952,0.25,0.64286,1.0,0.0,1.0,7,12,0,7,0,1,0,0,5,0,0,1,0,0,2,0,0,1,0,0,3,0,12],[16,93,0.172,0.42409,0.33404,0.14286,0.42857,0.60714,0.0,1.0,6,5,0,6,0,4,0,0,5,0,0,6,0,0,3,0,0,2,0,0,1,0,5],[20,93,0.2151,0.45089,0.40738,0.0,0.35714,0.85714,0.0,1.0,10,7,0,10,0,4,0,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,7],[24,93,0.2581,0.55357,0.33834,0.35715,0.42859,1.0,0.0,1.0,1,10,0,1,0,7,0,0,0,0,0,10,0,0,3,0,0,1,0,0,0,0,10],[28,93,0.3011,0.5357,0.36246,0.2857,0.4998,0.89286,0.0,1.0,5,8,0,5,0,2,0,0,5,0,0,4,0,0,3,0,0,2,0,0,3,0,8],[32,93,0.3441,0.51337,0.31915,0.2857,0.57121,0.71429,0.0,1.0,3,6,0,3,0,4,0,0,4,0,0,4,0,0,7,0,0,3,0,0,1,0,6],[36,93,0.3871,0.54908,0.31361,0.28571,0.57121,0.74996,0.0,1.0,2,6,0,2,0,4,0,0,4,0,0,4,0,0,5,0,0,5,0,0,2,0,6],[40,93,0.4301,0.47764,0.33236,0.14286,0.42857,0.71429,0.0,1.0,3,6,0,3,0,7,0,0,3,0,0,4,0,0,6,0,0,2,0,0,1,0,6],[44,93,0.4731,0.45089,0.3644,0.14286,0.42857,0.78571,0.0,1.0,6,8,0,6,0,4,0,0,5,0,0,6,0,0,2,0,0,1,0,0,0,0,8],[48,93,0.5161,0.52229,0.31865,0.2857,0.571,0.74996,0.0,1.0,5,4,0,5,0,1,0,0,3,0,0,6,0,0,5,0,0,4,0,0,4,0,4],[52,93,0.5591,0.47319,0.28444,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,6,0,0,8,0,0,3,0,0,3,0,0,3,0,3],[56,93,0.6022,0.42409,0.3204,0.14286,0.42857,0.60714,0.0,1.0,6,3,0,6,0,4,0,0,4,0,0,6,0,0,4,0,0,2,0,0,3,0,3],[60,93,0.6452,0.44195,0.34136,0.14286,0.42857,0.71429,0.0,1.0,5,5,0,5,0,6,0,0,3,0,0,7,0,0,1,0,0,3,0,0,2,0,5],[64,93,0.6882,0.37947,0.27342,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,4,0,0,4,0,0,9,0,0,2,0,0,5,0,0,1,0,1],[68,93,0.7312,0.42411,0.25874,0.28571,0.42857,0.57143,0.0,1.0,2,3,0,2,0,3,0,0,10,0,0,8,0,0,4,0,0,1,0,0,1,0,3],[72,93,0.7742,0.35715,0.25754,0.14286,0.35714,0.4286,0.0,1.0,5,1,0,5,0,5,0,0,6,0,0,9,0,0,3,0,0,1,0,0,2,0,1],[76,93,0.8172,0.45093,0.31157,0.14289,0.42859,0.71429,0.0,1.0,3,4,0,3,0,6,0,0,5,0,0,6,0,0,3,0,0,3,0,0,2,0,4],[80,93,0.8602,0.36605,0.28999,0.14286,0.35714,0.571,0.0,1.0,7,1,0,7,0,4,0,0,5,0,0,7,0,0,3,0,0,2,0,0,3,0,1],[84,93,0.9032,0.39284,0.27663,0.24999,0.42857,0.57143,0.0,1.0,6,2,0,6,0,2,0,0,7,0,0,6,0,0,5,0,0,4,0,0,0,0,2],[88,93,0.9462,0.35265,0.24215,0.14286,0.42857,0.571,0.0,0.857,6,0,0,6,0,4,0,0,5,0,0,8,0,0,5,0,0,3,0,0,1,0,0],[92,93,0.9892,0.19643,0.23077,0.0,0.14286,0.32143,0.0,1.0,14,1,0,14,0,5,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[93,93,1.0,0.14733,0.16554,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,10,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.21417,"x":0.7857,"p":[[0,46,0.0,0.21417,0.23143,0.0,0.14286,0.28571,0.0,0.857,11,0,0,11,0,9,0,0,5,0,0,2,0,0,3,0,0,1,0,0,1,0,0],[4,46,0.087,0.47325,0.39356,0.10714,0.42857,1.0,0.0,1.0,8,9,0,8,0,3,0,0,3,0,0,5,0,0,2,0,0,1,0,0,1,0,9],[8,46,0.1739,0.61607,0.35792,0.28571,0.71429,1.0,0.0,1.0,3,11,0,3,0,3,0,0,4,0,0,2,0,0,3,0,0,4,0,0,2,0,11],[12,46,0.2609,0.78116,0.30109,0.53572,1.0,1.0,0.14,1.0,0,18,0,0,0,3,0,0,1,0,0,4,0,0,1,0,0,2,0,0,3,0,18],[16,46,0.3478,0.62051,0.34554,0.28571,0.78564,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,4,0,0,2,0,0,3,0,0,1,0,0,8,0,8],[20,46,0.4348,0.66514,0.3085,0.42857,0.64286,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,2,0,0,4,0,0,6,0,0,3,0,0,1,0,12],[24,46,0.5217,0.63838,0.3499,0.28571,0.71429,1.0,0.0,1.0,2,10,0,2,0,4,0,0,4,0,0,1,0,0,2,0,0,4,0,0,5,0,10],[28,46,0.6087,0.7857,0.25506,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,6,0,0,3,0,15],[32,46,0.6957,0.54899,0.30339,0.28571,0.49979,0.85714,0.0,1.0,2,4,0,2,0,3,0,0,4,0,0,7,0,0,3,0,0,3,0,0,6,0,4],[36,46,0.7826,0.39275,0.27671,0.25,0.28571,0.57111,0.0,1.0,4,2,0,4,0,4,0,0,9,0,0,6,0,0,3,0,0,2,0,0,2,0,2],[40,46,0.8696,0.38837,0.22931,0.25,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,8,0,0,11,0,0,4,0,0,2,0,0,6,0,0,0,0,1],[44,46,0.9565,0.3214,0.17124,0.24999,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,6,0,0,13,0,0,5,0,0,5,0,0,1,0,0,0,0,0],[46,46,1.0,0.33026,0.2408,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,8,0,0,6,0,0,2,0,0,2,0,0,1,0,1]]}]},{"i":"e862ad8eca01b5f2","q":"Let  $A',\\,B',\\,C'$  be points, in which excircles touch corresponding sides of triangle  $ABC$ . Circumcircles of triangles  $A'B'C,\\,AB'C',\\,A'BC'$  intersect a circumcircle of  $ABC$  in points   $C_1\\ne C,\\,A_1\\ne A,\\,B_1\\ne B$  respectively. Prove that a triangle  $A_1B_1C_1$  is similar to a triangle, formed by points, in which incircle of  $ABC$  touches its sides.","t":[{"b":3,"e":1.0,"k":"rising","v":0.41735,"x":0.70089,"p":[[0,48,0.0,0.41735,0.31551,0.14286,0.42857,0.60714,0.0,1.0,4,3,3,4,1,7,0,0,3,0,0,4,0,0,5,0,0,3,0,0,2,0,3],[4,48,0.0833,0.57589,0.25874,0.39286,0.71429,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,3,0,0,3,0,0,13,0,0,3,0,2],[8,48,0.1667,0.6741,0.18638,0.67857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,19,0,0,2,0,3],[12,48,0.25,0.61159,0.23483,0.57132,0.71429,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,4,0,0,1,0,0,5,0,0,15,0,0,4,0,1],[16,48,0.3333,0.58024,0.23418,0.53539,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,2,0,0,2,0,0,2,0,0,4,0,0,18,0,0,2,0,0],[20,48,0.4167,0.55353,0.24419,0.39286,0.64286,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,3,0,0,5,0,0,13,0,0,2,0,1],[24,48,0.5,0.70089,0.14445,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,24,0,0,2,0,2],[28,48,0.5833,0.66292,0.18579,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,1,0,0,6,0,0,18,1,0,3,0,1],[32,48,0.6667,0.63391,0.23941,0.57143,0.71429,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,0,0,0,1,0,0,4,0,0,19,0,0,2,0,2],[36,48,0.75,0.61159,0.26302,0.57132,0.71429,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,2,0,0,1,0,0,2,0,0,19,0,0,2,0,2],[40,48,0.8333,0.60708,0.23691,0.571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,0,0,0,3,0,0,8,0,0,13,0,0,1,0,3],[44,48,0.9167,0.60706,0.22886,0.57143,0.71429,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,1,0,0,1,0,0,6,0,0,18,0,0,0,0,2],[48,48,1.0,0.63835,0.15563,0.57132,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,20,0,0,0,0,1]]},{"b":7,"e":0.71429,"k":"rising","v":0.36602,"x":0.70535,"p":[[0,52,0.0,0.36602,0.32325,0.14286,0.2143,0.57143,0.0,1.0,5,4,3,5,0,11,0,0,3,0,0,1,0,0,6,0,0,2,0,0,0,0,4],[4,52,0.0769,0.70535,0.26229,0.71429,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,0,0,0,1,0,0,15,0,0,4,0,7],[8,52,0.1538,0.59372,0.28818,0.28571,0.71429,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,5,0,0,0,0,0,5,0,0,12,0,0,1,0,5],[12,52,0.2308,0.43301,0.32435,0.14286,0.35714,0.71429,0.0,1.0,6,3,0,6,0,3,0,0,7,0,0,3,0,0,2,0,0,6,0,0,2,0,3],[16,52,0.3077,0.44643,0.26905,0.2857,0.42857,0.71429,0.0,0.85714,3,0,0,3,0,4,0,0,8,0,0,3,0,0,1,0,0,11,0,0,2,0,0],[20,52,0.3846,0.47321,0.2889,0.28571,0.28571,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,12,0,0,2,0,0,2,0,0,5,0,0,3,0,3],[24,52,0.4615,0.58927,0.20439,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,6,0,0,3,0,0,14,0,0,1,0,2],[28,52,0.5385,0.5089,0.24727,0.28571,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,5,0,0,3,0,0,5,0,0,12,0,0,2,0,0],[32,52,0.6154,0.49106,0.27184,0.28571,0.64264,0.71429,0.0,0.85714,4,0,0,4,0,1,0,0,7,0,0,3,0,0,1,0,0,14,0,0,2,0,0],[36,52,0.6923,0.59375,0.21461,0.42859,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,3,0,0,4,0,0,15,0,0,1,0,2],[40,52,0.7692,0.54017,0.24152,0.28571,0.64286,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,5,0,0,3,0,0,4,0,0,14,0,0,1,0,1],[44,52,0.8462,0.65174,0.19542,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,0,0,0,6,0,0,18,0,0,2,0,2],[48,52,0.9231,0.58258,0.21878,0.48215,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,2,0,1,4,0,0,18,0,0,0,0,1],[52,52,1.0,0.63821,0.14714,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,22,0,0,0,0,0]]}]},{"i":"e4d4d39d323960aa","q":"Let  $ABC$  be a triangle with  $\\angle B = 30^{\\circ }$ . We consider the closed disks of radius  $\\frac{AC}3$ , centered in  $A$ ,  $B$ ,  $C$ . Does there exist an equilateral triangle with one vertex in each of the 3 disks?\n\n*Radu Gologan, Dan Schwarz*","t":[{"b":1,"e":0.14286,"k":"falling","v":0.25444,"x":0.64059,"p":[[0,55,0.0,0.64059,0.28319,0.57132,0.57143,0.85714,0.0,1.0,3,6,1,3,0,0,0,0,1,0,1,1,0,0,11,0,0,4,0,0,5,0,6],[4,55,0.0727,0.42854,0.21427,0.2857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,11,0,0,5,0,0,9,0,0,1,0,0,0,0,2],[8,55,0.1455,0.37265,0.2518,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,3,0,0,5,0,1,5,0,0,9,0,0,2,0,0,0,0,1],[12,55,0.2182,0.25444,0.1914,0.14286,0.2857,0.42857,0.0,0.57143,7,0,0,7,0,8,0,0,6,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[16,55,0.2909,0.33033,0.24071,0.14286,0.28571,0.4642,0.0,1.0,5,1,0,5,0,6,0,0,8,0,0,5,0,0,5,0,0,2,0,0,0,0,1],[20,55,0.3636,0.38616,0.17208,0.28571,0.42857,0.4286,0.0,0.85714,1,0,0,1,1,1,0,0,12,0,0,10,0,0,5,0,0,1,0,0,1,0,0],[24,55,0.4364,0.38833,0.16853,0.2857,0.35714,0.57141,0.14,0.71429,0,0,0,0,0,5,0,0,11,0,0,6,0,0,8,0,0,2,0,0,0,0,0],[28,55,0.5091,0.37944,0.20703,0.14286,0.42857,0.571,0.0,0.71429,2,0,0,2,0,7,0,0,4,0,0,10,0,0,5,0,0,4,0,0,0,0,0],[32,55,0.5818,0.39717,0.24157,0.2857,0.42857,0.57111,0.0,1.0,4,1,0,4,0,3,0,0,8,0,0,4,0,0,9,0,0,3,0,0,0,0,1],[36,55,0.6545,0.39727,0.14603,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,4,0,0,8,0,0,12,0,0,7,0,0,1,0,0,0,0,0],[40,55,0.7273,0.35713,0.19231,0.2857,0.28571,0.57111,0.0,0.71429,3,0,0,3,0,3,0,0,12,0,0,5,0,0,7,0,0,2,0,0,0,0,0],[44,55,0.8,0.36382,0.22682,0.24999,0.35714,0.5711,0.0,1.0,4,1,0,4,0,4,0,0,8,0,0,6,0,1,7,0,0,1,0,0,0,0,1],[48,55,0.8727,0.40624,0.20237,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,9,0,0,6,0,0,8,0,0,2,0,0,0,0,1],[52,55,0.9455,0.35713,0.15566,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,12,0,0,9,0,0,5,0,0,1,0,0,0,0,0],[55,55,1.0,0.41073,0.16654,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,11,0,0,5,0,0,12,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.14955,"x":0.58036,"p":[[0,64,0.0,0.58036,0.34011,0.39285,0.57143,0.85714,0.0,1.0,4,7,0,4,0,3,0,0,1,0,0,3,0,1,6,0,0,2,1,0,4,0,7],[4,64,0.0625,0.48658,0.32509,0.2857,0.4998,0.75,0.0,1.0,5,4,0,5,0,1,0,0,8,0,0,2,0,0,6,0,0,2,0,0,4,0,4],[8,64,0.125,0.433,0.30405,0.24999,0.42857,0.57143,0.0,1.0,6,3,0,6,0,2,0,0,5,0,0,5,0,0,8,0,0,1,0,0,2,0,3],[12,64,0.1875,0.49106,0.20183,0.28571,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,2,0,0,6,0,0,5,0,0,10,0,0,7,0,0,1,0,0],[16,64,0.25,0.47093,0.23202,0.28571,0.57121,0.57143,0.0,1.0,2,1,0,2,0,3,0,1,5,0,0,1,0,0,14,0,0,5,0,0,0,0,1],[20,64,0.3125,0.51335,0.23106,0.42857,0.57143,0.60714,0.0,1.0,2,1,0,2,0,3,0,0,2,0,0,3,0,0,14,0,0,6,0,0,1,0,1],[24,64,0.375,0.39954,0.24344,0.24999,0.5712,0.57143,0.0,0.71429,6,0,0,6,0,2,0,0,5,0,0,2,0,0,13,1,0,3,0,0,0,0,0],[28,64,0.4375,0.38838,0.23209,0.24999,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,3,0,0,5,0,0,6,0,0,9,0,0,4,0,0,0,0,0],[32,64,0.5,0.39281,0.23139,0.2857,0.5005,0.57141,0.0,0.71429,6,0,0,6,0,1,0,0,6,0,0,3,0,0,14,0,0,2,0,0,0,0,0],[36,64,0.5625,0.36381,0.26686,0.10717,0.28571,0.57143,0.0,0.85714,8,0,0,8,0,1,0,0,8,0,0,2,0,0,7,1,0,4,0,0,1,0,0],[40,64,0.625,0.34372,0.28538,0.0,0.42857,0.57143,0.0,1.0,10,1,0,10,0,3,0,0,2,0,0,3,0,0,11,0,0,2,0,0,0,0,1],[44,64,0.6875,0.40179,0.26592,0.24999,0.42859,0.57143,0.0,1.0,6,1,0,6,0,2,0,0,6,0,0,4,0,0,8,0,0,5,0,0,0,0,1],[48,64,0.75,0.42186,0.25149,0.28571,0.57143,0.57143,0.0,0.71429,7,0,0,7,0,0,0,0,3,0,1,3,0,0,13,0,0,5,0,0,0,0,0],[52,64,0.8125,0.4464,0.25441,0.2857,0.57141,0.57143,0.0,0.85714,5,0,0,5,0,2,0,0,4,0,0,2,0,0,12,0,0,6,0,0,1,0,0],[56,64,0.875,0.36159,0.25996,0.14286,0.35714,0.57143,0.0,0.71429,7,0,0,7,0,4,0,0,5,0,0,2,0,0,9,0,0,5,0,0,0,0,0],[60,64,0.9375,0.14955,0.26867,0.0,0.0,0.17857,0.0,0.85714,23,0,0,23,0,1,0,0,1,0,0,2,0,1,1,0,0,1,0,0,2,0,0],[64,64,1.0,0.20088,0.22546,0.0,0.14286,0.2857,0.0,0.71429,14,0,0,14,0,4,0,0,8,0,0,1,0,0,3,0,0,2,0,0,0,0,0]]}]},{"i":"a63f621fed624bf1","q":"Let  $ABCD$  be a regular tetrahedron. Find the positions of point  $P$  on the edge  $BD$  such that the edge  $CD$  is tangent to the sphere with diameter  $AP$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.13839,"x":0.1875,"p":[[0,46,0.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.1875,0.11538,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,26,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,46,0.5217,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.16964,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.13839,"x":0.16965,"p":[[0,67,0.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.16965,0.10972,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,67,0.2985,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.16964,0.07524,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.16965,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8c871e6572efceb0","q":"Let  $G$  be a simple graph with  $n$  vertices and  $m$  edges. Two vertices are called *neighbours* if there is an edge between them. It turns out the  $G$  does not contain any cycles of length from 3 to  $2k$  (inclusive), where  $k\\geq2$  is a given positive integer. \n\na) Prove that it is possible to pick a non-empty set  $S$  of vertices of  $G$  such that every vertex in  $S$  has at least  $\\left\\lceil \\frac mn \\right\\rceil$  neighbours that are in  $S$ . ( $\\lceil x\\rceil$  denotes the smallest integer larger than or equal to  $x$ .)\n\nb) Suppose a set  $S$  as described in (a) is chosen. Let  $H$  be the graph consisting of the vertices in  $S$  and the edges between those vertices only. Let  $v$  be a vertex of  $H$ . Prove that at least  $\\left\\lceil \\left(\\frac mn -1\\right)^k \\right\\rceil$  vertices of  $H$  can be reached by starting at  $v$  and travelling across the edges of  $H$  for at most  $k$  steps. (Note that  $v$  itself satisfies this condition, since it can be reached by starting at  $v$  and travelling along the edges of  $H$  for 0 steps.)","t":[{"b":2,"e":0.85714,"k":"flat","v":0.63838,"x":0.79462,"p":[[0,25,0.0,0.63838,0.19557,0.5354,0.57143,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,4,0,0,4,0,4],[4,25,0.16,0.73212,0.16657,0.57143,0.71429,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,3,0,0,9,0,5],[8,25,0.32,0.7589,0.1871,0.57143,0.85707,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,3,0,0,10,0,7],[12,25,0.48,0.66072,0.17035,0.57143,0.57143,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,19,0,0,0,0,0,7,0,3],[16,25,0.64,0.67854,0.15154,0.57143,0.57143,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,3,0,0,6,0,3],[20,25,0.8,0.73659,0.17897,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,2,0,0,9,0,6],[24,25,0.96,0.79462,0.16346,0.57143,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,14,0,7],[25,25,1.0,0.69641,0.15872,0.57143,0.57143,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,9,0,3]]},{"b":6,"e":0.85714,"k":"flat","v":0.65622,"x":0.71872,"p":[[0,14,0.0,0.66514,0.23855,0.5354,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,6,0,0,13,0,0,0,0,0,2,0,9],[4,14,0.2857,0.71872,0.16937,0.57143,0.64286,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,0,10,0,4],[8,14,0.5714,0.65622,0.14224,0.57143,0.57143,0.85704,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,1,0,0,8,0,1],[12,14,0.8571,0.71429,0.16366,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,7,0,0,9,0,3],[14,14,1.0,0.70087,0.18338,0.57143,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,3,0,0,7,0,5]]}]},{"i":"56a4a2cd317783b5","q":"Let  $N$  a positive integer.\n\nIn a spaceship there are  $2 \\cdot N$  people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows:\n\n1) The first alien chooses any person as she wishes.\n\n2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn.\n\n3) The alien that can't play in her turn loses.\n\nProve that second player has a winning strategy *if, and only if*, the  $2 \\cdot N$  people can be divided in  $N$  pairs in such a way that two people in the same pair are friends.","t":[{"b":4,"e":1.0,"k":"rising","v":0.5223,"x":1.0,"p":[[0,23,0.0,0.5223,0.26392,0.28571,0.42857,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,14,0,0,4,0,0,5,0,0,1,0,0,4,0,4],[4,23,0.1739,0.90177,0.20343,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,25],[8,23,0.3478,0.92409,0.15149,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,24],[12,23,0.5217,0.83927,0.26428,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,4,0,0,0,0,22],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[23,23,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":6,"e":1.0,"k":"rising","v":0.46423,"x":0.98661,"p":[[0,29,0.0,0.46423,0.25998,0.2857,0.28571,0.57141,0.0,1.0,1,4,0,1,0,0,0,0,17,0,0,1,0,0,6,0,0,3,0,0,0,0,4],[4,29,0.1379,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[8,29,0.2759,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[12,29,0.4138,0.93737,0.11837,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[16,29,0.5517,0.93302,0.16363,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,27],[20,29,0.6897,0.93303,0.1287,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[24,29,0.8276,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,29,0.9655,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[29,29,1.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29]]}]},{"i":"806946c824b0c230","q":"Let  $N \\geq 2$  be an integer, and let  $\\mathbf a$   $= (a_1, \\ldots, a_N)$  and  $\\mathbf b$   $= (b_1, \\ldots b_N)$  be sequences of non-negative integers. For each integer  $i \\not \\in \\{1, \\ldots, N\\}$ , let  $a_i = a_k$  and  $b_i = b_k$ , where  $k \\in \\{1, \\ldots, N\\}$  is the integer such that  $i-k$  is divisible by  $n$ . We say  $\\mathbf a$  is  $\\mathbf b$ -*harmonic* if each  $a_i$  equals the following arithmetic mean: \\[a_i = \\frac{1}{2b_i+1} \\sum_{s=-b_i}^{b_i} a_{i+s}.\\]\nSuppose that neither  $\\mathbf a $  nor  $\\mathbf b$  is a constant sequence, and that both  $\\mathbf a$  is  $\\mathbf b$ -*harmonic* and  $\\mathbf b$  is  $\\mathbf a$ -*harmonic*. \n   \nProve that at least  $N+1$  of the numbers  $a_1, \\ldots, a_N,b_1, \\ldots, b_N$  are zero.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.23661,"x":0.53116,"p":[[0,66,0.0,0.23661,0.16982,0.14286,0.14286,0.28571,0.0,0.85714,3,0,1,3,0,14,0,0,10,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[4,66,0.0606,0.41517,0.32411,0.14286,0.35714,0.71429,0.0,1.0,3,4,0,3,0,11,0,0,2,0,0,4,0,0,3,0,0,4,0,0,1,0,4],[8,66,0.1212,0.39729,0.26899,0.14286,0.35714,0.60714,0.0,1.0,2,1,0,2,0,10,0,0,4,0,0,4,0,0,4,0,0,6,0,0,1,0,1],[12,66,0.1818,0.53116,0.32788,0.14289,0.57143,0.74996,0.0,1.0,2,5,0,2,0,7,0,0,2,0,0,4,0,0,2,0,0,7,0,0,3,0,5],[16,66,0.2424,0.45089,0.34462,0.14286,0.28571,0.85714,0.0,1.0,3,5,0,3,0,7,0,0,8,0,0,3,0,0,0,0,0,2,0,0,4,0,5],[20,66,0.303,0.38822,0.32986,0.14286,0.2143,0.71429,0.0,1.0,3,3,0,3,0,13,0,0,4,0,0,1,0,0,0,0,0,6,0,0,2,0,3],[24,66,0.3636,0.41964,0.3387,0.14286,0.28571,0.71429,0.0,1.0,2,5,0,2,0,11,0,0,7,0,0,1,0,0,1,0,0,3,0,0,2,0,5],[28,66,0.4242,0.44642,0.32291,0.1429,0.28571,0.71429,0.0,1.0,2,5,0,2,0,7,0,0,9,0,0,2,0,0,3,0,0,2,0,0,2,0,5],[32,66,0.4848,0.40179,0.27301,0.14289,0.28571,0.46431,0.0,1.0,1,2,0,1,0,8,0,0,9,0,0,6,0,0,1,0,0,2,0,0,3,0,2],[36,66,0.5455,0.38838,0.25563,0.14286,0.28571,0.57111,0.0,1.0,1,2,0,1,0,8,0,0,9,0,0,5,0,0,5,0,0,0,0,0,2,0,2],[40,66,0.6061,0.48213,0.31693,0.14286,0.42857,0.75,0.14286,1.0,0,5,0,0,0,10,0,0,5,0,0,2,0,0,6,0,0,1,0,0,3,0,5],[44,66,0.6667,0.39735,0.25687,0.2857,0.28571,0.46525,0.0,1.0,2,2,0,2,0,5,0,0,10,0,0,7,0,0,3,0,0,1,0,0,2,0,2],[48,66,0.7273,0.38821,0.27035,0.25,0.2857,0.4286,0.14,1.0,0,3,0,0,0,8,0,0,14,0,0,3,0,0,1,0,0,1,0,0,2,0,3],[52,66,0.7879,0.29901,0.22973,0.14286,0.21428,0.32143,0.0,0.85714,1,0,0,1,0,15,0,0,8,0,0,3,0,0,1,0,0,1,0,0,3,0,0],[56,66,0.8485,0.33929,0.21053,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,9,0,0,11,0,0,5,0,0,1,0,0,4,0,0,1,0,0],[60,66,0.9091,0.40624,0.30116,0.14286,0.2857,0.60714,0.14286,1.0,0,4,0,0,0,14,0,0,4,0,0,3,0,0,3,0,0,4,0,0,0,0,4],[64,66,0.9697,0.29911,0.2212,0.14286,0.2857,0.32143,0.14286,1.0,0,2,0,0,0,15,0,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,2],[66,66,1.0,0.36159,0.2201,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,10,0,0,8,0,0,8,0,0,2,0,0,2,0,0,1,0,1]]},{"b":3,"e":1.0,"k":"flat","v":0.30356,"x":0.63834,"p":[[0,74,0.0,0.30356,0.22515,0.14286,0.28571,0.42857,0.0,1.0,3,1,3,3,0,10,0,0,9,0,0,5,0,0,3,0,0,0,0,0,1,0,1],[4,74,0.0541,0.50445,0.32534,0.25,0.42857,0.75,0.0,1.0,2,6,0,2,0,6,0,0,4,0,0,6,0,0,3,0,0,3,0,0,2,0,6],[8,74,0.1081,0.58033,0.30291,0.39286,0.4998,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,4,0,0,8,0,0,2,0,0,3,0,0,5,0,6],[12,74,0.1622,0.57585,0.31029,0.39286,0.57121,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,3,0,0,5,0,0,4,0,0,4,0,0,7,0,4],[16,74,0.2162,0.55357,0.29179,0.28571,0.57143,0.71429,0.0,1.0,2,4,0,2,0,3,0,0,4,0,0,4,0,0,5,0,0,7,0,0,3,0,4],[20,74,0.2703,0.63834,0.29012,0.42857,0.71414,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,0,0,0,6,0,0,5,0,0,5,0,0,6,0,6],[24,74,0.3243,0.49997,0.31743,0.24999,0.4998,0.71429,0.0,1.0,2,5,0,2,0,6,0,0,5,0,0,3,0,0,5,0,0,4,0,0,2,0,5],[28,74,0.3784,0.58033,0.29437,0.39286,0.64286,0.85704,0.0,1.0,1,4,0,1,0,5,0,0,2,0,0,4,0,0,4,0,0,7,0,0,5,0,4],[32,74,0.4324,0.5625,0.28557,0.28571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,5,0,0,4,0,0,6,0,0,3,0,0,5,0,0,5,0,4],[36,74,0.4865,0.47768,0.29582,0.25,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,5,0,0,8,0,0,1,0,0,3,0,0,3,0,4],[40,74,0.5405,0.48214,0.27141,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,9,0,0,6,0,0,1,0,0,6,0,0,2,0,3],[44,74,0.5946,0.47321,0.30606,0.2857,0.28571,0.71429,0.0,1.0,2,4,0,2,0,4,0,0,11,0,0,1,0,0,3,0,0,5,0,0,2,0,4],[48,74,0.6486,0.46868,0.23207,0.2857,0.42857,0.571,0.0,1.0,1,2,0,1,0,1,0,0,10,0,0,8,0,0,6,0,0,2,0,0,2,0,2],[52,74,0.7027,0.40177,0.25111,0.2857,0.35714,0.46418,0.0,1.0,2,1,0,2,0,5,0,0,9,0,0,8,0,0,2,0,0,2,0,0,3,0,1],[56,74,0.7568,0.44195,0.25344,0.2857,0.42857,0.60714,0.0,1.0,2,2,0,2,0,3,0,0,8,0,0,9,0,0,2,0,0,5,0,0,1,0,2],[60,74,0.8108,0.41953,0.26956,0.2857,0.35714,0.4642,0.0,1.0,1,3,0,1,0,6,0,0,9,0,0,8,0,0,2,0,0,1,0,0,2,0,3],[64,74,0.8649,0.49553,0.30926,0.2857,0.42857,0.71429,0.0,1.0,2,7,0,2,0,2,0,0,9,0,0,8,0,0,2,0,0,2,0,0,0,0,7],[68,74,0.9189,0.42856,0.25253,0.28571,0.28571,0.57111,0.0,1.0,2,2,0,2,0,1,0,0,14,0,0,6,0,0,2,0,0,3,0,0,2,0,2],[72,74,0.973,0.49104,0.25984,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,13,0,0,4,0,0,4,0,0,3,0,0,3,0,3],[74,74,1.0,0.43748,0.23128,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,17,0,0,9,0,0,1,0,0,0,0,0,2,0,3]]}]},{"i":"c019d275cacef0fd","q":"Let  $O$  be the centre of the incircle of  $\\triangle ABC$ . Points  $K,L$  are the intersection points of the circles circumscribed about triangles  $BOC,AOC$  respectively with the bisectors of the angles at  $A,B$  respectively  $(K,L\\not= O)$ . Also  $P$  is the midpoint of segment  $KL$ ,  $M$  is the reflection of  $O$  with respect to  $P$  and  $N$  is the reflection of  $O$  with respect to line  $KL$ . Prove that the points  $K,L,M$  and  $N$  lie on the same circle.","t":[{"b":0,"e":1.0,"k":"flat","v":0.68745,"x":0.85712,"p":[[0,38,0.0,0.68745,0.28892,0.42859,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,6,0,0,2,0,0,6,0,0,2,0,0,4,0,11],[4,38,0.1053,0.7857,0.28349,0.57132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,0,0,0,3,0,18],[8,38,0.2105,0.808,0.23858,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,2,0,0,3,0,17],[12,38,0.3158,0.73658,0.2769,0.53539,0.857,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,6,0,0,2,0,0,3,0,0,4,0,0,4,0,13],[16,38,0.4211,0.808,0.26153,0.57132,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,1,0,0,0,0,20],[20,38,0.5263,0.81248,0.26593,0.67857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,3,0,0,2,0,19],[24,38,0.6316,0.77674,0.2811,0.571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,4,0,0,0,0,18],[28,38,0.7368,0.85712,0.23959,0.85711,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,5,0,20],[32,38,0.8421,0.77456,0.26792,0.57142,0.9643,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,4,0,0,1,1,16],[36,38,0.9474,0.72768,0.30589,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,6,0,0,2,0,0,4,0,0,2,0,0,1,0,16],[38,38,1.0,0.70978,0.31439,0.42857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,5,0,0,5,0,0,3,0,0,1,0,0,2,0,15]]},{"b":6,"e":0.857,"k":"flat","v":0.63392,"x":0.86604,"p":[[0,70,0.0,0.63392,0.25985,0.42857,0.57143,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,4,0,0,7,0,0,5,0,0,5,0,0,3,0,7],[4,70,0.0571,0.77677,0.25739,0.57132,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,1,0,0,3,0,16],[8,70,0.1143,0.81246,0.24078,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,0,0,0,6,0,16],[12,70,0.1714,0.7232,0.27185,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,7,0,0,6,0,0,2,0,0,1,0,14],[16,70,0.2286,0.71875,0.28679,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,4,0,0,1,0,14],[20,70,0.2857,0.74104,0.26352,0.5713,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,2,0,0,0,0,15],[24,70,0.3429,0.77679,0.25985,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,2,0,0,4,0,15],[28,70,0.4,0.86604,0.21413,0.82132,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,3,0,21],[32,70,0.4571,0.67856,0.29014,0.42859,0.57143,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,7,0,0,7,0,0,2,0,0,1,0,12],[36,70,0.5143,0.7232,0.3071,0.42857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,3,0,0,6,0,0,2,0,0,1,0,0,3,0,15],[40,70,0.5714,0.66071,0.33264,0.42857,0.71429,1.0,0.0,1.0,3,12,0,3,0,0,0,0,3,0,0,6,0,0,2,0,0,4,0,0,2,0,12],[44,70,0.6286,0.70978,0.32829,0.39286,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,6,0,0,1,0,0,5,0,0,1,0,0,1,0,16],[48,70,0.6857,0.78124,0.27662,0.53539,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,4,0,0,3,0,0,0,0,0,4,0,17],[52,70,0.7429,0.79018,0.2766,0.67857,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,3,0,0,3,0,0,1,0,0,5,0,0,1,0,18],[56,70,0.8,0.70982,0.29555,0.42857,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,5,0,0,4,0,0,4,0,0,1,0,0,4,0,13],[60,70,0.8571,0.77232,0.28763,0.57142,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,2,0,0,1,0,18],[64,70,0.9143,0.70533,0.30502,0.4286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,5,0,0,5,0,0,4,0,0,1,0,0,1,0,15],[68,70,0.9714,0.69188,0.26275,0.571,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,0,1,0,0,12,0,0,2,0,0,0,0,12],[70,70,1.0,0.74548,0.26666,0.571,0.85707,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,1,0,0,3,0,14]]}]},{"i":"6743881837fbe2d4","q":"Let  $\\Gamma$  be a semicircle with diameter  $AB$ . On this diameter is selected a  point  $C$ , and on the semicircle are selected points  $D$  and  $E$  so that  $E$  lies between  $B$  and  $D$ . It turned out that  $\\angle ACD = \\angle ECB$ . The intersection point of the tangents to  $\\Gamma$  at points  $D$  and  $E$  is denoted by  $F$ . Prove that  $\\angle EFD=\\angle  ACD+ \\angle ECB$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.34821,"x":0.58929,"p":[[0,108,0.0,0.40178,0.26108,0.28571,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,1,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[4,108,0.037,0.40178,0.29546,0.2857,0.28571,0.28571,0.0,1.0,2,6,0,2,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[8,108,0.0741,0.41964,0.28108,0.2857,0.28571,0.28571,0.14286,1.0,0,6,0,0,0,1,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[12,108,0.1111,0.48652,0.35339,0.28571,0.28571,1.0,0.0,1.0,2,10,0,2,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[16,108,0.1481,0.34821,0.19541,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[20,108,0.1852,0.57589,0.33785,0.28571,0.28571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,17,0,0,2,0,0,0,0,0,1,0,0,0,0,12],[24,108,0.2222,0.49553,0.34253,0.2857,0.28571,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[28,108,0.2593,0.49108,0.32915,0.2857,0.28571,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,9],[32,108,0.2963,0.4375,0.28558,0.2857,0.28571,0.35714,0.14286,1.0,0,6,0,0,0,1,0,0,23,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[36,108,0.3333,0.3973,0.26662,0.2857,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,2,0,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[40,108,0.3704,0.41071,0.28739,0.2857,0.2857,0.28571,0.0,1.0,1,6,0,1,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[44,108,0.4074,0.41508,0.28212,0.2857,0.28571,0.28571,0.14,1.0,0,6,0,0,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[48,108,0.4444,0.46875,0.30771,0.2857,0.28571,0.57143,0.2857,1.0,0,8,0,0,0,0,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[52,108,0.4815,0.53125,0.33926,0.2857,0.28571,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[56,108,0.5185,0.4375,0.3008,0.2857,0.28571,0.32143,0.14286,1.0,0,7,0,0,0,2,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[60,108,0.5556,0.45089,0.29257,0.2857,0.28571,0.42858,0.2857,1.0,0,7,0,0,0,0,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,7],[64,108,0.5926,0.54463,0.33586,0.2857,0.28571,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,11],[68,108,0.6296,0.50893,0.32329,0.28571,0.28571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,21,0,0,1,0,0,0,0,0,0,0,0,1,0,9],[72,108,0.6667,0.43303,0.28456,0.28571,0.28571,0.32143,0.14286,1.0,0,6,0,0,0,1,0,0,23,0,0,1,0,0,0,0,0,1,0,0,0,0,6],[76,108,0.7037,0.48214,0.31492,0.2857,0.28571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[80,108,0.7407,0.47767,0.33045,0.2857,0.28571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[84,108,0.7778,0.50446,0.335,0.2857,0.28571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[88,108,0.8148,0.58929,0.36377,0.28571,0.28571,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[92,108,0.8519,0.46428,0.31744,0.2857,0.28571,0.67857,0.0,1.0,1,8,0,1,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,8],[96,108,0.8889,0.375,0.23623,0.2857,0.28571,0.28571,0.2857,1.0,0,4,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[100,108,0.9259,0.50892,0.33108,0.2857,0.28571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[104,108,0.963,0.40177,0.2635,0.2857,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,1,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[108,108,1.0,0.43303,0.27776,0.28571,0.28571,0.32143,0.2857,1.0,0,6,0,0,0,0,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,6]]},{"b":6,"e":0.0,"k":"falling","v":0.04902,"x":0.45526,"p":[[0,118,0.0,0.38839,0.26782,0.2857,0.28571,0.28571,0.0,1.0,1,5,1,1,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,118,0.0339,0.39731,0.29824,0.2857,0.2857,0.28571,0.0,1.0,2,6,0,2,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[8,118,0.0678,0.41964,0.27649,0.28571,0.28571,0.32144,0.14286,1.0,0,5,0,0,0,2,0,0,22,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[12,118,0.1017,0.40625,0.33333,0.2857,0.28571,0.42857,0.0,1.0,4,7,0,4,0,3,0,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,7],[16,118,0.1356,0.33482,0.25407,0.2857,0.2857,0.28571,0.0,1.0,2,3,0,2,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[20,118,0.1695,0.37053,0.32313,0.14286,0.28571,0.42857,0.0,1.0,4,6,0,4,0,6,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[24,118,0.2034,0.375,0.31492,0.2857,0.28571,0.28571,0.0,1.0,4,6,0,4,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[28,118,0.2373,0.28125,0.21274,0.25,0.2857,0.28571,0.0,1.0,4,2,0,4,0,4,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[32,118,0.2712,0.44643,0.36026,0.2857,0.28571,1.0,0.0,1.0,4,9,0,4,0,2,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[36,118,0.3051,0.37946,0.32065,0.24999,0.28571,0.32143,0.0,1.0,4,6,0,4,0,4,0,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,6],[40,118,0.339,0.32588,0.28398,0.1429,0.2857,0.28571,0.0,1.0,5,3,0,5,0,5,0,0,16,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[44,118,0.3729,0.24545,0.28849,0.0,0.14286,0.28571,0.0,1.0,10,3,0,10,0,8,0,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[48,118,0.4068,0.35268,0.3369,0.14286,0.28571,0.39286,0.0,1.0,7,5,0,7,0,5,0,0,12,0,0,0,0,0,0,0,0,3,0,0,0,0,5],[52,118,0.4407,0.20089,0.24448,0.0,0.14286,0.28571,0.0,1.0,12,2,0,12,0,6,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[56,118,0.4746,0.38393,0.33012,0.14289,0.28571,0.42858,0.0,1.0,5,6,0,5,0,4,0,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,6],[60,118,0.5085,0.45526,0.3633,0.14289,0.28571,0.89275,0.0,1.0,3,8,0,3,0,6,0,0,12,0,0,0,0,0,0,0,0,2,0,0,1,0,8],[64,118,0.5424,0.37053,0.36221,0.14286,0.2857,0.39286,0.0,1.0,7,7,0,7,0,5,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[68,118,0.5763,0.30358,0.32684,0.0,0.2857,0.32143,0.0,1.0,10,4,0,10,0,5,0,0,9,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[72,118,0.6102,0.34821,0.2922,0.14286,0.2857,0.28571,0.0,1.0,1,5,0,1,0,10,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[76,118,0.6441,0.33911,0.28751,0.14289,0.28571,0.28571,0.0,1.0,3,4,0,3,0,6,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[80,118,0.678,0.29018,0.27776,0.14286,0.2857,0.28571,0.0,1.0,5,3,0,5,0,9,0,0,13,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[84,118,0.7119,0.25884,0.27769,0.14214,0.14286,0.28571,0.0,1.0,7,3,0,7,0,10,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[88,118,0.7458,0.24098,0.25366,0.14214,0.1429,0.2857,0.0,1.0,7,2,0,7,0,10,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[92,118,0.7797,0.22768,0.25966,0.0,0.14286,0.28571,0.0,1.0,11,2,0,11,0,6,0,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[96,118,0.8136,0.19196,0.15406,0.10714,0.14288,0.28571,0.0,0.71429,8,0,0,8,0,9,0,0,13,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[100,118,0.8475,0.1741,0.18808,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,10,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[104,118,0.8814,0.2634,0.30116,0.0,0.21435,0.28571,0.0,1.0,9,4,0,9,0,7,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[108,118,0.9153,0.16964,0.24598,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[112,118,0.9492,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,118,0.9831,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[118,118,1.0,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"91e285620cbef96a","q":"In  $\\vartriangle AB$ C,  $AB = 3$ ,  $AC = 6,$  and  $D$  is drawn on  $BC$  such that  $AD$  is the angle bisector of  $\\angle BAC$ .  $D$  is reflected across  $AB$  to a point  $E$ , and suppose that  $AC$  and  $BE$  are parallel. Compute  $CE$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.75446,"x":1.0,"p":[[0,31,0.0,0.75446,0.20277,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,10,0,0,6,0,0,4,0,10],[4,31,0.129,0.79018,0.20198,0.67857,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,0,5,0,12],[8,31,0.2581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,31,0.3871,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,31,0.5161,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[31,31,1.0,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]},{"b":5,"e":0.4286,"k":"flat","v":0.67857,"x":0.91964,"p":[[0,35,0.0,0.80804,0.19102,0.71429,0.85714,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,9,0,0,7,0,11],[4,35,0.1143,0.67857,0.19885,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,1,0,0,11,0,0,7,0,0,6,0,4],[8,35,0.2286,0.74553,0.20743,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,8,0,0,8,0,7],[12,35,0.3429,0.86161,0.23551,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,6,0,20],[16,35,0.4571,0.90625,0.18766,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,25],[20,35,0.5714,0.91964,0.18877,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,26],[24,35,0.6857,0.76785,0.23076,0.57143,0.85707,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,0,0,0,3,0,14],[28,35,0.8,0.85714,0.23958,0.78571,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,0,0,0,2,0,22],[32,35,0.9143,0.91964,0.17105,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,0,0,26],[35,35,1.0,0.87946,0.19597,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,8,0,19]]}]},{"i":"57a8c77cd5478ae1","q":"Let  $\\mathbb{N}_0$  denote the set of non-negative integers. Determine all non-negative integers  $k$  for which there exists a function  $f: \\mathbb{N}_0 \\to \\mathbb{N}_0$  such that  $f(2024) = k$  and  $f(f(n)) \\leq f(n+1) - f(n)$  for all non-negative integers  $n$ .","t":[{"b":4,"e":1.0,"k":"rising","v":0.73214,"x":0.99554,"p":[[0,105,0.0,0.73214,0.25191,0.53571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,4,0,0,3,0,12],[4,105,0.0381,0.94643,0.16269,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[8,105,0.0762,0.9375,0.18536,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[12,105,0.1143,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,105,0.1524,0.89286,0.20203,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,24],[20,105,0.1905,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[24,105,0.2286,0.93304,0.17122,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,27],[28,105,0.2667,0.91072,0.18471,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,25],[32,105,0.3048,0.9375,0.16728,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,27],[36,105,0.3429,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[40,105,0.381,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[44,105,0.419,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,105,0.4571,0.95089,0.14111,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[52,105,0.4952,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[56,105,0.5333,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[60,105,0.5714,0.91518,0.17807,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,25],[64,105,0.6095,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[68,105,0.6476,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[72,105,0.6857,0.95089,0.14556,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[76,105,0.7238,0.88839,0.21939,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,24],[80,105,0.7619,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[84,105,0.8,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,105,0.8381,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[92,105,0.8762,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[96,105,0.9143,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[100,105,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,105,0.9905,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[105,105,1.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30]]},{"b":7,"e":0.28571,"k":"flat","v":0.625,"x":0.95536,"p":[[0,77,0.0,0.76785,0.25692,0.42859,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,1,0,0,4,0,15],[4,77,0.0519,0.94642,0.14177,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[8,77,0.1039,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[12,77,0.1558,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[16,77,0.2078,0.90625,0.19434,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,24],[20,77,0.2597,0.90179,0.23266,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,27],[24,77,0.3117,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[28,77,0.3636,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[32,77,0.4156,0.91964,0.20183,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,27],[36,77,0.4675,0.92857,0.14286,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,24],[40,77,0.5195,0.91964,0.17835,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,26],[44,77,0.5714,0.94196,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[48,77,0.6234,0.85267,0.28006,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,23],[52,77,0.6753,0.90625,0.18073,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0,24],[56,77,0.7273,0.89732,0.23483,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,25],[60,77,0.7792,0.85714,0.24484,0.85711,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,21],[64,77,0.8312,0.83036,0.28669,0.67857,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,23],[68,77,0.8831,0.86607,0.22851,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,1,0,23],[72,77,0.9351,0.79017,0.27891,0.53539,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,5,0,0,1,0,0,4,0,0,0,0,19],[76,77,0.987,0.625,0.29179,0.42857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,5,0,0,11,0,0,0,0,0,4,0,0,1,0,10],[77,77,1.0,0.64286,0.3093,0.42857,0.42857,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,14,0,0,0,0,0,1,0,0,0,0,13]]}]},{"i":"8f4b457d4b2683b6","q":"Let  $a,b,c,\\alpha,\\beta,\\gamma \\in\\mathbb{R}$  such as  $a^2+b^2+c^2 \\neq 0 \\neq \\alpha\\beta\\gamma$  and  $24^{\\alpha}\\neq 3^{\\beta} \\neq 2012^{\\gamma} \\neq 24^{\\alpha}$ . Prove that the equation \\[  a \\cdot 24^{\\alpha x}+b \\cdot 3^{\\beta x} + c \\cdot 2012^{\\gamma x}=0 \\] has at most two real solutions.","t":[{"b":2,"e":1.0,"k":"flat","v":0.88839,"x":0.97768,"p":[[0,21,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,21,0.1905,0.88839,0.18466,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,21],[8,21,0.381,0.88839,0.19799,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0,23],[12,21,0.5714,0.93304,0.13825,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,25],[16,21,0.7619,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,21,0.9524,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[21,21,1.0,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.82143,"x":0.9375,"p":[[0,33,0.0,0.92411,0.13355,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,22],[4,33,0.1212,0.91964,0.17105,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,24],[8,33,0.2424,0.89286,0.18898,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,20],[12,33,0.3636,0.875,0.21943,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,4,0,21],[16,33,0.4848,0.89285,0.20825,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,22],[20,33,0.6061,0.9375,0.15542,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,27],[24,33,0.7273,0.89737,0.19627,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,1,0,24],[28,33,0.8485,0.89286,0.19562,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,0,1,0,23],[32,33,0.9697,0.82143,0.23419,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,6,0,0,0,0,19],[33,33,1.0,0.90179,0.1729,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,1,0,23]]}]},{"i":"5b556edd85831fb3","q":"Let  $f(x) = x^2 + 2018x + 1$ . Let  $f_1(x)=f(x)$  and  $f_k(x)=f(f_{k-1}(x))$  for all  $k\\geqslant 2$ . Prove that for any positive integer  $n{}$ , the equation  $f_n(x)=0$  has at least two distinct\u00a0real\u00a0roots.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.07143,"x":0.375,"p":[[0,20,0.0,0.375,0.18472,0.2857,0.28571,0.57143,0.0,0.57143,1,0,0,1,0,6,0,0,11,0,0,0,0,0,14,0,0,0,0,0,0,0,0],[4,20,0.2,0.37054,0.23107,0.14286,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,11,0,0,3,0,0,0,0,0,15,0,0,0,0,0,1,0,0],[8,20,0.4,0.37054,0.20782,0.14286,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,11,0,0,4,0,0,0,0,0,16,0,0,0,0,0,0,0,0],[12,20,0.6,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.125,0.13243,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,20,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[20,20,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.26337,"x":0.375,"p":[[0,13,0.0,0.375,0.20438,0.24999,0.35714,0.57143,0.0,0.57143,3,0,0,3,0,5,0,0,8,0,0,1,0,0,15,0,0,0,0,0,0,0,0],[4,13,0.3077,0.2857,0.21127,0.14286,0.2857,0.57143,0.0,0.57143,5,0,0,5,0,10,0,0,7,0,0,0,0,0,10,0,0,0,0,0,0,0,0],[8,13,0.6154,0.36159,0.17121,0.2857,0.28571,0.57143,0.14286,0.57143,0,0,0,0,0,7,0,0,13,0,0,0,0,0,12,0,0,0,0,0,0,0,0],[12,13,0.9231,0.26337,0.1601,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,9,0,0,15,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[13,13,1.0,0.29018,0.14054,0.24999,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,19,0,0,0,0,0,5,0,0,0,0,0,0,0,0]]}]},{"i":"89cc28f7d92703ca","q":"Let  $\\{a_n\\}$  be a sequence of integers satisfying  $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \\forall n\\ge 1$ . If  $2000|a_{1999}$ , find the smallest  $n\\ge 2$  such that  $2000|a_n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,199,0.0,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,27],[4,199,0.0201,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,199,0.0402,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,199,0.0603,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,199,0.0804,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,199,0.1005,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,199,0.1206,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,199,0.1407,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,199,0.1608,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,199,0.1809,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,199,0.201,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,199,0.2211,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[48,199,0.2412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,199,0.2613,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[56,199,0.2814,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,199,0.3015,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[64,199,0.3216,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,199,0.3417,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,199,0.3618,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,199,0.3819,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,199,0.402,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,199,0.4221,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,199,0.4422,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,199,0.4623,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[96,199,0.4824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,199,0.5025,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[104,199,0.5226,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,199,0.5427,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,199,0.5628,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[116,199,0.5829,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[120,199,0.603,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,199,0.6231,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[128,199,0.6432,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[132,199,0.6633,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[136,199,0.6834,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[140,199,0.7035,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[144,199,0.7236,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[148,199,0.7437,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[152,199,0.7638,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[156,199,0.7839,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[160,199,0.804,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[164,199,0.8241,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[168,199,0.8442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[172,199,0.8643,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[176,199,0.8844,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[180,199,0.9045,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[184,199,0.9246,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[188,199,0.9447,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[192,199,0.9648,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[196,199,0.9849,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[199,199,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.90179,"x":1.0,"p":[[0,109,0.0,0.90179,0.22711,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,25],[4,109,0.0367,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,109,0.0734,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,109,0.1101,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[16,109,0.1468,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,109,0.1835,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,109,0.2202,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,109,0.2569,0.98661,0.07457,1.0,1.0,1.0,0.5714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,109,0.2936,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,109,0.3303,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,109,0.367,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,109,0.4037,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[48,109,0.4404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,109,0.4771,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,109,0.5138,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[60,109,0.5505,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[64,109,0.5872,0.95089,0.15815,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[68,109,0.6239,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[72,109,0.6606,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[76,109,0.6972,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,109,0.7339,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,109,0.7706,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,109,0.8073,0.95527,0.15792,1.0,1.0,1.0,0.14,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[92,109,0.844,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[96,109,0.8807,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[100,109,0.9174,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,109,0.9541,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,109,0.9908,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[109,109,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"fac034062a3d7940","q":"Let  $m$  positive integers  $a_1, \\dots , a_m$  be given. Prove that there exist fewer than  $2^m$  positive integers  $b_1, \\dots , b_n$  such that all sums of distinct  $b_k$ \u2019s are distinct and all  $a_i \\  (i \\leq m)$  occur among them.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.05357,"x":0.45982,"p":[[0,55,0.0,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,1,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.45982,0.17399,0.39286,0.42857,0.46431,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,16,0,0,2,0,0,5,0,0,0,0,1],[8,55,0.1455,0.39284,0.12875,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,18,0,0,3,0,0,1,0,0,0,0,0],[12,55,0.2182,0.42857,0.17496,0.39286,0.42857,0.42858,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,18,0,0,1,0,0,4,0,0,1,0,0],[16,55,0.2909,0.4375,0.19541,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,12,0,0,10,0,0,4,0,0,2,0,0,3,0,0],[20,55,0.3636,0.44196,0.25344,0.2857,0.42857,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,8,0,0,9,0,0,3,0,0,2,0,0,4,0,1],[24,55,0.4364,0.41071,0.1915,0.28571,0.42857,0.42858,0.0,1.0,1,1,0,1,0,3,0,0,7,0,0,14,0,0,5,0,0,0,0,0,1,0,1],[28,55,0.5091,0.40179,0.22428,0.28571,0.42857,0.42858,0.0,1.0,2,1,0,2,0,3,0,0,9,0,0,11,0,0,3,0,0,1,0,0,2,0,1],[32,55,0.5818,0.39732,0.17399,0.28571,0.35714,0.42858,0.14286,0.85714,0,0,0,0,0,3,0,0,13,0,0,9,0,0,3,0,0,3,0,0,1,0,0],[36,55,0.6545,0.39284,0.1956,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,4,0,0,8,0,0,14,0,0,2,0,0,0,0,0,3,0,0],[40,55,0.7273,0.40625,0.23987,0.28571,0.42857,0.42857,0.0,1.0,2,1,0,2,0,3,0,0,10,0,0,11,0,0,0,0,0,2,0,0,3,0,1],[44,55,0.8,0.4107,0.26425,0.2857,0.35714,0.46429,0.0,1.0,3,1,0,3,0,3,0,0,10,0,0,8,0,0,1,0,0,2,0,0,4,0,1],[48,55,0.8727,0.36161,0.18552,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,11,0,0,12,0,0,1,0,0,0,0,0,1,0,1],[52,55,0.9455,0.44196,0.19352,0.28571,0.42857,0.46429,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,11,0,0,3,0,0,1,0,0,4,0,0],[55,55,1.0,0.37053,0.15093,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,12,0,0,15,0,0,1,0,0,0,0,0,0,0,1]]},{"b":2,"e":0.42857,"k":"rising","v":0.09813,"x":0.47767,"p":[[0,21,0.0,0.09813,0.11535,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.47767,0.17716,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,16,0,0,6,0,0,1,0,0,2,0,1],[8,21,0.381,0.39731,0.18116,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,5,0,0,9,0,0,11,0,0,3,0,0,3,0,0,1,0,0],[12,21,0.5714,0.38839,0.17215,0.28571,0.35714,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,14,0,0,13,0,0,0,0,0,0,0,0,3,0,0],[16,21,0.7619,0.36161,0.08737,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,14,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[20,21,0.9524,0.375,0.09943,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,10,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[21,21,1.0,0.38393,0.10971,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,7,0,0,19,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"f96ca58b7309b430","q":"Let  $n \\geqslant 2$  be an integer. There are  $n$  finite sets  ${A_1},{A_2},\\ldots,{A_n}$  which satisfy the condition \n\\[\\left| {{A_i}\\Delta {A_j}} \\right| = \\left| {i - j} \\right| \\quad \\forall i,j \\in \\left\\{ {1,2,...,n} \\right\\}.\\]\nFind the minimum of  $\\sum\\limits_{i = 1}^n {\\left| {{A_i}} \\right|} $ .","t":[{"b":2,"e":0.2857,"k":"falling","v":0.30804,"x":0.66076,"p":[[0,107,0.0,0.55354,0.22516,0.42857,0.57121,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,10,0,0,7,0,0,4,0,0,7,0,0],[4,107,0.0374,0.62499,0.23352,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,9,0,0,8,0,0,2,0,0,8,0,3],[8,107,0.0748,0.64732,0.20511,0.42857,0.57143,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,4,0,0,8,0,3],[12,107,0.1121,0.66076,0.21646,0.42965,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,4,0,0,4,0,6],[16,107,0.1495,0.60268,0.18808,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,10,0,0,11,0,0,4,0,0,3,0,3],[20,107,0.1869,0.58929,0.20124,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,13,0,0,5,0,0,5,0,0,5,0,2],[24,107,0.2243,0.54464,0.23538,0.42857,0.42857,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,13,0,0,6,0,0,3,0,0,3,0,3],[28,107,0.2617,0.52232,0.21011,0.42857,0.5,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,10,0,0,6,0,0,6,0,0,4,0,0],[32,107,0.2991,0.54464,0.19377,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,9,0,0,9,0,0,5,0,0,3,0,1],[36,107,0.3364,0.58928,0.23351,0.42857,0.57143,0.85704,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,12,0,0,6,0,0,3,0,0,7,0,2],[40,107,0.3738,0.60265,0.21939,0.42857,0.571,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,12,0,0,4,0,0,3,0,0,8,0,2],[44,107,0.4112,0.52679,0.24856,0.42857,0.42857,0.60714,0.0,1.0,2,4,0,2,0,0,0,0,3,0,0,13,0,0,6,0,0,3,0,0,1,0,4],[48,107,0.4486,0.63393,0.23128,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,7,0,0,4,0,0,7,0,0,7,0,3],[52,107,0.486,0.64286,0.29881,0.42857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,9,0,0,2,0,0,3,0,0,7,0,7],[56,107,0.5234,0.52677,0.24337,0.42857,0.49979,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,5,0,0,9,0,0,6,0,0,5,0,0,3,0,2],[60,107,0.5607,0.58481,0.20935,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,9,0,0,4,0,0,5,0,2],[64,107,0.5981,0.52679,0.21558,0.42857,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,9,0,0,9,0,0,2,0,0,6,0,0],[68,107,0.6355,0.56695,0.19393,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,10,0,0,8,0,0,4,0,0,5,0,1],[72,107,0.6729,0.46429,0.21724,0.28571,0.42859,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,5,0,0,8,0,0,10,0,0,2,0,0,3,0,0],[76,107,0.7103,0.52223,0.22207,0.42857,0.5,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,9,0,0,6,0,0,6,0,0,3,0,1],[80,107,0.7477,0.54017,0.26422,0.28571,0.42857,0.75,0.0,1.0,1,3,0,1,0,1,0,0,7,0,0,8,0,0,4,0,0,3,0,0,5,0,3],[84,107,0.785,0.57589,0.27545,0.42857,0.57143,0.85714,0.0,1.0,2,3,0,2,0,1,0,0,4,0,0,6,0,0,5,0,0,5,0,0,6,0,3],[88,107,0.8224,0.43303,0.20666,0.28571,0.42857,0.46429,0.0,1.0,1,1,0,1,0,1,0,0,11,0,0,11,0,0,2,0,0,4,0,0,1,0,1],[92,107,0.8598,0.5,0.2369,0.28571,0.42857,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,14,0,0,6,0,0,2,0,0,2,0,0,8,0,0],[96,107,0.8972,0.47766,0.18073,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,10,0,0,6,0,0,3,0,0,3,0,0],[100,107,0.9346,0.39732,0.18808,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,21,0,0,4,0,0,3,0,0,2,0,0,1,0,1],[104,107,0.972,0.31696,0.05906,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[107,107,1.0,0.30804,0.05187,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.51339,"x":0.71429,"p":[[0,123,0.0,0.51786,0.21354,0.42857,0.57141,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,3,0,0,10,0,0,10,0,0,2,0,0,5,0,0],[4,123,0.0325,0.64729,0.21125,0.571,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,5,0,0,8,0,0,8,0,0,7,0,2],[8,123,0.065,0.59821,0.23538,0.42857,0.57143,0.75,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,9,0,0,9,0,0,3,0,0,4,0,4],[12,123,0.0976,0.58036,0.23941,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,9,0,0,11,0,0,3,0,0,1,0,5],[16,123,0.1301,0.63839,0.19556,0.53571,0.57143,0.85704,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,7,0,0,10,0,0,5,0,0,7,0,2],[20,123,0.1626,0.53125,0.22084,0.42857,0.42857,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,1,0,0,14,0,0,9,0,0,1,0,0,2,0,3],[24,123,0.1951,0.60714,0.23419,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,14,0,0,4,0,0,3,0,0,7,0,3],[28,123,0.2276,0.61607,0.22428,0.42857,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,10,0,0,6,0,0,4,0,0,5,0,4],[32,123,0.2602,0.625,0.20124,0.42857,0.57143,0.85704,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,5,0,0,7,0,2],[36,123,0.2927,0.60713,0.22304,0.42857,0.57143,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,10,0,0,7,0,0,5,0,0,2,0,5],[40,123,0.3252,0.66518,0.19434,0.53571,0.64286,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,4,0,0,10,0,2],[44,123,0.3577,0.51339,0.21086,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,14,0,0,7,0,0,0,0,0,5,0,1],[48,123,0.3902,0.5625,0.22851,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,6,0,0,4,0,0,11,0,0,4,0,0,4,0,2],[52,123,0.4228,0.58036,0.23402,0.39286,0.57143,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,6,0,0,4,0,0,6,0,0,6,0,2],[56,123,0.4553,0.52232,0.21609,0.28571,0.42857,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,10,0,0,3,0,0,3,0,0,7,0,0],[60,123,0.4878,0.64284,0.25,0.42857,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,6,0,0,3,0,0,4,0,0,9,0,4],[64,123,0.5203,0.61606,0.17655,0.42857,0.57143,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,9,0,0,4,0,0,7,0,1],[68,123,0.5528,0.63839,0.19228,0.57143,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,3,0,0,15,0,0,7,0,0,3,0,3],[72,123,0.5854,0.70089,0.18336,0.57143,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,6,0,0,2,0,7],[76,123,0.6179,0.62054,0.12169,0.57143,0.57143,0.60714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,22,0,0,4,0,0,3,0,1],[80,123,0.6504,0.65624,0.15511,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,7,0,0,1,0,4],[84,123,0.6829,0.71429,0.17128,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,7,0,0,6,0,5],[88,123,0.7154,0.71429,0.19233,0.57143,0.64286,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,7,0,0,0,0,9],[92,123,0.748,0.65177,0.14699,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,7,0,0,2,0,3],[96,123,0.7805,0.71429,0.17496,0.57143,0.64286,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,5,0,0,5,0,6],[100,123,0.813,0.66518,0.16213,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,8,0,0,2,0,4],[104,123,0.8455,0.65179,0.17835,0.57143,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,6,0,0,1,0,5],[108,123,0.878,0.66518,0.17717,0.57143,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,6,0,0,2,0,5],[112,123,0.9106,0.68749,0.21261,0.57143,0.57143,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,17,0,0,5,0,0,2,0,7],[116,123,0.9431,0.61607,0.16536,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,19,0,0,1,0,0,3,0,3],[120,123,0.9756,0.66964,0.1729,0.57143,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,3,0,0,1,0,6],[123,123,1.0,0.67853,0.17499,0.57143,0.57143,0.75,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,6,0,0,3,0,5]]}]},{"i":"93c30f1d68f3a391","q":"24. C4 (GBR) Let $A$ be a set of $N$ residues $\\left(\\bmod N^{2}\\right)$. Prove that there exists a set $B$ of $N$ residues $\\left(\\bmod N^{2}\\right)$ such that the set $A+B=\\{a+b \\mid$ $a \\in A, b \\in B\\}$ contains at least half of all residues $\\left(\\bmod N^{2}\\right)$.","t":[{"b":0,"e":1.0,"k":"rising","v":0.80804,"x":1.0,"p":[[0,37,0.0,0.80804,0.37391,1.0,1.0,1.0,0.0,1.0,5,25,2,5,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,25],[4,37,0.1081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,37,0.2162,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,37,0.5405,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,37,0.6486,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,37,0.7568,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,37,0.8649,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,37,0.973,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[37,37,1.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,11,0.0,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,11,0.3636,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,11,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[11,11,1.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26]]}]},{"i":"5d7c67d1ba39e935","q":"Scalene triangle  $ABC$  satisfies  $\\angle A = 60^{\\circ}$ . Let the circumcenter of  $ABC$  be  $O$ , the orthocenter be  $H$ , and the incenter be  $I$ . Let  $D$ ,  $T$  be the points where line  $BC$  intersects the internal and external angle bisectors of  $\\angle A$ , respectively. Choose point  $X$  on the circumcircle of  $\\triangle IHO$  such that  $HX \\parallel AI$ . Prove that  $OD \\perp TX$ .","t":[{"b":4,"e":0.4286,"k":"falling","v":0.16071,"x":0.64286,"p":[[0,70,0.0,0.50451,0.33878,0.14286,0.42929,0.85714,0.0,1.0,2,4,0,2,0,7,0,0,6,0,0,2,0,0,2,0,0,2,0,0,7,0,4],[4,70,0.0571,0.64286,0.3312,0.39286,0.71429,1.0,0.0,1.0,2,10,0,2,0,2,0,0,4,0,0,4,0,0,2,0,0,4,0,0,4,0,10],[8,70,0.1143,0.39508,0.29503,0.14286,0.42857,0.57143,0.0,1.0,7,2,0,7,0,2,0,0,6,0,0,5,0,1,4,0,0,4,0,0,1,0,2],[12,70,0.1714,0.52231,0.36702,0.24999,0.4998,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,6,0,0,2,0,0,2,0,0,3,0,0,4,0,7],[16,70,0.2286,0.49095,0.37283,0.14286,0.571,0.85704,0.0,1.0,6,7,0,6,0,5,0,0,3,0,0,1,0,0,5,0,0,3,0,0,2,0,7],[20,70,0.2857,0.50893,0.37954,0.10714,0.57143,0.85714,0.0,1.0,8,5,0,8,0,3,0,0,0,0,0,3,0,0,5,0,0,1,0,0,7,0,5],[24,70,0.3429,0.60714,0.34442,0.42857,0.71429,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,2,0,0,6,0,0,1,0,0,6,0,0,4,0,8],[28,70,0.4,0.47768,0.38401,0.10714,0.42857,0.85714,0.0,1.0,8,5,0,8,0,4,0,0,1,0,0,5,0,0,0,0,0,3,0,0,6,0,5],[32,70,0.4571,0.33918,0.37076,0.0,0.14286,0.46418,0.0,1.0,10,6,0,10,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,0,1,0,6],[36,70,0.5143,0.33929,0.34022,0.0,0.28571,0.46431,0.0,1.0,10,3,0,10,0,5,0,0,4,0,0,5,0,0,1,0,0,1,0,0,3,0,3],[40,70,0.5714,0.36603,0.33678,0.0,0.35714,0.57111,0.0,1.0,9,3,0,9,0,6,0,0,1,0,0,5,0,0,4,0,0,2,0,0,2,0,3],[44,70,0.6286,0.35268,0.36767,0.0,0.14288,0.60712,0.0,1.0,9,6,0,9,0,8,0,0,3,0,0,3,0,0,1,0,0,2,0,0,0,0,6],[48,70,0.6857,0.32142,0.28346,0.14286,0.2857,0.42857,0.0,1.0,6,2,0,6,0,9,0,0,4,0,0,7,0,0,1,0,0,2,0,0,1,0,2],[52,70,0.7429,0.1875,0.26108,0.0,0.14286,0.17857,0.0,1.0,14,1,0,14,0,10,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,1],[56,70,0.8,0.22768,0.2854,0.0,0.14286,0.32143,0.0,1.0,13,2,0,13,0,7,0,0,4,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[60,70,0.8571,0.16071,0.19804,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[64,70,0.9143,0.29911,0.38525,0.0,0.14286,0.50002,0.0,1.0,14,6,0,14,0,7,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,6],[68,70,0.9714,0.17857,0.25505,0.0,0.14286,0.28571,0.0,1.0,15,2,0,15,0,7,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[70,70,1.0,0.29009,0.25129,0.14286,0.2857,0.28571,0.0,1.0,4,2,0,4,0,11,0,0,10,0,0,1,0,0,3,0,0,1,0,0,0,0,2]]},{"b":5,"e":0.42857,"k":"flat","v":0.37054,"x":0.68303,"p":[[0,67,0.0,0.44642,0.33264,0.14286,0.28571,0.75,0.0,1.0,2,3,1,2,0,11,0,0,4,0,0,1,0,0,3,0,0,3,0,0,5,0,3],[4,67,0.0597,0.63392,0.31122,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,2,0,0,4,0,0,3,0,0,6,0,0,4,0,8],[8,67,0.1194,0.61159,0.32387,0.42857,0.71429,0.85714,0.0,1.0,4,4,0,4,0,2,0,0,1,0,0,3,0,0,3,0,0,6,0,0,9,0,4],[12,67,0.1791,0.55802,0.39505,0.1429,0.57121,1.0,0.0,1.0,6,10,0,6,0,3,0,0,3,0,0,3,0,0,2,0,0,1,0,0,4,0,10],[16,67,0.2388,0.4955,0.35532,0.14289,0.49979,0.85714,0.0,1.0,5,6,0,5,0,4,0,0,5,0,0,2,0,0,4,0,0,3,0,0,3,0,6],[20,67,0.2985,0.68303,0.30875,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,2,0,0,4,0,0,3,0,0,1,0,0,6,0,0,5,0,10],[24,67,0.3582,0.61605,0.33964,0.39285,0.71429,0.89286,0.0,1.0,3,8,0,3,0,4,0,0,1,0,0,2,0,0,4,0,0,6,0,0,4,0,8],[28,67,0.4179,0.58482,0.32996,0.39286,0.64286,0.85714,0.0,1.0,5,4,0,5,0,1,0,0,2,0,0,2,0,0,6,0,0,4,0,0,8,0,4],[32,67,0.4776,0.58482,0.29743,0.39286,0.71429,0.75,0.0,1.0,1,5,0,1,0,5,0,0,2,0,0,4,0,0,3,0,0,9,0,0,3,0,5],[36,67,0.5373,0.48212,0.34763,0.14286,0.42859,0.74999,0.0,1.0,4,6,0,4,0,6,0,0,4,0,0,3,0,0,4,0,0,3,0,0,2,0,6],[40,67,0.597,0.59374,0.32948,0.28571,0.64286,1.0,0.0,1.0,2,9,0,2,0,3,0,0,4,0,0,5,0,0,2,0,0,6,0,0,1,0,9],[44,67,0.6567,0.63837,0.3009,0.42857,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,4,0,0,5,0,0,3,0,0,3,0,0,7,0,7],[48,67,0.7164,0.52228,0.32263,0.2857,0.571,0.75,0.0,1.0,3,6,0,3,0,3,0,0,6,0,0,3,0,0,6,0,0,3,0,0,2,0,6],[52,67,0.7761,0.4107,0.32092,0.14286,0.35714,0.71429,0.0,1.0,6,2,0,6,0,6,0,0,4,0,0,3,0,0,2,0,0,7,0,0,2,0,2],[56,67,0.8358,0.49554,0.36767,0.14286,0.57143,0.75,0.0,1.0,6,7,0,6,0,4,0,0,4,0,0,1,0,0,4,0,0,5,0,0,1,0,7],[60,67,0.8955,0.42402,0.33221,0.105,0.42857,0.71429,0.0,1.0,8,3,0,8,0,2,0,0,5,0,0,2,0,0,6,0,0,4,0,0,2,0,3],[64,67,0.9552,0.44196,0.35956,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,3,0,0,1,0,0,5,0,0,1,0,0,6,0,0,4,0,3],[67,67,1.0,0.37054,0.35689,0.0,0.28571,0.71429,0.0,1.0,12,2,0,12,0,2,0,0,4,0,0,1,0,0,2,0,0,6,0,0,3,0,2]]}]},{"i":"7f97d8374258ba30","q":"Let  $n$  be a fixed positive integer. Find the maximum possible value of \\[ \\sum_{1 \\le r < s \\le 2n} (s-r-n)x_rx_s, \\] where  $-1 \\le x_i \\le 1$  for all  $i = 1, \\cdots , 2n$ .","t":[{"b":4,"e":0.28571,"k":"falling","v":0.19643,"x":0.88393,"p":[[0,173,0.0,0.5,0.32341,0.28571,0.28571,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,17,0,0,0,0,0,2,0,0,1,0,0,1,0,8],[4,173,0.0231,0.88393,0.20652,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,5,0,21],[8,173,0.0462,0.80803,0.23313,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,3,0,0,9,0,13],[12,173,0.0694,0.79464,0.2765,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,6,0,16],[16,173,0.0925,0.81696,0.26302,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,3,0,0,4,0,18],[20,173,0.1156,0.8125,0.26592,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,2,0,0,6,0,17],[24,173,0.1387,0.70536,0.29437,0.42859,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,5,0,0,3,0,12],[28,173,0.1618,0.75892,0.25615,0.57132,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,1,0,0,6,0,13],[32,173,0.185,0.54911,0.3483,0.2857,0.35714,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,10,0,0,1,0,0,0,0,0,2,0,0,6,0,7],[36,173,0.2081,0.62946,0.31715,0.28571,0.64286,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,7,0,0,3,0,0,3,0,0,3,0,0,3,0,10],[40,173,0.2312,0.58927,0.30462,0.28571,0.49979,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,8,0,0,6,0,0,4,0,0,1,0,0,2,0,9],[44,173,0.2543,0.54911,0.36089,0.28571,0.42857,1.0,0.0,1.0,1,11,0,1,0,5,0,0,9,0,0,4,0,0,0,0,0,1,0,0,1,0,11],[48,173,0.2775,0.67411,0.29502,0.42857,0.64286,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,6,0,0,4,0,0,5,0,0,1,0,0,4,0,11],[52,173,0.3006,0.64731,0.30927,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,8,0,0,5,0,0,1,0,0,5,0,0,0,0,12],[56,173,0.3237,0.625,0.31894,0.28571,0.64286,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,9,0,0,4,0,0,1,0,0,1,0,0,6,0,9],[60,173,0.3468,0.55357,0.28959,0.39286,0.42859,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,4,0,0,9,0,0,6,0,0,0,0,0,2,0,7],[64,173,0.3699,0.62054,0.34369,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,6,0,0,4,0,0,1,0,0,1,0,0,4,0,11],[68,173,0.3931,0.61159,0.34485,0.28571,0.64286,1.0,0.0,1.0,2,9,0,2,0,3,0,0,5,0,0,4,0,0,2,0,0,1,0,0,6,0,9],[72,173,0.4162,0.57142,0.28794,0.28571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,6,0,0,3,0,0,7,0,0,2,0,0,5,0,5],[76,173,0.4393,0.56695,0.33973,0.28571,0.5712,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,6,0,0,2,0,0,3,0,0,3,0,0,2,0,9],[80,173,0.4624,0.58483,0.32802,0.28571,0.42857,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,11,0,0,5,0,0,1,0,0,0,0,0,3,0,10],[84,173,0.4855,0.38839,0.32387,0.14286,0.28571,0.50002,0.0,1.0,3,4,0,3,0,12,0,0,2,0,0,7,0,0,0,0,0,2,0,0,2,0,4],[88,173,0.5087,0.38393,0.27534,0.14286,0.28571,0.42858,0.0,1.0,2,2,0,2,0,8,0,0,8,0,0,7,0,0,1,0,0,1,0,0,3,0,2],[92,173,0.5318,0.39732,0.29175,0.14286,0.28571,0.57143,0.0,1.0,1,3,0,1,0,11,0,0,6,0,0,5,0,0,2,0,0,2,0,0,2,0,3],[96,173,0.5549,0.36161,0.27661,0.14286,0.28571,0.42857,0.0,1.0,1,4,0,1,0,10,0,0,10,0,0,6,0,0,0,0,0,1,0,0,0,0,4],[100,173,0.578,0.4017,0.34897,0.14286,0.28571,0.64286,0.0,1.0,3,6,0,3,0,11,0,0,6,0,0,3,0,0,1,0,0,0,0,0,2,0,6],[104,173,0.6012,0.41518,0.28652,0.14286,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,11,0,0,8,0,0,2,0,0,2,0,0,4,0,0,3,0,2],[108,173,0.6243,0.42857,0.26,0.28571,0.42857,0.42857,0.0,1.0,1,3,0,1,0,4,0,0,10,0,0,10,0,0,1,0,0,1,0,0,2,0,3],[112,173,0.6474,0.37053,0.24186,0.24999,0.28571,0.46429,0.0,0.85714,2,0,0,2,0,6,0,0,12,0,0,4,0,0,3,0,0,1,0,0,4,0,0],[116,173,0.6705,0.42857,0.26726,0.28571,0.35714,0.57143,0.0,1.0,1,2,0,1,0,6,0,0,9,0,0,6,0,0,3,0,0,2,0,0,3,0,2],[120,173,0.6936,0.4375,0.24206,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,7,0,0,7,0,0,7,0,0,2,0,0,1,0,2],[124,173,0.7168,0.51337,0.27399,0.28571,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,8,0,0,9,0,0,3,0,0,2,0,0,2,0,5],[128,173,0.7399,0.33928,0.17768,0.24999,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,8,0,0,13,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[132,173,0.763,0.47768,0.249,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,10,0,0,7,0,0,5,0,0,2,0,0,2,0,3],[136,173,0.7861,0.44196,0.24836,0.28571,0.42857,0.46429,0.14286,1.0,0,3,0,0,0,4,0,0,11,0,0,9,0,0,1,0,0,3,0,0,1,0,3],[140,173,0.8092,0.33036,0.19377,0.2857,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,5,0,0,16,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[144,173,0.8324,0.41072,0.27374,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,9,0,0,7,0,0,4,0,0,4,0,0,3,0,0,2,0,2],[148,173,0.8555,0.35268,0.1988,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,16,0,0,5,0,0,2,0,0,1,0,0,1,0,1],[152,173,0.8786,0.45982,0.28735,0.28571,0.35714,0.60714,0.14286,1.0,0,4,0,0,0,6,0,0,10,0,0,6,0,0,2,0,0,1,0,0,3,0,4],[156,173,0.9017,0.42857,0.22868,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,10,0,0,8,0,0,3,0,0,4,0,0,3,0,0],[160,173,0.9249,0.47765,0.22758,0.28571,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,12,0,0,5,0,0,4,0,0,4,0,0,5,0,0],[164,173,0.948,0.32589,0.22084,0.24999,0.28571,0.32143,0.0,1.0,2,2,0,2,0,6,0,0,16,0,0,3,0,0,3,0,0,0,0,0,0,0,2],[168,173,0.9711,0.36607,0.15542,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,4,0,0,13,0,0,11,0,0,2,0,0,1,0,0,1,0,0],[172,173,0.9942,0.24107,0.16146,0.14286,0.14286,0.32143,0.0,0.57143,3,0,0,3,0,15,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,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571,0.571,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,10,0,0,4,0,0,8,0,0,6,0,0,2,0,1],[176,187,0.9412,0.54464,0.27067,0.28571,0.42857,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,9,0,0,9,0,0,3,0,0,1,0,0,4,0,5],[180,187,0.9626,0.42857,0.20516,0.28571,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,16,0,0,6,0,0,4,0,0,2,0,0,2,0,1],[184,187,0.984,0.44643,0.18814,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,7,0,0,7,0,0,3,0,0,2,0,0],[187,187,1.0,0.37053,0.14223,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,15,0,0,7,0,0,6,0,0,1,0,0,0,0,0]]}]},{"i":"f097d5c2223f8bbc","q":"3. A3 (GRE) Let $a>2$ be given, and define recursively  $$ a_{0}=1, \\quad a_{1}=a, \\quad a_{n+1}=\\left(\\frac{a_{n}^{2}}{a_{n-1}^{2}}-2\\right) a_{n} $$  Show that for all $k \\in \\mathbb{N}$, we have  $$ \\frac{1}{a_{0}}+\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{k}}<\\frac{1}{2}\\left(2+a-\\sqrt{a^{2}-4}\\right) . $$","t":[{"b":6,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,41,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,41,0.0976,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,41,0.1951,0.91964,0.15542,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,25],[12,41,0.2927,0.93303,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[16,41,0.3902,0.94196,0.13767,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,27],[20,41,0.4878,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[24,41,0.5854,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,41,0.6829,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,41,0.878,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[41,41,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.80802,"x":0.96429,"p":[[0,35,0.0,0.95089,0.1411,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[4,35,0.1143,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[8,35,0.2286,0.89286,0.15567,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,20],[12,35,0.3429,0.90624,0.21611,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,25],[16,35,0.4571,0.96427,0.08754,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[20,35,0.5714,0.94643,0.13243,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[24,35,0.6857,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[28,35,0.8,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[32,35,0.9143,0.89286,0.18558,0.89286,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,24],[35,35,1.0,0.80802,0.20081,0.57143,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,3,0,15]]}]},{"i":"c9bb2e29ad63065c","q":"7. A7 (ARM) Let $f$ be a function from the set of real numbers $\\mathbb{R}$ into itself such that for all $x \\in \\mathbb{R}$, we have $|f(x)| \\leq 1$ and  $$ f\\left(x+\\frac{13}{42}\\right)+f(x)=f\\left(x+\\frac{1}{6}\\right)+f\\left(x+\\frac{1}{7}\\right) . $$  Prove that $f$ is a periodic function (that is, there exists a nonzero real number $c$ such that $f(x+c)=f(x)$ for all $x \\in \\mathbb{R})$.","t":[{"b":0,"e":1.0,"k":"rising","v":0.67857,"x":0.93973,"p":[[0,55,0.0,0.77232,0.2854,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,1,0,0,1,0,0,5,0,0,3,0,0,3,0,16],[4,55,0.0727,0.7857,0.27895,0.67857,0.85714,1.0,0.0,1.0,1,15,1,1,0,2,0,0,0,0,0,1,0,0,4,0,0,4,0,0,5,0,15],[8,55,0.1455,0.79464,0.25489,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,4,0,0,5,0,15],[12,55,0.2182,0.75,0.24223,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,4,0,0,8,0,0,4,0,0,2,0,13],[16,55,0.2909,0.67857,0.24223,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,9,0,0,7,0,0,1,0,0,7,0,7],[20,55,0.3636,0.82589,0.25187,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,0,5,0,17],[24,55,0.4364,0.73659,0.29039,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,4,0,0,5,0,0,2,0,0,2,0,15],[28,55,0.5091,0.70089,0.31412,0.42857,0.78571,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,0,0,0,4,0,0,4,0,0,3,0,0,3,0,13],[32,55,0.5818,0.75892,0.29545,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,0,0,0,1,0,0,6,0,0,3,0,0,2,0,16],[36,55,0.6545,0.76339,0.25407,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,1,0,0,6,0,13],[40,55,0.7273,0.79463,0.22571,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,2,0,0,5,0,14],[44,55,0.8,0.83482,0.2699,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,0,0,0,4,0,0,2,0,0,3,0,20],[48,55,0.8727,0.87946,0.24772,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,23],[52,55,0.9455,0.90847,0.20997,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,1,0,2,0,0,1,0,25],[55,55,1.0,0.93973,0.17045,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.78571,"x":0.90179,"p":[[0,58,0.0,0.80803,0.23585,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,2,0,0,3,0,17],[4,58,0.069,0.87499,0.18125,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,7,0,18],[8,58,0.1379,0.84821,0.18536,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,8,0,15],[12,58,0.2069,0.84821,0.18189,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,5,0,16],[16,58,0.2759,0.90179,0.21852,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,23],[20,58,0.3448,0.89286,0.21129,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,24],[24,58,0.4138,0.78571,0.22868,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,2,0,0,8,0,12],[28,58,0.4828,0.88839,0.18808,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,5,0,21],[32,58,0.5517,0.83482,0.20858,0.78571,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,0,0,0,9,0,15],[36,58,0.6207,0.87054,0.19019,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,17],[40,58,0.6897,0.84375,0.19352,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,4,0,17],[44,58,0.7586,0.89732,0.19638,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,22],[48,58,0.8276,0.8482,0.21412,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,3,0,19],[52,58,0.8966,0.87946,0.22619,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,23],[56,58,0.9655,0.89286,0.17857,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,21],[58,58,1.0,0.83482,0.26271,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,20]]}]},{"i":"6428b8fedd2b750e","q":"Quadrilateral $A P B Q$ is inscribed in circle $\\omega$ with $\\angle P=\\angle Q=90^{\\circ}$ and $A P=$ $A Q<B P$. Let $X$ be a variable point on segment $\\overline{P Q}$. Line $A X$ meets $\\omega$ again at $S$ (other than $A$ ). Point $T$ lies on arc $A Q B$ of $\\omega$ such that $\\overline{X T}$ is perpendicular to $\\overline{A X}$. Let $M$ denote the midpoint of chord $\\overline{S T}$. As $X$ varies on segment $\\overline{P Q}$, show that $M$ moves along a circle.","t":[{"b":3,"e":1.0,"k":"rising","v":0.48435,"x":0.98661,"p":[[0,91,0.0,0.48435,0.27124,0.28571,0.4643,0.60714,0.0,1.0,3,3,3,3,0,2,0,0,5,0,0,6,0,1,7,0,0,4,0,0,1,0,3],[4,91,0.044,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,91,0.0879,0.91071,0.24419,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,27],[12,91,0.1319,0.79909,0.28316,0.5354,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,0,2,0,19],[16,91,0.1758,0.83479,0.24774,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,0,4,0,19],[20,91,0.2198,0.74107,0.30606,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,5,0,0,0,0,0,2,0,0,3,0,16],[24,91,0.2637,0.71427,0.30515,0.42859,0.71429,1.0,0.0,1.0,1,15,0,1,0,0,0,0,5,0,0,3,0,0,4,0,0,4,0,0,0,0,15],[28,91,0.3077,0.76785,0.28065,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,4,0,0,0,0,0,4,0,0,3,0,16],[32,91,0.3516,0.72318,0.29869,0.4286,0.78571,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,6,0,0,5,0,0,2,0,0,1,0,15],[36,91,0.3956,0.8214,0.25756,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,4,0,18],[40,91,0.4396,0.90625,0.21609,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,26],[44,91,0.4835,0.82585,0.23623,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,0,3,0,18],[48,91,0.5275,0.7232,0.32915,0.4286,1.0,1.0,0.0,1.0,2,17,0,2,0,1,0,0,1,0,0,6,0,0,3,0,0,2,0,0,0,0,17],[52,91,0.5714,0.80802,0.24901,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,5,0,0,4,0,16],[56,91,0.6154,0.74552,0.24933,0.57143,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,7,0,0,1,0,13],[60,91,0.6593,0.77676,0.2574,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,4,0,14],[64,91,0.7033,0.6696,0.32231,0.39286,0.64286,1.0,0.0,1.0,1,13,0,1,0,1,0,0,6,0,0,3,0,0,5,0,0,1,0,0,2,0,13],[68,91,0.7473,0.7232,0.2922,0.5354,0.78571,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,4,0,0,4,0,0,4,0,0,3,0,13],[72,91,0.7912,0.76338,0.28929,0.4286,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,7,0,0,3,0,0,1,0,0,2,0,17],[76,91,0.8352,0.83927,0.25192,0.71429,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,2,0,0,3,0,20],[80,91,0.8791,0.81248,0.2156,0.67857,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,7,0,14],[84,91,0.9231,0.80802,0.25156,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,4,0,16],[88,91,0.967,0.70527,0.24732,0.571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,3,0,0,7,0,0,4,0,0,5,0,9],[91,91,1.0,0.67856,0.27199,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,9,0,0,3,0,0,5,0,0,1,0,11]]},{"b":7,"e":0.857,"k":"rising","v":0.375,"x":0.89286,"p":[[0,108,0.0,0.375,0.24679,0.14289,0.42857,0.57143,0.0,1.0,4,1,4,4,0,5,0,0,6,0,0,8,0,0,5,0,0,2,0,0,1,0,1],[4,108,0.037,0.88839,0.15866,0.82132,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,19],[8,108,0.0741,0.87499,0.24158,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,23],[12,108,0.1111,0.76783,0.33073,0.571,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,5,0,17],[16,108,0.1481,0.8839,0.19382,0.85711,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,4,0,21],[20,108,0.1852,0.82143,0.25754,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,3,0,0,1,0,20],[24,108,0.2222,0.86159,0.21574,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,4,0,20],[28,108,0.2593,0.85265,0.22445,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,0,0,21],[32,108,0.2963,0.83033,0.2434,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,5,0,18],[36,108,0.3333,0.89286,0.22016,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,3,0,24],[40,108,0.3704,0.75889,0.26832,0.571,0.85707,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,4,0,0,2,0,15],[44,108,0.4074,0.82142,0.23692,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,0,3,0,18],[48,108,0.4444,0.75441,0.26546,0.571,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,1,0,0,5,0,0,4,0,0,3,0,14],[52,108,0.4815,0.69192,0.2746,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,6,0,0,5,0,0,3,0,0,3,0,11],[56,108,0.5185,0.70531,0.2524,0.4286,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,5,0,0,2,0,11],[60,108,0.5556,0.70087,0.27284,0.42859,0.71414,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,11,0,0,2,0,0,3,0,0,1,0,13],[64,108,0.5926,0.66514,0.25657,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,4,0,0,6,0,7],[68,108,0.6296,0.71427,0.25254,0.5713,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,5,0,0,4,0,10],[72,108,0.6667,0.74998,0.25506,0.5354,0.85707,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,6,0,0,2,0,0,5,0,0,5,0,12],[76,108,0.7037,0.72316,0.22854,0.571,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,4,0,0,2,0,11],[80,108,0.7407,0.72321,0.3213,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,7,0,0,0,0,0,2,0,0,2,0,16],[84,108,0.7778,0.69195,0.29258,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,7,0,0,5,0,0,1,0,0,4,0,0,3,0,12],[88,108,0.8148,0.58921,0.23352,0.42857,0.571,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,10,0,0,7,0,0,7,0,0,0,0,5],[92,108,0.8519,0.62275,0.2836,0.42857,0.4998,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,11,0,0,1,0,0,2,1,0,5,0,7],[96,108,0.8889,0.71872,0.25875,0.4286,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,3,0,12],[100,108,0.9259,0.74996,0.24746,0.4286,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,2,0,0,6,0,12],[104,108,0.963,0.66515,0.22193,0.42857,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,7,0,0,4,0,6],[108,108,1.0,0.64285,0.24484,0.42859,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,8,0,0,7,0,0,3,0,0,5,0,6]]}]},{"i":"00a1635527f1fb44","q":"25. (KOR 3) Let $a, b$ be integers that are not perfect squares. Prove that if  $$ x^{2}-a y^{2}-b z^{2}+a b w^{2}=0 $$  has a nontrivial solution in integers, then so does  $$ x^{2}-a y^{2}-b z^{2}=0 $$","t":[{"b":0,"e":1.0,"k":"rising","v":0.66509,"x":1.0,"p":[[0,23,0.0,0.70981,0.36156,0.57132,0.85714,1.0,0.0,1.0,5,13,1,5,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,13],[4,23,0.1739,0.78125,0.37962,0.85714,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,21],[8,23,0.3478,0.66509,0.43548,0.14214,1.0,1.0,0.0,1.0,7,19,0,7,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,19],[12,23,0.5217,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,23,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.42857,"k":"flat","v":0.43301,"x":0.97321,"p":[[0,65,0.0,0.61159,0.40757,0.14286,0.78571,1.0,0.0,1.0,7,11,1,7,0,3,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,11],[4,65,0.0615,0.86607,0.30917,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[8,65,0.1231,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[12,65,0.1846,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[16,65,0.2462,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[20,65,0.3077,0.84821,0.31326,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,23],[24,65,0.3692,0.80357,0.3567,0.85714,1.0,1.0,0.0,1.0,3,21,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,21],[28,65,0.4308,0.85268,0.29984,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,23],[32,65,0.4923,0.89284,0.27201,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,26],[36,65,0.5538,0.87946,0.28818,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,25],[40,65,0.6154,0.78125,0.39121,0.89286,1.0,1.0,0.0,1.0,5,24,0,5,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,24],[44,65,0.6769,0.75893,0.38703,0.57143,1.0,1.0,0.0,1.0,5,21,0,5,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,21],[48,65,0.7385,0.88839,0.28512,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[52,65,0.8,0.56687,0.40962,0.14286,0.42857,1.0,0.0,1.0,3,14,0,3,0,7,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,14],[56,65,0.8615,0.62945,0.36746,0.24999,0.71429,1.0,0.0,1.0,2,12,0,2,0,6,0,0,1,0,0,3,0,0,4,0,0,0,0,0,4,0,12],[60,65,0.9231,0.56249,0.39113,0.14286,0.57121,1.0,0.0,1.0,3,11,0,3,0,7,0,0,4,0,0,1,0,0,2,0,0,1,0,0,3,0,11],[64,65,0.9846,0.43301,0.27076,0.25002,0.28571,0.60714,0.14286,1.0,0,2,0,0,0,8,0,0,9,0,0,4,0,0,3,0,0,3,0,0,3,0,2],[65,65,1.0,0.49996,0.2857,0.2857,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,6,0,0,4,0,0,3,0,0,6,0,0,3,0,3]]}]},{"i":"10997c496f732c49","q":"23. N6 (LAT) Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \\geq 2$, define $x_{n+1}=x_{n} x_{n-1}+1$. (a) Prove that for every $i>1$, there exists $j>i$ such that $x_{i}^{i}$ divides $x_{j}^{j}$. (b) Is it true that $x_{1}$ must divide $x_{j}^{j}$ for some $j>1$ ?","t":[{"b":5,"e":0.28571,"k":"flat","v":0.41509,"x":0.65177,"p":[[0,31,0.0,0.41509,0.19032,0.28571,0.42859,0.57143,0.0,0.71429,2,0,1,2,0,4,0,0,5,0,0,6,0,0,14,0,0,1,0,0,0,0,0],[4,31,0.129,0.63838,0.15966,0.57143,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,15,0,0,9,0,0,1,0,3],[8,31,0.2581,0.64057,0.19684,0.571,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,4,0,0,8,0,0,12,1,0,3,0,2],[12,31,0.3871,0.59158,0.243,0.42859,0.71429,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,1,0,0,5,0,0,5,0,0,10,1,0,5,0,1],[16,31,0.5161,0.65177,0.19784,0.57143,0.71429,0.71429,0.07143,1.0,0,2,0,0,1,1,0,0,0,0,0,3,0,0,7,1,0,13,0,0,4,0,2],[20,31,0.6452,0.58463,0.2535,0.42859,0.57143,0.74996,0.0,1.0,1,2,0,1,0,3,0,0,1,0,0,6,0,0,7,0,0,6,0,0,6,0,2],[24,31,0.7742,0.48647,0.31705,0.25,0.57141,0.71429,0.0,1.0,5,2,0,5,0,3,0,0,4,0,0,3,0,0,5,0,0,5,0,0,5,0,2],[28,31,0.9032,0.65177,0.18537,0.57143,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,13,0,0,2,0,3],[31,31,1.0,0.42646,0.16785,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,2,9,0,0,6,0,3,6,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.36159,"x":0.74107,"p":[[0,45,0.0,0.36159,0.20817,0.24999,0.42857,0.57143,0.0,0.57143,5,0,1,5,0,3,0,0,6,0,0,6,0,0,12,0,0,0,0,0,0,0,0],[4,45,0.0889,0.61158,0.16458,0.571,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,4,0,0,10,0,0,12,0,0,2,0,1],[8,45,0.1778,0.57145,0.24219,0.42965,0.57143,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,2,0,0,4,0,0,8,0,0,11,0,0,3,0,1],[12,45,0.2667,0.55356,0.18813,0.4286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,6,0,0,15,0,0,3,0,0,3,0,1],[16,45,0.3556,0.54018,0.20743,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,5,0,0,5,0,0,11,0,0,5,0,0,3,0,1],[20,45,0.4444,0.57369,0.19796,0.42857,0.57143,0.71429,0.14,1.0,0,1,0,0,0,1,0,0,3,0,0,8,0,0,9,0,0,5,1,0,4,0,1],[24,45,0.5333,0.53125,0.22934,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,7,0,0,9,0,0,6,0,0,3,0,1],[28,45,0.6222,0.58482,0.23786,0.42857,0.57143,0.85711,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,7,0,0,6,0,0,3,0,0,9,0,1],[32,45,0.7111,0.58038,0.22567,0.42859,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,4,0,0,11,0,0,4,0,0,5,0,2],[36,45,0.8,0.58702,0.16916,0.5354,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,1,5,0,0,14,0,0,5,0,0,5,0,0],[40,45,0.8889,0.66062,0.2546,0.5357,0.71429,0.85714,0.14,1.0,0,3,0,0,0,2,0,0,4,0,0,2,0,0,6,0,0,3,0,0,12,0,3],[44,45,0.9778,0.68514,0.18543,0.57143,0.71429,0.85714,0.1429,1.0,0,2,0,0,0,1,0,0,0,0,0,4,0,0,7,0,0,9,1,0,8,0,2],[45,45,1.0,0.74107,0.17382,0.625,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,5,1,0,8,1,0,10,0,4]]}]},{"i":"24d16b3e9d3f6a6a","q":"26. (GDR) ${ }^{\\text {IMO5 }}$ Let $a>0$ be a real number and $f(x)$ a real function defined on all of $\\mathbb{R}$, satisfying for all $x \\in \\mathbb{R}$,  $$ f(x+a)=\\frac{1}{2}+\\sqrt{f(x)-f(x)^{2}} . $$  (a) Prove that the function $f$ is periodic; i.e., there exists $b>0$ such that for all $x, f(x+b)=f(x)$. (b) Give an example of such a nonconstant function for $a=1$.  [^2]","t":[{"b":3,"e":0.28571,"k":"falling","v":0.43753,"x":0.96429,"p":[[0,24,0.0,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[4,24,0.1667,0.93304,0.19228,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[8,24,0.3333,0.95536,0.14032,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[12,24,0.5,0.8125,0.28221,0.64286,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,5,0,0,3,0,0,0,0,0,1,0,0,3,0,20],[16,24,0.6667,0.84375,0.26332,0.67857,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,23],[20,24,0.8333,0.46875,0.21793,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,5,0,0,5,0,0,4,0,0,1,0,2],[24,24,1.0,0.43753,0.18876,0.28571,0.42857,0.42893,0.2857,1.0,0,2,0,0,0,0,0,0,13,0,0,12,0,0,3,0,0,2,0,0,0,0,2]]},{"b":4,"e":0.57143,"k":"falling","v":0.44643,"x":0.99554,"p":[[0,39,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,39,0.1026,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[8,39,0.2051,0.89732,0.22934,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,25],[12,39,0.3077,0.86159,0.21276,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,2,0,21],[16,39,0.4103,0.91071,0.24157,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[20,39,0.5128,0.6339,0.3153,0.42857,0.57143,1.0,0.0,1.0,2,11,0,2,0,1,0,0,3,0,0,5,0,0,8,0,0,1,0,0,1,0,11],[24,39,0.6154,0.71427,0.26001,0.5713,0.57143,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,4,0,0,10,0,0,1,0,0,1,0,13],[28,39,0.7179,0.58928,0.29178,0.39293,0.57121,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,6,0,0,7,0,0,4,0,0,3,0,0,3,0,7],[32,39,0.8205,0.49107,0.24984,0.28571,0.42857,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,13,0,0,9,0,0,3,0,0,2,0,0,0,0,5],[36,39,0.9231,0.44643,0.13716,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,11,0,0,12,0,0,1,0,0,0,0,0],[39,39,1.0,0.50889,0.16725,0.42857,0.571,0.57143,0.143,1.0,0,1,0,0,0,1,0,0,4,0,0,10,0,0,13,0,0,2,0,0,1,0,1]]}]},{"i":"1843f96664f9c59f","q":"There are 9 black dots marked on a line. Monica chooses at least one of the black dots and erases the rest. Among the unerased dots, she then paints the leftmost one red and then pairs the remaining ones blue or green. If this procedure can be done in  $N$  different ways, what is the remainder when  $N$  is divided by 1000?","t":[{"b":0,"e":1.0,"k":"volatile","v":0.5,"x":1.0,"p":[[0,12,0.0,0.5,0.5,0.0,0.5,1.0,0.0,1.0,16,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,12,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.34375,"p":[[0,38,0.0,0.34375,0.47496,0.0,0.0,1.0,0.0,1.0,21,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b4332f50f7865181","q":"Let  $f$  be the function of the set of positive integers into itself, defi\fned by  $f(1) = 1$ , $f(2n) = f(n)$  and  $f(2n + 1) = f(n) + f(n + 1)$ . Show that, for any positive integer  $n$ , the\nnumber of positive odd integers m such that  $f(m) = n$  is equal to the number of positive\nintegers<span style=\"color:#0000FF\">**less or equal to**</span> $n$  and coprime to  $n$ .\n\n<span style=\"color:#FF0000\">[mod: the initial statement said less than  $n$ , which is wrong.]</span>","t":[{"b":2,"e":0.571,"k":"flat","v":0.72319,"x":0.89272,"p":[[0,26,0.0,0.79464,0.2549,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,6,0,0,3,0,0,2,0,17],[4,26,0.1538,0.78571,0.27433,0.57143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,1,0,0,0,0,19],[8,26,0.3077,0.72319,0.29439,0.571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,3,0,0,2,0,0,8,0,0,1,0,0,1,0,15],[12,26,0.4615,0.77229,0.28318,0.57132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,0,0,0,1,0,18],[16,26,0.6154,0.77231,0.24708,0.57143,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,4,0,0,0,0,16],[20,26,0.7692,0.85714,0.22016,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,3,0,0,1,0,21],[24,26,0.9231,0.85267,0.24869,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,21],[26,26,1.0,0.89272,0.20215,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,23]]},{"b":5,"e":1.0,"k":"rising","v":0.80354,"x":1.0,"p":[[0,16,0.0,0.80354,0.25193,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,1,0,0,2,0,18],[4,16,0.25,0.90625,0.17353,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,1,0,24],[8,16,0.5,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[12,16,0.75,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"221334c10c041642","q":"Prove that there are infinitely many pairs of positive integers $(x, y)$ such that\n\n$$\n\\frac{x+1}{y}+\\frac{y+1}{x}=4 .\n$$","t":[{"b":4,"e":1.0,"k":"rising","v":0.70536,"x":1.0,"p":[[0,16,0.0,0.82142,0.22589,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,9,0,0,1,0,17],[4,16,0.25,0.70536,0.31326,0.42857,0.78571,1.0,0.0,1.0,1,15,0,1,0,1,0,0,2,0,0,8,0,0,2,0,0,2,0,0,1,0,15],[8,16,0.5,0.82589,0.22513,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,2,0,18],[12,16,0.75,0.94197,0.15915,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"rising","v":0.75,"x":1.0,"p":[[0,23,0.0,0.75892,0.24074,0.42857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,7,0,0,1,0,14],[4,23,0.1739,0.75,0.22304,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,8,0,0,1,0,12],[8,23,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5cb87b03a1475b49","q":"A quadrilateral  $ABCD$  is inscribed in a circle. On each of the sides  $AB,BC,CD,DA$  one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to  $CD,DA,AB,BC,$  respectively. Prove that the centers of these four rectangles are vertices of a rectangle.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.11589,"x":0.19634,"p":[[0,75,0.0,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.14286,6,0,2,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.15607,0.05491,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,75,0.1067,0.18302,0.10244,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,27,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,75,0.16,0.15588,0.07456,0.14286,0.14286,0.14286,0.14,0.571,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,75,0.2133,0.17401,0.11141,0.14286,0.14286,0.14286,0.0,0.571,2,0,0,2,0,25,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,75,0.2667,0.15608,0.05491,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,75,0.32,0.19634,0.1777,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[28,75,0.3733,0.16491,0.10177,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,75,0.4267,0.16956,0.08331,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,28,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,75,0.48,0.16063,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.1784,0.07152,0.14286,0.14286,0.1429,0.14,0.4286,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.16054,0.09946,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,75,0.6933,0.16509,0.152,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,75,0.7467,0.1607,0.07777,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,75,0.8,0.16063,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,75,0.8533,0.16063,0.06918,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.18749,0.17652,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,26,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[72,75,0.96,0.18295,0.13942,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[75,75,1.0,0.1517,0.04974,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.12929,"x":0.1875,"p":[[0,74,0.0,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,74,0.0541,0.16948,0.08336,0.14286,0.14286,0.14287,0.0,0.4286,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,74,0.1081,0.165,0.11359,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,29,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,74,0.1622,0.16072,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,74,0.2162,0.165,0.08078,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,74,0.2703,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,74,0.3243,0.16071,0.05922,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,74,0.3784,0.15598,0.05493,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,74,0.4324,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.16054,0.05928,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,74,0.5405,0.18295,0.15666,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,74,0.5946,0.15179,0.04972,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,74,0.6486,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,74,0.7027,0.16956,0.07526,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,74,0.7568,0.14715,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.17847,0.14724,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[64,74,0.8649,0.1517,0.07937,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,74,0.9189,0.16509,0.10781,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,74,0.973,0.17411,0.11701,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,27,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[74,74,1.0,0.1875,0.1357,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"9658f65b93c39e3e","q":"Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive?    (4 points)\n                                                                                                                           \n                                                       Boris Frenkin","t":[{"b":1,"e":0.0,"k":"falling","v":0.125,"x":0.57589,"p":[[0,69,0.0,0.45982,0.37752,0.14286,0.42857,0.89286,0.0,1.0,4,8,2,4,0,10,0,0,1,0,0,5,0,0,2,0,0,0,0,0,2,0,8],[4,69,0.058,0.57143,0.3481,0.28571,0.5,1.0,0.0,1.0,1,9,0,1,0,5,0,0,7,0,0,3,0,0,2,0,0,1,0,0,4,0,9],[8,69,0.1159,0.57589,0.38213,0.24999,0.5,1.0,0.0,1.0,1,12,0,1,0,7,0,0,8,0,0,0,0,0,0,0,0,2,0,0,2,0,12],[12,69,0.1739,0.50446,0.34252,0.25002,0.42857,0.85714,0.0,1.0,1,7,0,1,0,7,0,0,7,0,0,5,0,0,1,0,0,0,0,0,4,0,7],[16,69,0.2319,0.44197,0.38855,0.14286,0.28571,1.0,0.0,1.0,3,10,0,3,0,11,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,10],[20,69,0.2899,0.47768,0.38566,0.14286,0.35714,0.89286,0.0,1.0,5,8,0,5,0,7,0,0,4,0,0,3,0,0,1,0,0,1,0,0,3,0,8],[24,69,0.3478,0.57143,0.33693,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,8,0,0,4,0,0,2,0,0,0,0,0,4,0,9],[28,69,0.4058,0.45089,0.33333,0.24999,0.28571,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,13,0,0,2,0,0,0,0,0,0,0,0,2,0,7],[32,69,0.4638,0.53125,0.36811,0.2857,0.28571,1.0,0.0,1.0,1,11,0,1,0,6,0,0,10,0,0,2,0,0,1,0,0,0,0,0,1,0,11],[36,69,0.5217,0.4375,0.32915,0.14289,0.28571,0.53574,0.14286,1.0,0,7,0,0,0,10,0,0,9,0,0,5,0,0,0,0,0,0,0,0,1,0,7],[40,69,0.5797,0.49991,0.37466,0.14286,0.35714,1.0,0.0,1.0,3,10,0,3,0,6,0,0,7,0,0,4,0,0,1,0,0,0,0,0,1,0,10],[44,69,0.6377,0.5267,0.349,0.2857,0.42857,1.0,0.14,1.0,0,10,0,0,0,7,0,0,8,0,0,5,0,0,1,0,0,0,0,0,1,0,10],[48,69,0.6957,0.38392,0.3719,0.14286,0.2857,0.74996,0.0,1.0,7,6,0,7,0,8,0,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,6],[52,69,0.7536,0.4017,0.36679,0.14286,0.28571,0.85714,0.0,1.0,3,7,0,3,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,0,2,0,7],[56,69,0.8116,0.38393,0.29545,0.14286,0.28571,0.32143,0.14286,1.0,0,4,0,0,0,10,0,0,14,0,0,1,0,0,0,0,0,1,0,0,2,0,4],[60,69,0.8696,0.41518,0.35778,0.14286,0.28571,0.85714,0.0,1.0,4,6,0,4,0,9,0,0,6,0,0,4,0,0,0,0,0,0,0,0,3,0,6],[64,69,0.9275,0.375,0.38091,0.14286,0.21428,0.57145,0.0,1.0,7,8,0,7,0,9,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,8],[68,69,0.9855,0.20536,0.3071,0.0,0.14286,0.28571,0.0,1.0,15,3,0,15,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[69,69,1.0,0.125,0.19805,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.30802,"x":0.63393,"p":[[0,83,0.0,0.35714,0.31339,0.14286,0.28571,0.57143,0.0,1.0,6,3,1,6,0,7,0,0,7,0,0,3,0,0,3,0,0,1,0,0,2,0,3],[4,83,0.0482,0.50893,0.37105,0.14286,0.28571,1.0,0.14286,1.0,0,10,0,0,0,10,0,0,8,0,0,2,0,0,0,0,0,0,0,0,2,0,10],[8,83,0.0964,0.53571,0.36422,0.2857,0.35714,1.0,0.0,1.0,2,10,0,2,0,4,0,0,10,0,0,3,0,0,0,0,0,1,0,0,2,0,10],[12,83,0.1446,0.46875,0.35217,0.14289,0.28571,0.89286,0.0,1.0,1,8,0,1,0,8,0,0,10,0,0,3,0,0,0,0,0,0,0,0,2,0,8],[16,83,0.1928,0.63393,0.37447,0.28571,0.78571,1.0,0.0,1.0,1,14,0,1,0,6,0,0,5,0,0,1,0,0,2,0,0,1,0,0,2,0,14],[20,83,0.241,0.52232,0.34369,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,14,0,0,3,0,0,0,0,0,0,0,0,1,0,10],[24,83,0.2892,0.46428,0.34626,0.24999,0.28571,0.85704,0.0,1.0,3,7,0,3,0,5,0,0,9,0,0,5,0,0,0,0,0,1,0,0,2,0,7],[28,83,0.3373,0.39732,0.30667,0.14286,0.28571,0.42857,0.0,1.0,1,5,0,1,0,11,0,0,6,0,0,7,0,0,1,0,0,0,0,0,1,0,5],[32,83,0.3855,0.56696,0.36854,0.2857,0.42857,1.0,0.0,1.0,1,12,0,1,0,5,0,0,9,0,0,3,0,0,0,0,0,1,0,0,1,0,12],[36,83,0.4337,0.49553,0.36243,0.24999,0.28571,0.89286,0.0,1.0,3,8,0,3,0,5,0,0,9,0,0,2,0,0,2,0,0,0,0,0,3,0,8],[40,83,0.4819,0.47768,0.29583,0.2857,0.28571,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,15,0,0,4,0,0,1,0,0,2,0,0,1,0,6],[44,83,0.5301,0.50884,0.32925,0.28571,0.42857,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,9,0,0,8,0,0,0,0,0,0,0,0,2,0,8],[48,83,0.5783,0.50893,0.29868,0.28571,0.35714,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,14,0,0,5,0,0,1,0,0,1,0,0,3,0,6],[52,83,0.6265,0.58035,0.30917,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,7,0,0,1,0,0,1,0,0,1,0,10],[56,83,0.6747,0.49553,0.32338,0.28571,0.28571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,9],[60,83,0.7229,0.55357,0.31894,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,12,0,0,5,0,0,2,0,0,0,0,0,2,0,9],[64,83,0.7711,0.57588,0.33784,0.28571,0.42857,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,9,0,0,5,0,0,1,0,0,0,0,0,3,0,10],[68,83,0.8193,0.43749,0.32328,0.14289,0.28571,0.60682,0.0,1.0,2,6,0,2,0,7,0,0,8,0,0,6,0,0,1,0,0,1,0,0,1,0,6],[72,83,0.8675,0.47768,0.33808,0.2857,0.28571,1.0,0.0,1.0,1,9,0,1,0,4,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,9],[76,83,0.9157,0.51785,0.3549,0.2857,0.42857,1.0,0.0,1.0,1,10,0,1,0,6,0,0,8,0,0,6,0,0,0,0,0,0,0,0,1,0,10],[80,83,0.9639,0.31696,0.21349,0.14286,0.28571,0.32143,0.14286,1.0,0,2,0,0,0,11,0,0,13,0,0,4,0,0,2,0,0,0,0,0,0,0,2],[83,83,1.0,0.30802,0.1561,0.14289,0.2857,0.42857,0.14286,0.85714,0,0,0,0,0,9,0,0,14,0,0,6,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"4531e894a2590100","q":"In an acute-angled and not isosceles triangle  $ABC,$  we draw the median   $AM$  and the height  $AH.$  \nPoints  $Q$  and  $P$  are marked on the lines  $AB$  and  $AC$ , respectively, so that the  $QM \\perp AC$  and  $PM \\perp AB$ .\nThe circumcircle of   $PMQ$  intersects the line  $BC$  for second time  at point  $X.$  Prove that  $BH = CX.$ M. Didin","t":[{"b":2,"e":0.57143,"k":"falling","v":0.5625,"x":0.84375,"p":[[0,88,0.0,0.84375,0.30589,0.85714,1.0,1.0,0.0,1.0,2,23,1,2,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,23],[4,88,0.0455,0.73212,0.30672,0.57143,0.85714,1.0,0.0,1.0,3,13,0,3,0,0,0,0,1,0,0,0,0,0,8,0,0,3,0,0,4,0,13],[8,88,0.0909,0.75893,0.2299,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,8,0,0,2,0,12],[12,88,0.1364,0.70308,0.24493,0.57143,0.71429,0.89286,0.07143,1.0,0,8,0,0,1,1,0,0,2,0,0,0,0,0,6,0,0,12,0,0,2,0,8],[16,88,0.1818,0.71204,0.27693,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,1,1,0,0,1,0,0,0,0,0,7,0,0,8,0,0,3,0,10],[20,88,0.2273,0.7098,0.26363,0.57132,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,2,0,0,3,0,0,8,0,0,5,0,0,2,0,11],[24,88,0.2727,0.6116,0.25563,0.42859,0.57143,0.71429,0.0,1.0,1,5,0,1,0,1,0,0,4,0,0,3,0,0,8,0,0,8,0,0,2,0,5],[28,88,0.3182,0.5625,0.2922,0.28571,0.71429,0.71429,0.0,1.0,4,2,0,4,0,1,0,0,4,0,0,1,0,0,3,0,0,14,0,0,3,0,2],[32,88,0.3636,0.61605,0.29545,0.39286,0.64286,0.85704,0.0,1.0,2,7,0,2,0,1,0,0,5,0,0,1,0,0,7,0,0,7,0,0,2,0,7],[36,88,0.4091,0.6674,0.25229,0.57143,0.57143,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,1,0,0,2,0,1,11,0,0,4,0,0,4,0,7],[40,88,0.4545,0.66071,0.27836,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,2,0,0,1,0,0,7,0,0,9,0,0,3,0,7],[44,88,0.5,0.63391,0.24207,0.57142,0.64286,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,5,0,0,0,0,0,10,0,0,9,0,0,2,0,5],[48,88,0.5455,0.65177,0.20184,0.57143,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,0,0,0,8,0,0,14,0,0,1,0,4],[52,88,0.5909,0.64731,0.22012,0.57143,0.64286,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,2,0,0,11,0,0,10,0,0,1,0,5],[56,88,0.6364,0.7165,0.19109,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,1,0,0,0,7,0,0,11,0,0,6,0,5],[60,88,0.6818,0.67856,0.20517,0.57143,0.71429,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,0,0,0,13,0,0,9,0,0,3,0,5],[64,88,0.7273,0.58481,0.18336,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,1,0,0,12,0,0,13,0,0,0,0,1],[68,88,0.7727,0.62052,0.15815,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,12,0,0,2,0,1],[72,88,0.8182,0.5892,0.1779,0.57143,0.57143,0.71429,0.14,1.0,0,2,0,0,0,1,0,0,3,0,0,2,0,0,15,0,0,9,0,0,0,0,2],[76,88,0.8636,0.64727,0.1184,0.57143,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,14,0,0,1,0,1],[80,88,0.9091,0.61161,0.22934,0.57143,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,4,0,0,0,0,0,9,0,0,13,0,0,1,0,3],[84,88,0.9545,0.59817,0.15336,0.5714,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,1,0,0,16,0,0,11,0,0,0,0,1],[88,88,1.0,0.63838,0.1287,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,20,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"falling","v":0.54673,"x":0.78571,"p":[[0,65,0.0,0.78571,0.35355,0.82132,1.0,1.0,0.0,1.0,3,20,3,3,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,20],[4,65,0.0615,0.75446,0.31791,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,4,0,0,0,0,0,5,0,0,2,0,0,2,0,17],[8,65,0.1231,0.6875,0.3102,0.57143,0.71429,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,0,2,0,0,8,0,0,4,0,0,0,0,13],[12,65,0.1846,0.60714,0.26486,0.53571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,4,0,0,1,0,0,9,0,0,8,0,0,2,0,5],[16,65,0.2462,0.64057,0.19025,0.57132,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,0,0,1,13,0,0,11,0,0,0,0,4],[20,65,0.3077,0.65399,0.24033,0.57132,0.71429,0.75,0.0,1.0,1,5,0,1,0,1,0,0,2,0,0,1,0,1,8,0,0,10,0,0,3,0,5],[24,65,0.3692,0.59374,0.21756,0.57143,0.64286,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,1,0,0,10,0,0,14,0,0,0,0,2],[28,65,0.4308,0.63167,0.24296,0.57132,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,3,0,1,1,0,0,1,0,0,9,0,0,11,0,0,1,0,5],[32,65,0.4923,0.63167,0.22097,0.571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,4,0,0,1,0,1,6,0,0,14,0,0,2,0,3],[36,65,0.5538,0.67186,0.24148,0.5713,0.71429,0.73214,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,3,0,0,4,0,0,13,1,0,0,0,7],[40,65,0.6154,0.64732,0.22299,0.57143,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,2,0,0,7,0,0,14,0,0,0,0,5],[44,65,0.6769,0.59821,0.22428,0.57143,0.64286,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,3,0,0,1,0,0,10,0,0,13,0,0,1,0,2],[48,65,0.7385,0.57587,0.24609,0.42857,0.71414,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,4,0,0,2,0,0,6,0,0,14,0,0,1,0,2],[52,65,0.8,0.60043,0.16353,0.48215,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,1,10,0,0,11,0,0,0,0,2],[56,65,0.8615,0.58704,0.15945,0.4286,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,6,0,1,8,0,0,13,0,0,0,0,1],[60,65,0.9231,0.6339,0.16729,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,2,0,0,8,0,0,19,0,0,0,0,1],[64,65,0.9846,0.54673,0.18769,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,1,4,0,0,9,0,0,13,0,0,0,0,0],[65,65,1.0,0.61157,0.10854,0.57142,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,11,0,0,1,0,0]]}]},{"i":"a069cc72d9032b52","q":"Let   $ G$  be a finite non-commutative group of order  $ t \\equal{} 2^nm$ , where  $ n, m$  are positive and  $ m$  is odd. Prove, that if the group contains an element of order  $ 2^n$ , then\r\n(i)  $ G$  is not simple;\r\n(ii)  $ G$  contains a normal subgroup of order  $ m$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13367,"x":0.18731,"p":[[0,29,0.0,0.13394,0.03458,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.17847,0.13831,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,26,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,29,0.4138,0.16054,0.08568,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,29,0.5517,0.18731,0.12597,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,26,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[20,29,0.6897,0.1517,0.07937,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,29,0.8276,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.14251,0.00095,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.13367,0.03452,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.571,"k":"rising","v":0.13394,"x":0.48197,"p":[[0,32,0.0,0.13394,0.03458,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.15616,0.0827,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,27,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.15607,0.08271,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,29,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,32,0.5,0.16063,0.07786,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.30356,0.20436,0.14286,0.1429,0.57111,0.14286,0.71429,0,0,0,0,0,18,0,0,3,0,0,2,0,0,7,0,0,2,0,0,0,0,0],[24,32,0.75,0.46874,0.18637,0.39286,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,2,0,0,5,0,0,15,0,0,4,0,0,0,0,0],[28,32,0.875,0.36603,0.21406,0.14286,0.35714,0.57143,0.0,0.71429,1,0,0,1,0,11,0,0,4,0,0,4,0,0,9,0,0,3,0,0,0,0,0],[32,32,1.0,0.48197,0.15886,0.42857,0.57121,0.57143,0.14,0.71429,0,0,0,0,0,4,0,0,3,0,0,3,0,0,21,0,0,1,0,0,0,0,0]]}]},{"i":"ff68f9afc837812a","q":"A polyhedron has $7 n$ faces. Show that there exist $n+1$ of the polyhedron's faces that all have the same number of edges.","t":[{"b":3,"e":0.42857,"k":"flat","v":0.52219,"x":0.6875,"p":[[0,27,0.0,0.67857,0.24743,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,8,0,0,5,0,0,5,0,0,4,0,8],[4,27,0.1481,0.61161,0.17941,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,10,0,0,7,0,0,9,0,0,3,0,2],[8,27,0.2963,0.52219,0.22206,0.42857,0.57071,0.71429,0.14,1.0,0,1,0,0,0,4,0,0,3,0,0,8,0,0,6,0,0,8,0,0,2,0,1],[12,27,0.4444,0.55357,0.21354,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,7,0,0,7,0,0,9,0,0,3,0,0,4,0,2],[16,27,0.5926,0.56246,0.17834,0.42857,0.4993,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,15,0,0,7,0,0,5,0,0,2,0,2],[20,27,0.7407,0.57589,0.22442,0.42857,0.5,0.60714,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,13,0,0,8,0,0,1,0,0,2,0,5],[24,27,0.8889,0.6875,0.23807,0.53571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,6,0,0,7,0,0,4,0,0,6,0,7],[27,27,1.0,0.65179,0.2111,0.53571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,7,0,0,10,0,0,4,0,0,6,0,4]]},{"b":7,"e":1.0,"k":"flat","v":0.55357,"x":0.69643,"p":[[0,19,0.0,0.6473,0.23415,0.42857,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,6,0,0,4,0,6],[4,19,0.2105,0.55357,0.1915,0.42857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,14,0,0,3,0,0,7,0,0,5,0,0],[8,19,0.4211,0.6116,0.21793,0.42857,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,8,0,0,6,0,0,3,0,4],[12,19,0.6316,0.65178,0.21409,0.42859,0.71429,0.85704,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,8,0,0,5,0,4],[16,19,0.8421,0.69643,0.23351,0.42859,0.78571,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,3,0,0,11,0,5],[19,19,1.0,0.67411,0.20589,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,6,0,0,7,0,4]]}]},{"i":"d1950bb2297049a2","q":"Let   $n\\ge 2$  is a given integer ,  $x_1,x_2,\\ldots,x_n $  be real numbers such that  $(1) x_1+x_2+\\ldots+x_n=0 $ , $(2) |x_i|\\le 1$   $(i=1,2,\\cdots,n)$ .\nFind the maximum of Min $\\{|x_1-x_2|,|x_2-x_3|,\\cdots,|x_{n-1}-x_n|\\}$ .","t":[{"b":4,"e":0.71429,"k":"falling","v":0.48659,"x":0.89284,"p":[[0,41,0.0,0.75445,0.18978,0.57143,0.71429,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,3,0,0,6,0,9],[4,41,0.0976,0.87946,0.24772,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,3,0,24],[8,41,0.1951,0.875,0.20124,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,9,0,18],[12,41,0.2927,0.85714,0.24484,0.82132,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,22],[16,41,0.3902,0.89284,0.2287,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,26],[20,41,0.4878,0.85714,0.22868,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,5,0,0,3,0,20],[24,41,0.5854,0.89284,0.24486,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,24],[28,41,0.6829,0.84375,0.25595,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,5,0,20],[32,41,0.7805,0.81695,0.27255,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,2,0,0,2,0,20],[36,41,0.878,0.60712,0.26964,0.28571,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,4,0,0,6,0,0,2,0,0,5,0,6],[40,41,0.9756,0.55803,0.27747,0.28571,0.42859,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,14,0,0,3,0,0,1,0,0,3,0,0,8,0,3],[41,41,1.0,0.48659,0.25966,0.28571,0.28571,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,18,0,0,1,0,0,4,0,0,4,0,0,1,0,4]]},{"b":5,"e":0.71429,"k":"flat","v":0.74995,"x":0.99554,"p":[[0,59,0.0,0.85708,0.18908,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,3,0,19],[4,59,0.0678,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,59,0.1356,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,59,0.2034,0.91516,0.21685,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,27],[16,59,0.2712,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,59,0.339,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,59,0.4068,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[28,59,0.4746,0.90621,0.15415,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,22],[32,59,0.5424,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[36,59,0.6102,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[40,59,0.678,0.9241,0.16746,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,25],[44,59,0.7458,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[48,59,0.8136,0.91964,0.15542,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,7,0,22],[52,59,0.8814,0.87943,0.17901,0.857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,6,0,19],[56,59,0.9492,0.91963,0.12344,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,20],[59,59,1.0,0.74995,0.22307,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,5,0,0,11,0,7]]}]},{"i":"0fc95550361f0675","q":"In the regular pentagon  $ABCDE$ , the perpendicular at  $C$  to  $CD$  meets  $AB$  at  $F$ . Prove that  $AE+AF=BE$ .\n\n*Proposed by Alireza Cheraghi*","t":[{"b":4,"e":0.42857,"k":"falling","v":0.46427,"x":0.95535,"p":[[0,100,0.0,0.66963,0.28221,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,7,0,0,6,0,0,1,0,0,4,0,10],[4,100,0.04,0.83927,0.24937,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,19],[8,100,0.08,0.80803,0.28484,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,1,0,0,2,0,20],[12,100,0.12,0.81696,0.2506,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,2,0,0,5,0,17],[16,100,0.16,0.88392,0.23809,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,24],[20,100,0.2,0.88839,0.19144,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,5,0,21],[24,100,0.24,0.83925,0.26669,0.857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,20],[28,100,0.28,0.87945,0.20239,0.85711,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,4,0,21],[32,100,0.32,0.9107,0.15875,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,22],[36,100,0.36,0.79908,0.2445,0.57132,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,1,0,0,3,0,17],[40,100,0.4,0.8616,0.23003,0.85711,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,0,0,0,4,0,21],[44,100,0.44,0.86167,0.21278,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,0,4,0,20],[48,100,0.48,0.87946,0.23176,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,6,0,21],[52,100,0.52,0.87052,0.17629,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,6,0,18],[56,100,0.56,0.81696,0.25313,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,1,0,0,4,0,18],[60,100,0.6,0.95535,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[64,100,0.64,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[68,100,0.68,0.92857,0.13363,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,23],[72,100,0.72,0.91071,0.15047,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,21],[76,100,0.76,0.9107,0.17408,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,21],[80,100,0.8,0.88837,0.18813,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,22],[84,100,0.84,0.82143,0.2369,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,0,3,0,18],[88,100,0.88,0.9241,0.18205,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,25],[92,100,0.92,0.875,0.21354,0.85711,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,3,0,22],[96,100,0.96,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[100,100,1.0,0.46427,0.25999,0.2857,0.4286,0.57143,0.0,1.0,1,2,0,1,0,6,0,0,4,0,0,7,0,0,8,0,0,1,0,0,3,0,2]]},{"b":7,"e":0.71429,"k":"flat","v":0.73215,"x":0.99106,"p":[[0,70,0.0,0.73215,0.29396,0.42859,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,7,0,0,2,0,0,1,0,0,3,0,15],[4,70,0.0571,0.83925,0.21654,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,1,0,0,6,0,17],[8,70,0.1143,0.90624,0.20707,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,25],[12,70,0.1714,0.91524,0.16701,0.85929,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,5,0,23],[16,70,0.2286,0.87946,0.23176,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,23],[20,70,0.2857,0.87052,0.22121,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,7,0,19],[24,70,0.3429,0.91071,0.19804,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,24],[28,70,0.4,0.90625,0.16213,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,21],[32,70,0.4571,0.99106,0.03461,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,70,0.5143,0.9732,0.08335,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[40,70,0.5714,0.92409,0.18555,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[44,70,0.6286,0.94187,0.15961,1.0,1.0,1.0,0.14,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[48,70,0.6857,0.89284,0.18901,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,20],[52,70,0.7429,0.94196,0.1551,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[56,70,0.8,0.90625,0.22759,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,25],[60,70,0.8571,0.94196,0.16312,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[64,70,0.9143,0.91517,0.21086,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,25],[68,70,0.9714,0.92856,0.189,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,26],[70,70,1.0,0.83035,0.29329,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,6,0,20]]}]},{"i":"810e49ff5af457e1","q":"An equilateral triangle is divided into  $25$  equal equilateral triangles labelled by  $1$  through  $25$ . Prove that one can find two triangles having a common side whose labels differ by more than  $3$ .","t":[{"b":1,"e":0.4286,"k":"rising","v":0.03125,"x":0.44195,"p":[[0,60,0.0,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,22,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,60,0.0667,0.26338,0.27223,0.0,0.14286,0.46418,0.0,1.0,12,1,0,12,0,6,0,0,1,0,0,5,0,0,6,0,0,1,0,0,0,0,1],[8,60,0.1333,0.27213,0.22977,0.14,0.2857,0.42857,0.0,1.0,7,1,0,7,0,7,0,0,9,0,0,3,0,0,5,0,0,0,0,0,0,0,1],[12,60,0.2,0.41963,0.22569,0.25001,0.42859,0.57143,0.0,0.85714,3,0,0,3,0,5,0,0,2,0,0,7,0,0,13,0,0,0,0,0,2,0,0],[16,60,0.2667,0.35264,0.24736,0.14286,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,4,0,0,3,0,0,6,0,0,10,0,0,1,0,0,1,0,0],[20,60,0.3333,0.44195,0.24577,0.39286,0.57143,0.57143,0.0,1.0,5,1,0,5,0,2,0,0,1,0,0,6,0,0,16,0,0,0,0,0,1,0,1],[24,60,0.4,0.32139,0.2082,0.14286,0.28571,0.571,0.0,0.57143,6,0,0,6,0,4,0,0,7,0,0,6,0,0,9,0,0,0,0,0,0,0,0],[28,60,0.4667,0.24554,0.24284,0.0,0.14286,0.57143,0.0,0.57143,13,0,0,13,0,4,0,0,3,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[32,60,0.5333,0.24999,0.27663,0.0,0.14286,0.57111,0.0,1.0,14,1,0,14,0,5,0,0,0,0,0,4,0,0,8,0,0,0,0,0,0,0,1],[36,60,0.6,0.27677,0.24467,0.0,0.28571,0.4642,0.0,0.71429,12,0,0,12,0,2,0,0,3,0,0,7,0,0,7,0,0,1,0,0,0,0,0],[40,60,0.6667,0.26337,0.24248,0.0,0.21429,0.42858,0.0,0.71429,12,0,0,12,0,4,0,0,1,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[44,60,0.7333,0.23661,0.21312,0.0,0.2857,0.42857,0.0,0.57143,12,0,0,12,0,3,0,0,5,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[48,60,0.8,0.29463,0.26228,0.0,0.35714,0.4642,0.0,1.0,11,1,0,11,0,3,0,0,2,0,0,8,0,0,7,0,0,0,0,0,0,0,1],[52,60,0.8667,0.23214,0.19805,0.10714,0.14286,0.42857,0.0,0.57143,8,0,0,8,0,11,0,0,2,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[56,60,0.9333,0.23214,0.1948,0.0,0.2857,0.42857,0.0,0.57143,10,0,0,10,0,5,0,0,7,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[60,60,1.0,0.20527,0.19215,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,9,0,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"rising","v":0.03125,"x":0.30802,"p":[[0,50,0.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,21,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,50,0.08,0.25893,0.21558,0.0,0.21428,0.42857,0.0,0.57143,9,0,0,9,0,7,0,0,3,0,0,7,0,0,6,0,0,0,0,0,0,0,0],[8,50,0.16,0.30802,0.26026,0.14286,0.2857,0.57143,0.0,1.0,7,1,0,7,0,8,0,0,5,0,0,1,0,0,9,0,0,1,0,0,0,0,1],[12,50,0.24,0.26784,0.24155,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,3,0,0,4,0,0,6,0,0,0,0,0,0,0,1],[16,50,0.32,0.20536,0.18877,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,13,0,0,5,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[20,50,0.4,0.20982,0.1988,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,10,0,0,3,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[24,50,0.48,0.27232,0.24836,0.10714,0.14286,0.46431,0.0,1.0,8,1,0,8,0,9,0,0,4,0,0,3,0,0,7,0,0,0,0,0,0,0,1],[28,50,0.56,0.21427,0.17125,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[32,50,0.64,0.23652,0.16607,0.14286,0.14286,0.32143,0.0,0.57143,4,0,0,4,0,14,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[36,50,0.72,0.19643,0.15872,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,17,0,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[40,50,0.8,0.16516,0.10774,0.14286,0.14286,0.14286,0.0,0.571,4,0,0,4,0,21,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,50,0.88,0.21864,0.12871,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,16,0,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[48,50,0.96,0.21429,0.07985,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.24107,0.06621,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10947e7fd9a6ca2e","q":"On a table there are several cards, some face up and others face down. The allowed operation is to choose 4 cards and turn them over. The goal is to get all the cards in the same state (all face up or all face down). Determine if the objective can be achieved through a sequence of permitted operations if initially there are:\na) 101 cards face up and 102 face down;\nb) 101 cards face up and 101 face down.","t":[{"b":0,"e":1.0,"k":"falling","v":0.71425,"x":0.96875,"p":[[0,55,0.0,0.94643,0.14174,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,28],[4,55,0.0727,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[8,55,0.1455,0.92857,0.13832,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,25],[12,55,0.2182,0.88393,0.16917,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,1,0,21],[16,55,0.2909,0.87499,0.17407,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,2,0,20],[20,55,0.3636,0.84817,0.19547,0.57143,1.0,1.0,0.57,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,2,0,19],[24,55,0.4364,0.83929,0.18472,0.67857,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,2,0,17],[28,55,0.5091,0.91071,0.15872,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,23],[32,55,0.5818,0.87946,0.17536,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,1,0,21],[36,55,0.6545,0.79909,0.2016,0.57143,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,2,0,15],[40,55,0.7273,0.79464,0.20806,0.57143,0.85714,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,0,0,16],[44,55,0.8,0.78567,0.1989,0.57143,0.71429,1.0,0.57,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,1,0,14],[48,55,0.8727,0.80804,0.18423,0.57143,0.78571,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,2,0,14],[52,55,0.9455,0.81696,0.1931,0.57143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,1,0,16],[55,55,1.0,0.71425,0.18561,0.57143,0.57143,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,5,0,0,0,0,9]]},{"b":5,"e":1.0,"k":"flat","v":0.96428,"x":1.0,"p":[[0,56,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,56,0.0714,0.96428,0.10102,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[8,56,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,56,0.2143,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[16,56,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,56,0.3571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,56,0.4286,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,56,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,56,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,56,0.6429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,56,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,56,0.7857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,56,0.8571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,56,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"4f6eb79cfcba165f","q":"A weighted complete graph with distinct positive wights is given such that in every triangle is *degenerate* that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.\n\n*Proposed by Morteza Saghafian*","t":[{"b":5,"e":0.14286,"k":"falling","v":0.1875,"x":0.48659,"p":[[0,40,0.0,0.40179,0.39357,0.0,0.14286,0.85714,0.0,1.0,9,5,0,9,0,8,0,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,5],[4,40,0.1,0.45089,0.36962,0.14286,0.28571,0.85714,0.0,1.0,5,5,0,5,0,8,0,0,4,0,0,2,0,0,1,0,0,2,0,0,5,0,5],[8,40,0.2,0.48659,0.36044,0.14286,0.42857,0.78571,0.0,1.0,4,8,0,4,0,6,0,0,5,0,0,2,0,0,4,0,0,3,0,0,0,0,8],[12,40,0.3,0.44196,0.38193,0.14286,0.28571,0.75,0.0,1.0,7,7,0,7,0,6,0,0,4,0,0,2,0,0,1,0,0,4,0,0,1,0,7],[16,40,0.4,0.32143,0.3481,0.0,0.21428,0.60714,0.0,1.0,12,2,0,12,0,4,0,0,6,0,0,0,0,0,2,0,0,2,0,0,4,0,2],[20,40,0.5,0.27679,0.29437,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,11,0,0,6,0,0,0,0,0,0,0,0,4,0,0,2,0,1],[24,40,0.6,0.28124,0.28455,0.14286,0.14286,0.28571,0.0,1.0,7,2,0,7,0,10,0,0,8,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[28,40,0.7,0.41518,0.31004,0.14286,0.28571,0.57143,0.0,1.0,2,4,0,2,0,9,0,0,7,0,0,3,0,0,4,0,0,1,0,0,2,0,4],[32,40,0.8,0.24107,0.24074,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,15,0,0,8,0,0,0,0,0,0,0,0,2,0,0,1,0,1],[36,40,0.9,0.1875,0.08328,0.14286,0.14286,0.1786,0.14286,0.4286,0,0,0,0,0,24,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.19642,0.09937,0.14286,0.14286,0.2857,0.14286,0.571,0,0,0,0,0,23,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.33929,"x":0.86161,"p":[[0,44,0.0,0.51339,0.36045,0.14286,0.35714,0.85714,0.0,1.0,3,6,0,3,0,6,0,0,7,0,0,1,0,0,0,0,0,4,0,0,5,0,6],[4,44,0.0909,0.33929,0.36202,0.0,0.14286,0.71429,0.0,1.0,10,3,0,10,0,8,0,0,3,0,0,1,0,0,1,0,0,2,0,0,4,0,3],[8,44,0.1818,0.3482,0.37445,0.0,0.14286,0.71429,0.0,1.0,13,3,0,13,0,4,0,0,2,0,0,1,0,0,3,0,0,2,0,0,4,0,3],[12,44,0.2727,0.56249,0.36759,0.14286,0.71429,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,2,0,0,2,0,0,1,0,0,7,0,0,4,0,7],[16,44,0.3636,0.36152,0.35717,0.0,0.21428,0.64286,0.0,1.0,9,3,0,9,0,7,0,0,3,0,0,3,0,0,2,0,0,0,0,0,5,0,3],[20,44,0.4545,0.41964,0.38785,0.0,0.28571,0.85714,0.0,1.0,9,6,0,9,0,5,0,0,3,0,0,3,0,0,1,0,0,2,0,0,3,0,6],[24,44,0.5455,0.5982,0.34523,0.2857,0.64286,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,6,0,0,2,0,0,2,0,0,1,0,0,7,0,8],[28,44,0.6364,0.76339,0.23313,0.57142,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,4,0,0,9,0,10],[32,44,0.7273,0.86161,0.145,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,10,0,13],[36,44,0.8182,0.83034,0.22144,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,10,0,13],[40,44,0.9091,0.72768,0.27747,0.53571,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,7,0,0,1,0,0,3,0,0,3,0,0,7,0,11],[44,44,1.0,0.67409,0.20277,0.5354,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,5,0,0,4,0,0,7,0,0,12,0,1]]}]},{"i":"61bedf0f04465cb0","q":"Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.","t":[{"b":1,"e":0.2857,"k":"rising","v":0.22759,"x":0.6875,"p":[[0,23,0.0,0.22759,0.14228,0.14286,0.2857,0.28571,0.0,0.57143,4,0,1,4,0,11,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,23,0.1739,0.61158,0.33926,0.28571,0.57143,1.0,0.0,1.0,1,11,0,1,0,2,0,0,10,0,0,0,0,0,4,0,0,2,0,0,2,0,11],[8,23,0.3478,0.6875,0.33012,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,1,0,0,4,0,14],[12,23,0.5217,0.58482,0.34876,0.2857,0.5,1.0,0.0,1.0,1,11,0,1,0,2,0,0,12,0,0,1,0,0,2,0,0,1,0,0,2,0,11],[16,23,0.6957,0.60714,0.34626,0.28571,0.57143,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,8,0,0,0,0,0,5,0,0,1,0,0,1,0,12],[20,23,0.8696,0.52678,0.31428,0.28571,0.35714,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,12,0,0,1,0,0,4,0,0,2,0,0,2,0,7],[23,23,1.0,0.51339,0.34967,0.2857,0.28571,1.0,0.14286,1.0,0,9,0,0,0,7,0,0,11,0,0,1,0,0,2,0,0,0,0,0,2,0,9]]},{"b":2,"e":0.2857,"k":"rising","v":0.29456,"x":0.76335,"p":[[0,22,0.0,0.29456,0.21416,0.14286,0.28571,0.32143,0.0,1.0,3,1,0,3,0,9,0,0,12,0,0,4,0,0,1,0,0,2,0,0,0,0,1],[4,22,0.1818,0.56248,0.34799,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,13,0,0,0,0,0,2,0,0,0,0,0,3,0,10],[8,22,0.3636,0.51339,0.3251,0.28571,0.28571,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,15,0,0,2,0,0,2,0,0,1,0,0,0,0,9],[12,22,0.5455,0.73659,0.30953,0.39288,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,7,0,0,1,0,0,2,0,0,3,0,0,2,0,16],[16,22,0.7273,0.64284,0.31135,0.28571,0.64286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,1,0,0,3,0,0,1,0,0,5,0,10],[20,22,0.9091,0.75008,0.30727,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,6,0,0,2,0,0,2,0,0,2,0,0,2,0,17],[22,22,1.0,0.76335,0.27577,0.57132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,4,0,0,0,0,0,7,0,0,1,0,0,4,0,15]]}]},{"i":"f6171a31959fb941","q":"Consider the acute-angled triangle  $ABC$ , with orthocentre  $H$  and circumcentre  $O$ .  $D$  is the intersection point of lines  $AH$  and  $BC$  and  $E$  lies on  $\\overline{AH}$  such that  $AE=DH$ .\nSuppose  $EO$  and  $BC$  meet at  $F$ . Prove that  $BD=CF$ .\n\n*(C\u0103lin Pop & Vlad Robu)*","t":[{"b":0,"e":0.0,"k":"falling","v":0.12054,"x":0.42853,"p":[[0,109,0.0,0.42853,0.18895,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,9,0,0,5,0,0,10,0,0,4,0,0,0,0,0],[4,109,0.0367,0.26338,0.19918,0.14286,0.28571,0.32143,0.0,0.71429,6,0,0,6,0,8,0,0,10,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[8,109,0.0734,0.2857,0.22014,0.14286,0.28571,0.32143,0.0,1.0,5,1,0,5,0,7,0,0,12,0,0,3,0,0,3,0,0,1,0,0,0,0,1],[12,109,0.1101,0.28569,0.18207,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,8,0,0,10,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[16,109,0.1468,0.3482,0.24205,0.14286,0.28571,0.57143,0.0,0.71429,5,0,0,5,0,5,0,0,9,0,0,3,0,0,4,0,0,6,0,0,0,0,0],[20,109,0.1835,0.32141,0.17494,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,8,0,0,9,0,0,6,0,0,7,0,0,0,0,0,0,0,0],[24,109,0.2202,0.4107,0.18122,0.28571,0.35714,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,12,0,0,4,0,0,8,0,0,4,0,0,0,0,0],[28,109,0.2569,0.36149,0.23421,0.14286,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,10,0,0,5,0,0,5,0,0,5,0,0,4,0,0,1,0,0],[32,109,0.2936,0.36604,0.21407,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,8,0,0,7,0,0,3,0,0,9,0,0,3,0,0,0,0,0],[36,109,0.3303,0.35712,0.17854,0.28571,0.28571,0.57111,0.0,0.57143,2,0,0,2,0,5,0,0,10,0,0,5,0,0,10,0,0,0,0,0,0,0,0],[40,109,0.367,0.33929,0.20438,0.25,0.28571,0.57143,0.0,0.71429,5,0,0,5,0,3,0,0,9,0,0,6,0,0,8,0,0,1,0,0,0,0,0],[44,109,0.4037,0.35705,0.19571,0.24999,0.28571,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,12,0,0,2,0,0,7,0,0,3,0,0,0,0,0],[48,109,0.4404,0.28122,0.20663,0.14286,0.28571,0.32143,0.0,0.71429,4,0,0,4,0,10,0,0,10,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[52,109,0.4771,0.28125,0.18723,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,9,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[56,109,0.5138,0.31247,0.24335,0.14286,0.28571,0.42858,0.0,1.0,5,1,0,5,0,7,0,0,10,0,0,3,0,0,3,0,0,3,0,0,0,0,1],[60,109,0.5505,0.29018,0.21275,0.14286,0.28571,0.46429,0.0,0.71429,4,0,0,4,0,11,0,0,7,0,0,2,0,0,6,0,0,2,0,0,0,0,0],[64,109,0.5872,0.20089,0.19186,0.10714,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,14,0,0,4,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[68,109,0.6239,0.20536,0.18877,0.14286,0.14286,0.2857,0.0,0.71429,7,0,0,7,0,14,0,0,6,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[72,109,0.6606,0.19634,0.1777,0.105,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,13,0,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[76,109,0.6972,0.25884,0.20346,0.14286,0.2143,0.42857,0.0,0.71429,6,0,0,6,0,10,0,0,6,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[80,109,0.7339,0.2365,0.15818,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,16,0,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[84,109,0.7706,0.20982,0.14718,0.14286,0.14288,0.2857,0.0,0.71429,5,0,0,5,0,12,0,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[88,109,0.8073,0.17857,0.18211,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,11,0,0,8,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[92,109,0.844,0.14277,0.10102,0.105,0.14286,0.17857,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,109,0.8807,0.14732,0.09094,0.14286,0.14286,0.14287,0.0,0.28571,6,0,0,6,0,19,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,109,0.9174,0.18295,0.07354,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,109,0.9541,0.13839,0.10999,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,109,0.9908,0.15625,0.10326,0.14286,0.14286,0.28571,0.0,0.28571,7,0,0,7,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[109,109,1.0,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.08482,"x":0.3482,"p":[[0,39,0.0,0.3482,0.17472,0.2857,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,5,0,0,10,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[4,39,0.1026,0.30794,0.22054,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,11,0,0,5,0,0,2,0,0,4,0,0,0,0,0],[8,39,0.2051,0.24107,0.20341,0.0,0.2857,0.28571,0.0,0.71429,9,0,0,9,0,5,0,0,11,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[12,39,0.3077,0.15179,0.13803,0.0,0.14286,0.2857,0.0,0.57143,10,0,0,10,0,13,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,39,0.4103,0.14732,0.16164,0.0,0.14286,0.2857,0.0,0.57143,14,0,0,14,0,8,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,39,0.5128,0.13384,0.10676,0.0,0.14286,0.1786,0.0,0.28571,10,0,0,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.14731,0.1939,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,8,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[28,39,0.7179,0.125,0.12753,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,14,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,39,0.8205,0.10714,0.12372,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.11598,0.12076,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1978914b68c6be46","q":"Assume that all angles of a triangle $A B C$ are acute. Let $D$ and $E$ be points on the sides $A C$ and $B C$ of the triangle such that $A, B, D$, and $E$ lie on the same circle. Further suppose the circle through $D, E$, and $C$ intersects the side $A B$ in two points $X$ and $Y$. Show that the midpoint of $X Y$ is the foot of the altitude from $C$ to $A B$.","t":[{"b":4,"e":0.28571,"k":"flat","v":0.07589,"x":0.42411,"p":[[0,88,0.0,0.21875,0.19227,0.14286,0.14286,0.2857,0.0,1.0,3,1,0,3,0,19,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[4,88,0.0455,0.18304,0.1197,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,19,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,88,0.0909,0.19187,0.13179,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,17,0,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,88,0.1364,0.16518,0.17169,0.14286,0.14286,0.14286,0.0,1.0,6,1,0,6,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,88,0.1818,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,88,0.2273,0.19643,0.18814,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,14,0,0,8,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[24,88,0.2727,0.19196,0.22192,0.10714,0.14286,0.17857,0.0,1.0,8,1,0,8,0,16,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[28,88,0.3182,0.10268,0.08917,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,88,0.3636,0.16964,0.16146,0.0,0.14286,0.28571,0.0,0.57143,11,0,0,11,0,9,0,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[36,88,0.4091,0.125,0.11152,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,20,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,88,0.4545,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,88,0.5,0.09375,0.10479,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,88,0.5455,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,88,0.5909,0.11607,0.10972,0.0,0.14286,0.14287,0.0,0.28571,13,0,0,13,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,88,0.6364,0.29464,0.20806,0.14286,0.28571,0.46431,0.0,0.57143,6,0,0,6,0,7,0,0,6,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[60,88,0.6818,0.42411,0.28231,0.14286,0.42857,0.57143,0.0,1.0,3,3,0,3,0,6,0,0,5,0,0,5,0,0,8,0,0,1,0,0,1,0,3],[64,88,0.7273,0.34373,0.2645,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,8,0,0,1,0,0,6,0,0,9,0,0,0,0,0,1,0,1],[68,88,0.7727,0.39729,0.23617,0.14286,0.571,0.57143,0.0,0.85714,3,0,0,3,0,8,0,0,2,0,0,2,0,0,15,0,0,1,0,0,1,0,0],[72,88,0.8182,0.37944,0.25655,0.14286,0.571,0.57143,0.0,0.85714,6,0,0,6,0,6,0,0,1,0,0,2,0,0,15,0,0,1,0,0,1,0,0],[76,88,0.8636,0.36606,0.27417,0.14286,0.28571,0.57143,0.0,1.0,6,1,0,6,0,5,0,0,7,0,0,0,0,0,10,0,0,2,0,0,1,0,1],[80,88,0.9091,0.28572,0.27199,0.10714,0.2143,0.42857,0.0,1.0,8,2,0,8,0,8,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,2],[84,88,0.9545,0.26339,0.23176,0.14286,0.14286,0.32143,0.0,1.0,5,1,0,5,0,12,0,0,7,0,0,4,0,0,2,0,0,0,0,0,1,0,1],[88,88,1.0,0.25445,0.23345,0.14286,0.14286,0.28571,0.0,0.85714,5,0,0,5,0,14,0,0,7,0,0,0,0,0,3,0,0,1,0,0,2,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.00893,"x":0.27232,"p":[[0,67,0.0,0.27232,0.23516,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,20,0,0,4,0,0,2,0,0,1,0,0,1,0,0,3,0,0],[4,67,0.0597,0.13839,0.08364,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.12937,0.09687,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.09822,0.09061,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.12045,0.08825,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,67,0.2985,0.11598,0.08325,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,67,0.4179,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,67,0.4776,0.10268,0.08917,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.11607,0.0974,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0848904bb01d4ecf","q":"Determine the polygons with $n$ sides $(n \\geq 4)$, not necessarily convex, which satisfy the property that the reflection of every vertex of polygon with respect to every diagonal of the polygon does not fall outside the polygon.\n\nNote: Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.","t":[{"b":3,"e":0.57143,"k":"rising","v":0.27679,"x":0.79464,"p":[[0,29,0.0,0.27679,0.25489,0.10714,0.14286,0.46429,0.0,0.71429,8,0,0,8,0,10,0,0,3,0,0,3,0,0,3,0,0,5,0,0,0,0,0],[4,29,0.1379,0.79464,0.21409,0.71429,0.85707,1.0,0.1429,1.0,0,12,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,9,0,0,5,0,12],[8,29,0.2759,0.76338,0.23314,0.57143,0.78564,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,7,0,0,4,0,12],[12,29,0.4138,0.78124,0.22585,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,7,0,11],[16,29,0.5517,0.6116,0.22083,0.57142,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,4,0,0,15,0,0,3,0,0,3,0,4],[20,29,0.6897,0.58481,0.12037,0.57143,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,28,0,0,0,0,0,0,0,2],[24,29,0.8276,0.50889,0.13331,0.571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,4,0,0,2,0,0,25,0,0,0,0,0,0,0,0],[28,29,0.9655,0.52674,0.16916,0.571,0.57143,0.57143,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,3,0,0,25,0,0,0,0,0,0,0,1],[29,29,1.0,0.52672,0.10371,0.571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,3,0,0,26,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"rising","v":0.27223,"x":0.84821,"p":[[0,41,0.0,0.27223,0.24322,0.105,0.14286,0.42857,0.0,0.85714,8,0,2,8,0,9,0,0,3,0,0,6,0,0,3,0,0,2,0,0,1,0,0],[4,41,0.0976,0.84821,0.17105,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,7,0,14],[8,41,0.1951,0.70534,0.23674,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,3,0,0,5,0,0,10,0,0,6,0,6],[12,41,0.2927,0.64283,0.26001,0.4286,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,5,0,0,3,0,0,5,0,0,9,0,0,3,0,6],[16,41,0.3902,0.58033,0.23402,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,1,0,0,8,0,0,9,0,0,7,0,0,2,0,3],[20,41,0.4878,0.52668,0.22157,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,4,0,0,5,0,0,8,0,0,10,0,0,2,0,0],[24,41,0.5854,0.62487,0.17793,0.571,0.57143,0.71429,0.14,1.0,0,3,0,0,0,1,0,0,0,0,0,6,0,0,10,0,0,12,0,0,0,0,3],[28,41,0.6829,0.54907,0.28146,0.39286,0.571,0.75,0.0,1.0,2,2,0,2,0,3,0,0,3,0,0,6,0,0,4,0,0,6,0,0,6,0,2],[32,41,0.7805,0.64279,0.25001,0.571,0.71429,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,3,0,0,7,0,0,8,0,0,7,0,3],[36,41,0.878,0.56247,0.19865,0.42859,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,6,0,0,7,0,0,14,0,0,1,0,0],[40,41,0.9756,0.60042,0.18887,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,2,0,0,14,0,0,8,1,0,4,0,0],[41,41,1.0,0.54012,0.21644,0.42859,0.57143,0.71407,0.0,0.85714,2,0,0,2,0,1,0,0,2,0,0,5,0,0,13,0,0,5,0,0,4,0,0]]}]},{"i":"3ed4fe9f3153e43a","q":"Determine all polynomials  $P(x)$  with integer coefficients such that, for any positive integer  $n$ , the equation  $P(x)=2^n$  has an integer root.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.29018,"x":0.41071,"p":[[0,39,0.0,0.3125,0.20025,0.14286,0.21429,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,1,0,0,11,0,0,3,0,0,0,0,0,0,0,1],[4,39,0.1026,0.31696,0.1665,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,9,0,0,14,0,0,4,0,0,4,0,0,0,0,0,1,0,0],[8,39,0.2051,0.3392,0.2224,0.14286,0.28571,0.42857,0.14,1.0,0,1,0,0,0,11,0,0,10,0,0,6,0,0,2,0,0,0,0,0,2,0,1],[12,39,0.3077,0.34374,0.20472,0.24999,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,8,0,0,15,0,0,3,0,0,3,0,0,0,0,0,3,0,0],[16,39,0.4103,0.29018,0.145,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,11,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[20,39,0.5128,0.32143,0.16752,0.24999,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,13,0,0,9,0,0,1,0,0,0,0,0,0,0,1],[24,39,0.6154,0.41071,0.22232,0.2857,0.42857,0.4643,0.14286,1.0,0,2,0,0,0,6,0,0,8,0,0,10,0,0,5,0,0,0,0,0,1,0,2],[28,39,0.7179,0.35715,0.16366,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,12,0,0,12,0,0,2,0,0,0,0,0,0,0,1],[32,39,0.8205,0.32134,0.11308,0.2857,0.28571,0.42857,0.14,0.57143,0,0,0,0,0,5,0,0,16,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[36,39,0.9231,0.35268,0.17852,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,11,0,0,9,0,0,4,0,0,0,0,0,0,0,1],[39,39,1.0,0.38842,0.10248,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,16,0,0,2,0,0,1,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"falling","v":0.00893,"x":0.43303,"p":[[0,39,0.0,0.33036,0.23266,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,15,0,0,3,0,0,10,0,0,1,0,0,1,0,0,0,0,2],[4,39,0.1026,0.43303,0.2382,0.28571,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,13,0,0,5,0,0,4,0,0,1,0,0,4,0,1],[8,39,0.2051,0.38393,0.25111,0.14286,0.28571,0.46429,0.14286,1.0,0,2,0,0,0,10,0,0,8,0,0,6,0,0,4,0,0,0,0,0,2,0,2],[12,39,0.3077,0.37045,0.23929,0.14286,0.28571,0.42858,0.14,1.0,0,1,0,0,0,11,0,0,7,0,0,7,0,0,2,0,0,2,0,0,2,0,1],[16,39,0.4103,0.27678,0.2257,0.14286,0.2857,0.32143,0.0,0.85714,5,0,0,5,0,10,0,0,9,0,0,2,0,0,4,0,0,0,0,0,2,0,0],[20,39,0.5128,0.26339,0.2055,0.14286,0.14286,0.32143,0.0,0.85714,3,0,0,3,0,14,0,0,7,0,0,5,0,0,1,0,0,0,0,0,2,0,0],[24,39,0.6154,0.24107,0.20652,0.10714,0.21429,0.42857,0.0,0.85714,8,0,0,8,0,8,0,0,7,0,0,6,0,0,2,0,0,0,0,0,1,0,0],[28,39,0.7179,0.20536,0.24984,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,11,0,0,5,0,0,0,0,0,3,0,0,0,0,0,1,0,1],[32,39,0.8205,0.12937,0.21236,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[36,39,0.9231,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4a819e3a889dd0f1","q":"A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.92856,"x":0.95982,"p":[[0,9,0.0,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[4,9,0.4444,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,9,0.8889,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[9,9,1.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.90178,"x":0.93304,"p":[[0,20,0.0,0.92408,0.07131,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[4,20,0.2,0.90178,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[8,20,0.4,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,20,0.6,0.93304,0.07129,0.85714,1.0,1.0,0.85714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[16,20,0.8,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[20,20,1.0,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12]]}]},{"i":"a563e85b14ce4618","q":"Does there exist a sequence  $a_1,a_2,a_3,\\ldots $  of positive integers such that the sum of every  $n$  consecutive elements is divisible by  $n^2$  for every positive integer  $n$ ?","t":[{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.48214,"p":[[0,43,0.0,0.34375,0.41166,0.0,0.0,0.75,0.0,1.0,17,6,0,17,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,0,2,0,6],[4,43,0.093,0.48214,0.48148,0.0,0.35714,1.0,0.0,1.0,15,14,0,15,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,14],[8,43,0.186,0.44196,0.45647,0.0,0.21429,1.0,0.0,1.0,15,10,0,15,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,10],[12,43,0.2791,0.42411,0.47176,0.0,0.0,1.0,0.0,1.0,17,12,0,17,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,12],[16,43,0.3721,0.36606,0.43877,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,8],[20,43,0.4651,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[24,43,0.5581,0.17857,0.34993,0.0,0.0,0.03571,0.0,1.0,24,4,0,24,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,4],[28,43,0.6512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.05357,0.21053,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[36,43,0.8372,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02679,"x":0.47321,"p":[[0,68,0.0,0.16071,0.30461,0.0,0.0,0.17857,0.0,1.0,23,2,0,23,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,2],[4,68,0.0588,0.47321,0.46075,0.0,0.35714,1.0,0.0,1.0,13,12,0,13,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,12],[8,68,0.1176,0.44643,0.47481,0.0,0.14286,1.0,0.0,1.0,15,12,0,15,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,12],[12,68,0.1765,0.29018,0.43811,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[16,68,0.2353,0.39732,0.48278,0.0,0.0,1.0,0.0,1.0,19,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12],[20,68,0.2941,0.27232,0.41243,0.0,0.0,0.71429,0.0,1.0,20,6,0,20,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,6],[24,68,0.3529,0.15625,0.32016,0.0,0.0,0.14286,0.0,1.0,23,3,0,23,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[28,68,0.4118,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,68,0.4706,0.11161,0.25935,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[36,68,0.5294,0.08927,0.25937,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[40,68,0.5882,0.07588,0.22861,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[44,68,0.6471,0.07589,0.2448,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,68,0.7059,0.11598,0.24855,0.0,0.0,0.14071,0.0,1.0,23,1,0,23,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[52,68,0.7647,0.08482,0.1984,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[56,68,0.8235,0.11607,0.2911,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[60,68,0.8824,0.05357,0.18123,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,68,0.9412,0.0625,0.1234,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,68,1.0,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4f3da746737924cf","q":"16. (GDR 4) Prove the following statement: If $r_{1}$ and $r_{2}$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_{1}, k_{2}}=k_{1} r_{1}+k_{2} r_{2}, k_{1}, k_{2}$ integers; i.e., for every real number $x$ and every positive real number $p$ two integers $k_{1}$ and $k_{2}$ can be found such that $\\left|x-\\left(k_{1} r_{1}+k_{2} r_{2}\\right)\\right|<p$.","t":[{"b":5,"e":0.85714,"k":"flat","v":0.87053,"x":0.9108,"p":[[0,31,0.0,0.87053,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[4,31,0.129,0.88393,0.10972,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,20,0,10],[8,31,0.2581,0.90177,0.06623,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[12,31,0.3871,0.9108,0.06909,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[16,31,0.5161,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[20,31,0.6452,0.90177,0.06623,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[24,31,0.7742,0.88391,0.05577,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[28,31,0.9032,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[31,31,1.0,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]},{"b":7,"e":0.85714,"k":"flat","v":0.86159,"x":0.93749,"p":[[0,27,0.0,0.86159,0.12619,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,23,0,7],[4,27,0.1481,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[8,27,0.2963,0.88838,0.05907,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[12,27,0.4444,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[16,27,0.5926,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[20,27,0.7407,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[24,27,0.8889,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[27,27,1.0,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16]]}]},{"i":"736027a7acf89bc9","q":"Determine all pairs  $(a, b)$  of integers that satisfy both: \n1.  $5 \\leq b < a$ 2. There exists a natural number  $n$  such that the numbers  $\\frac{a}{b}$  and  $a-b$  are consecutive divisors of  $n$ , in that order.**Note:** Two positive integers  $x, y$  are consecutive divisors of  $m$ , in that order, if there is no divisor  $d$  of  $m$  such that  $x < d < y$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.61155,"x":0.94642,"p":[[0,84,0.0,0.66066,0.19807,0.571,0.64286,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,6,0,0,7,0,3],[4,84,0.0476,0.9241,0.17491,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[8,84,0.0952,0.94642,0.08565,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[12,84,0.1429,0.83925,0.18817,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,7,0,15],[16,84,0.1905,0.81693,0.15667,0.71429,0.85714,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,8,0,10],[20,84,0.2381,0.82589,0.18808,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,10,0,0,6,0,13],[24,84,0.2857,0.76333,0.22196,0.57132,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,3,0,0,8,0,10],[28,84,0.3333,0.73654,0.24775,0.67846,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,1,0,0,4,0,0,9,0,0,7,0,8],[32,84,0.381,0.62942,0.23651,0.42857,0.64286,0.85704,0.14286,1.0,0,3,0,0,0,1,0,0,5,0,0,3,0,0,7,0,0,6,0,0,7,0,3],[36,84,0.4286,0.72755,0.26354,0.57143,0.71429,1.0,0.14,1.0,0,10,0,0,0,3,0,0,1,0,0,0,0,0,7,0,0,6,0,0,5,0,10],[40,84,0.4762,0.75889,0.21856,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,9,0,0,3,0,11],[44,84,0.5238,0.61155,0.23211,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,5,0,0,9,0,0,5,0,0,7,0,2],[48,84,0.5714,0.70083,0.23789,0.571,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,0,10,0,5],[52,84,0.619,0.75892,0.24857,0.5354,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,4,0,0,6,0,12],[56,84,0.6667,0.76781,0.14622,0.71429,0.71429,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,9,0,5],[60,84,0.7143,0.77678,0.21995,0.571,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,4,0,13],[64,84,0.7619,0.76779,0.19811,0.57132,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,10,0,8],[68,84,0.8095,0.69191,0.2503,0.5354,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,0,0,0,6,0,0,6,0,0,5,0,0,5,0,8],[72,84,0.8571,0.70976,0.20976,0.571,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,4,0,0,6,0,7],[76,84,0.9048,0.69645,0.28734,0.57132,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,3,0,0,2,0,0,4,0,0,7,0,0,5,0,9],[80,84,0.9524,0.69194,0.23988,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,5,0,0,6,0,7],[84,84,1.0,0.65174,0.17475,0.571,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,12,0,0,4,0,2]]},{"b":5,"e":1.0,"k":"flat","v":0.67399,"x":0.91964,"p":[[0,74,0.0,0.6874,0.19381,0.571,0.64271,0.857,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,7,0,0,3,0,6],[4,74,0.0541,0.91964,0.11259,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[8,74,0.1081,0.87943,0.16798,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,19],[12,74,0.1622,0.91517,0.12807,0.85711,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[16,74,0.2162,0.88838,0.13239,0.85711,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,16],[20,74,0.2703,0.8348,0.19922,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,0,6,0,15],[24,74,0.3243,0.81244,0.19384,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,5,0,0,6,0,13],[28,74,0.3784,0.78122,0.23688,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,10,0,10],[32,74,0.4324,0.79012,0.21429,0.57132,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,0,7,0,12],[36,74,0.4865,0.76782,0.21654,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,14,0,6],[40,74,0.5405,0.82587,0.20747,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,8,0,14],[44,74,0.5946,0.74997,0.25507,0.571,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,3,0,0,11,0,9],[48,74,0.6486,0.67399,0.20593,0.571,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,11,0,0,1,0,0,9,0,4],[52,74,0.7027,0.70534,0.22286,0.57132,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,6,0,0,6,0,7],[56,74,0.7568,0.75886,0.21858,0.571,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,4,0,0,7,0,10],[60,74,0.8108,0.75888,0.21263,0.57132,0.857,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,5,0,0,8,0,9],[64,74,0.8649,0.80353,0.22236,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,8,0,12],[68,74,0.9189,0.75439,0.23215,0.571,0.857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,2,0,0,6,0,11],[72,74,0.973,0.74105,0.22144,0.571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,7,0,0,6,0,9],[74,74,1.0,0.73655,0.20555,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,3,0,0,7,0,8]]}]},{"i":"2581ca4fbaa9495e","q":"Find all pairs of natural numbers  $(m, n)$  for which  $2^m3^n +1$  is the square of some integer.","t":[{"b":3,"e":0.57143,"k":"falling","v":0.75221,"x":0.91071,"p":[[0,67,0.0,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[4,67,0.0597,0.87052,0.11495,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,19,0,9],[8,67,0.1194,0.88392,0.09575,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,2,10],[12,67,0.1791,0.88837,0.07772,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[16,67,0.2388,0.89954,0.09255,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,0,15,0,13],[20,67,0.2985,0.88169,0.08859,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,1,9],[24,67,0.3582,0.8616,0.08364,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,21,0,6],[28,67,0.4179,0.85267,0.10403,0.857,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,18,0,7],[32,67,0.4776,0.85044,0.08057,0.85711,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,21,1,4],[36,67,0.5373,0.87498,0.1462,0.85714,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,16,0,12],[40,67,0.597,0.85713,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[44,67,0.6567,0.85713,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[48,67,0.7164,0.81696,0.13474,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,18,0,5],[52,67,0.7761,0.86146,0.12635,0.857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,15,0,10],[56,67,0.8358,0.86607,0.09407,0.85714,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,21,0,7],[60,67,0.8955,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7],[64,67,0.9552,0.79909,0.14666,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,20,0,3],[67,67,1.0,0.75221,0.1617,0.57143,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,6,1,0,3,0,0,17,0,2]]},{"b":5,"e":0.71429,"k":"flat","v":0.82587,"x":0.9375,"p":[[0,52,0.0,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[4,52,0.0769,0.83927,0.17767,0.857,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,8],[8,52,0.1538,0.83928,0.11152,0.82132,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,18,0,6],[12,52,0.2308,0.82587,0.1546,0.85711,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,19,0,6],[16,52,0.3077,0.87051,0.1033,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,9],[20,52,0.3846,0.88392,0.10374,0.85711,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,14,0,12],[24,52,0.4615,0.90177,0.07524,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[28,52,0.5385,0.85714,0.09449,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,6],[32,52,0.6154,0.87499,0.10565,0.85711,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,17,0,10],[36,52,0.6923,0.85267,0.08171,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,20,2,4],[40,52,0.7692,0.9174,0.07807,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,1,14],[44,52,0.8462,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[48,52,0.9231,0.91964,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,18],[52,52,1.0,0.87722,0.12782,0.857,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,14,1,12]]}]},{"i":"fddc5cdb616d1afc","q":"In the triangle  $ABC$  let  $AD$  be the interior angle bisector of  $\\angle ACB$ , where  $D\\in AB$ . The circumcenter of the triangle  $ABC$  coincides with the incenter of the triangle  $BCD$ . Prove that  $AC^2 = AD\\cdot AB$ .","t":[{"b":0,"e":0.57143,"k":"falling","v":0.49103,"x":0.8482,"p":[[0,39,0.0,0.8482,0.19867,0.57143,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,0,0,20],[4,39,0.1026,0.83929,0.23351,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,21],[8,39,0.2051,0.64284,0.26245,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,8,0,0,12,0,0,0,0,0,0,0,10],[12,39,0.3077,0.62499,0.25939,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,4,0,0,14,0,0,0,0,0,0,0,9],[16,39,0.4103,0.60268,0.2618,0.53572,0.57143,0.64286,0.0,1.0,1,7,0,1,0,2,0,0,1,0,0,4,0,0,16,0,0,0,0,0,1,0,7],[20,39,0.5128,0.49103,0.24726,0.28571,0.57143,0.57143,0.0,1.0,1,3,0,1,0,6,0,0,2,0,0,1,0,0,19,0,0,0,0,0,0,0,3],[24,39,0.6154,0.55356,0.24157,0.42857,0.57143,0.57143,0.0,1.0,1,4,0,1,0,3,0,0,1,0,0,4,0,0,17,0,0,1,0,0,1,0,4],[28,39,0.7179,0.56697,0.17672,0.57143,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,6,0,0,22,0,0,0,0,0,0,0,3],[32,39,0.8205,0.73653,0.22341,0.57143,0.57143,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,0,0,13],[36,39,0.9231,0.5446,0.23807,0.571,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,6,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,4],[39,39,1.0,0.67408,0.24547,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,1,0,0,19,0,0,0,0,0,0,0,10]]},{"b":1,"e":1.0,"k":"rising","v":0.81697,"x":1.0,"p":[[0,5,0.0,0.81697,0.2506,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,1,0,0,0,0,20],[4,5,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[5,5,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"51f417eeaa1c4735","q":"Find all sets  $X$  consisting of at least two positive integers such that for every two elements  $m,n\\in X$ , where  $n>m$ , there exists an element  $k\\in X$  such that  $n=mk^2$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.85714,"x":0.97767,"p":[[0,31,0.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,31,0.129,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[8,31,0.2581,0.87054,0.24578,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,23],[12,31,0.3871,0.92856,0.14289,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,24],[16,31,0.5161,0.88837,0.1417,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,18],[20,31,0.6452,0.87946,0.16793,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,2,0,19],[24,31,0.7742,0.87052,0.18683,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,18],[28,31,0.9032,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[31,31,1.0,0.85714,0.15972,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,7,0,14]]},{"b":4,"e":0.85714,"k":"flat","v":0.81249,"x":0.98661,"p":[[0,31,0.0,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[4,31,0.129,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,31,0.2581,0.9107,0.2075,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,25],[12,31,0.3871,0.90179,0.16917,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,22],[16,31,0.5161,0.8973,0.20899,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,24],[20,31,0.6452,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[24,31,0.7742,0.91517,0.18851,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[28,31,0.9032,0.85714,0.22588,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,4,0,0,5,0,19],[31,31,1.0,0.81249,0.24857,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,7,0,0,2,0,17]]}]},{"i":"ef5649a157119bcb","q":"Find the largest real number  $c$  such that \\[\\sum_{i=1}^{101}x_i^2\\geq cM^2\\] whenever  $x_1,\\ldots,x_{101}$  are real numbers such that  $x_1+\\cdots+x_{101}=0$  and  $M$  is the median of  $x_1,\\ldots,x_{101}$ .","t":[{"b":4,"e":0.71429,"k":"flat","v":0.89286,"x":0.99554,"p":[[0,69,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,69,0.058,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,69,0.1159,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[12,69,0.1739,0.94643,0.1915,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[16,69,0.2319,0.89286,0.19233,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,22],[20,69,0.2899,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[24,69,0.3478,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,69,0.4058,0.93304,0.14279,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,25],[32,69,0.4638,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[36,69,0.5217,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,69,0.5797,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[44,69,0.6377,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,69,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,69,0.7536,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[56,69,0.8116,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[60,69,0.8696,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[64,69,0.9275,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,69,0.9855,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[69,69,1.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,44,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,44,0.0909,0.94643,0.16269,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[8,44,0.1818,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,44,0.2727,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,44,0.3636,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[20,44,0.4545,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[24,44,0.5455,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[28,44,0.6364,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,44,0.9091,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,44,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a3b40d645aa984c0","q":"Find all positive integers  $m,n$  and prime numbers  $p$  for which  $\\frac{5^m+2^np}{5^m-2^np}$  is a perfect 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positive integer  $n$  we define  $f(n)$  as sum of all of its positive integer divisors (including  $1$  and  $n$ ). Find all positive integers  $c$  such that there exists strictly increasing infinite sequence of positive integers  $n_1, n_2,n_3,...$  such that for all  $i \\in \\mathbb{N}$  holds  $f(n_i)-n_i=c$","t":[{"b":0,"e":0.57143,"k":"falling","v":0.55354,"x":0.95535,"p":[[0,53,0.0,0.90625,0.16982,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,3,0,23],[4,53,0.0755,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[8,53,0.1509,0.95535,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[12,53,0.2264,0.92411,0.15966,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26],[16,53,0.3019,0.9375,0.13803,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,26],[20,53,0.3774,0.81253,0.22422,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,1,0,0,1,0,18],[24,53,0.4528,0.87052,0.20318,0.67857,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,1,0,22],[28,53,0.5283,0.92857,0.15152,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,26],[32,53,0.6038,0.82588,0.20745,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,2,0,0,1,0,18],[36,53,0.6792,0.72768,0.22406,0.57143,0.57143,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,1,0,0,3,0,11],[40,53,0.7547,0.61161,0.15251,0.57143,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,25,0,0,0,0,0,0,0,4],[44,53,0.8302,0.5714,0.13832,0.57143,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,0,1],[48,53,0.9057,0.55354,0.13243,0.57143,0.57143,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,0,0,1],[52,53,0.9811,0.56248,0.07085,0.57143,0.57143,0.57143,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0],[53,53,1.0,0.55796,0.04162,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.80802,"x":0.96427,"p":[[0,37,0.0,0.89731,0.17584,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,1,0,23],[4,37,0.1081,0.90177,0.19048,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,25],[8,37,0.2162,0.92856,0.15571,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[12,37,0.3243,0.92857,0.15567,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[16,37,0.4324,0.92411,0.14719,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,24],[20,37,0.5405,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[24,37,0.6486,0.89732,0.1931,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,1,0,24],[28,37,0.7568,0.91518,0.17807,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,26],[32,37,0.8649,0.96427,0.1072,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[36,37,0.973,0.80802,0.21313,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,1,0,0,1,0,17],[37,37,1.0,0.90625,0.16982,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,1,0,24]]}]},{"i":"e61d1fcbe81c9ea5","q":"For positive real numbers  $x,y,z$  with  $xy+yz+zx=1$ , prove that  $$ \\frac{2}{xyz}+9xyz \\geq 7(x+y+z) $$","t":[{"b":3,"e":0.85714,"k":"flat","v":0.22768,"x":0.41071,"p":[[0,62,0.0,0.25893,0.16536,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,3,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,62,0.0645,0.41071,0.25692,0.28571,0.28571,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,18,0,0,2,0,0,2,0,0,3,0,0,2,0,2],[8,62,0.129,0.33481,0.25154,0.2857,0.28571,0.32143,0.0,1.0,5,2,0,5,0,2,0,0,17,0,0,1,0,0,3,0,0,2,0,0,0,0,2],[12,62,0.1935,0.33036,0.2299,0.28571,0.28571,0.28571,0.0,1.0,4,1,0,4,0,1,0,0,21,0,0,0,0,0,3,0,0,0,0,0,2,0,1],[16,62,0.2581,0.34821,0.2257,0.28571,0.28571,0.28571,0.0,1.0,2,1,0,2,0,1,0,0,24,0,0,0,0,0,1,0,0,0,0,0,3,0,1],[20,62,0.3226,0.22768,0.12299,0.2857,0.28571,0.28571,0.0,0.42857,7,0,0,7,0,0,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.31696,0.25934,0.2857,0.28571,0.28571,0.0,1.0,6,1,0,6,0,1,0,0,20,0,0,0,0,0,0,0,0,1,0,0,3,0,1],[28,62,0.4516,0.3616,0.16746,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,22,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[32,62,0.5161,0.3125,0.19704,0.28571,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,1,0,0,24,0,0,1,0,0,0,0,0,0,0,0,3,0,0],[36,62,0.5806,0.30804,0.22619,0.28571,0.28571,0.28571,0.0,1.0,5,1,0,5,0,1,0,0,21,0,0,1,0,0,0,0,0,2,0,0,1,0,1],[40,62,0.6452,0.30357,0.15047,0.28571,0.28571,0.28571,0.0,1.0,2,1,0,2,0,0,0,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[44,62,0.7097,0.30357,0.13716,0.28571,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,0,0,0,26,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[48,62,0.7742,0.33036,0.18707,0.28571,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,2,0,0,25,0,0,0,0,0,1,0,0,0,0,0,3,0,0],[52,62,0.8387,0.27232,0.13054,0.28571,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,0,0,0,25,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[56,62,0.9032,0.25446,0.09268,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.32589,0.16457,0.28571,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[62,62,1.0,0.26786,0.13243,0.28571,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,1,0,0,24,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.85714,"k":"flat","v":0.25446,"x":0.40178,"p":[[0,74,0.0,0.25446,0.13709,0.24999,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,4,0,0,21,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,74,0.0541,0.26786,0.25692,0.0,0.28571,0.28571,0.0,1.0,9,1,0,9,0,2,0,0,17,0,0,0,0,0,0,0,0,1,0,0,2,0,1],[8,74,0.1081,0.30803,0.17169,0.28571,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,2,0,0,24,0,0,1,0,0,1,0,0,0,0,0,2,0,0],[12,74,0.1622,0.31696,0.23072,0.2857,0.28571,0.28571,0.0,1.0,5,1,0,5,0,1,0,0,19,0,0,3,0,0,1,0,0,0,0,0,2,0,1],[16,74,0.2162,0.29018,0.19719,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,2,0,0,22,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[20,74,0.2703,0.35268,0.23686,0.28571,0.28571,0.28571,0.0,1.0,2,2,0,2,0,1,0,0,24,0,0,0,0,0,1,0,0,0,0,0,2,0,2],[24,74,0.3243,0.30357,0.13243,0.28571,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,0,0,0,26,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[28,74,0.3784,0.38839,0.20897,0.28571,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,3,0,0,0,0,0,1,0,0,3,0,1],[32,74,0.4324,0.40178,0.24074,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,25,0,0,1,0,0,1,0,0,0,0,0,2,0,3],[36,74,0.4865,0.34375,0.26453,0.28571,0.28571,0.28571,0.0,1.0,4,3,0,4,0,2,0,0,19,0,0,2,0,0,1,0,0,0,0,0,1,0,3],[40,74,0.5405,0.29911,0.16506,0.28571,0.28571,0.28571,0.0,1.0,2,1,0,2,0,1,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[44,74,0.5946,0.375,0.19805,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,26,0,0,1,0,0,0,0,0,1,0,0,4,0,0],[48,74,0.6486,0.33482,0.19759,0.28571,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,0,0,0,26,0,0,0,0,0,0,0,0,1,0,0,3,0,0],[52,74,0.7027,0.26339,0.18935,0.2857,0.28571,0.28571,0.0,0.85714,6,0,0,6,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[56,74,0.7568,0.32143,0.19562,0.28571,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,0,0,0,24,0,0,2,0,0,0,0,0,0,0,0,3,0,0],[60,74,0.8108,0.26786,0.16269,0.28571,0.28571,0.28571,0.0,1.0,4,1,0,4,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,74,0.8649,0.2991,0.13997,0.28571,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[68,74,0.9189,0.34821,0.21706,0.28571,0.28571,0.28571,0.0,1.0,2,2,0,2,0,0,0,0,25,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[72,74,0.973,0.26339,0.15196,0.2857,0.28571,0.28571,0.0,0.57143,6,0,0,6,0,1,0,0,19,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[74,74,1.0,0.29018,0.16554,0.28571,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,1,0,0,22,0,0,2,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"7d7ec1a636e2b354","q":"For numbers  $a,b \\in \\mathbb{R}$  we consider the sets:  $$ A=\\{a^n | n \\in \\mathbb{N}\\} , B=\\{b^n | n \\in \\mathbb{N}\\} $$  Find all  $a,b > 1$  for which there exists two real , non-constant polynomials  $P,Q$  with positive leading coefficients st for each  $r \\in \\mathbb{R}$ :  $$  P(r) \\in A \\iff Q(r) \\in B $$","t":[{"b":4,"e":0.71429,"k":"flat","v":0.54461,"x":0.79016,"p":[[0,85,0.0,0.65173,0.19213,0.57143,0.71414,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,2,0,0,11,0,0,10,0,0,5,0,2],[4,85,0.0471,0.79016,0.25251,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,4,0,0,3,0,16],[8,85,0.0941,0.76338,0.24121,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,0,2,0,13],[12,85,0.1412,0.68748,0.26831,0.5354,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,4,0,0,3,0,0,4,0,0,7,0,0,5,0,8],[16,85,0.1882,0.76785,0.25939,0.67836,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,9,0,11],[20,85,0.2353,0.68301,0.29825,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,4,0,0,5,0,0,3,0,0,4,0,0,2,0,12],[24,85,0.2824,0.68748,0.21853,0.57132,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,8,0,0,3,0,7],[28,85,0.3294,0.7232,0.28781,0.57143,0.78571,1.0,0.0,1.0,2,11,0,2,0,0,0,0,2,0,0,2,0,0,5,0,0,5,0,0,5,0,11],[32,85,0.3765,0.73658,0.28148,0.53539,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,2,0,0,5,0,13],[36,85,0.4235,0.67411,0.30978,0.42859,0.71429,1.0,0.0,1.0,2,10,0,2,0,2,0,0,1,0,0,4,0,0,3,0,0,7,0,0,3,0,10],[40,85,0.4706,0.64718,0.2552,0.4286,0.71429,0.85714,0.14,1.0,0,6,0,0,0,2,0,0,3,0,0,4,0,0,6,0,0,7,0,0,4,0,6],[44,85,0.5176,0.5982,0.2976,0.39286,0.57143,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,5,0,0,4,0,0,5,0,0,5,0,0,4,0,6],[48,85,0.5647,0.66963,0.20652,0.57132,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,8,0,0,4,0,5],[52,85,0.6118,0.54461,0.31019,0.24999,0.57143,0.71429,0.0,1.0,2,5,0,2,0,6,0,0,1,0,0,3,0,0,7,0,0,6,0,0,2,0,5],[56,85,0.6588,0.74551,0.22229,0.571,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,7,0,0,3,0,11],[60,85,0.7059,0.558,0.29093,0.28571,0.57121,0.71429,0.0,1.0,1,6,0,1,0,4,0,0,4,0,0,4,0,0,6,0,0,7,0,0,0,0,6],[64,85,0.7529,0.59818,0.25615,0.4286,0.57143,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,2,0,0,4,0,0,8,0,0,8,0,0,4,0,3],[68,85,0.8,0.65622,0.27166,0.571,0.71429,0.85704,0.0,1.0,1,7,0,1,0,3,0,0,0,0,0,3,0,0,6,0,0,10,0,0,2,0,7],[72,85,0.8471,0.61158,0.24545,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,2,0,0,4,0,0,8,0,0,9,0,0,1,0,5],[76,85,0.8941,0.56249,0.27184,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,6,0,0,4,0,0,5,0,0,9,0,0,0,0,5],[80,85,0.9412,0.64061,0.23924,0.57143,0.71429,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,3,0,0,9,0,0,10,0,0,2,1,4],[84,85,0.9882,0.58475,0.19352,0.571,0.64286,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,1,0,0,4,0,0,9,0,0,15,0,0,1,0,0],[85,85,1.0,0.62497,0.12243,0.5713,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,19,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.41291,"x":0.82141,"p":[[0,50,0.0,0.66293,0.14965,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,8,1,0,13,0,0,3,0,2],[4,50,0.08,0.75892,0.27068,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,7,0,0,4,0,13],[8,50,0.16,0.82141,0.2287,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,4,0,0,8,0,14],[12,50,0.24,0.74999,0.26001,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,5,0,0,5,0,12],[16,50,0.32,0.70089,0.23787,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,1,0,0,8,0,0,6,0,0,6,0,7],[20,50,0.4,0.63381,0.29882,0.42857,0.64286,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,2,0,0,5,0,0,5,0,0,3,0,0,6,0,7],[24,50,0.48,0.69195,0.26272,0.53539,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,5,0,0,2,0,0,3,0,0,8,0,0,5,0,8],[28,50,0.56,0.62944,0.24964,0.5354,0.64286,0.75,0.14286,1.0,0,5,0,0,0,3,0,0,2,0,0,3,0,0,8,0,0,8,0,0,3,0,5],[32,50,0.64,0.62945,0.29421,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,2,0,0,6,0,0,2,0,0,6,0,0,6,0,6],[36,50,0.72,0.58926,0.25443,0.39286,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,6,0,0,4,0,0,6,0,0,5,0,0,6,0,3],[40,50,0.8,0.59147,0.22815,0.42857,0.571,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,7,0,0,6,0,0,6,0,0,3,1,3],[44,50,0.88,0.49105,0.29436,0.2857,0.4998,0.71429,0.0,1.0,3,4,0,3,0,3,0,0,6,0,0,4,0,0,5,0,0,7,0,0,0,0,4],[48,50,0.96,0.51337,0.2392,0.39285,0.57143,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,3,0,0,5,0,0,9,0,0,8,0,0,0,0,2],[50,50,1.0,0.41291,0.20801,0.2857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,3,0,1,8,0,0,5,0,0,8,0,0,5,0,0,0,0,0]]}]},{"i":"236abb2175c4bb2b","q":"Given triangle  $ABC$  and the points $D,E\\in \\left( BC \\right)$ ,  $F,G\\in \\left( CA \\right)$ ,  $H,I\\in \\left( AB \\right)$  so that  $BD=CE$ ,  $CF=AG$  and  $AH=BI$ . Note with  $M,N,P$  the midpoints of  $\\left[ GH \\right]$ ,  $\\left[ DI \\right]$  and  $\\left[ EF \\right]$  and with  ${M}'$  the intersection of the segments  $AM$ and  $BC$ .\na)\tProve that  $\\frac{B{M}'}{C{M}'}=\\frac{AG}{AH}\\cdot \\frac{AB}{AC}$ .\nb)\tProve that the segments $AM$ ,  $BN$  and  $CP$  are concurrent.","t":[{"b":2,"e":1.0,"k":"flat","v":0.90624,"x":0.99107,"p":[[0,23,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,23,0.1739,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[8,23,0.3478,0.90624,0.16216,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,23],[12,23,0.5217,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[16,23,0.6957,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,23,0.8696,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[23,23,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.78125,"x":1.0,"p":[[0,77,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,77,0.0519,0.78125,0.22011,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,2,0,0,3,0,14],[8,77,0.1039,0.89286,0.17857,0.82143,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,23],[12,77,0.1558,0.84375,0.19352,0.57143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,3,0,18],[16,77,0.2078,0.86161,0.20666,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,0,4,0,19],[20,77,0.2597,0.8482,0.19214,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,1,0,19],[24,77,0.3117,0.89285,0.16752,0.85713,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,7,0,19],[28,77,0.3636,0.83481,0.21757,0.57143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,1,0,19],[32,77,0.4156,0.87499,0.17405,0.82132,1.0,1.0,0.5714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,5,0,19],[36,77,0.4675,0.93748,0.13337,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[40,77,0.5195,0.9375,0.19865,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[44,77,0.5714,0.92409,0.13359,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,22],[48,77,0.6234,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[52,77,0.6753,0.9375,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[56,77,0.7273,0.91962,0.17477,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,24],[60,77,0.7792,0.96427,0.07145,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[64,77,0.8312,0.91963,0.13807,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,22],[68,77,0.8831,0.87945,0.21758,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0,4,0,22],[72,77,0.9351,0.87499,0.22233,0.85714,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,0,3,0,22],[76,77,0.987,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[77,77,1.0,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]}]},{"i":"1fc6617d56d2b5df","q":"If  $a, b, c$  are positive real numbers, prove that  $$ \\frac{a}{\\sqrt{(a + 2b)^3}}+\\frac{b}{\\sqrt{(b + 2c)^3}} +\\frac{c} {\\sqrt{(c + 2a)^3}} \\ge \\frac{1}{\\sqrt{a + b + c}} $$  Alexandru Mihalcu","t":[{"b":3,"e":0.1429,"k":"flat","v":0.28572,"x":0.98214,"p":[[0,118,0.0,0.41964,0.48173,0.0,0.07143,1.0,0.0,1.0,16,13,0,16,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[4,118,0.0339,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,118,0.0678,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,118,0.1017,0.89286,0.28571,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[16,118,0.1356,0.83036,0.3597,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[20,118,0.1695,0.83929,0.33646,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[24,118,0.2034,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[28,118,0.2373,0.87946,0.29904,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[32,118,0.2712,0.86161,0.32436,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[36,118,0.3051,0.9375,0.2257,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[40,118,0.339,0.88838,0.28289,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,26],[44,118,0.3729,0.82143,0.37287,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[48,118,0.4068,0.77679,0.3895,0.82143,1.0,1.0,0.0,1.0,3,24,0,3,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[52,118,0.4407,0.74107,0.41717,0.25,1.0,1.0,0.0,1.0,5,23,0,5,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[56,118,0.4746,0.6875,0.41101,0.24999,1.0,1.0,0.0,1.0,3,20,0,3,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,20],[60,118,0.5085,0.8125,0.35072,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,24],[64,118,0.5424,0.80804,0.36702,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[68,118,0.5763,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[72,118,0.6102,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[76,118,0.6441,0.82589,0.3655,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[80,118,0.678,0.79911,0.38276,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[84,118,0.7119,0.82589,0.36375,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[88,118,0.7458,0.62946,0.45156,0.14286,1.0,1.0,0.0,1.0,6,19,0,6,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[92,118,0.7797,0.74554,0.39566,0.35714,1.0,1.0,0.0,1.0,3,22,0,3,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,22],[96,118,0.8136,0.62946,0.45156,0.14286,1.0,1.0,0.0,1.0,6,19,0,6,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[100,118,0.8475,0.75,0.41033,0.46429,1.0,1.0,0.0,1.0,5,23,0,5,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,23],[104,118,0.8814,0.52677,0.46075,0.0,0.49979,1.0,0.0,1.0,10,15,0,10,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,15],[108,118,0.9153,0.38839,0.44925,0.0,0.14286,1.0,0.0,1.0,13,11,0,13,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[112,118,0.9492,0.49106,0.46144,0.0,0.2143,1.0,0.0,1.0,10,14,0,10,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,14],[116,118,0.9831,0.57134,0.46299,0.105,1.0,1.0,0.0,1.0,8,17,0,8,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17],[118,118,1.0,0.28572,0.41802,0.0,0.0,0.46427,0.0,1.0,17,8,0,17,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8]]},{"b":6,"e":0.4286,"k":"falling","v":0.02679,"x":0.96875,"p":[[0,147,0.0,0.36607,0.46282,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[4,147,0.0272,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[8,147,0.0544,0.80804,0.32656,0.67857,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,23],[12,147,0.0816,0.87946,0.28372,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,27],[16,147,0.1088,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,147,0.1361,0.85268,0.31029,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[24,147,0.1633,0.85714,0.33503,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[28,147,0.1905,0.88393,0.27534,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,27],[32,147,0.2177,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[36,147,0.2449,0.79018,0.3813,0.92857,1.0,1.0,0.0,1.0,4,24,0,4,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[40,147,0.2721,0.89732,0.27718,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[44,147,0.2993,0.71429,0.40248,0.28571,1.0,1.0,0.0,1.0,3,21,0,3,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,21],[48,147,0.3265,0.85715,0.32142,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[52,147,0.3537,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[56,147,0.381,0.86607,0.32328,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27],[60,147,0.4082,0.76777,0.40538,0.82143,1.0,1.0,0.0,1.0,5,24,0,5,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[64,147,0.4354,0.79911,0.3551,0.85715,1.0,1.0,0.0,1.0,1,24,0,1,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[68,147,0.4626,0.80802,0.34555,0.89275,1.0,1.0,0.0,1.0,2,24,0,2,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,24],[72,147,0.4898,0.77232,0.39747,0.82143,1.0,1.0,0.0,1.0,4,24,0,4,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[76,147,0.517,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[80,147,0.5442,0.82589,0.34577,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,25],[84,147,0.5714,0.88839,0.29824,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[88,147,0.5986,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[92,147,0.6259,0.79464,0.38949,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[96,147,0.6531,0.80804,0.34922,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[100,147,0.6803,0.69196,0.45752,0.0,1.0,1.0,0.0,1.0,9,22,0,9,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22],[104,147,0.7075,0.77232,0.39586,0.78571,1.0,1.0,0.0,1.0,3,24,0,3,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[108,147,0.7347,0.8125,0.3597,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,25],[112,147,0.7619,0.75,0.40248,0.25,1.0,1.0,0.0,1.0,3,23,0,3,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[116,147,0.7891,0.8125,0.35792,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[120,147,0.8163,0.89286,0.28571,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[124,147,0.8435,0.61161,0.44641,0.14286,1.0,1.0,0.0,1.0,5,18,0,5,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18],[128,147,0.8707,0.79018,0.38957,0.96429,1.0,1.0,0.0,1.0,5,24,0,5,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[132,147,0.898,0.79018,0.37284,0.85715,1.0,1.0,0.0,1.0,3,24,0,3,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[136,147,0.9252,0.64731,0.42406,0.14289,1.0,1.0,0.0,1.0,5,18,0,5,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,18],[140,147,0.9524,0.8125,0.35792,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[144,147,0.9796,0.24999,0.36769,0.0,0.14286,0.1786,0.0,1.0,15,5,0,15,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,5],[147,147,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5107510f0348371a","q":"In a park there are 23 trees  $t_0,t_1,\\dots,t_{22}$  in a circle and 22 birds  $b_1,n_2,\\dots,b_{22}.$  Initially, each bird is in a tree. Every minute, the bird  $b_i, 1\\leqslant i\\leqslant 22$  flies from the tree  $t_j{}$  to the tree  $t_{i+j}$  in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.","t":[{"b":1,"e":0.4286,"k":"falling","v":0.56695,"x":0.95982,"p":[[0,45,0.0,0.95981,0.12496,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[4,45,0.0889,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[8,45,0.1778,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[12,45,0.2667,0.91964,0.2257,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,28],[16,45,0.3556,0.92856,0.21131,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[20,45,0.4444,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[24,45,0.5333,0.93302,0.15969,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,27],[28,45,0.6222,0.86161,0.25376,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,23],[32,45,0.7111,0.9107,0.19152,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,25],[36,45,0.8,0.85712,0.25508,0.78571,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,1,0,23],[40,45,0.8889,0.71876,0.28003,0.42857,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,10,0,0,4,0,0,0,0,0,1,0,15],[44,45,0.9778,0.56695,0.18719,0.42857,0.4286,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,18,0,0,5,0,0,3,0,0,4,0,2],[45,45,1.0,0.58034,0.18877,0.42857,0.4998,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,15,0,0,5,0,0,4,0,0,6,0,1]]},{"b":6,"e":1.0,"k":"flat","v":0.97767,"x":0.99554,"p":[[0,18,0.0,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,18,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,18,0.4444,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,18,0.6667,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,18,0.8889,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[18,18,1.0,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31]]}]},{"i":"6755a3034267324e","q":"Show that there exists an integer $a$ for which $a^{3}-36 a^{2}+51 a-97$ is a multiple of $3^{2008}$.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.64732,"x":0.80803,"p":[[0,101,0.0,0.79018,0.19556,0.71429,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,5,0,0,16,0,6],[4,101,0.0396,0.70089,0.1488,0.57143,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,12,0,0,9,0,1],[8,101,0.0792,0.76786,0.15465,0.71429,0.78571,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,0,12,0,4],[12,101,0.1188,0.75,0.15152,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,12,0,0,11,0,3],[16,101,0.1584,0.77232,0.10012,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,14,0,1],[20,101,0.198,0.72321,0.13803,0.71429,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,16,0,0,11,0,0],[24,101,0.2376,0.71875,0.1838,0.67857,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,9,0,0,14,0,1],[28,101,0.2772,0.73214,0.17768,0.67857,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,6,0,0,17,0,1],[32,101,0.3168,0.77679,0.14698,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,17,0,2],[36,101,0.3564,0.75446,0.13939,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,13,0,2],[40,101,0.396,0.70534,0.14259,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,12,0,0,11,0,0],[44,101,0.4356,0.73213,0.18473,0.57143,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,5,0,0,17,0,1],[48,101,0.4752,0.73659,0.14774,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,9,0,0,12,0,2],[52,101,0.5149,0.64732,0.19228,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,12,0,0,9,0,0],[56,101,0.5545,0.73214,0.15047,0.67857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,11,0,0,11,0,2],[60,101,0.5941,0.72321,0.17474,0.57143,0.78571,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,5,0,0,14,0,2],[64,101,0.6337,0.73214,0.16656,0.67857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,0,13,0,2],[68,101,0.6733,0.71875,0.14934,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,10,0,1],[72,101,0.7129,0.76786,0.20124,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,6,0,0,15,0,5],[76,101,0.7525,0.70089,0.2,0.57143,0.78571,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,5,0,0,16,0,0],[80,101,0.7921,0.76786,0.15872,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,0,13,0,4],[84,101,0.8317,0.78125,0.14719,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,12,0,0,13,0,4],[88,101,0.8713,0.72321,0.11259,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,0,4,0,2],[92,101,0.9109,0.77232,0.10012,0.71429,0.78571,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,16,0,0],[96,101,0.9505,0.80803,0.14555,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,14,0,6],[100,101,0.9901,0.74554,0.12745,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,14,0,0,12,0,1],[101,101,1.0,0.75446,0.12492,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,14,0,0,13,0,1]]},{"b":7,"e":0.71429,"k":"flat","v":0.58929,"x":0.78571,"p":[[0,68,0.0,0.72768,0.17261,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,8,0,0,13,0,2],[4,68,0.0588,0.75,0.16751,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,7,0,0,19,0,0],[8,68,0.1176,0.74552,0.15867,0.67857,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,9,0,0,12,0,3],[12,68,0.1765,0.71429,0.16366,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,8,0,0,15,0,0],[16,68,0.2353,0.72321,0.15947,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,13,0,0,10,0,2],[20,68,0.2941,0.74107,0.17655,0.67857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,9,0,0,11,0,4],[24,68,0.3529,0.74107,0.18013,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,0,11,0,4],[28,68,0.4118,0.60714,0.26,0.42857,0.57143,0.85714,0.0,1.0,2,2,0,2,0,1,0,0,0,0,0,9,0,0,5,0,0,4,0,0,9,0,2],[32,68,0.4706,0.66071,0.25939,0.42857,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,7,0,0,4,0,0,5,0,0,8,0,5],[36,68,0.5294,0.70982,0.20973,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,5,0,0,10,0,5],[40,68,0.5882,0.62498,0.23891,0.42857,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,6,0,0,6,0,0,7,0,0,9,0,1],[44,68,0.6471,0.71429,0.24744,0.57143,0.78571,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,3,0,0,3,0,0,7,0,0,11,0,5],[48,68,0.7059,0.67856,0.20824,0.42857,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,7,0,0,1,0,0,9,0,0,12,0,1],[52,68,0.7647,0.6875,0.20025,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,10,0,0,8,0,3],[56,68,0.8235,0.65625,0.20473,0.42857,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,8,0,0,2,0,0,13,0,0,3,0,4],[60,68,0.8824,0.60267,0.24932,0.42857,0.71429,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,1,0,0,4,0,0,6,0,0,10,0,0,5,0,2],[64,68,0.9412,0.58929,0.25692,0.42857,0.64286,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,9,0,0,2,0,0,7,0,0,7,0,2],[68,68,1.0,0.78571,0.08748,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,18,0,0]]}]},{"i":"b9683d39e8a36825","q":"Peter has a deck of  $1001$  cards, and with a blue pen he has written the numbers  $1,2,\\ldots,1001$  on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card  $C$ , he took  $500$  consecutive cards following  $C$  (clockwise order), and denoted by  $f(C)$  the number of blue numbers written on those  $500$  cards that are greater than the blue number written on  $C$  itself. After all, he wrote this  $f(C)$  number on the top side of the card  $C$  with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.18304,"x":0.3884,"p":[[0,83,0.0,0.23652,0.30852,0.0,0.14286,0.42858,0.0,1.0,14,2,0,14,0,8,0,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,2],[4,83,0.0482,0.37947,0.37561,0.0,0.35714,0.60714,0.0,1.0,12,6,0,12,0,1,0,0,3,0,0,7,0,0,1,0,0,1,0,0,1,0,6],[8,83,0.0964,0.25,0.29014,0.0,0.14286,0.42857,0.0,1.0,12,3,0,12,0,5,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[12,83,0.1446,0.22321,0.2922,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,3,0,0,4,0,0,5,0,0,0,0,0,2,0,0,0,0,2],[16,83,0.1928,0.3482,0.2647,0.24999,0.28571,0.42857,0.0,1.0,6,3,0,6,0,2,0,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,3],[20,83,0.241,0.24999,0.20823,0.0,0.28571,0.42857,0.0,0.71429,11,0,0,11,0,1,0,0,9,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[24,83,0.2892,0.31696,0.34206,0.0,0.28571,0.42857,0.0,1.0,13,3,0,13,0,2,0,0,4,0,0,6,0,0,0,0,0,2,0,0,2,0,3],[28,83,0.3373,0.27679,0.24206,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,1,0,0,5,0,0,13,0,0,0,0,0,1,0,0,0,0,1],[32,83,0.3855,0.26786,0.24936,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,2,0,0,7,0,0,9,0,0,0,0,0,2,0,0,0,0,1],[36,83,0.4337,0.29017,0.30195,0.0,0.28571,0.42857,0.0,1.0,13,2,0,13,0,1,0,0,5,0,0,7,0,0,2,0,0,1,0,0,1,0,2],[40,83,0.4819,0.20534,0.24204,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,1,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[44,83,0.5301,0.20089,0.1984,0.0,0.21428,0.28571,0.0,0.71429,13,0,0,13,0,3,0,0,9,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[48,83,0.5783,0.20982,0.23954,0.0,0.07143,0.42857,0.0,0.71429,16,0,0,16,0,1,0,0,5,0,0,7,0,0,0,0,0,3,0,0,0,0,0],[52,83,0.6265,0.18304,0.24546,0.0,0.0,0.32164,0.0,1.0,17,1,0,17,0,3,0,0,4,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[56,83,0.6747,0.23214,0.20438,0.0,0.28571,0.42857,0.0,0.71429,12,0,0,12,0,2,0,0,6,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[60,83,0.7229,0.23661,0.20079,0.0,0.28571,0.42857,0.0,0.42857,13,0,0,13,0,0,0,0,4,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[64,83,0.7711,0.35268,0.18205,0.28571,0.42857,0.42857,0.0,1.0,3,1,0,3,0,3,0,0,6,0,0,19,0,0,0,0,0,0,0,0,0,0,1],[68,83,0.8193,0.37947,0.14987,0.28571,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,5,0,0,21,0,0,0,0,0,2,0,0,0,0,0],[72,83,0.8675,0.33482,0.1411,0.28571,0.42857,0.42857,0.0,0.4286,3,0,0,3,0,3,0,0,6,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[76,83,0.9157,0.32589,0.16065,0.2857,0.42857,0.42857,0.0,0.4286,5,0,0,5,0,2,0,0,4,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[80,83,0.9639,0.3884,0.1931,0.42857,0.42857,0.42857,0.0,1.0,4,1,0,4,0,1,0,0,1,0,0,24,0,0,0,0,0,1,0,0,0,0,1],[83,83,1.0,0.37054,0.1063,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,6,0,0,23,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"flat","v":0.16071,"x":0.30357,"p":[[0,37,0.0,0.21427,0.23417,0.14286,0.14286,0.17857,0.0,1.0,7,1,0,7,0,17,0,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,1],[4,37,0.1081,0.26786,0.23891,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,1,0,0,7,0,0,11,0,0,0,0,0,1,0,0,0,0,1],[8,37,0.2162,0.26786,0.20124,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,3,0,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[12,37,0.3243,0.22768,0.26693,0.0,0.14286,0.42857,0.0,1.0,14,2,0,14,0,3,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,2],[16,37,0.4324,0.27679,0.31326,0.0,0.28571,0.42857,0.0,1.0,14,3,0,14,0,0,0,0,8,0,0,5,0,0,0,0,0,2,0,0,0,0,3],[20,37,0.5405,0.19196,0.24643,0.0,0.0,0.32143,0.0,1.0,17,1,0,17,0,1,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[24,37,0.6486,0.24107,0.26592,0.0,0.28571,0.28571,0.0,1.0,12,2,0,12,0,3,0,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,2],[28,37,0.7568,0.16071,0.18814,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,6,0,0,5,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[32,37,0.8649,0.19196,0.23313,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,2,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[36,37,0.973,0.26785,0.1915,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,7,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[37,37,1.0,0.30357,0.27141,0.14286,0.21429,0.42857,0.0,1.0,7,1,0,7,0,9,0,0,2,0,0,9,0,0,1,0,0,1,0,0,2,0,1]]}]},{"i":"2d46c68951e532b5","q":"Suppose that  $m$  and  $k$  are non-negative integers, and  $p = 2^{2^m}+1$  is a prime number. Prove that**(a)**  $2^{2^{m+1}p^k} \\equiv 1$   $(\\text{mod } p^{k+1})$ ;**(b)**  $2^{m+1}p^k$  is the smallest positive integer  $n$  satisfying the congruence equation  $2^n \\equiv 1$   $(\\text{mod } p^{k+1})$ .","t":[{"b":4,"e":0.85714,"k":"rising","v":0.70979,"x":0.97321,"p":[[0,11,0.0,0.70979,0.27312,0.71429,0.857,0.85714,0.0,1.0,3,2,3,3,0,1,0,0,0,0,0,0,0,0,2,0,0,8,0,0,16,0,2],[4,11,0.3636,0.86606,0.11259,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,12,0,11],[8,11,0.7273,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[11,11,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.83481,"x":0.92857,"p":[[0,17,0.0,0.83481,0.27919,0.85714,0.85714,1.0,0.0,1.0,3,14,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,14],[4,17,0.2353,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[8,17,0.4706,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[12,17,0.7059,0.88392,0.11538,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,18,0,11],[16,17,0.9412,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[17,17,1.0,0.89732,0.10248,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14]]}]},{"i":"4b50a6951b767aac","q":"Suppose that  $x$ ,  $y$ ,  $z$  are real numbers  satisfying \\[x+y+z=12,\\qquad\\text{and}\\qquad x^2+y^2+z^2=54.\\] Prove that:[list](a) Each of the numbers  $xy$ ,  $yz$ ,  $zx$  is at least  $9$ , but at most  $25$ .\n(b) One of the numbers  $x$ ,  $y$ ,  $z$  is at most  $3$ , and another one is at least  $5$ .[/list]","t":[{"b":2,"e":0.42857,"k":"flat","v":0.53569,"x":0.69197,"p":[[0,29,0.0,0.59826,0.22425,0.42857,0.5007,0.60714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,16,0,0,8,0,0,1,0,0,0,0,7],[4,29,0.1379,0.62945,0.20782,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,11,0,0,10,0,0,4,0,0,1,0,6],[8,29,0.2759,0.6607,0.25443,0.42857,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,10,0,0,7,0,0,2,0,0,1,0,10],[12,29,0.4138,0.69197,0.22335,0.57142,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,12,0,0,1,0,0,3,0,9],[16,29,0.5517,0.61602,0.20342,0.42857,0.57143,0.64286,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,11,0,0,13,0,0,0,0,0,3,0,5],[20,29,0.6897,0.6696,0.2096,0.571,0.57143,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,7,0,0,12,0,0,5,0,0,0,0,8],[24,29,0.8276,0.53569,0.18898,0.42857,0.4286,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,15,0,0,7,0,0,4,0,0,0,0,3],[28,29,0.9655,0.55799,0.1868,0.42857,0.571,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,13,0,0,10,0,0,3,0,0,1,0,3],[29,29,1.0,0.59371,0.17169,0.42857,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,12,0,0,5,0,0,1,0,3]]},{"b":3,"e":0.71429,"k":"rising","v":0.53571,"x":0.88835,"p":[[0,63,0.0,0.53571,0.16751,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,19,0,0,8,0,0,1,0,0,2,0,2],[4,63,0.0635,0.6964,0.24421,0.571,0.64286,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,4,0,0,2,0,10],[8,63,0.127,0.71877,0.22999,0.5714,0.57143,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,0,0,0,1,0,12],[12,63,0.1905,0.67408,0.22085,0.42857,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,4,0,0,2,0,8],[16,63,0.254,0.64284,0.22868,0.42857,0.57143,0.85704,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,8,0,0,10,0,0,3,0,0,2,0,7],[20,63,0.3175,0.71873,0.21275,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,2,0,0,8,0,7],[24,63,0.381,0.70533,0.23404,0.53539,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,2,0,0,5,0,9],[28,63,0.4444,0.75,0.20516,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,6,0,0,6,0,9],[32,63,0.5079,0.7366,0.23448,0.57143,0.78571,1.0,0.286,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,1,0,0,5,0,11],[36,63,0.5714,0.71875,0.22441,0.53572,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,8,0,0,6,0,0,4,0,0,5,0,9],[40,63,0.6349,0.70979,0.21276,0.57132,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,6,0,0,2,0,9],[44,63,0.6984,0.72316,0.23132,0.571,0.64286,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,2,0,0,3,0,11],[48,63,0.7619,0.75443,0.24022,0.571,0.78564,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,4,0,0,3,0,13],[52,63,0.8254,0.77679,0.2141,0.57143,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,0,4,0,12],[56,63,0.8889,0.7946,0.2171,0.57143,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,5,0,0,0,0,16],[60,63,0.9524,0.88835,0.1627,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,1,0,21],[63,63,1.0,0.79908,0.18163,0.71429,0.857,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,6,0,11]]}]},{"i":"7dee3e1bb3c32153","q":"Solve the system of equations in real numbers:\n\\[ \\begin{cases} x-y+z-w=2  x^2-y^2+z^2-w^2=6  x^3-y^3+z^3-w^3=20  x^4-y^4+z^4-w^4=66 \\end{cases} \\]","t":[{"b":3,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,130,0.0,0.94643,0.1915,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[4,130,0.0308,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[8,130,0.0615,0.92857,0.21429,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[12,130,0.0923,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,130,0.1231,0.90625,0.21312,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,26],[20,130,0.1538,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[24,130,0.1846,0.86607,0.24206,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,23],[28,130,0.2154,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[32,130,0.2462,0.88839,0.23887,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,26],[36,130,0.2769,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,130,0.3077,0.94643,0.13243,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[44,130,0.3385,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[48,130,0.3692,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,26],[52,130,0.4,0.94196,0.15916,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,27],[56,130,0.4308,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[60,130,0.4615,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[64,130,0.4923,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[68,130,0.5231,0.92411,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[72,130,0.5538,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[76,130,0.5846,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[80,130,0.6154,0.92857,0.12877,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,24],[84,130,0.6462,0.91964,0.15542,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[88,130,0.6769,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[92,130,0.7077,0.92857,0.15972,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,26],[96,130,0.7385,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[100,130,0.7692,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[104,130,0.8,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[108,130,0.8308,0.92411,0.15146,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,25],[112,130,0.8615,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26],[116,130,0.8923,0.93304,0.15561,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,26],[120,130,0.9231,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[124,130,0.9538,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[128,130,0.9846,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[130,130,1.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26]]},{"b":4,"e":1.0,"k":"rising","v":0.76339,"x":0.99554,"p":[[0,140,0.0,0.76339,0.3722,0.5,1.0,1.0,0.0,1.0,2,22,0,2,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,22],[4,140,0.0286,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,140,0.0571,0.91963,0.15545,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,25],[12,140,0.0857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,140,0.1143,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,140,0.1429,0.94643,0.1915,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[24,140,0.1714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,140,0.2,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[32,140,0.2286,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,140,0.2571,0.95536,0.15746,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[40,140,0.2857,0.94196,0.15916,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[44,140,0.3143,0.9375,0.18877,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[48,140,0.3429,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[52,140,0.3714,0.94642,0.12756,1.0,1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number devil has coloured the integer numbers: every integer is coloured either black or white. \nThe number  $1$  is coloured white. For every two white numbers  $a$  and  $b$  ( $a$  and  $b$  are allowed to be equal) the numbers  $a-b$  and  $a + $ b have di\u000bfferent colours.\nProve that  $2011$  is coloured white.","t":[{"b":3,"e":0.42857,"k":"falling","v":0.34375,"x":0.96875,"p":[[0,101,0.0,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[4,101,0.0396,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[8,101,0.0792,0.8973,0.25062,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[12,101,0.1188,0.88393,0.19704,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,6,0,20],[16,101,0.1584,0.86606,0.2574,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,21],[20,101,0.198,0.93748,0.13337,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[24,101,0.2376,0.80578,0.28918,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,2,0,0,2,0,0,1,0,0,2,1,0,4,0,18],[28,101,0.2772,0.9241,0.15966,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[32,101,0.3168,0.78124,0.29877,0.5,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,6,0,17],[36,101,0.3564,0.76785,0.29179,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,6,0,15],[40,101,0.396,0.81682,0.28179,0.82132,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,8,0,16],[44,101,0.4356,0.75668,0.27944,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,1,0,0,2,0,0,4,0,0,4,1,0,6,0,12],[48,101,0.4752,0.63838,0.3813,0.28571,0.85714,1.0,0.0,1.0,5,12,0,5,0,0,0,0,6,0,0,1,0,0,1,0,0,2,0,0,5,0,12],[52,101,0.5149,0.83482,0.26027,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,4,0,19],[56,101,0.5545,0.7232,0.3008,0.42859,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,6,0,0,2,0,0,1,0,0,4,0,0,6,0,12],[60,101,0.5941,0.53124,0.34853,0.28571,0.35714,0.85714,0.0,1.0,3,7,0,3,0,2,0,0,11,0,0,1,0,0,1,0,0,3,0,0,4,0,7],[64,101,0.6337,0.59821,0.3597,0.2857,0.71429,1.0,0.0,1.0,4,9,0,4,0,0,0,0,9,0,0,0,0,0,2,0,0,3,0,0,5,0,9],[68,101,0.6733,0.69196,0.30952,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,8,0,0,1,0,0,3,0,0,5,0,0,0,0,14],[72,101,0.7129,0.6473,0.33117,0.28571,0.64286,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,11,0,0,2,0,0,2,0,0,1,0,0,2,0,13],[76,101,0.7525,0.66964,0.34151,0.28571,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,7,0,0,4,0,0,1,0,0,1,0,0,4,0,13],[80,101,0.7921,0.64284,0.31542,0.28571,0.78564,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,10,0,0,2,0,0,1,0,0,2,0,0,8,0,8],[84,101,0.8317,0.59821,0.29761,0.28571,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,12,0,0,3,0,0,4,0,0,1,0,0,4,0,8],[88,101,0.8713,0.58481,0.32411,0.28571,0.57143,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,12,0,0,1,0,0,2,0,0,3,0,0,4,0,8],[92,101,0.9109,0.50446,0.30511,0.28571,0.28571,0.85704,0.0,1.0,1,5,0,1,0,1,0,0,16,0,0,1,0,0,2,0,0,2,0,0,4,0,5],[96,101,0.9505,0.48212,0.28515,0.2857,0.28571,0.74996,0.2857,1.0,0,5,0,0,0,0,0,0,20,0,0,2,0,0,1,0,0,1,0,0,3,0,5],[100,101,0.9901,0.37053,0.15093,0.2857,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[101,101,1.0,0.34375,0.12299,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,20,0,0,8,0,0,2,0,0,1,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"flat","v":0.91071,"x":0.98661,"p":[[0,77,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,77,0.0519,0.96883,0.06888,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,77,0.1039,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,77,0.1558,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,77,0.2078,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,77,0.2597,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[24,77,0.3117,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[28,77,0.3636,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[32,77,0.4156,0.96874,0.06903,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[36,77,0.4675,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[40,77,0.5195,0.94642,0.14174,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[44,77,0.5714,0.91754,0.14397,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,1,0,2,0,23],[48,77,0.6234,0.91964,0.12846,0.85711,1.0,1.0,0.5714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[52,77,0.6753,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[56,77,0.7273,0.96874,0.08554,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[60,77,0.7792,0.91515,0.13303,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,8,0,20],[64,77,0.8312,0.91071,0.14617,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,21],[68,77,0.8831,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[72,77,0.9351,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[76,77,0.987,0.96428,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[77,77,1.0,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24]]}]},{"i":"cadb6cf2f6ff3226","q":"The sum of the squares of five real numbers  $a_1, a_2, a_3, a_4, a_5$  equals  $1$ . Prove that the least of the numbers  $(a_i - a_j)^2$ , where  $i, j = 1, 2, 3, 4,5$  and  $i \\neq j$ , does not exceed  $\\frac{1}{10}.$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.26786,"x":0.67857,"p":[[0,31,0.0,0.59821,0.43071,0.14286,1.0,1.0,0.0,1.0,2,17,0,2,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[4,31,0.129,0.67857,0.36943,0.25,0.92857,1.0,0.14286,1.0,0,16,0,0,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,16],[8,31,0.2581,0.66518,0.34183,0.28571,0.71429,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,8,0,0,1,0,0,3,0,0,2,0,0,0,0,15],[12,31,0.3871,0.61161,0.37327,0.28571,0.71429,1.0,0.14286,1.0,0,14,0,0,0,7,0,0,7,0,0,1,0,0,0,0,0,3,0,0,0,0,14],[16,31,0.5161,0.57143,0.39609,0.14286,0.57143,1.0,0.0,1.0,1,13,0,1,0,10,0,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,13],[20,31,0.6452,0.60714,0.40564,0.14286,0.85714,1.0,0.14286,1.0,0,16,0,0,0,11,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,16],[24,31,0.7742,0.375,0.33072,0.14286,0.2143,0.39286,0.14286,1.0,0,6,0,0,0,16,0,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,6],[28,31,0.9032,0.27232,0.24578,0.14286,0.14286,0.28571,0.14286,1.0,0,3,0,0,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[31,31,1.0,0.26786,0.23351,0.14286,0.14286,0.28571,0.14286,1.0,0,2,0,0,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,2]]},{"b":3,"e":0.14286,"k":"falling","v":0.17411,"x":0.72768,"p":[[0,36,0.0,0.55803,0.40933,0.14286,0.42857,1.0,0.0,1.0,1,14,0,1,0,12,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,14],[4,36,0.1111,0.72768,0.38855,0.14286,1.0,1.0,0.14286,1.0,0,21,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,21],[8,36,0.2222,0.61607,0.38206,0.25,0.71429,1.0,0.0,1.0,1,14,0,1,0,7,0,0,5,0,0,1,0,0,1,0,0,2,0,0,1,0,14],[12,36,0.3333,0.36607,0.34799,0.14286,0.14286,0.75,0.0,1.0,3,4,0,3,0,15,0,0,5,0,0,0,0,0,0,0,0,1,0,0,4,0,4],[16,36,0.4444,0.41964,0.39276,0.14286,0.2857,0.89286,0.0,1.0,6,8,0,6,0,9,0,0,5,0,0,1,0,0,1,0,0,0,0,0,2,0,8],[20,36,0.5556,0.47768,0.40344,0.14286,0.21429,1.0,0.0,1.0,3,11,0,3,0,13,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,11],[24,36,0.6667,0.36607,0.36411,0.14286,0.14286,0.64286,0.0,1.0,4,6,0,4,0,15,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,6],[28,36,0.7778,0.4017,0.37368,0.14286,0.14286,0.78571,0.0,1.0,1,8,0,1,0,18,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,8],[32,36,0.8889,0.24107,0.27067,0.14286,0.14286,0.14286,0.0,1.0,3,3,0,3,0,22,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[36,36,1.0,0.17411,0.21939,0.14286,0.14286,0.14286,0.0,1.0,5,2,0,5,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"7b0d72f717a1d19e","q":"Find the number of  $4$ -digit numbers (in base  $10$ ) having non-zero digits and which are divisible by  $4$  but not by  $8$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.80804,"x":1.0,"p":[[0,16,0.0,0.80804,0.26871,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,3,0,0,0,0,20],[4,16,0.25,0.8125,0.28446,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,18],[8,16,0.5,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[12,16,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,16,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.83929,"x":0.90625,"p":[[0,9,0.0,0.90625,0.17717,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,23],[4,9,0.4444,0.8616,0.09094,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,19,0,7],[8,9,0.8889,0.88839,0.10555,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,13],[9,9,1.0,0.83929,0.10564,0.71429,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,14,0,7]]}]},{"i":"62a6a719639825eb","q":"Let  $ABC$  be an acute triangle with  $\\angle ACB = 45^o$ ,  $G$  is the point of intersection of the medians, and  $O$  is the center of the circumscribed circle. If  $OG =1$  and  $OG \\parallel BC$ , find the length of  $BC$ .","t":[{"b":1,"e":0.57143,"k":"flat","v":0.49548,"x":0.66515,"p":[[0,49,0.0,0.66515,0.16983,0.57143,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,6,0,0,1,0,5],[4,49,0.0816,0.58477,0.15714,0.57143,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,26,0,0,0,0,0,0,0,3],[8,49,0.1633,0.55354,0.15872,0.57132,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,1,0,0,23,0,0,1,0,0,0,0,2],[12,49,0.2449,0.58478,0.15714,0.57143,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,26,0,0,0,0,0,0,0,3],[16,49,0.3265,0.56249,0.14698,0.57143,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,3,0,0,23,0,0,1,0,0,0,0,2],[20,49,0.4082,0.55352,0.13243,0.571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,1,0,0,24,0,0,2,0,0,0,0,1],[24,49,0.4898,0.5357,0.16751,0.57132,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,7,0,0,0,0,0,23,0,0,0,0,0,0,0,2],[28,49,0.5714,0.49548,0.13353,0.28571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,0,0,0,22,0,0,1,0,0,0,0,0],[32,49,0.6531,0.54017,0.16649,0.5354,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,2,0,0,21,0,0,1,0,0,0,0,2],[36,49,0.7347,0.60264,0.22228,0.571,0.57143,0.57143,0.2857,1.0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,19,0,0,1,0,0,0,0,6],[40,49,0.8163,0.50887,0.11809,0.571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,0,0,0,25,0,0,0,0,0,0,0,0],[44,49,0.898,0.49995,0.11842,0.4286,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,2,0,0,23,0,0,0,0,0,0,0,0],[48,49,0.9796,0.50444,0.1556,0.28571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,22,0,0,0,0,0,0,0,1],[49,49,1.0,0.51786,0.10564,0.5714,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,2,0,0,25,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.53121,"x":0.6964,"p":[[0,30,0.0,0.6964,0.17407,0.57143,0.57143,0.75,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,5,0,0,1,0,7],[4,30,0.1333,0.58926,0.12753,0.57143,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,25,0,0,2,0,0,0,0,2],[8,30,0.2667,0.53121,0.13939,0.571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,25,0,0,0,0,0,0,0,1],[12,30,0.4,0.53124,0.13474,0.57132,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,2,0,0,24,0,0,0,0,0,0,0,1],[16,30,0.5333,0.54908,0.07239,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0,0,0,0],[20,30,0.6667,0.54011,0.0855,0.571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,1,0,0,28,0,0,0,0,0,0,0,0],[24,30,0.8,0.55356,0.12752,0.57143,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,26,0,0,1,0,0,0,0,1],[28,30,0.9333,0.54909,0.12428,0.57143,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,27,0,0,0,0,0,0,0,1],[30,30,1.0,0.55354,0.06915,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0]]}]},{"i":"70e160f831a50b0e","q":"Let  $ABC$  be an acute traingle with  $AC \\neq BC$ . Point  $K$  is a foot of altitude through vertex  $C$ . Point  $O$  is a circumcenter of  $ABC$ . Prove that areas of quadrilaterals  $AKOC$  and  $BKOC$  are equal.","t":[{"b":1,"e":0.42857,"k":"falling","v":0.38838,"x":0.54463,"p":[[0,52,0.0,0.54463,0.25364,0.42857,0.49979,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,5,0,0,9,0,0,3,0,0,7,0,0,4,0,2],[4,52,0.0769,0.45089,0.21756,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,12,0,0,8,0,0,0,0,0,8,0,0,2,0,0],[8,52,0.1538,0.51338,0.24185,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,9,0,0,8,0,0,5,0,0,3,0,0,4,0,2],[12,52,0.2308,0.4464,0.18121,0.39286,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,14,0,0,5,0,0,4,0,0,1,0,0],[16,52,0.3077,0.43303,0.25376,0.28571,0.42857,0.57143,0.0,1.0,3,2,0,3,0,0,0,0,12,0,0,7,0,0,4,0,0,2,0,0,2,0,2],[20,52,0.3846,0.39285,0.15568,0.28571,0.35714,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,15,0,0,9,0,0,4,0,0,3,0,0,0,0,0],[24,52,0.4615,0.53125,0.22084,0.39286,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,10,0,0,2,0,0,7,0,0,4,0,1],[28,52,0.5385,0.53123,0.19638,0.42857,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,10,0,0,4,0,0,8,0,0,2,0,1],[32,52,0.6154,0.4375,0.17105,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,14,0,0,9,0,0,3,0,0,5,0,0,1,0,0],[36,52,0.6923,0.43303,0.15765,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,13,0,0,2,0,0,4,0,0,1,0,0],[40,52,0.7692,0.44641,0.20438,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,10,0,0,10,0,0,4,0,0,3,0,0,3,0,0],[44,52,0.8462,0.39286,0.14286,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,12,0,0,15,0,0,1,0,0,3,0,0,0,0,0],[48,52,0.9231,0.47321,0.16917,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,16,0,0,4,0,0,3,0,0,1,0,1],[52,52,1.0,0.38838,0.15662,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,15,0,0,9,0,0,2,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.39732,"x":0.57143,"p":[[0,38,0.0,0.57143,0.27894,0.28571,0.57143,0.75,0.0,1.0,1,6,0,1,0,0,0,0,9,0,0,4,0,0,6,0,0,4,0,0,2,0,6],[4,38,0.1053,0.46428,0.22015,0.28571,0.42857,0.46431,0.2857,1.0,0,2,0,0,0,0,0,0,13,0,0,11,0,0,2,0,0,1,0,0,3,0,2],[8,38,0.2105,0.44643,0.15465,0.39286,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,17,0,0,2,0,0,4,0,0,1,0,0],[12,38,0.3158,0.48214,0.24936,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,2,0,0,8,0,0,8,0,0,7,0,0,1,0,0,2,0,3],[16,38,0.4211,0.42411,0.20973,0.28571,0.42857,0.46429,0.0,1.0,1,2,0,1,0,0,0,0,14,0,0,9,0,0,5,0,0,0,0,0,1,0,2],[20,38,0.5263,0.44195,0.19018,0.28571,0.42857,0.46418,0.2857,1.0,0,2,0,0,0,0,0,0,13,0,0,11,0,0,4,0,0,2,0,0,0,0,2],[24,38,0.6316,0.49999,0.24484,0.28571,0.42857,0.60714,0.0,1.0,1,3,0,1,0,1,0,0,7,0,0,11,0,0,4,0,0,3,0,0,2,0,3],[28,38,0.7368,0.39732,0.17399,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,1,0,0,12,0,0,13,0,0,2,0,0,1,0,0,2,0,0],[32,38,0.8421,0.50444,0.21719,0.39286,0.42857,0.60714,0.0,1.0,1,2,0,1,0,0,0,0,7,0,0,10,0,0,6,0,0,5,0,0,1,0,2],[36,38,0.9474,0.47321,0.18364,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,8,0,0,8,0,0,3,0,0,1,0,1],[38,38,1.0,0.42857,0.14726,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,9,0,0,8,0,0,1,0,0,1,0,0]]}]},{"i":"4df2cb7a7f113465","q":"Given an integer  $n\\geq 2$ . There are  $N$  distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from  $X$ , players  $X$  and  $Y$  alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of  $n$  does  $Y$  have a winning strategy?","t":[{"b":3,"e":1.0,"k":"rising","v":0.28571,"x":0.77229,"p":[[0,28,0.0,0.39722,0.32294,0.105,0.42859,0.71429,0.0,1.0,8,2,1,8,0,4,0,0,3,0,0,4,0,0,4,0,0,5,0,0,2,0,2],[4,28,0.1429,0.31024,0.32467,0.0,0.2143,0.571,0.0,1.0,13,2,0,13,0,3,0,0,2,0,0,4,0,1,3,0,0,3,0,0,1,0,2],[8,28,0.2857,0.28571,0.3312,0.0,0.14286,0.46429,0.0,1.0,12,3,0,12,0,6,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,3],[12,28,0.4286,0.73658,0.26754,0.5354,0.78564,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,7,0,0,5,0,0,3,0,0,3,0,13],[16,28,0.5714,0.62049,0.29797,0.4286,0.64286,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,4,0,0,6,0,0,6,0,0,3,0,7],[20,28,0.7143,0.77229,0.25472,0.571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,3,0,0,3,0,15],[24,28,0.8571,0.69639,0.30253,0.4286,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,2,0,0,5,0,0,5,0,0,3,0,0,3,0,12],[28,28,1.0,0.76786,0.25187,0.571,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,2,0,0,1,0,16]]},{"b":7,"e":1.0,"k":"rising","v":0.29911,"x":0.75445,"p":[[0,37,0.0,0.29911,0.25843,0.10714,0.2857,0.57143,0.0,0.85714,8,0,0,8,0,7,0,0,5,0,0,3,0,0,5,0,0,3,0,0,1,0,0],[4,37,0.1081,0.41964,0.35702,0.0,0.42857,0.71429,0.0,1.0,10,3,0,10,0,2,0,0,1,0,0,7,0,0,1,0,0,4,0,0,4,0,3],[8,37,0.2162,0.46426,0.40563,0.0,0.42857,0.85714,0.0,1.0,11,7,0,11,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,0,4,0,7],[12,37,0.3243,0.34375,0.33476,0.0,0.35714,0.57143,0.0,1.0,11,3,0,11,0,4,0,0,1,0,0,7,0,0,2,0,0,3,0,0,1,0,3],[16,37,0.4324,0.48659,0.34229,0.14289,0.4286,0.75,0.0,1.0,6,5,0,6,0,3,0,0,1,0,0,8,0,0,3,0,0,3,0,0,3,0,5],[20,37,0.5405,0.67399,0.34867,0.42857,0.85714,1.0,0.0,1.0,4,11,0,4,0,1,0,0,1,0,0,4,0,0,2,0,0,3,0,0,6,0,11],[24,37,0.6486,0.4821,0.34394,0.24999,0.4286,0.75,0.0,1.0,6,6,0,6,0,2,0,0,3,0,0,8,0,0,3,0,0,2,0,0,2,0,6],[28,37,0.7568,0.6205,0.34922,0.39286,0.71429,0.85714,0.0,1.0,5,7,0,5,0,1,0,0,2,0,0,2,0,0,3,0,0,5,0,0,7,0,7],[32,37,0.8649,0.59373,0.35194,0.39286,0.57121,1.0,0.0,1.0,4,10,0,4,0,1,0,0,3,0,0,7,0,0,3,0,0,1,0,0,3,0,10],[36,37,0.973,0.75445,0.25314,0.4286,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,6,0,0,1,0,0,1,0,0,11,0,10],[37,37,1.0,0.70534,0.27879,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,7,0,0,1,0,0,6,0,0,5,0,10]]}]},{"i":"dc85f641e58c958e","q":"Let  $ABC$  be an equilateral triangle of side length  $15.$  Let  $A_b$  and  $B_a$  be points on side  $AB,$   $A_c$  and  $C_a$  be points on  $AC,$  and  $B_c$  and  $C_b$  be points on  $BC$  such that  $\\triangle{AA_bA_c}, \\triangle{BB_cB_a},$  and  $\\triangle{CC_aC_b}$  are equilateral triangles with side lengths  $3,4,$  and  $5,$  respectively. Compute the radius of the circle tangent to segments  $\\overline{A_bA_c}, \\overline{B_aB_c},$  and  $\\overline{C_aC_b}.$","t":[{"b":2,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,35,0.0,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[4,35,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,35,0.2286,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[12,35,0.3429,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,35,0.4571,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,35,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,35,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,35,0.9143,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[35,35,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,49,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,49,0.0816,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[8,49,0.1633,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,49,0.2449,0.96427,0.1429,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[16,49,0.3265,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,49,0.4082,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,49,0.4898,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,49,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,49,0.6531,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,49,0.7347,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,49,0.8163,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[44,49,0.898,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[48,49,0.9796,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[49,49,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"d667e3a24318e372","q":"Show that the set of all elements minus  $ 0 $  of a finite division ring that has at least  $ 4 $  elements can be partitioned into two nonempty sets  $ A,B $  having the property that  $$  \\sum_{x\\in A} x=\\prod_{y\\in B} y.  $$","t":[{"b":2,"e":0.28571,"k":"flat","v":0.67406,"x":0.90625,"p":[[0,45,0.0,0.78124,0.22864,0.67857,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,2,0,0,13,0,9],[4,45,0.0889,0.67406,0.26059,0.49968,0.85707,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,0,0,0,4,0,0,3,0,0,14,0,3],[8,45,0.1778,0.7857,0.18559,0.85714,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,24,0,2],[12,45,0.2667,0.79907,0.21089,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,2,0,0,14,0,9],[16,45,0.3556,0.84373,0.18683,0.85714,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,16,0,11],[20,45,0.4444,0.83482,0.25282,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,5,0,19],[24,45,0.5333,0.90625,0.1411,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,20],[28,45,0.6222,0.80804,0.26392,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,4,0,18],[32,45,0.7111,0.90624,0.16985,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,22],[36,45,0.8,0.90177,0.18015,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,3,0,23],[40,45,0.8889,0.87497,0.2308,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,2,0,23],[44,45,0.9778,0.85267,0.17674,0.85714,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,12,0,13],[45,45,1.0,0.7366,0.26513,0.53571,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,2,0,0,2,0,0,2,0,0,11,0,9]]},{"b":4,"e":0.2857,"k":"falling","v":0.58035,"x":0.74549,"p":[[0,36,0.0,0.74549,0.20745,0.57143,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,1,0,0,20,0,2],[4,36,0.1111,0.73214,0.22517,0.78571,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,24,0,0],[8,36,0.2222,0.65622,0.24965,0.28571,0.78571,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,3,0,0,4,0,0,15,0,1],[12,36,0.3333,0.74104,0.20959,0.57143,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,3,0,0,17,0,3],[16,36,0.4444,0.67855,0.25506,0.28571,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,0,0,0,2,0,0,0,0,0,21,0,0],[20,36,0.5556,0.68299,0.24417,0.571,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,0,0,0,5,0,0,0,0,0,20,0,0],[24,36,0.6667,0.5982,0.27067,0.28571,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,0,0,0,3,0,0,0,0,0,16,0,0],[28,36,0.7778,0.58481,0.29528,0.28571,0.85707,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,13,0,0,1,0,0,0,0,0,0,0,0,17,0,0],[32,36,0.8889,0.60264,0.27136,0.28571,0.78564,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,0,0,0,2,0,0,1,0,0,16,0,0],[36,36,1.0,0.58035,0.28107,0.28571,0.71421,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,16,0,0]]}]},{"i":"b22df3dd8b83db0f","q":"Let  $p,q, r, s$  be real numbers with  $q \\ne -1$  and  $s \\ne -1$ . Prove that the quadratic equations  $x^2 + px+q = 0$  and  $x^2 +rx+s = 0$  have a common root, while their other roots are inverse of each other, if and only if  $pr = (q+1)(s+1)$  and  $p(q+1)s = r(s+1)q$ .\n(A double root is counted twice.)","t":[{"b":0,"e":1.0,"k":"flat","v":0.80812,"x":0.96875,"p":[[0,65,0.0,0.95088,0.11072,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[4,65,0.0615,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[8,65,0.1231,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[12,65,0.1846,0.87053,0.19352,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,11,0,16],[16,65,0.2462,0.90625,0.18074,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,7,0,21],[20,65,0.3077,0.90178,0.12078,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,14,0,15],[24,65,0.3692,0.91517,0.14665,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,8,0,20],[28,65,0.4308,0.86605,0.17478,0.85714,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,9,0,16],[32,65,0.4923,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[36,65,0.5538,0.87486,0.15886,0.857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,10,0,15],[40,65,0.6154,0.89731,0.12496,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,16],[44,65,0.6769,0.92409,0.11285,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,18],[48,65,0.7385,0.88389,0.1492,0.857,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,17],[52,65,0.8,0.9241,0.1287,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,21],[56,65,0.8615,0.86156,0.14507,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,8,0,14],[60,65,0.9231,0.80812,0.1698,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,11,0,9],[64,65,0.9846,0.8973,0.17941,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,11,0,18],[65,65,1.0,0.89731,0.12996,0.82143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,18]]},{"b":2,"e":1.0,"k":"flat","v":0.94642,"x":0.99554,"p":[[0,52,0.0,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[4,52,0.0769,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,52,0.1538,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,52,0.2308,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,52,0.3077,0.94642,0.11156,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[20,52,0.3846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,52,0.4615,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,52,0.5385,0.97767,0.08074,1.0,1.0,1.0,0.5714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[32,52,0.6154,0.98661,0.04164,1.0,1.0,1.0,0.8571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,52,0.6923,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,52,0.7692,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,52,0.8462,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[48,52,0.9231,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[52,52,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"01184d696efcb9e3","q":"Solve in positive integers  $(a,b)$  such that  $3^a=b^2+2025$","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,91,0.0,0.96874,0.09938,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[4,91,0.044,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,91,0.0879,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,91,0.1319,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,91,0.1758,0.94629,0.13739,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[20,91,0.2198,0.94195,0.14226,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[24,91,0.2637,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,91,0.3077,0.9375,0.16342,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[32,91,0.3516,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,91,0.3956,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[40,91,0.4396,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,91,0.4835,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[48,91,0.5275,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[52,91,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,91,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,91,0.6593,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,91,0.7033,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[68,91,0.7473,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,91,0.7912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,91,0.8352,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,91,0.8791,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[84,91,0.9231,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[88,91,0.967,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[91,91,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.93304,"x":0.99107,"p":[[0,17,0.0,0.93304,0.13355,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,24],[4,17,0.2353,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[8,17,0.4706,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[12,17,0.7059,0.94196,0.15916,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,26],[16,17,0.9412,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[17,17,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"3549b3663c6c3984","q":"Let $A(n)$ denote the number of sequences $a_{1} \\geq a_{2} \\geq \\ldots \\geq a_{k}$ of positive integers for which $a_{1}+\\cdots+a_{k}=n$ and each $a_{i}+1$ is a power of two $(i=1,2, \\ldots, k)$. Let $B(n)$ denote the number of sequences $b_{1} \\geq b_{2} \\geq \\ldots \\geq b_{m}$ of positive integers for which $b_{1}+\\cdots+b_{m}=n$ and each inequality $b_{j} \\geq 2 b_{j+1}$ holds $(j=1,2, \\ldots, m-1)$.\n\nProve that $A(n)=B(n)$ for every positive integer $n$.","t":[{"b":5,"e":1.0,"k":"rising","v":0.6159,"x":0.82143,"p":[[0,31,0.0,0.6159,0.42041,0.14286,1.0,1.0,0.14,1.0,0,17,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,17],[4,31,0.129,0.71429,0.30929,0.42857,0.85714,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,3,0,0,6,0,0,3,0,0,2,0,0,0,0,16],[8,31,0.2581,0.82143,0.28122,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,4,0,0,2,0,20],[12,31,0.3871,0.81694,0.25815,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,2,0,0,2,0,19],[16,31,0.5161,0.73214,0.32488,0.42859,0.92857,1.0,0.0,1.0,1,16,0,1,0,2,0,0,3,0,0,4,0,0,0,0,0,4,0,0,2,0,16],[20,31,0.6452,0.71423,0.28574,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,3,0,0,4,0,0,6,0,0,4,0,11],[24,31,0.7742,0.62051,0.30849,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,6,0,0,1,0,0,5,0,0,3,0,0,6,0,0,3,0,8],[28,31,0.9032,0.71872,0.3164,0.42857,0.85707,1.0,0.0,1.0,1,14,0,1,0,3,0,0,0,0,0,6,0,0,1,0,0,4,0,0,3,0,14],[31,31,1.0,0.78125,0.31539,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,17]]},{"b":6,"e":1.0,"k":"volatile","v":0.35268,"x":0.97768,"p":[[0,36,0.0,0.53572,0.41649,0.14286,0.21445,1.0,0.14286,1.0,0,14,0,0,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,14],[4,36,0.1111,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,36,0.2222,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[12,36,0.3333,0.8482,0.25986,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,21],[16,36,0.4444,0.77232,0.28316,0.53571,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,3,0,0,2,0,17],[20,36,0.5556,0.48661,0.36746,0.14286,0.28571,1.0,0.14286,1.0,0,9,0,0,0,12,0,0,7,0,0,0,0,0,1,0,0,2,0,0,1,0,9],[24,36,0.6667,0.41962,0.3498,0.14286,0.2857,0.71429,0.0,1.0,1,7,0,1,0,14,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,7],[28,36,0.7778,0.35268,0.2696,0.14286,0.28571,0.46429,0.14286,1.0,0,2,0,0,0,14,0,0,9,0,0,1,0,0,1,0,0,4,0,0,1,0,2],[32,36,0.8889,0.7366,0.35012,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,5,0,17],[36,36,1.0,0.94643,0.10565,0.96429,1.0,1.0,0.5714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24]]}]},{"i":"45f02f2634638d88","q":"Let $A B$ and $A C$ be two distinct rays not lying on the same line, and let $\\omega$ be a circle with center $O$ that is tangent to ray $A C$ at $E$ and ray $A B$ at $F$. Let $R$ be a point on segment $E F$. The line through $O$ parallel to $E F$ intersects line $A B$ at $P$. Let $N$ be the intersection of lines $P R$ and $A C$, and let $M$ be the intersection of line $A B$ and the line through $R$ parallel to $A C$. Prove that line $M N$ is tangent to $\\omega$.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.51782,"x":0.91963,"p":[[0,59,0.0,0.51782,0.31287,0.28571,0.571,0.71429,0.0,1.0,4,4,1,4,0,2,0,0,5,0,0,3,0,0,6,0,0,5,0,0,3,0,4],[4,59,0.0678,0.87499,0.20749,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,0,0,22],[8,59,0.1356,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[12,59,0.2034,0.87946,0.1992,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,19],[16,59,0.2712,0.88839,0.21049,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,23],[20,59,0.339,0.8616,0.2004,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,3,0,19],[24,59,0.4068,0.87946,0.17168,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,19],[28,59,0.4746,0.76774,0.27859,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,0,0,0,2,0,0,2,0,0,8,0,0,3,0,14],[32,59,0.5424,0.84375,0.26088,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,0,2,0,20],[36,59,0.6102,0.77669,0.32346,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,0,0,0,2,0,0,0,0,0,6,0,0,3,0,17],[40,59,0.678,0.87946,0.22334,0.857,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,21],[44,59,0.7458,0.87052,0.20318,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,18],[48,59,0.8136,0.88839,0.18808,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,21],[52,59,0.8814,0.7723,0.30902,0.57132,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,2,0,0,1,0,0,3,0,0,4,0,0,1,0,18],[56,59,0.9492,0.83928,0.21651,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,7,0,0,4,0,17],[59,59,1.0,0.67855,0.29014,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,4,0,0,3,0,0,3,0,0,8,0,0,2,0,10]]},{"b":4,"e":0.4286,"k":"flat","v":0.57579,"x":0.89284,"p":[[0,40,0.0,0.5848,0.26088,0.42857,0.64286,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,4,0,0,7,0,0,3,0,0,9,0,0,4,0,3],[4,40,0.1,0.83482,0.25028,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,0,3,0,18],[8,40,0.2,0.87944,0.21465,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,21],[12,40,0.3,0.81247,0.26832,0.67846,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,3,0,0,2,0,19],[16,40,0.4,0.86149,0.23581,0.82143,1.0,1.0,0.14,1.0,0,22,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,1,0,0,2,0,22],[20,40,0.5,0.89284,0.19564,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,21],[24,40,0.6,0.78124,0.30927,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,0,0,0,0,0,0,3,0,0,7,0,0,0,0,18],[28,40,0.7,0.82588,0.26422,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,7,0,0,3,0,18],[32,40,0.8,0.87946,0.2372,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,22],[36,40,0.9,0.82589,0.27833,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,3,0,0,3,0,20],[40,40,1.0,0.57579,0.35091,0.28571,0.64286,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,2,0,0,5,0,0,2,0,0,5,0,0,3,0,8]]}]},{"i":"a449eefb24d7297a","q":"The points  $A,B,C,D$  lie, in this order, on a circle  $\\omega$ , where  $AD$  is a diameter of  $\\omega$ . Furthermore,  $AB=BC=a$  and  $CD=c$  for some relatively prime integers  $a$  and  $c$ . Show that if the diameter  $d$  of  $\\omega$  is also an integer, then either  $d$  or  $2d$  is a perfect square.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.35268,"x":0.37053,"p":[[0,9,0.0,0.37053,0.15093,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,5,0,0,2,0,0,2,0,0,1,0,0],[4,9,0.4444,0.37052,0.16696,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,0,0,0,3,0,0,3,0,0,1,0,0],[8,9,0.8889,0.36161,0.13825,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,6,0,0,2,0,0,1,0,0,1,0,0],[9,9,1.0,0.35268,0.11285,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,22,0,0,6,0,0,3,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.3125,"x":0.39732,"p":[[0,36,0.0,0.34375,0.1063,0.28571,0.28571,0.32143,0.2857,0.57143,0,0,0,0,0,0,0,0,24,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[4,36,0.1111,0.39732,0.198,0.2857,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,22,0,0,4,0,0,0,0,0,4,0,0,1,0,1],[8,36,0.2222,0.33928,0.12243,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[12,36,0.3333,0.33927,0.09276,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[16,36,0.4444,0.33482,0.10479,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,25,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[20,36,0.5556,0.33035,0.08328,0.28571,0.28571,0.32143,0.2857,0.57143,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[24,36,0.6667,0.33035,0.10374,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,26,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[28,36,0.7778,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.32589,0.08918,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[36,36,1.0,0.33482,0.07668,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"5ea11b7195cd8e8b","q":"The cubic equation  $x^3+2x-1=0$  has exactly one real root  $r$ .  Note that  $0.4<r<0.5$ .\n\n(a) Find, with proof, an increasing sequence of positive integers  $a_1 < a_2 < a_3 < \\cdots$ \u0001 \u0001 \u0001such that\n\\[\\frac{1}{2}=r^{a_1}+r^{a_2}+r^{a_3}+\\cdots.\\]\n(b) Prove that the sequence that you found in part (a) is the unique increasing sequence with the above property.","t":[{"b":1,"e":1.0,"k":"flat","v":0.85713,"x":0.89286,"p":[[0,16,0.0,0.88838,0.21648,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,25],[4,16,0.25,0.89286,0.18898,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,1,0,23],[8,16,0.5,0.88393,0.18707,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,0,0,22],[12,16,0.75,0.85713,0.21726,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,1,0,21],[16,16,1.0,0.875,0.20124,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,6,0,0,0,0,22]]},{"b":5,"e":1.0,"k":"flat","v":0.83482,"x":0.99107,"p":[[0,19,0.0,0.89732,0.1684,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,1,0,22],[4,19,0.2105,0.90179,0.19704,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,25],[8,19,0.4211,0.83482,0.2055,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,9,0,0,0,0,18],[12,19,0.6316,0.9375,0.13333,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25],[16,19,0.8421,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[19,19,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"54f745cf8a780f34","q":"Let  $ m,n$  be two natural nonzero numbers and sets  $ A \\equal{} \\{ 1,2,...,n\\}, B \\equal{} \\{1,2,...,m\\}$ . We say that subset  $ S$  of Cartesian product  $ A \\times B$  has property  $ (j)$  if  $ (a \\minus{} x)(b \\minus{} y)\\le 0$  for each pairs  $ (a,b),(x,y) \\in S$ . Prove that every set  $ S$  with propery  $ (j)$  has at most  $ m \\plus{} n \\minus{} 1$  elements.\n\n<span style=\"color:#FF0000\">The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into  $ (a \\minus{} x)(b \\minus{} y)\\ge 0$ , and that accounts for the following two posts.</span>","t":[{"b":3,"e":1.0,"k":"falling","v":0.7455,"x":0.97768,"p":[[0,17,0.0,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[4,17,0.2353,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,17,0.4706,0.90625,0.18074,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,24],[12,17,0.7059,0.9107,0.17408,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,24],[16,17,0.9412,0.75892,0.24338,0.5354,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,7,0,0,2,0,0,7,0,0,1,0,14],[17,17,1.0,0.7455,0.27138,0.57143,0.85707,1.0,0.0,1.0,2,9,0,2,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,10,0,9]]},{"b":5,"e":1.0,"k":"flat","v":0.86382,"x":0.96873,"p":[[0,10,0.0,0.86382,0.15193,0.76786,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,1,0,9,0,14],[4,10,0.4,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[8,10,0.8,0.94643,0.14174,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[10,10,1.0,0.96873,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28]]}]},{"i":"3ba80b0570563aa8","q":"Let  $ f,F:\\mathbb{R}\\longrightarrow\\mathbb{R} $  be two functions such that  $ f $  is nondecreasing,  $ F $  admits finite lateral derivates in every point of its domain,   $$  \\lim_{x\\to y^-} f(x)\\le\\lim_{x\\to y^-}\\frac{F(x)-F\\left( y \\right)}{x-y} ,\\lim_{x\\to y^+} f(x)\\ge\\lim_{x\\to y^+}\\frac{F(x)-F\\left( y \\right)}{x-y} ,   $$  for all real numbers  $ y, $  and  $ F(0)=0. $ Prove that  $ F(x)=\\int_0^x f(t)dt, $  for all real numbers  $ x. $","t":[{"b":5,"e":1.0,"k":"flat","v":0.78571,"x":0.98214,"p":[[0,38,0.0,0.78571,0.23146,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,0,10,0,10],[4,38,0.1053,0.9107,0.19807,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[8,38,0.2105,0.81697,0.33357,0.82143,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,22],[12,38,0.3158,0.87052,0.24838,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[16,38,0.4211,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,38,0.5263,0.875,0.29397,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,25],[24,38,0.6316,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[28,38,0.7368,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,38,0.8421,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[36,38,0.9474,0.94642,0.11714,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[38,38,1.0,0.91964,0.16342,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,23]]},{"b":6,"e":0.57143,"k":"falling","v":0.40625,"x":0.97768,"p":[[0,67,0.0,0.72767,0.31208,0.67857,0.857,1.0,0.0,1.0,3,11,0,3,0,1,0,0,1,0,0,0,0,0,3,0,0,7,0,0,6,0,11],[4,67,0.0597,0.9375,0.2257,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[8,67,0.1194,0.90178,0.23266,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,26],[12,67,0.1791,0.84375,0.32803,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[16,67,0.2388,0.90178,0.23266,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,25],[20,67,0.2985,0.84375,0.28652,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,1,0,23],[24,67,0.3582,0.85713,0.2396,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,22],[28,67,0.4179,0.87945,0.26514,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,25],[32,67,0.4776,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[36,67,0.5373,0.8125,0.34337,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,22],[40,67,0.597,0.92857,0.24223,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[44,67,0.6567,0.81696,0.34853,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[48,67,0.7164,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[52,67,0.7761,0.91518,0.19516,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[56,67,0.8358,0.83927,0.33073,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,25],[60,67,0.8955,0.64732,0.42255,0.14286,1.0,1.0,0.0,1.0,5,18,0,5,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,18],[64,67,0.9552,0.49107,0.43732,0.0,0.28571,1.0,0.0,1.0,9,11,0,9,0,5,0,0,3,0,0,0,0,0,0,0,0,2,0,0,2,0,11],[67,67,1.0,0.40625,0.41049,0.0,0.14286,0.85714,0.0,1.0,10,7,0,10,0,7,0,0,2,0,0,0,0,0,2,0,0,1,0,0,3,0,7]]}]},{"i":"b926befeb2faa11f","q":"3. A3 (CAN) Does there exist a function $s: \\mathbb{Q} \\rightarrow\\{-1,1\\}$ such that if $x$ and $y$ are distinct rational numbers satisfying $x y=1$ or $x+y \\in\\{0,1\\}$, then $s(x) s(y)=-1$ ? Justify your answer.","t":[{"b":0,"e":0.71429,"k":"rising","v":0.00446,"x":0.52232,"p":[[0,106,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,106,0.0377,0.45981,0.39079,0.0,0.42857,0.89286,0.0,1.0,10,8,0,10,0,0,0,0,5,0,0,2,0,0,5,0,0,1,0,0,1,0,8],[8,106,0.0755,0.42409,0.41107,0.0,0.28571,0.89286,0.0,1.0,12,8,1,12,0,2,0,0,3,0,0,0,0,0,5,0,0,1,0,0,1,0,8],[12,106,0.1132,0.52232,0.39222,0.14286,0.5,1.0,0.0,1.0,6,10,1,6,0,3,0,0,6,0,0,1,0,0,2,0,0,3,0,0,1,0,10],[16,106,0.1509,0.36606,0.3641,0.0,0.35714,0.57143,0.0,1.0,13,4,0,13,0,2,0,0,1,0,0,2,0,0,7,0,0,2,0,0,1,0,4],[20,106,0.1887,0.46875,0.42293,0.0,0.35716,1.0,0.0,1.0,11,11,0,11,0,0,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,11],[24,106,0.2264,0.42411,0.41724,0.0,0.28571,1.0,0.0,1.0,13,9,0,13,0,0,0,0,4,0,0,1,0,0,4,0,0,1,0,0,0,0,9],[28,106,0.2642,0.39731,0.40205,0.0,0.28571,0.71429,0.0,1.0,14,7,0,14,0,0,0,0,3,0,0,0,0,0,6,0,0,2,0,0,0,0,7],[32,106,0.3019,0.20088,0.27398,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,4,0,0,4,0,0,0,0,0,4,0,0,1,0,0,2,0,0],[36,106,0.3396,0.28124,0.34897,0.0,0.07143,0.57143,0.0,1.0,16,3,0,16,0,3,0,0,2,0,0,1,0,0,3,0,0,4,0,0,0,0,3],[40,106,0.3774,0.24106,0.31629,0.0,0.0,0.57143,0.0,1.0,18,2,0,18,0,2,0,0,1,0,0,0,0,0,9,0,0,0,0,0,0,0,2],[44,106,0.4151,0.31249,0.35613,0.0,0.21428,0.57143,0.0,1.0,15,4,0,15,0,1,0,0,4,0,0,1,0,0,5,0,0,2,0,0,0,0,4],[48,106,0.4528,0.23659,0.33043,0.0,0.0,0.46418,0.0,1.0,18,2,0,18,0,3,0,0,1,0,0,2,0,0,4,0,0,0,0,0,2,0,2],[52,106,0.4906,0.30801,0.37132,0.0,0.14288,0.57111,0.0,1.0,14,5,0,14,0,4,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,5],[56,106,0.5283,0.433,0.38874,0.0,0.28571,0.89286,0.0,1.0,9,8,0,9,0,2,0,0,7,0,0,1,0,0,4,0,0,0,0,0,1,0,8],[60,106,0.566,0.24552,0.34112,0.0,0.0,0.57143,0.0,1.0,19,3,1,19,0,1,0,0,2,0,0,0,0,0,6,0,0,1,0,0,0,0,3],[64,106,0.6038,0.20533,0.29434,0.0,0.0,0.32143,0.0,1.0,18,2,1,18,0,3,0,0,3,0,0,1,0,0,5,0,0,0,0,0,0,0,2],[68,106,0.6415,0.20536,0.24468,0.0,0.0,0.46429,0.0,0.57143,17,0,0,17,0,2,0,0,3,0,0,2,0,0,8,0,0,0,0,0,0,0,0],[72,106,0.6792,0.2812,0.2777,0.0,0.2857,0.571,0.0,1.0,12,1,0,12,0,3,0,0,6,0,0,0,0,0,9,0,0,1,0,0,0,0,1],[76,106,0.717,0.33036,0.36147,0.0,0.21428,0.57143,0.0,1.0,13,5,0,13,0,3,0,0,4,0,0,0,0,0,7,0,0,0,0,0,0,0,5],[80,106,0.7547,0.26339,0.32362,0.0,0.07143,0.46429,0.0,1.0,16,2,0,16,0,2,0,0,3,0,0,3,0,0,3,0,0,2,0,0,1,0,2],[84,106,0.7925,0.29911,0.32996,0.0,0.14286,0.57143,0.0,1.0,11,3,0,11,0,6,0,0,6,0,0,0,0,0,3,0,0,2,0,0,1,0,3],[88,106,0.8302,0.47766,0.32065,0.24999,0.57143,0.57143,0.0,1.0,6,4,0,6,0,2,0,0,4,0,0,0,0,0,13,0,0,1,0,0,2,0,4],[92,106,0.8679,0.45533,0.36672,0.0,0.4998,0.60714,0.0,1.0,9,7,0,9,0,1,0,0,3,0,0,3,0,0,8,0,0,1,0,0,0,0,7],[96,106,0.9057,0.28126,0.29984,0.0,0.28571,0.57143,0.0,1.0,14,2,0,14,0,1,0,0,5,0,0,2,0,0,8,0,0,0,0,0,0,0,2],[100,106,0.9434,0.4196,0.32129,0.0,0.571,0.57143,0.0,1.0,9,3,0,9,0,0,0,0,5,0,0,1,0,0,11,0,0,2,0,0,1,0,3],[104,106,0.9811,0.49106,0.37786,0.0,0.57143,0.78571,0.0,1.0,9,8,0,9,0,1,0,0,1,0,0,3,0,0,8,0,0,2,0,0,0,0,8],[106,106,1.0,0.45978,0.2806,0.28571,0.49979,0.57143,0.0,1.0,2,4,0,2,0,4,0,0,9,0,0,1,0,0,10,0,0,2,0,0,0,0,4]]},{"b":1,"e":0.57143,"k":"rising","v":0.03124,"x":0.74106,"p":[[0,89,0.0,0.03124,0.12228,0.0,0.0,0.0,0.0,0.571,30,0,1,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,89,0.0449,0.5625,0.38122,0.2857,0.57143,1.0,0.0,1.0,6,11,0,6,0,0,0,0,7,0,0,0,0,0,6,0,0,1,0,0,1,0,11],[8,89,0.0899,0.625,0.38755,0.2857,0.64286,1.0,0.0,1.0,6,13,0,6,0,1,0,0,2,0,0,1,0,0,6,0,0,1,0,0,2,0,13],[12,89,0.1348,0.48658,0.42387,0.0,0.571,1.0,0.0,1.0,12,10,0,12,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,0,1,0,10],[16,89,0.1798,0.5848,0.39505,0.14286,0.64286,1.0,0.0,1.0,7,11,0,7,0,2,0,0,1,0,0,1,0,0,5,0,0,3,0,0,2,0,11],[20,89,0.2247,0.49107,0.43439,0.0,0.57143,1.0,0.0,1.0,12,10,0,12,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,10],[24,89,0.2697,0.24999,0.32926,0.0,0.0,0.57141,0.0,1.0,18,2,0,18,0,1,0,0,3,0,0,0,0,0,6,0,0,1,0,0,1,0,2],[28,89,0.3146,0.3482,0.36932,0.0,0.28571,0.57143,0.0,1.0,13,5,0,13,0,2,0,0,4,0,0,1,0,0,6,0,0,0,0,0,1,0,5],[32,89,0.3596,0.37938,0.33614,0.0,0.28571,0.57143,0.0,1.0,10,4,0,10,0,2,0,0,5,0,0,0,0,0,10,0,0,1,0,0,0,0,4],[36,89,0.4045,0.36159,0.38296,0.0,0.28571,0.57143,0.0,1.0,14,6,0,14,0,0,0,0,5,0,0,0,0,0,6,0,0,1,0,0,0,0,6],[40,89,0.4494,0.32142,0.25753,0.10714,0.28571,0.57111,0.0,1.0,8,1,0,8,0,3,0,0,8,0,0,4,0,0,6,0,0,2,0,0,0,0,1],[44,89,0.4944,0.23213,0.33454,0.0,0.0,0.32143,0.0,1.0,18,4,0,18,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,4],[48,89,0.5393,0.38839,0.38504,0.0,0.28571,0.74996,0.0,1.0,14,3,0,14,0,0,0,0,3,0,0,0,0,0,5,0,0,2,0,0,5,0,3],[52,89,0.5843,0.42409,0.43665,0.0,0.28571,1.0,0.0,1.0,14,10,0,14,0,1,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,10],[56,89,0.6292,0.55356,0.38424,0.14286,0.71429,1.0,0.0,1.0,7,9,0,7,0,2,0,0,2,0,0,2,0,0,2,0,0,7,0,0,1,0,9],[60,89,0.6742,0.51783,0.41457,0.0,0.57121,1.0,0.0,1.0,10,10,0,10,0,1,0,0,1,0,0,2,0,0,5,0,0,1,0,0,2,0,10],[64,89,0.7191,0.44195,0.40932,0.0,0.35714,0.85714,0.0,1.0,11,7,0,11,0,3,0,0,2,0,0,1,0,0,3,0,0,2,0,0,3,0,7],[68,89,0.764,0.52676,0.3719,0.10714,0.57143,0.85714,0.0,1.0,8,7,0,8,0,1,0,0,2,0,0,0,0,0,8,0,0,4,0,0,2,0,7],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 $ n $  be a natural number, and  $ 2n $  nonnegative real numbers  $ a_1,a_2,\\ldots ,a_{2n} $  such that  $ a_1a_2\\cdots a_{2n}=1. $  Show that  $$  2^{n+1} +\\left( a_1^2+a_2^2 \\right)\\left( a_3^2+a_4^2 \\right)\\cdots\\left( a_{2n-1}^2+a_{2n}^2 \\right) \\ge 3\\left( a_1+a_2 \\right)\\left( a_3+a_4 \\right)\\cdots\\left( a_{2n-1}+a_{2n} \\right) ,  $$  and specify in which circumstances equality happens.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.08036,"x":0.16964,"p":[[0,141,0.0,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,141,0.0284,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,141,0.0567,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,141,0.0851,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,141,0.1135,0.13393,0.1234,0.14286,0.14286,0.14286,0.0,0.71429,7,0,0,7,0,23,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,141,0.1418,0.11607,0.08328,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,141,0.1702,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,141,0.1986,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,141,0.227,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,141,0.2553,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,141,0.2837,0.13393,0.15124,0.0,0.14286,0.14286,0.0,0.857,9,0,0,9,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,141,0.3121,0.12947,0.09008,0.10714,0.14286,0.14286,0.0,0.286,8,0,0,8,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,141,0.3404,0.09821,0.09062,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,141,0.3688,0.14286,0.07986,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,141,0.3972,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,141,0.4255,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,141,0.4539,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,141,0.4823,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,141,0.5106,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,141,0.539,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,141,0.5674,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,141,0.5957,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,141,0.6241,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,141,0.6525,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,141,0.6809,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,141,0.7092,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,141,0.7376,0.15178,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,141,0.766,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,141,0.7943,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,141,0.8227,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,141,0.8511,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,141,0.8794,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,141,0.9078,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,141,0.9362,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,141,0.9645,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[140,141,0.9929,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[141,141,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02679,"x":0.16518,"p":[[0,81,0.0,0.13839,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,81,0.0494,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,81,0.0988,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,81,0.1481,0.13839,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,81,0.1975,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,81,0.2469,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,81,0.2963,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,81,0.3457,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.125,0.05922,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,81,0.4444,0.16518,0.16409,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,81,0.4938,0.15179,0.09407,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,81,0.5432,0.12937,0.06544,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,81,0.642,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,81,0.7407,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,81,0.7901,0.08473,0.09348,0.0,0.14143,0.14286,0.0,0.42857,15,0,0,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b3803ce69dc41c0","q":"A simple graph is called \"divisibility\", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one.\n\nA simple graph is called \"permutationary\", if it's possible to put numbers  $1,2,...,n$  on its vertices and there is a permutation  $ \\pi $  such that there is an edge between vertices  $i,j$  if and only if  $i>j$  and  $\\pi(i)< \\pi(j)$  (it's not directed!)\n\nProve that a simple graph is permutationary if and only if its complement and itself are divisibility.\n\n*Proposed by Morteza Saghafian*\n.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.70089,"x":0.89282,"p":[[0,34,0.0,0.70089,0.15303,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,25,0,0,2,0,2],[4,34,0.1176,0.80801,0.23587,0.71429,0.92857,1.0,0.1429,1.0,0,16,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,8,0,0,2,0,16],[8,34,0.2353,0.79461,0.23676,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,6,0,0,5,0,14],[12,34,0.3529,0.70979,0.30407,0.5354,0.71429,1.0,0.0,1.0,1,12,0,1,0,2,0,0,3,0,0,2,0,0,2,0,0,7,0,0,3,0,12],[16,34,0.4706,0.79459,0.21115,0.71429,0.78564,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,3,0,13],[20,34,0.5882,0.83478,0.19924,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,8,0,14],[24,34,0.7059,0.73657,0.19271,0.67836,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,11,0,0,7,0,6],[28,34,0.8235,0.89282,0.17133,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,20],[32,34,0.9412,0.73657,0.18597,0.57142,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,7,0,0,10,0,5],[34,34,1.0,0.76783,0.19152,0.71429,0.78571,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,10,0,0,9,0,7]]},{"b":6,"e":1.0,"k":"flat","v":0.74552,"x":0.85268,"p":[[0,21,0.0,0.74552,0.13711,0.71429,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,24,0,0,1,0,5],[4,21,0.1905,0.83928,0.18813,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,9,0,0,3,0,16],[8,21,0.381,0.82589,0.18117,0.71429,0.857,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,12,0,0,3,0,14],[12,21,0.5714,0.81249,0.16538,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,1,0,13],[16,21,0.7619,0.85268,0.145,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,2,0,15],[20,21,0.9524,0.8348,0.14775,0.71429,0.78564,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,3,0,13],[21,21,1.0,0.83036,0.18013,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,11,0,0,4,0,14]]}]},{"i":"eb6bf2d78c490e4e","q":"Let  $ABC$  be a triangle with  $AB < AC$  and circumcenter  $O$ . The angle bisector of  $\\angle BAC$  meets the side  $BC$  at  $D$ . The line through  $D$  perpendicular to  $BC$  meets the segment  $AO$  at  $X$ . Furthermore, let  $Y$  be the midpoint of segment  $AD$ . Prove that points  $B, C, X, Y$  are concyclic.","t":[{"b":3,"e":0.0,"k":"flat","v":0.38391,"x":0.8817,"p":[[0,156,0.0,0.48656,0.1509,0.42857,0.571,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,10,0,0,17,0,0,0,0,0,1,0,0],[4,156,0.0256,0.78572,0.22016,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,0,9,0,10],[8,156,0.0513,0.81695,0.22655,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,10,0,13],[12,156,0.0769,0.7455,0.26902,0.57143,0.85712,1.0,0.0,1.0,1,10,0,1,0,2,0,0,0,0,0,1,0,0,6,0,0,4,0,0,8,0,10],[16,156,0.1026,0.8817,0.16426,0.82143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,5,1,18],[20,156,0.1282,0.80357,0.21053,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,11,0,10],[24,156,0.1538,0.72316,0.26714,0.57143,0.85714,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,0,0,0,0,0,0,8,0,0,4,0,0,10,0,7],[28,156,0.1795,0.75445,0.22085,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,3,0,0,12,0,7],[32,156,0.2051,0.77229,0.20784,0.57143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,1,0,0,11,0,9],[36,156,0.2308,0.80802,0.2276,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,14],[40,156,0.2564,0.8415,0.18878,0.71429,0.89286,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,6,1,15],[44,156,0.2821,0.83035,0.19377,0.82132,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,0,13,0,11],[48,156,0.3077,0.77902,0.21969,0.71429,0.85714,1.0,0.21429,1.0,0,10,0,0,0,0,0,1,0,0,0,5,0,0,1,0,0,6,0,0,9,0,10],[52,156,0.3333,0.83033,0.19705,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,8,0,14],[56,156,0.359,0.82141,0.18559,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,13],[60,156,0.3846,0.80583,0.19257,0.67857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,10,1,10],[64,156,0.4103,0.7321,0.21654,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,9,0,6],[68,156,0.4359,0.79018,0.22583,0.67857,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,11,0,10],[72,156,0.4615,0.81696,0.18638,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,8,0,12],[76,156,0.4872,0.74106,0.24074,0.57143,0.85707,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,4,0,0,7,0,0,3,0,0,8,0,9],[80,156,0.5128,0.67857,0.27893,0.4286,0.78564,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,0,0,0,6,0,0,6,0,0,1,0,0,9,0,7],[84,156,0.5385,0.77006,0.27361,0.67857,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,0,0,0,0,0,0,5,0,0,2,1,0,11,0,10],[88,156,0.5641,0.64286,0.33692,0.42857,0.78564,1.0,0.0,1.0,3,9,0,3,0,2,0,0,1,0,0,6,0,0,3,0,0,1,0,0,7,0,9],[92,156,0.5897,0.67853,0.26001,0.571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,1,0,0,4,0,0,8,0,0,4,0,0,6,0,7],[96,156,0.6154,0.58927,0.33264,0.39285,0.64286,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,2,0,0,3,0,0,5,0,0,6,0,0,3,0,7],[100,156,0.641,0.70979,0.28232,0.53539,0.78571,1.0,0.0,1.0,1,11,0,1,0,1,0,0,1,0,0,5,0,0,6,0,0,2,0,0,5,0,11],[104,156,0.6667,0.51782,0.30251,0.39286,0.42857,0.71429,0.0,1.0,4,4,0,4,0,1,0,0,3,0,0,10,0,0,2,0,0,5,0,0,3,0,4],[108,156,0.6923,0.63161,0.24165,0.42859,0.57121,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,2,0,0,7,0,0,8,0,0,4,0,0,5,1,4],[112,156,0.7179,0.67411,0.25812,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,7,0,0,4,0,0,4,0,0,8,0,6],[116,156,0.7436,0.6071,0.33312,0.39285,0.57143,0.85714,0.0,1.0,3,6,0,3,0,3,0,0,2,0,0,3,0,0,6,0,0,0,0,0,9,0,6],[120,156,0.7692,0.59375,0.26752,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,1,0,0,11,0,0,3,0,0,4,0,0,6,0,4],[124,156,0.7949,0.57132,0.29464,0.39286,0.57143,0.85714,0.0,1.0,2,4,0,2,0,3,0,0,3,0,0,4,0,0,6,0,0,5,0,0,5,0,4],[128,156,0.8205,0.55357,0.3531,0.25,0.57143,0.85714,0.0,1.0,5,6,0,5,0,3,0,0,1,0,0,5,0,0,4,0,0,2,0,0,6,0,6],[132,156,0.8462,0.5848,0.29528,0.42857,0.57143,0.85714,0.0,1.0,4,4,0,4,0,0,0,0,0,0,0,9,0,0,5,0,0,4,0,0,6,0,4],[136,156,0.8718,0.49542,0.25261,0.28571,0.571,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,4,0,0,6,0,0,8,0,0,6,0,0,1,0,2],[140,156,0.8974,0.46873,0.29065,0.39285,0.42857,0.60714,0.0,1.0,5,2,0,5,0,2,0,0,1,0,0,11,0,0,5,0,0,2,0,0,4,0,2],[144,156,0.9231,0.51337,0.35329,0.14286,0.49979,0.85704,0.0,1.0,6,5,0,6,0,3,0,0,1,0,0,6,0,0,3,0,0,3,0,0,5,0,5],[148,156,0.9487,0.56247,0.3193,0.42857,0.571,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,2,0,0,8,0,0,4,0,0,2,0,0,6,0,5],[152,156,0.9744,0.48211,0.27605,0.42857,0.4286,0.57143,0.0,1.0,5,2,0,5,0,1,0,0,0,0,0,11,0,0,8,0,0,2,0,0,3,0,2],[156,156,1.0,0.38391,0.22708,0.25001,0.42857,0.571,0.0,0.71429,5,0,0,5,0,3,0,0,3,0,0,12,0,0,4,0,0,5,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.08036,"x":0.81249,"p":[[0,46,0.0,0.5223,0.1365,0.42859,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,6,0,0,21,0,0,1,0,0,1,0,0],[4,46,0.087,0.77677,0.22571,0.57143,0.85707,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,6,0,11],[8,46,0.1739,0.64284,0.3312,0.5354,0.64286,1.0,0.0,1.0,3,9,0,3,0,3,0,0,0,0,0,2,0,0,8,0,0,2,0,0,5,0,9],[12,46,0.2609,0.76348,0.2566,0.57143,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,0,10,0,10],[16,46,0.3478,0.625,0.36726,0.35714,0.85707,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,0,0,0,3,0,0,1,0,0,3,0,0,10,0,7],[20,46,0.4348,0.7232,0.31327,0.57143,0.85714,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,1,0,0,7,0,0,1,0,0,8,0,11],[24,46,0.5217,0.66069,0.32878,0.42859,0.71429,1.0,0.0,1.0,3,9,0,3,0,2,0,0,1,0,0,3,0,0,4,0,0,4,0,0,6,0,9],[28,46,0.6087,0.81249,0.28222,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,19],[32,46,0.6957,0.75891,0.22143,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,2,0,0,13,0,7],[36,46,0.7826,0.71429,0.29451,0.57143,0.78571,1.0,0.0,1.0,1,11,0,1,0,2,0,0,2,0,0,2,0,0,4,0,0,5,0,0,5,0,11],[40,46,0.8696,0.55357,0.369,0.14286,0.57143,0.85714,0.0,1.0,6,6,0,6,0,3,0,0,0,0,0,5,0,0,4,0,0,0,0,0,8,0,6],[44,46,0.9565,0.48658,0.35688,0.10714,0.57121,0.85704,0.0,1.0,8,4,0,8,0,2,0,0,1,0,0,3,0,0,7,0,0,2,0,0,5,0,4],[46,46,1.0,0.08036,0.20183,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"5e4921c65650ebc2","q":"Let  $x_0,\\dots,x_{2017}$  are positive integers and  $x_{2017}\\geq\\dots\\geq x_0=1$  such that  $A=\\{x_1,\\dots,x_{2017}\\}$  consists of exactly  $25$  different numbers. Prove that  $\\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\\geq 623$ , and find the number of sequences that holds the case of equality.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.20982,"x":0.69643,"p":[[0,99,0.0,0.45982,0.29609,0.14286,0.57143,0.71429,0.0,0.85714,7,0,7,7,0,2,0,0,2,0,0,1,0,0,8,0,0,10,0,0,2,0,0],[4,99,0.0404,0.59822,0.23808,0.42859,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,2,0,0,8,0,0,10,0,0,1,0,4],[8,99,0.0808,0.48213,0.24679,0.28571,0.49979,0.60714,0.14286,1.0,0,3,0,0,0,3,0,0,12,0,0,1,0,0,8,0,0,5,0,0,0,0,3],[12,99,0.1212,0.49098,0.26241,0.28571,0.57143,0.71429,0.0,1.0,2,1,0,2,0,5,0,0,3,0,0,4,0,0,7,0,0,8,0,0,2,0,1],[16,99,0.1616,0.50897,0.26228,0.28571,0.50071,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,5,0,0,5,0,0,3,0,0,9,0,0,2,0,2],[20,99,0.202,0.60714,0.28794,0.39286,0.71429,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,5,0,0,4,0,0,3,0,0,8,0,0,3,0,6],[24,99,0.2424,0.63393,0.24727,0.53571,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,2,0,0,3,0,0,6,0,0,11,0,0,2,0,5],[28,99,0.2828,0.57589,0.28004,0.28571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,4,0,0,1,0,0,8,0,0,6,0,3],[32,99,0.3232,0.67411,0.1931,0.57143,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,18,0,0,1,0,4],[36,99,0.3636,0.63839,0.2525,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,1,0,0,6,0,0,4,0,0,9,0,0,4,0,5],[40,99,0.404,0.69643,0.19805,0.71429,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,18,0,0,3,0,4],[44,99,0.4444,0.67411,0.20277,0.4286,0.71429,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,7,0,0,1,0,0,14,0,0,4,0,4],[48,99,0.4848,0.66071,0.23891,0.53571,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,14,0,0,1,0,6],[52,99,0.5253,0.53571,0.23146,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,3,0,0,9,0,0,5,0,0,9,0,0,1,0,2],[56,99,0.5657,0.53125,0.23483,0.42857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,3,0,0,9,0,0,3,0,0,10,0,0,1,0,2],[60,99,0.6061,0.4375,0.24206,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,4,0,0,10,0,0,5,0,0,5,0,0,5,0,0,0,0,2],[64,99,0.6465,0.36606,0.21109,0.24999,0.28571,0.46418,0.14286,1.0,0,1,0,0,0,8,0,0,12,0,0,4,0,0,4,0,0,3,0,0,0,0,1],[68,99,0.6869,0.37499,0.25691,0.14286,0.28571,0.57143,0.0,1.0,3,1,0,3,0,8,0,0,6,0,0,5,0,0,3,0,0,6,0,0,0,0,1],[72,99,0.7273,0.34822,0.19865,0.25,0.28571,0.42857,0.0,1.0,1,1,0,1,0,7,0,0,10,0,0,10,0,0,1,0,0,2,0,0,0,0,1],[76,99,0.7677,0.35714,0.19885,0.24999,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,11,0,0,6,0,0,2,0,0,5,0,0,0,0,0],[80,99,0.8081,0.34374,0.21085,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,10,0,0,8,0,0,6,0,0,3,0,0,3,0,0,1,0,0],[84,99,0.8485,0.32143,0.19885,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,8,0,0,13,0,0,7,0,0,1,0,0,0,0,0,1,0,1],[88,99,0.8889,0.29463,0.20182,0.14286,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,9,0,0,12,0,0,3,0,0,1,0,0,4,0,0,0,0,0],[92,99,0.9293,0.20982,0.10705,0.14286,0.14286,0.28571,0.0,0.4286,2,0,0,2,0,16,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[96,99,0.9697,0.28571,0.20825,0.14286,0.2857,0.42857,0.0,1.0,2,1,0,2,0,13,0,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[99,99,1.0,0.27232,0.13997,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,11,0,0,13,0,0,4,0,0,3,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.13393,"x":0.4732,"p":[[0,153,0.0,0.44196,0.29957,0.10714,0.57143,0.60714,0.0,1.0,8,1,7,8,0,1,0,0,1,0,0,4,0,0,10,0,0,5,0,0,2,0,1],[4,153,0.0261,0.4732,0.26591,0.2857,0.42859,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,5,0,0,5,0,0,6,0,0,5,0,0,2,0,2],[8,153,0.0523,0.34374,0.21681,0.25,0.28571,0.57111,0.0,0.71429,5,0,0,5,0,3,0,0,10,0,0,5,0,0,6,0,0,3,0,0,0,0,0],[12,153,0.0784,0.36161,0.20511,0.28571,0.28571,0.46429,0.0,0.71429,3,0,0,3,0,4,0,0,10,0,0,7,0,0,4,0,0,4,0,0,0,0,0],[16,153,0.1046,0.43749,0.23673,0.28571,0.42857,0.57143,0.0,1.0,2,2,0,2,0,3,0,0,7,0,0,8,0,0,7,0,0,3,0,0,0,0,2],[20,153,0.1307,0.40178,0.25862,0.14286,0.35714,0.60714,0.0,0.85714,2,0,0,2,0,8,0,0,6,0,0,5,0,0,3,0,0,5,0,0,3,0,0],[24,153,0.1569,0.41518,0.25344,0.14289,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,9,0,0,8,0,0,3,0,0,5,0,0,5,0,0,0,0,2],[28,153,0.183,0.375,0.23623,0.14286,0.28571,0.57143,0.0,0.85714,3,0,0,3,0,6,0,0,8,0,0,5,0,0,4,0,0,5,0,0,1,0,0],[32,153,0.2092,0.29018,0.21275,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,6,0,0,12,0,0,4,0,0,4,0,0,0,0,0,0,0,1],[36,153,0.2353,0.35714,0.20825,0.24999,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,7,0,0,12,0,0,5,0,0,2,0,0,4,0,0,1,0,0],[40,153,0.2614,0.33481,0.24899,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,3,0,0,11,0,0,5,0,0,2,0,0,4,0,0,0,0,1],[44,153,0.2876,0.33036,0.22142,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,12,0,0,5,0,0,4,0,0,5,0,0,4,0,0,0,0,0],[48,153,0.3137,0.36161,0.2172,0.14286,0.28571,0.57143,0.0,0.71429,3,0,0,3,0,6,0,0,8,0,0,5,0,0,6,0,0,4,0,0,0,0,0],[52,153,0.3399,0.30804,0.1992,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,10,0,0,8,0,0,8,0,0,3,0,0,0,0,0,0,0,1],[56,153,0.366,0.35266,0.24479,0.14286,0.28571,0.46418,0.0,0.85714,5,0,0,5,0,5,0,0,7,0,0,7,0,0,2,0,0,5,0,0,1,0,0],[60,153,0.3922,0.28116,0.24092,0.14286,0.2857,0.42857,0.0,1.0,6,1,0,6,0,8,0,0,9,0,0,5,0,0,1,0,0,1,0,0,1,0,1],[64,153,0.4183,0.33927,0.24678,0.24999,0.28571,0.42857,0.0,1.0,5,2,0,5,0,3,0,0,12,0,0,6,0,0,3,0,0,1,0,0,0,0,2],[68,153,0.4444,0.31695,0.23617,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,6,0,0,10,0,0,5,0,0,2,0,0,2,0,0,2,0,0],[72,153,0.4706,0.375,0.24157,0.2857,0.28571,0.46429,0.0,0.85714,3,0,0,3,0,4,0,0,12,0,0,5,0,0,2,0,0,3,0,0,3,0,0],[76,153,0.4967,0.35714,0.22016,0.24999,0.28571,0.42858,0.0,1.0,3,1,0,3,0,5,0,0,9,0,0,8,0,0,4,0,0,2,0,0,0,0,1],[80,153,0.5229,0.31695,0.17397,0.24999,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,6,0,0,14,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[84,153,0.549,0.33482,0.25155,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,6,0,0,12,0,0,4,0,0,0,0,0,4,0,0,1,0,1],[88,153,0.5752,0.30357,0.22232,0.14286,0.28571,0.42858,0.0,0.71429,6,0,0,6,0,5,0,0,10,0,0,5,0,0,2,0,0,4,0,0,0,0,0],[92,153,0.6013,0.28571,0.19233,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,8,0,0,10,0,0,7,0,0,0,0,0,3,0,0,0,0,0],[96,153,0.6275,0.30803,0.20238,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,14,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[100,153,0.6536,0.26785,0.16656,0.14286,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,11,0,0,13,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[104,153,0.6797,0.30357,0.19805,0.14286,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,11,0,0,9,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[108,153,0.7059,0.20089,0.17445,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,12,0,0,10,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[112,153,0.732,0.23659,0.16599,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,6,0,0,12,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[116,153,0.7582,0.25893,0.16536,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,9,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[120,153,0.7843,0.30356,0.20437,0.14286,0.28571,0.32143,0.0,0.85714,2,0,0,2,0,9,0,0,13,0,0,4,0,0,1,0,0,1,0,0,2,0,0],[124,153,0.8105,0.23215,0.18123,0.10714,0.28571,0.32143,0.0,0.57143,8,0,0,8,0,7,0,0,9,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[128,153,0.8366,0.27678,0.15126,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,10,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[132,153,0.8627,0.24999,0.12874,0.14286,0.28571,0.28571,0.0,0.571,3,0,0,3,0,8,0,0,16,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[136,153,0.8889,0.17857,0.15972,0.0,0.14286,0.2857,0.0,0.57143,11,0,0,11,0,7,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[140,153,0.915,0.17848,0.1288,0.105,0.14286,0.28571,0.0,0.4286,8,0,0,8,0,10,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[144,153,0.9412,0.14723,0.13115,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,8,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[148,153,0.9673,0.13393,0.11811,0.0,0.14286,0.17857,0.0,0.42857,11,0,0,11,0,13,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[152,153,0.9935,0.16071,0.12752,0.0,0.14286,0.28571,0.0,0.42857,10,0,0,10,0,9,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[153,153,1.0,0.16964,0.14032,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,14,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"22d037b51dfeb737","q":"Let  $s(n)$  denote the sum of digits of a positive integer  $n$  in base  $10$ . If  $s(m)=20$  and  $s(33m)=120$ , what is the value of  $s(3m)$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,27,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,27,0.1481,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.91963,"x":0.96875,"p":[[0,16,0.0,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[4,16,0.25,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,16,0.5,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,16,0.75,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[16,16,1.0,0.91963,0.15129,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23]]}]},{"i":"ff4d1bf4b16b0986","q":"Let  $\\Gamma_1$  and  $\\Gamma_2$  be two circles of unequal radii, with centres  $O_1$  and  $O_2$  respectively, intersecting in two distinct points  $A$  and  $B$ . Assume that the centre of each circle is outside the other circle. The tangent to  $\\Gamma_1$  at  $B$  intersects  $\\Gamma_2$  again in  $C$ , different from  $B$ ; the tangent to  $\\Gamma_2$  at  $B$  intersects  $\\Gamma_1$  again at  $D$ , different from  $B$ . The bisectors of  $\\angle DAB$  and  $\\angle CAB$  meet  $\\Gamma_1$  and  $\\Gamma_2$  again in  $X$  and  $Y$ , respectively. Let  $P$  and  $Q$  be the circumcentres of triangles  $ACD$  and  $XAY$ , respectively. Prove that  $PQ$  is the perpendicular bisector of the line segment  $O_1O_2$ .\n\n*Proposed by Prithwijit De*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.17857,"x":0.70534,"p":[[0,64,0.0,0.21875,0.28118,0.0,0.14286,0.28571,0.0,1.0,12,3,0,12,0,7,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[4,64,0.0625,0.70534,0.31731,0.4286,0.78571,1.0,0.0,1.0,3,12,0,3,0,0,0,0,1,0,0,5,0,0,2,0,0,5,0,0,4,0,12],[8,64,0.125,0.6607,0.3531,0.39286,0.71429,1.0,0.0,1.0,2,13,0,2,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,0,2,0,13],[12,64,0.1875,0.558,0.36309,0.14286,0.71429,0.85714,0.0,1.0,6,7,0,6,0,3,0,0,1,0,0,1,0,0,4,0,0,8,0,0,2,0,7],[16,64,0.25,0.55802,0.34136,0.28571,0.71429,0.85714,0.0,1.0,5,6,0,5,0,1,0,0,4,0,0,4,0,0,1,0,0,8,0,0,3,0,6],[20,64,0.3125,0.62494,0.33024,0.42859,0.71429,0.875,0.0,1.0,4,7,0,4,0,2,0,0,0,0,1,2,0,0,4,0,0,8,0,0,3,1,7],[24,64,0.375,0.46427,0.34626,0.14286,0.4286,0.71429,0.0,1.0,7,5,0,7,0,4,0,0,0,0,0,6,0,0,3,0,0,7,0,0,0,0,5],[28,64,0.4375,0.50888,0.34056,0.25,0.4998,0.71429,0.0,1.0,6,6,0,6,0,2,0,0,1,0,0,7,0,0,4,0,0,5,0,0,1,0,6],[32,64,0.5,0.41069,0.24934,0.2857,0.42857,0.57111,0.0,1.0,5,1,0,5,0,1,0,0,6,0,0,10,0,0,3,0,0,6,0,0,0,0,1],[36,64,0.5625,0.38837,0.36636,0.0,0.28571,0.60714,0.0,1.0,10,6,0,10,0,3,0,0,4,0,0,4,0,0,3,0,0,2,0,0,0,0,6],[40,64,0.625,0.30799,0.32161,0.0,0.2857,0.57111,0.0,1.0,14,1,0,14,0,1,0,0,4,0,0,2,0,0,5,0,0,3,0,0,2,0,1],[44,64,0.6875,0.43299,0.37197,0.0,0.42859,0.71429,0.0,1.0,11,5,0,11,0,0,0,0,3,0,0,3,0,0,5,0,0,3,0,0,2,0,5],[48,64,0.75,0.2991,0.33948,0.0,0.14286,0.4642,0.0,1.0,16,3,0,16,0,0,0,0,1,0,0,7,0,0,2,0,0,3,0,0,0,0,3],[52,64,0.8125,0.47763,0.3285,0.24999,0.4286,0.60714,0.0,1.0,6,6,0,6,0,2,0,0,1,0,0,9,0,0,6,0,0,2,0,0,0,0,6],[56,64,0.875,0.17857,0.22016,0.0,0.0,0.42857,0.0,0.57143,19,0,0,19,0,0,0,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[60,64,0.9375,0.24106,0.23806,0.0,0.2143,0.4286,0.0,0.71429,14,0,0,14,0,2,0,0,2,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[64,64,1.0,0.23658,0.26146,0.0,0.07145,0.42858,0.0,0.71429,16,0,0,16,0,1,0,0,2,0,0,7,0,0,3,0,0,3,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"rising","v":0.1874,"x":0.5981,"p":[[0,69,0.0,0.1874,0.19045,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,9,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[4,69,0.058,0.58925,0.34209,0.28571,0.64264,1.0,0.0,1.0,4,9,0,4,0,1,0,0,4,0,0,4,0,0,3,0,0,6,0,0,1,0,9],[8,69,0.1159,0.5981,0.36158,0.28571,0.71429,1.0,0.0,1.0,5,9,0,5,0,2,0,0,2,0,0,2,0,0,4,0,0,5,0,0,3,0,9],[12,69,0.1739,0.33911,0.32695,0.0,0.2857,0.60714,0.0,1.0,10,2,0,10,0,4,0,0,6,0,0,2,0,0,2,0,0,4,0,0,2,0,2],[16,69,0.2319,0.33031,0.31827,0.0,0.28571,0.571,0.0,1.0,12,2,0,12,0,1,0,0,5,0,0,4,0,0,4,0,0,3,0,0,1,0,2],[20,69,0.2899,0.31247,0.3507,0.0,0.2143,0.571,0.0,1.0,14,3,0,14,0,2,0,0,4,0,0,3,0,0,2,0,0,2,0,0,2,0,3],[24,69,0.3478,0.46426,0.3234,0.2857,0.42857,0.71429,0.0,1.0,6,4,0,6,0,1,0,0,6,0,0,5,0,0,4,0,0,4,0,0,2,0,4],[28,69,0.4058,0.38838,0.34111,0.0,0.42857,0.71429,0.0,1.0,10,3,0,10,0,2,0,0,3,0,0,6,0,0,2,0,0,4,0,0,2,0,3],[32,69,0.4638,0.45531,0.3301,0.14286,0.4998,0.71429,0.0,1.0,7,4,0,7,0,2,0,0,4,0,0,3,0,0,5,0,0,7,0,0,0,0,4],[36,69,0.5217,0.36594,0.34247,0.0,0.28571,0.57143,0.0,1.0,11,2,0,11,0,2,0,0,5,0,0,1,0,0,6,0,0,1,0,0,4,0,2],[40,69,0.5797,0.26776,0.26669,0.0,0.2857,0.4286,0.0,1.0,12,1,0,12,0,3,0,0,5,0,0,6,0,0,3,0,0,2,0,0,0,0,1],[44,69,0.6377,0.50441,0.35352,0.21427,0.571,0.75,0.0,1.0,8,6,0,8,0,0,0,0,2,0,0,3,0,0,9,0,0,2,0,0,2,0,6],[48,69,0.6957,0.38389,0.33202,0.0,0.42857,0.57111,0.0,1.0,10,4,0,10,0,1,0,0,3,0,0,8,0,0,3,0,0,3,0,0,0,0,4],[52,69,0.7536,0.33471,0.30645,0.0,0.35714,0.57143,0.0,1.0,12,1,0,12,0,1,0,0,3,0,0,5,0,0,5,0,0,4,0,0,1,0,1],[56,69,0.8116,0.34374,0.37433,0.0,0.21428,0.57143,0.0,1.0,14,5,0,14,0,2,0,0,1,0,0,5,0,0,3,0,0,1,0,0,1,0,5],[60,69,0.8696,0.29909,0.34691,0.0,0.14288,0.57143,0.0,1.0,15,3,0,15,0,2,0,0,4,0,0,0,0,0,4,0,0,4,0,0,0,0,3],[64,69,0.9275,0.28567,0.29011,0.0,0.21428,0.57111,0.0,1.0,14,1,0,14,0,2,0,0,1,0,0,4,0,0,9,0,0,1,0,0,0,0,1],[68,69,0.9855,0.53553,0.34434,0.2857,0.571,0.75,0.0,1.0,6,7,0,6,0,1,0,0,2,0,0,4,0,0,7,0,0,4,0,0,1,0,7],[69,69,1.0,0.5312,0.34669,0.14289,0.571,0.75,0.0,1.0,5,7,0,5,0,4,0,0,0,0,0,4,0,0,7,0,0,4,0,0,1,0,7]]}]},{"i":"e42119c452c84f7d","q":"Let  $P$  be a point inside a triangle  $ABC$ . A line through  $P$  parallel to  $AB$  meets  $BC$  and  $CA$  at points  $L$  and  $F$ , respectively. A line through  $P$  parallel to  $BC$  meets  $CA$  and  $BA$  at points  $M$  and  $D$  respectively, and a line through  $P$  parallel to  $CA$  meets  $AB$  and  $BC$  at points  $N$  and  $E$  respectively. Prove\n\\begin{align*}\n[PDBL] \\cdot [PECM] \\cdot [PFAN]=8\\cdot [PFM] \\cdot [PEL] \\cdot\u0001 [PDN]  \\end{align*}\n\n*Proposed by Steve Dinh*","t":[{"b":3,"e":0.857,"k":"falling","v":0.7231,"x":1.0,"p":[[0,169,0.0,0.94196,0.17807,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[4,169,0.0237,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,169,0.0473,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,169,0.071,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[16,169,0.0947,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,169,0.1183,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,169,0.142,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,169,0.1657,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,169,0.1893,0.95313,0.15731,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,27],[36,169,0.213,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,169,0.2367,0.96428,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,169,0.2604,0.9598,0.1086,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[48,169,0.284,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,169,0.3077,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[56,169,0.3314,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,169,0.355,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[64,169,0.3787,0.93304,0.19556,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,27],[68,169,0.4024,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[72,169,0.426,0.98438,0.04966,1.0,1.0,1.0,0.78571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,29],[76,169,0.4497,0.91517,0.14667,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,23],[80,169,0.4734,0.96205,0.11845,1.0,1.0,1.0,0.357,1.0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,27],[84,169,0.497,0.96873,0.09275,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[88,169,0.5207,0.95536,0.1357,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[92,169,0.5444,0.9241,0.16361,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,25],[96,169,0.568,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[100,169,0.5917,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[104,169,0.6154,0.92857,0.19885,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[108,169,0.6391,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[112,169,0.6627,0.95982,0.09606,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[116,169,0.6864,0.91741,0.20133,1.0,1.0,1.0,0.1429,1.0,0,26,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,2,0,0,1,0,26],[120,169,0.7101,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[124,169,0.7337,0.90624,0.16985,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,22],[128,169,0.7574,0.9464,0.15878,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[132,169,0.7811,0.92409,0.16749,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,25],[136,169,0.8047,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[140,169,0.8284,0.95535,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[144,169,0.8521,0.93525,0.18157,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,1,27],[148,169,0.8757,0.95979,0.11435,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[152,169,0.8994,0.9107,0.19152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[156,169,0.9231,0.9375,0.15126,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,26],[160,169,0.9467,0.90847,0.22465,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,1,25],[164,169,0.9704,0.9442,0.12968,1.0,1.0,1.0,0.5,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,2,0,0,2,0,26],[168,169,0.9941,0.7231,0.20211,0.71429,0.71429,0.85714,0.14,1.0,0,4,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,14,0,0,8,0,4],[169,169,1.0,0.75442,0.10252,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,13,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,75,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,75,0.0533,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,75,0.1067,0.96875,0.12234,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[12,75,0.16,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,75,0.2133,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,75,0.2667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,75,0.32,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,75,0.3733,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,75,0.4267,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,75,0.48,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,75,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,75,0.5867,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,75,0.64,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,75,0.6933,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,75,0.7467,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,75,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,75,0.8533,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,75,0.9067,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,75,0.96,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[75,75,1.0,0.99777,0.01243,1.0,1.0,1.0,0.92857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31]]}]},{"i":"e5bdee4aa02fcbea","q":"Let $A B C$ be a triangle with $|A B|>|B C|$. Let $D$ be the midpoint of $A C$. Let $E$ be the intersection of the angle bisector of $\\angle A B C$ with the line $A C$. Let $F$ be a point on $B E$ such that $C F$ is perpendicular to $B E$. Let $G$ be the intersection of $C F$ and $B D$.\nProve that $D F$ bisects the line segment $E G$.","t":[{"b":4,"e":0.28571,"k":"flat","v":0.33929,"x":0.39732,"p":[[0,7,0.0,0.39732,0.13709,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,3,0,0,3,0,0,21,0,0,3,0,0,1,0,0,0,0,0],[4,7,0.5714,0.38839,0.09606,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.33929,0.10565,0.2857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,10,0,0,17,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.37951,"x":0.41518,"p":[[0,9,0.0,0.4107,0.13715,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,2,0,0,3,0,0,22,0,0,2,0,0,2,0,0,0,0,0],[4,9,0.4444,0.40623,0.10775,0.42857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,3,0,0,22,0,0,4,0,0,0,0,0,0,0,0],[8,9,0.8889,0.41518,0.07457,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[9,9,1.0,0.37951,0.10481,0.42857,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,5,0,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"515492d493126707","q":"Let $\\left(F_{k}\\right)_{k \\geqslant 0}$ be the sequence defined by $F_{0}=0, F_{1}=1$, and $F_{k+2}=F_{k}+F_{k+1}$ for all integers $k \\geqslant 0$. Let then $n \\geqslant 1$ be an integer. Prove that there are exactly $F_{n+1}$ ways to order the numbers $1,2, \\ldots, n$ to obtain an $n$-tuple $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ such that\n\n$$\na_{1} \\leqslant 2 a_{2} \\leqslant 3 a_{3} \\leqslant \\ldots \\leqslant n a_{n} .\n$$","t":[{"b":5,"e":0.42857,"k":"rising","v":0.45982,"x":0.65625,"p":[[0,24,0.0,0.48659,0.16697,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,10,0,0,8,0,0,7,0,0,0,0,0],[4,24,0.1667,0.45982,0.19475,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,6,0,0,0,0,0,16,0,0,1,0,0,9,0,0,0,0,0],[8,24,0.3333,0.48661,0.24707,0.25001,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,1,0,0,9,0,0,0,0,0,13,0,0,0,0,1],[12,24,0.5,0.47322,0.22711,0.35714,0.4286,0.71429,0.14286,0.71429,0,0,0,0,0,8,0,0,0,0,0,11,0,0,0,0,0,13,0,0,0,0,0],[16,24,0.6667,0.65625,0.11769,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,25,0,0,0,0,0],[20,24,0.8333,0.63393,0.14258,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,23,0,0,0,0,0],[24,24,1.0,0.65624,0.10631,0.67857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,24,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.41518,"x":0.69643,"p":[[0,51,0.0,0.41518,0.19678,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,8,0,0,9,0,0,4,0,0,6,0,0,0,0,0],[4,51,0.0784,0.52232,0.23585,0.42857,0.57144,0.71429,0.14286,1.0,0,1,0,0,0,6,0,0,1,0,0,9,0,0,0,0,0,15,0,0,0,0,1],[8,51,0.1569,0.56696,0.20973,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,5,0,0,0,0,0,5,0,0,3,0,0,19,0,0,0,0,0],[12,51,0.2353,0.51772,0.20736,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,15,0,0,0,0,0],[16,51,0.3137,0.59374,0.18595,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,3,0,0,0,0,0,7,0,0,1,0,0,21,0,0,0,0,0],[20,51,0.3922,0.64286,0.16752,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,24,0,0,1,0,0],[24,51,0.4706,0.66518,0.10479,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,26,0,0,0,0,0],[28,51,0.549,0.65177,0.14699,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,25,0,0,0,0,0],[32,51,0.6275,0.66518,0.12169,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,26,0,0,0,0,0],[36,51,0.7059,0.625,0.17768,0.67857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,0,24,0,0,0,0,0],[40,51,0.7843,0.65179,0.11811,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,25,0,0,0,0,0],[44,51,0.8627,0.65624,0.10631,0.67857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,24,0,0,0,0,0],[48,51,0.9412,0.69643,0.05923,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0],[51,51,1.0,0.69197,0.0724,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0]]}]},{"i":"b891ea463c6a2a24","q":"Let $ABC$ be a triangle and $\\Omega$ its circumcircle. We denote $A'$ as the point diametrically opposite to $A$ on the circle $\\Omega$. Let $I$ be the center of the incircle of triangle $ABC$, $E$ and $F$ the points of tangency of the incircle with the sides $AC$ and $AB$ respectively. The circumcircle of triangle $AEF$ intersects the circle $\\Omega$ at point $X$. Show that the points $A'$, $I$, and $X$ are collinear.","t":[{"b":0,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,26,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,26,0.1538,0.93304,0.22011,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[8,26,0.3077,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[12,26,0.4615,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.73214,"x":1.0,"p":[[0,18,0.0,0.73214,0.40681,0.28571,1.0,1.0,0.0,1.0,5,21,0,5,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,21],[4,18,0.2222,0.875,0.30252,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,27],[8,18,0.4444,0.875,0.27374,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,24],[12,18,0.6667,0.89723,0.24831,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,26],[16,18,0.8889,0.83036,0.35254,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,25],[18,18,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6882d97ac40deba2","q":"Let  $U$  be the incenter of a triangle  $ABC$  and  $O_{1}, O_{2}, O_{3}$  be the circumcenters of the triangles  $BCU, CAU, ABU$  , respectively. Prove that the circumcircles of the triangles  $ABC$  and  $O_{1}O_{2}O_{3}$  have the same center.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.70089,"x":0.89732,"p":[[0,25,0.0,0.89732,0.1439,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,21],[4,25,0.16,0.8125,0.1357,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,11],[8,25,0.32,0.75,0.20203,0.71429,0.71429,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,22,0,0,0,0,8],[12,25,0.48,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[16,25,0.64,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[20,25,0.8,0.70089,0.07457,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[24,25,0.96,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[25,25,1.0,0.70981,0.02485,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"falling","v":0.51786,"x":0.88839,"p":[[0,24,0.0,0.88839,0.13709,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,19],[4,24,0.1667,0.77232,0.20158,0.71429,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,0,0,10],[8,24,0.3333,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[12,24,0.5,0.73214,0.06916,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2],[16,24,0.6667,0.67411,0.2321,0.71429,0.71429,0.71429,0.0,1.0,3,3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,3],[20,24,0.8333,0.64719,0.24736,0.71429,0.71429,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,1,0,0,0,0,0,0,0,0,24,0,0,0,0,3],[24,24,1.0,0.51786,0.3004,0.2857,0.71429,0.71429,0.0,0.71429,7,0,0,7,0,0,0,0,3,0,0,0,0,0,0,0,0,22,0,0,0,0,0]]}]},{"i":"1084749d9bdc7c40","q":"Let $ABC$ be a triangle, with $AC > AB$, and $\\Gamma$ its circumcircle. Let $T$ be the intersection point of the tangent to $\\Gamma$ at $A$ with $(BC)$. Let $M$ be the midpoint of $[BC]$ and $R$ the symmetric point of $A$ with respect to $B$. Let $S$ be the point such that $SABT$ is a parallelogram. The parallel to $(AB)$ passing through $M$ intersects $(SB)$ at $P$. Suppose that $P$ is on $\\Gamma$, show that $(AC)$ is tangent to the circumcircle of $SRT$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.22768,"p":[[0,54,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.19196,0.16982,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,3,0,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[8,54,0.1481,0.17857,0.19233,0.0,0.21428,0.28571,0.0,0.71429,14,0,0,14,0,2,0,0,14,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[12,54,0.2222,0.16964,0.17655,0.0,0.28571,0.28571,0.0,0.71429,15,0,0,15,0,0,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,54,0.2963,0.22768,0.18509,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,0,0,0,19,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[20,54,0.3704,0.16062,0.17036,0.0,0.14143,0.28571,0.0,0.57143,15,0,0,15,0,2,0,0,13,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,54,0.4444,0.1607,0.16265,0.0,0.21428,0.28571,0.0,0.571,15,0,0,15,0,1,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,54,0.5185,0.13839,0.15355,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,3,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,54,0.5926,0.08929,0.13243,0.0,0.0,0.28571,0.0,0.28571,22,0,0,22,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.10268,0.12993,0.0,0.0,0.28571,0.0,0.28571,19,0,0,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.08929,0.16269,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,54,0.8889,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.27232,"p":[[0,77,0.0,0.03572,0.10102,0.0,0.0,0.0,0.0,0.42857,28,0,1,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.1875,0.15746,0.0,0.2857,0.28571,0.0,0.42857,12,0,0,12,0,2,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,77,0.1039,0.25,0.15567,0.2857,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,0,0,0,21,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[12,77,0.1558,0.27232,0.13533,0.2857,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,1,0,0,23,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,77,0.2078,0.19197,0.14111,0.0,0.28571,0.28571,0.0,0.42857,11,0,0,11,0,0,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.26339,0.17536,0.2857,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,0,0,0,20,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[24,77,0.3117,0.22768,0.17075,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,2,0,0,16,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[28,77,0.3636,0.19196,0.16602,0.0,0.2857,0.28571,0.0,0.4286,13,0,0,13,0,0,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,77,0.4156,0.16519,0.15613,0.0,0.28571,0.28571,0.0,0.57143,14,0,0,14,0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,77,0.4675,0.13839,0.16164,0.0,0.0,0.28571,0.0,0.5714,17,0,0,17,0,2,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,77,0.5195,0.1741,0.13709,0.0,0.2857,0.28571,0.0,0.28571,12,0,0,12,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,77,0.5714,0.14286,0.17128,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,1,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,77,0.6234,0.18304,0.18293,0.0,0.2143,0.28571,0.0,0.71429,13,0,0,13,0,3,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[52,77,0.6753,0.15177,0.15538,0.0,0.14286,0.28571,0.0,0.571,15,0,0,15,0,2,0,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,77,0.7273,0.05803,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.1183,0.13902,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,1,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78867e7f2e95549b","q":"Let $ABC$ be an acute triangle with $AB < AC < BC$ and let $\\Omega$ be its circumcircle. Let $D$ and $E$ be the points diametrically opposite to points $B$ and $C$ respectively in the circle $\\Omega$. The circle centered at $A$ with radius $AE$ intersects $[AC]$ at $K$. The circle centered at $A$ with radius $AD$ intersects $(BA)$ at $L$ (such that $A$ lies between $B$ and $L$). Show that the lines $(EK)$ and $(DL)$ intersect on the circle $\\Omega$.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.17412,"x":0.5,"p":[[0,58,0.0,0.49103,0.31528,0.28571,0.42857,0.57143,0.0,1.0,3,7,0,3,0,4,0,0,2,0,0,11,0,0,5,0,0,0,0,0,0,0,7],[4,58,0.069,0.49107,0.36932,0.14286,0.42857,1.0,0.0,1.0,6,9,0,6,0,3,0,0,2,0,0,10,0,0,0,0,0,2,0,0,0,0,9],[8,58,0.1379,0.38392,0.29545,0.14286,0.42857,0.4286,0.0,1.0,7,3,0,7,0,4,0,0,0,0,0,14,0,0,1,0,0,3,0,0,0,0,3],[12,58,0.2069,0.39284,0.31743,0.10714,0.42857,0.57111,0.0,1.0,8,4,0,8,0,3,0,0,1,0,0,11,0,0,3,0,0,2,0,0,0,0,4],[16,58,0.2759,0.35713,0.26485,0.14286,0.42857,0.42857,0.0,1.0,6,2,0,6,0,5,0,0,2,0,0,13,0,0,2,0,0,2,0,0,0,0,2],[20,58,0.3448,0.3973,0.30457,0.14286,0.42857,0.57111,0.0,1.0,7,3,0,7,0,3,0,0,3,0,0,9,0,0,4,0,0,2,0,0,1,0,3],[24,58,0.4138,0.35267,0.27428,0.10714,0.42857,0.4286,0.0,1.0,8,1,0,8,0,4,0,0,0,0,0,13,0,0,2,0,0,3,0,0,1,0,1],[28,58,0.4828,0.42407,0.33403,0.14286,0.42857,0.571,0.0,1.0,6,6,0,6,0,5,0,0,1,0,0,10,0,0,4,0,0,0,0,0,0,0,6],[32,58,0.5517,0.45086,0.3136,0.25,0.42857,0.60682,0.0,1.0,5,5,0,5,0,3,0,0,3,0,0,10,0,0,3,0,0,3,0,0,0,0,5],[36,58,0.6207,0.5,0.34253,0.14286,0.42929,0.74996,0.0,1.0,6,5,0,6,0,3,0,0,0,0,0,8,0,0,3,0,0,4,0,0,3,0,5],[40,58,0.6897,0.17412,0.23073,0.0,0.14286,0.32164,0.0,1.0,15,1,0,15,0,8,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[44,58,0.7586,0.35265,0.2258,0.14286,0.42857,0.4287,0.0,0.85714,6,0,0,6,0,4,0,0,1,0,0,14,0,0,5,0,0,1,0,0,1,0,0],[48,58,0.8276,0.49552,0.28343,0.42857,0.42857,0.60714,0.0,1.0,2,5,0,2,0,4,0,0,0,0,0,16,0,0,2,0,0,2,0,0,1,0,5],[52,58,0.8966,0.40621,0.27222,0.14286,0.42857,0.57111,0.0,1.0,6,2,0,6,0,3,0,0,1,0,0,11,0,0,6,0,0,3,0,0,0,0,2],[56,58,0.9655,0.3348,0.2958,0.14286,0.2857,0.4286,0.0,1.0,6,3,0,6,0,9,0,0,3,0,0,7,0,0,2,0,0,2,0,0,0,0,3],[58,58,1.0,0.22321,0.138,0.14286,0.14286,0.32143,0.0,0.571,1,0,0,1,0,21,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"falling","v":0.21429,"x":0.64729,"p":[[0,78,0.0,0.53569,0.27432,0.42857,0.42857,0.71429,0.0,1.0,2,5,1,2,0,2,0,0,0,0,0,15,0,0,4,0,0,2,0,0,2,0,5],[4,78,0.0513,0.47768,0.39222,0.0,0.42857,1.0,0.0,1.0,10,9,0,10,0,0,0,0,0,0,0,10,0,0,2,0,0,0,0,0,1,0,9],[8,78,0.1026,0.41517,0.32802,0.10714,0.42857,0.57111,0.0,1.0,8,4,0,8,0,2,0,0,1,0,0,12,0,0,2,0,0,1,0,0,2,0,4],[12,78,0.1538,0.48661,0.32313,0.14289,0.42857,0.71429,0.0,1.0,2,6,0,2,0,7,0,0,2,0,0,10,0,0,0,0,0,4,0,0,1,0,6],[16,78,0.2051,0.33927,0.33071,0.0,0.2857,0.42858,0.0,1.0,10,4,0,10,0,4,0,0,3,0,0,8,0,0,2,0,0,0,0,0,1,0,4],[20,78,0.2564,0.5848,0.31615,0.42857,0.4998,1.0,0.0,1.0,2,9,0,2,0,3,0,0,0,0,0,11,0,0,4,0,0,2,0,0,1,0,9],[24,78,0.3077,0.64729,0.3369,0.42857,0.571,1.0,0.0,1.0,1,14,0,1,0,3,0,0,2,0,0,9,0,0,2,0,0,1,0,0,0,0,14],[28,78,0.359,0.48657,0.33853,0.28571,0.42857,0.74996,0.0,1.0,6,6,0,6,0,1,0,0,3,0,0,10,0,0,2,0,0,2,0,0,2,0,6],[32,78,0.4103,0.54016,0.30458,0.42857,0.42857,0.74996,0.0,1.0,4,6,0,4,0,0,0,0,1,0,0,14,0,0,2,0,0,3,0,0,2,0,6],[36,78,0.4615,0.56694,0.32436,0.42857,0.42857,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,0,0,0,12,0,0,2,0,0,1,0,0,3,0,8],[40,78,0.5128,0.49551,0.34989,0.14286,0.4286,0.71429,0.0,1.0,7,6,0,7,0,2,0,0,0,0,0,8,0,0,3,0,0,5,0,0,1,0,6],[44,78,0.5641,0.35265,0.31739,0.0,0.42857,0.4642,0.0,1.0,10,3,0,10,0,3,0,0,1,0,0,10,0,0,3,0,0,1,0,0,1,0,3],[48,78,0.6154,0.37054,0.3251,0.0,0.42857,0.57143,0.0,1.0,10,4,0,10,0,2,0,0,1,0,0,10,0,0,4,0,0,1,0,0,0,0,4],[52,78,0.6667,0.30791,0.34278,0.0,0.14286,0.571,0.0,1.0,13,4,0,13,0,4,0,0,2,0,0,4,0,0,4,0,0,1,0,0,0,0,4],[56,78,0.7179,0.42411,0.36854,0.0,0.42857,0.71429,0.0,1.0,10,7,0,10,0,0,0,0,3,0,0,10,0,0,0,0,0,2,0,0,0,0,7],[60,78,0.7692,0.37945,0.24382,0.21429,0.42857,0.57111,0.0,0.71429,8,0,0,8,0,0,0,0,1,0,0,14,0,0,4,0,0,5,0,0,0,0,0],[64,78,0.8205,0.34371,0.36916,0.0,0.28571,0.571,0.0,1.0,14,5,0,14,0,2,0,0,0,0,0,6,0,0,3,0,0,2,0,0,0,0,5],[68,78,0.8718,0.4107,0.35668,0.0,0.42857,0.60714,0.0,1.0,10,5,0,10,0,1,0,0,2,0,0,9,0,0,2,0,0,1,0,0,2,0,5],[72,78,0.9231,0.40179,0.35614,0.0,0.42857,0.60714,0.0,1.0,11,5,0,11,0,0,0,0,2,0,0,9,0,0,2,0,0,2,0,0,1,0,5],[76,78,0.9744,0.25442,0.29389,0.0,0.07143,0.42857,0.0,1.0,16,2,0,16,0,1,0,0,0,0,0,10,0,0,3,0,0,0,0,0,0,0,2],[78,78,1.0,0.21429,0.24223,0.0,0.14286,0.42857,0.0,0.85714,15,0,0,15,0,4,0,0,1,0,0,8,0,0,3,0,0,0,0,0,1,0,0]]}]},{"i":"7eb460293c024359","q":"Let  $AEF$  be a triangle with  $EF = 20$  and  $AE = AF = 21$ . Let  $B$  and  $D$  be points chosen on segments  $AE$  and  $AF,$  respectively, such that  $BD$  is parallel to  $EF.$  Point  $C$  is chosen in the interior of triangle  $AEF$  such that  $ABCD$  is cyclic. If  $BC = 3$  and  $CD = 4,$  then the ratio of areas  $\\tfrac{[ABCD]}{[AEF]}$  can be written as  $\\tfrac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.70089,"x":1.0,"p":[[0,169,0.0,0.86161,0.19061,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,1,0,19],[4,169,0.0237,0.76339,0.17353,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,19,0,0,0,0,9],[8,169,0.0473,0.88839,0.19144,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,22],[12,169,0.071,0.87945,0.14775,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,19],[16,169,0.0947,0.80804,0.21902,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,14],[20,169,0.1183,0.70089,0.23244,0.57143,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,15,0,0,0,0,8],[24,169,0.142,0.85268,0.18029,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,11,0,0,0,0,18],[28,169,0.1657,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[32,169,0.1893,0.81696,0.21793,0.71429,0.78571,1.0,0.0,1.0,1,15,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,1,0,15],[36,169,0.213,0.84375,0.2,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,1,0,18],[40,169,0.2367,0.90624,0.13654,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[44,169,0.2604,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,169,0.284,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,169,0.3077,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[56,169,0.3314,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[60,169,0.355,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,169,0.3787,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,169,0.4024,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[72,169,0.426,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[76,169,0.4497,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[80,169,0.4734,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,169,0.497,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[88,169,0.5207,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,169,0.5444,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[96,169,0.568,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,169,0.5917,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,169,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[108,169,0.6391,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[112,169,0.6627,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[116,169,0.6864,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[120,169,0.7101,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[124,169,0.7337,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[128,169,0.7574,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[132,169,0.7811,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[136,169,0.8047,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[140,169,0.8284,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[144,169,0.8521,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[148,169,0.8757,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[152,169,0.8994,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[156,169,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[160,169,0.9467,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[164,169,0.9704,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[168,169,0.9941,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[169,169,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":0.71429,"k":"falling","v":0.45982,"x":0.92411,"p":[[0,618,0.0,0.82589,0.17399,0.71429,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,15,0,0,1,0,14],[4,618,0.0065,0.86161,0.14934,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,17],[8,618,0.0129,0.86161,0.15765,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0,17],[12,618,0.0194,0.90179,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0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,4,0,4,0,0,0,0,0,0,0,12,0,0,2,0,0,7,0,0,3,0,4],[560,618,0.9061,0.61607,0.2976,0.42857,0.64286,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,1,0,0,9,0,0,2,0,0,6,0,0,2,0,8],[564,618,0.9126,0.6384,0.33878,0.42859,0.71429,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,0,6,0,8],[568,618,0.9191,0.52679,0.36846,0.14286,0.57143,0.85714,0.0,1.0,7,7,0,7,0,2,0,0,2,0,0,2,0,0,5,0,0,5,0,0,2,0,7],[572,618,0.9256,0.51784,0.31894,0.25,0.57143,0.71429,0.0,1.0,5,3,0,5,0,3,0,0,1,0,0,5,0,0,3,0,0,9,0,0,3,0,3],[576,618,0.932,0.58929,0.30462,0.42857,0.57143,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,2,0,0,8,0,0,3,0,0,4,0,0,6,0,5],[580,618,0.9385,0.6116,0.31183,0.42857,0.64286,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,0,0,0,7,0,0,4,0,0,4,0,0,6,0,6],[584,618,0.945,0.59375,0.33141,0.42857,0.71429,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,1,0,0,9,0,0,0,0,0,7,0,0,2,0,8],[588,618,0.9515,0.54909,0.24772,0.42857,0.57143,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,1,0,0,9,0,0,4,0,0,9,0,0,4,0,1],[592,618,0.9579,0.59375,0.27225,0.42857,0.57143,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,2,0,0,9,0,0,3,0,0,8,0,0,1,0,6],[596,618,0.9644,0.50444,0.28117,0.42857,0.4998,0.71429,0.0,1.0,4,3,0,4,0,2,0,0,0,0,0,10,0,0,7,0,0,4,0,0,2,0,3],[600,618,0.9709,0.55357,0.37072,0.14286,0.71429,0.85714,0.0,1.0,6,7,0,6,0,3,0,0,1,0,0,4,0,0,1,0,0,6,0,0,4,0,7],[604,618,0.9773,0.49107,0.29001,0.39286,0.42857,0.71429,0.0,1.0,5,3,0,5,0,0,0,0,3,0,0,10,0,0,4,0,0,5,0,0,2,0,3],[608,618,0.9838,0.45982,0.38752,0.0,0.4286,0.75,0.0,1.0,11,6,0,11,0,0,0,0,2,0,0,4,0,0,2,0,0,5,0,0,2,0,6],[612,618,0.9903,0.55802,0.29093,0.42857,0.57121,0.75,0.0,1.0,2,5,0,2,0,3,0,0,1,0,0,9,0,0,5,0,0,4,0,0,3,0,5],[616,618,0.9968,0.51785,0.27374,0.42857,0.42859,0.71429,0.0,1.0,3,2,0,3,0,2,0,0,1,0,0,12,0,0,2,0,0,6,0,0,4,0,2],[618,618,1.0,0.50892,0.25738,0.42857,0.42857,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,2,0,0,11,0,0,6,0,0,4,0,0,3,0,2]]}]},{"i":"9b6ae4c0c88f7263","q":"Let $a_{0}<a_{1}<a_{2}<\\ldots$ be an infinite sequence of strictly positive integers. Prove that there exists a unique integer $n \\geqslant 1$ such that\n\n$$\na_{n}<\\frac{a_{0}+a_{1}+a_{2}+\\cdots+a_{n}}{n} \\leqslant a_{n+1} .\n$$","t":[{"b":1,"e":0.85714,"k":"flat","v":0.83035,"x":0.97768,"p":[[0,35,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,35,0.1143,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,35,0.2286,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[12,35,0.3429,0.9375,0.17835,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[16,35,0.4571,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[20,35,0.5714,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[24,35,0.6857,0.93749,0.10063,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[28,35,0.8,0.88393,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,18,0,10],[32,35,0.9143,0.85267,0.09094,0.857,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,6],[35,35,1.0,0.83035,0.06621,0.857,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,24,0,1]]},{"b":4,"e":0.85714,"k":"falling","v":0.80802,"x":0.97321,"p":[[0,86,0.0,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[4,86,0.0465,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,86,0.093,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,86,0.1395,0.96429,0.12877,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[16,86,0.186,0.94643,0.14173,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,26],[20,86,0.2326,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,26],[24,86,0.2791,0.91517,0.16314,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,6,0,22],[28,86,0.3256,0.89286,0.21429,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,4,0,23],[32,86,0.3721,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[36,86,0.4186,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[40,86,0.4651,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[44,86,0.5116,0.93304,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[48,86,0.5581,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[52,86,0.6047,0.91071,0.12242,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,17],[56,86,0.6512,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[60,86,0.6977,0.83482,0.13882,0.85714,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,22,0,5],[64,86,0.7442,0.84821,0.13803,0.85714,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,20,0,7],[68,86,0.7907,0.86607,0.10677,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,22,0,7],[72,86,0.8372,0.86607,0.09407,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,18,0,8],[76,86,0.8837,0.85714,0.11294,0.85714,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,20,0,7],[80,86,0.9302,0.8125,0.12595,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,19,0,3],[84,86,0.9767,0.83928,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,24,0,2],[86,86,1.0,0.80802,0.07672,0.71429,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,22,0,0]]}]},{"i":"27f9202e2fa79439","q":"Let $k$ be a natural number. Let $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a bijection such that for any two integers $i$ and $j$ for which $|i-j| \\leqslant k$ it holds that $|f(i)-f(j)| \\leqslant k$. Prove that for all $i, j \\in \\mathbb{Z}$ it holds that\n\n$$\n|f(i)-f(j)|=|i-j|\n$$\n\n(Miljan Kne\u017eevi\u0107)","t":[{"b":0,"e":1.0,"k":"flat","v":0.89719,"x":0.99107,"p":[[0,40,0.0,0.95535,0.14033,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[4,40,0.1,0.92857,0.14286,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,24],[8,40,0.2,0.89719,0.23494,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,25],[12,40,0.3,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[16,40,0.4,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[20,40,0.5,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,40,0.6,0.95088,0.13654,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[28,40,0.7,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,40,0.8,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[36,40,0.9,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,40,1.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]},{"b":6,"e":1.0,"k":"falling","v":0.7366,"x":0.93737,"p":[[0,26,0.0,0.93737,0.18552,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[4,26,0.1538,0.91964,0.17474,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,2,0,25],[8,26,0.3077,0.90179,0.24074,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,26],[12,26,0.4615,0.87946,0.16793,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,2,0,19],[16,26,0.6154,0.85267,0.18725,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,6,0,16],[20,26,0.7692,0.80803,0.22759,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,9,0,0,3,0,15],[24,26,0.9231,0.84821,0.18877,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,9,0,0,6,0,15],[26,26,1.0,0.7366,0.23175,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,11,0,0,8,0,7]]}]},{"i":"767c676348084c0b","q":"Let  $ABC$  be a triangle inscribed in a circle of center  $O$  and radius  $R$ . Let  $I$  be the incenter of  $ABC$ , and let  $r$  be the inradius of the same triangle,  $O\\neq I$ , and let  $G$  be its centroid. Prove that  $IG\\perp BC$  if and only if  $b=c$  or  $b+c=3a$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.83034,"x":1.0,"p":[[0,23,0.0,0.83034,0.20654,0.57143,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,19],[4,23,0.1739,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[8,23,0.3478,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,23,0.5217,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,23,0.6957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.78571,"x":0.97321,"p":[[0,33,0.0,0.9375,0.14258,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,26],[4,33,0.1212,0.78571,0.20516,0.57143,0.85707,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,3,0,14],[8,33,0.2424,0.84372,0.20938,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,0,0,0,3,0,19],[12,33,0.3636,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[16,33,0.4848,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[20,33,0.6061,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,33,0.7273,0.94642,0.12246,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[28,33,0.8485,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[32,33,0.9697,0.95982,0.08918,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[33,33,1.0,0.95535,0.11539,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27]]}]},{"i":"18ef9fde053b44ad","q":"Let  $I$  be the incenter of  $\\triangle ABC$ , and  $O$  be the excenter corresponding to  $B$ . If  $|BI|=12$ ,  $|IO|=18$ , and  $|BC|=15$ , then what is  $|AB|$ ? $ \\textbf{(A)}\\ 16\n\\qquad\\textbf{(B)}\\ 18\n\\qquad\\textbf{(C)}\\ 20\n\\qquad\\textbf{(D)}\\ 22\n\\qquad\\textbf{(E)}\\ 24\n$","t":[{"b":3,"e":0.42857,"k":"falling","v":0.36607,"x":0.54241,"p":[[0,42,0.0,0.51786,0.28738,0.39286,0.42857,0.71429,0.0,1.0,1,7,1,1,0,2,0,0,5,0,0,15,0,0,0,0,0,2,0,0,0,0,7],[4,42,0.0952,0.54018,0.23347,0.42857,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,19,0,0,0,0,0,3,0,0,1,0,5],[8,42,0.1905,0.50446,0.21424,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,20,0,0,0,0,0,3,0,0,0,0,4],[12,42,0.2857,0.54241,0.21119,0.42857,0.42857,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,1,19,0,0,0,0,0,6,0,0,0,0,4],[16,42,0.381,0.41071,0.12753,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,24,0,0,0,0,0,0,0,0,0,0,1],[20,42,0.4762,0.44643,0.17405,0.39286,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,20,0,0,0,0,0,2,0,0,0,0,2],[24,42,0.5714,0.42857,0.13363,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,25,0,0,0,0,0,1,0,0,0,0,1],[28,42,0.6667,0.39286,0.10101,0.39286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,23,0,0,0,0,0,1,0,0,0,0,0],[32,42,0.7619,0.42411,0.0977,0.42857,0.42857,0.42857,0.1429,0.71429,0,0,0,0,0,1,0,0,3,0,0,26,0,0,0,0,0,2,0,0,0,0,0],[36,42,0.8571,0.41741,0.10731,0.33918,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,1,19,0,0,2,0,0,2,0,0,0,0,0],[40,42,0.9524,0.42857,0.13832,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,17,0,0,0,0,0,5,0,0,0,0,0],[42,42,1.0,0.36607,0.10677,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,7,0,0,22,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.4375,"x":0.61607,"p":[[0,50,0.0,0.51786,0.29827,0.28571,0.42857,0.78571,0.14286,1.0,0,8,0,0,0,2,0,0,11,0,0,9,0,0,1,0,0,1,0,0,0,0,8],[4,50,0.08,0.57589,0.25874,0.42857,0.42857,0.78571,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,19,0,0,0,0,0,2,0,0,0,0,8],[8,50,0.16,0.59821,0.27994,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,14,0,0,0,0,0,4,0,0,0,0,9],[12,50,0.24,0.50893,0.23128,0.42857,0.42857,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,6,0,0,19,0,0,0,0,0,2,0,0,0,0,5],[16,50,0.32,0.51339,0.22829,0.42857,0.42857,0.42858,0.28571,1.0,0,5,0,0,0,0,0,0,5,0,0,20,0,0,0,0,0,2,0,0,0,0,5],[20,50,0.4,0.61607,0.29974,0.39286,0.42857,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,8,0,0,10,0,0,0,0,0,3,0,0,0,0,11],[24,50,0.48,0.48661,0.22829,0.42857,0.42857,0.42857,0.28571,1.0,0,5,0,0,0,0,0,0,7,0,0,20,0,0,0,0,0,0,0,0,0,0,5],[28,50,0.56,0.55357,0.26904,0.42857,0.42857,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,17,0,0,0,0,0,1,0,0,0,0,8],[32,50,0.64,0.54909,0.26752,0.42857,0.42857,0.67825,0.2857,1.0,0,8,0,0,0,0,0,0,6,0,0,17,0,0,1,0,0,0,0,0,0,0,8],[36,50,0.72,0.46875,0.18638,0.42857,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,23,0,0,0,0,0,1,0,0,0,0,3],[40,50,0.8,0.49554,0.18205,0.42857,0.42857,0.42857,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,24,0,0,1,0,0,2,0,0,0,0,3],[44,50,0.88,0.57589,0.24086,0.42857,0.42857,0.71429,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,21,0,0,0,0,0,3,0,0,0,0,7],[48,50,0.96,0.47321,0.20025,0.39286,0.42857,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,17,0,0,2,0,0,2,0,0,0,0,3],[50,50,1.0,0.4375,0.10677,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"7bdacc856ad6b4fd","q":"Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.","t":[{"b":5,"e":0.85714,"k":"flat","v":0.84821,"x":0.92857,"p":[[0,21,0.0,0.875,0.07784,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,7],[4,21,0.1905,0.87946,0.08073,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,8],[8,21,0.381,0.92857,0.09449,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,18],[12,21,0.5714,0.85714,0.18558,0.85714,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,21,0,9],[16,21,0.7619,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[20,21,0.9524,0.86607,0.11259,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,18,0,9],[21,21,1.0,0.84821,0.13333,0.85714,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,18,0,8]]},{"b":7,"e":0.57143,"k":"flat","v":0.85268,"x":0.96429,"p":[[0,27,0.0,0.875,0.1171,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,18,0,10],[4,27,0.1481,0.9241,0.07129,0.85714,0.85714,1.0,0.857,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[8,27,0.2963,0.89732,0.10248,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,13],[12,27,0.4444,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[16,27,0.5926,0.85268,0.13592,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,17,0,9],[20,27,0.7407,0.92857,0.08748,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[24,27,0.8889,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[27,27,1.0,0.93304,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20]]}]},{"i":"42d5d0e9430d41ae","q":"Let $n$ and $k$ be two strictly positive integers. Consider an assembly of $k$ people such that, for any group of $n$ people, there is a $(n+1)$-th person who knows them all (if $A$ knows $B$ then $B$ knows $A$).\n\n1) If $k=2n+1$, prove that there exists a person who knows everyone else.\n2) If $k=2n+2$, give an example of such an assembly in which no one knows everyone else.","t":[{"b":3,"e":0.4286,"k":"flat","v":0.41956,"x":0.52679,"p":[[0,61,0.0,0.4375,0.12846,0.42857,0.42857,0.42857,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,1],[4,61,0.0656,0.42856,0.13831,0.42857,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,2,0,0,0,0,0,24,0,0,4,0,0,0,0,0,1,0,0],[8,61,0.1311,0.48662,0.17807,0.42857,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,25,0,0,3,0,0,0,0,0,0,0,3],[12,61,0.1967,0.47326,0.19703,0.42857,0.42857,0.4286,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,24,0,0,3,0,0,0,0,0,0,0,3],[16,61,0.2623,0.4866,0.20472,0.42857,0.42857,0.4286,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,23,0,0,2,0,0,2,0,0,0,0,3],[20,61,0.3279,0.44643,0.11706,0.42857,0.42857,0.4286,0.14286,0.857,0,0,0,0,0,1,0,0,0,0,0,29,0,0,0,0,0,0,0,0,2,0,0],[24,61,0.3934,0.43742,0.18182,0.42857,0.42857,0.4286,0.0,1.0,3,1,0,3,0,0,0,0,0,0,0,22,0,0,5,0,0,1,0,0,0,0,1],[28,61,0.459,0.41956,0.05021,0.42857,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,61,0.5246,0.52679,0.20025,0.42857,0.42857,0.4286,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,25,0,0,1,0,0,1,0,0,1,0,4],[36,61,0.5902,0.47321,0.20958,0.42857,0.42857,0.42858,0.0,1.0,1,3,0,1,0,2,0,0,0,0,0,22,0,0,3,0,0,1,0,0,0,0,3],[40,61,0.6557,0.49121,0.171,0.42857,0.42857,0.4286,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,3],[44,61,0.7213,0.43308,0.15765,0.42857,0.42857,0.4286,0.0,1.0,2,1,0,2,0,0,0,0,0,0,0,27,0,0,1,0,0,1,0,0,0,0,1],[48,61,0.7869,0.44643,0.12242,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,29,0,0,0,0,0,1,0,0,0,0,1],[52,61,0.8525,0.43753,0.12338,0.42857,0.42857,0.42858,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,25,0,0,3,0,0,2,0,0,0,0,0],[56,61,0.918,0.44648,0.13716,0.42857,0.42857,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,28,0,0,1,0,0,1,0,0,0,0,1],[60,61,0.9836,0.49112,0.14696,0.42857,0.42857,0.42895,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,25,0,0,4,0,0,1,0,0,0,0,2],[61,61,1.0,0.49112,0.1634,0.42857,0.42857,0.46536,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,23,0,0,4,0,0,2,0,0,0,0,2]]},{"b":5,"e":0.43,"k":"flat","v":0.42858,"x":0.50445,"p":[[0,44,0.0,0.44643,0.12242,0.42857,0.42857,0.42857,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,27,0,0,2,0,0,1,0,0,1,0,0],[4,44,0.0909,0.50445,0.15965,0.42857,0.42857,0.4642,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,24,0,0,4,0,0,1,0,0,1,0,2],[8,44,0.1818,0.4866,0.16311,0.42857,0.42857,0.42858,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,26,0,0,1,0,0,1,0,0,1,0,2],[12,44,0.2727,0.45534,0.09057,0.42857,0.42857,0.42857,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0],[16,44,0.3636,0.49558,0.15964,0.42857,0.42857,0.4286,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,26,0,0,2,0,0,1,0,0,1,0,2],[20,44,0.4545,0.44642,0.18122,0.42857,0.42857,0.4286,0.0,1.0,2,2,0,2,0,0,0,0,0,0,0,26,0,0,2,0,0,0,0,0,0,0,2],[24,44,0.5455,0.45982,0.14167,0.42857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,25,0,0,4,0,0,1,0,0,0,0,1],[28,44,0.6364,0.43304,0.02486,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0],[32,44,0.7273,0.49563,0.19553,0.42857,0.42857,0.43,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,24,0,0,2,0,0,2,0,0,0,0,3],[36,44,0.8182,0.45982,0.11701,0.42857,0.42857,0.42858,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,27,0,0,2,0,0,1,0,0,0,0,1],[40,44,0.9091,0.46433,0.14285,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,2],[44,44,1.0,0.42858,1e-05,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2c36461820221607","q":"Two points  $K$  and  $L$  are chosen inside triangle  $ABC$  and a point  $D$  is chosen on the side  $AB$ . Suppose that  $B$ ,  $K$ ,  $L$ ,  $C$  are concyclic,  $\\angle AKD = \\angle BCK$  and  $\\angle ALD = \\angle BCL$ . Prove that  $AK = AL$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13393,"x":0.18304,"p":[[0,103,0.0,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,103,0.0388,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,103,0.0777,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,103,0.1165,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,103,0.1553,0.16518,0.12426,0.14286,0.14286,0.14286,0.14286,0.857,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,103,0.1942,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,103,0.233,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,103,0.2718,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,103,0.3107,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,103,0.3495,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,103,0.3883,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,103,0.4272,0.18304,0.16457,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[48,103,0.466,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,103,0.5049,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,103,0.5437,0.14286,2e-05,0.14286,0.14286,0.14286,0.14286,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,103,0.5825,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,103,0.6214,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,103,0.6602,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,103,0.699,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,103,0.7379,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,103,0.7767,0.15161,0.04975,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,103,0.8155,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,103,0.8544,0.16062,0.06919,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,103,0.8932,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,103,0.932,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,103,0.9709,0.14733,0.02485,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[103,103,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.13393,"x":0.15625,"p":[[0,69,0.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,69,0.2319,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.14286,1e-05,0.14286,0.14286,0.14286,0.14286,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9189e86eb4791089","q":"Let  $G$  be the centroid of a triangle  $ABC$ , and  $M$  be the midpoint of  $BC$ .  Let  $X$  be on  $AB$  and  $Y$  on  $AC$  such that the points  $X$ ,  $Y$ , and  $G$  are collinear and  $XY$  and  $BC$  are parallel.  Suppose that  $XC$  and  $GB$  intersect at  $Q$  and  $YB$  and  $GC$  intersect at  $P$ .  Show that triangle  $MPQ$  is similar to triangle  $ABC$ .","t":[{"b":5,"e":0.85714,"k":"flat","v":0.66518,"x":0.75892,"p":[[0,28,0.0,0.75887,0.11547,0.71429,0.85707,0.85714,0.57,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,17,0,0],[4,28,0.1429,0.75892,0.10376,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,15,0,0],[8,28,0.2857,0.70966,0.11567,0.57143,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,11,0,0,10,0,0],[12,28,0.4286,0.75,0.11845,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,12,0,1],[16,28,0.5714,0.68303,0.09933,0.57143,0.71429,0.71429,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,15,0,0,5,0,0],[20,28,0.7143,0.70088,0.0969,0.57143,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,17,0,0,6,0,0],[24,28,0.8571,0.66518,0.09852,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,13,0,0,4,0,0],[28,28,1.0,0.6875,0.0974,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,16,0,0,5,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.69642,"x":0.77232,"p":[[0,36,0.0,0.77232,0.12299,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,15,0,2],[4,36,0.1111,0.75446,0.10853,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,12,0,1],[8,36,0.2222,0.69642,0.11153,0.57143,0.71429,0.75,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,12,0,0,8,0,0],[12,36,0.3333,0.71428,0.10101,0.67857,0.71429,0.74996,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,16,0,0,8,0,0],[16,36,0.4444,0.70536,0.08702,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,20,0,0,5,0,0],[20,36,0.5556,0.70088,0.09691,0.57143,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,17,0,0,6,0,0],[24,36,0.6667,0.70533,0.1068,0.57143,0.71429,0.75,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,14,0,0,8,0,0],[28,36,0.7778,0.72767,0.09009,0.71429,0.71429,0.75,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,19,0,0,8,0,0],[32,36,0.8889,0.74107,0.10972,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,12,0,0,13,0,0],[36,36,1.0,0.73661,0.1017,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,15,0,0,11,0,0]]}]},{"i":"1659bfd14c71e4e8","q":"Two concentric spheres have radii $r$ and $R, r<R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $A B C$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \\leq 2 r$.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.88393,"x":0.95982,"p":[[0,19,0.0,0.88393,0.22142,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,23],[4,19,0.2105,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[8,19,0.4211,0.91518,0.14223,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,23],[12,19,0.6316,0.94642,0.12756,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[16,19,0.8421,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[19,19,1.0,0.91071,0.13716,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,22]]},{"b":7,"e":1.0,"k":"flat","v":0.71875,"x":0.88839,"p":[[0,28,0.0,0.84375,0.25843,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,20],[4,28,0.1429,0.71875,0.29771,0.49998,0.78571,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,8,0,0,0,0,0,6,0,0,2,0,0,1,0,15],[8,28,0.2857,0.88839,0.17762,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,2,0,21],[12,28,0.4286,0.84821,0.17835,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,2,0,17],[16,28,0.5714,0.83035,0.22428,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,3,0,18],[20,28,0.7143,0.85254,0.15777,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,7,0,14],[24,28,0.8571,0.77677,0.20185,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,4,0,0,6,0,11],[28,28,1.0,0.8125,0.18012,0.71429,0.85707,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,4,0,13]]}]},{"i":"00d1f7f6b666d060","q":"Let \\[ f(n)=  \\sum_{p|n , \\;p^{\\alpha} \\leq n < p^{\\alpha+1} \\ } p^{\\alpha} .\\] Prove that \\[ \\limsup_{n \\rightarrow \\infty}f(n) \\frac{ \\log \\log n}{n \\log n}=1 .\\] \r\n\r\n*P. Erdos*","t":[{"b":0,"e":1.0,"k":"rising","v":0.45982,"x":0.90179,"p":[[0,49,0.0,0.45982,0.22513,0.28571,0.28571,0.71429,0.0,0.85714,1,0,1,1,0,1,0,0,15,0,0,0,0,0,4,0,0,10,0,0,1,0,0],[4,49,0.0816,0.84374,0.18338,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,6,0,15],[8,49,0.1633,0.7857,0.22589,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,7,0,0,6,0,12],[12,49,0.2449,0.77676,0.22853,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,5,0,0,4,0,13],[16,49,0.3265,0.81249,0.2096,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,9,0,0,4,0,14],[20,49,0.4082,0.78568,0.19235,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,9,0,9],[24,49,0.4898,0.72767,0.20629,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,9,0,0,4,0,8],[28,49,0.5714,0.81695,0.21201,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,6,0,0,6,0,14],[32,49,0.6531,0.76338,0.21903,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,13,0,0,2,0,11],[36,49,0.7347,0.78571,0.23146,0.71429,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,11,0,0,2,0,13],[40,49,0.8163,0.81696,0.20589,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,8,0,0,8,0,12],[44,49,0.898,0.90179,0.13092,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,19],[48,49,0.9796,0.87946,0.15198,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,4,0,18],[49,49,1.0,0.83928,0.16269,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,8,0,12]]},{"b":1,"e":1.0,"k":"rising","v":0.48661,"x":0.87945,"p":[[0,36,0.0,0.48661,0.24707,0.28571,0.35714,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,13,0,0,1,0,0,3,0,0,7,0,0,5,0,0],[4,36,0.1111,0.87945,0.16411,0.82143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,7,0,17],[8,36,0.2222,0.79905,0.20788,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,8,0,12],[12,36,0.3333,0.83482,0.14773,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,8,0,11],[16,36,0.4444,0.81025,0.22982,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,3,1,0,7,0,14],[20,36,0.5556,0.84374,0.20002,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,6,0,16],[24,36,0.6667,0.78571,0.25754,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,8,0,0,3,0,15],[28,36,0.7778,0.77679,0.22286,0.67857,0.78571,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,8,0,0,4,0,12],[32,36,0.8889,0.78125,0.21124,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,8,0,0,9,0,9],[36,36,1.0,0.65177,0.24728,0.39286,0.71429,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,8,0,0,1,0,0,2,0,0,12,0,0,4,0,5]]}]},{"i":"7048fba3c55f90ad","q":"Let two circles  $A$  and  $B$  with unequal radii  $r$  and  $R$ , respectively, be tangent internally at the point  $A_0$ . If there exists a sequence of distinct circles  $(C_n)$  such that each circle is tangent to both  $A$  and  $B$ , and each circle  $C_{n+1}$  touches circle  $C_{n}$  at the point  $A_n$ , prove that \n\\[\\sum_{n=1}^{\\infty} |A_{n+1}A_n| < \\frac{4 \\pi Rr}{R+r}.\\]","t":[{"b":0,"e":0.0,"k":"volatile","v":0.34375,"x":0.95536,"p":[[0,15,0.0,0.95088,0.11637,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[4,15,0.2667,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[8,15,0.5333,0.7589,0.22144,0.57143,0.78571,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,0,5,0,11],[12,15,0.8,0.79017,0.23141,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,0,9,0,11],[15,15,1.0,0.34375,0.21975,0.14286,0.42857,0.42857,0.0,0.85714,6,0,0,6,0,3,0,0,4,0,0,13,0,0,4,0,0,1,0,0,1,0,0]]},{"b":7,"e":0.71429,"k":"falling","v":0.60267,"x":0.93304,"p":[[0,18,0.0,0.93304,0.1988,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,26],[4,18,0.2222,0.77232,0.23652,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,4,0,0,3,0,14],[8,18,0.4444,0.71428,0.25754,0.42857,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,2,0,0,3,0,12],[12,18,0.6667,0.62053,0.27107,0.42857,0.4998,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,13,0,0,5,0,0,0,0,0,2,0,9],[16,18,0.8889,0.60267,0.28956,0.42857,0.57143,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,1,0,0,6,0,0,8,0,0,3,0,0,4,0,6],[18,18,1.0,0.61158,0.16458,0.42859,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,10,0,0,8,0,0,4,0,1]]}]},{"i":"a32d91db7d9832ba","q":"a) Prove that in an infinite sequence  ${a_k}$  of integers, pairwise distinct and each member greater than  $1$ , one can find  $100$  members for which  $a_k > k$ . \n\nb) Prove that in an infinite sequence   ${a_k}$  of integers, pairwise distinct and each member greater than  $1$  there are infinitely many such numbers  $a_k$  such that  $a_k > k$ . \n\n(A Andjans, Riga)\n\nPS. (a) for juniors  (b) for seniors","t":[{"b":1,"e":0.42857,"k":"falling","v":0.45533,"x":0.9732,"p":[[0,39,0.0,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,39,0.1026,0.7857,0.29452,0.5354,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,3,0,0,1,0,19],[8,39,0.2051,0.66964,0.37701,0.39286,0.92857,1.0,0.0,1.0,2,16,0,2,0,5,0,0,1,0,0,5,0,0,1,0,0,0,0,0,2,0,16],[12,39,0.3077,0.7097,0.37554,0.42857,1.0,1.0,0.0,1.0,3,18,0,3,0,4,0,0,0,0,0,2,0,0,2,0,0,3,0,0,0,0,18],[16,39,0.4103,0.70087,0.33949,0.42857,0.85714,1.0,0.0,1.0,1,16,0,1,0,3,0,0,3,0,0,4,0,0,1,0,0,4,0,0,0,0,16],[20,39,0.5128,0.56248,0.3641,0.14289,0.4286,1.0,0.0,1.0,1,10,0,1,0,8,0,0,3,0,0,5,0,0,1,0,0,1,0,0,3,0,10],[24,39,0.6154,0.56679,0.3873,0.14286,0.57143,1.0,0.0,1.0,1,11,0,1,0,11,0,0,1,0,0,3,0,0,0,0,0,2,0,0,3,0,11],[28,39,0.7179,0.61148,0.3835,0.14286,0.64264,1.0,0.0,1.0,2,13,0,2,0,8,0,0,0,0,0,3,0,0,3,0,0,1,0,0,2,0,13],[32,39,0.8205,0.74545,0.37766,0.35716,1.0,1.0,0.0,1.0,1,21,0,1,0,7,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,21],[36,39,0.9231,0.57578,0.3738,0.14286,0.49979,1.0,0.14,1.0,0,12,0,0,0,10,0,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,12],[39,39,1.0,0.45533,0.34336,0.14286,0.28571,0.74996,0.0,1.0,1,7,0,1,0,11,0,0,5,0,0,3,0,0,3,0,0,1,0,0,1,0,7]]},{"b":7,"e":1.0,"k":"falling","v":0.68749,"x":0.9866,"p":[[0,45,0.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,45,0.0889,0.87945,0.27691,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,26],[8,45,0.1778,0.68749,0.38206,0.28571,1.0,1.0,0.0,1.0,2,18,0,2,0,4,0,0,4,0,0,2,0,0,0,0,0,2,0,0,0,0,18],[12,45,0.2667,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[16,45,0.3556,0.94643,0.18123,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[20,45,0.4444,0.80344,0.27399,0.67846,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,4,0,0,2,0,18],[24,45,0.5333,0.89283,0.19237,0.85711,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,23],[28,45,0.6222,0.79462,0.26951,0.571,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,6,0,0,2,0,0,4,0,0,1,0,18],[32,45,0.7111,0.76337,0.25906,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,7,0,0,3,0,0,4,0,0,2,0,15],[36,45,0.8,0.8437,0.21834,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,5,0,18],[40,45,0.8889,0.69644,0.32487,0.42859,0.78571,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,1,0,0,8,0,0,0,0,0,3,0,0,1,0,15],[44,45,0.9778,0.71417,0.28799,0.571,0.857,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,1,0,0,3,0,0,8,0,0,0,0,0,5,0,12],[45,45,1.0,0.72321,0.2922,0.42859,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,6,0,0,0,0,0,5,0,0,2,0,14]]}]},{"i":"6f4b6a07018f61a1","q":"21. (USS 1) Let $N$ be the number of integral solutions of the equation  $$ x^{2}-y^{2}=z^{3}-t^{3} $$  satisfying the condition $0 \\leq x, y, z, t \\leq 10^{6}$, and let $M$ be the number of integral solutions of the equation  $$ x^{2}-y^{2}=z^{3}-t^{3}+1 $$  satisfying the condition $0 \\leq x, y, z, t \\leq 10^{6}$. Prove that $N>M$.","t":[{"b":3,"e":1.0,"k":"rising","v":0.16072,"x":0.61159,"p":[[0,49,0.0,0.16072,0.14617,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,13,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.20312,0.23359,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,4,0,0,4,0,0,4,0,1,2,0,0,2,0,0,0,0,0],[8,49,0.1633,0.28124,0.34714,0.0,0.14286,0.42857,0.0,1.0,13,5,0,13,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,5],[12,49,0.2449,0.28562,0.26005,0.14214,0.14286,0.42858,0.0,1.0,7,1,0,7,0,10,0,0,4,0,0,4,0,0,3,0,0,3,0,0,0,0,1],[16,49,0.3265,0.42185,0.28027,0.14286,0.42857,0.57111,0.0,1.0,3,3,0,3,0,6,0,0,3,0,1,9,0,0,4,0,0,2,0,0,1,0,3],[20,49,0.4082,0.34375,0.19186,0.14286,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,3,0,0,14,0,0,3,0,0,2,0,0,0,0,0],[24,49,0.4898,0.42846,0.22597,0.28571,0.42857,0.57111,0.0,1.0,2,1,0,2,0,3,0,0,6,0,0,11,0,0,6,0,0,1,0,0,2,0,1],[28,49,0.5714,0.44197,0.24053,0.2857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,5,0,0,12,0,0,2,0,0,3,0,0,3,0,1],[32,49,0.6531,0.52678,0.25614,0.42857,0.4286,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,12,0,0,2,0,0,6,0,0,1,0,4],[36,49,0.7347,0.44866,0.24038,0.2857,0.42857,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,5,0,1,8,0,0,5,0,0,5,0,0,0,0,2],[40,49,0.8163,0.49106,0.20496,0.42857,0.42857,0.60714,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,13,0,0,5,0,0,5,0,0,3,0,0],[44,49,0.898,0.58927,0.23622,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,3,0,0,6,0,0,7,0,0,7,0,0,5,0,2],[48,49,0.9796,0.61159,0.24804,0.42857,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,6,0,0,4,0,0,5,0,0,9,0,2],[49,49,1.0,0.52676,0.23265,0.39286,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,9,0,0,6,0,0,3,0,0,4,0,2]]},{"b":4,"e":0.14286,"k":"flat","v":0.11607,"x":0.30356,"p":[[0,20,0.0,0.24767,0.17679,0.14286,0.2143,0.32143,0.0,0.71429,4,0,0,4,1,11,0,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[4,20,0.2,0.11607,0.18013,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,9,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,20,0.4,0.30356,0.23075,0.14286,0.35714,0.42858,0.0,0.71429,7,0,0,7,0,7,0,0,2,0,0,10,0,0,3,0,0,3,0,0,0,0,0],[12,20,0.6,0.20982,0.20512,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,7,0,0,2,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[16,20,0.8,0.29684,0.26967,0.10714,0.2857,0.42857,0.0,1.0,8,2,0,8,0,6,0,0,6,0,1,5,0,0,3,0,0,1,0,0,0,0,2],[20,20,1.0,0.17409,0.19142,0.0,0.14286,0.1786,0.0,0.71429,11,0,0,11,0,13,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"ca845c9332904873","q":"For each positive integer  $n$ , define  $$ P_n = (n+1)(n+2)(n+3)\\dots(n+2016) $$  and  $$ Q_n = \\text{lcm}(n+1, n+2, n+3, \\dots, n+2016), $$  meaning  $Q_n$  is the least common multiple of the numbers  $n+1, n+2, n+3, \\dots, n+2016$ . Determine whether or not there exists a constant  $C$  such that  $$ \\frac{P_n}{Q_n} < C, $$  for every positive integer  $n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.83482,"x":1.0,"p":[[0,26,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,26,0.1538,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,26,0.3077,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,26,0.4615,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[24,26,0.9231,0.87054,0.21237,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,1,0,22],[26,26,1.0,0.83482,0.24513,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,7,0,0,0,0,20]]},{"b":5,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,30,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,30,0.4,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,30,0.5333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,30,0.6667,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,30,0.8,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[28,30,0.9333,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[30,30,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"69aa7027f81a0199","q":"A field is made of  $2017 \\times 2017$  unit squares. Luffy has  $k$  gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a  $1500 \\times 1500$  area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the  $1500 \\times 1500$  area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of  $k$  in which this is possible.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,44,0.0,0.07143,0.15972,0.0,0.0,0.0,0.0,0.57143,25,0,19,25,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,44,0.0909,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,44,0.1818,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,44,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.4732,"p":[[0,75,0.0,0.06696,0.17122,0.0,0.0,0.0,0.0,0.85714,26,0,15,26,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,75,0.0533,0.30804,0.28146,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,0,0,0,9,0,0,5,0,0,2,0,0,3,0,0,1,0,1],[8,75,0.1067,0.30804,0.28818,0.0,0.28571,0.57143,0.0,0.85714,12,0,0,12,0,0,0,0,9,0,0,1,0,0,3,0,0,6,0,0,1,0,0],[12,75,0.16,0.29019,0.30615,0.0,0.2857,0.57143,0.0,0.85714,14,0,0,14,0,1,0,0,6,0,0,1,0,0,3,0,0,5,0,0,2,0,0],[16,75,0.2133,0.26339,0.33713,0.0,0.0,0.57143,0.0,1.0,17,1,0,17,0,2,0,0,3,0,0,1,0,0,2,0,0,3,0,0,3,0,1],[20,75,0.2667,0.4732,0.33963,0.0,0.57143,0.71429,0.0,1.0,9,1,0,9,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,0,5,0,1],[24,75,0.32,0.17409,0.21644,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,0,0,0,8,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[28,75,0.3733,0.10268,0.20896,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[32,75,0.4267,0.17857,0.27433,0.0,0.0,0.28571,0.0,0.85714,20,0,0,20,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,0,2,0,0],[36,75,0.48,0.13393,0.22851,0.0,0.0,0.2857,0.0,0.85714,21,0,0,21,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[40,75,0.5333,0.16071,0.25939,0.0,0.0,0.28571,0.0,1.0,20,1,0,20,0,2,0,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,1],[44,75,0.5867,0.15178,0.25985,0.0,0.0,0.2857,0.0,0.85714,22,0,0,22,0,0,0,0,5,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[48,75,0.64,0.23214,0.28291,0.0,0.0,0.42858,0.0,0.85714,17,0,0,17,0,0,0,0,6,0,0,2,0,0,2,0,0,4,0,0,1,0,0],[52,75,0.6933,0.19196,0.23038,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,1,0,0,6,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[56,75,0.7467,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,75,0.8,0.13384,0.24984,0.0,0.0,0.14071,0.0,0.85714,23,0,0,23,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,0],[64,75,0.8533,0.05804,0.16312,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[68,75,0.9067,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[75,75,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7dff52a77d165b1e","q":"1. A1 (KOR) ${ }^{\\mathrm{IMO} 4}$ Let $n \\geq 3$ be an integer and $t_{1}, t_{2}, \\ldots, t_{n}$ positive real numbers such that  $$ n^{2}+1>\\left(t_{1}+t_{2}+\\cdots+t_{n}\\right)\\left(\\frac{1}{t_{1}}+\\frac{1}{t_{2}}+\\cdots+\\frac{1}{t_{n}}\\right) . $$  Show that $t_{i}, t_{j}, t_{k}$ are the side lengths of a triangle for all $i, j, k$ with $1 \\leq i<j<k \\leq n$.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.4241,"x":0.87054,"p":[[0,77,0.0,0.4241,0.39038,0.14286,0.28571,0.78571,0.0,1.0,7,8,0,7,0,8,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,8],[4,77,0.0519,0.80356,0.27608,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,4,0,0,4,0,17],[8,77,0.1039,0.6964,0.32685,0.53539,0.71429,1.0,0.0,1.0,1,13,0,1,0,4,0,0,2,0,0,1,0,0,3,0,0,6,0,0,2,0,13],[12,77,0.1558,0.71875,0.35622,0.57143,0.92857,1.0,0.0,1.0,2,16,0,2,0,5,0,0,0,0,0,0,0,0,3,0,0,4,0,0,2,0,16],[16,77,0.2078,0.66516,0.2826,0.53539,0.71429,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,1,0,0,3,0,0,3,0,0,10,0,0,4,0,7],[20,77,0.2597,0.57143,0.31943,0.35718,0.71429,0.71429,0.0,1.0,3,5,0,3,0,5,0,0,0,0,0,3,0,0,3,0,0,11,0,0,2,0,5],[24,77,0.3117,0.65179,0.33869,0.53571,0.71429,1.0,0.0,1.0,2,11,0,2,0,5,0,0,0,0,0,1,0,0,6,0,0,5,0,0,2,0,11],[28,77,0.3636,0.75893,0.29545,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,2,0,0,1,0,0,7,0,0,1,0,16],[32,77,0.4156,0.62946,0.30064,0.5,0.71429,0.75,0.14286,1.0,0,7,0,0,0,7,0,0,1,0,0,0,0,0,3,0,0,13,0,0,1,0,7],[36,77,0.4675,0.54461,0.33963,0.14286,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,11,0,0,1,0,0,1,0,0,6,0,0,3,0,0,3,0,7],[40,77,0.5195,0.63839,0.37284,0.14286,0.78571,1.0,0.0,1.0,3,11,0,3,0,6,0,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,11],[44,77,0.5714,0.75444,0.19313,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,10,0,0,5,0,8],[48,77,0.6234,0.59354,0.32387,0.14286,0.71429,0.85714,0.0,1.0,2,4,0,2,0,7,0,0,0,0,0,0,0,0,3,0,0,10,0,0,6,0,4],[52,77,0.6753,0.62935,0.32129,0.53539,0.71429,0.89286,0.0,1.0,2,8,0,2,0,5,0,0,0,0,0,1,0,0,5,0,0,9,0,0,2,0,8],[56,77,0.7273,0.71429,0.28794,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,3,0,0,0,0,0,2,0,0,4,0,0,7,0,0,5,0,10],[60,77,0.7792,0.81249,0.25616,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,4,0,0,9,0,14],[64,77,0.8312,0.87054,0.10926,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,16,0,10],[68,77,0.8831,0.85712,0.13835,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,16,0,10],[72,77,0.9351,0.76338,0.20706,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,15,0,5],[76,77,0.987,0.7723,0.19841,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,5,0,0,19,0,3],[77,77,1.0,0.74999,0.15154,0.71429,0.78571,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,11,0,0,15,0,1]]},{"b":3,"e":1.0,"k":"rising","v":0.59374,"x":1.0,"p":[[0,72,0.0,0.59374,0.36441,0.28571,0.71429,1.0,0.0,1.0,3,10,0,3,0,4,0,0,4,0,0,3,0,0,1,0,0,4,0,0,3,0,10],[4,72,0.0556,0.7857,0.33313,0.67846,1.0,1.0,0.0,1.0,2,20,0,2,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,20],[8,72,0.1111,0.79464,0.29867,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,18],[12,72,0.1667,0.64732,0.38463,0.14286,0.85714,1.0,0.0,1.0,2,14,0,2,0,7,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,14],[16,72,0.2222,0.63839,0.37113,0.24999,0.71429,1.0,0.0,1.0,3,12,0,3,0,5,0,0,2,0,0,0,0,0,2,0,0,6,0,0,2,0,12],[20,72,0.2778,0.86607,0.22851,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,20],[24,72,0.3333,0.86161,0.28901,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[28,72,0.3889,0.92409,0.18555,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,25],[32,72,0.4444,0.92411,0.19557,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[36,72,0.5,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[40,72,0.5556,0.91964,0.12846,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[44,72,0.6111,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[48,72,0.6667,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,72,0.7222,0.96428,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[56,72,0.7778,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[60,72,0.8333,0.94643,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[64,72,0.8889,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[68,72,0.9444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,72,1.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26]]}]},{"i":"7dd1baa1228a32c5","q":"Let $n$ be a positive integer, and define $S_{n}=\\{1,2, \\ldots, n\\}$. Consider a non-empty subset $T$ of $S_{n}$. We say that $T$ is balanced if the median of $T$ is equal to the average of $T$. For example, for $n=9$, each of the subsets $\\{7\\},\\{2,5\\},\\{2,3,4\\},\\{5,6,8,9\\}$, and $\\{1,4,5,7,8\\}$ is balanced; however, the subsets $\\{2,4,5\\}$ and $\\{1,2,3,5\\}$ are not balanced. For each $n \\geq 1$, prove that the number of balanced subsets of $S_{n}$ is odd.\n\n(To define the median of a set of $k$ numbers, first put the numbers in increasing order; then the median is the middle number if $k$ is odd, and the average of the two middle numbers if $k$ is even. For example, the median of $\\{1,3,4,8,9\\}$ is 4 , and the median of $\\{1,3,4,7,8,9\\}$ is $(4+7) / 2=5.5$. $)$","t":[{"b":0,"e":1.0,"k":"rising","v":0.48655,"x":1.0,"p":[[0,32,0.0,0.49996,0.19231,0.49968,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,5,0,0,0,0,0,19,0,0,5,0,0,0,0,0],[4,32,0.125,0.49996,0.20824,0.28571,0.57143,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,8,0,0,0,0,0,16,0,0,4,0,0,2,0,0],[8,32,0.25,0.48655,0.22261,0.49968,0.57143,0.57143,0.0,0.85714,4,0,0,4,0,0,0,0,4,0,0,0,0,0,21,0,0,1,0,0,2,0,0],[12,32,0.375,0.84374,0.19019,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,8,0,15],[16,32,0.5,0.81692,0.18982,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,5,0,14],[20,32,0.625,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,32,0.75,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,32,1.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]},{"b":4,"e":0.28571,"k":"flat","v":0.48653,"x":0.60265,"p":[[0,24,0.0,0.53569,0.19884,0.53539,0.57143,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,6,0,0,1,0,0,20,0,0,1,0,0,1,0,2],[4,24,0.1667,0.48653,0.17072,0.28571,0.571,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,9,0,0,1,0,0,17,0,0,4,0,0,0,0,0],[8,24,0.3333,0.50885,0.17832,0.28571,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,2,0,0,15,0,0,2,0,0,3,0,0],[12,24,0.5,0.60265,0.14167,0.57143,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,1,0,0,17,0,0,8,0,0,3,0,0],[16,24,0.6667,0.54903,0.14333,0.571,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,6,0,0,0,0,0,20,0,0,5,0,0,1,0,0],[20,24,0.8333,0.5535,0.14616,0.571,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,1,0,0,16,0,0,9,0,0,0,0,0],[24,24,1.0,0.54014,0.16262,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,3,0,0,13,0,0,8,0,0,1,0,0]]}]},{"i":"b70cb5fbfae0d11c","q":"In triangle  $\\triangle ABC$ , the angle  $\\angle BAC$  is less than  $90^o$ . The perpendiculars from  $C$  on  $AB$  and from  $B$  on  $AC$  intersect the circumcircle of  $\\triangle ABC$  again at  $D$  and  $E$  respectively. If  $|DE| =|BC|$ , \ffind the measure of the angle  $\\angle BAC$ 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 $n$  be an odd natural number and  $A,B \\in \\mathcal{M}_n(\\mathbb{C})$  be two matrices such that  $(A-B)^2=O_n.$  Prove that  $\\det(AB-BA)=0.$","t":[{"b":4,"e":1.0,"k":"flat","v":0.80804,"x":1.0,"p":[[0,53,0.0,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20],[4,53,0.0755,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[8,53,0.1509,0.80804,0.23313,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,7,0,0,0,0,17],[12,53,0.2264,0.875,0.20748,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0,0,0,22],[16,53,0.3019,0.9375,0.1234,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,25],[20,53,0.3774,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[24,53,0.4528,0.95089,0.1365,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[28,53,0.5283,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[32,53,0.6038,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,28],[36,53,0.6792,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[40,53,0.7547,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[44,53,0.8302,0.90625,0.20079,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,25],[48,53,0.9057,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[52,53,0.9811,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[53,53,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.8482,"x":1.0,"p":[[0,49,0.0,0.87946,0.18935,0.71429,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,20],[4,49,0.0816,0.91071,0.13716,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,22],[8,49,0.1633,0.87054,0.15303,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,1,0,18],[12,49,0.2449,0.86607,0.15947,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,1,0,18],[16,49,0.3265,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[20,49,0.4082,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[24,49,0.4898,0.8482,0.19214,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,11,0,0,0,0,18],[28,49,0.5714,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,49,0.6531,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,49,0.7347,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,49,0.8163,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,49,0.898,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,49,0.9796,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[49,49,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"5a0808fdb98d881b","q":"Inside the circle $\\Gamma$ there are three circles $\\gamma_{1}, \\gamma_{2}$ and $\\gamma_{3}$, tangent to $\\Gamma$ at $A, B$ and $C$ respectively, all distinct. The circles $\\gamma_{2}$ and $\\gamma_{3}$ have a common point $K$ which lies on $[B C]$, the circles $\\gamma_{3}$ and $\\gamma_{1}$ have a common point $L$ which lies on $[C A]$, and the circles $\\gamma_{1}$ and $\\gamma_{2}$ have a common point $M$ which lies on $[A B]$.\n\nProve that the center of $\\Gamma$ belongs to $\\gamma_{1}, \\gamma_{2}$ and $\\gamma_{3}$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.83927,"x":0.97321,"p":[[0,77,0.0,0.83927,0.2918,0.85711,1.0,1.0,0.0,1.0,3,19,1,3,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,19],[4,77,0.0519,0.94196,0.18509,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[8,77,0.1039,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,77,0.1558,0.95088,0.11637,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[16,77,0.2078,0.92856,0.17128,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,25],[20,77,0.2597,0.96427,0.07145,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[24,77,0.3117,0.93303,0.10706,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[28,77,0.3636,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[32,77,0.4156,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[36,77,0.4675,0.9375,0.18189,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[40,77,0.5195,0.9241,0.18893,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[44,77,0.5714,0.87946,0.22619,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,1,0,22],[48,77,0.6234,0.90623,0.13179,0.82132,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[52,77,0.6753,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[56,77,0.7273,0.91964,0.17474,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,24],[60,77,0.7792,0.87946,0.13415,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,17],[64,77,0.8312,0.88839,0.12745,0.82143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16],[68,77,0.8831,0.89285,0.17128,0.85711,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,7,0,19],[72,77,0.9351,0.89286,0.13363,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,19],[76,77,0.987,0.90178,0.11538,0.85711,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,17],[77,77,1.0,0.92856,0.11294,0.85711,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22]]},{"b":2,"e":1.0,"k":"flat","v":0.77679,"x":0.99554,"p":[[0,127,0.0,0.77679,0.36932,0.71429,1.0,1.0,0.0,1.0,4,21,0,4,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,21],[4,127,0.0315,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[8,127,0.063,0.93749,0.10063,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[12,127,0.0945,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,127,0.126,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[20,127,0.1575,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,127,0.189,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,127,0.2205,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,127,0.252,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,127,0.2835,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[40,127,0.315,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,127,0.3465,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,127,0.378,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[52,127,0.4094,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,127,0.4409,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[60,127,0.4724,0.96875,0.11142,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[64,127,0.5039,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[68,127,0.5354,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[72,127,0.5669,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[76,127,0.5984,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[80,127,0.6299,0.96428,0.11294,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[84,127,0.6614,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[88,127,0.6929,0.9598,0.10254,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[92,127,0.7244,0.9598,0.09612,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[96,127,0.7559,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[100,127,0.7874,0.93747,0.14705,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,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every positive integer  $n$  let  $S(n)$  be the sum of its digits. We say  $n$  has a property  $P$  if all terms in the infinite secuence  $n, S(n), S(S(n)),...$  are even numbers, and we say  $n$  has a property  $I$  if all terms in this secuence are odd. Show that for,  $1 \\le n \\le 2017$  there are more  $n$  that have property  $I$  than those who have  $P$ 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the naked mole rat is hopping on the number line. He starts at  $0$  and jumps exactly  $2^{n}$  either forward or backward at random at time  $n$  seconds, his first jump being at time  $n = 0$ . What is the expected number of jumps James takes before he is on a number that exceeds  $8$ ?","t":[{"b":4,"e":0.14286,"k":"flat","v":0.125,"x":0.26786,"p":[[0,176,0.0,0.24107,0.14914,0.14286,0.28571,0.28571,0.0,0.85714,4,0,4,4,0,6,0,0,21,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,176,0.0227,0.23214,0.09279,0.14286,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,9,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,176,0.0455,0.20536,0.10062,0.14286,0.2143,0.28571,0.0,0.42857,3,0,0,3,0,13,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,176,0.0682,0.20089,0.1063,0.14286,0.21428,0.28571,0.0,0.42857,4,0,0,4,0,12,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,176,0.0909,0.25,0.08748,0.14286,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,8,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,176,0.1136,0.18974,0.09383,0.14286,0.14286,0.28571,0.0,0.357,3,0,0,3,0,16,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[24,176,0.1364,0.2008,0.08651,0.14286,0.14288,0.28571,0.0,0.28571,2,0,0,2,0,15,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,176,0.1591,0.21428,0.09449,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,13,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,176,0.1818,0.20089,0.11214,0.14286,0.2857,0.28571,0.0,0.42857,5,0,0,5,0,10,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,176,0.2045,0.22768,0.08645,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,12,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,176,0.2273,0.22768,0.10012,0.14286,0.2857,0.28571,0.0,0.4286,3,0,0,3,0,8,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,176,0.25,0.18303,0.11971,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,10,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,176,0.2727,0.22322,0.09407,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,11,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,176,0.2955,0.2008,0.08651,0.14286,0.14286,0.28571,0.0,0.28571,2,0,0,2,0,15,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,176,0.3182,0.23214,0.09279,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,176,0.3409,0.2366,0.09852,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[64,176,0.3636,0.21875,0.07973,0.14286,0.2857,0.2857,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,176,0.3864,0.26786,0.09279,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,5,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[72,176,0.4091,0.25892,0.05576,0.2857,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,176,0.4318,0.25,0.09449,0.14289,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,8,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,176,0.4545,0.25893,0.08328,0.24999,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,8,0,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[84,176,0.4773,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,6,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[88,176,0.5,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,176,0.5227,0.26777,0.09291,0.24999,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,8,0,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[96,176,0.5455,0.20536,0.07936,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,176,0.5682,0.1741,0.09268,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,17,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,176,0.5909,0.17634,0.09614,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,1,15,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,176,0.6136,0.19634,0.11156,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,14,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,176,0.6364,0.16964,0.06621,0.14286,0.14286,0.1429,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,176,0.6591,0.16964,0.10972,0.14286,0.14286,0.17857,0.0,0.57143,4,0,0,4,0,20,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[120,176,0.6818,0.16072,0.10564,0.14286,0.14286,0.28571,0.0,0.28571,7,0,0,7,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,176,0.7045,0.17411,0.09268,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,17,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,176,0.7273,0.19196,0.07667,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,176,0.75,0.20312,0.10275,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,1,17,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[136,176,0.7727,0.1875,0.0974,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[140,176,0.7955,0.16963,0.10967,0.14286,0.14286,0.17857,0.0,0.571,4,0,0,4,0,20,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[144,176,0.8182,0.14509,0.07677,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,1,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,176,0.8409,0.16964,0.08328,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,20,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,176,0.8636,0.15616,0.07458,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,176,0.8864,0.17857,0.09449,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,24,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[160,176,0.9091,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[164,176,0.9318,0.15179,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,176,0.9545,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[172,176,0.9773,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[176,176,1.0,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.14277,"x":0.20089,"p":[[0,22,0.0,0.20089,0.20472,0.0,0.14288,0.2857,0.0,1.0,10,1,8,10,0,7,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,22,0.1818,0.16964,0.06621,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.16519,0.06299,0.14286,0.14286,0.14286,0.0,0.286,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.18304,0.07349,0.14286,0.14286,0.17857,0.14286,0.42857,0,0,0,0,0,24,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"98d05f0c607fad38","q":"Find all pairs of real number  $x$  and  $y$  which simultaneously satisfy the following 2 relations: $x+y+4=\\frac{12x+11y}{x^2+y^2}$  $y-x+3=\\frac{11x-12y}{x^2+y^2}$","t":[{"b":0,"e":1.0,"k":"rising","v":0.49107,"x":0.97768,"p":[[0,149,0.0,0.49107,0.2111,0.42857,0.42857,0.57143,0.0,1.0,2,2,0,2,0,0,0,0,2,0,0,17,0,0,4,0,0,5,0,0,0,0,2],[4,149,0.0268,0.58927,0.21354,0.42857,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,16,0,0,3,0,0,7,0,0,0,0,5],[8,149,0.0537,0.60272,0.20429,0.42857,0.50071,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,3,0,0,7,0,0,2,0,4],[12,149,0.0805,0.56696,0.16553,0.42857,0.50001,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,15,0,0,3,0,0,11,0,0,1,0,1],[16,149,0.1074,0.5625,0.20183,0.42857,0.42859,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,18,0,0,3,0,0,6,0,0,0,0,4],[20,149,0.1342,0.76786,0.21053,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,12,0,0,1,0,12],[24,149,0.1611,0.91518,0.18851,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,25],[28,149,0.1879,0.85272,0.20963,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,3,0,19],[32,149,0.2148,0.86161,0.2004,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,4,0,19],[36,149,0.2416,0.92857,0.18211,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[40,149,0.2685,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[44,149,0.2953,0.92411,0.15966,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,25],[48,149,0.3221,0.85713,0.15972,0.82132,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,11,0,13],[52,149,0.349,0.90625,0.21609,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,25],[56,149,0.3758,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[60,149,0.4027,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[64,149,0.4295,0.91518,0.16311,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,8,0,21],[68,149,0.4564,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[72,149,0.4832,0.93304,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[76,149,0.5101,0.86605,0.21412,0.85711,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,19],[80,149,0.5369,0.91518,0.18161,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,22],[84,149,0.5638,0.92411,0.18552,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[88,149,0.5906,0.91517,0.17807,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[92,149,0.6174,0.95536,0.12078,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[96,149,0.6443,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[100,149,0.6711,0.93304,0.1636,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[104,149,0.698,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[108,149,0.7248,0.91518,0.17076,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,21],[112,149,0.7517,0.94643,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[116,149,0.7785,0.91517,0.16698,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,20],[120,149,0.8054,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[124,149,0.8322,0.95089,0.11071,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,24],[128,149,0.8591,0.91071,0.18472,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,5,0,23],[132,149,0.8859,0.90624,0.16985,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,22],[136,149,0.9128,0.94197,0.15915,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,26],[140,149,0.9396,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[144,149,0.9664,0.85268,0.29984,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,23],[148,149,0.9933,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[149,149,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":5,"e":0.42857,"k":"flat","v":0.4375,"x":0.64731,"p":[[0,210,0.0,0.60267,0.198,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,9,0,0,2,0,0,15,0,0,1,0,2],[4,210,0.019,0.54915,0.21159,0.42857,0.42859,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,18,0,0,3,0,0,6,0,0,1,0,3],[8,210,0.0381,0.59822,0.21852,0.42857,0.50001,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,15,0,0,2,0,0,8,0,0,2,0,4],[12,210,0.0571,0.57589,0.18029,0.42857,0.5,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,16,0,0,5,0,0,8,0,0,0,0,3],[16,210,0.0762,0.49999,0.19232,0.42857,0.42857,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,19,0,0,3,0,0,5,0,0,0,0,2],[20,210,0.0952,0.63393,0.20183,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,13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quadrilateral $A B C D$, it holds that $\\angle A=\\angle C=90^{\\circ}$. Let $E$ be a point inside the quadrilateral. Let $M$ be the midpoint of $B E$. Prove that $\\angle A D B=\\angle E D C$ if and only if $|M A|=|M C|$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.10713,"x":0.82587,"p":[[0,60,0.0,0.82587,0.2569,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,5,0,0,3,0,18],[4,60,0.0667,0.78569,0.31544,0.67846,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,18],[8,60,0.1333,0.7857,0.23959,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,5,0,0,6,0,13],[12,60,0.2,0.72767,0.34323,0.53571,0.85714,1.0,0.0,1.0,2,14,0,2,0,4,0,0,0,0,0,2,0,0,1,0,0,3,0,0,6,0,14],[16,60,0.2667,0.73211,0.27608,0.53539,0.78564,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,6,0,0,4,0,12],[20,60,0.3333,0.72764,0.3524,0.571,0.85714,1.0,0.0,1.0,2,14,0,2,0,5,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,14],[24,60,0.4,0.65168,0.36072,0.28571,0.71429,1.0,0.0,1.0,4,11,0,4,0,2,0,0,3,0,0,0,0,0,3,0,0,5,0,0,4,0,11],[28,60,0.4667,0.71875,0.3871,0.53572,1.0,1.0,0.0,1.0,5,18,0,5,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,18],[32,60,0.5333,0.71873,0.31438,0.53539,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,3,0,0,4,0,0,2,0,0,5,0,13],[36,60,0.6,0.78571,0.25754,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,1,0,0,2,0,0,6,0,0,8,0,12],[40,60,0.6667,0.75445,0.3467,0.67857,1.0,1.0,0.0,1.0,4,17,0,4,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,3,0,17],[44,60,0.7333,0.70088,0.33381,0.57143,0.78564,1.0,0.0,1.0,3,13,0,3,0,2,0,0,0,0,0,2,0,0,5,0,0,4,0,0,3,0,13],[48,60,0.8,0.50891,0.42248,0.0,0.57143,1.0,0.0,1.0,11,9,0,11,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,9],[52,60,0.8667,0.433,0.37538,0.0,0.4998,0.71429,0.0,1.0,11,4,0,11,0,2,0,0,1,0,0,2,0,0,3,0,0,7,0,0,2,0,4],[56,60,0.9333,0.52223,0.33438,0.28571,0.50001,0.74996,0.0,1.0,5,6,0,5,0,2,0,0,2,0,0,7,0,0,4,0,0,4,0,0,2,0,6],[60,60,1.0,0.10713,0.20822,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.571,"k":"falling","v":0.31247,"x":0.78556,"p":[[0,100,0.0,0.78556,0.26731,0.71321,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,0,0,0,0,0,0,4,0,0,5,0,0,7,0,13],[4,100,0.04,0.68748,0.38703,0.14289,0.85714,1.0,0.0,1.0,3,15,0,3,0,6,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,15],[8,100,0.08,0.74551,0.38256,0.571,1.0,1.0,0.0,1.0,5,19,0,5,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,19],[12,100,0.12,0.62944,0.38275,0.14289,0.78564,1.0,0.0,1.0,5,10,0,5,0,4,0,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,10],[16,100,0.16,0.55355,0.369,0.14286,0.57143,0.89286,0.0,1.0,4,8,0,4,0,5,0,0,3,0,0,2,0,0,3,0,0,3,0,0,4,0,8],[20,100,0.2,0.75446,0.31183,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,3,0,0,1,0,0,2,0,0,2,0,0,2,0,0,7,0,14],[24,100,0.24,0.49991,0.37805,0.14286,0.42857,0.85714,0.0,1.0,4,7,0,4,0,8,0,0,3,0,0,2,0,0,1,0,0,3,0,0,4,0,7],[28,100,0.28,0.66514,0.34184,0.4286,0.71429,1.0,0.0,1.0,2,13,0,2,0,4,0,0,0,0,0,4,0,0,4,0,0,4,0,0,1,0,13],[32,100,0.32,0.66518,0.35103,0.39285,0.85714,1.0,0.0,1.0,1,11,0,1,0,6,0,0,1,0,0,4,0,0,0,0,0,2,0,0,7,0,11],[36,100,0.36,0.5625,0.35703,0.14289,0.71429,0.85714,0.0,1.0,2,6,0,2,0,8,0,0,3,0,0,0,0,0,1,0,0,6,0,0,6,0,6],[40,100,0.4,0.55354,0.34579,0.14286,0.57143,0.85714,0.0,1.0,2,6,0,2,0,8,0,0,1,0,0,2,0,0,4,0,0,4,0,0,5,0,6],[44,100,0.44,0.61602,0.3489,0.2857,0.57143,1.0,0.0,1.0,2,10,0,2,0,5,0,0,2,0,0,2,0,0,6,0,0,1,0,0,4,0,10],[48,100,0.48,0.48211,0.35846,0.14286,0.571,0.71429,0.0,1.0,4,6,0,4,0,9,0,0,1,0,0,1,0,0,4,0,0,6,0,0,1,0,6],[52,100,0.52,0.51342,0.38606,0.14286,0.5005,1.0,0.0,1.0,3,10,0,3,0,10,0,0,1,0,0,2,0,0,4,0,0,1,0,0,1,0,10],[56,100,0.56,0.52232,0.33237,0.24999,0.57143,0.85714,0.0,1.0,1,6,0,1,0,7,0,0,7,0,0,0,0,0,4,0,0,4,0,0,3,0,6],[60,100,0.6,0.50446,0.36419,0.14286,0.42859,0.89286,0.0,1.0,3,8,0,3,0,8,0,0,3,0,0,3,0,0,2,0,0,4,0,0,1,0,8],[64,100,0.64,0.56249,0.37105,0.14289,0.64286,1.0,0.0,1.0,3,9,0,3,0,6,0,0,4,0,0,1,0,0,2,0,0,4,0,0,3,0,9],[68,100,0.68,0.5491,0.36089,0.14289,0.57121,0.89286,0.0,1.0,3,8,0,3,0,6,0,0,3,0,0,3,0,0,2,0,0,4,0,0,3,0,8],[72,100,0.72,0.56241,0.40247,0.14286,0.71429,1.0,0.0,1.0,4,10,0,4,0,8,0,0,2,0,0,0,0,0,0,0,0,4,0,0,4,0,10],[76,100,0.76,0.52224,0.37057,0.14286,0.57143,0.89286,0.0,1.0,2,8,0,2,0,11,0,0,0,0,0,2,0,0,4,0,0,2,0,0,3,0,8],[80,100,0.8,0.46871,0.38503,0.14286,0.42836,0.857,0.0,1.0,7,6,0,7,0,6,0,0,3,0,0,0,0,0,2,0,0,5,0,0,3,0,6],[84,100,0.84,0.41518,0.35419,0.14286,0.28571,0.71429,0.0,1.0,4,5,0,4,0,11,0,0,4,0,0,1,0,0,0,0,0,6,0,0,1,0,5],[88,100,0.88,0.41072,0.3567,0.14286,0.28571,0.71429,0.0,1.0,5,5,0,5,0,10,0,0,3,0,0,2,0,0,2,0,0,3,0,0,2,0,5],[92,100,0.92,0.31247,0.29107,0.14286,0.14286,0.46418,0.0,1.0,5,2,0,5,0,12,0,0,6,0,0,1,0,0,2,0,0,3,0,0,1,0,2],[96,100,0.96,0.39279,0.2901,0.14286,0.28571,0.60714,0.0,1.0,4,1,0,4,0,8,0,0,6,0,0,0,0,0,6,0,0,5,0,0,2,0,1],[100,100,1.0,0.31697,0.25187,0.14286,0.14286,0.4286,0.0,1.0,1,1,0,1,0,17,0,0,3,0,0,4,0,0,2,0,0,3,0,0,1,0,1]]}]},{"i":"ea8d6cca6efcc46e","q":"For some  $n \\geq 2$ , there is an increasing sequence  $(s_1, s_2, \\dots , s_n)$  of positive integers where for all sequences  $(t_1, t_2, \\dots , t_n)$  of positive integers for which  $s_j - s_i \\mid  t_j - t_i$  for all  $1 \\leq i < j \\leq n$ , there exists a polynomial  $P$  with integer coefficients such that  $P(s_i) = t_i$  for all  $1 \\leq i \\leq n$ . Find the maximum possible value of  $n$ , or show that no such maximum exists.\n\n*Proposed by Vikram Sarkar*","t":[{"b":1,"e":0.28571,"k":"volatile","v":0.13616,"x":0.77009,"p":[[0,48,0.0,0.2567,0.15242,0.19644,0.2857,0.42857,0.0,0.4286,6,0,5,6,0,2,0,6,7,0,1,10,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.58925,0.29234,0.42857,0.71429,0.73214,0.0,1.0,4,2,0,4,0,1,0,0,1,0,1,3,0,0,2,0,0,12,1,0,5,0,2],[8,48,0.1667,0.39508,0.30408,0.0,0.42857,0.60714,0.0,1.0,9,1,0,9,1,0,0,1,2,0,0,4,0,1,6,0,0,6,0,0,1,0,1],[12,48,0.25,0.19418,0.28019,0.0,0.0,0.42857,0.0,0.85714,20,0,0,20,0,0,0,1,2,0,0,3,0,0,3,0,0,1,0,0,2,0,0],[16,48,0.3333,0.13616,0.2515,0.0,0.0,0.17857,0.0,1.0,22,1,0,22,0,2,0,0,3,0,1,1,0,0,0,0,0,2,0,0,0,0,1],[20,48,0.4167,0.25883,0.33013,0.0,0.14143,0.32143,0.0,1.0,15,3,0,15,0,3,0,0,6,0,0,1,0,0,2,0,0,1,0,0,1,0,3],[24,48,0.5,0.7232,0.30344,0.57132,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,3,0,0,2,0,0,5,1,0,0,1,0,6,0,12],[28,48,0.5833,0.71429,0.32733,0.55357,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,4,0,0,0,0,1,4,1,0,0,2,0,3,0,14],[32,48,0.6667,0.77009,0.31122,0.55357,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,6,0,0,0,0,1,2,0,0,0,1,0,3,1,17],[36,48,0.75,0.30357,0.19805,0.2857,0.28571,0.28571,0.0,1.0,3,2,0,3,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,48,0.8333,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.28573,0.09449,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,1,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,48,1.0,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.06473,"x":0.55584,"p":[[0,36,0.0,0.26326,0.17265,0.14286,0.24999,0.42857,0.0,0.57143,6,0,1,6,1,2,0,7,4,0,0,9,0,2,1,0,0,0,0,0,0,0,0],[4,36,0.1111,0.55584,0.27991,0.42857,0.57143,0.71429,0.0,1.0,4,3,0,4,0,1,0,0,0,0,0,6,0,1,6,0,0,9,0,0,2,0,3],[8,36,0.2222,0.43079,0.35913,0.0,0.49979,0.71429,0.0,1.0,11,3,0,11,0,0,0,1,2,0,0,2,0,0,2,0,0,10,0,0,1,0,3],[12,36,0.3333,0.39953,0.29979,0.16071,0.42857,0.57143,0.0,1.0,8,2,0,8,0,0,0,2,3,0,2,4,0,1,5,0,0,4,0,0,1,0,2],[16,36,0.4444,0.41963,0.2671,0.2143,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,1,0,2,2,0,1,5,0,1,8,0,0,3,0,0,3,0,0],[20,36,0.5556,0.32366,0.32926,0.0,0.28571,0.57143,0.0,1.0,14,2,0,14,0,0,0,0,3,0,1,3,0,0,5,0,0,3,0,0,1,0,2],[24,36,0.6667,0.1875,0.27592,0.0,0.0,0.32144,0.0,0.78571,20,0,0,20,1,0,0,0,3,0,0,1,0,0,3,0,0,3,1,0,0,0,0],[28,36,0.7778,0.14062,0.23588,0.0,0.0,0.2857,0.0,0.78571,22,0,0,22,0,1,0,0,3,0,0,2,0,0,2,0,0,1,1,0,0,0,0],[32,36,0.8889,0.10045,0.17574,0.0,0.0,0.125,0.0,0.57143,23,0,0,23,1,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.06473,0.12288,0.0,0.0,0.0,0.0,0.357,25,0,0,25,0,0,0,0,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"36932de584a1eb64","q":"Find the smallest positive number  $\\lambda $  , such that for any complex numbers  ${z_1},{z_2},{z_3}\\in\\{z\\in C\\big| |z|<1\\}$  ,if   $z_1+z_2+z_3=0$ , then   $$ \\left|z_1z_2 +z_2z_3+z_3z_1\\right|^2+\\left|z_1z_2z_3\\right|^2 <\\lambda . $$","t":[{"b":3,"e":0.42857,"k":"flat","v":0.3125,"x":0.55357,"p":[[0,94,0.0,0.3125,0.14032,0.28571,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,4,0,0,19,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[4,94,0.0426,0.49554,0.1636,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,16,0,0,1,0,0,10,0,0,0,0,0],[8,94,0.0851,0.55357,0.1915,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,9,0,0,7,0,0,10,0,0,1,0,1],[12,94,0.1277,0.45536,0.16146,0.39286,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,14,0,0,4,0,0,6,0,0,0,0,0],[16,94,0.1702,0.46429,0.13363,0.42857,0.42857,0.4286,0.28571,0.71429,0,0,0,0,0,0,0,0,5,0,0,20,0,0,1,0,0,6,0,0,0,0,0],[20,94,0.2128,0.48661,0.15093,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,18,0,0,4,0,0,7,0,0,0,0,0],[24,94,0.2553,0.50893,0.17474,0.42857,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,15,0,0,4,0,0,6,0,0,1,0,1],[28,94,0.2979,0.49999,0.16366,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,15,0,0,2,0,0,10,0,0,0,0,0],[32,94,0.3404,0.51784,0.16656,0.42857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,14,0,0,4,0,0,9,0,0,1,0,0],[36,94,0.383,0.48214,0.15872,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,17,0,0,3,0,0,6,0,0,1,0,0],[40,94,0.4255,0.5223,0.12681,0.42857,0.42857,0.60714,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,6,0,0,8,0,0,0,0,0],[44,94,0.4681,0.46875,0.1439,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,18,0,0,3,0,0,6,0,0,0,0,0],[48,94,0.5106,0.51338,0.14664,0.42857,0.4286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,14,0,0,5,0,0,9,0,0,0,0,0],[52,94,0.5532,0.45536,0.16536,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,12,0,0,3,0,0,7,0,0,0,0,0],[56,94,0.5957,0.45982,0.15866,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,6,0,0,11,0,0,4,0,0,0,0,0],[60,94,0.6383,0.49106,0.19864,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,10,0,0,1,0,0,9,0,0,2,0,0],[64,94,0.6809,0.45088,0.15197,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,13,0,0,5,0,0,5,0,0,0,0,0],[68,94,0.7234,0.50002,0.15149,0.42857,0.5005,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,9,0,0,9,0,0,7,0,0,0,0,0],[72,94,0.766,0.47768,0.16982,0.28571,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,10,0,0,5,0,0,8,0,0,0,0,0],[76,94,0.8085,0.41517,0.15713,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,11,0,0,4,0,0,4,0,0,0,0,0],[80,94,0.8511,0.42856,0.16365,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,13,0,0,8,0,0,5,0,0,5,0,0,0,0,0],[84,94,0.8936,0.45088,0.13881,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,13,0,0,6,0,0,4,0,0,0,0,0],[88,94,0.9362,0.43302,0.14499,0.28571,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,11,0,0,5,0,0,4,0,0,0,0,0],[92,94,0.9787,0.46872,0.1249,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,6,0,0,14,0,0,9,0,0,3,0,0,0,0,0],[94,94,1.0,0.42411,0.15355,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,10,0,0,5,0,0,4,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.30804,"x":0.58036,"p":[[0,177,0.0,0.30804,0.12931,0.28571,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,4,0,0,19,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[4,177,0.0226,0.50892,0.15125,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,16,0,0,4,0,0,9,0,0,0,0,0],[8,177,0.0452,0.49554,0.16746,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,14,0,0,2,0,0,10,0,0,0,0,0],[12,177,0.0678,0.52679,0.16536,0.42857,0.42857,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,17,0,0,1,0,0,9,0,0,2,0,0],[16,177,0.0904,0.49107,0.17105,0.42857,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,13,0,0,2,0,0,10,0,0,0,0,0],[20,177,0.113,0.49105,0.18187,0.42857,0.42857,0.60714,0.0,0.857,1,0,0,1,0,0,0,0,6,0,0,11,0,0,6,0,0,7,0,0,1,0,0],[24,177,0.1356,0.48661,0.13767,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,18,0,0,3,0,0,7,0,0,0,0,0],[28,177,0.1582,0.53125,0.1684,0.42857,0.50001,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,13,0,0,7,0,0,8,0,0,0,0,1],[32,177,0.1808,0.46429,0.15152,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,15,0,0,4,0,0,6,0,0,0,0,0],[36,177,0.2034,0.49999,0.14725,0.42857,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,17,0,0,4,0,0,8,0,0,0,0,0],[40,177,0.226,0.50446,0.14279,0.42857,0.42857,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,15,0,0,5,0,0,8,0,0,0,0,0],[44,177,0.2486,0.51786,0.16269,0.42857,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,17,0,0,3,0,0,8,0,0,0,0,1],[48,177,0.2712,0.45536,0.19377,0.39286,0.42857,0.60714,0.0,0.71429,1,0,0,1,0,3,0,0,4,0,0,13,0,0,3,0,0,8,0,0,0,0,0],[52,177,0.2938,0.49107,0.16728,0.42857,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,18,0,0,1,0,0,7,0,0,0,0,1],[56,177,0.3164,0.52676,0.14031,0.42857,0.4286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,16,0,0,4,0,0,10,0,0,0,0,0],[60,177,0.339,0.49554,0.18893,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,3,0,0,13,0,0,7,0,0,6,0,0,0,0,1],[64,177,0.3616,0.44195,0.13996,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,19,0,0,1,0,0,5,0,0,0,0,0],[68,177,0.3842,0.5223,0.14554,0.42857,0.4286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,15,0,0,4,0,0,10,0,0,0,0,0],[72,177,0.4068,0.50005,0.14284,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,19,0,0,1,0,0,9,0,0,0,0,0],[76,177,0.4294,0.58036,0.14258,0.42857,0.64286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,0,16,0,0,0,0,0],[80,177,0.452,0.55804,0.15303,0.42857,0.50001,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,14,0,0,1,0,0,15,0,0,0,0,0],[84,177,0.4746,0.53125,0.1439,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,16,0,0,3,0,0,11,0,0,0,0,0],[88,177,0.4972,0.52231,0.1365,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,16,0,0,5,0,0,9,0,0,0,0,0],[92,177,0.5198,0.52679,0.14914,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,15,0,0,3,0,0,11,0,0,0,0,0],[96,177,0.5424,0.54911,0.13882,0.42857,0.50001,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,15,0,0,4,0,0,12,0,0,0,0,0],[100,177,0.565,0.54016,0.15865,0.42857,0.57121,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,11,0,0,6,0,0,10,0,0,1,0,0],[104,177,0.5876,0.50891,0.13333,0.42857,0.42857,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,18,0,0,4,0,0,8,0,0,0,0,0],[108,177,0.6102,0.50442,0.14277,0.42857,0.42857,0.60714,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,15,0,0,5,0,0,8,0,0,0,0,0],[112,177,0.6328,0.58034,0.15947,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,10,0,0,3,0,0,17,0,0,0,0,0],[116,177,0.6554,0.55356,0.15465,0.42857,0.57121,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,12,0,0,3,0,0,14,0,0,0,0,0],[120,177,0.678,0.54016,0.15865,0.42857,0.4998,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,12,0,0,3,0,0,13,0,0,0,0,0],[124,177,0.7006,0.53124,0.15663,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,16,0,0,2,0,0,12,0,0,0,0,0],[128,177,0.7232,0.49986,0.1182,0.42857,0.42857,0.57111,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,2,0,0,7,0,0,0,0,0],[132,177,0.7458,0.51338,0.13295,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,20,0,0,2,0,0,9,0,0,0,0,0],[136,177,0.7684,0.56696,0.145,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,14,0,0,2,0,0,15,0,0,0,0,0],[140,177,0.791,0.55356,0.13716,0.42857,0.4286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,2,0,0,13,0,0,0,0,0],[144,177,0.8136,0.58033,0.13333,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,15,0,0,0,0,0],[148,177,0.8362,0.53125,0.1394,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,18,0,0,2,0,0,11,0,0,0,0,0],[152,177,0.8588,0.52677,0.16145,0.42857,0.49979,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,14,0,0,6,0,0,10,0,0,0,0,0],[156,177,0.8814,0.49554,0.11285,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,3,0,0,6,0,0,0,0,0],[160,177,0.904,0.51784,0.13716,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,17,0,0,4,0,0,9,0,0,0,0,0],[164,177,0.9266,0.52228,0.13174,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,18,0,0,4,0,0,9,0,0,0,0,0],[168,177,0.9492,0.50446,0.18205,0.42857,0.42857,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,4,0,0,15,0,0,0,0,0,12,0,0,0,0,0],[172,177,0.9718,0.50893,0.14258,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,20,0,0,0,0,0,10,0,0,0,0,0],[176,177,0.9944,0.56695,0.14054,0.42857,0.49979,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,15,0,0,0,0,0],[177,177,1.0,0.49107,0.13333,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,19,0,0,3,0,0,7,0,0,0,0,0]]}]},{"i":"743835405e76bac8","q":"For which positive integers  $n\\geq4$  does there exist a convex  $n$ -gon with side lengths  $1, 2, \\dots, n$  (in some order) and with all of its sides tangent to the same circle?","t":[{"b":0,"e":0.71429,"k":"rising","v":0.44197,"x":0.82143,"p":[[0,68,0.0,0.44197,0.05486,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0],[4,68,0.0588,0.77678,0.20183,0.67857,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,0,2,0,12],[8,68,0.1176,0.82143,0.19885,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,8,0,0,7,0,13],[12,68,0.1765,0.74554,0.18808,0.67857,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,11,0,0,6,0,7],[16,68,0.2353,0.81248,0.15748,0.71429,0.85714,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,8,0,10],[20,68,0.2941,0.7857,0.19886,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,7,0,0,8,0,10],[24,68,0.3529,0.81695,0.15665,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,8,0,10],[28,68,0.4118,0.80802,0.15817,0.71429,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,6,0,10],[32,68,0.4706,0.74998,0.23146,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,7,0,0,4,0,11],[36,68,0.5294,0.8125,0.22428,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,5,0,14],[40,68,0.5882,0.79017,0.20513,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,8,0,0,6,0,11],[44,68,0.6471,0.80804,0.24643,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,3,0,16],[48,68,0.7059,0.73213,0.2468,0.57142,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,5,0,0,6,0,0,5,0,0,5,0,10],[52,68,0.7647,0.74554,0.18117,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,12,0,0,7,0,6],[56,68,0.8235,0.81247,0.17659,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,6,0,12],[60,68,0.8824,0.82142,0.18898,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,4,0,14],[64,68,0.9412,0.82143,0.20825,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,0,5,0,15],[68,68,1.0,0.7857,0.21726,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,3,0,0,10,0,10]]},{"b":7,"e":0.85714,"k":"rising","v":0.46429,"x":0.92397,"p":[[0,27,0.0,0.46429,0.12372,0.42857,0.42857,0.46431,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,23,0,0,5,0,0,3,0,0,0,0,0],[4,27,0.1481,0.82366,0.20123,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,9,1,0,1,0,16],[8,27,0.2963,0.85714,0.15152,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,5,0,15],[12,27,0.4444,0.80357,0.25692,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,5,0,14],[16,27,0.5926,0.89286,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[20,27,0.7407,0.875,0.12753,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,12,0,13],[24,27,0.8889,0.88837,0.14173,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,18],[27,27,1.0,0.92397,0.10731,0.85714,1.0,1.0,0.71,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20]]}]},{"i":"c6afa93ae56aaf20","q":"In a certain city there are $n$ straight streets, such that every two streets intersect, and no three streets pass through the same intersection. The City Council wants to organize the city by designating the main and the side street on every intersection. Prove that this can be done in such way that if one goes along one of the streets, from its beginning to its end, the intersections where this street is the main street, and the ones where it is not, will apear in alternating order.","t":[{"b":5,"e":0.0,"k":"flat","v":0.08482,"x":0.35713,"p":[[0,36,0.0,0.08482,0.17807,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,36,0.1111,0.26786,0.31491,0.0,0.14288,0.42858,0.0,1.0,13,2,0,13,0,5,0,0,5,0,0,2,0,0,1,0,0,3,0,0,1,0,2],[8,36,0.2222,0.31696,0.34206,0.0,0.2857,0.71429,0.0,1.0,13,3,0,13,0,2,0,0,6,0,0,2,0,0,0,0,0,6,0,0,0,0,3],[12,36,0.3333,0.35268,0.3589,0.0,0.28571,0.60714,0.0,1.0,10,5,0,10,0,4,0,0,7,0,0,2,0,0,1,0,0,2,0,0,1,0,5],[16,36,0.4444,0.35713,0.39769,0.0,0.2857,0.71429,0.0,1.0,14,7,0,14,0,1,0,0,5,0,0,2,0,0,1,0,0,2,0,0,0,0,7],[20,36,0.5556,0.24552,0.30352,0.0,0.14285,0.32143,0.0,1.0,16,2,0,16,0,0,0,0,8,0,0,2,0,0,1,0,0,3,0,0,0,0,2],[24,36,0.6667,0.24553,0.3199,0.0,0.14286,0.28571,0.0,1.0,15,3,0,15,0,2,0,0,9,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[28,36,0.7778,0.20088,0.2763,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,3,0,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[32,36,0.8889,0.23659,0.29365,0.0,0.21429,0.28571,0.0,1.0,15,2,0,15,0,1,0,0,10,0,0,0,0,0,3,0,0,0,0,0,1,0,2],[36,36,1.0,0.17411,0.20434,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,3,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.57143,"k":"flat","v":0.04911,"x":0.50442,"p":[[0,33,0.0,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.50442,0.38129,0.21427,0.571,0.89286,0.0,1.0,8,8,0,8,0,0,0,0,6,0,0,1,0,0,5,0,0,2,0,0,2,0,8],[8,33,0.2424,0.48661,0.38608,0.10714,0.42857,0.85714,0.0,1.0,8,6,0,8,0,2,0,0,6,0,0,0,0,0,1,0,0,5,0,0,4,0,6],[12,33,0.3636,0.43304,0.37878,0.0,0.28571,0.71429,0.0,1.0,9,7,0,9,0,2,0,0,6,0,0,2,0,0,2,0,0,4,0,0,0,0,7],[16,33,0.4848,0.35265,0.36592,0.0,0.28571,0.71429,0.0,1.0,12,4,0,12,0,2,0,0,7,0,0,0,0,0,2,0,0,3,0,0,2,0,4],[20,33,0.6061,0.35266,0.34989,0.0,0.28571,0.60714,0.0,1.0,11,4,0,11,0,2,0,0,7,0,0,2,0,0,2,0,0,3,0,0,1,0,4],[24,33,0.7273,0.32587,0.36109,0.0,0.21428,0.60714,0.0,1.0,14,4,0,14,0,2,0,0,4,0,0,1,0,0,3,0,0,4,0,0,0,0,4],[28,33,0.8485,0.22311,0.22287,0.0,0.2857,0.28571,0.0,1.0,10,1,0,10,0,5,0,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[32,33,0.9697,0.13393,0.14258,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,6,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.12937,0.13053,0.0,0.14143,0.2857,0.0,0.28571,15,0,0,15,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cb7f250ec6b2cfee","q":"How many pairs  $(a, b)$  of positive integers  $a,b$  solutions of the equation  $(4a-b)(4b-a )=1770^n$   exist , if  $n$  is a positive integer?","t":[{"b":0,"e":0.5714,"k":"flat","v":0.62497,"x":0.86606,"p":[[0,94,0.0,0.77231,0.19517,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,6,0,0,18,0,3],[4,94,0.0426,0.84373,0.14883,0.71429,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,12,0,11],[8,94,0.0851,0.85713,0.1288,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,18,0,9],[12,94,0.1277,0.79008,0.16396,0.71429,0.85714,0.85714,0.14,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,0,19,0,3],[16,94,0.1702,0.86606,0.07936,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,5],[20,94,0.2128,0.76786,0.1915,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,9,0,0,16,0,3],[24,94,0.2553,0.83929,0.13243,0.82143,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,16,0,8],[28,94,0.2979,0.81695,0.13942,0.71429,0.85714,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,16,0,6],[32,94,0.3404,0.81249,0.16536,0.82132,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,0,19,0,5],[36,94,0.383,0.86606,0.13803,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,16,0,11],[40,94,0.4255,0.81249,0.14916,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,17,0,6],[44,94,0.4681,0.83928,0.12752,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,15,0,8],[48,94,0.5106,0.82142,0.10714,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,18,0,4],[52,94,0.5532,0.79018,0.16746,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,0,17,0,4],[56,94,0.5957,0.79909,0.167,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,0,16,0,5],[60,94,0.6383,0.81248,0.19047,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,16,0,7],[64,94,0.6809,0.81696,0.16065,0.82132,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,19,0,5],[68,94,0.7234,0.83481,0.09522,0.82132,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,20,0,4],[72,94,0.766,0.77678,0.15947,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,0,15,0,3],[76,94,0.8085,0.77229,0.17809,0.67857,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,4,0,0,16,0,4],[80,94,0.8511,0.6964,0.17406,0.57143,0.71414,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,7,0,0,6,0,4],[84,94,0.8936,0.71875,0.20666,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,0,0,0,9,0,0,7,0,0,10,0,4],[88,94,0.9362,0.65624,0.23921,0.57143,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,3,0,0,11,0,2],[92,94,0.9787,0.62497,0.18814,0.57143,0.57143,0.74996,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,1,0,0,15,0,0,5,0,0,8,0,0],[94,94,1.0,0.67409,0.18978,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,12,0,0,8,0,0,6,0,3]]},{"b":5,"e":0.71429,"k":"flat","v":0.70086,"x":0.82143,"p":[[0,32,0.0,0.72763,0.14882,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,6,0,0,12,0,2],[4,32,0.125,0.76784,0.1915,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,6,0,0,14,0,5],[8,32,0.25,0.78571,0.19233,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,5,0,0,20,0,3],[12,32,0.375,0.82143,0.13363,0.71429,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,14,0,7],[16,32,0.5,0.79459,0.14704,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,0,18,0,4],[20,32,0.625,0.79463,0.14257,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,21,0,2],[24,32,0.75,0.75445,0.19311,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,6,0,0,16,0,3],[28,32,0.875,0.78568,0.15156,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,14,0,5],[32,32,1.0,0.70086,0.19679,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,7,0,0,12,0,2]]}]},{"i":"1ec0a8073d5612a7","q":"The four points of a circle are in the following order:  $A, B, C, D$ . Extensions of chord  $AB$  beyond point  $B$  and of chord   $CD$   beyond point  $C$  intersect at point  $E$ , with  $\\angle AED= 60^o$ . If  $\\angle ABD =3  \\angle  BAC$  , prove that  $AD$  is the diameter of the circle.","t":[{"b":1,"e":0.85714,"k":"falling","v":0.75445,"x":0.95536,"p":[[0,22,0.0,0.90624,0.13179,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,19],[4,22,0.1818,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[8,22,0.3636,0.91071,0.09942,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,16],[12,22,0.5455,0.88839,0.11701,0.82143,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,15],[16,22,0.7273,0.85714,0.11845,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,13,0,10],[20,22,0.9091,0.83929,0.09279,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,18,0,5],[22,22,1.0,0.75445,0.09609,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,13,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,44,0.0,0.89285,0.13363,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,16],[4,44,0.0909,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[8,44,0.1818,0.85714,0.18211,0.85714,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,11],[12,44,0.2727,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,44,0.3636,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,44,0.4545,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,44,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,44,0.6364,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,44,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,44,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,44,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,44,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"77a55f065ffbbe47","q":"Show that if $(2+\\sqrt{3})^{k}=1+m+n \\sqrt{3}$, for positive integers $m, n, k$ with $k$ odd, then $m$ is a perfect square.","t":[{"b":2,"e":0.0,"k":"falling","v":0.24106,"x":0.76786,"p":[[0,109,0.0,0.70536,0.36759,0.39286,1.0,1.0,0.0,1.0,2,17,0,2,0,5,0,0,1,0,0,1,0,0,1,0,0,5,0,0,0,0,17],[4,109,0.0367,0.70536,0.38455,0.42857,1.0,1.0,0.0,1.0,5,18,0,5,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,18],[8,109,0.0734,0.69643,0.35129,0.39286,0.78571,1.0,0.0,1.0,2,15,0,2,0,3,0,0,3,0,0,2,0,0,0,0,0,6,0,0,1,0,15],[12,109,0.1101,0.70982,0.35622,0.39288,0.92857,1.0,0.0,1.0,2,16,0,2,0,3,0,0,3,0,0,2,0,0,0,0,0,4,0,0,2,0,16],[16,109,0.1468,0.70536,0.34058,0.53571,0.71429,1.0,0.0,1.0,3,14,0,3,0,2,0,0,1,0,0,2,0,0,1,0,0,8,0,0,1,0,14],[20,109,0.1835,0.61605,0.36323,0.28571,0.71429,1.0,0.0,1.0,2,11,0,2,0,5,0,0,5,0,0,0,0,0,1,0,0,6,0,0,2,0,11],[24,109,0.2202,0.76786,0.34209,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,1,0,0,1,0,0,0,0,0,5,0,0,1,0,19],[28,109,0.2569,0.57143,0.40721,0.14286,0.71429,1.0,0.0,1.0,7,11,0,7,0,3,0,0,2,0,0,1,0,0,1,0,0,5,0,0,2,0,11],[32,109,0.2936,0.60268,0.39727,0.25,0.71429,1.0,0.0,1.0,6,13,0,6,0,2,0,0,3,0,0,1,0,0,2,0,0,5,0,0,0,0,13],[36,109,0.3303,0.56696,0.3838,0.21427,0.71429,1.0,0.0,1.0,8,9,0,8,0,0,0,0,2,0,0,2,0,0,2,0,0,8,0,0,1,0,9],[40,109,0.367,0.5625,0.35524,0.24999,0.5,1.0,0.0,1.0,2,10,0,2,0,6,0,0,3,0,0,5,0,0,1,0,0,5,0,0,0,0,10],[44,109,0.4037,0.61159,0.37836,0.28571,0.71429,1.0,0.0,1.0,6,10,0,6,0,1,0,0,2,0,0,3,0,0,1,0,0,5,0,0,4,0,10],[48,109,0.4404,0.57141,0.39286,0.10714,0.71429,1.0,0.0,1.0,8,10,0,8,0,1,0,0,1,0,0,2,0,0,1,0,0,9,0,0,0,0,10],[52,109,0.4771,0.58929,0.42969,0.14286,0.71429,1.0,0.0,1.0,7,14,0,7,0,4,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,14],[56,109,0.5138,0.67409,0.37156,0.39286,0.78571,1.0,0.0,1.0,4,15,0,4,0,2,0,0,2,0,0,1,0,0,4,0,0,3,0,0,1,0,15],[60,109,0.5505,0.65178,0.35703,0.42857,0.71429,1.0,0.0,1.0,4,13,0,4,0,1,0,0,2,0,0,4,0,0,3,0,0,4,0,0,1,0,13],[64,109,0.5872,0.61607,0.37019,0.25,0.71429,1.0,0.0,1.0,4,11,0,4,0,4,0,0,1,0,0,2,0,0,3,0,0,5,0,0,2,0,11],[68,109,0.6239,0.52678,0.37532,0.14286,0.57143,0.89286,0.0,1.0,5,8,0,5,0,5,0,0,3,0,0,2,0,0,2,0,0,5,0,0,2,0,8],[72,109,0.6606,0.60713,0.37287,0.28571,0.71429,1.0,0.0,1.0,5,11,0,5,0,2,0,0,2,0,0,3,0,0,3,0,0,4,0,0,2,0,11],[76,109,0.6972,0.70089,0.356,0.53571,0.85707,1.0,0.0,1.0,4,14,0,4,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,0,3,0,14],[80,109,0.7339,0.67857,0.3607,0.28571,0.85714,1.0,0.0,1.0,4,13,0,4,0,0,0,0,5,0,0,1,0,0,1,0,0,4,0,0,4,0,13],[84,109,0.7706,0.71874,0.31234,0.53571,0.78564,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,0,3,0,0,2,0,0,6,0,0,3,0,13],[88,109,0.8073,0.67409,0.33738,0.39286,0.71429,1.0,0.0,1.0,2,13,0,2,0,2,0,0,4,0,0,2,0,0,2,0,0,6,0,0,1,0,13],[92,109,0.844,0.47322,0.33012,0.28571,0.42857,0.71429,0.0,1.0,5,5,0,5,0,2,0,0,6,0,0,7,0,0,1,0,0,4,0,0,2,0,5],[96,109,0.8807,0.4241,0.35261,0.14289,0.28571,0.71429,0.0,1.0,7,6,0,7,0,3,0,0,7,0,0,4,0,0,1,0,0,4,0,0,0,0,6],[100,109,0.9174,0.35258,0.29453,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,2,0,0,6,0,0,5,0,0,4,0,0,3,0,0,2,0,1],[104,109,0.9541,0.45087,0.28371,0.28571,0.42857,0.60714,0.0,1.0,5,3,0,5,0,1,0,0,5,0,0,8,0,0,5,0,0,5,0,0,0,0,3],[108,109,0.9908,0.29015,0.31435,0.0,0.14286,0.57111,0.0,1.0,13,1,0,13,0,4,0,0,4,0,0,1,0,0,3,0,0,5,0,0,1,0,1],[109,109,1.0,0.24106,0.27992,0.0,0.14288,0.42857,0.0,1.0,14,1,0,14,0,3,0,0,6,0,0,4,0,0,1,0,0,2,0,0,1,0,1]]},{"b":5,"e":1.0,"k":"flat","v":0.52678,"x":0.83036,"p":[[0,58,0.0,0.59821,0.39518,0.28571,0.71429,1.0,0.0,1.0,5,12,0,5,0,2,0,0,6,0,0,1,0,0,0,0,0,3,0,0,3,0,12],[4,58,0.069,0.75,0.34442,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,3,0,0,2,0,0,0,0,0,0,0,0,7,0,0,0,0,18],[8,58,0.1379,0.6875,0.37019,0.64286,0.78571,1.0,0.0,1.0,5,13,0,5,0,2,0,0,0,0,0,1,0,0,0,0,0,8,0,0,3,0,13],[12,58,0.2069,0.83036,0.2976,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,22],[16,58,0.2759,0.74554,0.36724,0.42857,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,2,0,19],[20,58,0.3448,0.79018,0.30926,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,4,0,17],[24,58,0.4138,0.79911,0.29203,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,3,0,0,0,0,0,1,0,0,7,0,0,0,0,19],[28,58,0.4828,0.74552,0.33262,0.67846,0.92857,1.0,0.0,1.0,3,16,0,3,0,1,0,0,1,0,0,2,0,0,1,0,0,6,0,0,2,0,16],[32,58,0.5517,0.52678,0.36147,0.25,0.64286,0.85714,0.0,1.0,6,6,0,6,0,2,0,0,5,0,0,1,0,0,2,0,0,7,0,0,3,0,6],[36,58,0.6207,0.60714,0.3607,0.2857,0.71429,1.0,0.0,1.0,5,9,0,5,0,2,0,0,2,0,0,2,0,0,1,0,0,9,0,0,2,0,9],[40,58,0.6897,0.72768,0.34508,0.57143,0.92857,1.0,0.0,1.0,3,16,0,3,0,2,0,0,1,0,0,1,0,0,2,0,0,6,0,0,1,0,16],[44,58,0.7586,0.73661,0.33333,0.64286,0.85714,1.0,0.0,1.0,3,15,0,3,0,1,0,0,1,0,0,3,0,0,0,0,0,6,0,0,3,0,15],[48,58,0.8276,0.54464,0.35792,0.25,0.71429,0.71429,0.0,1.0,7,7,0,7,0,1,0,0,1,0,0,4,0,0,2,0,0,10,0,0,0,0,7],[52,58,0.8966,0.63839,0.30301,0.42857,0.71429,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,3,0,0,4,0,0,2,0,0,11,0,0,1,0,8],[56,58,0.9655,0.74554,0.27137,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,4,0,0,0,0,0,10,0,0,3,0,12],[58,58,1.0,0.66964,0.35072,0.39286,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,3,0,0,1,0,0,0,0,0,10,0,0,1,0,12]]}]},{"i":"74fb702359fec30e","q":"Find all triplets of real numbers  $(a,b,c)$  with  $a^2+b^2=c^2$  satisfying\n\n\\[2(a^3+b^3+c^3)=ab(3a+3b-4c)+bc(3b+3c-4a)+ca(3c+3a-4b)\\]\n\n*Proposed by Cody Johnson*","t":[{"b":0,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,177,0.0,0.96429,0.15567,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,177,0.0226,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,177,0.0452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,177,0.0678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,177,0.0904,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,177,0.113,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[24,177,0.1356,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,177,0.1582,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[32,177,0.1808,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[36,177,0.2034,0.95536,0.15746,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[40,177,0.226,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[44,177,0.2486,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[48,177,0.2712,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,177,0.2938,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[56,177,0.3164,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,177,0.339,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,177,0.3616,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,177,0.3842,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,177,0.4068,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,177,0.4294,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[80,177,0.452,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,177,0.4746,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,177,0.4972,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[92,177,0.5198,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[96,177,0.5424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,177,0.565,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,177,0.5876,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[108,177,0.6102,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,177,0.6328,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[116,177,0.6554,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[120,177,0.678,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[124,177,0.7006,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[128,177,0.7232,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[132,177,0.7458,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[136,177,0.7684,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[140,177,0.791,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[144,177,0.8136,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[148,177,0.8362,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[152,177,0.8588,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[156,177,0.8814,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[160,177,0.904,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[164,177,0.9266,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[168,177,0.9492,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[172,177,0.9718,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[176,177,0.9944,0.95536,0.13092,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[177,177,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":6,"e":0.85714,"k":"flat","v":0.875,"x":1.0,"p":[[0,523,0.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,523,0.0076,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,523,0.0153,0.9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04,523,0.3901,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[208,523,0.3977,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[212,523,0.4054,0.95088,0.11637,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[216,523,0.413,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[220,523,0.4207,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[224,523,0.4283,0.94643,0.15465,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,25],[228,523,0.4359,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[232,523,0.4436,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,25],[236,523,0.4512,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[240,523,0.4589,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[244,523,0.4665,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[248,523,0.4742,0.92411,0.17491,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[252,523,0.4818,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[256,523,0.4895,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[260,523,0.4971,0.9464,0.1276,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[264,523,0.5048,0.9375,0.19212,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[268,523,0.5124,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[272,523,0.5201,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[276,523,0.5277,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[280,523,0.5354,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[284,523,0.543,0.875,0.23351,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,22],[288,523,0.5507,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[292,523,0.5583,0.92854,0.17134,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,26],[296,523,0.566,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[300,523,0.5736,0.92411,0.21124,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,27],[304,523,0.5813,0.92409,0.15969,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,4,0,24],[308,523,0.5889,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[312,523,0.5966,0.9375,0.15542,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[316,523,0.6042,0.95982,0.13939,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[320,523,0.6119,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[324,523,0.6195,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[328,523,0.6272,0.91517,0.24964,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[332,523,0.6348,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[336,523,0.6424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[340,523,0.6501,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[344,523,0.6577,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[348,523,0.6654,0.95536,0.15746,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[352,523,0.673,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[356,523,0.6807,0.93749,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[360,523,0.6883,0.96429,0.10714,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[364,523,0.696,0.95536,0.10971,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[368,523,0.7036,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[372,523,0.7113,0.93749,0.14699,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,25],[376,523,0.7189,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[380,523,0.7266,0.9241,0.2172,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,27],[384,523,0.7342,0.91963,0.19544,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,26],[388,523,0.7419,0.94196,0.10012,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[392,523,0.7495,0.91964,0.19212,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,26],[396,523,0.7572,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[400,523,0.7648,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[404,523,0.7725,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[408,523,0.7801,0.94195,0.133,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[412,523,0.7878,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[416,523,0.7954,0.94196,0.17807,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,25],[420,523,0.8031,0.92411,0.17122,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,25],[424,523,0.8107,0.9375,0.12846,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,23],[428,523,0.8184,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[432,523,0.826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[436,523,0.8337,0.93302,0.18896,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[440,523,0.8413,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[444,523,0.8489,0.91964,0.17105,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,25],[448,523,0.8566,0.94643,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[452,523,0.8642,0.95535,0.13571,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[456,523,0.8719,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[460,523,0.8795,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[464,523,0.8872,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[468,523,0.8948,0.94197,0.13295,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[472,523,0.9025,0.92857,0.17857,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[476,523,0.9101,0.94196,0.16311,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[480,523,0.9178,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[484,523,0.9254,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[488,523,0.9331,0.94642,0.13244,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[492,523,0.9407,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[496,523,0.9484,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[500,523,0.956,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[504,523,0.9637,0.95981,0.12496,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[508,523,0.9713,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[512,523,0.979,0.91964,0.17835,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,23],[516,523,0.9866,0.93302,0.15969,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[520,523,0.9943,0.95088,0.12686,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[523,523,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b4b0e9277bb1ea74","q":"Suppose $a_{0}, \\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\\{0,1, \\ldots, 100\\}$ :\n\n$$\na_{100+k} x^{100}+100 a_{99+k} x^{99}+a_{98+k} x^{98}+a_{97+k} x^{97}+\\cdots+a_{2+k} x^{2}+a_{1+k} x+a_{k}\n$$\n\nwhere indices are taken modulo 101, i.e., $a_{100+i}=a_{i-1}$ for any $i$ in $\\{1,2, \\ldots, 100\\}$. Show that it is impossible that each of these 101 polynomials has all its roots real.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.17402,"x":0.54463,"p":[[0,84,0.0,0.17402,0.23888,0.0,0.07,0.28571,0.0,0.85714,16,0,0,16,0,7,0,0,2,0,0,4,0,0,0,0,0,2,0,0,1,0,0],[4,84,0.0476,0.4866,0.33854,0.2857,0.28571,0.85704,0.0,1.0,1,7,0,1,0,6,0,0,11,0,0,2,0,0,2,0,0,0,0,0,3,0,7],[8,84,0.0952,0.42856,0.31743,0.1429,0.28571,0.60714,0.0,1.0,3,5,0,3,0,6,0,0,8,0,0,5,0,0,2,0,0,2,0,0,1,0,5],[12,84,0.1429,0.45981,0.36723,0.14289,0.28571,0.85714,0.0,1.0,5,7,0,5,0,5,0,0,7,0,0,3,0,0,2,0,0,0,0,0,3,0,7],[16,84,0.1905,0.46875,0.36463,0.14286,0.28571,1.0,0.0,1.0,4,9,0,4,0,5,0,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,9],[20,84,0.2381,0.47768,0.32851,0.2857,0.35714,0.85714,0.0,1.0,1,7,0,1,0,6,0,0,9,0,0,6,0,0,1,0,0,0,0,0,2,0,7],[24,84,0.2857,0.43301,0.3302,0.14286,0.28571,0.60714,0.0,1.0,4,6,0,4,0,5,0,0,8,0,0,4,0,0,3,0,0,2,0,0,0,0,6],[28,84,0.3333,0.54016,0.33069,0.2857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,11,0,0,3,0,0,4,0,0,0,0,0,0,0,10],[32,84,0.381,0.34375,0.29203,0.14286,0.28571,0.42858,0.0,1.0,5,4,0,5,0,6,0,0,10,0,0,5,0,0,2,0,0,0,0,0,0,0,4],[36,84,0.4286,0.40177,0.2635,0.2857,0.28571,0.57111,0.0,1.0,2,3,0,2,0,3,0,0,15,0,0,3,0,0,4,0,0,1,0,0,1,0,3],[40,84,0.4762,0.48214,0.37072,0.14286,0.28571,1.0,0.0,1.0,3,9,0,3,0,7,0,0,7,0,0,3,0,0,1,0,0,1,0,0,1,0,9],[44,84,0.5238,0.49098,0.3589,0.2857,0.42857,1.0,0.0,1.0,4,9,0,4,0,3,0,0,7,0,0,8,0,0,0,0,0,0,0,0,1,0,9],[48,84,0.5714,0.48205,0.39253,0.14286,0.2857,1.0,0.0,1.0,3,11,0,3,0,9,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,11],[52,84,0.619,0.54463,0.34337,0.2857,0.42857,1.0,0.0,1.0,1,10,0,1,0,5,0,0,7,0,0,5,0,0,2,0,0,2,0,0,0,0,10],[56,84,0.6667,0.35266,0.26482,0.14286,0.28571,0.42858,0.0,1.0,3,2,0,3,0,7,0,0,11,0,0,4,0,0,3,0,0,0,0,0,2,0,2],[60,84,0.7143,0.39286,0.35175,0.14286,0.28571,0.57143,0.0,1.0,6,7,0,6,0,4,0,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,7],[64,84,0.7619,0.35715,0.31542,0.14286,0.28571,0.4286,0.0,1.0,6,3,0,6,0,8,0,0,4,0,0,7,0,0,0,0,0,2,0,0,2,0,3],[68,84,0.8095,0.47767,0.34921,0.14286,0.42859,0.85714,0.0,1.0,5,5,0,5,0,4,0,0,6,0,0,2,0,0,4,0,0,2,0,0,4,0,5],[72,84,0.8571,0.37944,0.31663,0.14286,0.28571,0.57111,0.0,1.0,6,4,0,6,0,6,0,0,5,0,0,5,0,0,5,0,0,0,0,0,1,0,4],[76,84,0.9048,0.32588,0.24019,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,8,0,0,10,0,0,6,0,0,2,0,0,1,0,0,0,0,2],[80,84,0.9524,0.22768,0.11769,0.14286,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,8,0,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[84,84,1.0,0.20089,0.12804,0.14286,0.1429,0.28571,0.0,0.571,5,0,0,5,0,12,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.1741,"x":0.62053,"p":[[0,52,0.0,0.21864,0.2448,0.0,0.14286,0.28571,0.0,1.0,11,1,0,11,0,8,0,0,6,0,0,4,0,0,1,0,0,0,0,0,1,0,1],[4,52,0.0769,0.59374,0.37134,0.28571,0.4998,1.0,0.0,1.0,5,12,0,5,0,0,0,0,4,0,0,7,0,0,1,0,0,2,0,0,1,0,12],[8,52,0.1538,0.5,0.35174,0.25,0.35714,0.78571,0.0,1.0,3,8,0,3,0,5,0,0,8,0,0,1,0,0,3,0,0,4,0,0,0,0,8],[12,52,0.2308,0.46874,0.3681,0.14286,0.42857,0.89286,0.0,1.0,6,8,0,6,0,3,0,0,6,0,0,5,0,0,2,0,0,1,0,0,1,0,8],[16,52,0.3077,0.62053,0.35104,0.28571,0.64286,1.0,0.0,1.0,2,12,0,2,0,2,0,0,6,0,0,5,0,0,1,0,0,2,0,0,2,0,12],[20,52,0.3846,0.57143,0.34069,0.28571,0.42857,1.0,0.0,1.0,1,10,0,1,0,3,0,0,9,0,0,4,0,0,2,0,0,1,0,0,2,0,10],[24,52,0.4615,0.52668,0.35263,0.28571,0.42857,1.0,0.0,1.0,2,10,0,2,0,4,0,0,9,0,0,3,0,0,3,0,0,1,0,0,0,0,10],[28,52,0.5385,0.36159,0.27195,0.14286,0.28571,0.46418,0.0,1.0,5,2,0,5,0,4,0,0,10,0,0,5,0,0,3,0,0,2,0,0,1,0,2],[32,52,0.6154,0.45981,0.29823,0.28571,0.42857,0.71429,0.0,1.0,2,4,0,2,0,5,0,0,7,0,0,7,0,0,2,0,0,3,0,0,2,0,4],[36,52,0.6923,0.40616,0.34469,0.14286,0.28571,0.75,0.0,1.0,5,4,0,5,0,8,0,0,5,0,0,4,0,0,1,0,0,1,0,0,4,0,4],[40,52,0.7692,0.375,0.30671,0.14286,0.28571,0.46431,0.0,1.0,4,4,0,4,0,7,0,0,9,0,0,4,0,0,2,0,0,1,0,0,1,0,4],[44,52,0.8462,0.33036,0.30606,0.14286,0.28571,0.57143,0.0,1.0,6,3,0,6,0,9,0,0,7,0,0,1,0,0,4,0,0,1,0,0,1,0,3],[48,52,0.9231,0.19196,0.11071,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,18,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.1741,0.18115,0.0,0.14286,0.2857,0.0,0.85714,9,0,0,9,0,14,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0]]}]},{"i":"3ace5992979df041","q":"The inscribed circle of the triangle  $ABC{}$  touches the sides  $BC,CA$  and  $AB{}$  at  $A',B'$  and  $C'{}$  respectively. Let  $I_a$  be the  $A$ -excenter of  $ABC{}.$  Prove that  $I_aA'$  is perpendicular to the line determined by the circumcenters of  $I_aBC'$  and  $I_aCB'.$","t":[{"b":4,"e":0.14286,"k":"flat","v":0.16509,"x":0.41515,"p":[[0,58,0.0,0.25882,0.25617,0.14286,0.14286,0.2857,0.0,1.0,3,2,0,3,0,19,0,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,2],[4,58,0.069,0.3883,0.24554,0.24999,0.28571,0.57143,0.0,1.0,2,1,0,2,0,6,0,0,9,0,0,6,0,0,4,0,0,2,0,0,2,0,1],[8,58,0.1379,0.36159,0.25249,0.14286,0.28571,0.57143,0.0,1.0,4,1,0,4,0,7,0,0,7,0,0,2,0,0,8,0,0,3,0,0,0,0,1],[12,58,0.2069,0.36161,0.29447,0.14286,0.28571,0.57143,0.0,1.0,6,2,0,6,0,5,0,0,9,0,0,2,0,0,3,0,0,4,0,0,1,0,2],[16,58,0.2759,0.31247,0.27065,0.0,0.28571,0.571,0.0,0.85714,9,0,0,9,0,4,0,0,7,0,0,3,0,0,3,0,0,5,0,0,1,0,0],[20,58,0.3448,0.33481,0.24118,0.14286,0.28571,0.57111,0.0,0.85714,5,0,0,5,0,7,0,0,6,0,0,5,0,0,5,0,0,3,0,0,1,0,0],[24,58,0.4138,0.35712,0.26724,0.14286,0.28571,0.57111,0.0,1.0,3,2,0,3,0,9,0,0,8,0,0,2,0,0,5,0,0,3,0,0,0,0,2],[28,58,0.4828,0.41515,0.30795,0.14286,0.28571,0.60714,0.0,1.0,5,2,0,5,0,5,0,0,7,0,0,1,0,0,6,0,0,3,0,0,3,0,2],[32,58,0.5517,0.28571,0.26486,0.10714,0.21428,0.46429,0.0,1.0,8,1,0,8,0,8,0,0,6,0,0,2,0,0,4,0,0,3,0,0,0,0,1],[36,58,0.6207,0.33929,0.31894,0.0,0.28571,0.71429,0.0,1.0,9,2,0,9,0,5,0,0,6,0,0,3,0,0,0,0,0,6,0,0,1,0,2],[40,58,0.6897,0.3214,0.29012,0.0,0.28571,0.57111,0.0,1.0,9,1,0,9,0,5,0,0,5,0,0,3,0,0,6,0,0,1,0,0,2,0,1],[44,58,0.7586,0.2723,0.26809,0.10714,0.14286,0.32143,0.0,1.0,8,1,0,8,0,9,0,0,7,0,0,1,0,0,3,0,0,2,0,0,1,0,1],[48,58,0.8276,0.39273,0.29239,0.14286,0.28571,0.57143,0.0,1.0,4,3,0,4,0,7,0,0,6,0,0,3,0,0,6,0,0,3,0,0,0,0,3],[52,58,0.8966,0.26785,0.26903,0.0,0.14288,0.4642,0.0,1.0,10,1,0,10,0,8,0,0,3,0,0,3,0,0,5,0,0,2,0,0,0,0,1],[56,58,0.9655,0.16509,0.24252,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,5,0,0,3,0,0,2,0,0,0,0,0,4,0,0,0,0,0],[58,58,1.0,0.19654,0.2468,0.0,0.14286,0.2857,0.0,0.85714,13,0,0,13,0,10,0,0,3,0,0,0,0,0,3,0,0,2,0,0,1,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.42856,"p":[[0,40,0.0,0.28562,0.26491,0.14286,0.14286,0.32143,0.0,1.0,3,2,0,3,0,16,0,0,5,0,0,3,0,0,1,0,0,1,0,0,1,0,2],[4,40,0.1,0.42856,0.3234,0.14286,0.35714,0.60714,0.0,1.0,5,3,0,5,0,5,0,0,6,0,0,3,0,0,5,0,0,1,0,0,4,0,3],[8,40,0.2,0.36158,0.27192,0.14286,0.28571,0.57111,0.0,1.0,3,2,0,3,0,8,0,0,10,0,0,1,0,0,5,0,0,2,0,0,1,0,2],[12,40,0.3,0.21865,0.23142,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,6,0,0,1,0,0,1,0,0,4,0,0,0,0,0],[16,40,0.4,0.3482,0.27183,0.14286,0.2857,0.57111,0.0,1.0,4,2,0,4,0,9,0,0,6,0,0,4,0,0,4,0,0,3,0,0,0,0,2],[20,40,0.5,0.24991,0.26004,0.0,0.21435,0.28571,0.0,1.0,11,1,0,11,0,5,0,0,9,0,0,0,0,0,4,0,0,2,0,0,0,0,1],[24,40,0.6,0.15625,0.17985,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,5,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[28,40,0.7,0.14286,0.19233,0.0,0.0,0.2857,0.0,0.57143,19,0,0,19,0,1,0,0,8,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[32,40,0.8,0.125,0.20748,0.0,0.0,0.14287,0.0,0.71429,20,0,0,20,0,6,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[36,40,0.9,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3f93881a71f08ba3","q":"Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns.\n\n1) On the preparatory turn (turn 0), Susana choose a positive integer  $n_0$  and writes the polynomial  $P_0(x)=n_0$ .\n\n2) On turn 1, Brenda choose a positive integer  $n_1$ , different from  $n_0$ , and either writes the polynomial  $$ P_1(x)=n_1x+P_0(x) \\textup{  or  } P_1(x)=n_1x-P_0(x) $$  3) In general, on turn  $k$ , the respective player chooses an integer  $n_k$ , different from  $n_0, n_1, \\ldots, n_{k-1}$ , and either writes the polynomial  $$ P_k(x)=n_kx^k+P_{k-1}(x) \\textup{  or  } P_k(x)=n_kx^k-P_{k-1}(x) $$  The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.","t":[{"b":2,"e":0.0,"k":"falling","v":0.02232,"x":1.0,"p":[[0,45,0.0,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,45,0.0889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,45,0.1778,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,45,0.2667,0.88393,0.29329,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,26],[16,45,0.3556,0.77679,0.41178,0.96429,1.0,1.0,0.0,1.0,7,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,24],[20,45,0.4444,0.45089,0.46854,0.0,0.21429,1.0,0.0,1.0,15,12,0,15,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,12],[24,45,0.5333,0.62946,0.46272,0.0,1.0,1.0,0.0,1.0,11,18,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,18],[28,45,0.6222,0.49107,0.47237,0.0,0.64286,1.0,0.0,1.0,15,13,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,13],[32,45,0.7111,0.46429,0.45876,0.0,0.49999,1.0,0.0,1.0,15,11,0,15,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,11],[36,45,0.8,0.28571,0.41955,0.0,0.0,0.71429,0.0,1.0,20,7,0,20,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,7],[40,45,0.8889,0.46875,0.47277,0.0,0.28573,1.0,0.0,1.0,15,12,0,15,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,12],[44,45,0.9778,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[45,45,1.0,0.14732,0.32436,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,3]]},{"b":7,"e":1.0,"k":"rising","v":0.8125,"x":1.0,"p":[[0,45,0.0,0.8125,0.35614,0.96429,1.0,1.0,0.0,1.0,4,24,2,4,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,24],[4,45,0.0889,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,45,0.1778,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[12,45,0.2667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,45,0.3556,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,45,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,45,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,45,0.7111,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,45,0.8,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[40,45,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[45,45,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"414ea473fb9c41cb","q":"The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ \u00b7 \u00b7 \u00b7 +2000\nFind all of the following positive integers that cannot be the value of n.","t":[{"b":4,"e":0.2857,"k":"falling","v":0.28571,"x":0.76786,"p":[[0,69,0.0,0.76786,0.28065,0.57143,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,4,0,0,0,0,0,4,0,0,5,0,0,3,0,15],[4,69,0.058,0.72321,0.30501,0.28571,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,6,0,0,1,0,15],[8,69,0.1159,0.50893,0.30917,0.28571,0.28571,0.75,0.14286,1.0,0,7,0,0,0,2,0,0,17,0,0,0,0,0,2,0,0,3,0,0,1,0,7],[12,69,0.1739,0.53571,0.29667,0.28571,0.35714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,15,0,0,2,0,0,2,0,0,3,0,0,3,0,6],[16,69,0.2319,0.54018,0.30249,0.28571,0.57143,0.75,0.0,1.0,2,5,0,2,0,1,0,0,10,0,0,2,0,0,4,0,0,5,0,0,3,0,5],[20,69,0.2899,0.47769,0.3001,0.28571,0.28571,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,15,0,0,2,0,0,1,0,0,4,0,0,3,0,4],[24,69,0.3478,0.44642,0.29179,0.2857,0.28571,0.60714,0.0,1.0,1,5,0,1,0,0,0,0,22,0,0,0,0,0,1,0,0,1,0,0,2,0,5],[28,69,0.4058,0.53125,0.29066,0.28571,0.28571,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,17,0,0,1,0,0,2,0,0,4,0,0,2,0,6],[32,69,0.4638,0.54464,0.29974,0.28571,0.28571,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,17,0,0,1,0,0,0,0,0,6,0,0,1,0,7],[36,69,0.5217,0.48214,0.27375,0.28571,0.28571,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,17,0,0,2,0,0,0,0,0,8,0,0,0,0,4],[40,69,0.5797,0.45089,0.25781,0.28571,0.28571,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,19,0,0,1,0,0,1,0,0,7,0,0,0,0,3],[44,69,0.6377,0.35714,0.21724,0.28571,0.28571,0.28571,0.0,1.0,1,2,0,1,0,1,0,0,25,0,0,0,0,0,0,0,0,3,0,0,0,0,2],[48,69,0.6957,0.41518,0.23517,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,23,0,0,2,0,0,0,0,0,4,0,0,0,0,3],[52,69,0.7536,0.38393,0.22142,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,26,0,0,1,0,0,0,0,0,1,0,0,2,0,2],[56,69,0.8116,0.37053,0.20473,0.2857,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,3,0,0,0,0,2],[60,69,0.8696,0.39285,0.2342,0.2857,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[64,69,0.9275,0.44643,0.27141,0.28571,0.28571,0.60714,0.2857,1.0,0,5,0,0,0,0,0,0,23,0,0,0,0,0,1,0,0,3,0,0,0,0,5],[68,69,0.9855,0.28571,1e-05,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.14268,"x":0.72321,"p":[[0,40,0.0,0.72321,0.26949,0.53571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,5,0,0,6,0,0,3,0,0,2,0,13],[4,40,0.1,0.47768,0.27108,0.28571,0.28571,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,17,0,0,2,0,0,1,0,0,7,0,0,0,0,4],[8,40,0.2,0.50446,0.29011,0.28571,0.28571,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,17,0,0,1,0,0,3,0,0,3,0,0,1,0,6],[12,40,0.3,0.49553,0.2923,0.28571,0.28571,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,18,0,0,2,0,0,0,0,0,3,0,0,3,0,5],[16,40,0.4,0.53571,0.29881,0.28571,0.42857,0.71429,0.14286,1.0,0,7,0,0,0,2,0,0,13,0,0,3,0,0,1,0,0,6,0,0,0,0,7],[20,40,0.5,0.375,0.26426,0.2857,0.28571,0.35714,0.0,1.0,1,3,0,1,0,6,0,0,17,0,0,0,0,0,3,0,0,1,0,0,1,0,3],[24,40,0.6,0.37053,0.22829,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,4,0,0,21,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[28,40,0.7,0.26339,0.20238,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,19,0,0,8,0,0,0,0,0,3,0,0,1,0,0,0,0,1],[32,40,0.8,0.2767,0.20811,0.14286,0.2143,0.28571,0.14,1.0,0,1,0,0,0,16,0,0,11,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[36,40,0.9,0.20089,0.17075,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[40,40,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7e3bc73e20817ee3","q":"There are  $n$  students each having  $r$  positive integers. Their  $nr$  positive integers are all different. Prove that we can divide the students into  $k$  classes satisfying the following conditions.\n\n(a)  $ k \\le 4r $ (b) If a student  $A$  has the number  $m$ , then the student  $B$  in the same class can't have a number  $l$  such that\n\\[ (m-1)! < l < (m+1)!+1  \\]","t":[{"b":3,"e":0.2857,"k":"rising","v":0.08704,"x":0.79683,"p":[[0,58,0.0,0.08704,0.17919,0.0,0.0,0.03571,0.0,0.71429,24,0,23,24,0,2,0,1,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,58,0.069,0.79683,0.27267,0.571,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,4,0,0,4,1,0,0,0,0,3,0,18],[8,58,0.1379,0.62941,0.27862,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,4,0,0,3,0,0,7,0,0,3,0,0,6,0,6],[12,58,0.2069,0.60044,0.32335,0.28571,0.71429,0.85714,0.0,1.0,1,5,0,1,0,4,0,1,5,0,0,2,0,0,1,0,0,4,0,0,9,0,5],[16,58,0.2759,0.63256,0.25182,0.42857,0.67857,0.85714,0.14286,1.0,0,4,0,0,0,3,0,1,0,0,0,6,0,0,3,3,0,5,2,0,5,0,4],[20,58,0.3448,0.52678,0.30813,0.1429,0.42859,0.85714,0.14286,1.0,0,4,0,0,0,9,0,0,1,0,0,7,0,0,2,0,0,4,0,0,5,0,4],[24,58,0.4138,0.50669,0.28424,0.28571,0.42857,0.75,0.14286,1.0,0,4,0,0,0,4,0,0,8,0,1,8,0,0,0,0,0,3,0,0,4,0,4],[28,58,0.4828,0.48657,0.34228,0.14286,0.4998,0.75,0.0,1.0,2,6,0,2,0,9,0,0,4,0,0,1,0,0,5,0,0,3,0,0,2,0,6],[32,58,0.5517,0.54239,0.26837,0.39286,0.4286,0.85714,0.0,1.0,1,2,0,1,0,2,0,1,4,0,0,9,0,0,3,0,0,3,0,0,7,0,2],[36,58,0.6207,0.5625,0.30027,0.28571,0.64286,0.75,0.0,1.0,2,4,0,2,0,3,0,1,3,0,0,4,0,1,2,0,0,8,0,0,4,0,4],[40,58,0.6897,0.59819,0.29111,0.42857,0.57121,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,5,0,0,7,0,0,3,0,0,5,0,0,4,0,6],[44,58,0.7586,0.58702,0.2671,0.42857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,0,0,1,3,0,0,6,0,0,5,0,0,8,0,0,3,0,4],[48,58,0.8276,0.5491,0.29202,0.28571,0.64286,0.71429,0.0,1.0,2,3,0,2,0,4,0,1,2,0,0,3,0,1,3,0,0,10,0,0,3,0,3],[52,58,0.8966,0.55132,0.24565,0.42857,0.4998,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,1,9,0,0,4,0,0,6,0,0,3,0,3],[56,58,0.9655,0.5982,0.24336,0.39286,0.71429,0.857,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,4,0,0,2,0,0,8,0,0,10,0,0],[58,58,1.0,0.66293,0.2463,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,7,0,0,4,1,0,3,0,0,10,0,4]]},{"b":7,"e":0.57143,"k":"rising","v":0.14286,"x":0.74999,"p":[[0,52,0.0,0.14286,0.22588,0.0,0.0,0.2857,0.0,1.0,19,1,19,19,0,4,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[4,52,0.0769,0.74999,0.2945,0.42857,0.85707,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,3,0,0,4,0,0,0,0,0,5,0,0,3,0,15],[8,52,0.1538,0.73884,0.27468,0.55357,0.78564,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,4,0,1,1,0,0,7,0,0,4,0,12],[12,52,0.2308,0.68748,0.26412,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,1,5,0,1,1,0,0,6,0,0,6,0,8],[16,52,0.3077,0.6897,0.31064,0.42857,0.85707,1.0,0.0,1.0,1,10,0,1,0,2,0,1,2,0,0,4,0,0,2,0,0,3,0,0,7,0,10],[20,52,0.3846,0.64731,0.3072,0.39286,0.64286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,6,0,0,5,0,0,3,0,0,3,0,0,2,0,11],[24,52,0.4615,0.62274,0.27096,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,8,0,0,4,1,0,5,0,0,1,0,8],[28,52,0.5385,0.62724,0.30185,0.42857,0.67864,1.0,0.14286,1.0,0,9,0,0,0,3,0,2,2,0,0,6,0,0,2,1,0,5,0,0,2,0,9],[32,52,0.6154,0.52228,0.29528,0.19644,0.571,0.80357,0.14286,1.0,0,2,0,0,0,8,0,1,3,0,0,2,0,0,5,0,0,4,1,0,6,0,2],[36,52,0.6923,0.58705,0.29164,0.42857,0.57143,0.85714,0.0,1.0,1,7,0,1,0,2,0,1,3,0,0,7,0,0,4,2,0,3,0,0,2,0,7],[40,52,0.7692,0.51561,0.29164,0.2857,0.46429,0.71429,0.14286,1.0,0,5,0,0,0,7,0,0,4,0,0,5,0,1,4,0,0,5,0,0,1,0,5],[44,52,0.8462,0.60712,0.31944,0.28571,0.64286,0.85714,0.0,1.0,1,7,0,1,0,4,0,0,4,0,0,4,0,0,3,0,0,3,0,0,6,0,7],[48,52,0.9231,0.45752,0.2486,0.26785,0.49979,0.57143,0.0,1.0,1,1,0,1,0,6,0,1,4,0,0,4,0,0,9,0,0,4,0,0,2,0,1],[52,52,1.0,0.45535,0.2641,0.14286,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,9,0,0,6,0,0,2,0,1,2,1,0,7,0,0,4,0,0]]}]},{"i":"7461fc9405bae363","q":"Let  $\\mathbb{N}$  denote the set of positive integers. For how many positive integers  $k\\le 2018$  do there exist a function  $f: \\mathbb{N}\\to \\mathbb{N}$  such that  $f(f(n))=2n$  for all  $n\\in \\mathbb{N}$  and  $f(k)=2018$ ?\n\n[i]Proposed by James Lin","t":[{"b":1,"e":0.4286,"k":"falling","v":0.54465,"x":0.86607,"p":[[0,55,0.0,0.80803,0.31463,0.67857,1.0,1.0,0.0,1.0,2,21,2,2,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,21],[4,55,0.0727,0.83915,0.15058,0.71429,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,4,0,13],[8,55,0.1455,0.86607,0.17105,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,1,0,18],[12,55,0.2182,0.75445,0.21794,0.67857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,11,0,0,2,0,11],[16,55,0.2909,0.82143,0.15972,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,0,1,0,13],[20,55,0.3636,0.77678,0.17835,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,17,0,0,1,0,10],[24,55,0.4364,0.67857,0.16366,0.67857,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,20,0,0,1,0,3],[28,55,0.5091,0.76338,0.18075,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,16,0,0,0,0,10],[32,55,0.5818,0.67856,0.20517,0.42859,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,14,0,0,1,0,6],[36,55,0.6545,0.71426,0.189,0.67857,0.71429,0.75,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,16,0,0,2,0,6],[40,55,0.7273,0.67856,0.18558,0.5354,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,15,0,0,1,0,5],[44,55,0.8,0.64286,0.19561,0.4286,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,8,0,0,2,0,0,16,0,0,0,0,4],[48,55,0.8727,0.64732,0.14719,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,22,0,0,0,0,1],[52,55,0.9455,0.60715,0.15567,0.4286,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,17,0,0,0,0,1],[55,55,1.0,0.54465,0.14479,0.42857,0.4286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,1,0,0,13,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.80803,"x":1.0,"p":[[0,109,0.0,0.90625,0.16602,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,23],[4,109,0.0367,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20],[8,109,0.0734,0.80803,0.19103,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,1,0,14],[12,109,0.1101,0.83036,0.19045,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,3,0,15],[16,109,0.1468,0.88839,0.15458,0.71429,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,4,0,19],[20,109,0.1835,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[24,109,0.2202,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,109,0.2569,0.95534,0.10976,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[32,109,0.2936,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22],[36,109,0.3303,0.93302,0.12368,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[40,109,0.367,0.91518,0.14664,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,22],[44,109,0.4037,0.92857,0.15568,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,1,0,25],[48,109,0.4404,0.9375,0.13333,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25],[52,109,0.4771,0.93304,0.13356,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,24],[56,109,0.5138,0.88839,0.20743,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,21],[60,109,0.5505,0.83927,0.18125,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,2,0,16],[64,109,0.5872,0.86161,0.2004,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,9,0,0,0,0,20],[68,109,0.6239,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[72,109,0.6606,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[76,109,0.6972,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[80,109,0.7339,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[84,109,0.7706,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[88,109,0.8073,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[92,109,0.844,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,109,0.8807,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[100,109,0.9174,0.94643,0.12752,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[104,109,0.9541,0.94195,0.11773,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[108,109,0.9908,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[109,109,1.0,0.95981,0.14394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29]]}]},{"i":"9f4510afd81aa643","q":"Let  $a\\geq 1$  a real number. Consider the squence  $(a_n)_{n \\ge 1}$  in which  $x_1=a$  and  $x_{n+1}=1+log_{2}{x_n}, \\forall n\\in\\mathbb{N^*}$ . Find the values of  $a$  such as all the terms of the sequence are rational numbers. \n\n*Mihail B\u0103lun\u0103*, Bucharest","t":[{"b":0,"e":0.85714,"k":"flat","v":0.97768,"x":1.0,"p":[[0,46,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,46,0.087,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[8,46,0.1739,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,46,0.2609,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,46,0.3478,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,46,0.4348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,46,0.5217,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,46,0.6087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,46,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,46,0.7826,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,46,0.8696,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,46,0.9565,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[46,46,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,42,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,42,0.0952,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,42,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,42,0.2857,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,42,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,42,0.4762,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,42,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,42,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,42,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[42,42,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"6feafe61e4c69b9e","q":"Let  $f(x)$  and  $g(x)$  be polynomials with non-negative integer coefficients, and let m be the largest coefficient of  $f.$  Suppose that there exist natural numbers  $a < b$  such that  $f(a) = g(a)$  and  $f(b) = g(b)$ . Show that if  $b > m,$  then  $f = g.$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.35713,"x":0.7857,"p":[[0,23,0.0,0.35713,0.34809,0.14286,0.14286,0.57143,0.0,1.0,7,5,6,7,0,11,0,0,0,0,0,4,0,0,3,0,0,2,0,0,0,0,5],[4,23,0.1739,0.71429,0.33312,0.42859,0.85714,1.0,0.14286,1.0,0,15,0,0,0,6,0,0,0,0,0,4,0,0,0,0,0,5,0,0,2,0,15],[8,23,0.3478,0.7857,0.33121,0.57132,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,0,3,0,20],[12,23,0.5217,0.74999,0.32538,0.57132,0.92857,1.0,0.0,1.0,2,16,0,2,0,2,0,0,1,0,0,2,0,0,2,0,0,4,0,0,3,0,16],[16,23,0.6957,0.58036,0.40238,0.14286,0.57143,1.0,0.0,1.0,3,14,0,3,0,8,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,14],[20,23,0.8696,0.54016,0.38254,0.14286,0.4998,1.0,0.0,1.0,3,9,0,3,0,9,0,0,0,0,0,4,0,0,2,0,0,1,0,0,4,0,9],[23,23,1.0,0.49554,0.32924,0.14286,0.42857,0.75,0.14286,1.0,0,7,0,0,0,10,0,0,3,0,0,7,0,0,1,0,0,3,0,0,1,0,7]]},{"b":5,"e":0.71429,"k":"rising","v":0.33929,"x":0.90178,"p":[[0,25,0.0,0.33929,0.41611,0.0,0.07143,0.85714,0.0,1.0,16,6,10,16,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,6],[4,25,0.16,0.67409,0.35217,0.42857,0.71429,1.0,0.0,1.0,3,13,0,3,0,3,0,0,1,0,0,2,0,0,3,0,0,5,0,0,2,0,13],[8,25,0.32,0.86603,0.25243,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,2,0,23],[12,25,0.48,0.88393,0.22711,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,21],[16,25,0.64,0.84375,0.24578,0.8214,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0,5,0,19],[20,25,0.8,0.81247,0.25367,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,0,7,0,15],[24,25,0.96,0.83482,0.17896,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,9,0,13],[25,25,1.0,0.90178,0.18364,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,22]]}]},{"i":"c9f5661042e962c6","q":"Prove that an $m \\times n$ rectangle is $(b, b)$-tileable if and only if $2 b \\mid m$ and $2 b \\mid n$.","t":[{"b":2,"e":1.0,"k":"volatile","v":0.34375,"x":0.86606,"p":[[0,15,0.0,0.34375,0.27862,0.14286,0.28571,0.42857,0.0,1.0,2,4,0,2,0,11,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,4],[4,15,0.2667,0.86606,0.27186,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,25],[8,15,0.5333,0.67415,0.30974,0.42857,0.57214,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,11,0,0,0,0,0,1,0,0,1,0,14],[12,15,0.8,0.7232,0.31327,0.42857,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,2,0,0,5,0,0,1,0,0,2,0,0,6,0,13],[15,15,1.0,0.65177,0.32131,0.42857,0.71429,1.0,0.0,1.0,2,11,0,2,0,1,0,0,3,0,0,7,0,0,1,0,0,5,0,0,2,0,11]]},{"b":3,"e":1.0,"k":"rising","v":0.38393,"x":0.97768,"p":[[0,24,0.0,0.38393,0.31831,0.14286,0.28571,0.4286,0.0,1.0,2,6,0,2,0,10,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,6],[4,24,0.1667,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[8,24,0.3333,0.92411,0.18893,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,27],[12,24,0.5,0.91071,0.19805,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,24],[16,24,0.6667,0.77678,0.26711,0.42859,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,6,0,0,1,0,0,1,0,0,6,0,15],[20,24,0.8333,0.89285,0.19885,0.85714,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,20],[24,24,1.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30]]}]}]);